QUANTITATIVE PSYCHOLOGY: Some Chosen Problems and New Ideas
ADVANCES IN PSYCHOLOGY 1s Editors
G . E. STELMACH P. A . VROON
NORTH-HOLLAND A M S T E R D A M . NEW YORK . O X F O R D
QUANTITATIVE PSYCHOLOGY: Some Chosen Problems and New Ideas
Maria NOWAKOWSKA Institute of Philosophy and Sociology Polish Academy of Sciences and
Machine Intelligence Institute Iona College
1983
NORTH-HOLLAND AMSTERDAM. NEW YORK . OXFORD
Elsevier Science Publishers B.V., 1983 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, o r transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner
ISBN: 0 444 86708 2
Publishers: ELSEVIER SCIENCE PUBLISHERS B.V. P.O.Box 1991 1000 B Z Amsterdam The Netherlands
Sole distributors for the U.S.A.and Canada: ELSEVIER SCIENCE PUBLISHING COMPANY, INC. 52, Vanderbilt Avenue NewYork,N.Y. 10017 U.S.A.
PRINTED IN T H E NETHERLANDS
TO THE MEMOR Y OF STANISLAW KO WAL
This Page Intentionally Left Blank
Vii
PREFACE
I n t h e p e r i o d of deepening s p e c i a l i z a t i o n and l e s s e n i n g , o f communication, n o t o n l y between d i s c i p l i n e s , b u t o f t e n a l s o b e t w e e n d o m a i n s o f t h e same d i s c i p l i n e , t h e book a t t e m p t s t o show n o t o n l y t h e c h o s e n p r o b l e m s o f q u a n t i t a t i v e psychology and i t s e x t e n s i o n s , b u t a l s o o u t l i n e some w i d e r p e r s p e c t i v e s . I t u n i f i e s t h e p s y c h o l o g i c a l t o p i c s , among o t h e r s by t h e o r i e s
o f observab-
i l i t y , s e m i o t i c s a n d knowledge s t r u c t u r e s , as w e l l as a c t i o n theory, connecting psychological, l i n g u i s t i c , l o g i c a l and d e c i s i o n a l a s p e c t s o f b e h a v i o u r . Q u a n t i t a t i v e psychology i s understood h e r e a s statistical
and m o d e l i n g a p p r o a c h e s . A s r e g a r d s t h e f i r s t ,
t h e book shows t h e o r y o f m e n t a l t e s t s c o r e s , w i t h t h e
s t r e s s p u t on new d e v e l o p m e n t s o f t h i s t h e o r y ( e . g . g e n e r a l i z a b i l i t y t h e o r y ) , a n d i t s p s y c h o l o g i c a l and m e t h o d o l o g i c a l problems. A s r e g a r d s t h e second approach, t h e book shows b a s i c p r o b l e m s o f measurement t h e o r y , and a s e r i e s o f new m o d e l s , e . g .
of r i s k , d e c i s i o n s ,
p e r c e p t i o n o f time, e t c . O r i g i n a l m a t e r i a l c o n s t i t u t e s o v e r one h a l f o f t h e b o o k . The r e f e r e n c e s a r e l i m i t e d t o t h o s e w o r k s , w h i c h
were a c t u a l l y u s e d . While t h e b o o k i s n o t a t e x t b o o k i n t h e s t r i c t s e n s e ,
i t may b e p r o f i t a b l y u s e d a s c o m p l e m e n t a r y r e a d i n g f o r
viii
PREFACE
s t u d e n t s i n t h e s o c i a l , e d u c a t i o n a l , m a n a g e r i a l , comp u t e r and i n f o r m a t i o n s c i e n c e s ( i n t e r e s t e d i n a r t i f i c i a l i n t e l l i g e n c e and p a t t e r n r e c o g n i t i o n ) , as w e l l as p h i l o s o p h y , l i n g u i s t i c and l o g i c . It w i l l a l l o w them t o deepen t h e view o f t h e i r own a r e a s , and devel o p new i n t u i t i o n s and i d e a s , The book, however, i s i n t e n d e d p r i m a r i l y f o r r e s e a r c h e r s i n t h e a r e a s menti o n e d . It assumes t h e e l e m e n t a r y knowledge o f s e t t h e o r y , f u z z y s e t t h e o r y , l o g i c , a s w e l l as p r o b a b i i t y and s t a t i s t i c s .
Maria Nowakowska
Houston, A p r i l 1983
CONTENTS
Preface Introduction Chapter 1: AN OUTLINE OF THE CONTEMPORARY THEORY OF PSYCHOLOGICAL TESTS 1. Object of Measurement in Psychology 2. Test as a Measurement Tool 3. Conceptual Foundations of the Classical Theory of Tests 3.1. The axiom system of Gulliksen 3.2. Discussion of the axioms 3.3. A critique of Gulliksen's approach 4. Conceptual Foundations of the Contemporary Theory of Tests 4.1. Propensity distribution, true and error scores 4.2. Linear experimental independence 4.3. The concept of parallel measurements
5. 6. 7. 8. 9.
Reliability Reliability of Composite Tests Homogeneity of a Test Validity Prediction 9.1. Models of regression 9.2. Construction of battery of predictors b y screening 9.3. The expected and actual quality of prediction 10. Some Problems of the Construction of Tests
vi i xvi i
1
3
4 4 8 9 11 11 17 21 24 32 41 43
47 47
54 57 59
X
CONTENTENTS
11. Further Directions of Development of Test Theory: Theory of Generic True Scores and Theory of Generalizability 11.1. Introductory remarks 11.2. Basic concepts of the theory of generic true scores 11.3. Schemes of experiments 12. Bias in Selection Procedures 13. Main Directions of Research in Test Theory
102
Bibliography to Chapter 1
109
Chapter 2: FACTOR ANALYSIS: ARBITRARY DECISIONS WITHIN MATHEMATICAL MODEL 1. Some Multivariate Techniques in Psychology 1.1. Factor analysis: the mathematical model 1.2. Discussion 1.2.1. Arbitrary decisions in applications of factor analysis 1.2.2. The problems of invariance 1.2.3. Interpretation of results of factor analysis 1.2.4. Cognitive limitations of factor analysis -- the subjective bias 1.2.5. The status of variables underlying factor analysis 1.3. Multidimensional scaling 1.3.1, Conceptual foundations 1.3.2. The main directions of development of the theory of multidimensional scaling 1.3.3. Multidimensional scaling and factor analysis 2. An Example of Application of Factor Analysis: Personality Theory of Cattell 2.1. Descriptive approach to personality theory 2.2. General characteristics of Cattell's approach 2.3. The basic hypothesis of Cattell 2.4. Some problems of research strategy of Cattell 2.4.1. Personality structure from &-data 2.4.2. Verification of the basic hypothesis of Cattell
67 67 70 76 93
115 115 121 121 127 130 131 135 138 138 142 144 145 145 148 152 153 153
156
CONTENTS
2.4.3. Cattell's explanation of difficulties in verifying his hypothesis 2.5. Discussion of hypothesis of indifference of media 2.6. General remarks about Cattell's approach Bibliography to Chapter 2 Chapter 3: SOME TSYCHOLOGICAL PROBLEMS OF CONSTRUCTION AND APPLICATION OF QUESTIONNAIRES 1. Models of Response to a Questionnaire Item 1.1. Variability and ambiguity 1.2. A fuzzification of the model 1.3. Psychometric paradox 1.3.1. Discriminativeness 1.3.2. Analysis of the paradox 1.4. Factors determining the answer to a questionnaire item and its variability 1.5. Analysis of the data 1.5.1. Factor analysis 1.5.2. Model of answering to a questionnaire item 1.5.3. Perception of items and variability of answers 1.6. Question answering as a multiple-criterion decision making: model MASIA 1.6.1. The model 1.6.2. The trial answers 1.6.3. The finaJ answer 1.6.4. Theoretical analysis 1.6.5. Some results 2. Some Alternative Ideas in Testing 2.1. Contextual tests 2.1.1. The foundations 2.1.2. The principle of estimation 2.1.3. Construction of the test 2.1.4. Some preliminary simulation results 2.2. Dynamic questionnaire 2.2.1. The formal scheme in general case 2.2.2. Random walk model
xi
160 161 163 174
183 184 191 193 194 197 205 207 207 210 213 224 225 226 229 232 240 242 243 243 247 249 252 256 256 259
xii
CONTENTS
3. Applications of Test Theory: Methodological Problems of Measurement of Fuzzy Concepts 3.1. Fuzzy sets and fuzzy identity 3.2. Structure of concepts 3.3. Measurement of concepts Bibliography to Chapter 3 Chapter 4: FORMAL SEMIOTICS: REPRESENTATIONS, OBSERVABILITY AND PERCEPTION 1. Introduction 2. Multimedia1 Communication 2.1. Formal approach to syntax of multimedia1 communication languages 2.1.1. Definition of multimedia1 languages 2.1.2. Probabilistic interpretation 2.2. Semantics of multimedia1 languages 2.3. Pragmatic semantics 2.4. Semantic languages 2.5. Graph representation of a unit 3. Control Processes in Perception 3.1. Process of spontaneous inspection 3.2. Purposeful inspection 3.3. Generation of meaning under inspection 3.4. Mechanism of transportation in comparisons 3.5. Events representation of an object: pre-events and memory 4. Observability and Change 4.1. Joint observability 4.2. Masks and filters 4.3. Operations on masks and filters 4.4. Observability of change 4.5. Semantic aspects of representations 4.5.1. Algebra of descriptions 4.5.2. Assertions and queries 4.5.3. Fuzzy attributes and verbal copies 5. Formal Semiotic Systems 5.1. Representations 5.2. Multiple representations 5.3. Fuzzy meanings 5.4. Relationships between structures
262 263 268 276 283
285 287 304 305 307 313 315 323 331 334
343 343 349 357 360
372 377 377 384
388 391 395 399 402
404 412 412 414 417 420
CONTENTS
5.4.1. Structure of represented objects 5.4.2. Homomorphisms 5.5. Signs and information 5.6. Statistical semiotics 5.7. Comments Bibliography to Chapter 4 Chapter 5: SELECTED TOPICS IN MEASUREMENT THEORY 1. Basic Procedures of Measurement 1.1. Ordinal measurement 1.2. Standard sequences 1.3. Solving inequalities 2. An Abstract Presentation of Measurement Problem 3. The Role and Character of Axioms 4. Extensive Measurement in Psychology 4.1. Measurement of risk 4.2. Qualitative scale of probability 5. Difference Measurement 5.1. The structure of algebraic differences 5.2. Bisections 5.3. The scale of just noticeable differences (jnd) 5.4. Semiorders 6. Conjoint Measurement 6.1. Necessary conditions 6.2. Sufficient conditions 6.3. Generalization: multinomial conjoint measurement 7. Utility and Subjective Probability 7.1. Primitive notions and their interpretation 7.2. The form of representation 7.3. The axiom system of Luce and Krantz 8. Some Problems in Utility Theory 8.1. Certain paradoxes of utility theory 8.2. A model of choice behaviour 8.2.1. General postulates 8.2.2. The general model 8.2.3. Some special cases 8.2.4. Discussion
xiii
421 425 428
434 437 440
443 445 445 449 455 456 461 466 468 475 482
484 486 488 490 493
497 504 505 516 516 518 519 531 532
535 540 542 552
557
x iv
CONTENTS
8.3. A decision model for social behaviour 8.3.1. The model 8.3.2. Analysis of the model 8.3.3. Discussion. Extension of the model
9. A New Theory of Time
561 561
563 567 575 575 577
9.1. The basic formal scheme 9.2. The objective cardinal and ordinal time 9.3. Subjective time and distortions of time perception 582 9.4. Time and frequency of events 590 10. Problems of Scaling 59 7 10.1. Introductory remarks 597 10.2. Relations to theories of measurement 598 10.3. Classification of scaling techniques 600 10.3.1. Pair Comparison 601 10.3.2. Magnitude scaling 613 10.3.3. Rating scales, and classification into equally spaced categories 619 10.4. Perception of scale of social power 10.4.1. The concept of power 10.4.2. Voting power 10.4.3. The hypotheses 10.4.4. The experimental material 10.4.5. The results 10.4.6. Discussion 11. Linguistic Measurement 11.1. Classification schemes 11.2. Dynamic classifications 11.3. Linguistic measurement 11.4. Construction of linguistic scales: analogies and metaphors 11.5. Dimensionality of descriptions 11.6. Measurement by analogy Bibliography to Chapter 5
625
626 628 631
637 653
660
661 661
667 672 680
683 685 689
Chapter 6: FORMAL THEORY OF ACTIONS
698 1. The Basic Scheme 1.1. The deterministic case 698 1.2. Language of actions 705 1.3. Algebraic approach to goal structure 707 1.4. A fuzzification of admissibility of actions718 1.5. Stochastic transitions 722
CONTENTS
xv
2. Temporal Relations between Events 727 2.1. The temporal truth systems 728 2.2. Events 729 2.3. Relative necessity of events 732 2.4. Knowledge representation 733 2.5. Consistency and adequacy of knowledge 740 3. Development 743 4. Multiple Goals 756 4.1. General structure of a goal 758 4.1.1. The case of one criterion. Additivity 761 4.1.2. Goals and their ideals 764 4.1.3. The effect of multidimensionality 765 4.2. The problems of aggregation 767 4.3. An algebra of criteria. Conflict theory 770 5. Verbal and Nonverbal Actions: Motivation and Choice 780 5.1. Linguistic representation of motivational space 782 5.2. Motivational calculus 789 5.3. Linguistic representation of time in motivational calculus 793 5.4. Motivational consistency and structural properties of sets of strings of actions which fulfill or break a promise 805 6. Structure of Sets of Time-Events 811 7. Group Actions 821 8. An Application to System Synthesis 834 9. Preference Systems and Relations to Decision Theory 845 10. Planning Actions and' Generative Grammars 851 11. An Application to Organization Theory 853 12. Ethical Valuations 859 13. Aggregation of Valuations 863 14. Applications: A Theory of Social Change 881 14.1. Structure of the society 882 14.2. Social change 885 14.3. A theory of freedom 888 14.4. Alienation and dynamics of social change 891 14.5. Sub,iective DerceDtion of one's l e v e l of alienatibn 896
xvi
CONTENTS
14.5.1. The model 14.5.2. Analysis 14.6. Some hypotheses 14.7. Some other mechanisms of social change 14.7.1. Group pressures and group choice 14.7.2. Communication networks 14.8. Concluding remarks. Some further perspectives 15. Examples of Application to Developmental Psychology 15.1. Stadia1 languages of development 15.2. Education and psychological development: modeling of the process Bibliography to Chapter 6
897 900 902 906 907 918 922 928 928
933 940
xvii
INTRODUCTION
T h i s book a t t e m p t s t o go beyond some s t e r e o t y p e s of q u a n t i t a t i v e p s y c h o l o g y . The q u a n t i t a t i v e a p p r o a c h e s a r e o f t e n r e g a r d e d as t o o n a r r o w , n o t i n s p i r i n g , and s u p e r f i c i a l i n t h e i r p s y c h o l o g i c a l c o n t e n t . The s t a t i s t i c a l t h e o r y o f t e s t s e n c o u n t e r e d even some r e s i s t a n c e and c e r t a i n l o s s o f s o c i a l c o n f i d e n c e , b e c a u s e o f
i t s application t o various selection procedures, t h a t b o t h i n t h e view o f t h e r e s e a r c h e r s and o f t h e u s e r s a r e b i a s e d , i . e . u n f a i r t o some s o c i a l g r o u p s . Moreover, t h e r e i s a c o n v i c t i o n t h a t t h e c o g n i t i v e weakness of t e s t s c a n n o t be overcome by even b e s t p s y c h o m e t r i c p a r a m e t e r s . The t e s t t h e o r y paradigm i s p e r c e i v e d , by many p s y c h o l o g i s t s , as s c i e n t i f i c a l l y i n f e r i o r . T h i s c a u s e d some l e s s e n i n g o f a t t r a c t i v e n e s s
o f t h i s a p p r o a c h , d e s p i t e t h e f a c t t h a t r e s e a r c h on f o r m a l models o f t e s t t h e o r y i s c o n t i n u i n g
w i t h new
i n t e r e s t i n g r e s u l t s . It seems v e r y d i f f i c u l t t o go beyond some i n h e r e n t c o g n i t i v e bounds o f t h i s a p p r o a c h . The q u e s t i o n a r i s e s how one can b r i n g some new i d e a s t o t h e domain and a p p l i c a t i o n of t e s t t h e o r y . I t seems t h a t t h e o r y may be d e v e l o p e d t o w a r d s dynamic a n a l y s i s and changes o f c o n c e p t s t r u c t u r e , o r more g e n e r a l l y
--
knowledge s t r u c t u r e . I n t h i s i n t e r p r e t a t i o n , o n e would have t o assume t h a t t h e b a s i c c o n s t r u c t o f t h e t h e o r y
xviii
INTRODUCTION
( t r a i t ) i m p l i e s some way o f v i e w i n g t h e w o r l d , or s t a t e o f knowledge a b o u t t h e w o r l d . The a n a l y s i s would conc e r n development o f v a r i o u s t y p e s o f r e p r e s e n t a t i o n of knowledge, t h e i r i n t e r a c t i o n , s u b s t i t u t a b i l i t y , d i s t o r t i o n s , e t c . It would b e i m p o r t a n t t o a n a l y s e knowledge s t r u c t u r e , as a c q u i r e d w i t h c o o p e r a t i o n w i t h a compute r a w h i c h i n some s e n s e i m p o s e s s e q u e n t i a l i z a t i o n a n d instructionalisrn i n thinking. Also, c o m p u t e r i z e d t e s t i n g o p e n s up p o s s i b i l i t i e s of new t y p e o f t e s t r e s e a r c h , i n c l u d i n g more c o n t r i b u t i o n o f e x p e r i m e n t a t i o n , more s t r e s s on i n t e r m e d i a l t e s t s , based on t h e i d e a o f s e m a n t i c e q u i v a l e n c e , and g e n e r a l l y , more c o n t r i b u t i o n o f l i n g u i s t i c and p e r c e p t u a l a s p e c t s . I n a r t i f i c i a l i n t e l l i g e n c e and computer s c i e n c e s , i n connection w i t h developing the so-called e x p e r t s y s t e m s or f u t u r e "smart c o m p u t e r s " , t h e r e i s g r e a t n e e d o f knowledge a b o u t knowledge s t r u c t u r e a n d i t s dynamics. E n g i n e e r s and computer s c i e n t i s t s n o t o n l y w a i t for new r e s u l t s i n t h i s d i r e c t i o n , b u t h a v e t h e i r own i d e a s ; t h e i d e a s o f p s y c h o l o g i s t s a r e o f t e n t o o s l o w i n c o m i n g , or b u i l t on i d e a s f o r m u l a t e d e a r l i e r i n computer s c i e n c e s . This concerns i n p a r t i c u l a r i n f o r m a t i o n p r o c e s s i n g , and has some n e g a t i v e c o n s e q u e n c e s for d e v e l o p m e n t o f new p s y c h o l o g i c a l i n t u i t i o n s , c o n c e p t s and t h e o r i e s . The m a i n s t r e s s o f p s y c h o l o g i c a l r e s e a r c h s h i f t e d some
time ago frqm measurement o f t r a i t s t o models of psy-
c h o l o g i c a l p r o c e s s e s . P a r t i c u l a r l y s u c c e s s f u l here a r e d e c i s i o n m o d e l s . However, i t a p p e a r s t h a t i d e a l i z a t i o n s o f t h e s e p r o c e s s e s a r e t o o s t r o n g , and t h e i n t u i t i o n s on w h i c h t h e m o d e l s a r e b a s e d a r e t o o f a r from t h e r e a l e x p e r i e n c e o f d e c i s i o n m a k e r s . For i n s t a n c e ,
INTRODUCTION
xix
i n p r o b l e m s o f r i s k , t o o much s t r e s s was p u t on i n t u i t i o n s o f "homo e c o n o m i c u s " , w i t h o u t t a k i n g i n t o a c c o u n t e t h i c a l v a l u a t i o n s o f a c t i o n s a n d d e c i s i o n s . One c a n show t h a t w i t h t h e s e f a c t o r s i n c l u d e d i n t h e r i s k mod e l , o n e g e t s new t y p e s o f i n t e r p r e t a t i o n a n d p r e d i c t i o n o f b e h a v i o u r . Also s t u d y o f s e q u e n t i a l d e c i s i o n s i n g r o u p c o n t e x t o p e n s new r e s e a r c h p o s s i b i l i t i e s ( s e e Chapter 5 , S e c t i o n 8 ) . Modeling o f i n t e r n a l d e c i s i o n s i n t h e p r o c e s s o f a n s w e r i n g q u e s t i o n n a i r e items g i v e s
a b r i d g e between d e c i s i o n t h e o r y and t e s t t h e o r y , i . e . i n t r o d u c e s d e c i s i o n m o d e l s i n t o t h e l a t t e r . Models o f i n t e r n a l d e c i s i o n s a l l o w t o c a p t u r e some c o g n i t i v e biases i n these d e c i s i o n s , and t o r e c o n s t r u c t t h e o r e t i c a l l y t h e s e b i a s e s , as o b s e r v e d i n a c t u a l r e s p o n s e s ( s e e C h a p t e r 3 , S e c t i o n 1 . 6 ) . It a p p e a r e d , m o r e o v e r , t h a t it i s e s s e n t i a l t o b u i l d i n t e r a c t i o n a l d e c i s i o n models for some c l a s s e s o f s i t u a t i o n s ( s e e C h a p t e r 6 , S e c t ion 14). Finally,
one can b u i l d a r i c h t h e o r y d e s c r i b i n g t h e
s t r u c t u r a l a s p e c t s o f s i t u a t i o n s and d e c i s i o n s , which i s a n i m p o r t a n t complement o f d e c i s i o n a l a p p r o a c h e s (Chapter 6 ) . P r o g r e s s i n s p e c i a l i z a t i o n i n psychology , and slow d i f f u s i o n o f c o n c e p t s f r o m o n e domain t o a n o t h e r , c a u s e s for i n s t a n c e , t h a t i n e x p e r i m e n t a l r e s e a r c h on s u b j e c t i v e p r o b a b i l i t y , t h e r e a r e a l m o s t no s t u d i e s on s t a b i l i t y and v a r i a b i l i t y o f a s s e s s m e n t s . T h i s i s j u s t o n e o f t h e many e x a m p l e s o f n o t i n f r e q u e n t p s y c h o m e t r i c "under- i n s t r u m e n ta i t i o n " o f psycho l o g i c a l r e s e a r c h
.
Measurement t h e o r y s u p p l i e s c o g n i t i v e l y i m p o r t a n t models , o f f o u n d a t i o n a l c h a r a c t e r f o r q u a n t i t a t i v e
INTROD UCTlON
xx
psychology.
Here a l s o w e have a c o g n i t i v e paradigm,
of c o n s i d e r a b l e m a t h e m a t i c a l e l e g a n c e , and v e r y v a l u a b l e r e s u l t s , n o t o n l y for q u a n t i t a t i v e p s y c h o l o g y , but also c o n s t i t u t i n g a c o n t r i b u t i o n t o m a t h e m a t i c s . The c o g n i t i v e s p e c t r u m of t h e m o d e l s o f c l a s s i c a l meas u r e m e n t t h e o r y i s n o t t o o w i d e . The t h q o r y d e t e r m i n e s m a i n l y t h e c o n d i t i o n s w h i c h must b e m e t , i n o r d e r f o r t h e e x i s t e n c e o f a g i v e n t y p e o f r e p r e s e n t a t i o n of a
s e t o f o b j e c t s a n d r e l a t i o n s on t h e m . I t h a s , h o w e v e r , a d e s c r i p t i v e r a t h e r t h a n e x p l a n a t o r y c h a r a c t e r . It does not i n t r o d u c e deeper c o g n i t i v e models. I n p a r t i c u l a r , it does not t a k e i n t o account c e r t a i n p e r c e p t u a l , memory o r n e u r o b i o l o g i c a l v a r i a b l e s . The c o r e o f measurement t h e o r y paradigm i s f i n d i n g a numerical r e p r e s e n t a t i o n for a s e t of e m p i r i c a l r e l a t i o n s . The p r o p e r t i e s of t h e l a t t e r allow t o determine t h e c l a s s o f t r a n s -
f o r m a t i o n s under which t h e r e p r e s e n t a t i o n ( t h e s c a l e v a l u e s ) remain i n v a r i a n t ; t h i s c l a s s determines a l s o t h e set of meaningful a s s e r t i o n s about s c a l e v a l u e s w h i c h o n e c a n make. T h i s seems t o c o v e r t h e e s s e n t i a l a s p e c t s o f m e a n i n g f u l n e s s ; however, t h e scope
of t h e
definition is rather limited. I n p h i l o s o p h y of measurement t h e o r y , p e r c e p t u a l b i a s e s are a n a l y s e d by v a r i o u s a s s u m p t i o n s a b o u t e r r o r s i n c l a s s i f i c a t i o n s and o r d e r i n g s . The c e n t r a l r o l e i s p l a y e d by d i s c r i m i n a t i v e a b i l i t y . I t a p p e a r s t h a t t h e s t u d y o f d i f f e r e n t f o r m s o f s u b j e c t i v e time a n d i t s d i s t o r t i o n s , t h e t o p i c t h u s f a r unexplored i n measurement t h e o r y ,
a l l o w s t h e a n a l y s i s o f new a s p e c t s o f
s t r u c t u r e and f u n c t i o n of c o g n i t i v e p r o c e s s e s ( s e e Chapter 5 , S e c t i o n
9).
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I n m e a s u r e m e n t t h e o r y , t h e o b j e c t s may b e a r b i t r a r y , and t h e o n l y - r e q u i r e m e n t i s t h a t t h e e m p i r i c a l r e l a t i o n s s a t i s f y t h e a p p r o p r i a t e axioms , implying t h e e x i s t e n c e
of a s c a l e of t h e a t t r i b u t e i n question. I n conjoint m e a s u r e m e n t , t h e scheme i s t h e same, e x c e p t t h a t t h e r e l a t i o n a l s y s t e m i n v o l v e s o b s e r v a t i o n s o f two (or mor e ) a t t r i b u t e s . P e r h a p s some e x t e n s i o n s c o u l d b e obt a i n e d h e r e by c o n s i d e r i n g o b j e c t s o f complex s t r u c t u r e ( m u l t i d i m e n s i o n a l u n i t s or s t i m u l i ) . T h i s would req u i r e a n e x t e n s i o n and a new f o r m o f r e p r e s e n t a t i o n s , new f o r m s o f i n v a r i a n c e , and c h a n g e o f t h e c r i t e r i a of meaningfulness. An e x t e n s i o n o f m e a s u r e m e n t t h e o r y t o w a r d s w e a k e r f o r m s o f m e a s u r e m e n t was i n t r o d u c e d i n C h a p t e r 5 , S e c t i o n 11, d e a l i n g w i t h l i n g u i s t i c measurement, comprising a l s o m e a s u r e m e n t by a n a l o g y a n d by m e t a p h o r s . I n t h e same s e c t i o n , t h e dynamic c l a s s i f i c a t i o n was i n t r o d u c e d , where t h e c l a s s i f i c a t i o n schemes a r e s a m p l e d f r o m o c c a s i o n t o o c c a s i o n , a n d m e a s u r e m e n t i s made a c c o r d i n g t o t h e sampled scheme. It seems t h a t t h e o r y o f m e a s u r e m e n t c o n s i d e r s o n l y
o n e c l a s s o f c o n s t r a i n t s , i . e . t h o s e i n n u m e r i c a l dom a i n , i n t o w h i c h t h e e m p i r i c a l r e l a t i o n s a r e mapped. One c a n i m a g i n e a s t u d y o f o t h e r t y p e s o f c o n s t r a i n t s , c o n n e c t e d w i t h t h e mechanisms o f t r a n s p o r t a t i o n , v e r y b a s i c i n comparison o f a t t r i b u t e s o f s t i m u l i . The l a t t e r i s o n e o f t h e key mechanisms u s e d i n dy-
namic m o d e l s o f p e r c e p t i o n ( s e e C h a p t e r 4 , S e c t i o n 3 ) . It c o u l d p e r h a p s s e r v e as a new s t a r t i n g p o i n t f o r p o s s i b l e d e v e l o p m e n t s o f measurement t h e o r y . Maybe t h e o t h e r mechanisms w i l l h a v e s i m i l a r u s e f u l n e s s ,
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namely t h e mechanisms o f s p o n t a n e o u s and p u r p o s e f u l i n s p e c t i o n , a n d c o n n e c t e d w i t h them r a n d o m n e s s a n d p a r t i a l c o n t r o l o f p e r c e p t i o n . The l a t t e r i m p l i e s t h e e x i s t e n c e o f s t r a t e g i e s and o p t i m a l i z a t i o n o f p e r c e p t i o n p r o c e s s . The p e r c e p t u a l mechanisms c o u l d s e r v e a s a b a s i s o f new a x i o m s y s t e m s i n measurement t h e o r y , s i m i l a r l y a s masks a n d f i l t e r s ( s e e C h a p t e r 4) c o u l d a l l o w t h e s t u d y of new t y p e s o f c o n s t r a i n t s i n t h e o b s e r v a t i o n a l domain. A s may be s e e n from t h e a b o v e o u t l i n e , t h e a i m o f t h e
book was a c r i t i c a l p r e s e n t a t i o n o f some c h o s e n p r o b lems i n q u a n t i t a t i v e p s y c h o l o g y . T h u s , t h e book s t a r t s f r o m a p r e s e n t a t i o n o f t h e c o n t e m p o r a r y t h e o r y o f psyc h o l o g i c a l t e s t s . It covers a l l b a s i c concepts, includi n g t h e t h e o r y of g e n e r a l i z a b i l i t y . P a r t i c u l a r a t t e n t i o n h e r e i s d i r e c t e d a t t h e p r o b l e m s of u s a g e o f t e s t s for s e l e c t i o n , a n d b i a s e s i n s u c h p r o c e d u r e s . C h a p t e r 2 c o n c e r n s f a c t o r a n a l y s i s , as a n e x a m p l e o f
a model i n w h i c h a r b i t r a r i n e s s o f d e c i s i o n s a f f e c t s t h e r e s u l t s . The c h a p t e r a l s o shows p r o b l e m s o f m u l t i d i m e n sional scaling, i n particular i t s difficulties with t h e i n t e r p r e t a t i o n . A s a n e x a m p l e o f a t h e o r y b a s e d on
f a c t o r analysis, Cattell's theory of personality i s shown, and c o g n i t i v e beound imposed by t h e method a r e discussed. C h a p t e r 3 d e a l s w i t h p s y c h o l o g i c a l p r o b l e m s o f cons t r u c t i o n a n d a p p l i c a t i o n o f q u e s t i o n n a i r e s . Here main a t t e n t i o n i s c o n c e n t r a t e d on f a c t o r s d e t e r m i n i n g t h e an s wers , and p o s s i b l e b i a s e s and d i s t o r t i o n s o f t h e a n s w e r s . I n p a r t i c u l a r , it i s shown t h a t a m b i g u i t y of q u e s t i o n s a n d i t s e m p i r i c a l c o u n t e r p a r t -- v a r i a b i l i t y o f a n s w e r s -- i s o n e o f t h e most i m p o r t a n t f a c t o r s
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a f f e c t i n g t h e r e s p o n s e . It i s a l s o e x p l a i n e d why w e l l d i s c r i m i n a t i n g i t e m s a r e h i g h l y v a r i a b l e ( t h e phenomenon c a l l e d p s y c h o m e t r i c p a r a d o x ) . Next, t h e c h a p t e r e x p l o r e s an i m p o r t a n t c o n n e c t i o n between t e s t t h e o r y and t h e o r y o f fuzzy s e t s ; t h e f o r mer may namely p r o v i d e a n e m p i r i c a l a c c e s s t o measurement o f membership f u n c t i o n s f o r c o n c e p t s o f s p e c i a l structure. A t h e o r e t i c a l model o f a n s w e r i n g q u e s t i o n n a i r e i t e m s ,
based on e m p i r i c a l d a t a , i s shown. T h i s model a l l o w s f o r q u a l i t a t i v e p r e d i c t i o n o f v a r i a b i l i t y and s t a b i l i t y o f i t e m s , given t h e p s y c h o l o g i c a l c h a r a c t e r i s t i c s of these items.
Further
t h e o r e t i c a l e x t e n s i o n o f t h i s model is t h e
s t o c h a s t i c d e c i s i o n model M A S I A , which a l l o w s f o r a r e c o n s t r u c t i o n o f e m p i r i c a l l y observed f r e q u e n c i e s o f answers. F i n a l l y , t h i s c h a p t e r a l s o shows two a l t e r n a t i v e models o f t e s t i n g , namely c o n t e x t u a l t e s t s and dynamic q u e s t i o n n a i r e s . The f i r s t i s b a s e d on t h e i d e a o f u n f o l d i n g s c a l e s , w i t h s p e c i a l s t r u c t u r e o f complex s t i m u l i . The second c o n s i d e r s dynamic q u e s t i o n n a i r e s , where t h e dec i s i o n r e g a r d i n g t h e c h o i c e o f n e x t q u e s t i o n (or dec i s i o n about t e r m i n a t i o n o f t e s t ) i s r e a c h e d s e q u e n t i a l l y , depending on t h e answers t o t h e p r e c e d i n g i t e m s . C h a p t e r 4 p r e s e n t s a n o u t l i n e o f f o r m a l s e m i o t i c s . The s t a r t i n g p o i n t o f t h i s t h e o r y i s a r e l a t i o n between an o b j e c t ( r e a l o r imagined, random o r n o t ) , i d e n t i f i e d w i t h a c e r t a i n r e l a t i o n a l s t r u c t u r e , and i t s v a r i o u s v e r b a l , p i c t o r i a l , symbolic, n u m e r i c a l , e t c . r e p r e s e n t -
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a t i o n s . The main c o n c e p t o f t h e t h e o r y i s t h a t o f m u l t i d i m e n s i o n a l u n i t ( a composite s t i m u l u s ) , and multimed-
i a l languages, allowing p a r a l l e l description of s t r i n g s o f p h y s i c a l or m e n t a l a c t i o n s . Elements o f p r o b a b i l i s t i c i n t e r p r e t a t i o n o f u n i t s and m u l t i m e d i a 1 l a n g u a g e s a r e a l s o shown. T h i s a l l o w s t o i n t r o d u c e random o b j e c t s t o t h e t h e o r y o f o b s e r v a b i l i t y and change, and t o c o n s i d e r j o i n t o b s e r v a b i l i t y , as w e l l as t e m p o r a l t r a c e s o f o b s e r v a t i o n s , c a l l e d m a s k s . I n o t h e r words, o b s e r v a b i l i t y t h e o r y d e a l s w i t h v a r i o u s c o n s t r a i n t s on o b s e r v a b i l i t y , e x p r e s s e d by c o n c e p t s o f m a s k s and f i l t e r s . System o f models of v a r i o u s p e r c e p t i o n mechanisms p r e s e n t e d i n t h i s c h a p t e r was i n t r o d u c e d f o r e x p l a i n i n g the p a r t i a l control process of perception of objects. The s y s t e m d e p a r t s s u b s t a n t i a l l y from t h e m a t h e m a t i c a l a p p r o a c h e s i n p a t t e r n r e c o g n i t i o n and a r t i f i c i a l i n t e l l i g e n c e advanced t h u s f a r . The s e c t i o n on s e m i o t i c s y s t e m s , c o v e r i n g a l s o e l e m e n t s of s t a t i s t i c a l semiotics, closes t h e chapter. The s e t o f i n t e r r e l a t e d t h e o r i e s o f C h a p t e r 4 a l l o w s f o r t r e a t i n g i n a u n i f i e d way d i f f e r e n t p e r c e p t u a l , cognit i v e , l i n g u i s t i c and p h i l o s o p h i c a l problems. I t a l s o a l l o w s t o s e e t h e o r y o f t e s t s and t h e o r y o f measurement i n a wider p e r s p e c t i v e . The l a t t e r i s c o v e r e d i n C h a p t e r 5 , which c o n s i s t s o f 11 s e c t i o n s . The f i r s t e i g h t d e a l w i t h c l a s s i c a l problems: e x t e n s i v e , d i f f e r e n c e and c o n j o i n t measurements, u t i l i t y and s u b j e c t i v e p r o b a b i l i t y . S e c t i o n 8 shows some new problems o f u t i l i t y t h e o r y , namely a model o f
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c h o i c e d e c i s i o n u n d e r r i s k , a n d a model o f s o c i a l b e h a v i o u r i n g r o u p c o n t e x t s . A s r e g a r d s t h e f i r s t , it i s d e s i g n e d to c o v e r t h e s i t u a t i o n s i n w h i c h t h e SEU model d o e s n o t a p p l y ; i t t a k e s i n t o a c c o u n t some p s y c h o l o g i c a l f e a t u r e s of t h e m o t i v a t i o n model o f A t k i n s o n . The main r e s u l t o f t h i s g e n e r a l model o f c h o i c e c o n c e r n s t h e e x i s t e n c e o f non-normal r i s k a r e a s , w h i c h e x p l a i n some a p p a r e n t l y i r r a t i o n a l b e h a v i o u r of c e r t a i n d e c i s i o n makers. It a l s o e x p l a i n s t h e assymmetry b e t w e e n t h e r o l e s o f f e a r o f f a i l u r e a n d a c h i e v e m e n t m o t i v e i n dec i s i o n m a k i n g . I n t h i s model o n e a b a n d o n s a n a r r o w way of l o o k i n g a t s u c c e s s i n p u r e l y e c o n o m i c a l t e r m s , and t h i n k s o f r e w a r d s a l s o i n m o r a l , e t h i c a l or i d e o l o g i c a l terms. T h e s e c o n d model s t a r t s f r o m t h e a s s u m p t i o n t h a t i n
m a k i n g t h e c h o i c e o f a c t i o n , p e o p l e t e n d t o maximize b o t h t h e e x p e c t e d d i r e c t reward a n d a l s o i n d i r e c t r e w a r d ,
i n t e r m s of a c c e p t a n c e o r r e j e c t i o n by o t h e r s i n t h e r e f e r e n c e g r o u p . The a s s u m p t i o n s on u t i l i t y f u n c t i o n l e a d h e r e t o a s e r i e s of q u a l i t a t i v e h y p o t h e s e s and p r e d i c t i o n s o f b e h a v i o u r , d e p e n d i n g on t h e e v a l u a t i o n s . of t h e g r o u p . An e x t e n s i o n of t h e m a i n i d e a s o f t h i s model t o a g e n e r a l d e c i s i o n model ( n o t r e s t r i c t e d t o t h e c a s e of s o c i a l d e c i s i o n s a n d u t i l i t y i n d e x c o n n e c t e d w i t h t h e j u d g m e n t s ) was a l s o d i s c u s s e d . A p p l i c a t i o n s o f t h e i d e a s o f measurement t h e o r y t o cons t r u c t i o n of a t h e o r y of t i m e , b o t h o b j e c t i v e a n d subj e c t i v e , i s shown i n S e c t i o n 9 . The t h e o r y i s b a s e d on fuzzy r e l a t i o n d e s c r i b i n g t h e s u b j e c t i v e perception o f time. Some h y p o t h e s e s o f p s y c h o l o g i c a l c h a r a c t e r about t h e d i s t o r t i o n s o f p e r c e p t i o n o f t i m e are formulated.
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A f t e r s h o w i n g b a s i c p r o b l e m s o f s c a l i n g ( S e c t i o n lo), a p p l i c a t i o n s o f t h i s t h e o r y t o s c a l i n g s o c i a l power
a r e d i s c u s s e d . The a t t e n t i o n i.s c o n c e n t r a t e d on t h e f a c t o r s accounting f o r t h e d i s t o r t i o n s of t h e p e r c e p t i o n o f s o c i a l power. The c h a p t e r e n d s w i t h a n e x t e r . s i o n o f c o n c e p t i o n o f measurement t h e o r y t o i t s weak-est f o r m
--
linguistic
m e a s u r e m e n t . The l a t t e r seems t o b e t h e most n a t u r a l and f l e x i b l e f o r m o f measurement a t human d i s p o s a l . T h i s t h e o r y i s r e l a t e d i n a n e s s e n t i a l way w i t h t o p i c s o f p e r c e p t i o n , d e s c r i p t i o n s , v e r b a l c o p i e s and i d e n t i f -
i c a t i o n of Chapter of motivation i n
4, as w e l l a s t h o s e o f r e p r e s e n t a t i o n Chapter 6.
The book e n d s w i t h t h e c h a p t e r d e v o t e d t o s t r u c t u r a l p r o p e r t i e s o f d e c i s i o n s and a c t i o n s , and a g e n e r a l d e s c r i p t i o n o f human b e h a v i o u r i n a v e r y f l e x i b l e f o r m a l f r a m e w o r k . The b a s i c n o t i o n s h e r e a r e l a n g u a g e o f a c t i o n s , an a l g e b r a i c s t r u c t u r e of g o a l s , fuzzy admissib i l i t y o f a c t i o n s , and s t o c h a s t i c t r a n s i t i o n s . N e x t s e c t i o n a n a l y s e s a n important a t t r i b u t e o f an e v e n t , namely t h e t i m e o f i t s o c c u r r e n c e , a n d i n t r o d u c e s t h e c o n c e p t o f h i s t o r y a n d t e m p o r a l t r u t h s y s t e m s . The e v e n t s a r e d e f i n e d a s s e t s o f h i s t o r i e s , and t h e n o t i o n o f n e c e s s i t y o f an e v e n t i s i n t r o d u c e d .
The knowledge i s i d e n t i f i e d w i t h a s e t o f a s s e r t i o n s about t h e h i s t o r y of t h e s y s t e n . I t s t r u t h , b e l i e v a b i l i t y , c e r t a i n t y , e t c . are generalized t o t h e notion of 4-bility,
and t h e c o n d i t i o n s o f adequacy, c o n s i s t e n c y ,
t r u t h and p r e c i s i o n a r e d i s c u s s e d . S p e c i a l k i n d of h i s t o r i e s , r e p . r e s e n t i n g d e v e l o p m e n t , a r e t h e n i n t r o d u c e d and a n a l y s e d t'hrough t h e c o n c e p t o f
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p h a s e s and t h e i r l a n g u a g e . Section
4
d e a l s w i t h g o a l s o f a c t i o n s , e s p e c i a l l y mul-
t i p l e and d i v e r s e g o a l s , and t h e n w i t h t h e i r a g g r e g a t i o n , as w e l l as with t h e theory
of c o n f l i c t s a r i s i n g
i n c a s e o f j o i n t l y u n a t t a i n a b l e g o a l s . The l a t t e r t h e ory
i s o f p a r t i c u l a r importance f o r d e c i s i o n s i t u a t -
ions. R e l a t i o n s between v e r b a l and n o n v e r b a l a c t i o n s ( S e c t i o n
5 ) i s b a s e d on t h e e x p e c t a t i o n s o f c o n s i s t e n c y between u t t e r e d p l a n s , i n t e n t i o n s and a c t i o n s . The a n a l y s i s o f u t t e r a n c e s i n which a p e r s o n p r e s e n t s h i s p l a n s , j u s t i f i e s or e x p l a i n s h i s g o a l s , e t c . a l l o w s t o introduce a l i n g u i s t i c representation of motivation, while l o g i c a l a n a l y s i s of t h e s e u t t e r a n c e s l e a d s t o m o t i v a t i o n a l c a l c u l u s , b e i n g a s e t of i n f e r e n c e r u l e s b a s e d on t h e n o t i o n o f s e m a n t i c i m p l i c a t i o n . T h i s c a l c u l u s was l a t e r e n r i c h e d by i n c l u s i o n o f t e m p o r a l v a r i a b l e s , l e a d i n g , among o t h e r s , t o d e f i n i t i o n s o f some quantifiers. R e l a t i o n s between m o t i v a t i o n a l c o n s i s t e n c y and s t r u c t u r -
a l p r o p e r t i e s o f s e t s o f s t r i n g s o f a c t i o n s were cons i d e r e d on example o f p r o m i s e s . The t h e o r e m s h e r e r e f l e c t c e r t a i n g e n e r a l r e g u l a r i t i e s i n t r e a t m e n t o f norms of s o c i a l o r d e r , namely r e l a t i v e u n i v e r s a l i t y o f cont e x t s t o which t h e y a p p l y . These r e s u l t s may be o f some i n t e r e s t for deontic logic. E v e n t s o c c u r i n t i m e , and t h e t e m p o r a l p a t t e r s o f t h e i r o c c u r r e n c e , p e r i o d i c i t y , h o r i z o n e t c and d e a l t with i n Section 6 .
Next, t h e system i s extended t o
t h e c ase of simultaneous a c t i o n s o f s e v e r a l persons, s o
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a s t o c a p t u r e two a s p e c t s : j o i n t p e r f o r m a n c e , a n d i n t e r a c t i o n s between s t a t e s o f t h e systems c o r r e s p o n d i n g t o d i f f e r e n t persons. T h i s g e n e r a l i z e s t h e concept o f m u l t i m e d i a 1 u n i t s o f communication language o f Ch a p te r
4,
and a l s o l e a d s t o g e n e r a l problems o f s y s t e m s s y n th e -
s i s ( S e c t i o n 8 ) , i . e . a g g r e g a t i o n of simple s y s t e m s t o c o m p o s i t e o n e s , u n d e r some c o n s t r a i n t s . Here t h e n o t i o n s of p o s i t i v e and n e g a t i v e i n t e r f e r e n c e and synergy a r e i n t r o d u c e d and a n a l y s e d . Connections w i t h d e c i s i o n t h e o r y and p r e f e r e n c e s y s t e m s a r e shown i n S e c t i o n 9; it i s a r g u e d t h a t a c t i o n t h e o r y
i s i n f a c t a theory of predecisional situations. In the s e q u e l i t i s shown how g e n e r a t i v e grammars may be u s e d f o r f o r m a l i z a t i o n o f t h e t h e o r y o f p l a n s and d e s i g n s ( S e c t i o n lo), a n d how a c t i o n t h e o r y may b e a p p l i e d t o o r g a n i z a t i o n t h e o r y ( S e c t i o n 11). Here t h e a c t i o n s y s -
t e m i s e n r i c h e d by t h e n o t i o n o f a n i n s t r u c t i o n , w h i c h m o d i f i e s t h e a d m i s s i b i l i t y of s t r i n g s o f a c t i o n s . E t h i c a l v a l u a t i o n s o f a c t i o n s a r e d i s c u s s e d i n n e x t two s e c t i o n s . Thus, S e c t i o n 1 2 concerns t h e problems o f e t h i c a l v a l u a t i o n s from m e a s u ~ e m e n t - t h e o r e t i c a l viewp o i n t . N o t i o n s o f l i b e r a l and p u r i s t i c a g g r e g a t i o n s
are introduced, generalized next t o aggregations of v a l u e s o b t a i n e d from some e v a l u a t i o n s w i t h r e s p e c t t o different criteria. The c h a p t e r e n d s w i t h v a r i o u s e x a m p l e s o f a p p l i c a t i o n o f some c o n c e p t s o f t h e t h e o r y ; i n p a r t i c u l a r , S e c t i o n
1 4 d e a l s w i t h a theory of s o c i a l change, w i t h b a s i c n o t i o n s o f f r e e d o m a n d a l i e n a t i o n , combined w i t h t h e c o n s i d e r a t i o n s of s u b j e c t i v e p e r c e p t i o n of o n e ' s a l i e n a t i o n . This p e r c e p t i o n changes w i t h s e q u e n t i a l l y acquire d knowledge o f t h e s i t u a t i o n of o t h e r s .
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To d e e p e n t h e a n a l y s i s , a model o f g r o u p p r e s s u r e a n d g r o u p c h o i c e i s shown. T h i s model c o n c e r n s s i t u a t i o n s i n w h i c h a g r o u p makes a d e c i s i o n f o r o r a g a i n s t a n i s s u e . The main r e s u l t a l l o w s t o o p t i m a l l y a l l o c a t e t h e propaganda t a s k s . An e s s e n t i a l complement o f t h e model o f s o c i a l c h a n g e i s a t h e o r y of communication n e t w o r k s , b e i n g models f o r s p r e a d and s c o p e o f d i s s e m i n a t i o n o f i n f o r m a t i o n i n a group. A n o t h e r example o f a p p l i c a t i o n o f
a c t i o n s t h e o r y con-
c e r n s d e v e l o p m e n t a l p s y c h o l o g y , namely m u l t i d i m e n s i o n a l s t a d i a 1 languages of development, e s p e c i a l l y convenient f o r d e s c r i p t i o n o f e a r l y s t a d i a o f development o f a child. The b o o k c l o s e s w i t h an o u t l i n e o f a p p l i c a t i o n o f a c t i o n t h e o r y t o e d u c a t i o n a l psychology, and mention numerous o t h e r a p p l i c a t i o n s , e . g .
of
t o theory of d i a l o -
gues, e t c .
It i s w o r t h t o s t r e s s t h a t t h e f o r m a l t h e o r y o f a c t i o n s i s t h e f i r s t i n t h e l i t e r a t u r e system t h a t connects p s y c h o l o g i c a l , l l n g u i s t i c , l o g i c a l and d e c i s i o n a l a s p e c t s of behaviour. The above s k e t c h o f t h e c o n t e n t o f t h e book shows t h e d i r e c t i o n s o f r e s e a r c h and s o l u t i o n s , b o t h
i n the
t r a d i t i o n a l p r o b l e m s o f q u a n t i t a t i v e p s y c h o l o g y , as
w e l l as i n t h e a t t e m p t s o f i t s e x t e n s i o n s . The g o a l o f t h e book w i l l be met, i f i t d i r e c t s t h e a t t e n t i o n o f p s y c h o l o g i s t s a n d s o c i a l r e s e a r c h e r s t o some new
p o s s i b i l i t i e s o f t h e development o f t h e domain, and i f i t s h o u l d s u p p l y new i n t u i t i o n s and m o d e l s t o r e -
xxx
INTRODUCTION
l a t e d d i s c i p l i n e s , such a s a r t i f i c i a l i n t e l l i g e n c e
and computer s c i e n c e s , management s c i e n c e s , as w e l l a s p h i l o s o p h y and l o g i c .
1
CHAPTER 1
AN OUTLINE OF THE CONTEMPORARY THEORY
OF PSYCHOLOGICAL TESTS
1. OBJECT OF MEASUREMENT I N PSYCHOLOGY
The o b j e c t o f any measurement a r e s u c h p r o p e r t i e s o f o b j e c t s , w h i c h t h e l a t t e r may p o s s e s s i n v a r y i n g d e g -
r e e ; s u c h a t t r i b u t e s a r e s a i d t o be m o d a l . E a c h measur e m e n t i s s u b j e c t t o c e r t a i n e r r o r s , and t h e b a s i c p r o b lem i n any m e a s u r e m e n t t h e o r y i s t h e i n f e r e n c e a b o u t t h e v a l u e s m e a s u r e d on t h e b a s i s o f t h e v a l u e s o b s e r v ed.
I n p s y c h o l o g y , w h e r e t h e main o b j e c t o f s t u d y a r e humans, o n e i s i n t e r e s t e d i n t h e v a l u e s o f t h e i r p s y c h o l o g i c a l t r a i t s . The l a t t e r n o t i o n l e d t o numerous c o n t r o v e r s i e s ; for t h e p u r p o s e o f m e a s u r e m e n t , h o w e v e r , i t
i s n o t n e c e s s a r y t o d w e l l on t h e s e c o n t r o v e r s i e s , and t a k e as a s t a r t i n g p o i n t o f t h e d e f i n i t i o n o f a t r a i t t h e f a c t of t h e e x i s t e n c e of s t a t i s t i c a l r e g u l a r i t i e s o f b e h a v i o u r . The v a l u e o f a t r a i t f o r a g i v e n p e r s o n
i s simply a c e r t a i n parameter of t h e p r o b a b i l i t y d i s t r i -
bution of c e r t a i n s p e c i f i c behaviours of t h i s person i n specific situations. Such an approach does n o t r e q u i r e d i s c u s s i n g t h e prob-
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CHAPTER 1
l e m how " r e a l " i s t h e t r a i t , i . e . w h e t h e r i t h a s a n y p h y s i c a l or p h y s i o l o g i c a l c o u r t e r p a r t s . I n p s y c h o l o g y , and i n p a r t i c u l a r i n p s y c h o m e t r i c s w h i c h d e a l s w i t h measurements of v a l u e s o f t r a i t s
,
t h e only e s s e n t i a l
f a c t i s t h a t p e o p l e d o b e h a v e as i f t h e s e t r a i t s r e a l l y e x i s t e d , or more p r e c i s e l y , tl-at d i f f e r e n t p e r s o n s behave as i f t h e y had t h e s e t r a i t s i n d i f f e r e n t d e g r e e s ( p s y c h o m e t r i c s d e a l s o n l y w i t h modal t r a i t s ) . T h u s , from a f o r m a l p o i n t o f v i e w a t r a i t ( a n d more p r e c i s e l y , i t s v a l u e ) i s a c e r t a i n parameter of t h e f r e q u e n c y d i s t r i b u t i o n o f some b e h a v i o u r s . A s a cons e q u e n c e , l e a r n i n g t h i s v a l u e a l l o w s a s t a t i s t i c a l pred i c t i o n of such behaviours. From t h e p o i n t o f view o f p s y c h o l o g y , a t r a i t may b e d e f i n e d as a t e rm t h a t g e n e r a l i z e s s u c h s e t s o f b e h a v i o u r s which have a t tendency t o a p p e a r and vary j o i n t l y . I n o t h e r words, a t r a i t i s a n a b s t r a c t i o n o f conc r e t e behaviours ( o r other t r a i t s ) . The m a i n o b j e c t o f i n t e r e s t i n p s y c h o l o g y , h o w e v e r ,
are t h o s e t r a i t s which have e x p l a n a t o r y c h a r a c t e r : t h e y a r e c a l l e d c o n s t r u c t s , or t h e o r e t i c a l n o t i o n s . The e x p l a n a t i o n h a s t h e f o r m o f a c e r t a i n h y p o t h e s i s or a s e t o f h y p o t h e s e s , c o n c e r n i n g c a u s a l i n t e r p r e t a t i o n of j o i n t a p p e a r e n c e of t h o s e b e h a v i o u r s which a r e t h e d e s c r i p t i o n o f t h e t r a i t . One o f t h e m a i n g o a l s of such explanation i s t o provide t h e o r e t i c a l premises
f o r p r e d i c t i o n s . For i n s t a n c e , w i t h t h e term s u c h as " i n t r o v e r s i o n " one a s s o c i a t e s n o t o n l y a s e t o f behavi o u r s w h i c h a r e c h a r a c t e r i s t i c f o r i t , b u t a l s o some hypotheses which e x p l a i n t h e i n t e r r e l a t i o n s between these behaviours,
t h e i r o r i g i n , d y n a m i c s , and g e n e r a l
CONTEMPORARY THEORY OF PSYCHOLOGICAL TESTS
3
p r e d i c t i o n of b e h a v i o u r o f i n t r o v e r t i c p e r s o n s . The t h e o r e t i c a l n o t i o n s a r e c h a r a c t e r i z e d by t h e f a c t t h a t t h e i r c o n t e n t i s u s u a l l y f u z z y and n o t u n i q u e l y d e f i n e d ; i n o t h e r words, t h e same t e r m , w i t h t h e same e x p l a n a t o r y h y p o t h e s e s , may be a s s o c i a t e d w i t h sev e r a l s e t s of b e h a v i o u r s , e a c h c o n s t i t u t i n g a good des c r i p t i o n of t h i s terms. The f l e x i b i l i t y o f t h e c o r r e s pondence between t h e name of t h e t r a i t and i t s d e s c r i p t i o n i s c a l l e d s u r p l u s meaning. A s a consequence o f surp l u s meaning, w e may c o n s t r u c t s e v e r a l e q u a l l y good d e f i n i t i o n s o f t h e same t r a i t , and a l s o have s e v e r a l measurement t o o l s o f i t .
2 . TEST AS A MEASUREMENT TOOL
A f t e r d e f i n i n g t h e o b j e c t of measurement i n p s y c h o l o g y ,
i t i s now n e c e s s a r y t o d e t e r m i n e t h e methodology o f s u c h measurements, and a l s o t o p r o v i d e t h e r u l e s o f i n f e r e n c e about t r u e v a l u e s of t r a i t s from t h e o b s e r v e d v a l u e s o f measurement. The s o l u t i o n h e r e i s d e t e r m i n e d by t h e a c c e p t e d s t a t i s t i c a l d e f i n i t i o n of p s y c h o l o g i c a l t r a i t , i d e n t i f i e d w i t h a p a r a m e t e r of a f r e q u e n c y d i s t r i b u t i o n o f b e h a v i o u r s which s e r v e sa a d e s c r i p t i o n of t h e given t r a i t . This parameter, c a l l e d t h e t r u e v a l u e of a t r a i t ( f o r a g i v e n s u b j e c t ) i s estimat e d from a sample of b e h a v i o u r s . F o r m a l l y , a psychologi c a l t e s t i s j u s t a t o o l which a l l o w s u s t o c o l l e c t s u c h a sample o f b e h a v i o u r s (which a r e a d e s c r i p t i o n o f a g i v e n t r a i t ) , and a l s o a l l o w s us t o d e t e r m i n e t h e n u m e r i c a l v a l u e of t h e e s t i m a t e , g i v e n t h e b e h a v i o u r s . The n u m e r i c a l v a l u e i n q u e s t i o n i s o f t e n c a l l e d t h e t e s t score.
4
CHAPTER 1
To f o r m a l l y d e s c r i b e t h i s s i t u a t i o n , l e t A d e n o t e t h e p o p u l a t i o n u n d e r s t u d y , and l e t G be t h e s e t o f m e a s u r ement t o o l s u n d e r c o n s i d e r a t i o n . I n p a r t i c u l a r c a s e , G may c o n s i s t o f j u s t one t o o l . F o r a C A and g t. G , a n a p p l i c a t i o n of g t o a g i v e s a n u m e r i c a l r e s u l t , which The l a t t e r w i l l be r e g a r d e d ga’ as a v a l u e o f a c e r t a i n random v a r i a b l e X The d i s ga * t r i b u t i o n o f t h i s random v a r i a b l e ,
w i l l be d e n o t e d by x
i s c a l l e d the EropensiQ
d i s t r i b u t i o n of person a w i t h
r e s p e c t t o t e s t g. The p r o p e n s i t y d i s t r i b u t i o n i s n o t o b s e r v a b l e , s i n c e i n p r a c t i c e , t h e r e i s no way o f o b t a i n i n g a s u f f i c i e n t l y l o n g s e q u e n c e of i n d e p e n d e n t o b s e r v a t i o n s o f t h e ( e . g . b e c a u s e of l e a r n i n g e f f e c t ) . random v a r i a b l e X ga I t i s assumed t h a t t h e v a l u e s o f t h e random v a r i a b l e
a r e e x p r e s s e d on a s c a l e o f a t l e a s t i n t e r v a l t y p e , ga s o t h a t one may compute t h e e x p e c t a t i o n and v a r i a n c e .
X
3. CONCEPTUAL FOUNDATIONSS OF THE CLASSICAL THEORY OF TE:STS
3.1. The axiom s y s t e m o f G u l l i k s e n The f i r s t f o u n d a t i o n s o f t h e t h e o r y o f t e s t s were g i v e n i n 1950 by G u l l i k s e n ( G u l l i k s e n 1 9 5 0 ) ; t h i s t h e o r y was s u b s e q u e n t l y m o d i f i e d by N0vic.k ( 1 9 6 6 ) a n d Lord a n d
CONTEMPORARY THEORY OF PSYCHOLOGICAL TESTS
5
Novick ( 1 9 6 8 ) . G u l l i k s e n a d a p t e d t o psychology t h e model o f measurement i n p h y s i c a l s c i e n c e s . He assumed t h a t e a c h p e r s o n has, a t any g i v e n t i m e , some v a l u e s of p s y c h o l o g i c a l t r a i t s , i n analogy w i t h p h y s i c s , where e a c h o b j e c t h a s some w e l l d e f i n e d v a l u e s o f i t s a t t r i b u t e s , r e g a r d l e s s
of t h e a c t o f measurement and t h e c h o i c e o f measurement t o o l . F o r m a l l y , t o e a c h p e r s o n a 6 A and t e s t g C G t h e r e corresponds a value T equal t o t h e value of ga ' t h e t r a i t measured by g f o r p e r s o n a . Next, a g a i n i n a n a l o g y w i t h p h y s i c a l s c i e n c e s , G u l l i k s en assumed t h a t when a measurement i s performed, t h e n t h e observed v a l u e ' e q u a l s t o t h e t r u e v a l u e p l u s a r a n dom e r r o r . Thus, t h e measurement e r r o r i s t h e d i f f e r e n c e
or
(3.1) A s a c o n s e q u e n c e , b o t h t h e t r u e v a l u e and t h e e r r o r
o f measurement a r e n o t o b s e r v a b l e : i n formula ( 3 . 1 ) onl y t h e l e f t hand s i d e i s o b s e r v a b l e , w h i l e t h e compon e n t s o f t h e r i g h t hand s i d e a r e n o t . I n o r d e r t o enable t h e a p p l i c a t i o n s of s t a t i s t i c a l methods o f e s t i m a t i o n of t h e t r u e v a l u e on t h e b a s i s o f t h e o b s e r v e d v a l u e s , i t was n e c e s s a r y t o a c c e p t some a s s u m p t i o n s c o n c e r n i n g t h e randomness of t h e o b s e r v e d
6
CHAPTER 1
scores. It was namely assumed t h a t t h e randomness o f t h e o b s e r v -
ed s c o r e i s of a "double" c h a r a c t e r : i t s f i r s t s o u r c e i s t h e randomness o f t h e measurement p r o c e s s , and t h e second s o u r c e i s t h e randomness o f t h e c h o i c e of t h e p e r s o n from t h e p o p u l a t i o n . The f i r s t t y p e of randomness i s f o r m a l i z e d by assuming
i s a r e a l i z a t i o n of t h e ga with the propensity d i s t r i b u t i o n random v a r i a b l e X ga' F The c h a r a c t e r of t h e second s o u r c e of randomness ga' r e q u i r e s more d e t a i l e d e x p l a n a t i o n , because of t h e
t h a t t h e observed s c o r e x
f a c t t h a t i t i s s p e c i f i c f o r mental t e s t theory, with no c o r r e s p o n d i n g c o u n t e r p a r t i n t h e p h y s i c a l s c i e n c e s . I t i s namely assumed t h a t t h e p e r s o n i n q u e s t i o n i s s e l e c t e d a t random from t h e p o p u l a t i o n under s t u d y . The l a t t e r p o p u l a t i o n c o n s i s t s , roughly speaking, of a l l p e r s o n s who a r e i d e n t i c a l w i t h t h e s u b j e c t from t h e p o i n t o f view o f t h e c r i t e r i a which d e c i d e d on t h e c h o i c e o f t h i s p a r t i c u l a r p e r s o n . For i n s t a n c e , i f one c h o o s e s a s a i l o r ( s a y ) t o measure h i s t r a i t s u c h a s i n t r o v e r s i o n , t h e n i t i s assumed t h a t t h e t r u e s c o r e o f i n t r o v e r s i o n o f t h i s s a i l o r i s a v a l u e of a random v a r i a b l e , whose d i s t r i b u t i o n i s t h e d i s t r i b u t i o n o f t h e t r u e s c o r e s of i n t r o v e r s i c n i n t h e p o p u l a t i o n of all sailors. F o r m a l l y , i t i s assumed t h a t t h e s u b j e c t s from t h e pop u l a t i o n A a r e sampled a c c o r d i n g t o some p r o b a b i l i t y d i s t r i b u t i o n . It i s n o t n e c e s s a r y t o s p e c i f y t h i s d i s t r i b u t i o n ; g e n e r a l l y , one assumes ( i n most c a s e s ) t h a t i t i s uniform on a c e r t a i n s u b s e t of A . A t any r a t e , as t h e r e s u l t of t h i s a s s u m p t i o n , t h e i n d e x a i s random,
CONTEMPORARY THEORY OF PSYCHOLOGICAL TESTS
7
S y m b o l i c a l l y , i t w i l l be c o n v e n i e n t t o i n t r o d u c e t h e notation X for t h e random v a r i a b l e , whose r e a l i z a t i o n s g* a r e t h e v a l u e s i n t e s t g for a randomly s e l e c t e d p e r s o n . S i m i l a r l y , t h e t r u e s c o r e f o r a randomly s e l e c t e d p e r -
son w i l l be t r e a t e d a s a random v a r i a b l e T
g*
’
whose va-
l u e s are t h e t r u e s c o r e s i n t e s t g , and t h e e r r o r s of measurements are t h e r e a l i z a t i o n s of t h e random v a r i a b le I g*
.
Thus, e q u a t i o n ( 3 . 1 ) w i l l now t a k e t h e form X
g*
= T
g*
+
I
g*
.
(3.2)
The n e c e s s i t y o f s u c h a c o n s t r u c t i o n i s connected c l o s e l y w i t h t h e f a c t t h a t i n p r a c t i c e one cannot o b t a i n s u f f i c i e n t numbers of i n d e p e n d e n t o b s e r v a t i o n s for t h e same p e r s o n . The p o i n t i s , t h a t for a s i n g l e p e r s o n a , t h e i n f e r e n c e on h i s t r u e s c o r e i: i s based on t h e ga e s t i m a t i o n of e r r o r e For t h e l a t t e r , i n t u r n , one ga would have t o p e r f o r m many i n d e p e n d e n t o b s e r v a t i o n s of t h e random v a r i a b l e X (for f i x e d p e r s o n a ) . ga
.
S i n c e i n p r a c t i c e one can c o u n t o n l y f o r two (or a t there ga’ i s no way o f e s t i m a t i n g t h e v a r i a n c e of e Thus, t h e ga’ (which i s t h e e s t i m a t i o n c o n c e r n s t h e v a r i a n c e of I g* a v e r a g e v a r i a n c e of e r r o r s of measurements). b e s t s e v e r a l ) independent o b s e r v a t i o n s o f X
A s a r e s u l t , one g e t s t h e p o s s i b i l i t y of e s t i m a t i n g t h e
t r u e s c o r e o f a f i x e d p e r s o n on t h e b a s i s of h i s o b s e r v ed s c o r e . N a t u r a l l y , f o r s u c h an e s t i m a t i o n t o be p o s s i b l e , i t i s n e c e s s a r y t o make some p o s t u l a t e s a b o u t t h e errors I i n s u c c e s s i v e measurements, and about t h e g*
8
CHAPTER i
-
r e l a t i o n between G and t h e t r u e s c o r e T These posg* g* t u l a t e s , c o n s t i t u t i n g t h e axioms of t h e t e s t t h e o r y o f G u l l i k s e n , a r e as f o l l o w s : I . The e x p e c t e d e r r o r s c o r e i s z e r o : E(1
g*
) = 0;
11. The e r r o r s c o r e and t h e t r u e s c o r e a r e u n c o r r e l a -
ted: E(1
T
g* g*
) = 0;
111. E r r o r s c o r e s i n two d i s t i . n e t measurements a r e un-
correlated
3 . 2 . D i s c u s s i o n o f t h e axioms The f i r s t and t h e t h i r d axiom a r e t a k e n d i r e c t l y from t h e t h e o r y o f p h y s i c a l measurements. I n d e e d , axiom I s t a t e s t h a t measurements a r e u n b i a s e d , w h i l e axiom I11 i s a weakened v e r s i o n of t h e assumption t h a t e r r o r s i n s u c c e s s i v e measurements a r e i n d e p e n d e n t . However, axiom I1 has no c o u n t e r p a r t i n p h y s i c a l s c i e n c e s , s i n c e i n
physics
t h e t r u e v a l u e i s n o t a random v a r i a b l e .
By axiom 11, t h e v a r i a n c e of t h e observed s c o r e i s t h e
sum o f two components: v a r i a n c e of t r u e s c o r e and v a r i a n c e o f e r r o r ' ) ' The o b j e c t i s t o e s t i m a t e t h e second ') F o r m a l d e r i v a t i o n s w i l l be p r e s e n t e d l a t e r i n t h i s chapter.
CONTEMFORARY THEORY OFPSYCHOLOGIC4L TESTS
9
o f t h e s e v a r i a n c e s . The r e a s o n i n g l e a d i n g t o d e t e r m i n a t i o n o f t h e v a r i a n c e of e r r o r s c o r e i s a s f o l l o w s . I f t h e v a r i a n c e of t r u e s c o r e i s l a r g e i n comparison w i t h t h e v a r i a n c e o f t h e o b s e r v e d s c o r e (hence i f t h e v a r i a n c e o f e r r o r s c o r e i s small as compared w i t h t h e v a r i a n c e o f observed s c o r e ) , t h e n -- b e c a u s e of axiom I11 a s s e r t i n g l a c k o f c o r r e l a t i o n i n two s u c c e s s i v e measurements -- t h e observed s c o r e s i n two measurements w i l l d i f f e r l i t t l e , hence t h e y w i l l be h i g h l y c o r r e l a t e d . On t h e o t h e r hand, i f t h e e r r o r v a r i a n c e i s l a r g e i n comparison w i t h t h e v a r i a n c e o f o b s e r v e d s c o r e , t h e measurements i n two t r i a l s w i l l have low c o r r e l a t i o n . F o r m a l l y , i t f o l l o w s from t h e axioms o f G u l l i k s e n , t h a t t h e c o e f f i c i e n t of c o r r e l a t i o n between two s u c c e s s i v e measurements i s e q u a l t o t h e s o - c a l l e d r e l i a b i l i t y of measurements, d e f i n e d as t h e r a t i o of t h e v a r i a n c e o f t r u e s c o r e s t o t h e v a r i a n c e of o b s e r v e d s c o r e s . Thus, t h e knowledge o f r e l i a b i l i t y , a c q u i r e d by e s t i m a t i n g t h e c o r r e l a t i o n c o e f f i c i e n t between t h e r e s u l t s of two measurements o f t h e same group of s u b j e c t s by t h e same t e s t , combined w i t h t h e knowledge of t h e v a r i a n c e of t h e observed s c o r e , allows us t o e s t i m a t e t h e v a r i a n c e of t r u e s c o r e , hence a l s o t h e e r r o r v a r i a n c e . I n t h i s way, i f one a c c e p t s t h e axioms o f G u l l i k s e n , one g e t s a p o s s i b i l i t y of e s t i m a t i o n of e r r o r v a r i a n c e , hence also c o n s t r u c t i o n of c o n f i d e n c e i n t e r v a l s f o r t h e unknown t r u e s c o r e of a g i v e n p e r s o n .
3.3. A c r i t i q u e of G u l l i k s e n ' s approach The axiom s y s t e m of G u l l i k s e n was c r i t i c i z e d ; t h e main
10
CHAPTER i!
o b j e c t i o n was t h e assumption t h a t t h e t r u e s c o r e e x i s t s i n d e p e n d e n t l y o f t h e f a c t o f measurement. F o r i n s t a n c e , Thorndike (1964) claimed t h a t s i n c e t h e t r u e scores a r e never observable d i r e c t l y (they a r e s u b j e c t t o measurement e r r o r s ) , , t h e n o t i o n of t r u e scor e i s a m y t h , which c a n n o t have any t h e o r e t i c a l s i g n i f i c a n c e . On t h e o t h e r hand, Loevinger ( 1 9 5 7 ) claimed t h a t t h i s c o n c e p t cannot have any p r a c t i c a l s i g n i f i c a n c e . According t o h e r , i t makes s e n s e t o d e a l o n l y w i t h observed s c o r e s . W i t h t h e p r e s e n t s t a t u s of s t a t i s t i c a l knowledge
, both
t h e s e o b j e c t i o n s a r e u n t e n a b l e , , s i n c e t h e f a c t of i m p o s s i b i l i t y of l e a r n i n g t h e e x a c t v a l u e of a p a r a m e t e r s h o u l d n o t be an argument a g a i n s t i n t r o d u c i n g t h i s parameter. A more s e r i o u s o b j e c t i o n a g a i n s t t h e axioms o f G u l l i k -
s e n , and i n p a r t i c u l a r a g a i n s t t h e concept o f t h e t r u e s c o r e , l i e s i n t h e f a c t t h a t G u l l i k s e n was f o r c e d t o a c c e p t p o s t u l a t e s f o r which t h e r e a r e no e m p i r i c a l p r o c e d u r e s of t h e i r v e r i f i c a t i o n . T h i s may be s e e n as f o l l o w s . If we assume, as i n p h y s i c a l s c i e n c e s , t h a t t h e t r u e v a l u e o f a t r a i t of a g i v e n p e r s o n e x i s t s i n d e p e n d e n t l y o f t h e f a c t o f measurement , we may d e f i n e t h e e r r o r s c o r e , simply as t h e d i f f e r e n c e between t h e o b s e r v e d and t r u e s c o r e s . To c o n s t r u c t a t h e o r y which would l e a d t o methods of i n f e r a n c e , G u l l i k s e n was f o r ced t o make some a s s u m p t i o n s about t h e e r r o r s c o r e s ( s u c h as t h e i r u n b i a s e d n e s s and l a c k of c o r r e l a t i o n i n c o n s e c u t i v e measurements). ‘Phis amounts t o assuming t h a t t h e t o o l of measurement i s u n b i a s e d : i t s a v e r a g e e r r o r s c o r e i s z e r o . Now, i n p h y s i c a l s c i e n c e s , s u c h
CONTEMPORARY THEORY OF PSYCHOLOGICAL TESTS
11
t y p e o f assumption may b e v e r i f i e d e x p e r i m e n t a l l y , simpl y by m e a s u r i n g many t i m e s t h e same known q u a n t i t y . On
t h e o t h e r hand, i n t h e t h e o r y o f t e s t s , t h e r e e x i s t no procedure f o r v e r i f y i n g t h e unbiasedness o f t h e t o o l s of measurement. The same a p p l i e s a l s o t o o t h e r p o s t u l a t e s of G u l l i k s e n .
4. CONCEPTUAL FOUNDATIONS
OF THE CONTEMPORARY
THEORY OF TESTS
4 . 1 . P r o p e n s i t y d i s t r i b u t i o n , t r u e and e r r o r s c o r e s D e s p i t e t h e above o b j e c t i o n s , t h e r e i s c o n s i d e r a b l e i n t u i t i v e a p p e a l o f t h e c o n c e p t s o f t r u e and e r r o r scor e s , based on s t r o n g a n a l o g i e s w i t h t h e p h y s i c a l measur e m e n t , and a l s o on t h e u t i l i t y o f t o o l s based on t h i s t h e o r y . T h i s caused Lord and Novick (1968) t o c o n s t r u c t a new a x i o m a t i z a t i o n of t h e t h e o r y , s u c h t h a t a l l axioms of G u l l i k s e n would become t h e o r e m s .
The s t a r t i n g p o i n t o f t h e c o n s t r u c t i o n o f Lord and Nov i c k (1968) i s t h e s y s t e m
where A i s t h e s e t o f p e r s o n s ( p o p u l a t i o n of s u b j e c t s ) , G i s t h e c l a s s o f t e s t s under s t u d y , ' r i s a p r o b a b i l i t y d i s t r i b u t i o n on A , and X i s a f a m i l y of random vaga = t ] i s t o be i n t e r p r e t e d as r i a b l e s . The e v e n t $ X ga "the subject a obtained the score t i n t e s t g".
12
CHAPTER 1
It i s assumed t h a t i f a # a ' , t h e n f o r a r b i t r a r y g , g '
i n G ( d i s t i n c t o r n o t ) , t h e random v a r i a b l e s X
and gYa X a r e i n d e p e n d e n t . T h i s means t h a t t h e r e s u l t s of g ' ,a' measurements of d i f f e r e n t s u b j e c t s a r e i n d e p e n d e n t . Let u s d e n o t e
(4.2) t h i s distribution function i s called the propensity
d i s t r i b u t i o n of s u b j e c t a i n t e s t g . Thus, t h e t e s t s c o r e of p e r s o n a i n t e s t g i s t r e a t e d as a r e a l i z a t i o n of a random v a r i a b l e , whose p o s s i b l e values are a l l potentially possible r e s u l t s of t h i s t e s t f o r t h e g i v e n p e r s o n , and whose p r o b a b i l i t y d i s t r i b u t i o n i s c h a r a c t e r i s t i c f o r t h i s t e s t and p e r s o n . Because of t h e a l r e a d y mentioned i m p o s s i b i l i t y o f p e r f o r m i n g i n d e p e n d e n t o b s e r v a t i o n s of t h e same t e s t f o r t h e same p e r s o n , t h e p r o p e n s i t y d i s t r i b u t i o n i s n o t o b s e r v a b l e . Thus, i t i s a c e r t a i n t h e o r e t i c a l c o n s t r u c t . I t s a c c e p t a n c e i s m o t i v a t e d by t h e e m p i r i c a l f a c t o f v a r i a b i l i t y of t e s t r e s u l t s f o r t h e same p e r s o n ; t h u s , i t i s a much s t r o n g e r argwnent t h a n t h e a n a l o g i e s w i t h p h y s i c a l measurement, used by G u l l i k s e n . POSTULATE 1. F o r e v e r y a 6 A
distribution F 2
d (Xga)
ga
g
GG
the propensity
has f i n i t e v a r i a n c e
= Var
(xga) c
03
.
From P o s t u l a t e 1 i t f o l l o w s i n p a r t i c u l a r t h a t propens i t y d i s t r i b u t i o n s have a l s o f i n i t e e x p e c t a t i o n s , t o be d e n o t e d by
CONTEMPORARY THEORY OF PSYCHOLOGICAL TESTS
= E(Xga).
et.
13
(4.3)
ga
DEFINITION. The e x p e c t a t i o n ( 4 . 3 ) i s c a l l e d t h e t r u e s c o r e of person a i n t e s t g. Next, d e n o t e 1 ga
= x
(4.4)
- T
ga
ga'
We have t h e r e f o r e E(1
since E ( I
ga
ga
) = 0
) = E(X
ga
)
(4.5)
- E(Tga)
Thus, t r u e s c o r e i s d e f i n e d i n t e r m s of p r o p e n s i t y d i s t r i b u t i o n as i t s e x p e c t a t i , o n , w h i l e t h e e r r o r s c o r e ( 4 . 4 ) i s d e f i n e d as t h e d i f f e r e n c e between t h e o b s e r v e d and t r u e s c o r e s . Thus, e r r o r s c o r e i s s t i l l unobservabl e , as i n t h e t h e o r y of G u l l i k s e n ; however, f o r e v e r y person, t h e expected e r r o r s c o r e i s z e r o i n v i r t u e of t h e d e f i n i t i o n , and n o t as a p o s t u l a t e , as i n t h e t h e o r y of Gulliksen.
It i s w o r t h t o m e n t i o n t h a t P o s t u l a t e 1 o f f i n i t e n e s s of v a r i a n c e i s a u t o m a t i c a l l y s a t i s f i e d i n a p p l i c a t i o n s of t e s t t h e o r y , s i n c e as a r u l e , e v e r y t e s t i s c o n s t r u c t e d i n s u c h a way t h a t i t s p o s s i b l e r e s u l t s a r e bounded; t h i s i mplies f i n i t e n e s s o f v a r i a n c e . F o r s y s t e m ( 4 . 1 ) we a c c e p t t h e f o l l o w i n g CONSTRAINT ON OBSERVABILITY. F o r any a one may make o n l y one o b s e r v a t i o n o f X
A and g 6 G ga'
14
CHAPTER 1
T h i s c o n s t r a i n t i m p l i e s t h e i m p o s s i b i l i t y of e s t i m a t i n g
variance cf2(Xga)
f o r any f i x e d g and a .
I n c i d e n t a l l y , t h e e l e m e n t s of t h e s e t G a r e t o be i n t e r p r e t e d n o t as t e s t s , b u t as d i s t i n c t c a s e s o f use of a s p e c i f i c t e s t . I n t h e s e q u e l , t o d e s c r i b e a r e p e a t ed a p p l i c a t i o n o f t h e same t e s t , t o t h e same p e r s o n , w e s h a l l use n o t a t i o n s s u c h as X and Xha, and we s h a l l ga make some assumptions about t h e r e l a t i o n s between t h e s e random v a r i a b l e s . The c o n s i d e r a t i o n s t h u s f a r d i d n o t use t h e d i s t r i b u t i o n ?T a c c o r d i n g t o which t h e p e r s o n s a r e sampled from p o p u l a t i o n A . The randomness of' c h o i c e of p e r s o n a e A caiises t h e t r u e s c o r e 'T t o be a random v a r i a b l e . I n ga t h e s e q u e l , t h i s random v a r i a b l e w i l l be d e n o t e d b y T * i t s v a l u e s are, t h e r e f o r e , t h e unobservable t r u e g*' scores 7 and i t s d i s t r i b u t i o n depends on t h e a c c e p t ga' ed s a m p l i n g scheme, i . e . t h e distribution r. Similarly, X w i l l d e n o t e t h e random v a r i a b l e , whose g* v a l u e s a r e t h e observed s c o r e s o f t h e p e r s o n s sampled. Thus, randomness o f Xg, i s o f a "double" c h a r a c t e r : f i r s t one samples t h e p e r s o n a , hence ( f o r f i x e d g) h i s
propensity d i s t r i b u t i o n F and t h e n one samples X ga ' ga according t o t h i s propensity d i s t r i b u t i o n . I n a s i m i l a r way one d e f i n e s t h e e r r o r s c o r e I g* f o r e v e r y p e r s o n a c A we have
Since
t h e same r e l a t i o n h o l d s f o r any sampling scheme%; we
have t h e r e f o r e
CONTEMPORARY THEORY OF PSYCHOLOGICAL TESTS
15
THEOREM 4 . 1 .
X
g*
= T
g*
+ I
g*
For s u b s e q u e n t c o n s i d e r a t i o n s , i t i s n e c e s s a r y t o s t r e n g h t e n somewhat P o s t u l a t e 1, namely assume POSTULATE 1 ' .
Observe t h a t P o s t u l a t e 1' d o e s n o t f o l l o w from Postul a t e 1 of f i n i t e n e s s of v a r i a n c e for e v e r y a . However, s u c h an i m p l i c a t i o n h o l d s i n t h e most i m p o r t a n t c a s e when a l l v a r i a n c e s V a r (X ) a r e bounded b y a common c o n s t a n t , i . e . when C2(X K for a l l a E A . ga
7'
<
From P o s t u ; a t e 1' and theorem 4 . 1 i t f o l l o w s t h a t we have COROLLARY. The v a r i a n c e
variance
2
(i
(I
g*
G 2(T
g*
) of t r u e s c o r e and t h e
) o f e r r o r s c o r e a r e bounded.
We s h a l l now p r e s e n t some f u r t h e r consequences o f t h e d e f i n i t i o n o f t h e t r u e s c o r e and e r r o r s c o r e , under P o s t u l a t e 1'; i n p a r t i c u l a r , w e s h a l l d e r i v e t h e p o s t u l a t e s of G u l l i k s e n as t h e o r e m s . F i r s t o f a l l , s i n c e for all a
A we have E ( I
) = 0,
ga t h e same remains t r u e r e g a r d l e s s of t h e way o f s e l e c t i n g p e r s o n s a t o t h e sample. We have t h e r e f o r e
THEOREM 4.2. E(1
g*
= 0,
(4.8)
16
CHAPTER i
hence t h e e x p e c t e d e r r o r s c o r e i s z e r o . Thus, t h e f i r s t axiom of G u l l i k s e n becomes a t h e o r e m i n t h e t h e o r y of Lord and Novick. N e x t , for any f i x e d p e r s o n a G A we have
(4.9) s i n c e E ( T I ) = T E ( I ) = 0 . S i n c e t h e same r e l a t i o n ga g a ga ga h o l d s under any s a m p l i n g scheme, w e have THEOREM 4 . 3 . E(T
I ) = 0. gn g*
(4.10)
I ) - E(T )E(Ign) g* g* g* and I ( t r u e and g* g* e r r o r s c o r e ) a r e u n c o r r e l a t e d . T h i s i s t h e second axiom
C o n s e q u e n t l y , Cov ( T g n , I g t )
= E(T
= 0 , i . e . t h e random v a r i a b l e s T
of G u l l i k s e n .
From t h e o r e m 4 . 3 and f o r m u l a ( 4 . 6 ) i t f o l l o w s t h a t we have COROLLARY.
(4.11) T o sum up t h e c o n s i d e r a t i o n s t h u s f a r , t h e o b s e r v e d s c o -
was r e p r e s e n t e d a d d i t i v e l y as a sum of t r u e and ga e r r o r s c o r e s , w i t h t h e terms 'being u n c o r r e l a t e d . More-
re X
o v e r , t h e e x p e c t a t i o n of t h e e r r o r s c o r e i s e q u a l z e r o . T h i s makes t h e f i r s t two axioms of G u l l i k s e n t h e theorems
of t h e t h e o r y of Lord and Novick.
CONTEMPORARY THEORY OF PSYCHOLOGICAL TESTS
4.2.
17
L i n e a r e x p e r i m e n t a l independence
We s h a l l now d i s c u s s t h e n o t i o n s and a s s u m p t i o n s conc e r n i n g t h e r e l a t i o n s between two d i s t i n c t measurements for t h e same p e r s o n . Let a ; ; A and l e t g , h G: G . It i s c l e a r , t h a t t h e s t r o n g e s t a s s u m p t i o n h e r e would be t h e l a c k o f any r e l a t i o n s h i p between t h e random v a r i a b l e s X and X h a , r e p r e s e n t €3 i n g t h e r e s u l t s o f measurement o f p e r s o n a w i t h t e s t s g and h . T h i s independence means t h a t t h e j o i n t d i s t r i and Xha bution of X ga distributions :
i s t h e p r o d u c t of t h e m a r g i n a l
S t a t i s t i c a l independence ( 4 . 1 2 ) i m p l i e s t h a t t h e o b s e r v ed s c o r e i n one t e s t does n o t p r o v i d e any i n f o r m a t i o n a b o u t t h e s c o r e on t h e o t h e r t e s t . Formally, t h e conditionsal d i s t r i b u t i o n P(Xga 5
XIXha
= y)
d o e s n o t depend on y , and e q u a l s t o t h e u n c o n d i t i o n a l d i s t r i b u t i o n F ( x ) (and s i m i l a r l y w i t h g and h i n t e r ga changed)
.
F o r t h e b a s i c theorems of t e s t t h e o r y , t h e a s s u m p t i o n
(4.12) i s t o o s t r o n g ; i t t u r n s o u t t h a t i t i s s u f f i c i e n t t o assume o n l y t h e s e c a l l e d l i n e a r e x p e r i m e n t a l i n d e pendence, which r e q u i r e s t h e c o n d i t i o n a l and uncondi-
18
CHAPTER I
t i o n a l e x p e c t a t i o n s to c o i n c i d e . More p r e c i s e l y , we s t a r t from t h e f o l l o w i n g d e f i n i t i o n : DEFINITION. The random v a r i a b l e s X and Xha ga ed l i n e a r l y e x p e r i m e n t a l l y i n d e p e n d e n t , i f
are call-
and
for any x. I f t h e above c o n d i t i o n s h o l d f o r e v e r y a , t h e measurements g and h a r e c a l l e d d i s t i n c t .
It i s e a s y to show t h a t l i n e a r e x p e r i m e n t a l independen-
c e i s weaker t h a n independence. I n d e e d , suppose t h a t t h e random v a r i a b l e s X and Xha have j o i n t d i s t r i b u t ga i o n which i s uniform on t h e domain ABCD ( s e e F i g . 4 . 1 ) .
I Example o f random v a r i a b l e s which a r e l i n e a r l y experimentally independent, but not s t o c h a s t i c a l l y independent.
Fig. 4.1.
CONTEMPORARY THEORY OF PSYCHOLOGICAL TESTS
19
These random v a r i a b l e s a r e n o t i n d e p e n d e n t : t o s e e t h i s ,
i t s u f f i c e s t o observe t h a t i n f o r m a t i o n t h a t X = x ga a l l o w s t o narrow down t h e p o s s i b l e v a l u e s of X to ha the i n t e r v a l [-q,ql, while the information t h a t X ga y narrows t h e s e v a l u e s down o n l y t o [ - p , p l .
=
On t h e o t h e r hand, t h e s e random v a r i a b l e s a r e l i n e a r l y e x p e r i m e n t a l l y i n d e p e n d e n t : i n d e e d , because of t h e symmetry, we have h e r e E ( X every x , that X ga ric with every x.
) = E(Xha) = 0 , w h i l e f o r ga t h e c o n d i t i o n a l d i s t r i b u t i o n of Xha, g i v e n = x i s uniform on an i n t e r v a l which i s symmetr e s p e c t t o 0 , hence E ( X h a l X g a = x ) = 0 f o r Similar reasoning applies a l s o t o X ga'
The a s s u m p t i o n of l i n e a r e x p e r i m e n t a l independence p l a y s c r u c i a l r o l e i n t e s t theory: i t replaces c e n t r a l i n s t a t i s t i c a l r e a s o n i n g a s s u m p t i o n o f i n d e p e n d e n c e , and i s a s p e c i f i c a s s u m p t i o n of t e s t t h e o r y a s s e r t i n g t h e r e g u l a r i t i e s of r e p e a t e d measurements for t h e same p e r son. N a t u r a l l y , t h e a s s u m p t i o n s of independence,
or linear
e x p e r i m e n t a l independence, a r e n o t t e s t a b l e e m p i r i c a l l y . To t e s t s u c h a n a s s u m p t i o n , one would have t o know t h e j o i n t p r o p e n s i t y d i s t r i b u t i o n f o r e a c h p e r s o n , and e a c h p a i r of measurements. A s a l r e a d y e x p l a i n e d , however, the propensity d i s t r i b u t i o n i s not a t t a i n a b l e empirically. D e s p i t e t h a t , t h e assumption o f l i n e a r e x p e r i m e n t a l independence was a c c e p t e d i n t e s t t h e o r y , as t h e weake s t among u n v e r i f i a b l e a s s u m p t i o n s , under which one may s t i l l o b t a i n a l l t h e major theorems o f t e s t t h e o r y . The v a l u e o f t h i s a s s u m p t i o n l i e s p r e c i s e l y i n t h e f a c t t h a t i t i s t h e weakest p o s s i b l e , hence g i v i n g h i g h e s t
20
CHAPTER 1
“ch.&nces” of adequacy of t h e t h e o r y
.
Under t h e assumption of l i n e a r e x p e r i m e n t a l independenand Xha one may p r o v e t h a t e r r o r s c o r e s I ce of X ga g* and Ihn a r e u n c o r r e l a t e d (which i s t h e t h i r d axiom i n G u l l i k s e n t h e o r y ) . We have namely THEOREM 4 . 4 .
If t h e measureme:=
g
and
h are d i s t i n c t ,
then E(1
(4.13)
g+I hn ) = 0.
Proof. F o r e v e r y f i x e d p e r s o n a we have E(Xga[Xha
=
x ) = E(X
ga
),
For a f i x e d Xha = x, t h e e r r o r Iha i s a c o n s t a n t , e q u a l x - Tha, s o - t h a t we may w r i t e , for any f i x e d a C A :
= EIIhaE(IgalXha
= x ) ] = EIIhaE(X
ga
) ] = E(Iha-O)
= 0.
for any f i x e d a , t h e same i s t r u e under any s a m p l i n g scheme, which p r o v e s ( 4 . 1 3 ) . Since E ( I g a I h a )
= 0
To swn up, Lord and Novick a c c e p t e d such a d e f i n i t i o n
of t h e t r u e s c o r e , under which t h e p r o p o s i t i o n which a s s e r t s t h a t error s c o r e has e x p e c t a t i o n z e r o i s a n a n a l y t i c a l s e n t e n c e , i . e . a s e n t e n c e whose t r u t h follows from t h e d e f i n i t i o n o n l y . The j u s t i f i c a t i o n o f t h i s p r o p o s i t i o n i s s i m p l e : for any f i x e d p e r s o n , t h e e x p e c t a t i o n of h i s e r r o r s c o r e i s z e r o , as t h e e x p e c t a t i o n of t h e d e v i a t i o n from t h e mean. C o n s e q u e n t l y , t h e
CONTEMPORARY THEORY OF PSYCHOLOGICAL TESTS
21
same r e m a i n s t r u e under any s a m p l i n g scheme.
The same h o l d s f o r t h e second axiom of G u l l i k s e n : f o r any f i x e d p e r s o n , t h e e x p e c t e d p r o d u c t of h i s t r u e and e r r o r s c o r e i s z e r o , simply because h i s t r u e s c o r e i s a c e r t a i n c o n s t a n t , and h i s e x p e c t e d e r r o r s c o r e is 0 . S i n c e t h i s i s v a l i d a l s o under any s a m p l i n g scheme, t h e t r u e and e r r o r s c o r e s a r e u n c o r r e l a t e d . The f i r s t two axioms of G u l l i k s e n r e q u i r e d o n l y t h e a s s u m p t i o n of f i n i t e n e s s of v a r i a n c e s . . H o w e v e r , t o obt a i n t h e t h i r d axiom o f G u l l i k s e n as a theorem, i t i s n e c e s s a r y t o assume t h a t measurements a r e d i s t i n c t , i . e . t h a t t h e measurements f o r any p e r s o n a r e l i n e a r l y exp e r i m e n t a l l y independent. Thus, w h i l e i n G u l l i k s e n ' s t h e o r y t h e t r u e s c o r e has a n " a b s o l u t e " c h a r a c t e r , and i s e x t e r i o r w i t h r e s p e c t t o measurement, i n t h e t h e o r y o f Lord and Novick t h e t r u e score i s defined through t h e ppopensity d i s t r i b u t i o n connected w i t h a g i v e n measurement t o o l , which s e r v e s as a s o u r c e o f s t a t i s t i c a l d a t a on b e h a v i o u r s . Because o f t h e s e d i f f e r e n c e s o f i n t e r p r e t a t i o n of t h e n o t i o n o f t r u e s c o r e , t h e G u l l i k s e n ' s approach was termed P l a t o n i a n , and t h e a p p r o a c h o f Lord and Novick statistical.
4 . 3 . The c o n c e p t o f p a r a l l e l measurements Most methods o f m a t h e m a t i c a l s t a t i s t i c s , i n p a r t i c u l a r e s t i m a t i o n methods, a r e developed for t h e c a s e when t h e o b s e r v a t i o n s are r e g a r d e d a s a r e a l i z a t i o n o f a m u l t i d i m e n s i o n a l random v a r i a b l e , whose components a r e
CHAPTER I
22
i n d e p e n d e n t and i d e n t i c a l l y d i s t r i b u t e d . A s a l r e a d y mentioned, t h e assumption of independence i s
r e p l a c e d i n t e s t t h e o r y by t h e a s s u m p t i o n of l i n e a r e x p e r i m e n t a l independence. Now we s h a l l i n t r o d u c e t h e assumption which s e r v e s i n t e s t t h e o r y as a c o u n t e r p a r t o f t h e a s s u m p t i o n of i d e n t i c a l d i s t r i b u t i o n s . S i n c e most c o n c e p t s and theorems o f t e s t t h e o r y c o n c e r n s only t h e f i r s t two moments of t h e observed s c o r e s , i . e . t h e i r means and ( c o ) v a r i a n c e s , t h e a s s u m p t i o n s about i d e n t i c a l d i s t r i b u t i o n s a r e n o t n e c e s s a r y ; i t i s enough t o assume i d e n t i t y of t h e f i r s t two moments. A c c o r d i n g l y , we i n t r o d u c e t h e f o l l o w i n g d e f i n i t i o n . D E F I N I T I O N . Two d i s t i n c t measurements g and h are c a l l -
ed : and
( a ) p a r a l l e l , i f f o r e v e r y a E A we have T = 2 2 ga 6 (X
ga
rha
) = d (Xha);
(b) 2-equivalent
(c) essentially f o r every a 6 A .
,
if
Tga = Zha f o r e v e r y a ir A ;
T - e q u i v a l e n t , if
ga = c gh + r ha
T
The i n t u i t i v e c o n t e n t o f t h e s e n o t i o n s i s a f o r m a l exp r e s s i o n of t h e r e q u i r e m e n t t h a t t h e two t e s t s measure t h e same t h i n g . Using t h e n o t i o n of t r u e s c o r e , as t h e e x p e c t a t i o n o f t h e p r o p e n s i t y d i s t r i b u t i o n , i t seems n a t u r a l t o a c c e p t t h a t two t e s t s measure t h e same t h i n g i f t h e i r e x p e c t a t i o n s ( f o r e v e r y s u b j e c t ) are t h e same. I n a d d i t i o n , one assumes also t h a t t h e measurement a r e d i s t i n c t ( l i n e a r l y experimentally independent). This,
CONTEMPORARY THEORY OF PSYCHOLOGICAL TESTS
23
i n a s e n s e , g u a r a n t e e s t h a t e a c h new measurement p r o v i d e s some new i n f o r m a t i o n about t h e t r u e s c o r e . F o r m a l l y , two t e s t s a r e 7 - e q u i v a l e n t , i f t h e y a r e lin e a r l y e x p e r i m e n t a l l y i n d e p e n d e n t , and f o r e a c h p e r s o n , t h e i r t r u e s c o r e s a r e t h e same; i n o t h e r words, i f f o r e a c h p e r s o n h i s e x p e c t e d r e s u l t i n one t e s t does n o t depend on t h e r e s u l t i n a n o t h e r t e s t , and t h e e x p e c t a t i o n s i n b o t h t e s t s a r e t h e same. On t h e one hand, t h i s d e f i n i t i o n may b e s t r e n g h t e n e d , by r e q u i r i n g i n a d d i t i o n t h e e q u a l i t y of v a r i a n c e s ; t h i s g i v e s t h e n o t i o n of p a r a l l e l measurements. Thus, two measurements a r e p a r a l l e l , i f t h e y a r e l i n e a r l y exp e r i m e n t a l l y i n d e p e n d e n t , and f o r e a c h p e r s o n , t h e prop e n s i t y d i s t r i b u t i o n s i n b o t h t e s t s have t h e same means and v a r i a n c e s ( s o t h a t t h e t e s t s measure t h e same w i t h t h e same p r e c i s i o n ) . On t h e o t h e r hand, one may a l s o weaken t h e d e f i n i t i o n , t o e s s e n t i a l T - e q u i v a l e n c e , by r e q u i r i n g t h a t t h e exp e c t e d s c o r e s i n b o t h t e s t s f o r any g i v e n p e r s o n d i f f e r by a c o n s t a n t which i s i n d e p e n d e n t on t h e p e r s o n . T h i s g i v e s t h e d e f i n i t i o n of e s s e n t i a l Z e q u i v a l e n c e , when t h e two measurements a r e l i n e a r l y e x p e r i m e n t a l l y i n d e pendent and t h e i r t r u e s c o r e s d i f f e r by a c o n s t a n t . Two e s s e n t i a l l y 2 - e q u i v a l e n t t e s t s , for which t h e cons t a n t a p p e a r i n g i n d e f i n i t i o n i s z e r o a r e a l s o T-equiv a l e n t ; i n t u r n , two T - e q u i v a l e n t t e s t s f o r which t h e v a r i a n c e s o f p r o p e n s i t y d i s t r i b u t i o n s c o i n c i d e f o r each person a r e p a r a l l e l . Consequently, p a r a l l e l t e s t s a r e a l s o 2 - e q u i v a l e n t , and T - e q u i v a l e n t t e s t s a r e a l s o e s s e n t i a l l y T-equivalent
.
24
CHAPTER .f
5. RELIABILITY
Let u s c o n s i d e r now a f i x e d t e s t g , and t h e o b s e r v e d
.
ard e r r o r score I One true score T score X g*’ g* g* o f t h e b a s i c n o t i o n s of t e s t t h e o r y i s t h a t of r e l i a b i l i t y , d e f i n e d as f o l l o w s .
DEFINITION. The r e l i a b i l i t y
of t e s t g E G i s d e f i n e d
as t h e r a t i o r
= G 2 (T
g
)/!3 2 ( X g * ) .
g*
(5.1)
2 2 2 S i n c e G (X ) = rS ( T ) t rJ ( I ) , r e l i a b i l i t y may be g* g* g* e x p r e s s e d by means o f e a c h o f t h e two f o r m u l a s below:
The n o t a t i o n r f o r r e l i a b i l i t y i s somewhat m i s l e a d i n g : g i t s u g g e s t s t h a t r e l i a b i l i t y depends o n l y on t h e t e s t g under c o n s i d e r a t i o n . I n f a c t , r e l i a b i l i t y depends on t h e v a r i a n c e of t r u e s c o r e , which measures t h e s p r e a d o f t r u e s c o r e s i n t h e p o p u l a t i o n under s t u d y . Thus, r e l i a b i l i t y depends i n a s i g n i f i c a n t way on t h e p o p u l a t i o n , or (more p r e c i s e l y ) , on t h e s a m p l i n g scheme ‘TT.
We s h a l l prove below some b a s i c theorems which c h a r a c t e r i z e r e l i a b i l i t y as a t h e o r e t i c a l p a r a m e t e r ; t h e f i r s t theorem r e l a t e s r e l i a b i l i t y w i t h c o r r e l a t i o n b e t ween t r u e and o b s e r v e d s c o r e s . The n e x t t h e o r e m s w i l l e x p r e s s some u n o b s e r v a b l e v a l u e s ( c o n c e r n i n g t r u e and
25
CONTEMPORARY THEORY OF PSYCHOLOGICAL TESTS
e r r o r s c o r e s ) i n terms o f observed v a l u e s . L e t us f i r s t d e t e r m i n e t h e c o v a r i a n c e between t h e t r u e
and o b s e r v e d s c o r e . O m i t t i n g f o r s i m p l i c i t y t h e i n d e x e s g and *, and w r i t i n g X, T and I f o r X T and I g*’ g* g* we have COV
(X,T) = E(XT)
-
=
E[(T+I)T]
=
E(T2)
+
E(X)E(T)
- E(TtI)E(T) E(IT) -[E(T)]*
- E(T)E(I)
or
where i n t h e d e r i v a t i o n we used t h e f a c t s t h a t E(1) = 0 and E(T1) = 0 . Since
s q u a r i n g b o t h s i d e s and u s i n g ( 5 . 3 ) we g e t
p2(X,T) =
(TI -- - d2 = o~(x)G~(T) d2(x)(i2(~) u2(x) Cov 2 (X,T)
--
d4(T)
rT
where rT s t a n d s f o r r e l i a b i l i t y of measurement X w i t h t r u e s c o r e T. We have t h e r e f o r e proved
26
CHAPTER I
THEOREM 5 . 1 .
i . e . r e l i a b i l i t y of a t e s t e q u a l s t h e s q u a r e o f t h e coef f i c i e n t o f c o r r e l a t i o n between t h e t r u e and o b s e r v e d s cores. R e w r i t i n g ( 5 . 5 ) i n an e q u i v a l e n t form p2(X,T)
= C2(T)/d2(X)
(5.6)
w e s e e t h a t theorem 5 . 1 g i v e s two n u m e r i c a l l y equiva-
l e n t ways of e x p r e s s i n g t h e r e l a t i o n between t h e o b s e r v ed and t r u e s c o r e s . The n e x t theorem c o n n e c t s r e l i a b i l i t y w i t h some p o t e n t i a l l y observable e n t i t i e s .
L e t X , X ' d e n o t e two p a r a l l e l measurements. From t h e e q u a l i t y o f t r u e s c o r e s and v a r i a n c e s of e r r o r s c o r e s f o r e a c h p e r s o n i t f o l l o w s t h a t t h e t r u e s c o r e s T and T 1 of measurements X and X I a r e i d e n t i c a l l y e q u a l , 2 2 i . e . T 2 TI, hence a l s o d ( T ) = 6 (T'). S i m i l a r l y , w e 2 2 2 have 0 ( I ) = d ( I 1 ) ,and t h e r e f o r e a l s o d ( X ) =
c2(X1).
I n o t h e r words, two p a r a l l e l measurements have t h e same v a r i a n c e s o f t r u e , e r r o r and o b s e r v e d s c o r e s . Let u s c a l c u l a t e t h e c o v a r i a n c e between X and X':
CONTEMPORARY THEORY OF PSYCHOLOGICAL TESTS
=
E(TT') t E(T1')
- E(T)E(I') =
E(TT')
-
-
-t
E(T'1) t E(I1')
E(T')E(I)
-
27
E(T)E(T')
- E(I)E(I')
E(T)E(T'),
s i n c e a l l o t h e r terms v a n i s h by t h e assumed l i n e a r exp e r i m e n t a l independence. Since T tain
Z
TI, hence TT' = T2 , and
D i v i d i n g b o t h s i d e s by \Id 2(X)d2(XT)
E(T) = E(T'), w e ob-
= 5 2 X), we g e t
(5.7) t h e c o r r e l a t i o n c o e f f i c i e n t between two p a r a l l e l measurements e q u a l s t o t h e r e l i a b i l i t y o f each o f t h e s e measurements. T h i s theorem p r o v i d e s u s w i t h a p o t e n t i a l p o s s i b i l i t y
of estimation of r e l i a b i l i t y ; it i s only necessary t o
have p a r a l l e l measurements. Methods o f a c h i e v i n g p a r a l l e l measurements w i l l be d i s c u s s e d i n n e x t s e c t i o n . A t p r e s e n t w e may s a y t h a t i f t h e r e s u l t s o f a two-fold a p p l i c a t i o n of t h e same t e s t c o n s t i t u t e p a r a l l e l measurements, t h e n t h e r e l i a b i l i t y o f t h e t e s t e q u a l s t o t h e c o e f f i c i e n t of c o r r e l a t i o n between two s u c h measurements.
28
CHAPTER 1
Theorems 5 . 1 and 5 . 2 l e a d t o e q u a l i t y
where X and X I a r e p a r a l l e l measurements. Each o f t h e t e r m s of ( 5 . 8 ) may s e r v e as a d e f i n i t i o n o f r e l i a b i l i t y of measurement. L e t u s r e w r i t e ( 5 . 7 ) i n t h e form
A f t e r s i m p l e t r a n s f o r m a t i o n s , 'we o b t a i n COROLLARY:
(5.9) or
The above f o r m u l a s e x p r e s s t h e s t a n d a r d d e v i a t i o n o f
e r r o r s c o r e t h r o u g h two p o t e n t i a l l y o b s e r v a b l e v a l u e s : s t a n d a r d d e v i a t i o n of o b s e r v e d s c o r e s , and r e l i a b i l i t y o f t h e t e s t (or: c o r r e l a t i o n between two p a r a l l e l r e p e t i t i o n s o f t h e same measurement). The formula ( 5 . 1 0 ) p l a y s c r u c i a l r o l e i n a p p l i c a t i o n s of t e s t t h e o r y , by p r o v i d i n g means of e s t i m a t i n g t h e e r r o r of measurement, hence a l s o p r o v i d i n g means for i n f e r e n c e about t h e t r u e s c o r e of t h e s u b j e c t , e . g . b y c o n s t r u c t i o n of confidence i n t e r v a l s f o r i t . F i n a l l y , t h e t h i r d theorem on r e l i a b i l i t y i s t h e s o - c a l l -
CONTEMM>RARY THEORY OF PSYCHOLOGICAL TESTS
29
ed a t t e n u a t i o n f o r m u l a , which p r o v i d e s an answer t o a n i m p o r t a n t t h e o r e t i c a l l y and p r a c t i c a l l y q u e s t i o n a b o u t t h e r e l a t i o n between t h e t r u e s c o r e s i n two d i s t i n c t measurements. To d e r i v e t h i s f o r m u l a , l e t X and Y d e n o t e two d i s t i n c t
measurements, w i t h t r u e s c o r e s r e s p e c t i v e l y T and T y , X and e r r o r s c o r e s I x , I y . The problem l i e s i n d e t e r m i n i n g t h e c o r r e l a t i o n ; j ( T T ) X’ Y between t h e t r u e s c o r e s i n t e r m s of o b s e r v a b l e q u a n t i t i e s : c o r r e l a t i o n between o b s e r v e d s c o r e s , and r e l i a b i l i t i e s of measurements X and Y . L e t u s f i r s t d e t e r m i n e t h e c o v a r i a n c e between t h e observed scores: COV ( X , Y )
= E(XY)
- E(X)E(Y)
s i n c e a l l terms i n t h e penultimate expression, except t h o s e which e n t e r i n t o c o v a r i a n c e between TX and Ty a r e z e r o . Thus, we proved COROLLARY:
If X and
Y are distinct
measurements, t h e n
30
CHAPTER 1
To d e t e r m i n e t h e c o e f f i c i e n t of c o r r e l a t i o n between TX and T y , l e t us d i v i d e b o t h s i d e s o f ( 5 . 1 1 ) by & ( T X ) d ( T y ) and l e t u s w r i t e
We o b t a i n e d t h e r e f o r e THEOREM 5 . 3 .
where rx and Y.
and
( A t t e n u a t i o n formula)
ry a r e r e l i a b i l i t i e s of measurements X
I f XI and Y' a r e measurements p a r a l l e l t o X and Y , t h e n by theorem 5 . 3 we may w r i t e
(5.13)
The meaning o f t h e s e f o r m u l a s i s as f o l l o w s : t h e y a l l o w t o estimate t h e c o r r e l a t i o n between two ( u n o b s e r v a b l e ) c o n s t r u c t s TX and Ty, u s i n g t h e o b s e r v a b l e c o e f f i c i e n t s o f c o r r e l a t i o n between measures X and Y o f t h e s e const r u c t s , and r e l i a b i l i t i e s o f measurements X and Y.
CONTEMDORARY THEORY OF PSYCHOLOGICAL TESTS
31
The p r a c t i c a l importance o f t h e n o t i o n o f r e l i a b i l i t y i s c o n n e c t e d p r i m a r i l y w i t h two t h e o r e m s . Theorem 5 . 1 ( o r , more p r e c i s e l y , formula ( 5 . 9 ) ) c o n n e c t s t h e e r r o r s c o r e v a r i a n c e w i t h t h e v a r i a n c e of o b s e r v e d s c o r e and r e l i a b i l i t y ( w i t h r e s p e c t t o t h e same p o p u l a t i o n ) . Theorem 5 . 3 ( A t t e n u a t i o n f o r m u l a ) corinects t h e c o r r e l a t i o n between t r u e s c o r e s o f two t e s t s w i t h c o r r e l a t i o n b e t ween o b s e r v e d s c o r e s i n t h e s e t e s t s and t h e i r r e l i a b i lities. G e n e r a l l y s p e a k i n g , i n b o t h t h e s e theorems a p a r a m e t e r which i s i m p o r t a n t f o r i n f e r e n c e i s e x p r e s s e d t h r o u g h some o t h e r p a r a m e t e r s , f o r which one may p r o v i d e a n e m p i r i c a l method of e v a l u a t i o n . I n t h e f i r s t c a s e , t h e parameter i n q u e s t i o n i s t h e v a r i a n c e o f e r r o r s c o r e , needed f o r b u i l d i n g c o n f i d e n c e i n t e r v a l s f o r t r u e s c o r e , e t c . I n t h e second c a s e , t h e p a r a m e t e r i n q u e s t i o n i s t h e c o r r e l a t i o n between t r u e s c o r e s i n two t e s t s , hence a measure o f t h e e x t e n t t o which t h e s e t e s t s measure t h e same t r a i t . I n b o t h c a s e s , t h e main p o i n t l i e s i n e s t i m a t i o n of r e l i a b i l i t y . Under t h e assumption t h a t r e p e a t e d measure-
ments w i t h t h e same t e s t a r e p a r a l l e l , a n e s t i m a t e of r e l i a b i l i t y i s o b t a i n e d t h r o u g h t h e c o r r e l a t i o n between measurements. T h i s , however, r e q u i r e s two measurements o f t h e same p o p u l a t i o n , which may be sometimes i n c o n v e n i e n t . Moreover, t h e i n f l u e n c e o f f a c t o r s s u c h as l e a r n i n g , memory, p o s s i b l e d i f f e r e n c e s i n t e s t i n g cond i t i o n s , e t c . may c a u s e t h e a s s u m p t i o n o f p a r a l l e l i s m t o be i n a d e q u a t e . Thus, t h e r e a r i s e s t h e problem of f i n d i n g theorems which could s e r v e a s a b a s i s f o r cons t r u c t i o n o f e s t i m a t o r s o f r e l i a b i l i t y which u s e t h e r e s u l t s of one measurement o n l y . T h i s problem i s c l o -
32
CHAPTER 1
s e l y r e l a t e d t o other i s s u e s concerning r e l i a b i l i t y , of g r e a t p r a c t i c a l and t h e o r e t i c a . 1 i m p o r t a n c e , namely t h e r e l a t i o n s between t h e l e n g t h o f a t e s t (number of i t e m s ) and i t s r e l i a b i l i t y .
6 . R E L I A B I L I T Y OF COMPOSITE TESTS
I n t h e c o n s i d e r a t i o n s t h u s f a r we d i d n o t t a k e i n t o account t h e i n t e r n a l s t r u c t u r e c f t e s t s : t h e y were t r e a t ed as w h o l e s . I n p r a c t i c e , p s y c h o l o g i c a l t e s t s a r e u su a l l y composed o f a s e r i e s of s u b t e s t s ( i n p a r t i c u l a r , i t e m s ) , whose r e s u l t s a r e t r e a t e d a d d i t i v e l y , g i v i n g t h e f i n a l score. I n such cases it i s n a t u r a l t o a s k f o r t h e r e l a t i o n s between t h e p a r a m e t e r s o f t h e t e s t as a whole, and t h e c o r r e s p o n d i n g p a r a m e t e r s of s u b t e s t s . O f p a r t i c u l a r i m p o r t a n c e h e r e a r e t h e f o r m u l a s which e x p r e s s t h e r e l a t i o n between r e l i a b i l i t y o f t h e t e s t as
a f u n c t i o n o f r e l i a b i l i t i e s o f i t s components. We s h a l l c a r r y on t h e d e t a i l e d c a l c u l a t i o n s for t h e
s p e c i a l c a s e o f two s u b t e s t s ; t h e g e n e r a l f o r m u l a s f o r a n a r b i t r a r y number of s u b t e s t s w i l l be g i v e n w i t h o u t proof. L e t us t h e r e f o r e c o n s i d e r t h e s i m p l e s t s i t u a t i o n , when
1 + Y 2’ where Y1 and Y2 a r e d i s t i n c t ( l i n e a r l y e x p e r i m e n t a l l y i n d e p e n d e n t ) measurements, w i t h t r u e s c o r e s T and T 2 and e r r o r s c o r e s I1 and I*. T h e r e f o r e 1
X
= Y
Y1 and
= T1
+
11, Y2 = T2
+
I
2
33
CONTEMPORARY THEORY OF PSYCHOLOGICAL TESTS
X + T + I ,
where T = T1
+
T2,
I = I1
+
I
2'
L e t us f i r s t d e t e r m i n e t h e v a r i a n c e o f t h e o b s e r v e d ,
t r u e and e r r o r s c o r e s f o r X . We have h e r e
G 2 ( T ) = 62 (T1 = G
2
+
T2) = a2(Tl)
t
2( T 2 )
+
2Cov(T1,T2)
2 (T,) + G ( T 2 ) + 2p(T1,T2)r(T1)G(T2) ( 6 . 2 )
(6.3) where t h e l a s t formula f o l l o w s from t h e f a c t t h a t t h e e r r o r s c o r e s i n d i s t i n c t measurements a r e u n c o r r e l a t e d . I n a p a r t i c u l a r c a s e when Y1 and Y2 a r e p a r a l l e l mea2 2 2 s u r e m e n t s , we have d (Y,) = d ( Y 2 ) , T1 = T 2 , G (T,) = 2 2 2 ( I 2 ) .D e n o t i n g f o r s i m p l i c i t y Y, 0 ( T 2 ) and G (I1) = and Y2 by Y and Y', f o r m u l a s ( 6 . 1 ) - ( 6 . 3 ) assume a
-
s i m p l e r form
34
CHAPTER 1
(6.6) where t o o b t a i n t h e f o r m u l a ( 6 . 5 ) from ( 6 . 2 ) w e used t h e f a c t t h a t T1- T 2 , hence p ( T 1 , T 2 ) = 1. It i s t o be remarked t h a t t h e v a r i a n c e o f t r u e s c o r e i n c r e a s e d by t h e f a c t o r of 4, w h i l e t h e v a r i a n c e o f t h e e r r o r s c o r e only d o u b l e d . A s a r e s u l t , t h e r e l i a b i l i t y of t h e composite t e s t exceeds t h e r e l i a b i l i t y of e a c h o f i t s components. To s e e t h i s , i t s u f f i c e s t o d e t e r m i ne t h e r e l i a b i l i t y of t e s t X ( u n d e r t h e a s s u m p t i o n t h a t t h e components Y and Y ' a r e p a r a l l e l ) :
s i n c e C 2 ( T , ) / S 2 ( Y ) i s t h e r e l i a b i l i t y of e a c h o f t h e components Y , Y ' , hence e q u a l s P ( Y , Y ' ) by theorem 5 . 2 , t h e e q u a l i t y ( 6 . 7 ) may b e r e w r i t t e n as
(6.8)
If X and X '
a r e p a r a l l e l measurements, t h e l a s t formu l a may be w r i t t e n as
(6.9)
CONTEMPORARY THEORY OFPSYCHOLOGICAL TESTS
35
c e e d s 1, hence
which p r o v e s t h a t t h e r e l i a b i l i t y of a t e s t composed o f two p a r a l l e l s u b t e s t s e x c e e d s t h e r e l i a b i l i t y o f e a c h o f t h e components. Analogous c o n s i d e r a t i o n s a p p l y to t h e c a s e of a t e s t composed o u t o f n p a r a l l e l components, One may namely prove t h e f o l l o w i n g theorem.
THEOREM 6 . 1 . L e t t h e measurements. Y1, r a l l e l , and l e t
x
= Y
1
+
Y
2
Y2,
...,
Y
n be pa-
*
t
...
+ Yn;
moreover, l e t P d e n o t e t h e c o e f f i c i e n t o f c o r r e l a t i o n between p a r a l l e l measurements Y1, Y i . Then
(6.11) (6.12) A s may be s e e n , t h e v a r i a n c e o f t r u e s c o r e grows f a s t e r
than the variance of e r r o r score with t h e increase of t h e number of components n . A s a r e s u l t , t h e r e l i a b i l i t y of a t e s t i n c r e a s e s w i t h i t s ' l e n g t h . It i s e a s y t o compute t h e r e l i a b i l i t y of t e s t X composed o u t o f n p a r a l l e l components; i t e q u a l s t o t h e f o l l o w i n g expression:
36
CHAPTER 1
using t h e f a c t t h a t C2(T1)/Z2(Y
) =,4Y 1
we o b t a i n
THEOREM 6.2. (Spearman-Brown). The r e l i a b i l i t y of a
t e s t c o n s i s t i n g o f n p a r a l l e l components e q u a l s
(6.13)
T h i s formula c o n n e c t s t h e r e l i a b i l i t y of a t e s t w i t h
t h e r e l i a b i l i t i e s of i t s components, and a l s o shows t h a t w i t h t h e i n c r e a s e o f n ( i . e . w i t h t h e i n c r e a s e of t h e number o f components) t h e r e l i a b i l i t y of a t e s t i n c r e a s e s t o 1. Formula ( 6 . 1 3 ) i s a s p e c i a l c a s e of a more g e n e r a l t h e orem which g i v e s t h e r e l a t i o n between t h e r e l i a b i l i t y o f a t e s t and p a r a m e t e r s of i t s components ( w i t h o u t assumption of p a r a l l e l i s m ) . W e have namely t h e f o l l o w i n g theorem (proved by Guttman, 1 9 4 5 ) , which w i l l be p r e s e n t e d here w i t h o u t p r o o f : THEOREM 6 . 3 .
Let
Y2,..., Y n be d i s t i n c t measurements, and l e t X = Y1 t Y2 t ... t Y n . Then t h e r e l i a Y1,
b i l i t y of t h e t e s t X s a t i s f i e s t h e i n e q u a l i t y
(6.14) T h i s theorem a l l o w s u s t o e s t i m a t e from below t h e rel i a b i l i t y o f a t e s t , knowing o n l y t h e v a r i a n c e s of i t s
components and t h e v a r i a n c e o f t h e composite t e s t (hen-
37
CONTEMPORARY THEORY OF PSYCHOLOGICAL TESTS
ce knowing t h e v a l u e s a t t a i n a b l e on t h e b a s i s o f one measurement o n l y )
.
The q u a n t i t y which a p p e a r s on t h e r i g h t hand s i d e o f (6,14), namely
(6.15) i s called the coefficient
.J(
of Cronbach ( s e e Cronbach
1951). The c o e f f i c i e n t & may be e t e r m i n e d e m p i r i c a l l y on t h e b a s i s o f one a p p l i c a t i o n o f a t e s t ; n a t u r a l l y , i t s val u e depends on tHe p a r t i t i o n of t h e t e s t X i n t o compon e n t s Y1,.
. . , 'n
*
Let u s imagine t h a t t h e t e s t c o n s i s t s o f 2n items (subOn t e s t s ) ; one may t h e n d e t e r m i n e t h e v a l u e d = 6 2n ' t h e o t h e r hand, one may a l s o c o n s i d e r v a r i o u s p a r t i t i o n s of t h e t e s t X i n t o two components, e a c h comprisi n g n i t e m s ( s u b t e s t s ) . F o r e a c h s u c h p a r t i t i o n , one may d e t e r m i n e t h e v a l u e d = d 2 . The f o l l o w i n g i m p o r t a n t theorem p r o v i d e s r e l a t i o n between p o s s i b l e v a l u e s o f d and t h e v a l u e d 2 n . 2
THEOREM 6 . 4 .
If t h e two p a r t s of t h e t e s t a r e s e l e c t e d
a t random, t h e n
I n o t h e r words, under a random p a r t i t i o n i n t o two p a r t s of n i t e m s e a c h , t h e e x p e c t e d v a l u e o f t h e c o e f f i c i e n t o(
o!
computed f o r s u c h a p a r t i t i o n e q u a l s t h e c o e f f i c i e n t f o r t h e p a r t i t i o n i n t o 2n i t e m s .
38
CHAPTER 1
Theorem 6 . 3 a s s e r t s t h a t t h e r e l i a b i l i t y o f a t e s t i s
.
a t l e a s t as h i g h as t h e c o e f f i c i e n t d This leads t o t h e q u e s t i o n under which c o n d i t i o n s we have t h e e q u a l i t y s i g n i n formula ( 6 . 1 4 ) .
I t i s e a s y t o s e e t h a t t h i s i s t h e c a s e when t h e compoY a r e p a r a l l e l , s i n c e t h e n t h e Cronbach's nents Y1,..., n e q u a l s t h e r i g h t hand s i d e of t h e f o r m u l a of S p e a r -
man-Brown. T o prove i t , l e t us c a l c u l a t e t h e v a l u e
4 i n c a s e when
2
...
t h e t e s t s a r e p a r a l l e l . We have t h e n G ( Y , ) = = nc2(Y1). N e x t , d 2 (X) = n C 2 ( Y l ) r n [ 1 t ( n - l ) p l . Consequently ,
V 2 ( Y ) and Z G 2 ( Y i )
The a s s u m p t i o n t h a t Y1,
...,Y n
a r e p a r a l l e l i s n o t nece-
s s a r y for e q u a l i t y i n formula ( 6 . 1 4 ) . t h a t we have t h e f o l l o w i n g theorem: THEOREM 6 . 5 .
I t a p p e a r s namely
I n order t h a t t h e equality sign holds i n
( 6 . 1 4 ) i t i s n e c e s s a r y and s u f f i c i e n t t h a t t h e measurements Y
-
1'""
Y
n
be e s s e n t i a l 1 . y 9 - e q u i v a l e n t .
Thus, for t e s t s which a r e composed o u t o f n e s s e n t i a l l y c - e q u i v a l e n t components Y 1 , , , , Y n , t h e r e l i a b i l i t y
equals
.
39
CONTEMPORARY THEORY OF PSYCHOLOGICAL TESTS
I n t h e p a r t i c u l a r c a s e when t h e s u b t e s t s Y1, dichotomous i t e m s , t h e l a s t f o r m u l a becomes
where pi and "no" known as addition then
...,
Yn a r e
and qi a r e t h e p r o b a b i l i t i e s o f answers "yes" r e s p e c t i v e l y i n i - t h i t e m . T h i s formula i s t h e Kuder-Richardson f o r m u l a KR 2 0 . When i s a l l items have t h e same mean ( p r o b a b i l i t y p i ) ,
which i s t h e Kuder-Richardson formu s KR and R i c h a r d s o n 1937).
1 ( s e e Kuder
The above a n a l y s i s o f r e l i a b i l i t y o f composite t e s t s may b e summarized as f o l l o w s . By d e f j n i t i o n , t h e t r u e s c o r e of s u c h a t e s t i s t h e sum o f t r u e s c o r e s of' t h e components, and t h e same i s t r u e f o r t h e e r r o r s c o r e s . The main problem l i e s i n d e t e r m i n i n g t h e r e l i a b i l i t y of t h e t e s t , g i v e n i n f o r m a t i o n a b o u t t h e parameters o f s u b t e s t s . The main r e s u l t h e r e i s t h e i n e q u a l i t y ( 6 . 1 4 ) which a s s e r t s t h a t t h i s r e l i a b i l i t y i s not l e s s t h a n some q u a n t i t y which i n v o l v e s o n l y t h e number o f s u b t e s t s , t h e i r v a r i a n c e s and t h e v a r i a n c e o f t h e composite t e s t . This i n e q u a l i t y does n o t allow us t o , d e t e r m i n e t h e rel i a b i l i t y , b u t p r o v i d e s a lower bound f o r i t . However, a n o t h e r theorem ( o f Spearman-Brown) a l l o w s u s t o b u i l d an e s t i m a t o r of r e l i a b i l i t y ; t h i s formula i s a l s o c a l l e d sometimes t h e s p l i t - h a l f f o r m u l a , and i s based on t h e
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assumption t h a t t h e p a r t s a r e p a r a l l e l . C o n s e q u e n t l y , i f only one knows how t o p a r t i t i o n t h e t e s t i n t o two p a r a l l e l s u b t e s t s , one may e s t i m a t e t h e r e l i a b i l i t y , knowing o n l y t h e v a r i a n c e s o f s u b t e s t s and t h e v a r i a n c e o f t h e t e s t as a whole. The t r o u b l e l i e s , of course, i n obtaining p a r a l l e l subtests. I n p r a c t i c e , a n e s t i m a t e o f r e l i a b i l i t y by t h i s method i s o b t a i n e d by p a r t i t i o n i n g t h e t e s t a t random ( e . g . by t a k i n g odd numbered i t e m s as one s u b t e s t , and even numb e r e d i t e m s as a n o t h e r s u b t e s t ; o r a l t e r n a t i v e l y , one may group items i n p a i r s w i t h e q u a l means, and a l l o c a t e members o f p a i r s randomly t o s u b t e s t s ) . The i d e a u n d e r l y i n g random p a r t i t i o n i s t h a t whatever t r a i t i s measured by t h e t e s t , i t must a l s o be measured by i t s s u b t e s t s , i f t h e y a r e randomly chosen. The method of e s t i m a t i n g r e l i a b i l i t y based on t h i s i d e a i s c a l l e d s p l i t - h a l f r e l i a b i l i t y e s t i m a t i o n . N a t u r a l l y , i t has been g e n e r a l i z e d i n t o p a r t i t i o n i n t o more t h a n two subtests. Such an e s t i m a t i o n d o e s n o t r e q u i r e two measurements, and i s t h e r e f o r e f r e e from b i a s e s due t o l e a r n i n g , memory e f f e c t , changes of t e s t i n g c o n d i t i o n s , e t c . However, t h e d i s a d v a n t a g e l i e s i n t h e f a c t t h a t i f t h e c o n d i t i o n o f p a r a l l e l i s m i s n o t met, one o b t a i n s an underestimate of r e l i a b i l i t y . B e s i d e s t h e method o f e s t i m a t i o n , t h e theorem o f Spearman-Brown g i v e s a v e r y i m p o r t a n t t h e o r e t i c a l conseque n c e , namely i t i m p l i e s t h a t r e l i a b i l i t y of a composit e t e s t i n c r e a s e s t o 1 w i t h t h e i n c r e a s e o f t h e number o f s u b t e s t s . T h i s f o l l o w s from t h e f a c t t h a t t h e v a r -
CONTEMPORARY THEORY OF PSYCHOLOGICAL TESTS
41
ia nce of t r u e scor e i n c r e a s e s f a s t e r t h a n t h e variance of e r r o r s c o r e . G e n e r a l l y , r e l i a b i l i t y as a p a r a m e t e r depends on t h e p o p u l a t i o n under s t u d y , and i t r e p r e s e n t s p r e c i s i o n o f measurement w i b h r e s p e c t t o t h i s p o p u l a t i o n . More p r e c i s e l y , r e l i a b i l i t y depends on t h e s a m p l i n g scheme, i . e . on t h e p r o b a b i l i t y d i s t r i b u t i o n a c c o r d i n g t o which t h e s u b j e c t s are sampled. A s a r u l e , t h e s e p r o b a b i l i t i e s c o n s t i t u t e a uniform d i s t r i b u t i o n on some s e t of p e r s o n s , and a r e z e r o f o r all o t h e r p e r s o n s ; t h e r e s u l t s , however, do n o t depend on t h i s p a r t i c u l a r form of sampling d i s t r i b u t i o n . A s a t h e o r e t i c a l p a r a m e t e r , r e l i a b i l i t y i t s e l f needs
t o be e s t i m a t e d on t h e b a s i s o f a sample, u s i n g t h e a p p r o p r i a t e e s t i m a t o r s . I n s h o r t , r e l i a b i l i t y i s a parameter of a c e r t a i n e s t i m a t o r , and i t s e l f i t must be e s t i m a t e d . Both t h e p r e s e n t e d methods o f e s t i m a t i o n ( " s p l i t - h a l f " and " c o r r e l a t i o n a l " ) have t h e i r w e l l defined s t a t i s t i c a l properties, leading t o determination of b i a s and s t a n d a r d e r r o r of r e l i a b i l i t y e s t i m a t e s , and c o n s e q u e n t l y , a l l o w i n g t h e c o n s t r u c t i o n o f c o n f i dence i n t e r v a l s .
7 . HOMOGENEITY OF A TEST
T e s t s f o r which t h e i r components Y 1 , . . . ,
..
yn
a r e essen-
t i a l l y c;: - e q u i v a l e n t a r e c a l l e d homogeneous. I n t u i t i v e l y , a t e s t i s homogeneous, i f e a c h o f i t s components
measure t h e same t r a i t ( u p t o an a d d i t i v e c o n s t a n t ) , though p e r h a p s w i t h v a r y i n g a c c u r a c y .
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CHAPTER I
Theorem 6 . 5 a s s e r t s t h a t a t e s t composed o f e s s e n t i a l l y T - e q u i v a l e n t components h a s t h e s a m a l l e s t p o s s i b l e rel i a b i l i t y , e q u a l t o t h e c o e f f i c i e n t .\ of Cronbach; conv e r s e l y , a t e s t whose r e l i a b i l i t y e q u a l s t h e c o e f f i c i e n t OC i s homogeneous. I n f o r m a l l y , homogeneity i s a concept which c o n c e r n s t h e r e l a t i o n between t h e t e s t as a whole, and i t s components (subtests), so t h a t it is r e l a t i v e with respect t o a p a 3 i t i o n o f the t e s t .
G e n e r a l l y , homogeneity means
t h a t e a c h o f t h e s u b t e s t s measures t h e same t r a i t . I n p r a c t i c e , homogeneity may be a t t a i n e d by c h o o s i n g i t e m s with appropriate content.
The t h e o r y o f t e s t s g i v e s a c r i t e r i o n f o r v e r i f y i n g whether a g i v e n t e s t i s homogeneous: t h e r e l i a b i l i t y of t h i s t e s t s h o u l d be e q u a l t o C r o n b a c h ' s
o(
,
hence
t o a q u a n t i t y which depends on t h e v a r i a n c e s o f subt e s t s and t h e v a r i a n c e o f t h e t e s t as a whole. The p r a c t i c a l a p p l i c a b i l i t y o f t h i s c r i t e r i o n r e l i e s on t h e p o s s i b i l i t y o f e s t i m a t i n g r e l i a b i l i t y and GA s e p a r a t e l y . The f i r s t may be e s t i m a t e d t h r o u g h t h e coe f f i c i e n t o f c o r r e l a t i o n i n two a p p l i c a t i o n s o f t h e
t e s t ( n a t u r a l l y , i n t h i s c a s e one s h o u l d a v o i d u s i n g t h e s p l i t - h a l f e s t i m a t o r s , s i n c e t h e y a r e based on t h e assumption o f e s s e n t i a l z - e q u i v a l e n c e , which i s t h e p r o p e r t y a b o u t which one t r i e s t o make i n f e r e n c e ) . The p o i n t i s t h a t homogeneity i s r e l a t i v e t o t h e p a r t i t i o n o f t e s t i n t o s u b t e s t s . To u n d e r s t a n d t h a t , i t i s enough t o r e a l i z e t h a t t h e r e may e x i s t t e s t s which a r e homogeneous under one p a r t i t i o n , and n o t homogeneous under a n o t h e r p a r t i t i o n . Imagine a t e s t c o n s i s t i n g o f
CONTEMPORARY THEORY OF PSYCHOLOGICAL TESTS
43
1 0 0 ( s a y ) i t e m s , which i s homogeneous under a p a r t i t i o n
i n t o 5 s u b t e s t s o f 20 i t e m s e a c h . I t i s e v i d e n t , howe v e r t h a t t h e s e 2 0 - i t e m s t e s t s need n o t be homogeneous under t h e i r p a r t i t i o n i n t o two 1 0 - i t e m t e s t s : t h e exp e c t a t i o n s i n t h e s e s u b t e s t s may be d i f f e r e n t , and onl y t h e i r sums c o r r e s p o n d i n g t o 2 0 - i t e m s u b t e s t s must be i d e n t i c a l . Thus, t h e o r i g i n a l 1 0 0 i t e m t e s t need n o t be homogeneous under t h e p a r t i t i o n i n t o 1 0 s u b t e s t s o f 10 items e a c h . F i n a l l y , i t i s w o r t h s t r e s s i n g t h a t homogeneity, unders t o o d a s "measuring t h e same" by s u b t e s t s may be f o r m a l l y e x p l i c a t e d i n v a r i o u s o t h e r w a y s , e . g . by r e q u i -
rement t h a t t h e s u b t e s t s be p a r a l l e l , e t c .
a.
VALIDITY
G e n e r a l l y s p e a k i n g , v a l i d i t y of a t e s t e x p r e s s e s t h e d e g r e e t o which t h e t e s t measures o r p r e d i c t s a g i v e n c r i t e r i o n . C o n s e q u e n t l y , v a l i d i t y of a t e s t ( o r a b a t t e r y of t e s t s ) i s r e l a t i v e t o a c r i t e r i o n , and a t e s t would t y p i c a l l y have many v a l i d i t i e s . Each o f t h e m , i n t u r n , may be e x p r e s s e d i n v a r i o u s w a y s as a c o e f f i c i e n t of some s o r t . G e n e r a l l y , one t a k e s h e r e t h e c o r r e l a t i o n c o e f f i c i e n t , which r e p r e s e n t s t h e d e g r e e o f l i n e a r p r e d i c t a b i l i t y of a g i v e n c r i t e r i o n by t h e r e s u l t s o f t h e t e s t or battery of t e s t s . The c r i t e r i a which a p p e a r h e r e may be p a r t i t i o n e d i n t o two c a t e g o r i e s : t h o s e c r i t e r i a which have t h e form o f t h e o r e t i c a l c o n c e p t s ( c o n s t r u c t s ) , and t h o s e which have t h e form o f o b s e r v a b l e v a r i a b l e s , connected w i t h t h e b e h a v i o r o f t h e s u b j e c t i n some t i m e i n t e r v a l .
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Thus, t h e knowledge of v a l i d i t i e s o f t h e t e s t w i t h r e s p e c t t o v a r i o u s c r i t e r i a a l l o w s u s t o d e t e r m i n e t h e val u e of t h e t e s t as a p s y c h o l o g i c a l t o o l . One can d i s t i n g u i s h two t y p e s of s t r a t e g i e s which l e a d t o d e t e r m i n i n g t h e psychologucal v a l u e of a t e s t . The f i r s t c l a s s of s t r a t e g i e s i s c o n s i s t e n t w i t h t h e
c o n c e p t of measurement, as e x p l i c a t e d i n t h e c l a s s i c a l t e s t t h e o r y . For a g i v e n t h e o r e t i c a l c o n c e p t , t h e cons t r u c t o r of a t e s t forms i t s d e s c r i p t i o n , i . e . b u i l d s a c e r t a i n t e s t . The b a s i c q u e s t i o n which a r i s e s i s whether or n o t t h e t e s t i n q u e s t i o n i s i n d e e d a t o o l for measurement o f t h e i n t e n d e d c o n s t r u c t ( i . e . whether i t i s v a l i d w i t h r e s p e c t t o t h e c o n s t r u c t which i t i s i n t e n d e d t o m e a s u r e ) . S i n c e c o n s t r u c t s are n o t d i r e c t l y m e a s ~ r a b 1 . e t~o f i n d t h i s v a l i d i t y ( c a l l e d c o n s t r u c t v a l i d i t y ) , one has t o use i n d i r e c t methods. The s u c c e s s i v e s t e p s of t h e p r o c e d u r e may b e r o u g h l y summarized as f o l l o w s . The f i r s t s t e p i s c o l l e c t i n g , on t h e ground of some p s y c h o l o g i c a l t h e o r y , o f a s e t o f a s s e r t i o n s about : ( a ) r e l a t i o n s o f t h e c o n s t r u c t under s t u d y w i t h some other constructs; ( b ) l a c k o f r e l a t i o n between t h e c o n s t r u c t under s t u d y
w i t h some o t h e r c o n s t r u c t s ;
( c ) r e l a t i o n s between t h e c o n s t r u c t under s t u d y w i t h observable v a r i a b l e s . The second s t e p c o n s i s t s o f t h e c h o i c e of t e s t s , which
are a c c e p t a b l e as measures o f c o n s t r u c t s which a p p e a r i n ( a ) and ( b ) .
CONTEMPORARY THEORY OF PSYCHOLOGICAL TESTS
45
The t h i r d s t e p c o n s i s t s o f f o r m u l a t i n g , on t h e b a s i s of
(a), (b) and ( c ) , of a s e t o f h y p o t h e s e s c o n c e r n i n g t h e measure o f t h e c o n s t r u c t ( t e s t ) . These h y p o t h e s e s a s s e r t t h a t i f t h e t e s t i s i n d e e d an a d e q u a t e measure o f t h e c o n s t r u c t which i t i n t e n d s t o measure, t h e n i t ought t o : ( A ) c o r r e l a t e h i g h l y w i t h t h e measures o f t h o s e c o n s t r u c t s which - a c c o r d i n g t o t h e t h e o r y -- s h o u l d be r e l a t ed t o t h e c o n s t r u c t measured ( t h i s t y p e o f v a l i d i t y i s r e f e r r e d t o as convergent v a l i d i t y ) ;
(B) n o t c o r r e l a t e w i t h t h e measures o f t h o s e c o n s t r u c t s which a r e -- a c c o r d i n g t o t h e t h e o r y -- n o t r e l a t e d t o the construct i n question ( t h i s i s called discriminant v a l i d i t y ) ;
(C) a l l o w a v a l i d p r e d i c t i o n o f t h o s e o b s e r v a b l e behavi o u r s which -- a c c o r d i n g t o t h e t h e o r y -- are r e l a t e d t o t h e construct i n question. Generally, t h e l a t t e r v a l i d i t i e s a r e r e f e r r e d t o as e m p i r i c a l , and d e p e n d i n g on w h e t h e r t h e y concern a c r i t e r i o n which i s r e l a t e d w i t h t h e p r e s e n t or f u t u r e b e h a v i o u r , t h e y are c a l l e d d i a g n o s t i c or p r o g n o s t i c . Finally, the fourth s t e p i s the empirical v e r i f i c a t i o n o f t h e s e t o f h y p o t h e s e s ( A ) , (B) and (C) by an approp r i a t e e x p e r i m e n t and s t a t i s t i c a l p r o c e d u r e s . It i s c l e a r , t h a t i n p a r t i c u l a r c a s e s some o f t h e s e t s ( A ) , ( B ) and ( C ) may be empty. A t any r a t e , a t l e a s t one o f t h e s e t s ( A ) and (C) must be nonempty. The reas o n i s t h a t a c o n s t r u c t about which one cannot a s s e r t a n y t h i n g p o s i t i v e c o n c e r n i n g i t s r e l a t i o n t o o t h e r con-
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CHAPTER f
s t r u c t s o r o b s e r v a b l e v a r i a b l e s , ( o r about which one can make only n e g a t i v e a s s e r t i o n s o f t h e t y p e i n (B)) cannot be t h e o r e t i c a l l y f e r t i l e .
,
A p o s i t i v e v e r i f i c a t i o n of h y p o t h e s e s ( A ) ,
(B) and ( C ) i s a premise f o r t h e h y p o t h e s i s t h a t t h e t e s t i n q u e s t i o n i s a n a d e q u a t e measure of t h e c o n s t r u c t .
Consequently, t h e f i r s t t y p e o f s t r a t e g y l e a d s t o det e r m i n a t i o n of p s y c h o l o g i c a l v a l i d i t y of a t e s t t h r o u g h more o r l e s s p r e c i s e d e t e r m i n a t i o n of i t s s e m a n t i c s . The second t y p e o f s t r a t e g y i s o f p r i m a r i l y p r a g m a t i c c h a r a c t e r . The s t a r t i n g p o i n t h e r e i s t h a t a t e s t det e r m i n e s a c e r t a i n a p r i o r i unknown s e t of c r i t e r i a , namely t h o s e f o r which i t may p r o v i d e a good p r e d i c t i o n . T h i s t y p e of s t r a t e g y c o n s i s t s o f d e t e r m i n i n g t h i s s e t of c r i t e r i a , by method of s u c c e s s i v e t e s t i n g whether o r n o t a g i v e n v a r i a b l e may be a d e q u a t e l y p r e d i c t e d by t h e r e s u l t s of t h e t e s t . I n o t h e r words, one t r i e s t o det e r m i n e e m p i r i c a l v a l i d i t y ( d i a g n o s t i c or p r o g n o s t i c ) w i t h respect t o various c r i t e r i a . In t h i s case, the p s y c h o l o g i c a l v a l u e o f t h e test, depends on t h e t y p e o f c r i t e r i a which i t p r e d i c t s . It i s c l e a r , t h a t t h i s a p p r o a c h may a l s o lead t o det e r m i n i n g t h e s e m a n t i c s o f t h e t e s t , t h r o u g h t h e psychol o g i c a l a n a l y s i s o f t h e c o n t e n t of t h e set o f p r e d i c t e d criteria. I n p r a c t i c e , t h e c r i t e r i a a r e n o t chosen a t random from some " u n i v e r s e of c r i t e r i a " , b u t a r e s e l e c t e d s y s t e m a t i c a l l y , a c c o r d i n g to t h e g o a l o f r e s e a r c h and t h e assumed c o n t e n t o f t h e t e s t ( t h e s o - c a l l e d f a c e validity).
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To sum up, t h e two t y p e s of s t r a t e g i e s o u t l i n e d above
c o n s t i t u t e two complementary s o u r c e s o f i n f o r m a t i o n about t h e psychological value of t e s t s . I n b o t h c a s e s , t h e v a l u e of t h e t e s t o r b a t t e r y of t e s t s , i s a s s e s s e d t h r o u g h some c o e f f i c i e n t s which e x p r e s s t h e v a l i d i t i e s of t h e t e s t w i t h r e s p e c t t o v a r i ous c r i t e r i a .
9 . PREDICTION
9 . 1 . Models of r e g r e s s i o n The problem o f d e t e r m i n i n g t h e v a l i d i t y of a t e s t or b a t t e r y of t e s t s w i t h r e s p e c t t o a g i v e n c r i t e r i o n may be r e p r e s e n t e d i n an a b s t r a c t way as t h e q u e s t i o n o f t h e d e g r e e t o which a g i v e n random v a r i a b l e ( t e s t ) or a v e c t o r o f random v a r i a b l e s ( b a t t e r y ) a l l o w s t o p r e d i c t t h e v a l u e s of some o t h e r random v a r i a b l e ( c r i t e r i o n ) . It s h o u l d be s t r e s s e d , t h a t t h e t o p i c of v a l i d i t y t r e a t -
ed as "goodness" o f p r e d i c t i o n , i s c o n s i d e r e d h e r e for a f i x e d s e t of p r e d i c t o r s and a f i x e d c r i t e r i o n . The l o g i c a l f o u n d a t i o n f o r p r e d i c t i o n i s t h e e x i s t e n c e
of s t a t i s t i c a l r e l a t i o n s h i p s between t h e p r e d i c t o r s and t h e c r i t e r i o n . The c o n s t r u c t i o n of p r e d i c t o r s i s based on t h e knowledge of t h e s e r e l a t i o n s h i p s , and t h e a c t u a l q u a l i t y o f p r e d i c t i o n depends on whether t h e o b s e r v a t i o n s a r e s u b j e c t t o t h e same laws which s e r v e d as a basis f o r d e t e r m i n i n g t h e p r e d i c t o r s ( i . e . finding the optimal prognosis).
48
I n general, t o a s e t of T h i s number r i o n . Thus,
CHAPTER 1
p r e d i c t i o n c o n s i s t s o f a s s i g n i n g one number v a l u e s o f t h e o b s e r v e d random v a r i a b l e s . i s r e f e r r e d to as p r e d i c t o r f o r t h e c r i t e t h e problem i s t o d e t e r m i n e a f u n c t i o n ,
which c o u l d s e r v e as p r e d i c t o r . To f o r m a l l y d e s c r i b e t h e s i t u a t i o n , l e t us f i r s t con-
s i d e r t h e s i m p l e s t c a s e : l e t u s namely imagine t h a t t h e p r e d i c t i o n s a r e t o concern t h e v a l u e s o f a random v a r i a b l e Y , and a r e based on o b s e r v a t i o n s of a random v a r iable X. A s already stated, i n order f o r prediction t o b e p o s s i b l e , i t i s n e c e s s a r y t h a t X and Y a r e s t a tistically related. The p r o g n o s i s i s a c e r t a i n f u n c t i o n u: i f we o b s e r v e t h e v a l u e X = x, t h e n we t a k e u ( x ) as t h e p r e d i c t e d v a l u e of Y . S i n c e t h e c r i t e r i o n Y i s random, i t assumes v a r i o u s val u e s d e p e n d i n g on chance. C o n s e q u e n t l y , t h e p r e d i c t i o n is subject t o an e r r o r ( i . e . the predicted value u(x> d i f f e r s from t h e a c t u a l v a l u e t a k e n up by Y). To d e t e r m i n e t h e o p t i m a l p r e d i c t i o n , i t i s n e c e s s a r y t o
d e f i n e t h e s o - c a l l e d loss f u n c t i o n , which e x p r e s s e s num e r i c a l l y t h e consequences o f a s s e r t i n g u = u ( x ) i f i n reality Y = y. L e t u s d e n o t e by L t h e loss f m c t i o n ,
d e p e n d i n g on two v a r i a b l e s : t h e p r o g n o s i s u , and t h e v a l u e y o f t h e c r i t e r i o n . The problem of f i n d i n g t h e o p t i m a l p r e d i c t o r
may t h e n be f o r m u l a t e d as t h a t o f f i n d i n g t h e f u n c t i o n
u f o r which
EL[Y, u ( X ) ] = min.
CONTEMPORARY THEORY OF PSYCHOLOGICAL TESTS
49
Here t h e e x p e c t e d v a l u e is c a l c u l a t e d w i t h r e s p e c t t o t h e j o i n t d i s t r i b u t i o n o f X and Y . I n g e n e r a l , t h e loss f u n c t i o n depends on t h e e r r o r o f p r e d i c t i o n , t h a t i s , on t h e d i f f e r e n c e y - u ( x ) . The b e s t developed mathematically i s h e r e t h e p r e d i c t i o n t h e o r y i n which t h e l o s s f u n c t i o n i s d e f i n e d as t h e s q u a r e o f e r r o r ; t h e n t h e p a r a m e t e r which e x p r e s s e s t h e goodness of p r e d i c t i o n i s t h e mean s q u a r e e r r o r of p r o g n o s i s . T h i s t h e o r y i s known under t h e name r e g r e s s ion theory. Formally, l o s s f u n c t i o n
equals here
hence EL[Y,
u(X)]
= E[Y
-
u(X)I2.
R e g r e s s i o n t h e o r y i s based on t h e theorem which a s s e r t s t h a t i n p r e d i c t i n g a v a l u e o f a random v a r i a b l e , i n t h e s e n s e o f minimal s q u a r e e r r o r , t h e o p t i m a l p r e d i c t i o n i s a l w a y s t h e e x p e c t a t i o n of t h i s random v a r i a b l e . T h i s theorem a l l o w s t h e r e f o r e t o d e s c r i b e t h e f u n c t i o n which i s t h e o p t i m a l p r e d i c t i o n i n t h e c l a s s of a l l f u n c t i o n s ; t h i s function i s called t h e t r u e regression. For a given set of p r e d i c t o r s , t h e v a l u e of t h i s f u n c t i o n e q u a l s simply t h e e x p e c t e d v a l u e o f t h e c r i t e r i o n , g i v en t h e observations. Thus, t h e f u n c t i o n u f o r which t h e e x p e c t e d s q u a r e e r r o r i s minimal i s d e f i n e d by t h e r e l a t i o n u(x) = E(Y(X
=
x).
CHAPTER I
50
I n g e n e r a l c a s e , when t h e p r e d i c t i o n i s based on observa t i o n o f n random v a r i a b l e s X1,
.. ., X n ,
t h e problem i s
t o f i n d a f u n c t i o n u of n v a r i a b l e s s u c h t h a t ECY
-
u(X1,.
. ., X n ) l
2
=
miri.
I n a s i m i l a r w a y , t h e o p t i m a l p r o g n o s i s , which minimizes t h e mean s q u a r e e r r o r , has t h e form
The f u n c t i o n s o d e f i n e d i s c a l l e d t h e t r u e r e g r e s s i o n
.
( o f Y on (Xl,.. , X n ) ) .
A p p l i c a t i o n o f t r u e r e g r e s s i o n as a f u n c t i o n of p r e d i c t i o n l e a d s , n a t u r a l l y , t o t h e smallest mean s q u a r e e r r o r of p r e d i c t i o n , i n d e p e n d e n t l y o f t h e v a l u e s of p r e d i c t o r s ; because o f t h a t , t h e t r u e r e g r e s s i o n i s t h e b e s t p o s s i b l e p r e d i c t o r of t h e c r i t e r i o n . However, t o det e r m i n e t h e t r u e r e g r e s s i o n r e q u i r e s t h e knowledge o f t h e j o i n t d i s t r i b u t i o n of t h e v a r i a b l e s Xi and t h e c r i t e r i o n Y , and t h e f o r m u l a s a r e u s u a l l y r a t h e r cwnbersome. Because o f t h a t , i n p r a c t i c e one u s e s o f t e n t h e s o - c a l l e d l i n e a r r e g r e s s i o n , c a l l e d a l s o t h e regr e s s i o n of t h e second k i n d . It i s o b t a i n e d by r e s t r i c t i o n of t h e c l a s s o f p r e d i c t i o n f u n c t i o n s t o l i n e a r f u n c t i o n s , i . e . t o l i n e a r c o m b i n a t i o n s of t h e o b s e r v e d v a l u e s . By c h o o s i n g t h e o p t i m a l f u n c t i o n from t h i s c l a s s ( i . e . by c h o o s i n g t h e l i n e a r f u n c t i o n which give s t h e minimal mean s q u a r e e r r o r o f p r e d i c t i o n ) one o b t a i n s t h e l i n e a r r e g r e s s i o n f u n c t i o n . I n o t h e r words, l i n e a r r e g r e s s i o n i s b e s t i n t h e c l a s s of a l l l i n e a r p r e d i c t o r s . The w e i g h t s o f p r e d i c t o r s i n t h e r e g r e s s i o n
CONTEMPORARY THEORY OF PSYCHOLOGICAL TESTS
51
e q u a t i o n , and t h e f r e e t e r m , a r e c a l l e d t h e r e g r e s s i o n coefficients. F o r m a l l y , t h e c o n s i d e r a t i o n s are r e s t r i c t e d t o f u n c t i o n s o f t h e form
. ,xn)
u(xl,..
= a0
+ alxl +
... +
anxn.
The mean s q u a r e e r r o r e q u a l s t h e n E ( Y - a
0
- a X 1 1
... -
a X ) n n
2
and t h e problem r e d u c e s t o d e t e r m i n i n g t h e r e g r e s s i o n * * * c o e f f i c i e n t s , i . e . t h e v a l u e s a Q , al, ..., an which m i nimize t h e above e x p r e s s i o n . A s opposed t o t h e c a s e o f t r u e r e g r e s s i o n , d e t e r m i n i n g
t h e r e g r e s s i o n c o e f f i c i e n t s does n o t p r e s e n t t e c h n i c a l d i f f i c u l t i e s , and r e q u i r e s t h e knowledge o n l y o f t h e means, v a r i a n c e s and c o v a r i a n c e s between p r e d i c t o r s and between p r e d i c t o r s and t h e c r i t e r i o n (and n o t of t h e f u l l knowledge o f t h e ' j o i n t d i s t r i b u t i o n ) . F o r a n example, l e t u s c o n s i d e r t h e problem o f f i n d i n g t h e o p t i m a l l i n e a r p r e d i c t i o n o f Y on t h e b a s i s o f one random v a r i a b l e X . The p r e d i c t i o n f u n c t i o n i s i n t h i s c a s e of t h e form
a0
+ alX,
and t h e r e g r e s s i o n c o e f f i c i e n t s a r e t o be d e t e r m i n e d so that
E(Y
-
a.
-
alx)'
= min.
52
CHAPTER 1
One can c a l c u l a t e ( s e e e . g . Lord and Novick, 1 9 6 8 ) t h e o p t i m a l v a l u e s o f c o e f f i c i e n t s , as
a.* = E(Y)
-
E(X)
p(X,Y)-
6(X)
and
*
-
"1 -
5(Y) (j-0
JXX,Y)
*
*
*
The numbers a and a l a r e r e g r e s s i o n c o e f f i c i e n t s of 0 Y on X , and t h e o p t i m a l l i n e a r p r e d i c t i o n i s of t h e form 'predicted
*
= a.
t
*
alX.
After determining t h e optimal I-inear prediction, t h e r e a r i s e s t h e problem o f e v a l u a t i n g t h e q u a l i t y o f t h i s p r e d i c t i o n . The b a s i c q u a n t i t y which c h a r a c t e r i z e s t h e q u a l i t y of p r e d i c t i o n i s t h e c o r r e l a t i o n between t h e c r i t e r i o n and t h e o p t i m a l l i n e a r p r e d i c t i o n ( i . e . t h a t l i n e a r combination of p r e d i c t o r s which y i e l d s minimal mean s q u a r e e r r o r ) . T h i s c o e f f i c i e n t is c a l l e d t h e mult i p l e correlation coefficient. I n c a s e when t h e p r e d i c t o r s a r e t e s t s , t h e m u l t i p l e c o r r e l a t i o n c o e f f i c i e n t i s t a k e n as a measure o f t h e v a l i d i t y of t e s t s f o r t h e c r i t e r i o n i n q u e s t i o n . To sum up, t h e l i n e a r r e g r e s s i o n , as opposed t o t r u e r e g r e s s i o n , does n o t have t o b e t h e b e s t among a l l p r e d i c t o r s ; i n o t h e r words, t h e l i n e a r and t r u e r e g r e s s i o n d o n o t need t o c o i n c i d e . The m o t i v a t i o n f o r u s e o f t h e l i n e a r r e g r e s s i o n l i e s , however, i n t h e f a c t ( b e s i d e s t h e a l r e a d y mentioned s i m p l i c i t y ) t h a t i n t h e p r a c t i -
CONTEMPORARY THEORY OF PSYCHOLOGICAL TESTS
53
c a l l y most i m p o r t a n t c a s e when t h e j o i n t d i s t r i b u t i o n o f t h e p r e d i c t o r s and t h e c r i t e r i o n i s normal, t h e lin e a r and t r u e r e g r e s s i o n c o i n c i d e . I n o t h e r words, i f t h e j o i n t d i s t r i b u t i o n i s normal, t h e t r u e r e g r e s s i o n is linear. A t t h e e n d , i t i s w o r t h t o s t r e s s t h a t t h e use of t h e m u l t i p l e c o r r e l a t i o n c o e f f i c i e n t as a measure o f t h e q u a l i t y of p r e d i c t i o n i s s p e c i f i c a l l y r e l a t e d t o t h e use o f mean s q u a r e e r r o r as t h e l o s s f u n c t i o n . I n t h e o r y , one c a n i n t r o d u c e o t h e r loss f u n c t i o n s as measures o f e r r o r ; i n s u c h a c a s e , t h e c o e f f i c i e n t of v a l i d i t y would b e some number, s p e c i f i c a l l y r e l a t e d t o loss function, other than m u l t i p l e correlation coefficient.
P r e d i c t i n g a c r i t e r i o n i s o n l y one of t h e p o s s i b l e t a s k s of a p s y c h o l o g i s t u s i n g t e s t s o r a b a t t e r y o f t e s t s as a s o u r c e of i n f o r m a t i o n a b o u t t h e s u b j e c t . These t a s k s may b e v a r i e d : i t may b e , f o r i n s t a n c e , a c l i n i c a l d i a g n o s i s , an o p i n i o n about t h e i n t e l l i g e n c e l e v e l , a d v i c e on c h o o s i n g a p r o f e s s i o n , p r e d i c t i o n of f u t u r e succ e s s e s or f a i l u r e s , and s o on. I n e a c h c a s e , we have a s e t o f p o s s i b l e d e c i s i o n s , and t h e p s y c h o l o g i s t must make one of them. For i n s t a n c e , i n c a s e of c l i n i c a l d i a g n o s i s , t h e d e c i s i o n s a r e simply t h e c a t e g o r i e s ; i n c a s e of d e t e r m i n n g t h e l e v e l of i n t e l l i g e n c e , t h e d e c i s i o n s may be s i m p l y t h e v a l u e s of a c e r t a i n i n d e x , s u c h a s I&. I n o t h e r c a s e s , t h e d e c i s i o n s e t may c o n t a i n j u s t two e l e m e n t s , e . g . "send t o a s p e c i a l s c h o o l " a g a i n s t " n o t t o send t o a s p e c i a l school". The p s y c h o l o g i s t ' s o b j e c t i v e i s t o choose t h e d e c i s i o n
which i s o p t i m a l w i t h r e s p e c t t o t h e g i v e n s u b j e c t .
54
CHAPTER I
O p t i m a l i t y depends h e r e , of c o u r s e , on t h e a c c e p t e d c r i t e r i o n . It may g e n e r a l l y be assumed t h a t t h i s c r i t e r i o n has t h e form of a c e r t a i n v a r i a b l e , whose v a l u e i s unknown a t t h e time of d e c i s i o n making. A d e c i s i o n r u l e i s a r u l e , which c o n n e c t s d e c i s i o n s w i t h t h e observed v a l u e s of v a r i a b l e s . A r u l e l e a d s t o
e x p e c t e d l o s s connected w i t h u s i n g i t , s i n c e any r u l e may sometimes l e a d t o wrong d e c i s i o n s . An o p t i m a l de-
c i s i o n r u l e i s such which minimizes t h e e x p e c t e d loss, and t h e l a t t e r may s e r v e as a measure of v a l i d i t y o f t h e v a r i a b l e s which s e r v e f o r making d e c i s i o n ( p r e d i c t -
ors) w i t h r e s p e c t to t h e g i v e n c r i t e r i o n .
9.2. C o n s t r u c t i o n o f b a t t e r y o f p r e d i c t o r s by s c r e e n i n g
Thus f a r , we d i s c u s s e d t h e problem of d e t e r m i n i n g t h e v a l i d i t y of a t e s t o r a b a t t e r y of t e s t s w i t h r e s p e c t
to a g i v e n c r i t e r i o n . L e t us c o n s i d e r now, for a g i v e n c r i t e r i o n , t h e problem o f c h o i c e o f a b a t t e r y o f t e s t s ( f r o m a g i v e n p o o l ) which would be most v a l i d for t h i s c r i t e r i o n . More p r e c i s e l y : t h e p o o l of t e s t s and t h e c r i t e r i o n are f i x e d , and t h e problem c o n s i s t s e i t h e r on f i n d i n g a b a t t e r y w i t h t h e g i v e n s i z e and h i g h e s t v a l i d i t y , or a b a t t e r y w i t h g i v e n v a l i d i t y and s m a l l e s t possible size. Thus f a r , t h e s e problems have n o t been s o l v e d t h e o r e t i c a l l y i n a s a t i s f a c t o r y way, i . e . i n a way which would a l l o w p r a c t i c a l i m p l e m e n t a t i o n . The o b v i o u s sug g e s t i o n of i n s p e c t i n g a l l p o s s i b l e s e t s of a g i v e n s i z e and f i n d i n g t h e o p t i m a l one i s i m p r a c t i c a l , even
CONTEMPORARY THEORY OF PSYCHOLOGICAL TESTS
55
w i t h t h e use of computers, s i n c e t h e number o f s e t s t o be i n s p e c t e d i s e x c e e d i n g l y l a r g e .
I n p r a c t i c e , two approaches have been s u g g e s t e d ; none o f t h e m , however, g u a r a n t e e s t h e o p t i m a l c h o i c e . The f i r s t a p p r o a c h t e l l s t o choose a c c o r d i n g t o t h e following principles. Include i n the battery: ( a ) t h o s e t e s t s which c o r r e l a t e h i g h l y w i t h t h e c r i t e r i o n , and have p o s s i b l y l o w e s t i n t e r c o r r e l a t i o n s ; ( b ) t h o s e t e s t s which c o r r e l a t e low w i t h t h e c r i t e r i o n ,
but c o r r e l a t e h i g h l y w i t h t h e t e s t s o f t h e b a t t e r y . The i n t u i t i v e j u s t i f i c a t i o n of t h e s e p r i n c i p l e s i s as f o l l o w s . 8 ~r e g a r d s ( a ) , t h e recommendation t h a t t h e t e s t s s h o u l d c o r r e l a t e h i g h l y w i t h t h e c r i t e r i o n requi r e s no comments. On t h e o t h e r hand, t h e recommendation t h a t t h e t e s t s s h o u l d have low i n t e r c o r r e l a t i o n s i s connected w i t h t h e i n t e n t i o n of h a v i n g t e s t s , e a c h o f which s u p p l i e s new i n f o r m a t i o n about t h e c r i t e r i o n . I n d e e d , i f t h e s e t e s t s had h i g h i n t e r c o r r e l a t i o n s , t h e i n f o r m a t i o n s u p p l i e d by them would be o v e r l a p p i n g . Recommendation ( b ) i s connected w i t h t h e p o s s i b i l i t y o f t h e f o l l o w i n g phenomenon. The i n t e r c o r r e l a t i o n s o f t h e v a r i a b l e s which have h i g h c o r r e l a t i o n w i t h t h e c r i t e r i o n may be d u e , a t l e a s t i n p a r t , t o t h e f a c t t h a t t h e y measure n o t only t h e c r i t e r i o n , b u t a l s o som e o t h e r v a r i a b l e , u n r e l a t e d t o i t . The t e s t s chosen a c c o r d i n g t o recommendation ( b ) , c a l l e d s u p p r e s s o r var i a b l e s , are aimed a t e l i m i n a t i n g (by s u b t r a c t i o n ) o f t h i s u n r e l a t e d v a r i a b l e , t h u s enhancing the v a l i d i t y
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o f t h e b a t t e r y as a whole. It i s w o r t h s t r e s s i n g t h a t i n p r a c t i c e , good s u p p r e s s o r v a r i a b l e s are h a r d t o f i n d . Moreover, some of t h e v a r i a b l e s which were t h o u g h t t o be s u p p r e s s o r s , l o s t t h i s q u a l i t y u n d e r i n c r e a s e of t h e sample. The second a p p r o a c h t o t h e c o n s t r u c t i o n of a b a t t e r y of t e s t s by s c r e e n i n g , i s based on t h e n o t i o n of i n c r e mental v a l i d i t y . T h i s v a l i d i t y concerns a s i n g l e t e s t w i t h r e s p e c t t o a g i v e n c r i t e r i o n and b a t t e r y and i s d e f i n e d as t h e d e g r e e t o which i n c l u s i o n of t h i s t e s t
t o t h e b a t t e r y would i n c r e a s e t h e v a l i d i t y o f t h e b a t te r y
.
The programs of c h o i c e of t e s t s t o t h e b a t t e r y a r e b u i l t
on t h e b a s i s o f t h i s n o t i o n . These programs e i t h e r i n clude a
t e s t which would yie1.d h i g h e s t i n c r e m e n t a l va-
l i d i t y of a b a t t e r y w h i c h h a s alrieady been s e l e c t e d , o r p r o c e e d "backwards", t h r o u g h a n a l y s i s of t h e l o s s i n v a l i d i t y which would r e s u l t . from removing a g i v e n t e s t from t h e b a t t e r y . A s a l r e a d y mentioned, n e i t h e r o f t h e s e programs g u a r a n t e e s t h e o p t i m a l c h o i c e . The r e a s o n s a r e c o n n e c t e d w i t h t h e f o l l o w i n g phenomenon. It may namely happen t h a t i f we t a k e t h e o p t i m a l b a t t e r y of s i z e k , and add t o i t t h e t e s t w i t h h i g h e s t i n cremental v a l i d i t y , tnen t h e r e s u l t i n g b a t t e r y of s i z e ktl need n o t be o p t i m a l . Already i n t h e s i m p l e s t c a s e , i f we t a k e t h e o p t i m a l t e s t arid add t o i t t h e t e s t w i t h h i g h e s t i n c r e m e n t a l v a l i d i t y , we may o b t a i n a p a i r which i s n o t o p t i m a l . The f i n a l number of t e s t s which a r e n e c e s s a r y f o r a p r e d i c t i o n w i t h s p e c i f i e d l e v e l of v a l i d i t y cannot be
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d e t e r m i n e d a p r i o r i , and depends on t h e c o m p l e x i t y o f the t e s t s andthe criterion. I n g e n e r a l , t h e t o p i c s of o p t i m a l c h o i c e of b a t t e r y of t e s t s c o n t a i n s t i l l many open problems.
9 . 3 . The e x p e c t e d and a c t u a l q u a l i t y o f p r e d i c t i o n Both t h e o p t i m a l p r e d i c t i o n of t h e c r i t e r i o n f o r a f i x ed t e s t (or b a t t e r y of t e s t s ) , and t h e c h o i c e o f a bat t e r y from a p o o l of t e s t s , are based on t h e knowledge of means, v a r i a n c e s and c o v a r i a n c e s between t e s t s , and between t e s t s and c r i t e r i o n . I n p r a c t i c e , t h e s e p a r a m e t e r s a r e seldom known e x a c t l y , and have t o be e s t i mated from t h e sample. S i n c e t h e a c t u a l c h o i c e o f t e s t s t o t h e b a t t e r y , and d e t e r m i n a t i o n of t h e r e g r e s s i o n coe f f i c i e n t s i s based on e s t i m a t e d v a l u e s , t h e r e a l goodn e s s of p r e d i c t i o n ( t h e v a l u e of t h e m u l t i p l e c o r r e l a t i o n c o e f f i c i e n t ) d i f f e r s from t h e e s t i m a t e d o n e , and t h e l a t t e r a r e always h i g h e r t h a n t h e f o r m e r . Empirica l l y , t h i s means t h a t when t h e o b t a i n e d p r e d i c t i o n equa t i o n i s a p p l i e d t o a new sample, i t t u r n s o u t t h a t i t s goodness i s always lower t h a n e x p e c t e d . T h i s phenomenon i s c a l l e d s h r i n k a g e , and may o c c a s i o n a l l y be s o s e v e r e t h a t t h e p r e d i c t i o n t u r n s o u t t o be w o r t h l e s s . A famous example o f s u c h s h r i n k a g e a r e t h e r e s u l t s of
M o s t e l l e r and Wallace (1964), where t h e v a l i d i t y of p r e d i c t i o n for t h e sample was 0 . 7 , and when t h e same p r e d i c t o r s were a p p l i e d t o a d i f f e r e n t sample, t h e v a l i d i t y dropped down t o about 0 . 0 4 . The cause o f t h i s phenomenon i s t h e f a c t t h a t b o t h t h e
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c h o i c e of t e s t s t o t h e b a t t e r y and d e t e r m i n a t i o n o f r e g r e s s i o n c o e f f i c i e n t s ( w e i g h t s of p r e d i c t o r s ) a r e bas e d n o t on t h e t r u e v a l u e s of means, v a r i a n c e s and cov a r i a n c e s , but on t h e v a l u e s o b s e r v e d i n t h e sample. F i r s t l y , t h e c h o i c e of t e s t s t o t h e b a t t e r y ( c h o i c e o f p r e d i c t o r s ) uses t y p i c a l l y t h e h i g h e s t observed c o r r e l a t i o n s between t e s t s and t h e c r i t e r i o n . However, bec a u s e o f random f l u c t u a t i o n s from sample t o sample, t h e v a l u e s which were o b s e r v e d as h i g h e s t i n one p a r t i c u l a r sample need n o t be h i g h e s t i n r e a l i t y . Thus, t h e s e l e c t e d b a t t e r y need n o t be o p t i m a l , w i t h t h e d a n g e r o f c h o o s i n g a s u b o p t i m a l b a t t e r y bee-Qming h i g h e r w i t h i n c r e a s e of t h e s i z e of t h e p o o l o f t e s t s , d e c r e a s e o f t h e s i z e o f t h e b a t t e r y and d e c r e a s e o f t h e sample s i z e . The c h o i c e o f t h e b a t t e r y based on t h e o b s e r v e d , i n s t e a d of t r u e , v a l u e s o f c o r r e l a t i o n c o e f f i c i e n t s i s b i a s e d by what Lord and Novick c a l l c a p i t a l i z a t i o n on chance. The second cause of s h r i n k a g e i s connected w i t h t h e f a c t t h a t t h e c a l c u l a t i o n of r e g r e s s i o n c o e f f i c i e n t s i s a l s o based on t h e o b s e r v e d v a l u e s o f c o r r e l a t i o n s , means and v a r i a n c e s , i n s t e a d of t h e t r u e v a l u e s (which a r e n o t known). A s a r e s u l t , t h e o b t a i n e d p r e d i c t i o n , b e i n g o p t i m a l for t h e v a l u e s f o r which i t was o b t a i n e d , i s suboptimal f o r o t h e r v a l u e s . This implies t h a t t h e a c t u a l q u a l i t y o f p r e d i c t i o n w i l l a l w a y s be lower t h a n t h e expected q u a l i t y . I n p r a c t i c e , t o d e c r e a s e t h e phenomenon o f s h r i n k a g e , t h e f o l l o w i n g r o u t i n e p r o c e d u r e i s a c c e p t e d . The f i r s t s t e p i s s c r e e n i n g , or t h e c h o i c e o f t e s t s from t h e p o o l t o t h e b a t t e r y , based on t h e r e s u l t s o b t a i n e d i n t h e
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s c r e e n i n g sample. Next, on t h e b a s i s o f a n o t h e r sample ( t h e s o - c a l l e d c a l i b r a t i o n s a m p l e ) , one c a l c u l a t e s t h e r e g r e s s i o n c o e f f i c i e n t s . F i n a l l y , t h e v a l i d i t y of p r e d i c t i o n i s t e s t e d on s t i l l a n o t h e r sample, c a l l e d t h e v a l i d a t i o n sample
.
I t i s t o be s t r e s s e d t h a t s u c h a p r o c e d u r e i s v e r y i m p o r t a n t as a p r o c e d u r e f o r c h o o s i n g p r e d i c t o r s ; i n f a c t , b e c a u s e of l a c k of s a t i s f a c t o r y t h e o r e t i c a l s o l u t i o n s , i t i s e s s e n t i a l f o r a t t a i n i n g meaningful p r e d i c t i o n s . I n c a s e of b r e a k i n g of t h e s e r u l e s , t h e p r e d i c t i o n obt a i n e d on t h e b a s i s o f one sample o n l y has only i l l u sory v a l i d i t y .
1 0 . SOME PROBLEMS OF THE CONSTRUCTION OF TESTS
I n most c a s e s o c c u r r i n g i n p r a c t i c e , t e s t s a r e b u i l t o u t of items ( t e s t p r o b l e m s , q u e s t i o n s , e t c . ) and t h e t e s t s c o r e i s t h e sum of t h e s c o r e s i n p a r t i c u l a r i t e m s . There a r i s e s t h e r e f o r e t h e problem of f i n d i n g r e l a t i o n s between a p p r o p r i a t e p a r a m e t e r s of i t e m s and p a r a m e t e r s of t h e t e s t as a whole. The p r a c t i c a l i m p o r t a n c e o f t h i s problem ' l i e s i n t h e f a c t t h a t one u s u a l l y s t a r t s w i t h a p o o l of i t e m s which a r e s u b j e c t t o some s t a t i s t i c a l o b s e r v a t i o n s , and t h e n one chooses some o f t h e se items t o t h e t e s t , It i s t h e r e f o r e e s s e n t i a l t o c o n s i d e r only t h o s e p a r a m e t e r s o f i t e m s which a r e i n v a r i a n t w i t h r e s p e c t t o
t h e c h o i c e o f t h e sample o f s u b j e c t s . I n o t h e r words, t h e p o i n t i s n o t t o c o n s i d e r p a r a m e t e r s which would be s p e c i f i c f o r a g i v e n group of s u b j e c t s o n l y .
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are t h e o b s e r v e d s c o r e s f o r t h e i t e m s which e n t e r t h e t e s t , t h e n
If Y l a ,
Y 2 a a * - * , Yna
(10.1)
i s t h e o b s e r v e d s c o r e of p e r s o n a . Taking e x p e c t a t i o n s
w i t h r e s p e c t t o a ( i . e . w i t h r e s p e c t t o t h e sampling d i s t r i b u t i o n ) , we o b t a i n
C o n s e q u e n t l y , t h e e x p e c t e d val.ue f o r a randomly s e l e c t ed p e r s o n e q u a l s n
= E(Y ) i s t h e e x p e c t e d s c o r e o f t h e randomg g* l y chosen p e r s o n i n i t e m g . The p a r a m e t e r v i s c a l l e d g sometimes t h e d i f f i c u l t y o f i t e m g .
where
v
If yga d e n o t e s t h e o b s e r v e d s c o r e ( i n a n n element sam-
p l e ) of t h e p e r s o n a i n i t e m g , t h e n t h e u n b i a s e d e s t i mate o f t h e d i f f i c u l t y v i s g
<-
t h a t i s , t h e a v e r a g e o b s e r v e d s c o r e i n i t e m g from t h e sample. S i m i l a r l y , t h e e s t i m a t o r of i s t h e average observed s c o r e i n t h e sample, t h a t i s n
mx
=
2 Pg' g=1
L e t us c o n s i d e r now t h e v a r i a n c e o f t h e t e s t s c o r e X :
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(10.2)
where (;g d e n o t e s t h e s t a n d a r d d e v i a t i o n of t h e random and f h i s t h e c o r r e l a t i o n between Y variable Y g*’ lg g* and Y h t . Thus, t h e v a r i a n c e of t h e t e s t s c o r e , one o f t h e most i m p o r t a n t parameters o f t h e t e s t , which i n t e r v e n e s i n t h e f o r m u l a f o r r e l i a b i l i t y o f t h e t e s t , depends on t h e v a r i a n c e s of i t e m s , and on t h e i r i n t e r c o r r e l a t i o n s . I n t h e p a r t i c u l a r c a s e of dichotomous i t e m s , t h e v a r i ance a2 e q u a l s g
For c o n s t r u c t i n g a t e s t from t h e i n i t i a l p o o l o f i t e m s which were s u b j e c t p r e v i o u s l y t o i n t r o d u c t o r y s t a t i s t i c a l a n a l y s i s , one chooses u s u a l l y t h o s e items which have h i g h v a r i a n c e . The r e a s o n i s t h a t t h e i t e m s w i t h small v a r i a n c e cannot have t o o h i g h e f f e c t on t h e v a r attains i a n c e F z o f t h e t e s t . For dichotomous i t e m s , g maximum when v = 2 , hence f o r items o f a v e r a g e d i g f f i c u l t y ( t h e items w i t h small v a r i a n c e 6* have t h e g d i f f i c u l t y v c l o s e e i t h e r t o 0 o r t o 1, s o t h a t t h e g answers t e n d t o be t h e same i n t h e whole p o p u l a t i o n and t h e items s u p p l y l i t t l e i n f o r m a t i o n a b o u t t h e subjects.
2
i s , technically ’ gh s p e a k i n g , n o t a parameter o f an i t e m , s i n c e i t depends on two i t e m s , g and h. Its knowledge i s , however, esThe c o e f f i c i e n t o f c o r r e l a t i o n p
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s e n t i a l for t e s t c o n s t r u c t i o n , i f only b o t h i t e m s g and h a r e t o be i n c l u d e d i n t h e t e s t . To i n c r e a s e t h e v a r i a n c e o f t h e t e s t ( s e e formula ( 1 0 . 2 ) ) , i t i s r e a s o n a b l e t o t a k e s u c h i t e m s which have p o s s i b l y h i g h e s t i n t e r ( a s w i l l be shown, though, t o i n c r e a s e correlation n 1-gh t h e v a l i d i t y of t h e t e s t , one s h o u l d t a k e i t e m s which have low i n t e r c o r r e l a t i o n ) . To s t u d y more e x a c t l y t h e e f f e c t o f i n t e r c o r r e l a t i o n between i t e m s on t h e p a r a m e t e r s of t h e t e s t , l e t u s compute a g a i n t h e v a r i a n c e o f t h e t e s t s c o r e , i n a somewhat d i f f e r e n t w a y :
which y i e l d s (7x -
n
'j- GgfgX'
(10.3)
g= 1
Thus, t h e s t a n d a r d d e v i a t i o n of t h e t e s t s c o r e e q u a l s t h e weighted a v e r a g e of c o r r e l a t i o n s between i t e m s and i s sometimes c a l l e d t h e t h e t e s t . The c o r r e l a t i o n ,P gx d i s c r i m i n a t i n g power of t h e i t e m . Let u s now c a l c u l a t e t h e d i s c r i m i n a t i n g power , e x p r e s s i n g i t t h r o u g h p a r a m e t e r s of t h e i t e m s :
.-
CONTEMPORARY THEORY OF PSYCHOLOGICAL TESTS
63
hence t h e d i s c r i m i n a t i n g power i s t h e weighted a v e r a g e of i t e m intercorrelations. We may now d e t e r m i n e t h e lower bound for r e l i a b i l i t y o f a composite t e s t , e x p r e s s i n g i t t h r o u g h i t e m p a r a m e t e r s ( u s i n g t h e f o r m u l a of Kuder-Richardson and f o r m u l a (10.3) ) :
Here g i v e s t h e lower bound f o r r e l i a b i l i t y of t h e t e s t X ; from t h i s f o r m u l a i t i s c l e a r t h a t w i t h f i x e d r e l i a b i l i t y increases with the ini t e m v a r i a n c e s 0’ g’ between i t e m s and t h e t e s t , c r e a s e of c o r r e l a t i o n s gx i . e . w i t h t h e i n c r e a s e of d i s c r i m i n a t i n g powers of items.
P
L e t u s now c o n s i d e r v a l i d i t y o f t h e t e s t X w i t h r e s p e c t t o some e x t e r n a l c r i t e r i o n z , d e f i n e d as t h e c o r r e l a t i o n between X and z . Let us f i r s t d e t e r m i n e t h e c o v a r i a n c e between X and z :
n
=
2 !Tgcj-zpgz ,
g= 1 where D 1
rion z .
gz
i s t h e c o r r e l a t i o n between i t e m g and c r i t e -
C o n s e q u e n t l y , u s i n g ( 1 0 . 3 ) we may w r i t e
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NX,Z)
= cov ( x , z ) / , 5 x g z
Formula (10.4) e x p r e s s e s t h e v a l i d i t y o f t e s t X w i t h r e s p e c t t o a n e x t e r n a l c r i t e r i o n z t h r o u g h parameters o f i t e m s . T h i s formula may be also w r i t t e n i n t h e following form, u s i n g ( 1 0 . 3 ) and ( 1 0 . 2 ) : . n ("(X,Z)
=
n r--
n
n
(10.5)
From t h e l a s t formula i t f o l l o w s t h a t to maximize v a l i d i t y ( u n d e r f i x e d i t e m v a r i a n c e s ) , one s h o u l d choose i t e m s which (1) c o r r e l a t e h i g h l y w i t h t h e c r i t e r i o n , and ( 2 ) have low, or p e r h a p s even n e g a t i v e i n t e r c o r r e l a t i o n s . T h i s shows a c e r t a i n n e g a t i v e r e l a t i o n b e t ween v a l i d i t y and r e l i a b i l i t y of t e s t measurement: m a -
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65
x i m i z a t i o n o f v a l i d i t y r e q u i r e s p o s s i b l y low ( o r negat i v e ) i n t e r c o r r e l a t i o n s between i t e m s , w h i l e maximizat i o n of r e l i a b i l i t y r e q u i r e s c h o o s i n g i t e m s which c o r r e l a t e h i g h l y w i t h t h e t e s t s c o r e X , and a l s o among themselves. N a t u r a l l y , i n p r a c t i c a l s i t u a t i o n s t h e above dilemma c a n n o t a p p e a r t o o s h a r p l y : one may namely show t h a t i f the i t e m s correlate highly w i t h the external c r i t e r i o n , t h e n i t i s n o t p o s s i b l e f o r a l l i n t e r c o r r e l a t i o n s t o be n e g a t i v e . I n d e e d , s i n c e C ( X , z ) C 1, we may w r i t e , u s i n g formula ( 1 0 . 5 ) :
I f t h e r i g h t hand s i d e i s p o s i t i v e (which w i l l happen i f t h e items correlate highly w i t h the c r i t e r i o n ) , w e
may s q u a r e b o t h s i d e s , o b t a i n i n g
and i t f o l l o w s t h a t i t i s i m p o s s i b l e t h a t a l l negative.
egh a r e
To complete t h e c o n s i d e r a t i o n s o f t h i s s e c t i o n , i t i s w o r t h w h i l e t o p o i n t o u t t h a t t h e problem of f i n d i n g t h e o p t i m a l t e s t i s o f t e n r e d u c i b l e t o t h e p.roblem o f c h o o s i n g a b e s t b a t t e r y of p r e d i c t o r s f o r a g i v e n c r i t e r i o n . T h i s o c c u r s a l w a y s when t h e t e s t i s t o b e c o n s t r u c t e d as t h e most v a l i d f o r a g i v e n a p r i o r i ext e r n a l c r i t e r i o n . The d i f f e r e n c e l i e s i n t h e f a c t t h a t
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t y p i c a l l y i n t e s t c o n s t r u c t i o n one does n o t t r y t o determine the regression c o e f f i c i e n t s , i . e . optimal weights o f i t e m s . It i s simply assumed t h a t a l l i t e m s have t h e same w e i g h t , and t h e r e f o r e t h e t e s t s c o r e i s t h e sum of s c o r e s on i t e m s . It f o l l o w s a l s o from t h i s remark t h a t i n t h e c o n s t r u c t -
i o n o f a t e s t based on i n t r o d u c t o r y s t a t i s t i c a l a n a l y s i s o f i t e m s , t h e r e o c c u r s t h e phenomenon o f c a p i t a l i z a t i o n on c h a n c e , d i s c u s s e d i n t h e p r e c e d i n g s e c t i o n i n c o n n e c t i o n w i t h r e g r e s s i o n methods. I n d e e d , t h e p a r a m e t e r s of t h e t e s t , s u c h a s r e l i a b i l i t y o r v a l i d i t i e s , depend on t h e t r u e v a l u e s o f p a r a m e t e r s s u c h as G g y fgh’ e t c . , which a r e unknown. I n t e s t c o n s t r u c t i o n , one s e l e c t s i t e m s for which t h e sample v a l u e s o f t h e s e coe f f i c i e n t s are b e s t i n some a p p r o p r i a t e s e n s e . A s a r e s u l t , t h e a c t u a l q u a l i t y of t h e t e s t ( i t s r e l i a b i l i t y or v a l i d i t i e s ) w i l l , i n g e n e r a l , be lower t h a n exp e c t e d or: t h e b a s i s of n u m e r i c a l c a l c u l a t i o n s . T h i s phenomenon may be c a l l e d “ s h r i n k a g e “ of t h e q u a l i t y of the t e s t
.
These remarks a p p l y n o t o n l y i n t h e c a s e when one t r i e s t o c o n s t r u c t a t e s t which would b e most v a l i d w i t h r e s p e c t t o a given e x t e r n a l c r i t e r i o n , but a l s o t o t h e c a s e (which i s q u i t e f r e q u e n t ) , when one c o n s t r u c t s a t e s t u s i n g t h e known s t a t i s t i c a l p r o p e r t i e s o f i t e m s , w i t h o u t r e f e r e n c e t o any e x t e r n a l c r i t e r i o n . I n t h i s c a s e , t h e e x p e r i m e n t e r t r i e s t o choose i t e m s which have d e s i r a b l e s t a t i s t i c a l p r o p e r t i e s , and a l s o he uses h i s p s y c h o l o g i c a l knowledge and i n t u i t i o n c o n c e r n i n g p o s s i b l e r e l a t i o n s of t h e i t e m s w i t h t h e c o n s t r u c t which h e i n t e n d s t o measure.
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I n s u c h s i t u a t i o n s , t h e c h o i c e i s based on a c e r t a i n (sometimes q u i t e s u b s t a n t i a l ) s u b j e c t i v i t y , i . e . t h e c o n v i c t i o n o f t h e p s y c h o l o g i s t as r e g a r d s t h e s o - c a l l e d f a c e v a l i d i t y o f a n i t e m f o r a g i v e n c o n s t r u c t . The val i d i t y o f t h e t e s t as a whole i s v e r i f i e d l a t e r , by s t u d y i n g i t s r e l a t i o n s t o v a r i o u s c o n s t r u c t s from d i f f e r e n t t h e o r i e s , as a l r e a d y e x p l a i n e d a t t h e b e g i n n i n g of t h e p r e c e d i n g s e c t i o n .
11. FURTHER DIRECTIONS OF DEVELOPMENT OF TEST THEORY: THEORY OF G E N E R I C TRUE SCORES AND THEORY
OF GENERALIZABILITY
11.1. I n t r o d u c t o r y remarks
One o f t h e r e c e n t d i r e c t i o n s o f development o f t e s t t h e o r y i s t h e t h e o r y o f g e n e r i c t r u e s c o r e s , and more p r e c i s e l y , g e n e r a l i z a b i l i t y t h e o r y . The l a t t e r was i n t r o d u c e d and d e v e l o p e d mainly by Cronbach, G l e s e r , Nanda and R a j a r a t n a m ( 1 9 7 2 ) .
The s t a r t i n g p o i n t o f t h e t h e o r y o f g e n e r i c t r u e s c o r e s i s t h e o f t e n encountered p r a c t i c a l s i t u a t i o n connected w i t h a p p l i c a t i o n of t e s t s t o s e l e c t i o n . I f t h e t e s t i s t o be used many t i m e s ( e . g . i n p r o c e s s i n g j o b a p p l i c a t i o n s , w r i t t e n exams f o r d r i v e r ' s l i c e n c e , e t c . ) , i t i s n e c e s s a r y t o have a s e r i e s of v e r s i o n s o f t h e t e s t , o f which one randomly chosen i s a d m i n i s t e r e d t o a g i v e n c a n d i d a t e . N a t u r a l l y , i t would be i d e a l i f t h e v e r s i o n s used were p a r a l l e l ; i n s u c h a c a s e i t would n o t m a t t e r a t a l l which v e r s i o n was a d m i n i s t e r e d . Howe v e r , c o n s t r u c t i o n of a s u f f i c i e n t number o f p a r a l l e l
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t e s t s i s o f t e n d i f f i c u l t , if n o t i m p o s s i b l e . Thus, one
u s u a l l y has a number o f v e r s i o n s of t h e t e s t which a r e n o t e n t i r e l y p a r a l l e l , w h i l e b e i n g s u f f i c i e n t l y paral l e l f o r p r a c t i c a l p u r p o s e s , s o t h a t from t h e p o i n t o f view o f t h e d e c i s i o n maker i t does n o t m a t t e r which v e r s i o n of t h e t e s t was a c t u a l l y u s e d . Such t e s t s a r e c a l l e d n o m i n a l l y p a r a l l e l . The l a t t e r n o t i o n i s n o t def i n e d m a t h e m a t i c a l l y ; i n t u i t i v e l y , i t means s i m p l y a s u f f i c i e n t l y high degree of p a r a l l e l i s m . I n such s i t u a t i o n s , t h e experimenter i s not i n t e r e s t e d i n t h e t r u e s c o r e of a g i v e n p e r s o n i n t h e p a r t i c u l a r v e r s i o n of t h e t e s t u s e d , b u t r a t h e r i n t h e e x p e c t e d s c o r e i n a randomly chosen v e r s i o n o f t h e t e s t . I n o t h e r words, i f T d e n o t e s t h e t r u e s c o r e of p e r s o n a i n ga v e r s i o n g of t h e t e s t , t h e i n t e r e s t i n g q u a n t i t y i s t h e me a n
c a l l e d t h e g e n e r i c t r u e s c o r e . I n c a s e when t h e t e s t has n v e r s i o n s which a r e chosen a t random, we have
The t h e o r y o f g e n e r i c t r u e s c o r e s , t o be s k e t c h e d be-
low, a n a l y s e s t h e means of e s t i m a t i n g t h e e r r o r and g e n e r i c t r u e s c o r e on t h e b a sis o f t h e o b s e r v e d s c o r e . T h i s t h e o r y may be g e n e r a l i z e d , s t a r t i n g from t h e f o l -
lowing a n a l o g y . I n t u i t i v e l y i t i s c l e a r t h a t t h e d i f f e r e n c e s i n t r u e s c o r e s 7 i n p a r t i c u l a r v e r s i o n s of t h e ga test w i l l cause an i n c r e a s e of t h e e r r o r v a r i a n c e : p a r t of t h i s v a r i a n c e w i l l be r e l a t e d t o t h e d e v i a t i o n s
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of t h e o b s e r v e d s c o r e s y from t h e t r u e s c o r e s T and ga ga' p a r t w i l l be r e l a t e d t o t h e randomness of t h e c h o i c e of t h e t e s t , hence w i t h randomness o f $ ga *
I t t u r n s o u t t h a t by an a p p r o p r i a t e e x p e r i m e n t i t i s
p o s s i b l e t o e s t i m a t e t h a t p o r t i o n o f t h e v a r i a n c e which i s due t o t h e c h o i c e of t h e t e s t , and t h a t p a r t which
is r e l a t e d t o t h e randomness of t h e answer. I n o t h e r words, t h e components of t h e v a r i a n c e a r i s i n g from t h e two s o u r c e s mentioned may be e s t i m a t e d s e p a r a t e l y . S t a r t i n g from t h i s remark, one may c o n s i d e r a l s o o t h e r s o u r c e s of randomness ( e r r o r ) , n o t o n l y c o n n e c t e d w i t h t h e c h o i c e o f t h e v e r s i o n o f t h e t e s t , but a l s o connected w i t h o t h e r f a c t o r s , s u c h as age o r s e x of t h e subj e c t s , t y p e o f i n s t r u c t i o n o r t r a i n i n g , t e s t i n g condit i o n s , t i m e of t e s t i n g , and s o f o r t h . According t o t h e g e n e r a l l y a c c e p t e d t e r m i n o l o g y , we d e a l h e r e w i t h " f a c t o r s " (which a r e n o t t o be confused w i t h f a c t o r s , as a n a l y s e d i n f a c t o r a n a l y s i s ) . These f a c t o r s may o p e r a t e on v a r i o u s l e v e l s . F o r i n s t a n c e , one o f t h e f a c t o r s i s t h e v e r s i o n o f t h e t e s t , and t h i s f a c t o r may have as many l e v e l s as t h e r e a r e v e r s i o n s . Another f a c t o r may be t h e s e x of t h e s u b j e c t ; h e r e we have two l e v e l s , e t c . It s h o u l d be remarked t h a t " l e v e l " need n o t have any n u m e r i c a l r e p r e s e n t a t i o n , n o r need t h e l e v e l s be o r d e r e d i n any n a t u r a l way. The o n l y r e q u i r e m e n t i s t h a t one may d i s t i n g u i s h a w e l l d e f i n e d s e t of observable conditions. The g e n e r a l i z a t i o n d i s c u s s e d h e r e , based on t h e methods of v a r i a n c e a n a l y s i s , i s c a l l e d t h e t h e o r y of g e n e r a l i z a b i l i t y . I n s t e a d of g e n e r i c t r u e s c o r e , b e i n g t h e ave-
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r a g e t r u e s c o r e under t h e random c h o i c e o f t h e v e r s i o n o f t h e t e s t , one c o n s i d e r s t h e s o - c a l l e d u n i v e r s a l scor e s , d e f i n e d as t h e c o r r e s p o n d i n g means under a random c h o i c e of l e v e l s of f a c t o r s ( i t f o l l o w s t h a t t h e c o n c e p t o f u n i v e r s a l s c o r e i s r e l a t i v e t o t h e f a c t o r s under consideration). The g o a l of a n a l y s i s of v a r i a n c e i s t o f i n d experimenta l schemes which a l l o w t o e s t i m a t e t h e e f f e c t s of v a r i o u s f a c t o r s on t h e v a r i a b i l i t y of o b s e r v e d s c o r e , i . e . e s t i m a t e t h e components o f v a r i a n c e . The s t u d i e s l e a d i n g t o d e t e r m i n a t i o n of v a r i a n c e comp o n e n t s a r e c a l l e d by Cronbach e t a l . (1972) t h e G s t u d i e s ( f r o m g e n e r a l i z a b i l i t y s t u d i e s ) , as d i s t i n c t from t h e s o - c a l l e d D s t u d i e s (from: d e c i s i o n s t u d i e s ) , a i m ed a t making a d e c i s i o n on t h e b a s i s o f t h e o b s e r v e d s c o r e ( e . g . a c c e p t i n g or r e j e c t i n g a c a n d i d a t e , o p i n i o n about t h e v a l u e o f h i s t r a i t , a s s e r t i o n about t h e d i f f e r e n c e between two groups o f s u b j e c t s , e t c . )
1 1 . 2 . B a s i c concept of t h e t h e o r y o f g e n e r i c t r u e s c o -
res To i l l u s t r a t e how one may a p p l y t h e model of a n a l y s i s
of v a r i a n c e t o s t u d y t h e e f f e c t s of v a r i o u s f a c t o r s , i t i s most c o n v e n i e n t t o r e c o n s i d e r t h e model o f t h e c l a s s i c a l t e s t t h e o r y , and t h e n t r y some n a t u r a l generalizations. The s t a r t i n g p o i n t o f t h e c l a s s i c a l t e s t t h e o r y i s t h e identity
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where X i s t h e observed s c o r e o f p e r s o n a i n t e s t g . ga T h i s i d e n t i t y i f , o f c o u r s e , v a l i d under any c h o i c e o f T If we t a k e T = E(X ) , and d e n o t e I = X -'% ga * ga ga ga ga ga' t h e n t h e l a s t f o r m u l a takes t h e form
where E ( I ) = 0 . Thus, t h e o b s e r v e d s c o r e was r e p r e €9 s e n t e d as a sum o f two e f f e c t s : t h e t r u e s c o r e z and ga error I where t h e e x p e c t e d e r r o r i s z e r o . I f now ga ' t h e p e r s o n a i s sampled a t random, t h e t r u e s c o r e becomes a random v a r i a b l e ( d e n o t e d b y T ) , and t h e e r r o r g* s c o r e becomes t h e random v a r i a b l e I with T and I g* g* g* ' b e i n g u n c o r r e l a t e d , and s a t i s f y i n g t h e e q u a t i o n X = g* T + I g* g*
.
A s a r e s u l t , t h e v a r i a n c e of t h e o b s e r v e d s c o r e , G z ( X ) g* p a r t i t i o n s i n t o t h e v a r i a n c e s o f t r u e s c o r e and of e r r o r score:
L e t u s now c a r r y on t h e a n a l o g o u s r e a s o n i n g u n d e r t h e a s s u m p t i o n t h a t t h e t e s t g i s chosen a t random from a c e r t a i n universe of t e s t s . Let X d e n o t e , as b e f o r e , ga t h e s c o r e o f p e r s o n a i n t e s t g. The a n a l o g u e o f t h e b a s i c i d e n t i t y of t e s t t h e o r y i s h e r e
) i s t h e expectedscore of person a ga i n a randomly chosen v e r s i o n o f t h e t e s t ( t h e g e n e r i c t r u e s c o r e ) , T = E ( X ) i s t h e expected s c o r e of a
where z a = E (X g
€5
a
ga
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randomly chosen p e r s o n i n t e s t g ( t h e s o - c a l l e d d i f f i c u l t y of t h e t e s t ) , m = E CE ( X > I i s t h e g e n e r a l a g ga d e n o t e s t h e i n t e r a c t i o n between t h e t e s t mean, and d ga g and p e r s o n a ( t h i s term may be s e p a r a t e d from e r r o r o n l y i n c a s e of r e p e a t e d measurements). T o e x p l a i n t h e meaning of i n t e r a c t i o n , o b s e r v e t h a t
t h e model w i t h o u t i n t e r a c t i o n assumes s i m p l e a d d i t i v i i s t h e sum 01: t h e t e r m e x p r e s s i n g ga t h e i n f l u e n c e of a, a t e r m e x p r e s s i n g t h e i n f l u e n c e of g , and e r r o r . A d i s t o r t i o n of t h i s a d d i t i v i t y i s c a l l e d i n t e r a c t i o n . For i n s t a n c e , some s u b j e c t s may have s p e c i a l a b i l i t i e s w i t h respect t o c e r t a i n types of t e s t s , e t c . which g i v e s i n t e r a c t i o n between t h e s u b j e c t s and t e s t s . t y : the score X
A s a consequence of t h e l a s t f o r m u l a , t h e e r r o r of
measurement of t h e g e n e r i c t r u e s c o r e z a , i . e . t h e d i f f e r e n c e Xga - z a , e q u a l s
where I i s , as b e f o r e , t h e e r r o r i n measurement o f ga t h e t r u e s c o r e of p e r s o n a i n t e s t g . The e x p e c t a t i o n of t h e g e n e r i c e r r o r s c o r e for p e r s o n
a , w i t h a random c h o i c e of t e s t , i s
hence for e a c h p e r s o n t h e e x p e c t e d v a l u e of t h e e r r o r is zero. score E ga
If b o t h t h e p e r s o n a and t e s t g a r e s e l e c t e d a t random,
CONTEMPORARY THEORY OFPSYCHOLOGICAL TESTS
13
t h e n t h e observed s c o r e , d e n o t e d by X,,, is represented as a sum of random v a r i a b l e s , as b e f o r e , which a c c o u n t f o r t h e e ' f f e c t s o f t h e c h o i c e o f t e s t g , c h o i c e of t h e p e r s o n a , i n t e r a c t i o n o f p e r s o n and t e s t , and e r r o r . One may p r o v e ( s e e Lord and Novick 1 9 6 8 , c h a p t e r 8 ) t h a t t h e v a r i a n c e of e r r o r &, ( i . e . e r r o r E f o r the ranga domly chosen t e s t and p e r s o n ) e q u a l s
where t h e t e r m s on t h e r i g h t are r e s p e c t i v e l y : 1) I n t e r t e s t v a r i a n c e ; 2) i n t e r a c t i o n v a r i a n c e ; 2 2 3 ) v a r i a n c e cr (I,,) = E [a ( I )I, i . e . t h e a v e r a g e (ung g* d e r random c h o i c e of t e s t ) of t h e s p e c i f i c e r r o r v a r i a n ces. I t f o l l o w s from t h e l a s t f o r m u l a t h a t t h e g e n e r i c e r r o r 2 i s a l w a y s a t l e a s t as l a r g e as t h e mev a r i a n c e r~ ( L , , ) a n o f v a r i a n c e s o f s p e c i f i c e r r o r s , and w i l l a l w a y s be s t r i c t l y l a r g e r , i f only t h e v e r s i o n s o f t h e t e s t vary 2 i n t h e i r d i f f i c u l t y ( s o t h a t G (a,) > 0 ) , or i f t h e r e e x i s t s a n i n t e r a c t i o n between t e s t s and p e r s o n s ( i . e .
The example below, t a k e n from Lord and Novick (1968, pp.
179-180) i l l u s t r a t e s t h e s i g n i f i c a n c e of t h e g e n e r i c e r r o r v a r i a n c e . Imagine t h a t i n a c c e p t i n g c a n d i t a t e s f o r a c e r t a i n p o s i t i o n , one a p p l i e s a t e s t which has a number o f nominally p a r a l l e l v e r s i o n s . The v e r s i o n s are sampled a t random, and i t i s known t h a t t h e r e s u l t s have a p p r o x i m a t e l y normal d i s t r i b u t i o n w i t h s t a n d a r d d e v i a t i o n 3 . 0 0 . T o become a c c e p t e d , t h e c a n d i d a t e must s c o r e more t h a n 70.
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L e t u s imagine t h a t M r . Smith s c o r e d 6 5 , hence was n o t a c c e p t e d . Suppose t h a t he d e c i d e s t o t r y a g a i n , a f t e r two weeks ( s a y ) ; we assume t h a t ; d u r i n g t h e s e two weeks h i s t r a i t d o e s n o t change. What; i s t h e p r o b a b i l i t y t h a t he w i l l be a c c e p t e d on t h e second a t t e m p t ? To answer t h i s q u e s t i o n , l e t u s o b s e r v e t h a t t h e d i f f e r -
e n c e between two s c o r e s (assuming t h a t t h e y were chosen a t random from t h e p o p u l a t i o n of s c o r e s w i t h unknown mean and s t a n d a r d d e v i a t i o n 3 . 0 0 , and t h a t t h e r e s u l t s a r e i n d e p e n d e n t ) i s a random v r a r i a b l e w i t h mean 0 and variance 3
2
t
32
=
18, hence w i t h s t a n d a r d d e v i a t i o n
equal 3 6 . Consequently , t h e p r o b a b i l i t y t h a t t h i s d i f f e r e n c e w i l l exceed 5/(3'@) = 1.18 i s a b o u t 0 . 2 4 . Thus, w i t h p r o b a b i l i t y 0 . 1 2 M r . Smith w i l l s c o r e o v e r 7 0 on t h e second t r i a l , and w i t h t h e same p r o b a b i l i t y 0 . 1 2 he w i l l g e t a s c o r e below 6 0 . The o u t l i n e d here t h e o r y of g e n e r i c t r u e s c o r e s (which i s a s p e c i a l c a s e o f t h e t h e o r y o f g e n e r a l i z a b i l i t y ) desp i t e a n a l o g i e s w i t h t h e c l a s s i c a l theory of t e s t s , d i f f e r s i n many e s s e n t i a l a s p e c t s from t h e l a t t e r . The most i m p o r t a n t among t h e s e d i f f e r e n c e s c o n s i s t s o f t h e f a c t t h a t now t h e g e n e r i c e r r o r s f o r d i f f e r e n t p e r s o n s a r e correlated. I n s t e a d of d e r i v i n g t h e a p p r o p r i a t e f o r m u l a s , i t w i l l be i n t u i t i v e l y e x p l a i n e d why t h i s phenomenon o c c u r s . Suppose t h a t we c o i l s i d e r a s e t o f n o m i n a l l y p a r a l l e l v e r s i o n s of t h e t e s t , c o n s i s t i n g of ( s a y ) t h r e e v e r s i o n s , and t h a t one of them i s chosen a t random f o r a d m i n i s t r a t i o n . Suppose t h a t f o r p e r s o n a t h e t r u e s c o r e s i n v e r s i o n s A ,
CONTEMPORARY THEORY OF PSYCHOLOGICAL TESTS
75
= 87 ( s o t h a t , aA = 80, TaB = 85 and ‘taC a t l e a s t f o r p e r s o n a , v e r s i o n A i s t h e most d i f f i c u l t , and v e r s i o n C i s t h e e a s i e s t ) . The g e n e r i c t r u e s c o r e , b e i n g t h e a v e r a g e of t r u e s c o r e s , e q u a l s h e r e ( 8 0 + 85 + 8 7 ) / 3 = 84.
B and C a r e 7
C o n s e q u e n t l y , one may e x p e c t t h a t i f t h e v e r s i o n C i s c h o s e n , t h e g e n e r i c e r r o r s ( d e v i a t i o n s of observed s c o r e from t h e g e n e r i c t r u e s c o r e 84) w i l l t e n d t o be p o s i t i v e , while f o r v e r s i o n A they w i l l t e n d t o be n e g a t i v e . Suppose t h a t a n a l o g o u s s i t u a t i o n h o l d s f o r p e r s o n b ; l e t t h e t r u e s c o r e s be e q u a l h e r e ZbA = 30, ‘rbB = 38 and ‘Cbc = 43 (which g i v e s t h e g e n e r i c t r u e s c o r e 3 7 ) . Here a g a i n v e r s i o n A i s t h e most d i f f i c u l t , and v e r s i o n C i s the e a s i e s t ) . It i s c l e a r t h a t i f v e r s i o n C i s c h o s e n , t h e g e n e r i c
e r r o r s w i l l be p o s i t i v e b o t h for p e r s o n a and b y w h i l e under c h o i c e o f v e r s i o n A , b o t h e r r o r s w i l l t e n d t o be n e g a t i v e . G e n e r a l l y , one may prove t h a t f o r a f i x e d v e r s i o n o f t h e t e s t , t h e g e n e r i c e r r o r s of two p e r s o n s i n t h i s v e r s i o n w i l l t e n d t o be p o s i t i v e l y c o r r e l a t e d ( t h e r e w i l l be a tendency f o r t h e e r r o r s c o r e s t o b e o f t h e same s i g n ) . The d i f f e r e n c e s , as compared w i t h t h e c l a s s i c a l t e s t t h e o r y , c o n c e r n i n g a s i n g l e t e s t , and n o t a s e t o f nominally p a r a l l e l t e s t s , are as follows:
For a f i x e d t e s t g: 1) t h e g e n e r i c e r r o r s
# 0).
C
ga
may be b i a s e d ( i . e . E( Cga)
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2 ) t h e g e n e r i c e r r o r s c o r e s may be c o r r e l a t e d w i t h t h e
generic true scores;
3 ) The g e n e r i c e r r o r s c o r e s o f one v e r s i o n o f t h e t e s t may be c o r r e l a t e d w i t h g e n e r i c e r r o r s c o r e s i n a n o t h e r version;
4) t h e g e n e r i c e r r o r s c o r e s of a p e r s o n , under r e p e t i t i o n of a t e s t ( u s i n g a n o t h e r nominally p a r a l l e l v e r s i o n ) may be l i n e a r l y e x p e r i m e n t a l l y d e p e n d e n t . These f a c t s , n a t u r a l l y , c o m p l i c a t e t h e t h e o r y c o n s i d e r a b l y . It i s w o r t h t o m e n t i o n , however, t h a t t h e s e com-
p l i c a t i o n s make t h e t o those situations ricians: situations a r e b i a s e d , and n o t under r e p e t i t i o n .
theory of generic scores applicable which a r e t r o u b l i n g t h e psychometi n which t h e e r r o r s o f measurements l i n e a r l y e x p e r i m e n t a l l y independent
11.3. Schemes of e x p e r i m e n t s The c o n s i d e r a t i o n s t h u s f a r concerned o n l y t h e i n f l u -
e n c e o f v a r i o u s v e r s i o n s o f t h e t e s t on t h e s c o r e and on t h e v a r i a n c e of e r r o r s c o r e . To i l l u s t r a t e t h e r i c h n e s s o f t h e schemes of a n a l y s i s of v a r i a n c e , i t i s worth w h i l e t o s y s t e m a t i c a l l y l i s t a l l p o s s i b i l i t i e s which a r i s e i n t a k i n g i n t o a c c o u n t two f a c t o r s . Thus, imagine t h a t w e c o n s i d e r a group o f p e r s o n s , d e n o t e d by 1 , 2 , N , and we a n a l y s e t h e i n f l u e n c e o f two f a c t o r s , w i t h f a c t o r I o p e r a t i n g on l e v e l s A , B , C , and In special f a c t o r I1 o p e r a t i n g on l e v e l s ~ , b , c , may d e n o t e v e r s i o n s o f t h e t e s t , s i t u a t i o n s , A,B, while a , b , c , may d e n o t e e x p e r i m e n t a l c o n d i t i o n s ( e . g .
...,
...
.. .
... .
...
CONTEMPORARY THEORY OF PSYCHOLOGICAL TESTS
77
t h e y may s t a n d for a r b i t e r s , d u r a t i o n s o f t e s t i n g , lighting conditions, e t c . ) . G e n e r a l l y , one may d i s t i n g u i s h two c a t e g o r i e s of t h e w a y s i n which t h e p e r s o n s and l e v e l s of f a c t o r s a r e combined: c r o s s e d d e s i g n and n e s t e d d e s i g n . I f e a c h p e r s o n i s t e s t e d on e a c h l e v e l of a g i v e n f a c -
t o r , we s p e a k of a c r o s s e d d e s i g n of p e r s o n s w i t h t h i s f a c t o r ; s i m i l a r l y , i f e a c h l e v e l o f one f a c t o r i s comb i n e d w i t h e a c h l e v e l of a n o t h e r f a c t o r , we s a y t h a t these factors are crossed. If e a c h o f t h e p e r s o n s i s t e s t e d under some l e v e l s of
a f a c t o r ( n o t a l l ) , or i f t h e l e v e l s o f one f a c t o r a r e combined w i t h some l e v e l s of t h e o t h e r f a c t o r o n l y , w e speak of a n e s t e d d e s i g n . For one f a c t o r , t h e s e p o s s i b i l i t i e s a r e b e s t i l l u s t r a t ed on t h e a p p r o p r i a t e diagrams ( s e e F i g . 11.1 and 1 1 . 2 ) . Factor A
B
C
D
Persons
F i g . 11.1.
Crossed d e s i g n
E
F
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Factor A
B
C
D
E
F
G
H
I
F i g . 1 1 . 2 . Nested d e s i g n
The g r a p h i c a l r e p r e s e n t a t i o n of' s e v e r a l p o s s i b l e exper i m e n t a l schemes i n c a s e of two f a c t o r s i s g i v e n on
Factor I A
B
C
D
E
F
Factor
Persons
Fig.
11.3. F u l l c r o s s e d d e s i g n for four p e r s o n s and two f a c t o r s
CONTEMPORARY THEORY OF PSYCHOLOGICAL TESTS
Factor I
B
A
F a c t o r I1
a
c
b
d
19
C
e
f
g
h
i
I
I
1
I
Persons
4
Fig. 11.4.
N e s t i n g o f F a c t o r , I1 i n F a c t o r I , c r o s s ed w i t h p e r s o n s .
Factor I
A
F a c t o r I1
a
B
b
c
d
I
1
Persons
Fig.
-t
1 1 . 5 . A mixed d e s i g n : c r o s s i n g o f f a c t o r I1 a n d persons, nested with F a c t o r I.
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A c o n v e n i e n t s y s t e m a t i z a t i o n of' schemes o f e x p e r i m e n t s
may be o b t a i n e d when one i n t r o d u c e s t h e f o l l o w i n g n o t a t i o n ( s e e Cronbach e t a l . , 1 9 7 2 ) : t h e s e t of p e r s o n s , as w e l l as s e t s of l e v e l s of f a c t o r s , a r e r e p r e s e n t e d by c e r t a i n a r e a s , w i t h t h e c r o s s e d d e s i g n d e n o t e d as
on F i g . 1 1 . 6 , and a n e s t e d d e s i g n - as on F i g . 1 1 . 7 .
F i g . 1 1 . 6 . Symbolic representation of
F i g . 1 1 . 7 . Symbolic r e p r e s e n t a t i o n of
a crossed design
a nested design
D e n o t i n g by c o n t i n u o u s l i n e t h e s e t of p e r s o n s , and by dashed and d o t t e d l i n e s f a c t o r s I and 11, v a r i o u s schemes may be r e p r e s e n t e d by c o n f i g u r a t i o n s o f domains on t h e p l a n e .
For i n s t a n c e , t h e f i r s t scheme, when a l l f a c t o r s a r e c r o s s e d , i s shown on F i g . 11. 8. The second scheme i s p r e s e n t e d on F i g . 1 1 . 9 . The convenience h e r e c o n s i s t s p r i m a r i l y of t h e f a c t t h a t one can r e a d immediately which components of v a r i a n c e o f t h e o b s e r v e d s c o r e can be e s t i m a t e d from d a t a a r i s i n g from a g i v e n e x p e r i m e n t a l scheme.
81
CONTEMPORARY THEORY OFPSYCHOLOGICAL TESTS
/---
/
-
I
/
I
/
/ \
i
\ I
F i g . 1 1 . 8 . Schematic r e p r e s e n t a t i o n of s y s t e m from F i g . 11.3
F i g . 1 1 . 9 . Schematic representation of s y s t e m from F i g . 1 1 . 4
I n g e n e r a l , under s u c h g r a p h i c a l r e p r e s e n t a t i o n , a n a r e a which b e l o n g s t o one domain o n l y r e p r e s e n t s t h e p o s s i b i l i t y of e s t i m a t i n g t h e e f f e c t of t h e f a c t o r which i s r e p r e s e n t e d by t h i s domain. Also, a n a r e a which bel o n g s t o two domains, r e p r e s e n t s t h e e f f e c t of i n t e r a c t i o n of t h e c o r r e s p o n d i n g f a c t o r s , w h i l e t h e i n t e r s e c t i o n o f a l l t h r e e domains r e p r e s e n t s t h e i n t e r a c t i o n of a l l t h r e e f a c t o r s p l u s e r r o r . D e n o t i n g by S t h e i n f l u e n c e of t h e s u b j e c t , by I and I1 t h e e f f e c t s of p a r t i c u l a r f a c t o r s , by e t h e e r r o r of measurement, and by p a i r s o f t h e form ( S , I ) , (1,II) e t c . t h e c o r r e s p o n d i n g i n t e r a c t i o n s , one may e s t i m a t e a l l 7 components i n t h e complete c r o s s e d d e s i g n ( s e e Fig. 11.10). I n o t h e r words, a complete c r o s s e d d e s i g n a l l o w s t o estimate t h e e f f e c t s from p e r s o n s (S), o f f a c t o r s I and 11, i n t e r a c t i o n of p e r s o n s and f a c t o r I , e t c . The t r i p -
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l e i n t e r a c t i o n ( S , I , I I ) may be s e p a r a t e d from t h e e r r o r
e only i n c a s e o f r e p e a t e d measurements. If f o r e v e r y p e r s o n and e v e r y combination of l e v e l s of f a c t o r s I and I1 one makes o n l y one measurement, t h e t r i p l e i n t e r a c t i o n cannot be s e p a r a t e d from e r r o r of measurement. I n t h e c a s e o f t h e second scheme under c o n s i d e r a t i o n , t h e s i t u a t i o n i s as on F i g . 1.11.
/--
'\
S
1 I
_ -
\
--. \
,,
.
/
IJ
F i g . 1 1 . 1 0 . Components o f v a r i a n c e i n a complete c r o s s e d d e s i g n I n t h i s scheme one may s e p a r a t e t h e e f f e c t s of p e r s o n s S , f a c t o r I , and i n t e r a c t i o n between S and I , but one
CONTEMPORARY THEORY OFPSYCHOLOGICAL TESTS
83
.-
,/'
, .,
,
.*
.
1
,
I'
\ \
/
i \
i
(S,
F i g . 11.11. Components o f v a r i a n c e i n nested design
c a n n o t e s t i m a t e s e p a r a t e l y t h e e f f e c t s of f a c t o r I1 and
i n t e r a c t i o n between 1 and I1 ( t h e s e a r e n o t s e p a -
r a b l e ) . Thus , t h i s scheme o f e x p e r i m e n t s a l l o w s t h e p a r t i t i o n o f t h e v a r i a n c e of o b s e r v e d s c o r e i n t o 5 components. N a t u r a l l y , for e a c h e x p e r i m e n t a l scheme t h e r e e x i s t comp u t a t i o n a l f o r m u l a s f o r e s t i m a t i o n o f d i f f e r e n t compon e n t s o f v a r i a n c e ; t h e s e f o r m u l a s w i l l be p r e s e n t e d h e r e o n l y f o r t h e s i m p l e s t c a s e o f one f a c t o r and f u l l crossed design. To simplify t h e formulations,
l e t u s assume t h a t t h e l e v e l s of t h e f a c t o r a r e v e r s i o n s of t h e t e s t , g = 1,
...,
2, n , and t h a t t h e r e a r e N s u b j e c t s ( a =,l,...,N). We s h a l l c o n s i d e r two c a s e s : e x p e r i m e n t w i t h o u t r e p e t i t i o n s , and e x p e r i m e n t w i t h r e p e t i t i o n s (r t i m e s , where r > 1). I n o t h e r words, each person i s connected w i t h
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e a c h Level of t h e f a c t o r ( v e r s i o n o f t h e t e s t ) , w i t h e i t h e r one o r r measurements. I n t h e f i r s t c a s e , of scheme w i t h o u t r e p e t i t i o n s , we d e a l t h e r e f o r e w i t h a scheme p r e s e n t e d on F i g . 1 1 . 1 2 . Versions of t h e t e s t
1
2
...
3
...
...
n-1
n
... ...
F i g . 1 1 . 1 2 . Scheme of e x p e r i m e n t s w i t h o u t repetitions I n c a s e of r e p e t i t i o n s , t h e y may be t r e a t e d as l e v e l s of some f a c t o r , and t h e scheme o f e x p e r i m e n t s looks as on F i g . 1 1 . 1 3 . Thus, i n t h i s case each person responds r times t o each
v e r s i o n o f t h e t e s t ( i n g e n e r a l , r = 2 , as a l r e a d y mentioned).
a) ga
Scheme w i t h o u t r e p e t i t i o n s . N o t a t i o n s :
-
t h e s c o r e of p e r s o n a i n t e s t g;
1 N yg+ = R a=1 Yga
z
-
mean s c o r e i n t e s t g;
CONTEMPORARY THEORY OF PSYCHOLOGICAL TESTS
1
n
85
N
- t h e g e n e r a l mean s c o r e . g=l a=l
F i n a l l y , l e t G d e n o t e t h e number of t e s t s used i n t h e e x p e r i m e n t , of which n a r e s e l e c t e d a t random ( i n p a r t i c u l a r c a s e we may have G = n , when a l l v e r s i o n s a r e a n a l y s e d , of G = , which c o r r e s p o n d s t o t h e s i t u a t i o n when t h e r e a r e i n f i n i t e l y many v e r s i o n s ) . XJ
The e s t i m a t o r s o f components of v a r i a n c e a r e t h e n r e p r e s e n t e d i n T a b l e 11.1. The f o u r t h column of t h i s t a b l e c o n t a i n s q u a n t i t i e s , whose e s t i m a t o r s a p p e a r i n t h e 2 second column. Here B z i s t h e v a r i a n c e o f t h e g e n e r i c 2 t r u e s c o r e , 6 (I**) i s t h e v a r i a n c e of t h e g e n e r i c e r i s t h e v a r i a n c e due t o i n t e r a c t i o n , and : 0 is r o r , c2 $54 t h e v a r i a n c e connected w i t h d i f f e r e n c e s o f d i f f i c u l t y of v a r i o u s v e r s i o n s . A s may be s e e n , scheme w i t h o u t r e p e t i t i o n does n o t a l 2 2
low t o e s t i m a t e s e p a r a t e l y
G
(I**) and
r-
.
b ) Scheme w i t h r e p e t i t i o n s . N o t a t i o n s
'gak
-
t h e s c o r e of p e r s o n a i n v e r s i o n g of t h e t e s t ,
a t k - t h r e p e t i t i o n ( k = 1,. 'gar
- 1' - - kz' Ygak -- mean s c o r e of p e r s o n a i n v e r s i o n g; =l N
Yg**
.., r ) ;
-k
rN
r
2 2 a=l
k=l
Ygak
--
mean s c o r e i n t e s t g;
86
CHAPTER 1
.
n
N
r
The r e m a i n i n g n o t a t i o n s a r e as b e f o r e . A s may be s e e n from T a b l e 1 1 . 2 , i n t h i s c a s e i t i s pos s i b l e to s e p a r a t e t h e e f f e c t o f i n t e r a c t i o n from t h e v a r i a n c e of e r r o r s c o r e . Repetitions
n Ver s i ons of t e s t
1
2
/
r
,
... +
! . ,'1. . /. . . /.ii
i
2
,
,/ .
I
1
...
~
.
,
... ...
1
Persons
1
.
/
r
i
/ / I
i '
11
' '
I
,i
i
* ,,
,,
.. .. ... I
N
...
F i g . 1 1 . 1 3 . The scheme of e x p e r i m e n t w i t h
r repetitions
v n
+ v
H
n
+ **
c \
H
**
v
+ **
I
I
ri
G
I
ri
n
z
V
n
N
(d
** +h * h I * M M
(d
h I
I
h
**
n
N
z r:
v
ri
n
b
cu
H
b
N
r-! ri
I
r:
I
z
cu \3
V
CONTEMPORARY THEORY OF PSYCHOLOGICAL TESTS
c
0 .ri
P V
d
P
a,
a x
cii
k 0
~~
cu n
N n
I
h
* *
rn
I
h
**
k 5
a,
0
(d
v)
0
k
5
v)
87
T a b l e 1 1 . 2 . A n a l y s i s of v a r i a n c e i n c r o s s e d d e s i g n with r e p e t i t l o n s
(from L o r d and Novick, 1 9 6 8 )
'
S o u r c e of variability
Sum o f s q u a r e s
1
DegEFes freedom
I
' %
Between subjects
c N ( Y * ~ +- y***
rn
T-
a=1 n
I 1 I
Between
tests
I
Repetitions
r
1
N-1
!
i n-1 I
N 2 L ( Y ~ ~ * - Y ~ * ~ - Y * ~ , + Y , * *) (N-l)(n-l) g=l a = l
i'
n N -
2-
I-'
r i (ygak
g=l a=l k=l I
- Y*** ) 2
1
n
Interaction
I
z (Yg** g= 1
rN
I
i
)2
-
yga* ) *
nN(r-1)
Expect a t i o n
CONTEMPORARY THEORY OF PSYCHOLOGICAL TESTS
89
For n e s t e d d e s i g n s , t h e f o r m u l a s f o r t h e c o r r e s p o n d i n g e s t i m a t o r s a r e s i m i l a r , and w i l l n o t b e r e p r o d u c e d h e r e . A t t h e end i t i s w o r t h w h i l e t o p o i n t o u t t h e f a c t t h a t
w h i l e t h e complete c r o s s e d d e s i g n a l l o w s f o r e s t i m a t i n g a l l components o f v a r i a n c e , i n c l u d i n g a l l i n t e r a c t i o n s , t h i s i s a t t a i n e d a t t h e c o s t of performing a d i f f i c u l t e x p e r i m e n t : e a c h p e r s o n must be t e s t e d a t e v e r y combination of t h e l e v e l s o f f a c t o r s . Nested d e s i g n s are much e a s i e r t o implement i n p r a c t i c e , b u t t h e y p r o v i d e l e s s i n f o r m a t i o n , by n o t a l l o w i n g
t o s e p a r a t e t h e e f f e c t s o f c e r t a i n f a c t o r s from some interactions. I n g e n e r a l , one c o n s i d e r s two t y p e s of f a c t o r s . Thus, w e speak of f i x e d e f f e c t s , i f t h e l e v e l s of f a c t o r a r e d e t e r m i n e d b y t h e e x p e r i m e n t e r i n some s y s t e m a t i c w a y , and e x h a u s t a l l p o s s i b i l i t i e s . S e c o n d l y , w e s p e a k o f random e f f e c t s , if t h e l e v e l s are s e l e c t e d a t random from a c e r t a i n u n i v e r s e . T h i s may b e s t be s e e n from t h e f o r m u l a s p r e s e n t e d i n T a b l e s 11.1 and 1 1 . 2 . If t h e e x p e r i m e n t e r h a s G v e r s i o n s o f t h e t e s t a t h i s
d i s p o s a l , and p r o c e e d s t o a n a l y s e a l l o f them s y s t e m a t i c a l l y ( n = G ) , t h e n v e r s i o n of t h e t e s t i s a f a c t o r w i t h c o n s t a n t e f f e c t s . Such a p r o c e d u r e i s p o s s i b l e o n l y f o r r e l a t i v e l y small G . If t h e s e t o f t e s t s Is numerous, s o t h a t i t i s impossib l e t o a n a l y s e a l l of them, one samples n t e s t s f o r t h e s t u d y . We d e a l i n s u c h a c a s e w i t h random e f f e c t s . I n p a r t i c u l a r , one may assume G = aand t r e a t t h e n t e s t s as s e l e c t e d a t random from a c e r t a i n i n i f i n i t e "uni-
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v e r s e " of t e s t s p o s s i b l e i n a given s i t u a t i o n . G e n e r a l l y s p e a k i n g , i n t e s t i n g problems, one d e a l s most o f t e n w i t h random e f f e c t s . It i s n o t p o s s i b l e to g i v e t h e b e s t e x p e r i m e n t a l schem e : t h e c h o i c e h e r e depends on t h e g o a l of s t u d y , and
a l s o on t h e s u b j e c t i v e c o n v i c t i o n o f t h e e x p e r i m e n t e r t h a t s u c h and s u c h i n t e r a c t i o n s a r e z e r o . I f he t h i n k s t h a t c e r t a i n i n t e r a c t i o n s v a n i s h , he may choose t h e corresponding nested design, thus simplifying t h e s t u dy. F i n a l l y , i t i s w o r t h to m e n t i o n t h a t t h e t h e o r y of gen e r a l i z a b i l i t y , while leading i n general t o a decrease o f e r r o r o f measurement (by s e p a r a t i n g v a r i o u s compon e n t s of v a r i a n c e ) , p r o v i d e s o f t e n u n s a t i s f a c t o r y e s t i mation of t h e t r u e s c o r e . To i l l u s t r a t e t h i s phenomenon, and a l s o i n o r d e r to s k e t c h new a p p r o a c h e s to e s t i m a t i o n o f t h e t r u e s c o r e , i t i s b e s t t o use a n example. I n g e n e r a l , under t h e a s s u m p t i o n t h a t t h e d i s t r i b u t i o n of e r r o r s c o r e s , i.e. d e v i a t i o n s between t h e t r u e and observed s c o r e i s normal, one c o n s t r u c t s c o n f i d e n c e i n t e r v a l s f o r t h e unknown t r u e s c o r e .
For i n s t a n c e , i n c a s e o f c o n f i d e n c e l e v e l 0 . 9 5 , t h e c o n f i d e n c e i n t e r v a l (X - 1 . 9 6 6 , X t 1.966) w i l l c o v e r t h e unknown t r u e s c o r e w i t h p r o b a b i l i t y 0 . 9 5 . C o n s t r u c t i o n of confidence i n t e r v a l s i s of g r e a t pract i c a l i m p o r t a n c e i n t h e p h y s i c a l s c i e n c e s , where t y p i c a l l y , t h e e r r o r o f measurement i s small as compared
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w i t h t h e w i d t h o f " u n c e r t a i n t y band", i n which
i n g the t h e experimenter which he m e a s u r e s .
--
--
accordl i e s t h e unknown t r u e v a l u e
For example, i n t h e p h y s i c a l s c i e n c e s i t i s s e n s i b l e t o a s s e r t t h a t t h e t r u e weight of a g i v e n o b j e c t e q u a l s 15.03 ? 0 . 0 2 grams, where t h e v a l u e 0 . 0 2 was o b t a i n e d from p r e v i o u s s t u d i e s o f t h e mean s q u a r e e r r o r of t h e measurement t o o l (one u s u a l l y o m i t s h e r e t h e c o n f i d e n c e l e v e l ) . Such a n a s s e r t i o n c o n t a i n s s i g n i f i c a n t informa t i o n : b e f o r e measurement was t a k e n , from an i n s p e c t i o n of t h e o b j e c t , knowledge, e x p e r i e n c e , and s o f o r t h , one c o u l d a s s e r t o n l y ( s a y ) t h a t i t s weight l i e s w i t h i n t h e l i m i t s 1 2 and 18 grams. The p o i n t i s t h a t i n most c a s e s , t h e w i d t h of " u n c e r t a i n t y band" ( 1 2 - 18 grams) i s c o n s i d e r a b l y l a r g e r t h a n t h e w i d t h o f t h i s band a f t e r t h e measurmement was t a k e n . I n p s y c h o l o g y , t h e s i t u a t i o n i s , a s a r u l e , q u i t e oppos i t e : t h e w i d t h of " u n c e r t a i n t y band" a p r i o r i ( b e f o r e t h e measurement was t a k e n ) i s n o t much l a r g e r t h a n t h e w i d t h o f t h e band a f t e r measurement.
For i n s t a n c e , imagine t h a t we measure I&of a psychol o g y j u n i o r . Without any measurement, we may r e a s o n a b l y e x p e c t h i s I& t o be somewhere between 1 1 0 and 130: t h e r e w i l l be o n l y few s t u d e n t s w i t h I&lower t h a n 1 1 0 , and a l s o few w i t h I& e x c e e d i n g 130. Suppose t h a t t h e measuremer.t gave t h e r e s u l t 118, and w e a l s o know t h a t t h e s t a n d a r d d e v i a t i o n i s 5 . Then t h e c o n f i d e n c e i n t e r v a l , w i t h c o n f i d e n c e l e v e l 0 . 9 5 , w i l l be 118 ? 1.96.5 which i s about ( 1 0 8 , 1 2 8 ) . The p r o f i t from measurement i s h e r e r a t h e r mediocre: a f t e r measurement w e know, i n e s s e n c e , as much as we knew b e f o r e anyway.
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Such a s i t u a t i o n , o f r a t h e r c o n s i d e r a b l e p r i o r knowledge and l a r g e e r r o r of measurement, i s t y p i c a l i n measurement i n psychology, and t h e s o c i a l s c i e n c e s i n gen e r a l . It c a s t s some d o u b t s on t h e s e n s e o f u s i n g confidence intervals. A p o s s i b l e a p p r o a c h h e r e i s t o use B a y e s i a n e s t i m a t o r s
( s e e e . g . Novick 1969, 1971; Novick and G r i z z l e 1965; Novick and J a c k s o n 1970, o r Novick, Jackson and T h a y e r 1 9 7 1 ) . These e s t i m a t o r s u s e t h e a p r i o r i knowledge, r e p r e s e n t e d i n form o f an a p p r o p r i a t e p r i o r d i s t r i b u t i o n , and c o n s i s t o f t a k i n g t h e mode of t h e p o s t e r i o r d i s t r i b u t i o n ( g i v e n t h e observed s c o r e ) as a n e s t i m a t e of t h e t r u e s c o r e . F o r m a l l y , l e t f ( T ) be t h e p r i o r d e n s i t y of t h e t r u e s c o r e , and l e t g ( X [ T ) be t h e d e n s i t y o f t h e o b s e r v e d s c o r e X f o r g i v e n t r u e s c o r e T . Then
i s t h e c o n d i t i o n a l d e n s i t y of t h e t r u e s c o r e T g i v e n
t h e observed s c o r e X . A s B a y e s i a n e s t i m a t o r , one u s u a l l y t a k e s t h e mode o f the d i s t r i b u t i o n h ( X / T ) , t h a t i s t h e v a l u e of T (de-
p e n d i n g on X ) which m a x i m i z e s h ( T / X ) . The problem, n a t u r a l l y , l i e s i n f i n d i n g t h e p r i o r d i s -
t r i b u t i o n which would b e i n t u i t i v e l y a c c e p t a b l e as a f a i t h f u l r e p r e s e n t a t i o n o f t h e c o n v i c t i o n s and knowledge o f t h e e x p e r i m e n t e r , and a t t h e same t i m e which would lead t o r e l a t i v e l y s i m p l e c o m p u t a t i o n a l f o r m u l a s f o r estimators.
CONTEMPORARY THEORY OF PSYCHOLOGICAL TESTS
93
1 2 . B I A S I N SELECTION PROCEDURES
The use o f t e s t s i s o f t e n s e v e r e l y c r i t i c i z e d because of t h e p o s s i b i l i t y o f d i s c r i m i n a t i o n i n s o c i a l p o l i c i e s . We b e g i n w i t h p r e s e n t i n g t h e f o r m u l a s f o r t h e b e s t lin e a r e s t i m a t o r o f t h e t r u e s c o r e of t h e s u b j e c t , given h i s observed s c o r e X .
Y = Yp i s t h e mean s c o r e i n t h e p o p u l a t i o n s u b j e c t , and w = wp i s t h e r e l i a b i l i t y o f t h e p o p u l a t i o n P , t h e n t h e r e g r e s s i o n estimate o f s c o r e o f t h e s u b j e c t who s c o r e d X on t h e t e s t If
z
=
wx t ( 1 - w ) Y
P of the test in the true is (12.1)
( s e e Lord and Novick 1 9 6 8 ) . Formula ( 1 2 . 1 ) i m p l i e s t h a t i f t h e t e s t i s h i g h l y r e l i a b l e ( w i s c l o s e t o l), t h e n t h e o b s e r v e d s c o r e X i s g i v e n much weight i n t h e assessment o f t h e t r u e s c o r e . I f t h e t e s t i s n o t h i g h l y r e l i a b l e , l e s s weight i s g i v e n t o t h e o b s e r v e d s c o r e X , and more t o t h e g e n e r a l i n f o r m a t i o n about t h e s u b j e c t , namely t h a t he i s a member o f a p o p u l a t i o n whose e x p e c t e d t r u e score i s Y . T h i s appears r e a s o n a b l e and c o n v i n c i n g ; y e t , i t may l e a d
t o somewhat p a r a d o x i c a l r e s u l t s . Suppose, f o r i n s t a n c e , t h a t t h e t e s t i s used t o s e l e c t b e s t c a n d i d a t e s from a number o f groups ( e . g . s c h o o l s , or c l a s s e s w i t h i n t h e same s c h o o l , s o c i a l g r o u p s , e t c . ) . To f i x i d e a s , s u p p o s e t h a t a f o u r t h - g r a d e r z1 from some s c h o o l s c o r e d 4 5 on t h e t e s t i n q u e s t i o n , and t h e same s c o r e ( 4 5 ) was o b t a i n e d by a fif'th-grader z 2 . Suppose t h a t t h e r e l i a b i l i t i e s o f t h e t e s t i n t h e p o p u l a t i o n s of f o u r t h - g r a d -
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e r s and i n t h e p o p u l a t i o n o f f i f t h - g r a d e r s a r e t h e same and e q u a l 0 . 8 0 . F i n a l l y , suppose t h a t t h e mean s c o r e i n t h e p o p u l a t i o n of f o u r t h g r a d e r s i s Y = 4 0 , and i n t h e population of f i f t h graders it i s Y = 50. I n t h i s case, the estimator (12.1) yields f o r subject
z1 ( a f o u r t h - g r a d e r ) t h e e s t i m a t e d t r u e s c o r e Z = 0 . 8 . 4 5 + 0.2-40 = 44. S i m i l a r l y , for t h e f i f t h - g r a d e r z2 w e get Z = 0.8.45
+
t r u e s c o r e of z
1
0.2.50
=
46. We s e e t h a t t h e e s t i m a t e d
i s lower t h a n t h a t of z2, d e s p i t e t h e
f a c t t h a t z1 s c o r e d above a v e r a g e i n h i s r e f e r e n c e g r o u p , w h i l e z2 s c o r e d below a v e r a g e i n h i s r e f e r e n c e g r o u p . I n a sense, z
1 i s b e i n g "punished" f o r membership i n a group w i t h low a v e r a g e , w h i l e z2 i s b e i n g "rewarded" for membership i n a group w i t h h i g h a v e r a g e . I n a w i d e r c o n t e x t , t h e problems i l l u s t r a t e d h e r e conc e r n b i a s e s i n t e s t i n g and s e l e c t i o n p r o c e d u r e s . F o r a v e r y t h o r o u g h and e x h a u s t i v e d i s c u s s i o n of t h i s c r u c i a l t o p i c , and a d e t a i l e d a n a l y s i s o f t h e e m p i r i c a l d a t a collected thus far, see Jensen (1980). F i r s t l y , t h e s e l e c t i o n i s based u s u a l l y on p r e d i c t e d val u e s of some c r i t e r i o n U ( e . g . f u t u r e p e r f o r m a n c e , e t c . ) T y p i c a l l y , U i s r e l a t e d t o t h e t r u e s c o r e X on t h e t e s t i n q u e s t i o n ; t h u s , t e s t s c o r e X i s f i r s t used t o o b t a i n
a p r e d i c t i o n U = U(X) o f t h e c r i t e r i o n v a r i a b l e U, and t h e n t h e o b t a i n e d v a l u e s U a r e used f o r s e l e c t i o n procedure: t h e c a n d i d a t e s a r e o r d e r e d a c c o r d i n g t o t h e est i m a t e d v a l u e s of U , and t h e n t h e s e l e c t i o n p r o c e e d s from t h e t o p , u n t i l e i t h e r a r e q u i r e d number o f candidat e s i s f o u n d , o r u n t i l t h e v a l u e s o f U become lower t h a n some p r e - a s s i g n e d t h r e s h o l d .
The example above shows
t h a t ( i n t h e p a r t i c u l a r c a s e when t h e e s t i m a t e d t r u e
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CONTEMPORARY THEORY OF PSYCHOLOGICAL TESTS
s c o r e i s t h e v a r i a b l e used as a c r i t e r i o n ) , s u c h a proc e d u r e would d e c r e a s e c h a n c e s o f s e l e c t i o n f o r c a n d i d a t e s from g r o u p s w i t h l o w e r means ( i n c a s e of t e s t s w i t h r e l i a b i l i t y less than one). The p r e d i c t i o n o f U on t h e b a s i s o f X h a s t h e form o f regression equation U = a t b X
(12.2)
where t h e c o e f f i c i e n t s a and b ( t h e i n t e r c e p t and t h e s l o p e ) , as w e l l a s t h e e r r o r o f t h e e s t i m a t e , may depend on t h e r e f e r e n c e g r o u p . Suppose now t h a t we have two g r o u p s o f s u b j e c t s , s a y A and B , and t h e s e l e c t i o n i s t o be a p p l i e d t o b o t h
g r o u p s j o i n t l y . The c o n c e p t s r e l e v a n t h e r e a r e t h o s e o f b i a s o f t h e t e s t , and o f f a i r n e s s of i t s u s e .
for d e f i n i n g t h e n o t i o n o f b i a s o f a t e s t ( w i t h
Firstly,
r e s p e c t t o t h e two g r o u p s i n q u e s t i o n ) , l e t
U
=
a 1 t blX,
U = a2
+
b2X
(12.3)
be t h e r e g r e s s i o n e q u a t i o n s f o r p r e d i c t i n g U on t h e b a s i s
o f X i n g r o u p s A and B r e s p e c t i v e l y . The s t a n d a r d e r r o r o f p r e d i c t i o n e q u a l s
(12.4) where r Y 2 i s t h e v a r i a n c e of t h e c r i t e r i o n U , and rXU U i s t h e c o e f f i c i e n t o f c o r r e l a t i o n between t e s t and c r i t e r i o n . N a t u r a l l y , we have h e r e two v a l u e s , s and s2, 1
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connected w i t h t h e f a c t t h a t t h e c o r r e l a t i o n c o e f f i c i e n t between X and U , and v a r i a n c e o f U , may d i f f e r i n groups A and B . We may now s a y t h a t t e s t X i s u n b i a s e d f o r U , w i t h r e s p e c t t o groups A and B , i f
a l = a 2,
bl
= b2,
s
1
= s
2'
(12.5)
N a t u r a l l y , t h e above c o e f f i c i e n t s a r e t h e o r e t i c a l param e t e r s , which must be e s t i m a t e d from t h e sample. Consequently, i n practice the t e s t i s biased, i f there i s a s t a t i s t i c a l l y s i g n i f i c a n t d i f f e r e n c e between a t l e a s t one o f t h e p a i r s o f v a l u e s i n ( 1 2 . 5 ) . Also, one s h o u l d t a k e i n t o a c c o u n t some a p p r o p r i a t e c o r r e c t i o n s f o r r e l i a b i l i t y o f t e x t X ( i . e . requirement ( 1 2 . 5 ) a p p l i e s t o t h e i d e a l i z e d c a s e of p e r f e c t l y r e l i a b l e t e s t s ) . Observe t h a t t h i s d e f i n i t i o n o f u n b i a s e d n e s s i s e n t i r e l y u n r e l a t e d t o f a i r n e s s , t h e l a t t e r being a concept pert a i n i n g t o u s e o f t h e t e s t . B i a s and u n b i a s e d n e s s a r e purely "neutral" s t a t i s t i c a l notions. Observe a l s o t h a t t h e d e f i n i t i o n o f u n b i a s e d n e s s d o e s n o t r e q u i r e t h a t groups A and B have t h e same means o r v a r i a n c e s i n t e s t X . Note, t o o , t h a t t h e f a c t t h a t t h e t e s t X i s b i a s e d does n o t , by i t s e l f , d i s q u a l i f y i t from u s e . An example g i v e n by J e n s e n ( 1 9 8 0 ) might c l a r i f y t h i s point. C o n s i d e r t h e groups A and B d e t e r m i n e d b y s e x , and l e t U be t h e p r e d i c t e d h e i g h t a t mature a g e . A s a p r e d i c t o r X we t a k e t h e mean h e i g h t o f p a r e n t s , i . e . t h e m i d p o i n t between t h e h e i g h t o f f a t h e r and m o t h e r . I n t h i s
CONTEMPORARY THEORY OF PSYCHOLOGICAL TESTS
97
c a s e , t h e r e g r e s s i o n o f U on X i s d i f f e r e n t f o r men and f o r women, simply because t h e f o r m e r t e n d t o b e , on t h e a v e r a g e , t a l l e r t h a n women. Thus, t h e a v e r a g e h e i g h t of p a r e n t s i s a b i a s e d p r e d i c t o r o f t h e h e i g h t a t mature a g e ; t h i s f a c t , however, d o e s n o t d i s q u a l i f y t h e use o f X . To a p p r e c i a t e t h e i n t u i t i v e j u s t i f i c a t i o n o f r e q u i r e m e n t (12.5) f o r unbiasedness, it i s best t o consider the consequences o f v i o l a t i n g t h e e q u a l i t i e s i n ( 1 2 . 5 ) . I f a1 # a2 a n d / o r b1
#
b 2 , t h e n f o r t h e same o b s e r v e d
s c o r e X , t h e p r e d i c t e d v a l u e s would, i n g e n e r a l , be d i f f e r e n t f o r groups A and B . C o n s e q u e n t l y , t h e e s t i m a t e of t h e c r i t e r i o n i n one group i s h i g h e r t h a n i n t h e other: t h e c r i t e r i o n i s underestimated o r overestimated (or both). If a l = a*, b1
--
b2, but
s1 # s2, t h e b i a s l i e s i n t h e
f a c t t h a t w h i l e t h e same s c o r e X y i e l d s t h e same p r e d i c t ed c r i t e r i o n , t h e p r o b a b i l i t y o f p r e d i c t i o n b e i n g i n e r r o r by a g i v e n amount i s d i f f e r e n t i n group A and i n group B . Now, as r e g a r d s t h e problem o f s e l e c t i o n ( w h e t h e r w i t h a b i a s e d o r u n b i a s e d t e s t ) , one would l i k e i t t o meet t h e c o n d i t i o n s of f a i r n e s s . The l a t t e r n o t i o n may be e x p l i c a t e d i n v a r i o u s w a y s . Hun-
t e r and Schmidt ( 1 9 7 6 ) c l a s s i f y a l l methods of s e l e c t i o n s u g g e s t e d t h u s f a r i n t o t h r e e broad c a t e g o r i e s , c a l l e d u n q u a l i f i e d i n d i v i d u a l i s m , q u a l i f i e d i n d i v i d u a l i s m , and q u o t a s y s t e m . These t h r e e methods w i l l be o u t l i n e d below, f o r t h e c a s e o f t w o g r o u p s , A and B.
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According t o t h e p r i n c i p l e o f u n q u a l i f i e d i n d i v i d u a l i s m , t h e c a n d i d a t e s s h o u l d be chosen a c c o r d i n g t o t h e b e s t a v a i l a b l e p r e d i c t o r o f t h e i r f u t u r e performance U . It i s not r e q u i r e d t h a t t h e same p r e d i c t o r s be a p p l i e d t o any p e r s o n , as l o n g one u s e s t h e p r e d i c t o r which i n a g i v e n i n s t a n c e has t h e h i g h e s t p o s s i b l e v a l i d i t y . The v a r i a b l e s which e n t e r t h e p r e d i c t o r may t h e r e f o r e comprise t h e s c o r e s on v a r i o u s t e s t s , as w e l l as group membership (e.g. sex, race, e t c . ) . This principle guarantees
t h e h i g h e s t a v e r a g e perform-
ance l e v e l among t h e chosen c a n d i d a t e s , and a l s o minimiz e s t h e e x p e c t e d number o f e r r o r s ( a c c e p t i n g a c a n d i d a t e who l a t e r f a i l s t o meet t h e r e q u i r e d performance l e v e l , and r e j e c t i n g a c a n d i d a t e who would meet t h i s l e v e l i f a c c e p t e d ) . The c h o i c e i s f a i r , i n t h e s e n s e t h a t no p e r s o n i s s e l e c t e d o r r e j e c t e d because o f h i s group membership. The main o b j e c t i o n a g a i n s t t h e u s e o f t h i s p r i n c i p l e l i e s i n t h e f a c t t h a t t h e e x i s t i n g p r e d i c t o r s may have d i f f e r e n t v a l i d i t i e s f o r d i f f e r e n t g r o u p s . I n such a cas e , p e r s o n s from t h e group w i t h p r e d i c t o r s o f a lower v a l i d i t y may have lower chances o f b e i n g s e l e c t e d . The second p r i n c i p l e , of q u a l i f i e d i n d i v i d u a l i s m , d i f f e r s
from t h e p r e c e d i n g one i n one r e s p e c t , namely t h a t one i s n o t a l l o w e d t o use t h e group membership as a v a r i a b l e entering t h e predictors. In t h i s case therefore, t h e same v a r i a b l e s s h o u l d b e used i n b o t h g r o u p s , and t h e same common e q u a t i o n f o r performance p r e d i c t i o n U has t o be u s e d .
I n t h i s c a s e , t h e a v e r a g e l e v e l o f performance among t h e
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s e l e c t e d p e r s o n s w i l l , i n g e n e r a l , be l o w e r t h a n i n t h e f i r s t case, i f t h e predictor
has d i f f e r e n t v a l i d i t y i n
c a s e of d i f f e r e n t g r o u p s . T h i s i s compensated by t h e f a c t t h a t no one can c l a i m t h a t he was t r e a t e d i n any s p e c i a l ( f a v o u r i n g o r d i s c r i m i n a t i n g ) way b e c a u s e of h i s group membership. However, i n a n a t t e m p t t o i n c r e a s e t h e v a l i d i t y of t h e p r e d i c t o r s u s e d , one o f t e n u s e s v a r i a b l e which c o r r e l a t e h i g h l y b o t h w i t h t h e performance l e v e l and w i t h t h e group membership. Thus, group membership e n t e r s i n a d i s g u i s e d way ( e . g . t h r o u g h s o c i o e c o n o m i c a l v a r i a b l e s ) , and i t i s n o t e a s y t o d e t e r m i n e which v a r i a b l e s may and which may n o t be used i n p r e d i c t i o n . F i n a l l y , i n t h e q u o t a system,
one r e s i g n s from t h e
p r i n c i p l e of m i n i m i z i n g t h e e r r o r , or maximizing t h e l e v e l of f u t u r e p e r f o r m a n c e , and a l l o c a t e s some q u o t a s t o g r o u p s A and B. Thus, w i t h i n t h e groups t h e c a n d i d a t e s a r e s e l e c t e d by t h e i r p r e d i c t e d l e v e l o f performanc e , b u t it may happen t h a t a lower s c o r i n g c a n d i d a t e w i l l be s e l e c t e d and a h i g h e r s c o r i n g c a n d i d a t e w i l l be
r e j e c t e d i f t h e y a r e from v a r i o u s g r o u p s . I n more g e n e r a l s e t u p , by q u o t a s y s t e m one means any method o f s e l e c t i o n under which a g i v e n c a n d i d a t e w i t h lower p r e d i c t e d performance may be c h o s e n o v e r a c a n d i d a t e w i t h h i g h e r p r e d i c t e d performance. Quota s y s t e m s are t h o u g h t o f as means of improving t h e e x i s t i n g s o c i a l s t r u c t u r e s and undoing some p r e v i o u s i n j u s t i c e . The d e c i s i o n as t o t h e c h o i c e of q u o t a s i s a m a t t e r of s o c i a l p o l i c y . Once t h i s i s d e c i d e d , however, one may s u g g e s t numerous s e l e c t i o n schemes s o as t o o p t i m i z e
CHAPTER I
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c e r t a i n c r i t e r i a , meeting t h e q u o t a c o n s t r a i n t s . A t t h e e n d , w e s h a l l p r e s e n t some of t h e s u g g e s t e d s o l -
utions i n the quota system. C o n s i d e r one g r o u p , and assume t h a t t h e j o i n t d i s t r i b u t i o n o f t h e c r i t e r i o n U and t e s t X i s normal. Note t h a t U i s n o t o b s e r v a b l e a t t h e time of d e c i s i o n making.
Suppose ( s e e F i g . 1 2 . 1 ) t h a t U
*
i s the required l e v e l n of performance (for a g i v e n g r o u p ) , and t h a t X i s t h e * c u t - o f f p o i n t f o r t h i s group (so t h a t X i s t h e c o n t r o l
X* F i g . 1 2 . 1 . Choice a r e a s i n q u o t a s y s t e m v a r i a b l e ) . The a r e a s A , B , C
and D on F i g . 1 2 . 1 r e p r e s e n t
respectively: A
--
c a n d i d a t e s who a r e a c c e p t e d and whose performance exceeds t h e r e q u i r e d l e v e l ( c o r r e c t acceptance)
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B
--
c a n d i d a t e s who are r e j e c t e d , b u t whose l e v e l o f performance would meet t h e r e q u i r e d l e v e l (wrong rejection);
C
--
c a n d i d a t e s who a r e r e j e c t e d , and whose l e v e l o f performance d o e s n o t meet t h e s t a n d a r d ( c o r r e c t rejections) ;
D
--
c a n d i d a t e s who a r e a c c e p t e d , b u t whose l e v e l o f performance d o e s n o t meet t h e r e q u i r e d s t a n d a r d (wrong a c c e p t a n c e ) .
To g i v e j u s t few examples o f t h e s u g g e s t e d s e l e c t i o n p r o c e d u r e s : one may r e q u i r e t h a t i n e a c h group, t h e rat i o o f t h o s e a c c e p t e d t o t h o s e r e j e c t e d s h o u l d be t h e same. T h i s s e l e c t i o n p r o c e d u r e r e q u i r e s t h e r e f o r e t h a t * t h e c u t o f f p o i n t X (and hence a l s o t h e r e q u i r e d p e r * formance l e v e l U ) a r e s e t s o t h a t t h e r a t i o ( A + D ) / ( B - k C s h o u l d be t h e same i n e a c h group.
Thorndike ( 1 9 7 1 ) proposed a n o t h e r model o f s e l e c t i o n , a c c o r d i n g t o which t h e r a t i o o f a c c e p t a n c e t o s u c c e s s s h o u l d be t h e same i n e a c h group, i . e . he p o s t u l a t e d t h a t ( A t D ) / ( A t B ) s h o u l d be t h e same i n e a c h group. I n s t i l l a n o t h e r p r o c e d u r e , Cole
(1973) s u g g e s t e d t h a t
p r o b a b i l i t y o f s e l e c t i o n among t h o s e who succeed ( a r e * above U ) s h o u l d be t h e same i n b o t h g r o u p s . T h i s r e q u i r e s e q u a l i z i n g t h e r a t i o A / ( A + B) i n b o t h g r o u p s . F i n a l l y , P e t e r s e n and Novick (19762 a r g u e t h a t a l l acc e p t e d a p p l i c a n t s s h o u l d have t h e same c h a n c e s of b e i n g s u c c e s s f u l ; t h i s , i n t u r n , means t h a t t h e r a t i o s A / ( A + B should be e q u a l i n b o t h groups.
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13. M A I N DIRECTIONS OF RESEARCH I N TEST THEORY
It a p p e a r s t h a t t h e development o f t e s t t h e o r y i n t h e
l a s t t e n or s o y e a r s p r o c e e d e d i n t h r e e main d i r e c t i o n s : d e v e 1o pme n t o f
I'
c 1as s i c a 1 t e s t t he o r y , g e n e r a 1i z a b i 1i t y
t h e o r y , and t h e o r y o f l a t e n t t r a i t s . A s r e g a r d s t h e development of' c l a s s i c a l t e s t t h e o r y , t h e most i m p o r t a n t c o n t r i b u t i o n s a p p e a r t o i n v o l v e t h e a p p l i c a t i o n of t h e t h e o r y o f s t o c h a s t i c p r o c e s s e s . To mention j u s t few e x a m p l e s , t h e t h e o r e m s on r e l a t i o n s between r e l i a b i l i t y and l e n g t h of t h e t e s t may be r e i n t e r p r e t e d a s s t a t e m e n t s about t h e p r o p e r t i e s of s t o c h a s t i c processes occurring i n continuous time, representi n g t h e f r a c t i o n o f i t e m s answered. Such a n a p p r o a c h h a s been s u g g e s t e d by Novick 1 9 7 6 . A n o t h e r p o s s i b i l i t y i s c o n n e c t e d w i t h t h e problem o f d e t e r m i n i n g t h e b e s t i n t e r v a l o f time between t e s t i n g and r e - t e s t i n g ,
so as t o o b t a i n t h e b e s t e s t i m a t e of
r e l i a b i l i t y by t e s t - r e t e s t c o r r e l a t i o n ( s e e M o r r i s o n
1 9 8 1 ) . Here t h e main i s s u e i s t h e dilemma: i f t h e second t e s t i n g comes t o o e a r l y , t h e memory e f f e c t s may be a p p r e c i a b l e . On t h e o t h e r hand, i n c a s e o f t e s t i n g t o o f a r a p a r t i n t i m e , t h e t r u e s c o r e may undergo a p p r e c i a b l e c h a n g e s . Thus, M o r r i s o n s u g g e s t s a model o f b o t h memory f a d i n g and c h a n g e s of t r u e s c o r e , and d e r i v e s t h e f o r mulas for t h e o p t i m a l t i m e s e p a r a t i n g t e s t i n g and r e t e s t ing. A s regards generalizability theory, several directions
of r e s e a r c h a r e w o r t h n o t i n g . F i r s t l y , i t o u g h t t o be o b s e r v e d t h a t t h e a p p l i e d methods o f e s t i m a t i o n of t h e
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components o f v a r i a n c e are based on f o r m u l a s which
ex-
p r e s s t h e s e components as sums o f o t h e r components ( s e e T a b l e s i n S e c t i o n 11). C o n s e q u e n t l y , a n e s t i m a t e o f a v a r i a n c e component i s , t y p i c a l l y , a l i n e a r combination of sums o f s q u a r e s o f o b s e r v a t i o n s , w i t h some c o e f f i c i e n t s b e i n g n e g a t i v e . S i n c e t h e e s t i m a t e s a r e s u b j e c t t o sampl i n g e r r o r s , i t may o c c u r t h a t t h e e s t i m a t e s y i e l d negat ive values. Moreover, t y p i c a l l y t h e more i m p o r t a n t and i n t e r e s t i n g v a r i a n c e components w i l l have h i g h e r s a m p l i n g e r r o r s , which a f f e c t s p r e c i s i o n of e s t i m a t i o n and a l s o makes them more prone t o assume n e g a t i v e v a l u e s . One of t h e a p p r o a c h e s i s t o r e p l a c e n e g a t i v e e s t i m a t e s by z e r o , and p r o c e e d w i t h f u r t h e r e s t i m a t e s u s i n g t h e o b t a i n e d v a l u e s . However, as p o i n t e d o u t by ScheffC ( 1 9 5 9 ) , t h e s a m p l i n g d i s t r i b u t i o n of e s t i m a t o r s which i n v o l v e s u b s t i t u t i o n of e s t i m a t o r s o b t a i n e d by s u c h r e p l a c e m e n t s o f n e g a t i v e v a l u e s by z e r o , i s q u i t e complicated. A p r o m i s i n g s o l u t i o n h e r e i s t h e use of Bayesian methods,
imposing a n o n n e g a t i v i t y c o n s t r a i n t ( s e e e . g . Novick and J a c k s o n 1 9 7 4 o r Box and T i a o 1 9 7 3 ) . Numerous s i m u l a t i o n s t u d i e s ( s e e e . g . Smith 1978) seem t o i n d i c a t e t h a t sampling e r r o r s i n g e n e r a l i z a b i l i t y s t u d i e s a r e t y p i c a l l y q u i t e l a r g e , and t h a t i n o r d e r t o g e t s t a b l e e s t i m a t e s one needs u n i v e r s e s w i t h l a r g e numbers o f f a c e t s . The s i t u a t i o n becomes much more c o m p l i c a t e d when t h e d e s i g n o f e x p e r i m e n t i n unbalanced ( i . e . t h e numbers o f o b s e r v a t i o n s i n c e l l s do n o t meet p r o p o r t i o n a l i t y
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conditions). A s another l i n e of r e s e a r c h i n g e n e r a l i z a b i l i t y t h e o r y
one s h o u l d mention h e r e t h e a t t e m p t s t o d e v e l o p a m u l t i d i m e n s i o n a l v a r i a n t of t h e t h e o r y ( s e e e . g . J o e and Woodward 1 9 7 6 ) . For a thorough review of g e n e r a l i z a b i l i t y t h e o r y , i t s a p p l i c a t i o n , and a n e x t e n s i v e b i b l i o g r a p h y of t h e subj e c t , s e e S h a v e l s o n and Webb ( 1 9 8 1 ) . F i n a l l y , t h e t h i r d l i n e o f r e s e a r c h concerns t h e theory o f l a t e n t t r a i t s , s u g g e s t e d o r i g i n a l l y be L a z a r s f e l d (1959, 1 9 6 0 ) . T h i s t h e o r y d e a l s w i t h s i m u l t a n e o u s measurement by a
number o f t o o l s , s u c h as i t e m s ( i n c a s e o f a t e s t ) . The c e n t r a l i d e a i s t o use t h e n o t i o n o f l o c a l c o n d i t i o n a l independence ( c a l l e d a l s o symmetric dependence, o r e x c h a n g e a b i l i t y ; see F e l l e r 1966) : a l a t e n t s t r u c t u r e i s c o m p l e t e , if i n any group o f p e r s o n s i d e n t i c a l w i t h respect t o a l l l a t e n t t r a i t s , t h e observable v a r i a b l e s ( i t e m r e s p o n s e s , f o r i n s t a n c e ) a r e independent. It f o l l o w s i n p a r t i c u l a r , t h a t t h e o b s e r v e d v a r i a b l e s
a r e i n d e p e n d e n t f o r e a c h p e r s o n s e p a r a t e l y , s i m p l y bec a u s e a s i n g l e p e r s o n i s a s e p e c i a l c a s e o f a group of p e r s o n s w i t h i d e n t i c a l l a t e n t t r a i t s . The a s s u m p t i o n o f c o n d i t i o n a l independence o f r e s p o n s e s i s not v e r i f i a b l e , because of t h e d i f f i c u l t y i n g e t t i n g i n d e p e n d e n t r e p e t i t i o n s o f measurements o f t h e same p e r s o n . I n t h i s s e n s e , t h i s a s s u m p t i o n p l a y s t h e same r o l e as a s s u m p t i o n o f l i n e a r e x p e r i m e n t a l independence
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i n classical t e s t theory. The a s s u m p t i o n a b o u t independence of o b s e r v a t i o n s i n
a group o f p e r s o n s w i t h t h e same v a l u e s of l a t e n t t r a i t s s e r v e s as a b a s i s f o r i d e n t i f i c a t i o n of s u c h g r o u p s . To d e t e r m i n e t h e l a t e n t s t r u c t u r e of a p o p u l a t i o n , i t s u f f i c e s t o p a r t i t i o n t h i s p o p u l a t i o n i n t o groups, such t h a t measurements a r e i n d e p e n d e n t w i t h i n g r o u p s . These groups , by d e f i n i t i o n , a r e g r o u p s w i t h i d e n t i c a l v a l u e s of l a t e n t t r a i t s . I n o t h e r words, t h e p o i n t i s t o p a r t i t i o n t h e p o p u l a t i o n i n t o g r o u p s , s a y G1, G 2 , with t h e following property. L e t Xs, Y s , be t h e random v a r i a b l e s d e s c r i b i n g t h e measurements of p e r s o n s w i t h v a r i o u s t o o l s ( e . g . t h e s e random v a r i a b l e s may be r e s p o n s e s t o v a r i o u s i t e m s ) . Then, g i v e n s c G t h e random v a r i a b l e s X s , Y s , . . . are i’ independent, s o t h a t
...
...
Consequently, g i v e n t h e i n f o r m a t i o n t h a t s i s i n Gi ’ t h e knowledge of h i s r e s u l t s i n some t e s t s d o e s n o t g i v e any i n f o r m a t i o n a b o u t h i s v a l u e s on o t h e r t e s t s . I n s t i l l o t h e r words, t h e u n d e r l y i n g i n t u i t i v e i n t e r p r e t a t i o n i s t h a t a l l p s y c h o l o g i c a l i n f e r e n c e a b o u t t h e subj e c t i s t o b e based only on t h e i n f o r m a t i o n about h i s group membership; once t h i s i s known, f u r t h e r i n f o r m a t i o n a b o u t t h e v a l u e s of v a r i o u s measurements i s psychologically irrelevant. N a t u r a l l y , when a p a r t i t i o n w i t h t h i s p r o p e r t y i s obt a i n e d , t h e r e i s s t i l l a t a s k of i t s p s y c h o l o g i c a l i n terpretation.
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D e s p i t e g r e a t g e n e r a l i t y and c o n c e p t u a l s i m p l i c i t y , t h e theory of l a t e n t t r a i t s encounters i n p r a c t i c a l applica t i o n s c o n s i d e r a b l e t h e o r e t i c a l and t e c h n i c a l d i f f i c u l t i e s . A s a t i s f a c t o r y development i s r e s t r i c t e d t o oned i m e n s i o n a l t r a i t s , which means ‘chat p s y c h o l o g i c a l a p p l i c a b i l i t y r e d u c e s m o s t l y t o a b i l i t i e s and a t t i t u d e s , which o f t e n may be t a k e n as one-dimensional. However, even i n t h e one-dimensional c a s e t h e l a t e n t t r a i t t h e o r y enounters c e r t a i n d i f f i c u l t i e s . To e x p l a i n t h e s i t u a t i o n , l e t u s o u t l i n e t h e s i m p l e s t c a s e . Imagine t h a t we have r b i n a r y i t e m s , which may be answered c o r r e c t l y o r i n c o r r e c t l y . L e t u be t h e l a t e n t t r a i t . I f X . i s t h e random v a r i a b l e which assumes t h e v a l u e 1 1
i t e m i s c o r r e c t and v a l u e 0 i f t h e answer i s i n c o r r e c t , t h e n t h e f u n c t i o n i f t h e answer t o i - t h
P(Xi
=
llu) =
fib)
i s c a l l e d t h e item c h a r a c t e r i s t i c curve ( I C C ) of i - t h i t e m . Given t h e assumption of l o c a l c o n d i t i o n a l i n d e pendence, t h e p r o b a b i l i t y of r e s p o n s e = (xl, ...,x r ) on a l l r items b y a p e r s o n w i t h v a l u e u o f t h e t r a i t i s r p ( x l u ) = l-7 P ( X i = X i J U ) i=l
x
and e a c h t e r m o f t h i s p r o d u c t i s e i t h e r f i ( u ) o r l - f i ( u ) . T o e n a b l e t h e s t a t i s t i c a l , and hence a l s o p s y c h o l o g i c a l ,
i n f e r e n c e about t h e s u b j e c t , one i n t r o d u c e s a p a r a m e t r i c form o f f i ( u ) . The b e s t known model h e r e i s t h a t o f l o g i s t i c i t e m c h a r a c t e r i s t i c c u r v e (Rasch 1960; s e e a l s o Birnbaum 1 9 6 8 ) . I n t h i s model i t i s assumed t h a t
CONTEMPORARY THEORY OF PSYCHOLOGICAL TESTS
utk Ce
fib) = 1
+
107
i
u+ki Ce
where ki i s a c o n s t a n t connected w i t h i - t h i t e m , C i s a s c a l i n g c o n s t a n t , and u i s a one-dimensional t r a i t r e p r e s e n t i n g a b i l i t y ( s a y ) . When u i n c r e a s e s , t h e prob a b i l i t y f i ( u ) o f c o r r e c t answer t e n d s t o 1, w h i l e w i t h a decrease of u, t h i s probability decreases t o zero. Also, f o r a g i v e n u, t h e p r o b a b i l i t y o f c o r r e c t r e s p o n s e i s l a r g e r f o r l a r g e r k i , s o t h a t - k i i s a measure o f i t e m difficulty. The g e n e r a l problem i s now t o e s t i m a t e t h e v a l u e s ul, * ,un f o r n i n d i v i d u a l s and v a l u e s k l , . . . , k r for r i t e m s , g i v e n t h e d a t a on r e s p o n s e s o f t h e s e i n d i v i d u a l s . F o r t h e e s t i m a t i o n p r o c e d u r e s , goodness o f f i t o f t h e model, e t c . s e e f o r i n s t a n c e Anderson 1973, Cohen 1979, G o l d s t e i n 1979, G u s t a f s s o n 1980 or Sanathanan and Blument ha1 197 8 .
..
The d i f f i c u l t i e s connected w i t h Rasch model, and a l s o w i t h some o t h e r l a t e n t t r a i t models, were poJnted o u t by G o l d s t e i n ( 1 9 8 0 ) . These d i f f i c u l t i e s a r e r e l a t e d t o t h e f a c t t h a t t h e model p r e s u p p o s e s t h a t u i s measured on some s c a l e w i t h known p r o p e r t i e s , w h i l e i n f a c t t h e a s s u m p t i o n s about t h e s c a l e a r e n o t t e s t a b l e . The p r a c t i c a l consequences of t h i s i s t h a t t h e a b i l i t i e s are d i s c r i m i n a t e d d i f f e r e n t l y on t h e ends of t h e s c a l e t h a n i n t h e m i d d l e , which l e a d s t o some s c a l e v a l u e s a p p e a r i n g c o n t r a r y t o i n t u i tion. F o r a t h o r o u g h r e v i e w o f r e c e n t development o f l a t e n t
108
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t r a i t t h e o r y s e e Hambleton e t a l . 1978. I n a d d i t i o n , one s h o u l d mention a l s o t h e newest r e s u l t s o f Holland
(1981), who i n t r o d u c e s n o t i o n s somewhat weaker t h a n t h a t of l o c a l c o n d i t i o n a l i n d e p e n d e n c e , and p r o v i d e s i n t h i s way c e r t a i n means o f t e s t i n g e m p i r i c a l l y for t h i s i n d e pendence. Comments. The p s y c h o l o g i c a l t h e o r y o f t r a i t s , b e i n g t h e
f o u n d a t i o n o f t e s t t h e o r y , only i n d i r e c t l y takes i n t o a c c o u n t s i t u a t i o n a l f a c t o r s . I n most r e c e n t development o f t e s t theory ( i n p a r t i c u l a r g e n e r a l i z a b i l i t y ) , t h e r e s e a r c h goes t o w a r d s r e f i n e d r e s p o n s e a n a l y s i s , s o as t o be a b l e t o d i s t i n g u i s h t h e f a c t o r s c o n n e c t e d w i t h t h e s e t of t r a i t s , as w e l l as t h e s e t o f s i t u a t i o n a l c h a r a c t e r i s t i c s . The s i t u a t i o n , however, i s u n d e r s t o o d i n a r a t h e r narrow s e n s e , and w i t h o u t a w i d e r t h e o r y ( u s i n g mainly c o r r e l a t i o n a l m e t h o d s ) . To deepen t h e p s y c h o l o g i c a l p i c t u r e s u p p l i e d by t e s t s , i t would b e f r u i t f u l t o t a k e i n t o account t h e o r y o f s i t u a t i o n s , u n d e r s t o o d as i t s s t r u c t u r e and knowledge a b o u t i t . It seems t h a t t e s t t h e o r y would p r o f i t from r e - i n t e r p r e t a t i o n o f i t s b a s i c n o t i o n s , s u c h as t r a i t . P e r h a p s some r e f r e s h i n g o f t e s t t h e o r y , and more u s e f u l n e s s i n p r e s e n t l y d e v e l o p i n g o t h e r domains ( s u c h as e . g . c o g n i t i v e s c i e n c e s , a r t i f i c i a l i n t e l l i g e n c e ) , c o u l d be o b t a i n e d by some change o f i n t e r p r e t a t i o n of t h e c o n c e p t o f t r a i t , t r e a t i n g i t as a c e r t a i n c o n f i g u r a t i o n of d i s p o s i t i o n s , g e n e t i c a l l y d e t e r m i n e d and e n v i r o n m e n t a l l y m o d i f i e d , t o (1) some t y p e s o f p e r c e p t i o n o f s i t u a t i o n and s e n s i t i v i t y t o some e l e m e n t s o f t h e i r s t r u c t u r e , ( 2 ) a b i l i t y t o a c q u i r i n g some t y p e s o f knowledge s t r u c t u r e and ( 3 ) c h o i c e of a c t i o n s imposed by g i v e n t y p e of p e r c e p t i o n and knowledge. I n o t h e r words, more s t r e s s would b e p u t on knowledge s t r u c t u r e i n some c l a s s e s o f s i t u a t -
CONTEMPORARY THEORY OFPSYCHOLOGICAL TESTS
109
i o n s , which would i n v o l v e g r e a t e r c o n t r i b u t i o n o f l i n g u i s t i c s and t e x t t h e o r y on t h e one hand, and c o g n i t i v e p r o c e s s e s ( i n p a r t i c u l a r p e r c e p t i o n models) on t h e o t h e r . Subsequent c h a p t e r s w i l l show some p o s s i b l e e x t e n s i o n s o f t h e s e ideas.
110
CHAPTER 1
BIBLIOGRAPHY TO CHAPTER 1
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Approximate E x p r e s s i o n s for Parameter E s t i m a t e s i n t h e R a s c h Model. B r i t i s h J o u r n a l of Mathe-
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B i a s i n S e l e c t i o n . J o u r n a l of E d u c a t i o n a l Meas u r e m e n t . 1 0 ; 237-255.
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An I n t r o d u c t i o n t o P r o b a b i l i t y Theory a n d I t s A p p l i c a t i o n s . V o l . 2 . N e w York. W i l e y .
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C o n s e q u e n c e s o f U s i n g t h e R a s c h Model for Edu-
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CONTEMPORARY THEORY OF PSYCHOLOGICAL TESTS
c a t i o n a l A s s e s s m e n t . B r i t i s h E d u c a t i o n a l Resea r c h J o u r n a l . 5; 211-220. 1980
D i m e n s i o n a l i t y , B i a s , I n d e p e n d e n c e and Measurement S c a l e P r o b l e m s i n L a t e n t T r a i t T e s t S c o r e Models. B r i t i s h J o u r n a l of M a t h e m a t i c a l a n d S t a -
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Developments i n L a t e n t T r a i t T h e o r y : M o d e l s , T e c h n i c a l I s s u e s a n d A p p l i c a t i o n s . Review of E d u c a t i o n a l R e s e a r c h . 48;
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B i a s " . P s y c h o l o g i c a l B u l l e t i n . 83; 1053-1071. JENSEN, A . R . 1980
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1976 Some Developments i n M u l t i v a r i a t e G e n e r a l i z a b i l i t y . Psychometrika. RICHARDSON, M . W .
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L a t e n t S t r u c t u r e A n a l y s i s and T e s t T h e o r y . I n H : G u l l i k s e n , S . Messick ( e d s . ) P s y c h o l o g i c a l S c a l i n g . Theory and A p p l i c a t i o n s .
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CHAPTER I
SCHEFFE, H.
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S a m p l i n g Errors of V a r i a n c e Components i n S m a l l Sample M u l t i f a c e t G e n e r a l i z a b i l i t y S t u d i e s . J o u r n a l of E d u c a t i o n a l S t a t i s t i c s . 3 ; 3 1 9 - 3 4 6 .
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a l Testing Service. C o n c e p t s of C u l t u r e F a i r n e s s . J o u r n a l o f Educ a t i o n a l M e a s u r e m e n t . 8; 63-70.
115
CHAPTER 2 FACTOR A N A L Y S I S : ARBITRARY D E C I S I O N S W I T H I N M A T H E M A T I C A L MODEL
The aim of t h e p r e s e n t c h a p t e r i s a d i s c u s s i o n o f t h o se a s p e c t s of f a c t o r a n a l y s i s which may l e a d t o method o l o g i c a l r e s e r v a t i o n s . The l a t t e r a r e c o n n e c t e d , f i r s t l y , w i t h making m u l t i p l e a r b i t r a r y d e c i s i o n s i n apply-
ing factor chological cessity of t e r of t h e
a n a l y s i s , and s e c o n d l y ( e s p e c i a l l y i n p s y and s o c i o l o g i c a l a p p l i c a t i o n s ) w i t h t h e nemaking s t r o n g a s s u m p t i o n s about t h e c h a r a c t s c a l e s on which one measures t h e u n d e r l y i n g
variables. S e c t i o n 1 d e a l s w i t h a p r e s e n t a t i o n and d i s c u s s i o n of t h e model o f f a c t o r a n a l y s i s and o f t h e a l t e r n a t i v e model o f m u l t i d i m e n s i o n a l s c a l i n g . Section 2 presents a discussion of an a p p l i c a t i o n of f a c t o r a n a l y s i s t o t h e c o n s t r u c t i o n of C a t t e l l ' s persona l i t y theory.
1. SOME M U L T I V A R I A T E T E C H N I Q U E S I N PSYCHOLOGY
1.1. F a c t o r a n a l y s i s : t h e m a t h e m a t i c a l model
L e t us c o n s i d e r t h e s i t u a t i o n i n which for e a c h o f t h e
116
CHAPTER 2
e l e m e n t s under s t u d y ( i t i s assumed t h a t t h e s e e l e m e n t s form a sample from some p o p u l a t i o n ) one makes a c e r t a i n number n of measurements. I n t y p i c a l p s y c h o l o g i c a l a p p l i c a t i o n s t h e e l e m e n t s o f t h e sample a r e s u b j e c t s , s e l e c t e d a t random from some p o p u l a t i o n ; t h e r e s u l t s o f measurement would t y p i c a l l y be s c o r e s on some t e s t s , coded answers t o v a r i o u s q u e s t i o n s , e v a l u a t i o n s of beh a v i o u r o f s u b j e c t s , a s g i v e n by j u d g e s ( f r o m o u t s i d e t h e s a m p l e ) , o r even c e r t a i n a n t h r o p o l o g i c a l measurements, age of t h e s u b j e c t s , e t c . I n a p p l i c a t i o n s from o u t s i d e p s y c h o l o g y , t h e o b j e c t s s t u d i e d may be samples o f s o i l a n a l y s e d from t h e p o i n t o f view o f a number of c h a r a c t e r i s t i c s , m u n i c i p a l counc i l s a n a l y s e d from t h e p o i n t of view of v a r i o u s c r i t e r i a , and s o f o r t h .
G e n e r a l l y , l e t x j s ) d e n o t e t h e r e s u l t o f measurement o f i - t h v a r i a b l e f o r t h e s - t h element o f t h e sample; h e r e t h e lower i n d e x r e f e r s t o t h e measurement o f t h e c h a r a c t e r i s t i c under s t u d y , and t h i s i n d e x may assume as many v a l u e s as t h e r e a r e c h a r a c t e r i s t i c s . The upper i n d e x r e f e r s t o t h e element i n t h e sample, and may assume as many v a l u e s as t h e r e a r e e l e m e n t s . Thus, t h e raw d a t a may be r e p r e s e n t e d i n form o f a m a t r i x
.................
(1.1)
where n i s t h e number of c h a r a c t e r i s t i c measured, and
FACTOR ANALYSIS
117
k i s t h e s i z e o f t h e sample. Thus, e v e r y row c o r r e s p o n d s
t o a s e t of measurements o f a n element o f t h e sample, and e a c h column c o r r e s p o n d s t o measurements o f one char a c t e r i s t i c i n t h e sample. The a s s u m p t i o n t h a t t h e sample r e p r e s e n t s t h e p o p u l a t i o n under s t u d y may b e e x p r e s s e d as s a y i n g t h a t e a c h row of t a b l e (1.1) i s a r e a l i z a t i o n o f a c e r t a i n n-dim e n s i o n a l random v a r i a b l e
x'
= (xl,
...,
Xn).
Here d e n o t e s t r a n s p o s i t i o n , i . e . change o f rows t o columns. I n t h i s c h a p t e r , v e c t o r s ( d e n o t e d b y a n underl i n e d l e t t e r , w i t h o u t t h e s i g n * ) w i l l be r e g a r d e d as columns. For g r a p h i c a l c o n v e n i e n c e , t h e y a r e w r i t t e n as t r a n s p o s i t i o n s , i . e . row v e c t o r s . The a i m o f f a c t o r a n a l y s i s i s t o decompose t h e random
v a r i a b l e 2 i n t o some o t h e r random v a r i a b l e s . It i s assumed t h a t 5 has f i n i t e moments o f t h e second o r d e r ( h e n c e v a r i a n c e s and c o v a r i a n c e s e x i s t ) . L e t
...,
m n ) . Thus, mi i s t h e e x p e c t e d s c o r e and m* = ( m l , on i - t h measurement i n t h e p o p u l a t i o n under s t u d y . Next (1.3) Thus, x;
5
i s t h e c o v a r i a n c e m a t r i x of t h e random v e c t o r
on t h e i n t e r s e c t i o n o f i - t h row and j - t h column of - m i ) ( x j - m . ) = cov ( x i , x j ) .
t h i s m a t r i x we have E ( x i
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In particular, on the diagonal we have the elements of the form E(xi - mi) c , hence variances of the components of the vector 5 . r
Both the vector m and the covariance matrix may be estimated from the sample (1.1). In particular, the estimator of m i is
while the estimator of the covariance Cov (xi,x.) is J
-1
k
(xit) - Ci)(x (t)
t=l
j
-
m.). J
(1.5)
In particular,
the estimator of the variance of the random variable x is i
-1 E. (Xi(t)
- * mi) 2 .
(1.6)
t=l
h
h
In the sequel, g and will denote the estimates of the expectation fi and covariance matrix El. The model of factor analysis may be formulated as follows: for the vector 2 of the observed random variables we l o o k for a representation in the form x =
g + Af
+
g,
(1.7)
where f ' = (fl,..., f ) is a certain r-dimensional ranr dom variable, whose components are called common factors, & is a matrix of dimension n x r (with n rows and r columns), called the matrix of factor loadings, and = (ul, un ) is an n-dimensional random variable, whose components are called specific factors.
...,
119
FACTOR ANALYSIS
The a s s u m p t i o n s c o n c e r n i n g t h e r e p r e s e n t a t i o n ( 1 . 7 ) a r e as f o l l o w s :
which means t h a t t h e number of common f a c t o r s i s l e s s t h a n t h e number o f v a r i a b l e s ; (b)
E(2lf)
=
9
f o r a l l v a l u e s of
1,
t h i s assumption i m p l i e s t h a t t h e r e g r e s s i o n of 5 on
t h e f a c t o r f i s l i n e a r . To prove i t , i t s u f f i c e s t o w r i t e ( 1 . 7 ) i n t h e form
and t a k i n g e x p e c t a t i o n s c o n d i t i o n a l on f one o b t a i n s
or
From a s s u m p t i o n ( b ) i t f o l l o w s a l s o t h a t E ( 2 ) = 0, s i n - = E [ E ( u i f ) ] = E ( 2 ) = 2, and t h a t t h e random v a r ce E(u) i a b l e s ui ( i = l,.. ,n) and f. (j = l,.. . , r ) are unJ correlated (since E(u.f.) = E[E(u.f.lf)l = E[fjE((uilf)l
.
1 J
= 0).
The n e x t assumption has t h e form
1 3
120
CHAPTER 2
D
(c)
=
F(=')
i s a diagonal m a t r i x .
T h i s assumption means t h a t t h e s p e c i f i c f a c t o r s u
are m u t u a l l y u n c o r r e l a t e d ( t h e y a r e a l s o , as shown, not c o r r e l a t e d w i t h common f a c t o r s ) . j
We assume a l s o t h a t ; (d)
H = E (ff?)
d
i s not s i n g u l a r .
T h i s a s s u m p t i o n means t h a t common f a c t o r s fl,...,f
1"
are
l i n e a r l y i n d e p e n d e n t , and c o n s e q u e n t l y , t h e r e e x i s t s no r e p r e s e n t a t i o n ( 1 . 7 ) w i t h a smaller number of f a c tors. F i n a l l y , t h e l a s t assumption a s s e r t s t h a t (e)
h2 = 1 f o r e a c h j, where h2 i s t h e d i a g o n a l jj jj
element of m a t r i x
3.
T h i s a s s u m p t i o n , l e a s t i m p o r t a n t among t h e a s s u m p t i o n s ,
s t a t e s t h a t v a r i a n c e s of common f a c t o r s are 1, i . e . t h a t common f a c t o r s are s t a n d a r d i z e d . T h i s may a l w a y s be a c h i e v e d by a change of f a c t o r l o a d i n g s i n m a t r i x
A.
From t h e above a s s u m p t i o n s one may d e r i v e t h e f o l l o w i n g important r e s u l t concerning t h e decomposition of t h e of vector 5 : covariance matrix
( s e e Lord and Novick 1968, p . 5 3 3 ) . I n d e e d , from t h e d e f i n i t i o n we have 5 = E ( q - E ) ? , hence i n view of x - m = Af + u ( s e e f o r m u l a ( 1 . 7 ) ) one g e t s
m)(x
FACTOR ANALYSIS
= E[A ff'A' = AE(E')A'
t
Afu
t Uf'A'
-
t E (u u ' ) = AHA'
121
&'I t D.
I n p a r t i c u l a r , i f t h e common f a c t o r s a r e u n c o r r e l a t e d , then = 2, where 2 i s t h e i d e n t i t y m a t r i x of s i z e ~ x r , hence
R
=
t
D.
(1.10)
Such a d e c o m p o s i t i o n o f m a t r i x zation.
i s called its factori-
1.2. Discussion
1.2.1.
Arbitrary decisions i n applications of factor
a n a l y s i s . To sum up t h e c o n t e n t o f t h e p r e c e d i n g s e c t i o n , one may s a y t h a t t h e model of f a c t o r a n a l y s i s cons i s t s of e x p l a i n i n g t h e whole r e l a t i o n s h i p between t h e o b s e r v e d v a r i a b l e s 5 by t h e a p p e a r e n c e of common f a c t ors: a f t e r s u b t r a c t i n g t h e a p p r o p r i a t e weighted a v e r a g e s o f t h e s e f a c t o r s (and s u b t r a c t i o n o f t h e common - m - Af which are mean) t h e r e remain r e s i d u a l s 1 = mutually uncorrelated.
x
The b a s i c theorems ( f o r m u l a s ( 1 . 9 ) and ( 1 . 1 0 ) ) a s s e r t
t h a t under a s s u m p t i o n s ( a )
rix
-
( e ) , the covariance m a t may b e decomposed i n t o t h e m a t r i x of c o v a r i a n c e s
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H , m a t r i x of f a c t o r l o a d i n g s A , and t h e d i a g o n a l m a t -
r i x D of v a r i a n c e s of s p e c i f i c f a c t o r s . The f o l l o w i n g q u e s t i o n s , b a s i c f o r t h e model o f f a c t o r a n a l y s i s , may be a s k e d . 1) Does a d e c o m p o s i t i o n g i v e n by ( 1 . 9 ) a l w a y s e x i s t ?
I n o t h e r words, can e v e r y c o v a r i a n c e m a t r i x be decomposed? The answer t o t h i s q u e s t i o n i s p o s i t i v e : f o r e v e r y covariance matrix that the matrix
E
t h e r e e x i s t m a t r i c e s g and A s u c h - A H A ' i s d i a g o n a l . What i s more,
one may prove ( s e e Lord and Novick 1968, p . 5 3 3 ) t h a t t h e rank of matrix i s u n i q u e l y d e t e r m i n e d . Thus, t h e covariance matrix common f a c t o r s .
d e t e r m i n e s u n i q u e l y t h e number o f
T h i s f a c t h a s , however, o n l y a l i m i t e d p r a c t i c a l con-
sequence, s i n c e t h e covariance m a t r i x known; one knows o n l y i t s e s t i m a t e
n
i s u s u a l l y un-
E.
The f a c t t h a t i n a p p l i c a t i o n s o f f a c t o r a n a l y s i s one d e a l s w i t h t h e e s t i m a t e s o f c o v a r i a n c e m a t r i x has a l s o more s e r i o u s consequences t h a n i n c o m p l e t e n e s s o f knowl e d g e of p a r a m e t e r s i n seemingly s i m i l a r s t a t i s t i c a l problems. The p o i n t i s t h a t i n o r d e r t o a p p l y t h e s t a n d a r d e s t i m a t i o n t e c h n i q u e s , s u c h a s maximum l i k e l i h o o d ,
i t i s n e c e s s a r y t o have t h e c o r r e s p o n d i n g e q u a t i o n s , i n which t h e p a r a m e t e r s a p p e a r as unknowns. I n c a s e o f f a c t o r a n a l y s i s , w r i t i n g such equations i s impossible ( u n l e s s o f assumes a p r i o r i t h e number o f f a c t o r s , which i s an unknown). There i s no need t o p r e s e n t h e r e t h e t e c h n i c a l l y com-
123
FACTOR ANALYSIS
p l i c a t e d f o r m u l a s . G e n e r a l l y , t h e s o l u t i o n s c o n s i s t of a p p l y i n g t h e known t e c h n i q u e s o f f a c t o r e x t r a c t i o n u n t i l the matrix (1.11) w i l l be " s u f f i c i e n t l y c l o s e " t o a d i a g o n a l m a t r i x , i n
t h e s e n s e o f a c r i t e r i o n , e x p r e s s e d t h r o u g h a n appropr i a t e f u n c t i o n o f non-diagonal e l e m e n t s of (1.11). One may a l s o a p p l y t h e s c r e e t e s t s u g g e s t e d by C a t t e l l for t h e number of f a c t o r s ( s e e C a t t e l l 1 9 6 6 a ) : f o r t e s t i n g t h e number o f f a c t o r s i n t h e m a t r i x 5 C a t t e l l s u g g e s t s u s i n g t h e i n t u i t i v e e x p e c t a t i o n t h a t i n s u c c e s s i v e matrices - =' for i n c r e a s i n g number of f a c t o r s , t h e r e s h o u l d be a r a p i d i n c r e a s e of t h e q u a l i t y of a p p r o x i m a t i o n t o a d i a g o n a l m a t r i x when t h e t r u e number of f a c t o r s i s exceeded. For o t h e r t e c h n i q u e s o f d e t e r m i n i n g t h e number of f a c t o r s , s e e e . g . C a t t e l l 1958 o r Henrysson 1957. I t s h o u l d , however, be mentioned t h a t t h e s c r e e t e s t was c r i t i c i z e d , e . g . by G u i l f o r d ( s e e G u i l f o r d and Hoepfner 1971 ) .
Another methods a r e based on assuming a g i v e n number of f a c t o r s ( s a y , t w o ) . Then one may d e v i s e t e c h n i q u e s , based e . g . on maximum l i k e l i h o o d p r i n c i p l e , t o e s t i m a t e t h e m a t r i x o f f a c t o r l o a d i n g s ( s e e e . g . Harman 1976) I t seems t h e r e f o r e t h a t t h e d e c i s i o n c o n c e r n i n g t h e number o f f a c t o r s i s t h e f i r s t i n a s e r i e s o f d e c i s i o n s
which a r e made i n a more or l e s s a r b i t r a r y way, hence d e c i s i o n s which b i a s t h e r e s u l t s o f f a c t o r a n a l y s i s by s u b j e c t i v i t y . The n e x t s u c h d e c i s i o n s concern r o t a t i o n
-
124
CHAPTER 2
and i n t e r p r e t a t i o n . I n d e e d , one has t o d e c i d e e i t h e r t o
use a program which e x t r a c t s a f i x e d number of f a c t o r s , o r a program which has a " b u i l t i n " s t o p p i n g r u l e , which t e r m i n a t e s t h e e x t r a c t i o n when a n a p p r o p r i a t e c r i t e r i o n i s m e t . N a t u r a l l y , a f t e r s u c h a program i s s e l e c t e d , t h e r e s u l t s of f a c t o r a n a l y s i s are o b j e c t i v e , i n t h e s e n s e t h a t t h e y a r e d e t e r m i n e d by t h e d a t a . The p o i n t i s , however, t h a t t h e c h o i c e o f t h e program i s l e f t t o the experimenter's discretion. The n e x t q u e s t i o n connected w i t h t h e model of f a c t o r analysis i s t h e following: 2 ) I s t h e f a c t o r i z a t i o n o f t h e c o v a r i a n c e m a t r i x uni-
quely determined? It t u r n s o u t t h a t h e r e t h e answer i s n e g a t i v e : t h e r e a l w a y s e x i s t i n f i n i t e l y many f a c t o r i z a t i o n s o f m a t r i x R , b o t h o f t h e form (1.9), as w e l l as of t h e form ( 1 . 1 0 ) . The c h o i c e o f one p a r t i c u l a r d e c o m p o s i t i o n i s c a l l e d t h e problem of r o t a t i o n ( o f t h e e x t r a c t e d s t r u c t u r e ) . To show t h a t f a c t o r i z a t i o n i s n e v e r u n i q u e , s u p p o s e that = m t Af t g, s o t h a t R = AHA' t D i s a f a c t o r i n t o t h e m a t r i x of i z a t i o n of t h e c o v a r i a n c e m a t r i x covariances between common f a c t o r s , m a t r i x 4 o f f a c t o r l o a d i n g s , and m a t r i x g o f v a r i a n c e s of s p e c i f i c f a c t o r s . Suppose t h a t 5 i s a m a t r i x of dimension r x r , and l e t 2 d e n o t e a n a r b i t r a r y n o n - s i n g u l a r m a t r i x of t h e s i z e r x r . L e t us d e f i n e t h e new v e c t o r of common -1 f a c t o r s g b y t h e f o r m u l a g = Tf, i . e . t = T. g. Then x = m t g - ' g t g, and t h e c o v a r i a n c e m a t r i x I3 w i l l be * -1 * R = -AT _ H- ( A T - l ) ' t ll, where = E(gg') = E ( T f ) ( T f ) ' - 1- w e = E(T ff " T ' ) = EE(ff')T' = E'.D e n o t i n g = AT
x
125
FACTOR ANALYSIS
obtain another factorization
E
*
= BH B' t
g.
I n a s i m i l a r way one may show t h a t f a c t o r i z a t i o n i s n o t u n i q u e , even i f one r e s t r i c t s t h e c o n s i d e r a t i o n t o o r t h o gonal s t r u c t u r e s only. i . e . t o f a c t o r i z a t i o n s of t h e form ( 1 . 1 0 ) . I n d e e d , i f 5 = g t Af + 2 where E (ff') = I , i . e . E = g't g, t h e h one may t a k e g = T f , where T i s an a r b i t r a r y orthogonal matrix, i . e . such t h a t -1 * = T - I . Then 2 = m t 2 g t 2, as b e f o r e , and g = E(=') = E E ( f f * ) T ' = TIT' = I . Consequently, = BB' + D, with = AT-l-is a f a c t o r i z a t i o n of m a t r i x g o f t h e form ( 1 . 1 0 ) .
r'
I n p r a c t i c e , t h e problem o f r o t a t i o n i s u s u a l l y s o l v e d i n s e v e r a l c o n s e c u t i v e s t e p s , e a c h of them r e q u i r i n g an arbitrary choice. Naturally, i n these choices t h e e x p e r i m e n t e r i s guided by h i s knowledge, i n t u i t i o n , p r i o r a s s u m p t i o n s about t h e phenomenon s t u d i e d , common s e n s e , and f i n a l l y - tendency t o parsimony of t h e obtained structure. Thus, t h e f i r s t d e c i s i o n c o n c e r n s t h e c h o i c e between a n o r t h o g o n a l s t r u c t u r e , and one o u t o f i n f i n i t e l y many o b l i q u e s t r u c t u r e s ( f o r m a l l y s p e a k i n g , t h i s c h o i c e does not concern r o t a t i o n , s i n c e t h e e x t r a c t e d s t r u c t u r e , o r t h o g o n a l or n o t , must s t i l l be r o t a t e d ) . Here t h e r e s e a r c h e r must t a k e i n t o a c c o u n t t h e f a c t t h a t , on t h e one hand, o r t h o g o n a l s t r u c t u r e s are e a s i e r t o h a n d l e by s t a t i s t i c a l t e c h n i q u e s , and a r e i n some sense more p a r s i m o n i o u s t h a n o b l i q u e s t r u c t u r e s . On t h e o t h e r hand, t h e r e e x i s t s i t u a t i o n s i n which t h e phenomenon a n a l y s e d i s a l r e a d y d e s c r i b e d , a t l e a s t p a r t i a l ly, i n terms of some f a c t o r s w i t h f i x e d and w e l l e s tablished t r a d i t i o n . If these factors are correlated,
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CHAPTER 2
i t i s u s u a l l y b e t t e r t o use a n o b l i q u e s t r u c t u r e , w i t h correlated factors. An i l l u m i n a t i n g example h e r e i s g i v e n by T h u r s t o n e . It i s narhely known t h a t weight and h e i g h t a r e c o r r e l a t e d . One can c e r t a i n l y f i n d two l i n e a r c o m b i n a t i o n s o f we--i g h t and h e i g h t which would b e u n c o r r e l a t e d , b u t i t d o e s n o t seem r e a s o n a b l e t o u s e s u c h l i n e a r c o m b i n a t i o n s i n practice. N e x t , as r e g a r d s a r o t a t i o n of t h e e x t r a c t e d s t r u c t u r e , a l l known t e c h n i q u e s seem t o be b a s e d on t h e p r i n c i p l e o f t h e s o - c a l l e d s i m p l e s t r u c t u r e , s u g g e s t e d by Thurstone (1947). T h i s p r i n c i p l e i s based on a s u p p o s i t i o n t h a t i n t h e
phenomenon under s t u d y e a c h of t h e v a r i a b l e s i s l e s s complex f a c t o r i a l l y t h a n t h e whole phenomenon; if s o , one should be a b l e t o o b t a i n a m a t r i x o f f a c t o r l o a d i n g s s u c h t h a t t h e v a r i a b l e s would c o r r e l a t e h i g h l y w i t h few f a c t o r s and have low c o r r e l a t i o n s w i t h t h e r e m a i n i n g f a c t o r s . I n s u c h a c a s e , i n e a c h row o f t h e m a t r i x A of f a c t o r l o a d i n g s t h e r e s h o u l d be p o s s i b l y many z e r o s . Moreover, t o f a c i l i t a t e i n t e r p r e t a t i o n , one imposes t h e a d d i t i o n a l c o n d i t i o n , t h a t p o s s i b l y many z e r o s s h o u l d a l s o a p p e a r i n e a c h column ( s o t h a t e a c h f a c t o r would be r e p r e s e n t e d by few v a r i a b l e s o n l y ) ; f o r e a c h two columns t h e r e s h o u l d e x i s t v a r i a b l e s which have h i g h c o r r e l a t i o n s i n one column and low c o r r e l a t i o n s i n a n o t h e r ( s o as t o e n s u r e o r t h o g o n a l i t y o f f a c t o r s ) ; moreover, t h e r e s h o u l d b e many v a r i a b l e s w i t h z e r o ( l o w ) c o r r e l a t i o n s i n b o t h columns, and only few w i t h h i g h c o r r e l a t i o n s i n b o t h columns. These q u a l i t a t i v e conditions characterize simple s t r u c t u r e s .
FACTOR ANALYSIS
121
It i s w o r t h w h i l e t o p o i n t o u t h e r e t h a t t h e e x i s t e n -
ce of a s i m p l e s t r u c t u r e i s a c e r t a i n h y p o t h e s i s ; i t may p e r h a p s be r e a s o n a b l y e x p e c t e d t o h o l d i n most cases, but it i s not automatically s a t i s f i e d . T h i s p r i n c i p l e i s v e r y g e n e r a l , and i n p a r t i c u l a r i n -
s t a n c e s i t must b e a p p r o p r i a t e l y s p e c i f i e d , s i n c e t h e r e may e x i s t many r o t a t i o n s which l e a d t o m a t r i x o f f a c t or l o a d i n g s s a t i s f y i n g t h e r e q u i r e m e n t s of s i m p l e s t r u c t u r e . Thus, e a c h program f o r a computer which e x t r a c t s a s i m p l e s t r u c t u r e i s based on a c e r t a i n c r i t e r i o n , which r e q u i r e s m a x i m i z a t i o n of m i n i m i z a t i o n o f a c e r t a i n f u n c t i o n ( e . g . programs V a r i m a x , Q u a r t i m a x , Q u a r t imin, e t c ) . There e x i s t s a r a t h e r wide s p r e a d c o n v i c t i o n t h a t t h e results of factor analysis are objective (at least with r e s p e c t t o t h e r o t a t i o n ) , s i n c e t h e y a r e o b t a i n e d as a r e s u l t o f a c e r t a i n o b j e c t i v e p r o c e d u r e . It i s c l e a r , t h a t when t h e program i s s e l e c t e d , t h e r e s u l t s a r e dat a - d e t e r m i n e d ; t h e p o i n t i s , however, t h a t t h e c h o i c e o f t h e program i s t o a l a r g e e x t e n t s u b j e c t i v e : one c h o o s e s , as a r u l e , s u c h a program which l e a d s t o a s t r u c t u r e which i s e a s i e s t t o i n t e r p r e t .
The problems o f i n v a r i a n c e . I n o r d e r t o be a b l e t o a s s i g n meaningful i n t e r p r e t a t i o n t o f a c t o r s , t h e l a t t e r cannot be a c c i d e n t a l , i . e . s p e c i f i c f o r t h e g i v e n c o v a r i a n c e m a t r i x . The same f a c t o r s s h o u l d have a r e a s o n a b l e chance o f a p p e a r i n g i n o t h e r s e t s o f d a t a . T h i s r e q u i r e m e n t , o f t h e e x t r a c t e d and r o t a t e d c o n f i g u r a t i o n t o be r e p e t i t i v e , i s u s u a l l y c a l l e d t h e r e q u i r e 1.2.2.
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ment of i n v a r i a n c e . I t would p r e s e n t l i t t l e s t a t i s t i c a l d i f f i c u l t i e s i f i t were r e s t r i c t e d t o r e p e t i t i o n s o f c o n f i g u r a t i o n s of f a c t o r s from s e t s o f d a t a o b t a i n e d by u s i n g a n a l o g o u s measurement t o o l s f o r o t h e r samp l e s from t h e same p o p u l a t i o n . I n d e e d , a f a c t o r may be r e g a r d e d a s a f u n c t i o n d e f i n e d on t h e s e t o f v a r i a b l e s , namely as a v e c t o r of f a c t o r l o a d i n g s . F o r t h e same domain, i . e . f o r t h e same s e t o f v a r i a b l e s , two f a c t o r s are i d e n t i c a l , i f t h e c o r r e s p o n d i n g f u n c t i o n s c o i n c i d e , i . e . i f t h e y have t h e
same v e c t o r s o f f a c t o r l o a d i n g s . I n t h i s c a s e one may r e l a t i v e l y easily construct a c r i t e r i o n f o r t e s t i n g t h e d i f f e r e n c e between two f a c t o r s . Such t e s t s may b e based on t h e l o c a t i o n o f maxima ( t h e s o - c a l l e d marker v a r i a b l e s ) , o r on t h e s u i t a b l y d e f i n e d d i s t a n c e between v e c t o r s of f a c t o r l o a d i n g s ( c o m p a r i s o n of p r o f i l e s ) . The problem, however, becomes more c o m p l i c a t e d i f one requires a higher order of generality of r e s u l t s , i . e . i f one r e q u i r e s i n v a r i a n c e a c r o s s p o p u l a t i o n s , o r
--
which i s even more d i f f i c u l t -- i n v a r i a n c e w i t h r e s p e c t t o a p a r t i a l o r complete change o f t o o l s of measurement. The l a t t e r problem was c o n s i d e r e d by C a t t e l l ( s e e Catt e l l 1957, 1 9 6 1 , 196613; C a t t e l l and Digman 1 9 6 4 ; C a t t e l l , Eber and T a t s u o k a 1970) i n c o n n e c t i o n w i t h h i s t h e o r y
of p e r s o n a l i t y . C a t t e l l d i s t i n g u i s h e d namely t h r e e s o r c e s o f i n f o r m a t i o n about p e r s o n a l i t y : s e l f - e v a l u a t i o n , c a l l e d Q d a t a (from: q u e s t i o n n a i r e d a t a ) , L d a t a , ( l i f e d a t a ) , b e i n g t h e e v a l u a t i o n s by o t h e r s , and T d a t a ( t e s t d a t a ) , from o b j e c t i v e t e s t s . C a t t e l l formula t e d a n i n t e r e s t i n g h y p o t h e s i s , which a s s e r t s t h a t t h e f a c t o r s t r u c t u r e e x t r a c t e d from e a c h of t h e s e s e t s of
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d a t a s h o u l d be t h e same. The i n t u i t i v e j u s t i f i c a t i o n
of t h i s h y p o t h e s i s i s t h a t e a c h o f t h e s e s e t s o f d a t a c o n t a i n s i n f o r m a t i o n about t h e same phenomenon, namel y p e r s o n a l i t y . If t h i s h y p o t h e s i s should t u r n o u t t o be t r u e , i t would mean t h a t one c o u l d p r e d i c t b e h a v i o u r of a s u b j e c t on t h e b a s i s of one t y p e o f d a t a o n l y . T h i s e x p l a i n s t h e name used b y C a t t e l l : h y p o t h e s i s o f i n d i f f e r e n c e o f media. I t i s w o r t h t o m e n t i o n , i n c i d e n t a l l y , t h a t t h e hypothe-
s i s o f i n d i f f e r e n c e o f media was n e v e r f u l l y confirme d . T h i s f a i l u r e o f C a t t e l l may be r e g a r d e d as one o f h i s most i n t e r e s t i n g r e s u l t s ( s e e Nowakowska 1 9 7 3 % and a l s o t h e subsequent s e c t i o n ) . The methodology f o r t e s t i n g i n v a r i a n c e , c r e a t e d mainly by C a t t e l l i n o r d e r t o t e s t h i s h y p o t h e s i s of i n d i f f e r n e c e of media, l i e s on t h e b o r d e r l i n e of s t a t i s t i c a l i n f e r e n c e and p s y c h o l o g i c a l a n a l y s i s of t h e c o n t e n t o f e x t r a c t e d and r o t a t e d f a c t o r s . From t h e s t a t i s t i c a l vie.wpoint, t h e methodology of t e s t i n g i n v a r i a n c e c o n s i s t s o f s i m u l t a n e o u s f a c t o r anal y s i s of t h e j o i n t covariance m a t r i x f o r various sets of d a t a ( s e e Tucker 1958; Browne 1979, 1 9 8 0 ) . Roughly, two f a c t o r s from two domains may be r e g a r d e d as i d e n t i c a l , i f t h e y c o r r e l a t e h i g h l y among t h e m s e l v e s , and a l s o have i d e n t i c a l c o r r e l a t i o n s w i t h o t h e r v a r i a b l e s , t a k e n as c r i t e r i a . T h i s r e q u i r e m e n t , however, i s onl y necessary: i t s f u l f i l l m e n t i s not s u f f i c i e n t f o r i n v a r i a n c e . To show t h a t two f a c t o r s a r e i d e n t i c a l n o t only s t a t i s t i c a l l y but a l s o p s y c h o l o g i c a l l y , i t i s necessary t o analyse t h e content of t h e f a c t o r s .
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1 . 2 . 3 . I n t e r p r e t a t i o n of r e s u l t s of f a c t o r a n a l y s i s . If a g i v e n c o n f i g u r a t i o n o f f a c t o r s s a t i s f i e s t h e cond i t i o n s of i n v a r i a n c e , one may a s s i g n meanings t o f a c t o r s . Roughly s p e a k i n g , two s t r a t e g i e s a r e c o n s i d e r e d h e r e . F i r s t l y , f a c t o r s may be r e g a r d e d merely as means of c l a s s i f i c a t i o n . I n t h i s c a s e , t h e i n t e r p r e t a t i o n c o n s i s t s s i m p l y of c h o o s i n g a n a p p r o p r i a t e name f o r t h e s e t o f v a r i a b l e s r e p r e s e n t i n g i t (i.e. s e t o f v a r i a b l e s with highest f a c t o r loadings), so as t o generalize t h e s e v a r i a b l e s ( s e e e . g . Horst 1 9 6 5 ) . Secondly ( s e e e . g . T h u r s t o n e 1947), f a c t o r s may be assi g n e d n o t o n l y names, b u t a l s o e x p l a n a t o r y h y p o t h e s e s , which a c c o u n t f o r t h e c o v a r i a n c e s between v a r i a b l e s . I n s u c h a c a s e , f a c t o r i s t r e a t e d as a d e t e r m i n a n t o f c o - v a r i a b i l i t y ; i n t h i s s e n s e , i t becomes a c o n s t r u c t , endowed w i t h s u r p l u s meaning. Thus, w h i l e b e f o r e i n t e r p r e t a t i o n f a c t o r c o n s t a i n s no more i n f o r m a t i o n t h a n t h e c o v a r i a n c e m a t r i x , t h e expl a n a t o r y h y p o t h e s e s , as i n v o l v i n g s u r p l u s meaning, p r o v i d e more i n f o r m a t i o n t h a n c o n t a i n e d i n t h e c o v a r i a n c e m a t r i x . Such an e x p l a n a t o r y h y p o t h e s i s i s , however, t o a l a r g e e x t e n t s u b j e c t i v e ; t h i s i s t h e p r i c e which one has t o pay f o r p a s s i n g from a r i g i d and p u r e l y d e s c r i p t i v e i n t e r p r e t a t i o n t o an explanator y i n t e r p r e t a t i o n , which a l l o w s t o l o c a t e t h e f a c t o r i n t h e network o f o t h e r t h e o r e t i c a l n o t i o n s . I n t h i s approach, t h e successive steps of f a c t o r analysis a r e i n f a c t steps i n t h e i n d u c t i o n p r o c e s s . The p r e l i m i n a r y h y p o t h e s i s , b e f o r e invariance i s established, yields an i n t e r p r e t a t i o n which may be confirmed or r e j e c t e d by f u r t h e r s t u d i e s concerning invariance.
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C o g n i t i v e l i m i t a t i o n s o f f a c t o r a n a l y s i s -- the s u b j e c t i v e b i a s . A s e x p l a i n e d above, t h e r e s u l t s o f f a c t o r a n a l y s i s a r e b i a s e d i n a t l e a s t two p l a c e s . F i r s t l y , t h e r e u s u a l l y e x i s t s a number o f v a r i o u s programs f o r e x t r a c t i o n and r o t a t i o n of f a c t o r s , e a c h l e a d i n g t o a s i m p l e s t r u c t u r e , which a l l o w s a c l e a r i n t e r p r e t ation. 1.2.4.
Secondly ( t h i s remark a p p l i e s when f a c t o r a n a l y s i s i s used n o t o n l y as a t a x o n o m i c a l method, b u t a l s o as a s o u r c e of e x p l a n a t o r y h y p o t h e s e s ) , s u b j e c t i v i t y i n t e r venes i n i n t e r p r e t a t i o n o f f a c t o r s , i . e . i n a s s i g n i n g t o them meanings c o n s i s t e n t w i t h known t h e o r i e s . Looking somewhat s u p e r f i c i a l l y one could s a y t h a t t h e same o b j e c t i o n s ( a t l e a s t t h e f i r s t o n e ) a p p l y a l s o t o o t h e r m a t h e m a t i c a l models. I n d e e d , one could c l a i m t h a t if someone a p p l i e s a model of P o i s s o n p r o c e s s t o t h e phenomenon o f r a d i o a c t i v e d e c a y , or t o t h e work o f a t e l e p h o n e exchange, he makes a s u b j e c t i v e c h o i c e o f one of many a l t e r n a t i v e models. T h i s , o f c o u r s e , i s t r u e , but t h e b a s i c d i f f e r e n c e l i e s i n t h e f a c t t h a t i n such a p p l i c a t i o n s t h e c o g n i t i v e r e s t r i c t i o n s a r e e i t h e r made precise a t t h e beginning, o r a r e studied empirically. No one would c l a i m ( a s i t was done by t h e p r o p o n e n t s of f a c t o r a n a l y s i s ) t h a t P o i s s o n p r o c e s s i s t h e d e s s r i p t i o n o f t h e " t r u e " and "unique" n a t u r e o f r a d i o a c t i v e p r o c e s s , or t h e work o f a t e l e p h o n e exchange. One c o u l d a l s o d e f e n d t h e o b j e c t i v l t y o f f a c t o r a n a l y s i s by u s i n g t h e argument o f v a l i d i t y o f p r e d i c t i o n s o f v a r i o u s e x t e r n a l c r i t e r i a , and c l a i m t h a t t h e same arguments a r e u s e d f o r t h e d e f e n s e o f P o i s s o n p r o c e s s as a model f o r r a d i o a c t i v e d e c a y . A moment o f r e f l e c t -
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i o n s u f f i c e s , however, t o r e a l i z e t h a t v a l i d i t y of p r e d i c t i o n shows o n l y t h e p r a g m a t i c v a l u e o f t h e model, but n o t i t s u n i q u e n e s s . I n f a c t , t h e p o s s i b i l i t y o f p r e d i c t i o n of one or more e x t e r n a l c r i t e r i a on t h e basis of a g i v e n s e t o f v a r i a b l e s depends o n l y on t h e j o i n t p r o b a b i l i t y d i s t r i b u t i o n of t h e s e v a r i a b l e s and c r i t e r i a . The o p t i m a l p r e d i c t i o n , g i v e n by t h e r e g r e s s i o n of t h e f i r s t k i n d ( s e e p r e c e d i n g c h a p t e r ) i s , in c a s e o f f a c t o r a n a l y s i s , irnpossible, since t h e data collected f o r f a c t o r a n a l y s i s do n o t u s u a l l y p r o v i d e t h e p o s s i b i l i t y of e s t i m a t i n g the j o i n t probability distribution. There remains t h e p o s s i b i l i t y o f u s i n g t h e methods of l i n e a r r e g r e s s i o n , where one t a k e s a s p r e d i c t o r s t h e a p p r o p r i a t e l i n e a r c o m b i n a t i o n s o f v a r i a b l e s ( t h e coe f f i c i e n t s b e i n g d e t e r m i n e d s o as t o minimize t h e mean s q u a r e e r r o r o f p r e d i c t i o n ) . Such a p r o c e d u r e r e q u i r e s o n l y t h e knowledge o f means, v a r i a n c e s and c o v a r i a n c e s of t h e v a r i a b l e s and t h e c r i t e r i o n , hence r e q u i r e s t h e d a t a which a r e u s u a l l y a c c e s s i b l e from t h e r e s u l t s of f a c t o r a n a l y s i s . Moreover, l i n e a r r e g r e s s i o n has t h e property t h a t it coincides with t r u e regression i n c a s e o f j o i n t normal d i s t r i b u t i o n o f v a r i a b l e s and c r i teria. D e t e r m i n i n g t h e l i n e a r combinations of v a r i a b l e s des t r o y s p a r t i a l l y t h e i n f o r m a t i o n c o n t a i n e d i n t h e values o f t h e s e v a r i a b l e s ( s i m p l y because t h e knowledge of t h e v a l u e o f l i n e a r combination does n o t s u f f i c e t o r e c o n s t r u c t t h e v a l u e s o f v a r i a b l e s ) . T h i s l o s s 01’ i n f o r m a t i o n may be more or l e s s s e v e r e , d e p e n d i n g on t h e joint probability distribution.
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I n c a s e of f a c t o r a n a l y s i s t h e s i t u a t i o n i s more comp l e x , s i n c e f a c t o r s are l i n e a r combinations of v a r i a b l e s . I f t h e problem c o u l d be r e d u c e d t o p r e d i c t i n g of one c r i t e r i o n o n l y , i t would p r o b a b l y b e b e t t e r t o b a s e t h e p r e d i c t i o n s on t h e v a r i a b l e s d i r e c t l y , i n s t e a d of u s i n g f a c t o r s which d e s t r o y p a r t i a l l y t h e i n f o r m a t i o n . The p o i n t i s , however, t h a t a s a r u l e one wants t o use t h e same f a c t o r s f o r p r e d i c t i n g v a r i o u s c r i t e r i a . I n s u c h c a s e s , i t seems r e a s o n a b l e t o f i n d a "good" i n i t i a l r e d u c t i o n of d a t a t o a s m a l l e r number o f v a r i a b l e s ( f a c t o r s ) -- b u t s o as n o t t o l o s e t o o much informa t i o n . T h i s i s t h e b a s i c i d e a of f a c t o r a n a l y s i s . However, even i f s u c h a r e d u c t i o n p r o v i d e s a s e t o f v a r i a b l e s which a r e u s e f u l i n p r e d i c t i n g v a r i o u s c r i t e r i a , a l l one may s a y i s t h a t t h e f a c t o r s a r e w e l l c h o s e n , u s i n g v a l i d i t y o f p r e d i c t i o n s as a n argument. There i s s t i l l no ground t o a s s e r t t h a t t h e c h o i c e leads t o t h e unique t r u e s t r u c t u r e o f t h e phenomenon, as some f a c t o r a n a l y s t s t e n d t o c l a i m , s i m p l y because there i s no r e a s o n why some o t h e r c h o i c e could n o t be b e t t e r . S e c o n d l y , even i f u n i q u e n e s s c o u l d be proved i n some way, i t would be u n i q u e n e s s w i t h r e s p e c t t o t h e s e l e c t ed s e t of v a r i a b l e s , and n o t " u n i v e r s a l " u n i q u e n e s s w i t h r e s p e c t t o t h e phenomenon. G e n e r a l l y , even i f t h e problems o f u n i q u e n e s s and i n v a r i a n c e c o u l d be s a t i s f a c t o r i l y r e s o l v e d , i t c o u l d a t b e s t s e r v e as a n argument t h a t f a c t o r a n a l y s i s a l l o w s u s t o e x t r a c t t h e t r u e n a t u r e o f t h e phenomenon as i t was d e f i n e d by t h e v a r i a b l e s chosen; i f some a s p e c t s a r e o m i t t e d i n t h e i n i t i a l s e t of v a r i a b l e s , t h e y w i l l not appear i n f a c t o r analysis.
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C o n s e q u e n t l y , i n p l a n n i n g a f a c t o r i a l e x p e r i m e n t , one s h o u l d choose t h e i n i t i a l s e t o f v a r i a b l e s i n an exhaus t i v e w a y , i n c l u d i n g a l l v a r i a b l e s which may c o n t r i b u t e t o the description.
I t is worth t o mention h e r e t h a t t h e r e e x i s t two " s t y l e s " of a p p l y i n g t h e model o f f a c t o r a n a l y s i s . F i r s t l y , one may p e r f o r m f a c t o r a n a l y s i s on a s e t o f d a t a w i t h o u t any p r i o r p r e - c o n c e p t i o n s
and h y p o t h e s e s a b o u t t h e
n a t u r e of t h e phenomenon under i n v e s t i g a t i o n . Simply , u s i n g s u f f i c i e n t l y many v a r i a b l e s which " c o v e r " t h e phenomenon, and p e r f o r m i n g f a c t o r a n a l y s i s , one o b t a i n s f a c t o r s r e g a r d e d as i n t r o d u c t o r y h y p o t h e s e s , which one needs t o f u r t h e r v e r i f y . S e c o n d l y , one may u s e f a c t o r a n a l y s i s as a t o o l f o r t e s t i n g i n t r o d u c t o r y hypotheses about t h e f a c t o r i a l s t r u c t u r e of t h e phenomenon ( f o r a d i s c u s s i o n o f t h e two s t r a t e g i e s , s e e e . g . Henrysson
1957).
These two s t y l e s , or s t r a t e g i e s , o f r e s e a r c h i n f a c t or a n a l y s i s a r e sometimes r e f e r r e d t o as e x p l o r a t o r y factor analysis
,
and c o n f i r m a t o r y f a c t o r a n a l y s i s . I t
ought t o be mentioned t h a t t h e f i r s t d o m i n a t e s t h e s e cond, b o t h as r e g a r d s t h e number o f t h e o r e t i c a l contr i b u t i o n s f o r e x p l a n a t o r y v s . c o n f i r m a t o r y f a c t o r anal y s i s , as w e l l as w i t h r e g a r d s t o t h e number o f empi-
r i c a l s t u d i e s . The same two s t r a t e g i e s c a r r y o v e r t o t h e t i o p i c s o f multidimensional s c a l i n g , a s e t of techn i q u e s w i t h aims l a r g e l y t h e same as f a c t o r a n a l y s i s , but based on weaker a s s u m p t i o n s a b o u t t h e d a t a . A s r e g a r d s f a c t o r a n a l y s i s , t h e b e s t known c o n f i r m a t o r y
t e c h n i q u e i s t h a t o f J d r e s k o g (1967), b a s e d on maximum
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l i k e l i h o o d p r i n c i p l e ( s e e a l s o Lee 1 9 8 1 ) . A t t h i s p l a c e i t i s w o r t h t o mention t h e work o f Davison (1980) who g i v e s a method o f t e s t i n g some h y p o t h e s e s a b o u t t h e o r d e r of items ( t e s t s , s t i m u l i , e t c . ) . Roughly s p e a k i n g , t h e s i t u a t i o n i s as f o l l o w s . Conside r t h e m a t r i x (1.1) o f r e s u l t s o f measurements, and suppose t h a t t h e i t e m s ( a s r e p r e s e n t e d b y columns) a r e dichotomous, s o t h a t x j j ) i s t h e r e s p o n s e ( s a y 1 o r 0 ) of j - t h p e r s o n t o i - t h i t e m . Suppose t h a t t h e items a r e o r d e r e d i n s u c h a way t h a t any p e r s o n who g i v e s answer " y e s " t o any i t e m s h o u l d a l s o g i v e answer " y e s " t o a l l i t e m s which follow i t i n t h e o r d e r i n g . T h i s a l lows us t o d i s t i n g u i s h t h e c l a s s of a d m i s s i b l e and t h e c l a s s o f i n a d m i s s i b l e r e s p o n s e p a t t e r n s . The problem l i e s i n t e s t i n g t h e h y p o t h e s i s about t h e o r d e r o f i t e m s i n t h e p r e s e n c e o f e r r o r s , i . e . assuming some d e g r e e of randomness i n r e s p o n s e s . Also, i t i s allowed t h a t some r e s p o n s e p a t t e r n s a r e e x c l u d e d by t h e c o n s t r a i n t s of t h e experiment.
1.2.5. The s t a t u s o f t h e v a r i a b l e s u n d e r l y i n g f a c t o r a n a l y s i s . T y p i c a l l y , i n p s y c h o l o g i c a l and s o c i o l o g i c a l a p p l i c a t i o n s o f f a c t o r a n a l y s i s , t h e v a r i a b l e s used a r e t h e r e s u l t s o f some t e s t s ( o r s i n g l e i t e m s ) f o r p a r t i c u l a r s u b j e c t s . To a p p l y t h e programs for e x t r a c t i n g f a c t o r s i t i s n e c e s s a r y t h a t t h e v a r i a b 3 e s a r e meas u r e d on a t l e a s t a n i n t e r v a l s c a l e , s o t h a t one may j u s t i f i a b l y c a l c h l a t e means., v a r i a n c e s and c o v a r i a n c e s . Moreover, f o r i n t e r p r e t a t i o n o f r e s u l t s , i t i s n e c e s s a r y t h a t one knows which t r a i t s a r e measured b y p a r t i cular tests (items).
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T h e o r e t i c a l l y , t o v e r i f y t h e s e two a s s u m p t i o n s , i t i s n e c e s s a r y f i r s t t o d e t e r m i n e t h e s t a t u s o f t h e measured t r a i t from t h e m e a s u r e m e n t - t h e o r e t i c a l p o i n t o f view, and t h e n check t h a t t h e i t e m i n q u e s t i o n p r o v i d e s a n a d e q u a t e measure of t h e t r a i t . To d e t e r m i n e t h e s t a t u s o f a g i v e n t r a i t from t h e p o i n t of view of measurement t h e o r y ( s e e e . g . Suppes and Z i nnes 1963; P f a n z a g l 1968, E l l i s 1966, Coombs, Dawes and Tversky 1970, Krantz e t a l . , 1970, o r R o b e r t s 1979), i t i s n e c e s s a r y t o a s s i g n t o t h i s t r a i t a c e r t a i n empir i c a l r e l a t i o n a l s y s t e m , and v e r i f y t h a t i t s a t i s f i e s t h e a p p r o p r i a t e axioms i m p l y i n g t h e e x i s t e n c e o f measurement on a s c a l e of a g i v e n t y p e ( s e e C h a p t e r 4 ) . I n c a s e o f i n t e r v a l s c a l e s , t h e r e must e x i s t a n u m e r i c a l r e p r e s e n t a t i o n of t h e t r a i t unique up t o a p o s i t i v e l i n e a r transformation. Another a p p r o a c h , a p p l i e d i n m e n t a l t e s t t h e o r y ( s e e Lord and Novick 1968) c o n s i s t s o f simply assuming t h a t t e s t r e s u l t s ( s c o r e s ) c o n s t i t u t e measurement on a t least i n t e r v a l t y p e scale. I n f a c t , one assumes even more, namely c o m p a r a b i l i t y of u n i t s o f measurement. The u s e of t h i s a s s u m p t i o n i s a p p a r e n t a l r e a d y on t h e l e v e l o f q u e s t i o n n a i r e i t e m s . I n d e e d , i f someone answers "yes" t o two q u e s t i o n s , s u c h as "DO you o f t e n f e e l t i r e d i n t h e morning?'' and "DO you t h i n k t h a t some numbers a r e u n l u c k y ? " , t h e n h i s s c o r e i n t h e " s u b t e s t " c o n s i s t i n g o f t h e above two i t e m s e q u a l s 2 . I f d i f f e r e n t w e i g h t s were a s s i g n e d to t h e s e items ( a s i t i s sometimes done i n c a s e o f t e s t s ) , i t would amount merely t o changing o f t h e u n i t s o f measurement.
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I n any c a s e , f o r t h e c o n d i t i o n s a l l o w i n g i n t e r p r e t a t i o n of t h e e x t r a c t e d and r o t a t e d f a c t o r s , i t i s e s s e n t i a l t h a n one knows what i s b e i n g measured by a g i v e n item (variable)
.
Theoretically, f o r t r a i t s defined through appropriate e m p i r i c a l systems i m p l y i n g i n t e r v a l t y p e s c a l e , s u c h a v e r i f i c a t i o n i s p o s s i b l e , i f o n l y t h e r e e x i s t s a way of m e a s u r i n g a g i v e n t r a i t w i t h o u t u s i n g t h e t e s t . Such an i d e a l s i t u a t i o n i f v e r y d i f f i c u l t , i f n o t i m possible t o a t t a i n i n practical situations, especially i n p s y c h o l o g i c a l r e s e a r c h , because t h e v a r i a b l e s a r e u s u a l l y on t o o h i g h l e v e l o f g e n e r a l i t y . I n p r a c t i c e , t o check t h a t a g i v e n p s y c h o l o g i c a l t o o l X ( t e s t , s a y ) measures a s e l e c t e d t r a i t Y, one has t o a p p l y t h e methods o u t l i n e d i n C h a p t e r 1: X s h o u l d c o r r e l a t e h i g h l y w i t h t h o s e t r a i t s which ( a c c o r d i n g t o p s y c h o l o g i c a l t h e o r y ) a r e r e l a t e d t o Y , and s h o u l d n o t c o r r e l a t e w i t h t h o s e t r a i t s which a r e n o t r e l a t e d t o Y. A t t h i s p l a c e i t i s worth t o mention t h a t a l a r g e
group o f p s y c h o m e t r i c i a n s ( e . g . Cowen 1963, Edwards 1957, Damarin and Messick 1965, Rorer 1965, E i s e n b e r g 1 9 4 1 , E i s e n b e r g and Wesnman 1941, Goldberg 1963) s p e n t a c o n s i d e r a b l e amount of e f f o r t on t e s t i n g t h e hypot h e s i s o f r e s p o n s e b i a s . G e n e r a l l y , i t was a r g u e d t h a t t e s t s measure n o t s o much t h e b a s i c d e t e r m i n a n t s of b e h a v i o u r ( t r a i t s ) , but s u c h v a r i a b l e s as r e s p o n s e s t y l e , e . g . p r e f e r e n c e towards some c a t e g o r i e s of a n s w e r s , o r t h e so-called s o c i a l d e s i r a b i l i t y .
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1 . 3 . Multidimensional s c a l i n g 1 . 3 . 1 . C o n c e p t u a l f o u n d a t i o n s . We s h a l l now mention b r i e f l y t h e group o f t e c h n i q u e s known under t h e name " m u l t i d i m e n s i o n a l s c a l i n g " , which a r e f r e e o f c e r t a i n o f t h e r e s t r i c t i v e assumptions o f f a c t o r a n a l y s i s . The b a s i c i d e a o f m u l t i d i m e n s i o n a l s c a l i n g may be p r e s e n t e d as f o l l o w s . We c o n s i d e r a c e r t a i n c l a s s o f obj e c t s ; t h e s e may be t e s t s , s u b j e c t s , or i n some c a s e s t h i s c l a s s may c o n s i s t o f b o t h s u b j e c t s and s t i m u l i ( e . g . o b j e c t s w i t h r e s p e c t to which one e x p r e s s e s p r e f e r e n ces). I n t h e s e t under c o n s i d e r a t i o n one o b s e r v e s c e r t a i n emp i r i c a l r e l a t i o n s . Depending on t h e c o n t e x t , t h e s e may be o r d e r i n g r e l a t i o n s o f " d i s t a n c e s " between t h e obj e c t s , t h e n o t i o n of " d i s t a n c e " o r "proximity" b e i n g a s s e s s e d s u b j e c t i v e l y , or measured by some means, e . g . c o r r e l a t i o n c o e f f i c i e n t ( i n c a s e o f t e s t s ) , and s o on. I n o t h e r s i t u a t i o n s , t h e o r d e r i n g s may be p r e f e r e n t i a l o r d e r i n g o f o b j e c t s e l i c i t e d from t h e s u b j e c t s . I n e a c h c a s e , t h e t e c h n i q u e of m u l t i d i m e n s i o n a l s c a l i n g c o n s i s t s of finding a geometrical representation of t h e s e t of o b j e c t s , i . e . l o c a t i n g these objects i n s p a c e w i t h p o s s i b l y l o w number of d i m e n s i o n s , i n s u c h a way t h a t m u t u a l l o c a t i o n s o f t h e p o i n t s r e p r e s e n t t h e r e l a t i o n s h i p s between o b j e c t s g i v e n i n t h e d a t a . To u s e a n example, two of t h e s i t u a t i o n s may be p r e s e n t -
ed i n an a b s t r a c t way as f o l l o w s . ( a ) Nonmetric a n a l y s i s o f s i m i l a r i t i e s . L e t us c o n s i -
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d e r t h e r e l a t i o n a l system o f t h e form
where A i s a c e r t a i n s e t , and S i s a r e l a t i o n which o r d e r s A X A . Let t h e symbol ( a i , a . ) S ( a k y a m ) mean t h a t J t h e p a i r ( a i , a . ) a p p e a r s i n t h e o r d e r i n g of S X S e a r l J i e r (not l a t e r ) than t h e p a i r (akyam)W . e assume t h a t from S we have f o r a l l ai yaj
* * *
The f i r s t t h r e e p o s t u l a t e s a s s e r t simply t h a t S i s a weak o r d e r i n g o f t h e s e t A x A , i . e . t h a t S i s r e f l e x i v e , t r a n s i t i v e and c o n n e c t e d . P o s t u l a t e ( d ) a s s e r t s t h a t p a i r s which d i f f e r o n l y by o r d e r of t h e i r e l e m e n t s do n o t p r e c e d e one a n o t h e r i n t h e o r d e r i n g . F i n a l l y , p o s t u l a t e ( e ) r e q u i r e s t h a t p a i r s w i t h i d e n t i c a l terms precede a l l o t h e r p a i r s . The above p o s t u l a t e s c o n c e r n f i n i t e s e t s A .
In case of i n f i n i t e s e t s , one imposes s t i l l more requirements ( s e e B e a l s , Krantz and Tversky 1968), c o n c e r n i n g continuity, etc.
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The problem l i e s i n c o n t r u c t i o n o f a s y s t e m
where Er i s t h e r - d i m e n s i o n a l s p a c e , d i s a c e r t a i n metric (not necessarily Euclidean) i n t h i s space, i . e .
a r e a l - v a l u e d f u n c t i o n d e f i n e d on E r x E' s u c h t h a t d(u,v,) 0 , w i t h d ( u , v ) = 0 i f f u = v , and a l s o s a t i s f y i n g d ( u , v ) = d ( v , u ) and t r i a n g l e i n e q u a l i t y d ( u , w ) ( d(u,v) t d(v,w).
>
-
Finally, f : A Er i s a n assignment of p o i n t s i n Er t o e l e m e n t s o f t h e s e t A , i . e . a mapping which t o ever y a 6 A a s s i g n s t h e p o i n t f ( a ) E Er s u c h t h a t
Moreover, i t i s r e q u i r e d t h a t r i s t h e s m a l l e s t p o s s i b l e number of' d i m e n s i o n s , i . e . t h a t t h e r e i s no system <En,d,f ) w i t h n
<
r satisfying (*).
I n t u i t i v e l y , t h e problem i s t o f i n d a g e o m e t r i c r e p r e s e n t a t i o n of o b j e c t s i n space, such t h a t t h e r e l a t i o n of i n e q u a l i t y f o r geometric d i s t a n c e s correspond t o t h e o r d e r i n g of t h e i n i t i a l r e l a t i o n S. T y p i c a l l y , r e l a t i o n S i s o b t a i n e d by s u b j e c t i v e judgments o f s i m i l a r i t i e s or d i f f e r e n c e s between o b j e c t s ( i . e . answers t o q u e s t i o n s of t h e t y p e : "Is t h e o b j e c t a more s i m i l a r t o o b j e c t b t h a n o b j e c t c i s s i m i l a r t o o b j e c t d ? " . A s mentioned, i n c a s e of t e s t s , one can measure d i s t a n c e s by t h e i r i n t e r c o r r e l a t i o n s , i . e .
( a i , a . ) s ( a k , a m ) if p ( a i , a j ) J
the correlation coefficient.
2 p ( a k , a m ) , where p i s
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( b ) Coombs’ t e c h n i q u e o f u n f o l d i n g s c a l e s . A s a n o t h e r
example o f t h e t e c h n i q u e s o f m u l t i d i m e n s i o n a l s c a l i n g , c o n s i d e r a system (A,
(1.12)
B, { . R b , b 2 B f )
where A and B a r e two s e t s , c a l l e d r e s p e c t i v e l y t h e s e t
of o b j e c t s and s e t o f s u b j e c t s , w h i l e e a c h Rb i s a weak o r d e r of e l e m e n t s o f A , i . e . a r e l a t i o n s u c h t h a t f o r a l l a i , a j , a k i A and b t B we have
(a)
a R a
(b)
If aiRbaj,
j b j
(reflexivity); then a R a
j b k’
t h e n aiRbak ( t r a n s i t i v -
ity);
(c)
e i t h e r aiRbaj or a R a
j b i
(connectedness).
Observe t h a t i t i s n o t r e q u i r e d t h a t A and B be d i s j o i n t ; for i n s t a n c e , J o n e s and Young ( 1 9 7 2 ) c a r r i e d o u t a s t u d y i n which t h e o b j e c t s were a l s o t h e e v a l u a t i n g persons. Thus, t h e r e l a t i o n a l system ( 1 . 1 2 ) has t h e form o f a s e t of o r d e r i n g s o f t h e s e t A , one o r d e r i n g for e a c h o f t h e s u b j e c t s i n B . A t y p i c a l c a s e here would be a p r e f e r e n t i a l ordering of o b j e c t s i n A. The g o a l i s t o f i n d a system <Er,
d, f
>
where Er and d have t h e same meaning as b e f o r e , w h i l e
142
f: A
CHAPTER 2
B
--+
E’
i s a n a l l o c a t i o n o f b o t h o b j e c t s and
s u b j e c t s i n s p a c e Er , s u c h t h a t ( * * ) f o r e a c h b F- B and a i , a
J t
A the condition a R a
h o l d s i f and o n l y i f d [ f ( b ) , f ( a i ) ]
i b j
< d [ f ( b ) , f ( a5 ) I .
Moreover, as b e f o r e , one r e q u i r e s t h a t t h e d i m e n s i o n r be s m a l l e s t p o s s i b l e , s o t h a t t h e r e d o e s n o t e x i s t a system <En,d,f> s a t i s f y i n g ( * * ) w i t h n < r .
1 . 3 . 2 . The main d i r e c t i o n s o f d e v e l o p m e n t o f t h e t h e o r y o f m u l t i d i m e n s i o n a l s c a l i n g . The t h e o r e t i c a l r e s e a r c h i n m u l t i d i m e n s i o n a l s c a l i n g may be r o u g h l y d i v i d ed i n t o two g r o u p s . One o f them c o m p r i s e s t h e s o l u t i o n s , u s u a l l y accompanied w i t h c o m p u t a t i o n a l p r o g r a m s , which a l l o w us t o d e t e r m i n e , f o r g i v e n d a t a , t h e r e p r e s e n t a t i o n f . I n o t h e r words, here t h e problem l i e s i n f i n d i n g , f o r a given set of data, t h e r e p r e s e n t a t i o n f m e e t i n g u s u a l l y some a d d i t i o n a l c o n s t r a i n t s . F o r a r e v i e w and taxonomy o f v a r i o u s t y p e s o f p r o b l e m s h e r e , s e e Shepard 1972, 1974. F o r t h e main r e s u l t s i n t h e a r e a , see e . g . S h e p a r d 1 9 6 2 a , b . K r u s k a l 1 9 6 4 a , b ; Torg e r s o n 1952; L i n g o e s 1965; Guttman 1968; C a r r o l l and Chang 1970; B e n t l e r and Weeks 1978; Bloxom 1978; Weeks and B e n t l e r 1979, or Miyano a n d I n u k a i 1982. The s e c o n d d i r e c t i o n o f r e s e a r c h , much l e s s f r e q u e n t ,
c o n c e n t r a t e s on t h e p r o b l e m o f e x i s t e n c e o f t h e soluu t i o n s i n t h e two main g r o u p s ( a ) and ( b ) as d e s c r i b ed i n t h e p r e c e d i n g s e c t i o n . The main i s s u e i s t o f i n d empirically testable conditions f o r the existence
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of an a p p r o p r i a t e r e p r e s e n t a t i o n . I n c a s e o f problem ( a ) , of s c a l i n g s i m i l a r i t i e s , s u c h c r i t e r i a were formu l a t e d b y B e a l s , Krantz and Tversky (1968), who gave a n e m p i r i c a l l y t e s t a b l e axiom system f o r t h e e x i s t e n c e of g e n e r a l c l a s s o f m u l t i d i m e n s i o n a l r e p r e s e n t a t i o n s ( i n c a s e of i n f i n i t e s e t s of o b j e c t s ) . I n c a s e of t e c h n i q u e s of u n f o l d i n g s c a l e s ( b ) , t h e c r i t e r i a a r e based on t h e a n a l y s i s of a d m i s s i b l e permuta t i o n s o f o b j e c t s o f t h e s e t A . It a p p e a r s t h a t t h e number of a d m i s s i b l e p e r m u t a t i o n s i s c l o s e l y r e l a t e d t o t h e dimension of t h e s p a c e ( s e e e . g . H a y s and Benn e t t 1961). For i n s t a n c e , i n case of 4 o b j e c t s , there e x i s t s o n l y 7 p e r m u t a t i o n s f o r one-dimensional r e p r e s e n t a t i o n . I n consequence, i f t h e o b s e r v e d s e t of p e r m u t a t i o n s c o n t a i n s more t h a n s e v e n e l e m e n t s , l i n e a r r e p r e s e n t a t i o n ( r = 1) i s n o t p o s s i b l e . It i s a l s o w o r t h m e n t i o n i n g t h a t t h e e m p i r i c a l r e s u l t s
which s e r v e as a basis f o r m u l t i d i m e n s i o n a l s c a l i n g a r e o f t e n u n s t a b l e b e c a u s e of f l u c t u a t i o n o f a t t e n t i o n ( S h e p a r d 1 9 6 4 ) , and c o n t e x t u a l e f f e c t s ( T o r g e r s o n 1 9 6 5 ) . T h i s l a c k o f s t a b i l i t y i s n o t e a s i l y remedied w i t h i n t h e s t a t i c model of m u l t i d i m e n s i o n a l s c a l i n g . It seems t h a t i n o r d e r t o o b t a i n r e p r e s e n t a t i o n s which a r e p s y c h o l o g i c a l l y s e n s i b l e i t i s n e c e s s a r y t o impose c o n s t r a i n t s on t h e c l a s s of e x p e r i m e n t s . One s h o u l d a l s o mention some s i m u l a t i o n s t u d i e s ( e . g . Cohen and J o n e s 1973) f o r t e s t i n g . h y p o t h e s e s a b o u t s t a b i l i t y o f r e s u l t s under v a r i o u s a s s u m p t i o n s about f l u e t u a t i o n of a t t e n t i o n ( e . g . b a s i n g t h e e v a l u a t i o n s on some s e l e c t e d d i m e n s i o n s , which change from o c c a s i o n t o occasion).
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1 . 3 . 3 . M u l t i d i m e n s i o n a l s c a l i n g and f a c t o r a n a l y s i s . The a d v a n t a g e of t e c h n i q u e s of m u l t i d i m e n s i o n a l s c a l i n g o v e r f a c t o r a n a l y s i s c o n s i s t s n o t only on g r e a t e r a p p l i c a b i l i t y o f t h e f o r m e r . The main d i f f e r e n c e cons i s t s , f i r s t o f a l l , on t h e f a c t t h a t m u l t i d i m e n s i o n a l s c a l i n g i s based on weaker a s s u m p t i o n s , u s i n g o n l y t h e o r d e r i n f o r m a t i o n c o n t a i n e d i n t h e d a t a , and n o t t h e n u m e r i c a l v a l u e s (which are u s u a l l y s u b j e c t t o much h i g h e r e r r o r ) . Moreover, f a c t o r a n a l y s i s i s based on t h e a s s u m p t i o n t h a t t h e r e s u l t s f o r p a r t i c u l a r v a r i a b l e s may be r e p r e s e n t e d a d d i t i v e l y , as weighted sums o f f a c t o r s . T h i s a s s u m p t i o n h a s no c o n v i n c i n g s u p p o r t i n form of p s y c h o l o g i c a l h y p o t h e s e s . While i t i s r e l a t i v e l y e a s y to j u s t i f y , on t h e ground on some psychol o g i c a l t h e o r y , t h a t s u c h and s u c h f a c t o r s i n f l u e n c e the analysed reaction, the a s s e r t i o n t h a t t h i s influence i s e x p r e s s e d a d d i t i v e l y i s much h a r d e r t o j u s t i f y . A j u s t i f i c a t i o n would u s u a l l y r e q u i r e c h e c k i n g t h e axioms f o r c o n j o i n t measurement ( s e e C h a p t e r 4) , and such a procedure i s s u c c e s s f u l only i n rare i n s t a n c e s . A s opposed t o t h a t , t h e t e c h n i q u e s o f m u l t i d i m e n s i o n a l
s c a l i n g a r e based only on o r d e r r e l a t i o n s i n e m p i r i c a l data. A s r e g a r d s t h e i n t e r p r e t a t i o n of r e s u l t s , i . e . o f f a c -
tors i n c a s e of f a c t o r a n a l y s i s , or dimensions i n c a s e of m u l t i d i m e n s i o n a l s c a l i n g , t h e s i t u a t i o n i s s i m i l a r : i n both cases, it r e f e r s t o t h e psychological content of t h e o b j e c t s analysed ( t e s t s , items, e t c . ) . One c a n , however, h o l d t h e o p i n i o n t h a t t h e d i s a d v a n t a g e s of f a c t o r a n a l y s i s do n o t overweight t h e advantages: i f f a c t o r a n a l y s i s i s used as a s t a t i s t i c a l
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t e c h n i q u e w i t h a p p r o p r i a t e c h o i c e of d a t a , i t may s e r v e
as p o w e r f u l means of f o r m u l a t i n g h y p o t h e s e s and t h e o r e t i c a l l y f e r t i l e notions. Since t h i s i s , i n e f f e c t , one of t h e main o b j e c t i v e s o f any s c i e n t i f i c r e s e a r c h , any means of a t t a i n i n g i t s h o u l d b e r e g a r d e d as v a l u able.
I n f a c t , one c a n e n c o u n t e r two o p p o s i n g a t t i t u d e s t o wards f a c t o r a n a l y s i s among p s y c h o l o g i s t s : some are i n c l i n e d t o a t t a c h more s i g n i f i c a n c e t o c e r t a i n n o t i o n s b e c a u s e t h e y r e s u l t from a p p l c c a t i o n o f f a c t o r a n a l y sis, while others tend t o disregards these notions f o r p r e c i s e l y t h e same r e a s o n . Both a t t i t u d e s here seem somewhat e x a g g e r a t e d ; i t may be argued t h a t i f a n o t i o n s e r v e s w e l l t h e r o l e f o r which i t was c r e a t e d , i t matt e r s l i t t l e whether i t was o b t a i n e d from f a c t o r a n a l y sis o r not.
2 . AN EXAMPLE OF APPLICATION OF FACTOR ANALYSIS:
PERSONALITY THEORY OF CATTELL 2 . 1 . D e s c r i p t i v e approach t o p e r s o n a l i t y t h e o r y
The p e r s o n a l i t y t h e o r y p r e s e n t e d i n t h i s s e c t i o n i s p e r h a p s t h e b e s t example o f a n a p p l i c a t i o n o f f a c t o r a n a l y s i s , as i t a l l o w s t o t r a c e a l l c o g n i t i v e l i m i t a t i o n s o f t h i s method. The concept o f p e r s o n a l i t y i s o n e . o f t h e b a s i c c o n s t -
r u c t s of contemporary p s y c h o l o g y . I n most g e n e r a l t e r m s , i n a d e s c r i p t i v e a p p r o a c h , i t i s d e f i n e d t h r o u g h some o t h e r c o n s t r u c t s , c a l l e d p s y c h o l o g i c a l t r a i t s . The org a n i z a t i o n of t h e s e t o f t r a i t s , i . e . d e t e r m i n i n g t h e i r
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number, p s y c h o l o g i c a l c o n t e n t , and mutual r e l a t i o n s , l i e s a t t h e f o u n d a t i o n of e v e r y p e r s o n a l i t y t h e o r y which must a c c o u n t f o r t h e o r g a n i z a t i o n o f t h e s e t o f t r a i t s i n a s p e c i f i c w a y . Thus, a p s y c h o l o g i c a l t r a i t i s t r e a t e d as a h y p o t h e t i c a l component o f p e r s o n a l i t y , o b t a i n e d by a s s i g n i n g a name which g e n e r a l i z e s , and sometimes e x p l a i n s , t h e p s y c h o l o g i c a l c o n t e n t of a s e t of b e h a v i o u r s which have a tendency t o a p p e a r j o i n t l y . The f o l l o w i n g two premises l i e a t t h e f o u n d a t i o n o f
t h e d e s c r i p t i v e approach t o p e r s o n a l i t y : ( a ) o b s e r v a t i o n s i n d i c a t e a c e r t a i n s t a b i l i t y o f some e l e m e n t s of b e h a v i o u r i n s i m i l a r c l a s s e s of s i t u a t i o n s ; ( b ) r e g u l a r i t y and s i m i l a r i t y o f b e h a v i o u r s of d i f f e r -
e n t p e r s o n s a l l o w us t o assume t h a t t h e s e p e r s o n s have some common t r a i t s , and a l l o w u s t o d e s c r i b e t h e s e p e r s o n s i n t e r m s of t h e s e t r a i t s . I n o t h e r words, i n t h e b e h a v i o u r o f a s i n g l e p e r s o n , and a l s o i n t h e b e h a v i o u r o f a group o f p e r s o n s , one may d e t e c t r e g u l a r i t i e s o f s t a t i s t i c a l c h a r a c t e r , s u f f i c i e n t l y w e l l pronounced t o j u s t i f y , on t h e one hand, a s s i g n i n g a common name g e n e r a l i z i n g t h e c o n t e n t of t h e s e b e h a v i o u r s , and on t h e o t h e r hand -- s u f f i c i e n t t o allow f o r p r e d i c t i o n of f u t u r e behaviours. R e l a t i v e s t a b i l i t y of b e h a v i o u r s ( p r e m i s e ( a ) ) means h i g h p r o b a b i l i t y of a p p e a r i n g o f a g i v e n b e h a v i o u r or b e h a v i o u r s i n s i m i l a r s i t u a t i o n s , and i t s e r v e s as a basis f o r p r e d i c t i o n s o f f u t u r e b e h a v i o u r s , N a t u r a l l y , t h e s e p r e d i c t i o n s a r e n o t i n f a l l i b l e , and t h e s o u r c e of e r r o r s i s t w o f o l d . F i r s t l y , we d e a l here w i t h t h e
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r e g u l a r i t i e s of a s t a t i s t i c a l c h a r a c t e r , s o t h a t r a n domness i s i n h e r e n t t o t h e phenomenon. S e c o n d l y , t h e p r e d i c t i o n s a r e t y p i c a l l y based on o b s e r v a t i o n s o f a s e l e c t e d sample of b e h a v i o u r s , w h i l e t h e y concern o t h e r b e h a v i o u r s , n o t observed i n t h e sample. We d e a l h e r e , therefore, with a certain extrapolation. A s e x p l a i n e d i n C h a p t e r 1, t h e v a l u e o f a t r a i t o f a
g i v e n p e r s o n i s d e f i n e d as t h e e x p e c t a t i o n o f h i s prop e n s i t y d i s t r i b u t i o n , which c h a r a c t e r i z e s t h e s t a t i s t i c a l r e g u l a r i t i e s o f b e h a v i o u r s of t h i s p e r s o n w i t h r e s p e c t t o a g i v e n t e s t . A t any r a t e , t h e r e f o r e , t h e p r e d i c t i o n c a n be o n l y as good as t h e r e g r e s s i o n equat i o n s allow. The second p r e m i s e i s based on i n t u i t i o n s c o n c e r n i n g t h e n o t i o n o f a common t r a i t . I t i s namely assumed t h a t b o t h t h e s p e c i f i c and common t r a i t s d e v e l o p as a r e s u l t o f g e n e t i c and e n v i r o n m e n t a l f a c t o r s , w i t h common t r a i t s d e v e l o p i n g from c o n s t a n t e x p o s u r e t o s i m i l a r c o n f i g u r a t i o n s of e x t e r n a l s t i m u l i . These t r a i t s r e f l e c t s t a t i s t i c a l r e g u l a r i t i e s o f b e h a v i o u r i n groups o f s u b j e c t s , and t h e r e f o r e may be d e t e c t e d by s t u d y i n g groups i n s i milar s t i m u l u s s i t u a t i o n s . The p r e d i c t i o n s based on common t r a i t s a r e n e c e s s a r i l y l e s s p r e c i s e t h a n t h o s e based on b o t h s p e c i f i c and common t r a i t s . However, by r e s t r i c t i n g t h e c o n s i d e r a t i o n t o common t r a i t s o n l y one g e t s t h e p o s s i b i l i t y of comp a r i n g groups of s u b j e c t s . The b a s i c g o a l of a l l d e s c r i p t i v e t h e o r i e s o f p e r s o n a l i t y which r e l y on t h e n o t i o n o f common t r a i t i s : 1) t o d e t e r m i n e which, and how many t r a i t s a r e s u f f i c -
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i e n t t o describe personality; 2 ) t o c o n s t r u c t methods o f measurement of v a l u e s o f
t r a i t s f o r p a r t i c u l a r persons. The second c o n d i t i o n i s c o n n e c t e d w i t h t h e f a c t t h a t i n r e s t r i c t i n g t h e c o n s i d e r a t i o n s t o common t r a i t s o n l y , i n o r d e r t o maximize t h e v a l i d i t y o f p r e d i c t i o n s and a l s o t o be a b l e t o compare p e r s o n s , i t i s e s s e n t i a l t o have p o s s i b l y most e x a c t and s u b t l e q u a n t i t a t i v e methods of measurement.
2.2.
G e n e r a l c h a r a c t e r i s t i c s of C a t t e l l ' s a p p r o a c h
I n most g e n e r a l t e r m s , t h e a i m of C a t t e l l was t o make t h e d e s c r i p t i v e t h e o r y of p e r s o n a l i t y p r e c i s e and obj e c t i v e . P o s t u l a t i n g t h e e x i s t e n c e o f common t r a i t s , he d e c i d e d t o s e a r c h f o r a n e m p i r i c a l answer t o t h e q u e s t i o n about t h e number and c o n t e n t o f common t r a i t s . T h i s i s e q u i v a l e n t t o b u i l d i n g a taxonomy o f p e r s o n a l ity. It i s w o r t h t o s t r e s s h e r e t h a t t h e o r e t i c i a n s do n o t a g r e e as t o t h e number o f t r a i t s s u f f i c i e n t t o d e s c r i be p e r s o n a l i t y , n o r as t o t h e s t a b i l i t y and g e n e r a l i t y of t h e s e t r a i t s ( s e e e . g . F r e n c h 1953 o r C a t t e l l , Wagn e r and C a t t e l l 1970; t h e l a s t p a p e r c o n t a i n s a l s o C a t t e l l ' s remarks c o n c e r n i n g v a r i o u s p e r s o n a l i t y t h e o r i e s , e s p e c i a l l y t h a t o f Eysenck)
.
R e t u r n i n g t o C a t t e l l ' s o b j e c t i v e : i t was a c o n s t r u c t i o n o f a p r e c i s e , o b j e c t i v e , and i n d u c t i v e l y d e r i v e d from t h e d a t a taxonomy of p e r s o n a l i t y .
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D e s p i t e a p p a r e n t l y extreme e m p i r i c i s m , i . e . r e j e c t i o n o f any i n t r o d u c t o r y h y p o t h e s e s about t h e c o n t e n t and number of t r a i t s , i n o r d e r t o d e t e r m i n e t h e s e t of v a r i a b l e s t o b e measured, C a t t e l l had t o g i v e some v e r y g e n e r a l d e f i n i t i o n of p e r s o n a l i t y . He namely assumed t h a t p e r s o n a l i t y i s a l l , which a l l o w s t o p r e d i c t t h e b e h a v i o u r of a p e r s o n i n a g i v e n s i t u a t i o n ( C a t t e l l 1 9 5 0 ) . S i n c e any b e h a v i o u r may have p r o g n o s t i c v a l u e f o r some o t h e r b e h a v i o u r , t h e s e t of b e h a v i o u r s which d e t e r m i n e s p e r s o n a l i t y i s v e r y l a r g e -- it comprises s i m p l y a l l b e h a v i o u r s . To a p p l y some measurement proc e d u r e s , C a t t e l l had t o r e s t r i c t t h i s s e t . A s b r i e f l y mentioned i n t h e p r e c e d i n g s e c t i o n , i n t r y -
i n g t o c o v e r t h e s p h e r e of p e r s o n a l i t y i n a complete f a s h i o n , C a t t e l l s e l e c t e d t h r e e sources of information about t h e s u b j e c t s , which he c a l l e d t h e domains Q , L and T . The domain Q ( f r o m : Q u e s t i o n n a i r e d a t a ) was cons t r u c t e d on t h e assumption t h a t a l l e s s e n t i a l a s p e c t s of b e h a v i o u r a r e r e f l e c t e d i n t h e n a t u r a l l a n g u a g e . T h e Q domain, which s e r v e s as t h e main s o u r c e of i n f o r m a t i o n , was o b t a i n e d by c h o o s i n g t h e l i s t o f 1 7 1 adj e c t i v e s ( w i t h no synonyms) g i v i n g t h e complete s e t of terms r e l a t e d t o p e r s o n a l i t y . These a d j e c t i v e s were t h e n r e p r e s e n t e d as q u e s t i o n n a i r e items, l e a d i n g event u a l l y t o c o n s t r u c t i o n of C a t t e l l ' s p e r s o n a l i t y questionnaire. The domain L (from: L i f e d a t a ) c o n s i s t s of a s e r i e s o f e v a l u a t i o n s c a l e s , f o r a s s e s s i n g t h e behaviour of t h e s u b j e c t by j u d g e s . F i n a l l y , domain T ( f r o m : Test d a t a ) comprises a v e r y r i c h s e t of l a b o r a t o r y r t e e h n i q u e s , which c o u l d p o t e n t i a l l y s u p p l y i n f o r m a t i o n about t h e subject I s personality.
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For a d e s c r i p t i o n of t h e s e t e c h n i q u e s , s e e " O b j e c t i v e A n a l y t i c ( 0 - A ) P e r s o n a l i t y B a t t e r y " ( C a t t e l l 1957 , p p . 58-60, 225-275, and C a t t e l l and Warburton 1 9 6 7 ) . The o b j e c t i v e t e s t s a r e d e s i g n e d as s t a n d a r d i z e d s t i m u l u s s i t u a t i o n s . O b j e c t i v i t y of t h e s e t e s t s i s understood as o b j e c t i v i t y o f t h e d a t a , i . e . t h e i r independence f r D m t h e c o n s c i o u s or s u b c o n s c i o u s tendency t o t h e des i r e d r e s u l t s , and on t h e o t h e r hand -- as o b j e c t i v i t y of s c o r i n g t e c h n i q u e s , i . e . independence from t h e i n f l u e n c e of p e r s o n s who s c o r e , e v a l u a t e and a s s e s s t h e behaviour. Because o f a l a r g e number of v a r i a b l e s i n d a t a Q , L and T , i t was n e c e s s a r y t o r e s i g n from t h e c l a s s i c a l t w o - f a c t o r experiment aimed a t d e t e r m i n i n g t h e f u n c t i o n a l r e l a t i o n s between v a r i a b l e s , and u s e a n o t h e r model o f e x p e r i m e n t , namely m u l t i v a r i a t e d e s i g n , where one o b s e r v e s a l a r g e number of v a r i a b l e s f o r a l a r g e group of s u b j e c t s . Under t h i s d e s i g n one c a n n o t g e t informa t i o n about t h e f u n c t i o n a l r e l a t i o n s between v a r i a b l e s , and one g e t s o n l y t h e measures of l i n e a r r e l a t i o n s h i p s between v a r i a b l e s , a s e x p r e s s e d b y c o r r e l a t i o n coefficients. T h i s l o s s o f i n f o r m a t i o n may be q u i t e s e v e r e . F i r s t l y , i f t h e r e e x i s t s a c a u s a l r e l a t i o n between v a r i a b l e s ,
i t i s n o t r e c o v e r a b l e from t h e d a t a o f m u l t i v a r i a t e e x p e r i m e n t . A t b e s t , one may hope t o be a b l e t o deduce t h e e x i s t e n c e of s u c h a r e l a t i o n s h i p i n d i r e c t l y . Sec o n d l y , i f t h e r e l a t i o n s h i p s between t h e v a r i a b l e s a r e n o n l i n e a r , t h e y may be m i s r e p r e s e n t e d by c o r r e l a t i o n coefficients. Moreover, t h e use o f c o r r e l a t i o n c o e f f i c i e n t s as t h e
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b a s i s f o r i n f e r e n c e r e q u i r e s t h a t t h e y b e known e x a c t l y . T h i s , i n t u r n , i m p l i e s t h e u s e o f l a r g e and r e p r e -
s e n t a t i v e s a m p l e s , s e l e c t e d a t random from a g i v e n pop u l a t i o n . These d i s a d v a n t a g e s o f m u l t i v a r i a t e d e s i g n a r e compensated b y t h e p o s s i b i l i t y o f s i m u l t a n e o u s est i m a t i o n of a l a r g e number of p a r a m e t e r s . The r e s u l t s o f a m u l t i v a r i a t e a n a l y s i s a r e t y p i c a l l y p r e s e n t e d i n form o f a c o r r e l a t i o n m a t r i x . The e l e m e n t s of t h e m a t r i x may b e c o r r e l a t i o n s between v a r i a b l e s o b t a i n e d by u s l n g l a r g e number of t o o l s for a g i v e n group. T h i s i s t h e c o - c a l l e d t e c h n i q u e R , l e a d i n g t o common f a c t o r s i n t h e group. Another p o s s i b i l i t y i s t o t a k e c o r r e l a t i o n c o e f f i c i e n t s between measurements of t h e same p e r s o n a t d i f f e r e n t moments. T h i s i s t h e P technique, suggested by C a t t e l l , t o e x t r a c t t h e f a c t o r s s p e c i f i c f o r a given person.
One s h o u l d m e n t i o n , however, t h a t a p o s s i b l e o b j e c t i o n a g a i n s t h e use o f t e c h n i q u e P i s as f o l l o w s . I f t h i s t e c h n i q u e i s t o l e a d us t o e x t r a c t i o n of i n d i v i d u a l t r a i t s o f t h e s u b j e c t , i t i s n e c e s s a r y to assume t h a t t h e v a l u e s o f t h e t r a i t change f a s t enough, s o t h a t t h e c o v a r i a t i o n p a t t e r n s c o u l d a p p e a r as f a c t o r s . But i n s u c h a c a s e , a t l e a s t some o f t h e numbers p u b l i s h e d as r e l i a b i l i t i e s o f t h e t o o l s o f measurement o f t h e t r a i t s i n q u e s t i o n are n o t r e l i a b i l i t i e s , s i n c e what was t r e a t e d as e r r o r of measurement c o n t a i n s a compon e n t connected w i t h v a r i a b i l i t y of t h e t r a i t . A s a l r e a d y remarked, t h e main t e c h n i q u e used by C a t t e l l
i s f a c t o r a n a l y s i s : C a t t e l l t r e a t s t h e e x t r a c t e d fact o r s as a p p r o x i m a t i o n s t o p e r s o n a l i t y t r a i t s . A s a r e s u l t of f a c t o r a n a l y s i s , C a t t e l l d i s c o v e r e d a
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c e r t a i n number o f common f a c t o r s , which he c a l l e d s o u r c e t r a i t s . The 1 6 most u n i v e r s a l and most i m p o r t a n t among them were t h e n chosen, and a measurement t o o l w a s c o n s t r u c t e d f o r them (The S i x t e e n P e r s o n a l i t y F a c t o r Q u e s t i o n n a i r e , o r s h o r t l y 1 6 PF, p u b l i s h e d f i r s t i n
1 9 4 9 , and t h e n m o d i f i e d s e v e r a l times i n s u b s e q u e n t e d i t i o n s ) . According t o C a t t e l l , t h e s e 1 6 f a c t o r s cons t i t u t e t h e b a s i c taxonomical u n i t s of p e r s o n a l i t y . Thus, e a c h p e r s o n may be d e s c r i b e d and compared w i t h o t h e r s i n terms o f t h e 1 6 f a c t o r s , and t h e knowledge of t h e v a l u e s o f h i s f a c t o r s a l l o w s t h e p r e d i c t i o n and i n t e r p r e t a t i o n of h i s behaviour. I n c i d e n t a l l y , a f t e r 2 0 y e a r s of r e s e a r c h , t h e number of f a c t o r s was i n c r e a s e d t o 23, p l u s some p a t h o l o g i c a l d i m e n s i o n s ( s e e C a t t e l l , Eber and Tatsuoka 1 9 7 0 , pp. 258-259). P e r f o r m i n g f a c t o r a n a l y s i s on t h e m a t r i x o f i n t e r c o r r e l a t i o n s between t h e b a s i c 1 6 f a c t o r s , C a t t e l l d i s covered more g e n e r a l second-order f a c t o r s , t h e most i m p o r t a n t among them b e i n g t h e f a c t o r o f i n t r o v e r s i o n e x t r a v e r s i o n , and a n x i e t y - i n t e g r a t i o n . D e r i v a t i o n of f u n c t i o n a l u n i t y o f f a c t o r s of t h e second o r d e r , w i t h t h e p o s s i b i l i t y of s e p a r a t i n g them i n t o d e e p e r r e a c h i n g b a s i c f a c t o r s i s -- a c c o r d i n g t o C a t t e l l -- one o f h i s most i m p o r t a n t a c h i e v e m e n t s .
2 . 3 , The b a s i c h y p o t h e s i s o f C a t t e l l S t a r t i n g from t h e assumption t h a t f a c t o r a n a l y s i s l e a d s t o e x t r a c t i o n of t h e “ t r u e f a c t o r s t r u c t u r e ” of t h e phenomenon under s t u d y e x p l a i n i n g t h e o b s e r v e d r e l a t -
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i o n s h i p s between t h e v a r i a b l e s , C a t t e l l f o r m u l a t e d t h e f o l l o w i n g h y p o t h e s i s : s i n c e t h e v a r i a b l e s i n each of t h e t h r e e d i s t i n g u i s h e d domains ( Q , L and T ) c o n c e r n t h e same o b j e c t , namely p e r s o n a l i t y , t h e r e s u l t s o f f a c t o r a n a l y s i s performed on t h e s e t h r e e c l a s s e s o f v a r i a b l e s s h o u l d y i e l d t h e same f a c t o r s t r u c t u r e o f p e r s o n a l i t y . F o r i n s t a n c e ( s e e C a t t e l l 1957, p . 3 0 2 ) s c h i z o t h y m i a ought t o m a n i f e s t s i t s e l f i n o b s e r v a t i o n a l d a t a (L) as " k e e p i n g away from p e o p l e l ' a n d a c e r t a i n r i g i d i t y o f b e h a v i o u r ; i n s e l f - o b s e r v a t i o n d a t a (Q) -- as tendency t o l o n e l i n e s s and p r o f e s s i o n s which do n o t r e q u i r e c o n t a c t s w i t h p e o p l e ; f i n a l l y , i n o b j e c t i v e t e s t s (T)
--
as " p e r c e p t i o n of t h r e a t " , d i f f i c u l t y i n s e l f - d e s c r i p t i o n and l a c k o f s e l f - a s s u r a n c e i n new s i t u a t i o n s . I f t h e same f a c t o r s t r u c t u r e s c o u l d be o b t a i n e d from a l l t h r e e domains, i t would mean t h a t t h e assumed model o f p e r s o n a l i t y i s s u f f i c i e n t l y g e n e r a l and r e f l e c t s t h e " t r u e " n a t u r e o f t h e phenomenon. The l a t t e r -though viewed from d i f f e r e n t a n g l e s -- a p p e a r s i n v a r i a n t and l a r g e l y i n d e p e n d e n t from t h e i n i t i a l d a t a . I n t h i s s e n s e , o b s e r v a t i o n , s e l f - o b s e r v a t i o n and obj e c t i v e t e s t s would be simply t h r e e d i f f e r e n t w a y s o f c o l l e c t i n g i n f o r m a t i o n about t h e same p e r s o n a l i t y t r a i t s , u n d e r l y i n g t h e whole b e h a v i o u r . The v e r i f i c a t i o n o f t h i s h y p o t h e s i s was n o t s u c c e s s f u l ; i n t h e s e q u e l , t h e arguments w i l l b e p r e s e n t e d f o r t h e a s s e r t i o n t h a t t h i s p a r t i a l f a i l u r e of Cattell i s one o f t h e most i n t e r e s t i n g and v a l u a b l e c o n t r i b u t i o n s of h i s r e s e a r c h s t r a t e g y . 2.4. 2.4.1.
Some problems o f r e s e a r c h s t r a t e g y o f C a t t e l l P e r s o n a l i t y s t r u c t u r e from &-data. A s a l r e a d y
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mentioned, f a c t o r a n a l y s i s i s t h e main t o o l i n C a t t e l l ' s t h e o r y ; c o n s e q u e n t l y , a l l l i m i t a t i o n s and m e r i t s o f f a c t t o r analysis discussed i n t h e preceding sections proj e c t on t h e c o g n i t i v e c o n t e n t o f C a t t e l l ' s s y s t e m . A s a r e s u l t of f a c t o r a n a l y s e s o f e x p e r i m e n t a l d a t a ,
16 f a c t o r s were d i s c o v e r e d (as n o t e d , a t p r e s e n t t h e t h e o r y c o n t a i n s 2 3 f a c t o r s ; t h e d i s c u s s i o n below w i l l c o n c e r n , however, t h e c a s e o f 1 6 f a c t o r s measured by
16 PF). Most o f t h e f a c t o r s were d i s c o v e r e d i n 4 d a t a . According t o C a t t e l l , t h e f a c t o r s measured b y 1 6 PF sat i s f y t h e c r i t e r i o n of i n v a r i a n c e when t h e q u e s t i o n n a i -
r e i s a p p l i e d i n c r o s s - c u l t u r a l comparisons. Regarding v a l i d i t y , t h e 16 PF has been checked w i t h r e s p e c t t o c l i n i c a l , e d u c a t i o n a , i n d u s t r i a l and s o c i a l r e a l l i f e c r i t e r i a ( s e e e . g . C a t t e l l , Eber and T a t s u o k a 1 9 7 0 ) . T h i s s e t o f v a l i d a t i o n s s e r v e s n o t o n l y p r a c t i c a l purp o s e , b u t a l s o p r o v i d e s C a t t e l l w i t h an i n d i r e c t v e r i f i c a t i o n of h i s hypotheses r e g a r d i n g t h e i n t e r p r e t a t i o n o f f a c t o r s ( i t s h o u l d be remarked, t h a t C a t t e l l d i d n o t v e r i f y h y p o t h e s e s about f a c t o r s by c o n t r o l l e d b i v a r i a t e e x p e r i m e n t s , a s o r i g i n a l l y planned ( s e e C a t t e l l 1957), r e l y i n g m o s t l y on c h e c k i n g t h e v a l i d i t i e s of 1 6 PF. However, d e s p i t e a l l t h e above m e r i t s o f t h e 16 PF, nam e l y i n v a r i a n c e and v a l i d i t y , t h e y c a n n o t be t a k e n a s
a complete p r o o f of C a t t e l l ' s c l a i m (see C a t t e l l , Eber and T a t s u o k a 1 9 7 0 , p . 1 3 ) t h a t he d i s c o v e r e d t h e "uniquel y d e f i n a b l e " s o u r c e t r a i t s ; and even l e s s s o c a n t h e y p r o v i d e a n argument f o r c a u s a l i n t e r p r e t a t i o n o f f a c t o r s . Indeed, o b j e c t i o n s a g a i n s t f a c t o r a n a l y s i s , notably s u b j e c t i v e b i a s and r e l a t i v i t y o f t h e r e s u l t s t o t h e
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i n i t i a l s e t of v a r i a b l e s , as w e l l as u n c l e a r ( f r o m t h e measurement t h e o r y v i e w p o i n t ) s t a t u s of t h e s e v a r i a b l e s , a p p l y t o t h e C a t t e l l ' s 1 6 f a c t o r s . Even assuming t h a t t h e v a r i a b l e s chosen as t h e " p e r s o n a l i t y s p h e r e " i n f a c t exhaust a l l t h e r e l e v a n t a s p e c t s of p e r s o n a l i t y and have unambiguous s t a t u s , i t i s s t i l l c o n c e i v a b l e t h a t one c o u l d f i n d 1 6 , or some o t h e r number, of f a c t o r s ( w i t h i n t h e same p e r s o n a l i t y s p h e r e ) p r o v i d i n g e q u a l l y good, o r even b e t t e r , d e s c r i p t i o n o f p e r s o n a l i t y s a t i s f y i n g t h e c r i t e r i a of i n v a r i a n c e , v a l i d i t y and e x p l a n a t o r y i n t e r p r e t a t i o n . Thus, t h e u n i q u e n e s s c l a i m cannot be def'ended.
C a t t e l l a s s e r t s ( s e e C a t t e l l , Wagner and C a t t e l l 1 9 7 0 , or C a t t e l l , Eber and T a t s u o k a 19701, t h a t t h e p e r s o n a l i t y s t r u c t u r e as measured by 1 6 PF i s b e t t e r t h a n t h e a l t e r n a t i v e d e s c r i p t i o n s g i v e n b y , s a y , Eysenck o r G u i l f o r d and Zimmermann. But even i f t h e 16 PF s h o u l d g i v e a b e t t e r d e s c r i p t i o n of p e r s o n a l i t y t h a n any of t h e p r e s e n t l y e x i s t i n g q u e s t i o n n a i r e s , i t would s t i l l p r o v i d e no l o g i c a l argument f o r t h e u n i q u e n e s s of Cattell's structure. Now, i f t h e s t r u c t u r e i s n o t u n i q u e , one cannot a t t a c h t o i t any " a b s o l u t e " meaning. Thus, a l l t h a t i s shown by i n v a r i a n c e and v a l i d i t y i s t h a t i n d i f f e p e n t popul a t i o n s t h e r e e x i s t similar p a t t e r n s i n answering t h e same s e t o f i t e m s , and t h a t u s i n g t h i s f a c t , one can c o n s t r u c t a s e t of v a r i a b l e s s e r v i n g f a i r l y w e l l f o r p r e d i c t i n g b e h a v i o u r . T h i s shows t h e p r a g m a t i c v a l u e of t h e 1 6 P F . Moreover, one ought t o p o i n t o u t t h a t C a t t e l l ' s f a c t o r s p r o v i d e a r i c h and i n t e r e s t i n g desc r i p t i o n o f p e r s o n a l i t y , a r e c o n s i s t e n t w i t h psycholog i c a l knowledge and i n t u i t i o n , and t h e i r i n t e r p r e t a t i o n
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r e v e a l s g r e a t p s y c h o l o g i c a l i n s i g h t and e r u d i t i o n of t h e i r author .
2.4.2.
V e r i f i c a t i o n of t h e b a s i c h y p o t h e s i s of C a t t e l l .
A s a l r e a d y mentioned, C a t t e l l d i d n o t succeed i n a com-
plete
v e r i f i c a t i o n o f h i s fundamental h y p o t h e s i s o f
i d e n t i t y of s t r u c t u r e s i n t h e t h r e e s o u r c e s of d a t a : o b s e r v a t i o n s ( L ) , s e l f - o b s e r v a t i o n s (Q), and o b j e c t i v e tests (T). The f a i l u r e o f t h e v e r i f i c a t i o n o f t h i s h y p o t h e s i s may be r e g a r d e d as one of t h e v e r y v a l u a b l e and i n t e r e s t i n g r e s u l t s of C a t t e l l . I n t u i t i v e l y , C a t t e l l ' s h y p o t h e s i s may a t f i r s t g l a n c e a p p e a r q u i t e p l a u s i b l e : 5.t a s s e r t s t h a t i d e n t i c a l f a c t o r s t r u c t u r e s can be found i n a l l t h r e e c l a s s e s o f p e r s o n a l i t y v a r i a b l e s . I n o t h e r words, t h e phenomenon, though viewed from d i f f e r e n t s i d e s , remains e s s e n t i a l l y t h e same. I d e a l l y , t h e n , i t s h o u l d be p o s s i b l e t o c o n s t r u c t t h r e e t o o l s of measurement of t h e same common f a c t o r s t r u c t u r e i n d a t a Q , L and T ; a q u e s t i o n n a i r e , a method of s c o r i n g o f s u i t a b l e o b s e r v a t i o n a l d a t a , and a method of s c o r i n g t h e r e s u l t s o f o b j e c t i v e t e s t s . These t o o l s o f measurement, when a p p l i e d t o a g i v e n i n d i v i d u a l , s h o u l d p r o v i d e t h r e e i d e n t i c a l ( w i t h i n t h e e r r o r s of measurement) " p e r s o n a l i t y p r o f i l e s " . I f t h i s were t h e c a s e , t h e n whatever p r e d i c t i o n s c o u l d be made on t h e ground of a p e r s o n a l i t y p r o f i l e d e r i v e d from one domain of d a t a , t h e y c o u l d a l s o be made on t h e ground of prof i l e s d e r i v e d f r o m t h e o t h e r two domains;
moreover,
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as ( i d e a l l y ) t h e s e p r o f i l e s ought t o be i d e n t i c a l , i n
p r a c t i c a l s i t u a t i o n s one s h o u l d be a b l e t o p r e d i c t a p r o f i l e o f a g i v e n p e r s o n i n one domain, knowing h i s p r o f i l e i n a n o t h e r domain. I n s h o r t , t h e r e s u l t s of e m p i r i c a l v e r i f i c a t i o n o f C a t t e l l ' s h y p o t h e s i s were as f o l l o w s ( s e e C a t t e l l 1 9 5 7 , p p . 324-326; a l s o S c h e i e r and C a t t e l l 1958, and S c h a i e (1962) : (1) F o r e a c h c l a s s o f p e r s o n a l i t y v a r i a b l e s , one can
f i n d f a c t o r s which have no c o r r e s p o n d i n g f a c t o r s i n other classes; ( 2 ) sometimes t h e f a c t o r s from d i f f e r e n t c l a s s e s of
variables d i f f e r i n their generality level; for instanc e , some second o r d e r f a c t o r s i n o b j e c t i v e t e s t d a t a (T) c o r r e s p o n d t o t h e f i r s t o r d e r f a c t o r s i n t h e ques t i o n n a i r e d a t a (9). ( 3 ) regarding i d e n t i f i c a t i o n , Cattell claims t h a t i t i s p o s s i b l e t o i d e n t i f y 1 2 f a c t o r s from two o r t h r e e c l a s s e s o f p e r s o n a l i t y v a r i a b l e s ; as a r u l e , f a c t o r s from o b s e r v a t i o n a l and q u e s t i o n n a i r e d a t a were e a s i e r t o match one w i t h a n o t h e r t h a n f a c t o r s from t h e o b j e c t i v e t e s t data.
Moreover, t h e r e e x i s t e x t e n s i v e d a t a c o n c e r n i n g t h e v a l i d i t i e s of 16 PF w i t h r e s p e c t t o v a r i o u s s o c i a l r e a l - l i f e c r i t e r i a . While p r o v i d i n g arguments for t h e e x i s t e n c e o f r e l a t i o n s between domains L and Q , t h e y n e v e r t h e l e s s l a c k t h e f o r c e of a d i r e c t v e r i f i c a t i o n . Theoretically, t h e following conditions constitute t h e
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necessary requirements f o r a p o s i t i v e v e r i f i c a t i o n of t h e h y p o t h e s i s o f i n d i f f e r e n c e o f media:
(a) t h e f a c t o r analyses of t h e t h r e e sets of data should l e a d t o e x t r a c t i o n of f a c t o r s t r u c t u r e s which c o u l d be r o t a t e d t o p o s i t i o n s a l l o w i n g i d e n t i f i c a t i o n o f t h e same f a c t o r s i n a l l t h r e e d o m a i n s . The s c o r e s i n t h e same f a c t o r i n a l l t h r e e domains s h o u l d be h i g h l y c o r -
related. Given t h a t t h e above r e q u i r e m e n t i s s a t i s f i e d , i t i s a l s o necessary t h a t : ( b ) when t h e f a c t o r a n a l y s i s i s p e r f o r m e d on t h e j o i n t m a t r i x o f i n t e r c o r r e l a t i , o n s f o r two or t h r e e d o m a i n s , i t o u g h t t o b e p o s s i b l e to e x t r a c t a s t r u c t u r e w i t h f a c t o r s c o m p r i s i n g a l l v a r i a b l e s , which a p p e a r w i t h h i g h l o a d i n g s i n t h e same f a c t o r from a l l d o m a i n s . I n a d d i t i o n , i f t h e f a c t o r s t r u c t u r e s a r e to b e i d e n t i c a l , one s h o u l d a l s o have: ( c ) e v e r y f a c t o r i n a g i v e n domain s h o u l d m a t c h w i t h e x a c t l y o n e f a c t o r i n e a c h of t h e o t h e r two d o m a i n s . I n o t h e r w o r d s , l e t u s s u p p o s e t h a t by s e p a r a t e f a c t o r a n a l y s e s o f Q , L and T d a t a , one o b t a i n s f a c t o r s , p re -
...
sumed t o b e i d e n t i c a l , m e a s u r e d by v a r i a b l e s q 1 , q 2 , i n Q d a t a , b y v a r i a b l e s 11, 12, i n L d a t a , a n d by
...
...
variables tl, t2, i n T d a t a . It o u g h t t h e n to b e p o s s i b l e t o c o n s t r u c t l i n e a r combinations of v a r i a b l e s q l , q2,..., 11' 12'". and t l ' t 2 , w h i c h would b e
...,
h i g h l y c o r r e l a t e d . M o r e o v e r , when t h e j o i n t d a t a m a t r i x
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i t ought t o be p o s s i b l e t o do i t i n s u c h a way as t o i n c l u d e a l l v a r i a b l e s q 1 , q 2 , . ,11’12 ’ ,tl,t2, i n one f a c t o r .
i s factorized,
...
...
..
The e m p i r i c a l r e s u l t s do n o t s a t i s f y t h e s e r e q u i r e m e n t s : a n a l y s i n g t h e m a t r i c e s o f i n t e r c o r r e l a t i o n s between var i a b l e s i n d i f f e r e n t domains, one n o t e s t h a t t h e c o r r e l a t i o n s between v a r i a b l e s m e a s u r i n g t h e same f a c t o r a r e s u r p r i s i n g l y low ( h i g h e s t o f t h e o r d e r 0 . 2 0 , o f t e n as low as 0 . 0 2 ; s e e C a t t e l l 1957, Appendix). I n t e r e s t i n g l y enough, such v a r i a b l e s w i t h almost z e r o c o r r e l a t i o n appear o f t e n w i t h h i g h e s t l o a d i n g s i n a given f a c t o r , hence p r o v i d e i t s supposedly b e s t measure. I n view of t h e above, i t would a p p e a r t h a t t h e hypothes i s cannot be m a i n t a i n e d and has t o be m o d i f i e d : s i n c e t h e v a r i a b l e s i n presumably m a t c h i n g f a c t o r s have v e r y low i n t e r c o r r e l a t i o n s a c r o s s t h e domains, i t i s c e r t a i n l y i m p o s s i b l e f o r t h e f a c t o r s c o r e s on t h e same f a c t o r i n d i f f e r e n t domains t o be h i g h l y c o r r e l a t e d . T h i s make s t h e p r e d i c t i o n from one domain t o a n o t h e r p r a c t i c a l l y impossible. I n addition, t h e f a c t o r structures i n d i f f e r e n t domains a r e n o t i d e n t i c a l , as e a c h domain has some f a c t o r s s p e c i f i c f o r t h i s domain o n l y . Neverthel e s s , C a t t e l l ( s e e C a t t e l l , E b e r and T a t s u o k a 1970, p . 7 ) c l a i m s t h a t s o u r c e t r a i t s c a n be measured n o t o n l y by a q u e s t i o n n a i r e , b u t a l s o by o b j e c t i v e l a b o r a t o r y kind o f t e s t s . I n view of t h e above f i n d i n g s , c o n t r a r y t o C ’ a t t e l l ’ s c l a i m , i t seems t h a t t h e b e s t one c o u l d s a y i s t h a t t h e f a c t o r s t r u c t u r e s i n t h e t h r e e domains r e v e a l a c e r t a i n d e g r e e of s i m i l a r i t y .
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2.4.3.
C a t t e l l ' s e x p l a n a t i o n of d i f f i c u l t i e s i n v e r i f y -
ing h i s hypothesis.
I n s p i t e o f t h e above d e s c r i b e d
r e s u l t s o f e x p e r i m e n t s , C a t t e l l m a i n t a i n s t h a t h i s hypot h e s i s i s t r u e ; he e x p l a i n s t h e p a r t i a l f a i l u r e o f ver i f i c a t i o n by t h e p r e s e n c e o f d i s t u r b i n g f a c t o r s , and p r e d i c t s t h a t i n t h e f u t u r e one w i l l be a b l e t o c o n s t r u c t s p e c i a l e x p e r i m e n t s and t e c h n i q u e s which would e i t h e r e l i m i n a t e t h e d i s t u r b i n g f a c t o r s , or a l l o w one t o evaluate t h e i r influence, leading ultimately t o tools which would y i e l d t h e p o s i t i v e v e r i f i c a t i o n o f h i s hypothesis. Among f o u r s o u r c e s of d i s t u r b a n c e mentioned by C a t t e l l ( s e e C a t t e l l 1 9 6 1 ; C a t t e l l and Digman 1 9 6 4 ) t h e most i n t e r e s t i n g seem t o be i n s t r u m e n t a l f a c t o r s , and d i f f e r ences i n "density" of representation o f v a r i a b l e s ( t h e q u o t a t i o n mark i s used by C a t t e l l t o s t r e s s t h e i n t u i t i v e meaning of t h e term)- C a t t e l l c l a i m s t h a t i n s t rumental f a c t o r s play a c r u c i a l r o l e i n d i s t u r b i n g t h e i d e n t i f i c a t i o n of f a c t o r s from t h e Q , L and T d a t a . I n most g e n e r a l terms, t h e s e f a c t o r s a r e c o n n e c t e d w i t h t h e form o f t o o l , i . e . t h e y are s p e c i f i c f o r t h e t o o l
as a whole. I n t u i t i v e l y , due t o a f o r m a l s i m i l a r i t y o f v a r i a b l e s c o n s t i t u t i n g a g i v e n tool, t h e i n s t r u m e n t a l f a c t o r d i s t o r t s u n i f o r m l y a l l p e r s o n a l i t y f a c t o r s meas u r e d by t h i s t o o l . Thus, f o r e v e r y f o r m a l l y homogeneous c l a s s of p e r s o n a l i t y v a r i a b l e s , one c a n e x p e c t t h e e x i s t e n c e of i n s t r u m e n t a l f a c t o r s c h a r a c t e r i s t i c f o r t h i s c l a s s o f v a r i a b l e s . Such f a c t o r s can be d i s c o v e r e d by comparing t h e f a c t o r s t r u c t u r e s from f o r m a l l y d i f f e r e n t t o o l s w i t h t h e same p s y c h o l o g i c a l c o n t e n t . Regarding t h e " d e n s i t y " of r e p r e s e n t a t i o n , C a t t e l l maint a i n s t h a t t o e n s u r e t h e adequacy o f measurement o f t h e
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same t r a i t i n d i f f e r e n t c l a s s e s o f p e r s o n a l i t y v a r i a b l e s , t h e measures ought t o b e based on d i f f e r e n t numb e r s of v a r i a b l e s . I n d e e d , i t i s known t h a t q u e s t i o n n a i r e d a t a have a h i g h e r l e v e l o f g e n e r a l i t y ( t h e subj e c t answers how he u s u a l l y b e h a v e s , i n a b s t r a c t i o n from a s p e c i f i c s i t u a t i o n ) t h a n t h e L d a t a ( t h e o b s e r v e r r e c o r d s what he s e e s ) , and i n p a r t i c u l a r t h a n t h e T data ( t h e t e s t r e f l e c t s t h e behaviour a t a given moment )
.
C o n s e q u e n t l y , a l r e a d y a r e l a t i v e l y small number o f variables i n Q i s s u f f i c i e n t t o d e f i n e a given t r a i t , w h i l e i n o r d e r t o d e f i n e i t i n terms o f v a r i a b l e s i n L , or e s p e c i a l l y i n T , one needs t o use more v a r i a b l e s . I n C a t t e l l ’ s words, t h e v a r i a b l e s i n Q have h i g h e r “dens i t y ” t h a n t h e v a r i a b l e s i n L or T . According t o C a t t e l l t h i s e x p l a i n s why f o r matching f a c t o r s i t was n e c e s s a r y t o i d e n t i f y t h e second o r d e r f a c t o r s from t h e T d a t a w i t h t h e f i r s t o r d e r f a c t o r s from t h e Q d a t a : t o a t t a i n t h e l e v e l o f g e n e r a l i t y of t h e l a t t e r , i t was n e c e s s a r y t o go t o a h i g h e r l e v e l o f a b s t r a c t i o n w i t h t h e t e s t d a t a , s u b p l i e d by t h e second o r d e r f a c t o r a n a l y s i s .
2 . 5 . D i s c u s s i o n o f h y p o t h e s i s o f i n d i f f e r e n c e of media The above e x p l a n a t i o n o f C a t t e l l a p p e a r s q u i t e convinci n g and i s i n l i n e w i t h t h e contemporary t e n d e n c i e s towards s t u d y i n g t h e f a c t o r s d i s t u r b i n g t h e measurements by p s y c h o m e t r i c t o o l s . However, by f a r t h e most obvious e xpl anation of t h e p a r t i a l f a i l u r e i n v e r i f i c a t i o n o f t h e h y p o t h e s i s i s simply t h a t t h e h y p o t h e s i s
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may b e f a l s e , a t l e a s t r e g a r d i n g t h o s e o f i t s a s s e r t -
i o n s which f a i l e d t o be v e r i f i e d ( e . g . about t h e i d e n t i t y of s t r u c t u r e s ; t h i s removed from t h e h y p o t h e s i s , and i t i s t h a t t h e s t r u c t u r e s a r e not i d e n t i c a l ,
the assertion r e q u i r e m e n t was now r e c o g n i z e d s i n c e e a c h con-
t a i n s f a c t o r s s p e c i f i c t o t h e g i v e n domain, s u c h as f a c t o r s Q1
- Q4).
The p o i n t o f view t h a t t h e h y p o t h e s i s may be p a r t i a l l y f a l s e i s s t r o n g l y s u p p o r t e d when one looks a t t h e matr i c e s o f i n t e r c o r r e l a t i o n s f o r v a r i a b l e s from d i f f e r e n t domains, mentioned i n t h e p r e c e d i n g s e c t i o n . I t a p p e a r s t h a t t h e covariational p a t t e r n s a r e not s u f f i c i e n t l y pronounced t o y i e l d p s y c h o l o g i c a l l y e q u i v a l e n t and h i g h l y c o r r e l a t e d f a c t o r s i n a l l t h r e e media.
C o n s e q u e n t l y , i f f a c t o r a n a l y s i s i s t r e a t e d p u r e l y as a s t a t i s t i c a l t e c h n i q u e l e a d i n g t o e x t r a c t i o n of i n t e r pretable covariational p a t t e r n s , then the hypothesis s h o u l d be l a r g e l y m o d i f i e d ( b y a s s e r t i n g merely t h e e x i s t e n c e o f some i n t e r r e l a t i o n s between v a r i a b l e s def i n a b l e i n two o r t h r e e d o m a i n s ) . I f t h i s h y p o t h e s i s i s t o be r e t a i n e d , t h e n n o t o n l y i t i s n e c e s s a r y t o p u t t h e blame f o r i n c o m p l e t e n e s s o f i t s v e r i f i c a t i o n on d i s t u r b i n g f a c t o r s , b u t one has t o t r e a t f a c t o r a n a l y s i s ( a s d o e s C a t t e l l ) as a much more powerful t o o l t h a n i t a c t u a l l y i s , namely as a t o o l which can r e a c h i n t o t h e c a u s a l s t r u c t u r e of t h e phenomenon, r e g a r d l e s s o f how d i s t o r t e d t h e c o v a r i a t i o n a l p a t t e r n may b e . The c r i t i q u e o f C a t t e l l ’ s e x p l a n a t i o n d o e s n o t , o f cou r s e , e x t e n d t o t h e c o n c e p t s which he used f o r i t . I t seems t h a t b o t h t h e i n s t r u m e n t a l f a c t o r s and t h e “dens i t y ” ( i . e . l e v e l of g e n e r a l i t y o f v a r i a b l e s ) a r e of
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p r i m a r y i m p o r t a n c e i n t h e t h e o r y o f m u l t i v a r i a t e experiments. F i n a l l y , i n connection w i t h t h e v e r i f i c a t i o n o f t h e h y p o t h e s i s , i t i s w o r t h s t r e s s i n g t h a t one can d e s i g n r e l a t i v e l y s i m p l e and s t r a i g h t f o r w a r d t e c h n i q u e s f o r a s s e s s i n g t h e e x t e n t t o which t h e h y p o t h e s i s may be r e t a i n e d . The idea i s t o u s e i n f o r m a t i o n t h e o r y and eval u a t e t h e amount o f i n f o r m a t i o n c a r r i e d by v a r i a b l e s i n one domain about t h e v a r i a b l e s i n o t h e r domains. Whatever p r e d i c t i o n s , l i n e a r o r n o t , a r e t o be made a c r o s s t h e domains, t h e i r p r e c i s i o n i s bounded above by t h e a p p r o p r i a t e amounts o f i n f o r m a t i o n . The u s e o f i n f o r m a t i o n t h e o r y has a n a d d i t i o n a l a d v a n t a g e o v e r t h e c o r r e l a t i o n methods, namely t h a t no assumptions are needed a b o u t t h e n a t u r e o f s c a l e s of measurement ( a l l measurements are t r e a t e d as i f t h e y were on nominal scale). Knowing t h e " i n f o r m a t i o n a l : s t r u c t u r e " o f t h e t h r e e domains, one c o u l d p r o c e e d w i t h b u i l d i n g a p p r o p r i a t e f u n c t i o n s o f t h e v a r i a b l e s ( l i n e a r or n o t ) s u p p l y i n g as much i n f o r m a t i o n about one a n o t h e r as p o s s i b l e ; t h e s e f u n c t i o n s would p l a y t h e r o l e o f f a c t o r s , p r o v i d ed a n i n t e r p r e t a t i o n i s a s s i g n e d t o them.
2.6.
G e n e r a l remarks about C a t t e l l ' s a p p r o a c h
Looking on C a t t e l l ' s r e s e a r c h program and h i s r e s u l t s from a somewhat more g e n e r a l p o i n t o f view, one may i n t e r p r e t i t as an a t t e m p t t o c o n s t r u c t a language f o r a domain o f psychology, namely p e r s o n a l i t y t h e o r y . The a n a l o g i e s between C a t t e l l ' s s y s t e m and l i n g u i s t i c con-
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c e p t s o r t h e o r i e s may be drawn i n s e v e r a l d i f f e r e n t w a y s . F i r s t of a l l , l o o k i n g r a t h e r s u p e r f i c i a l l y o n l y on t h e methods a p p l i e d , one may n o t e t h a t C a t t e l l ' s program f o r s t u d y i n g t h e s t r u c t u r e o f p e r s o n a l i t y i s i n many r e s p e c t s a n a l o g o u s t o t h e program o f Osgood ( s e e Osgood,
S u c i and Tannenbaum 1 9 5 7 ) f o r s t u d y i n g t h e
s t r u c t u r e o f meaning. I n some s i m p l i f i c a t i o n , one may
s a y t h a t i n b o t h c a s e s t h e dimensions o f t h e phenomenon under i n v e s t i g a t i o n were o b t a i n e d by i n t e r p r e t i n g s u i t able c o v a r i a t i o n a l p a t t e r n s i n s e l e c t e d s e t s of a d j e c t i v e s . The c o v a r i a t i o n s were o b t a i n e d by e v a l u a t i n g , i n t e r m s of t h e c o n s i d e r e d a d j e c t i v e s , t h e samples o f obj e c t s s e l e c t e d from a p p r o p r i a t e p o p u l a t i o n s ( t h e obj e c t s b e i n g i n d i v i d u a l s i n C a t t e l l ' s a p p r o a c h , or nouns i n O s g o o d ' s ) . N a t u r a l l y , t h e s e t s o f a d j e c t i v e s were d i f f e r e n t ; a l s o d i f f e r e n t were t h e d e t a i l s o f c o n s t r u c t i o n o f t h e t e c h n i q u e s o f e v a l u a t i n g an o b j e c t w i t h resp e c t t o a given a d j e c t i v e . A n a l y s i n g t h e s t r u c t u r a l p r o p e r t i e s of t e n d e n c i e s o f some a d j e c t i v e s t o a p p e a r t o g e t h e r when t h e y a r e used t o d e s c r i b e i n d i v i d u a l s , one a r r i v e s ( b y f a c t o r a n a l y s i s ) a t t h e s t r u c t u r e of p e r s o n a l i t y . On t h e o t h e r hand, a n a l y s i n g t h e s t r u c t u r a l p r o p e r t i e s o f t h e t e n d e n c i e s of some a d j e c t i v e s t o a p p e a r t o g e t h e r when t h e y are used t o d e s c r i b e nouns (more p r e c i s e l y , p r o p e r t i e s o f o b j e c t s d e s i g n a t e d by t h e s e n o u n s ) , one a r r i v e s , a l s o by f a c t o r a n a l y s i s , a t t h e s t r u c t u r e o f meaning. The g e o m e t r i c d i s t a n c e between two words i n t h e o b t a i n ed s p a c e i s t h e n r e l a t e d t o t h e a c c e p t a b i l i t y o f a comb i n a t i o n o f t h e s e two words. T h i s analogy between C a t t e l l ' s and Osgood's r e s e a r c h programs c a n n o t , however, be pursued t o o far. C a t t e l l
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c o n t i n u e s t h e r e s e a r c h towards checking t h e i n v a r i a n c e o f h i s s t r u c t u r e and i n v e s t i g a t i n g t h e v a l i d i t i e s w i t h
r e s p e c t t o v a r i o u s c r i t e r i a on t h e one hand, and on t h e o t h e r , t oward s t e st i n g t h e hypo t he s i s o f i n d i f f e r e n c e o f media. Osgoodls r e s e a r c h seems t o be d e v e l o p i n g i n a d i r e c t i o n away from f a c t o r a n a l y t i c models (which p r o v i d e s o l u t i o n s n o t v e r y a t t r a c t i v e from t h e l i n g u i s t i c v i e w p o i n t ) t o w a r d s h i e r a r c h i c a l taxonomies under which e a c h noun i s c a t e g o r i z e d w i t h r e s p e c t t o s u c c e s s i v e dependent c a t e g o r i e s . T h i s g i v e s a b e t t e r a c c e s s t o s p e c i f i c a t i o n of t h e l i n g u i s t i c r u l e s f o r s e l e c t i o n ( t h e t e c h n i q u e s i n v o l v e c a t e g o r i z a t i o n o f nouns by v e r b s , and n o t by a d j e c t i v e s ; s e e Osgood 1970, and a l s o Noordman and L e v e l t 1970). Now, l o o k i n g n o t o n l y f o r a n a l o g i e s i n methods, b u t a l s o a t t h e o b t a i n e d r e s u l t s , C a t t e l l ' s a p p r o a c h may be i n t e r p r e t e d as a n a t t e m p t t o c o n s t r u c t a c e r t a i n l a n g u a g e , o r more p r e c i s e l y , a d i c t i o n a r y , whose t e r m s ( e n t r i e s ) a r e o b t a i n e d a c c o r d i n g t o t h e r u l e s determined by t h e c h o i c e o f program o f f a c t o r a n a l y s i s . T h e scheme o f a d e f i n i t i o n o f a term i s , r o u g h l y s p e a k i n g , g i v e n by a p r o p o s i t i o n a l f u n c t i o n o f t h e form " f a c t o r
x i s t h e c o n j u n c t i o n of a , b , c ,... ' I . More p r e c i s e l y , "N has t r a i t x" means "N i s a , and N i s b, and . . . ' I .
...
Here a , b , c , need n o t be words i n t h e n a t u r a l langua g e ; t h e y may be t a k e n as c e r t a i n b e h a v i o u r s ( t r e a t i n g a s e l f - e v a l u a t i o n on a q u e s t i o n n a i r e item a l s o a s a
...
form of b e h a v i o u r ) . I f i n p l a c e of a , b , c , one subs t i t u t e s t h e w o r d - v a r i a b l e s which s e r v e d as t h e b a s i s
for f a c t o r a n a l y s i s ,
one o b t a i n s a d e f i n i t i o n o f t h e
...
one subf i r s t order factor; i f i n place of a,b,c, s t i t u t e s t h e f i r s t o r d e r f a c t o r s , one o b t a i n s t h e de-
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f i n i t i o n o f a second o r d e r f a c t o r , e t c . A s a r e s u l t , t h e t e r m s o f t h e language s o c o n s t r u c t e d form a c e r t a i n h i e r a r c h y , i n which t o t e r m s on one l e v e l t h e r e c o r r e s pond s e t s o f terms on lower l e v e l s . T e c h n i c a l l y , a l l d e f i n i t i o n s of t e r m s i n t h i s language a r e unambiguous ( a t l e a s t to t h e e x t e n t o f Unambiguity of t h e i n i t i a l terms a , b , c , t a k e n as a f o u n d a t i o n of t h e c o n s t r u c t i o n of t h e language).
...
However, t o r e l a t e terms o f t h i s l a n g u a g e w i t h t h e p r e s e n t p s y c h o l o g i c a l knowledge and i n t u i t i o n , C a t t e l l o f t e n a s s i g n e d t o f a c t o r s ( i . e . s e t s of e l e m e n t s a , b , c , . . . ) names o f t r a i t s e x i s t i n g i n psychology. T h i s may o c c a s i o n a l l y l e a d t o a c e r t a i n a m b i g u i t y , due t o t h e p o s s i b l e d i f f e r e n c e s i n o p i n i o n as t o t h e c o r r e c t n e s s o f s p e c i f i c c h o i c e s . Thus, w h i l e a t r a n s l a t i o n o f t e r m s o b t a i n e d by C a t t e l l i n t o t h e e x i s t i n g l a n g u a g e of psychology i n c r e a s e s t h e M r e a d Z & b i l i t y o" f C a t t e l l I
s
t e r m s , a t t h e same t i m e it t a k e s o u t some o f t h e p r e c i s i o n o f u s a g e , i n view o f t h e s u r p l u s meaning u s u a l l y connected w i t h t h e s e t e r m s . The d e f i n i t i o n s r e m a i n , o f c o u r s e , p r e c i s e and unambiguous; i t i s o n l y t h a t t h e t e r m s which are b e i n g d e f i n e d have a l r e a d y - e s t a b l i s h e d i n t u i t i v e meanings, a l t h o u g h n o t n e c e s s a r i l y p r e c i s e o n e s , or i d e n t i c a l f o r d i f f e r e n t p s y c h o l o g i s t s . To use an example, when C a t t e l l d e f i n e s t e r m s s u c h a s , s a y , P r e m s i a , t h e s e a r e new t e r m s , and everyone who u s e s them must do s o i n t h e s e n s e as d e v i s e d by C a t t e l l . When, however, C a t t e l l u s e s for h i s f a c t o r s t e r m s l i k e " a n x i e t y " o r "superego" , t h e n -- i n d e p e n d e n t l y o f t h e s u g g e s t e d d e f i n i t i o n s -- t h e p s y c h o l o g i s t s u s i n g Catt e l l ' s t e s t s would o f t e n r e l y on t h e i r own i n t u i t i o n s
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concerning these concepts. Nevertheless, t h e very e x i s t ence o f r i g i d d e f i n i t i o n s o f c o n c e p t s r e d u c e s t h e l o o s e n e s s o f t h e i r usage, t h u s l e a d i n g towards g r e a t e r p r e c i s i o n of p s y c h o l o g i c a l language. Now, c r e a t i n g a s t r u c t u r e d d i c t i o n a r y o f p s y c h o l o g i c a l t e r m s , w h e t h e r new or p r e v i o u s l y known, i . e . a c e r t a i n system o f i n t e r r e l a t e d c o n c e p t s , amounts t o e s t a b l i s h i n g a s e m a n t i c s o f t h e language o f p e r s o n a l i t y . I f one a g r e e s t o t r e a t t h e u n d e r l y i n g v a r i a b l e s i n t h e domains Q , L and T as s e m a n t i c components ( i n t h e s e n s e of B i e r w i s c h 1969; s e e a l s o Katz and Fodor 1963), t h e n C a t t e l l ' s d i c t i o n a r y of p e r s o n a l i t y p r o v i d e s each l e x i c a l e n t r y t o g e t h e r w i t h i t s semantic decomposition. These e n t r i e s form a h i e r a r c h i c a l s t r u c t u r e , i n t h e s e n se t h a t e n t r i e s on t h e lower l e v e l o f a b s t r a c t i o n form p a r t s o f e n t r i e s on h i g h e r l e v e l s of a b s t r a c t i o n . Thus, " p e r s o n a l i t y " i s t h e e n t r y whose meaning i s d e f i n a b l e i n t e r m s o f h i g h e s t o r d e r t r a i t s ( s e c o n d , o r even t h i r d order f a c t o r s ) ; these i n t u r n a r e definable i n terms of lower o r d e r t r a i t s ( f i r s t o r d e r f a c t o r s ) , and s o o n , down t o t h e l e v e l o f b e h a v i o u r a l v a r i a b l e s , t h a t i s , t h e s e m a n t i c components t o g e t h e r w i t h t h e r e l a t i o n s between them g i v e n by t h e m a t r i x o f i n t e r c o r r e l a t i o n s . The fundamental t y p e of p r o p o s i t i o n s which can be formu l a t e d i n t e r m s o f t h e language c r e a t e d o r f o r m a l i z e d by C a t t e l l , a r e t h o s e which a s s e r t t h e d e g r e e t o which f a c t o r x may be a s c r i b e d t o a given' p e r s o n N . The i n t e n s i o n a l i t y i s d e t e r m i n e d by a p p r o p r i a t e measurement procedures, thus providing t h e p o s s i b i l i t y of expressi n g i t much more a c c u r a t e l y t h a n w i t h t h e use o f i n t e n s i o n a l f u n c t o r s such as "very", " h i g h l y " , e t c . i n t h e
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n a t u r a l l a n g u a g e . The b a s i c t y p e s o f p r o p o s i t i o n s have t h e form "N h a s t r a i t x i n d e g r e e d " ; a c c o r d i n g t o C a t t e l l , t h e c o n j u n c t i o n of 16 such p r o p o s i t i o n s , f o r t h e
16 p e r s o n a l i t y t r a i t s ( t h e s o - c a l l e d p e r s o n a l i t y p r o f i l e ) c o n t a i n s most o f t h e p s y c h o l o g i c a l l y r e l e v a n t i n f o r m a t i o n about t h e s u b j e c t ; t h i s , i n t u r n , a l l o w s one t o make p r e d i c t i o n s about t h e f u t u r e b e h a v i o u r o f t h e subject. I n t h e c o n s i d e r e d l i n g u i s t i c i n t e r p r e t a t i o n , t h e fundamental h y p o t h e s i s o f i n d i f f e r e n c e of media a s s e r t s t h a t i f one t a k e s as i n i t i a l v a r i a b l e s a , b , c , the variabl e s from t h e domains Q , L and T , t h e o b t a i n e d language
...
w i l l c o n t a i n synonyms. More p r e c i s e l y , t h i s h y p o t h e s i s
a s s e r t s t h a t for e x p r e s s i o n s o b t a i n e d by s u b s t i t u t i n g i n p l a c e of a , b , c t h e v a r i a b l e s from t h e c l a s s Q , t h e r e e x i s t e x p r e s s i o n s w i t h t h e same meaning o b t a i n e d by s u b s t i t u t i n g t h e v a r i a b l e s from c l a s s e s L o r T . Equiv a l e n t l y , one may s a y t h a t t h e e x p r e s s i o n s whose s e -
,...
m a n t i c components i n v o l v e t h e v a r i a b l e s from one domain only may be p a r a p h r a s e d t o e x p r e s s i o n s i n v o l v i n g t h e components i n o t h e r domains. A s shown e m p i r i c a l l y , however, n o t a l l terms i n s u c h
a language have synonyms: t h e r e e x i s t e x p r e s s i o n s char a c t e r i s t i c f o r one domain of v a r i a b l e s o n l y , which have no c o r r e s p o n d i n g e x p r e s s i o n s from o t h e r domains. I n c o n n e c t i o n w i t h t h e above i n t e r p r e t a t i o n o f C a t t e l l ' s r e s u l t s as a l a n g u a g e o f p e r s o n a l i t y , one c a n pose a q u e s t i o n t o which e x t e n t t h e m u l t i v a r i a t e methodology and e s p e c i a l l y f a c t o r a n a l y s i s , c o n t r i b u t e d t o c r e a t i n g a u n i v e r s a l b e h a v i o u r a l language o f d e s c r i p t i v e psychol o g y . The b a s i c r e s u l t s a r e i n t h e domains s u c h as p e r -
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s o n a l i t y and m o t i v a t i o n ( C a t t e l l 1957; C a t t e l l and Schei e r 1961; G u i l f o r d 1959; Eysenck 1953) and i n t e l l i g e n ce ( G u i l f o r d 1956, 1 9 7 1 ) . There a r e a l s o a p p l i c a t i o n s t o a p t i t u d e s , a t t i t u d e s , s o c i a l behaviour, learning, p e r c e p t i o n , motor s k i l l , e t c . C e r t a i n a t t e m p t s o f a s y n t h e s i s were g i v e n by French (1951, 1953) and Royce ( 1 9 7 0 ) . A s y n t h e s i s from a m e t a s c i e n t i f i c p o i n t o f view ( o f t h e o r i e s of m o t i v a t i o n , n o t o n l y b a s e d on f a c t o r a n a l y s i s ) was g i v e n b y Madsen ( 1 9 6 8 ) . Here one s h o u l d a l s o mention t h e program o f Royce ( 1 9 6 5 ) who wanted t o c o n s t r u c t a u n i f y i n g m u l t i - f a c t o r t h e o r y o f b e h a v i o u r a l v a r i a b i l i t y , comprising not only s o c i a l and c u l t u r a l , b u t a l s o g e n e t i c and b i o l o g i c a l d e t e r m i n a n t s . H e wanted t o l i n k t h e m u l t i p l e f a c t o r t h e o r y o f p e r s o n a l i t y w i t h t h e m u l t i p l e f a c t o r theory of g e n e t i c s . When one a t t e m p t s t o s y n t h e s i s e t h e r e s u l t s o f v a r i o u s a u t h o r s , or ( a s i n t h e c a s e of Royce 1965) when one s t a r t s i n a s e n s e a f r e s h w i t h a l a r g e number o f i n i t i a l v a r i a b l e s c o v e r i n g a wide domain, t h e b a s i c d i f f i c u l t y l i e s i n t h e problem o f i d e n t i f i c a t i o n o f f a c t o r s w i t h i n one domain or a c r o s s d i f f e r e n t subdomains. A s shown on t h e example o f C a t t e l l ' s h y p o t h e s i s o f i n d i f f e r e n c e o f media, s u c h a m a t c h i n g , b e i n g i n f a c t e q u i v a l e n t t o c o n s t r u c t i n g a s y s t e m o u t of more or l e s s l o o s e l y connected concepts, i s not easy t o achieve. By means o f f a c t o r a n a l y s i s , one o b t a i n s a h i e r a r c h i c a l
system, t h e h i e r a r c h y b e i n g induced by t h e r e l a t i o n of p a r t i a l o r d e r i n g ( " i s a n element o f " ) . The matching c o n s i s t s o f l o c a t i n g nodes w i t h i d e n t i c a l meanings i n t h e o b t a i n e d graph. N o w , a l l nodes of t h i s h i e r a r c h i c a l t r e e , e x c e p t t h o s e
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a t t h e l o w e s t l e v e l of o b s e r v a t i o n a l v a r i a b l e s , r e p r e s e n t e i t h e r i n t e r v e n i n g v a r i a b l e s ( i f c l o s e enough t o t h e l e v e l of o b s e r v a t i o n s ) , or c o n s t r u c t s (Royce 1 9 6 7 ) . The p a r t o f t h e t r e e below a g i v e n node r e p r e s e n t i n g
a c o n s t r u c t provides a decomposition of t h i s c o n s t r u c t i n t h e domain of v a r i a b l e s l o c a t e d a t t h e l o w e s t l e v e l o f t h i s s u b t r e e ( s e r v i n g , i n f a c t , as a p o s s i b l e d e f i n i t i o n o f t h i s c o n s t r u c t ) . A s e a c h c o n s t r u c t becomes t h u s r e p r e s e n t e d i n form o f one o r more t r e e s ( i f d i f f e r e n t domains a r e u s e d ) one can e a s i l y o b t a i n a taxonomy of c o n s t r u c t s ( o r , more p r e c i s e l y , o f t h e i r r e p r e s e n t a t i o n s ) i n t e r m s o f c e r t a i n p a r a m e t e r s ( d i m e n s i o n s ) . Thus, f o r example, one can t h i n k o f "depth" o f a c o n s t r u c t , d e f i n e d as t h e number o f l e v e l s s e p a r a t i n g i t from t h e l e v e l of observable v a r i a b l e s ( t h i s corresponds t o t h e o r d e r o f f a c t o r s , t h a t i s , t o t h e i r l e v e l of g e n e r a l i i t y ) . Another p o s s i b l e parameter may be t h e "width" o f t h e c o n s t r u c t , d e f i n e d as t h e number o f nodes a t t h e f i r s t l e v e l of t h e t r e e , c o u n t i n g from t h e t o p ( t h i s may p e r h a p s be connected w i t h t h e scope o f meaning o f t h e c o n s t r u c t ) . F i n a l l y , s t i l l a n o t h e r p a r a m e t e r may be t h e number o f nodes a t t h e l o w e s t l e v e l . Depending on t h e r e l a t i v e l e v e l s of g e n e r a l i t y o f t h e u n d e r l y i n g v a r i a b l e s used f o r t h e r e p r e s e n t a t i o n of t h e same cons t r u c t i n d i f f e r e n t domains, t h e s e numbers may v a r y ; i t e x p l a i n s why t h e same f a c t o r s i n d i f f e r e n t domains i n C a t t e l l ' s r e s u l t s had b o t h d i f f e r e n t " d e p t h s " and " d e n s i t i e s " , t h e l a t t e r c o r r e s p o n d i n g t o t h e numbers o f nodes a t t h e l o w e s t l e v e l . C a t t e l l ' s e x p e r i e n c e shows t h a t t h e above dimensions
of c o n s t r u c t s p l a y an importabt r o l e i n matching f a c t o r s a c r o s s t h e domains; t h u s , t h e y can be o f some method-
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o l o g i c a l i n t e r e s t f o r t h e m u l t i v a r i a t e approach t o psychology. A s a l r e a d y p o i n t e d out, t h e main g o a l i n t h e d i s c u s s e d
" f a c t o r i a l g e n e r a t i o n " of a language o f a g i v e n domain
i s to c h a r a c t e r i z e and a n a l y s e i t s s e m a n t i c s . Here t h e t e r m "language" i s i n t e r p r e t e d l i t e r a l l y as a c o l l e c t i o n o f words whose i n t e r r e l a t e d meanings a r e e x p r e s s i b l e i n b e h a v i o u r a l t e r m s . One c a n , however, u s e t h e l i n g u i s t i c techniques d i r e c t l y f o r t h e analysis of t h e b e h a v i o u r i t s e l f ( s e e c h a p t e r on language o f a c t i o n s ) , by t r e a t i n g t h e b e h a v i o u r as a c e r t a i n "langua g e " , and a p p l y i n g t h e methods o f m a t h e m a t i c a l l i n g u i s t i c s to a n a l y s e i t s s y n t a x and s e m a n t i c s . A s d e f i n e d f o r m a l l y , language i s a s e t of f i n i t e s e -
quences ( s t r i n g s ) formed out of e l e m e n t s o f a c e r t a i n " a l p h a b e t " , o r "vocabulary". These e l e m e n t s may be l e t t e r s , words, o r a b s t r a c t symbols ( a s i n t h e c a s e o f l a n g u a g e s of some l o g i c a l s y s t e m s , o r computer l a n g u a g e s ) . The s t r i n g s i n t h e language a r e c a l l e d "words", " s e n t e n c e s " , " w e l l formed f o r m u l a s " , "programs", e t c . , d e p e n d i n g on t h e chosen i n t e r p r e t a t i o n . Roughly, by a n a l y s i s o f a s y n t a x o f a g i v e n l a n g u a g e ( s e e , f o r i n s t a n c e , Chomsky 1963, o r Ginsburg 1966), one means t h e c o n s t r u c t i o n o f a f i n i t e s e t o f w e l l def i n e d r u l e s of g e n e r a t i o n , s u c h t h a t by a p p l y i n g t h e s e r u l e s one o b t a i n s a l l s t r i n g s of t h i s l a n g u a g e and o n l y t h o s e s t r i n g s . Languages may t h e n be c l a s s i f i e d a c c o r d i n g to t h e c o m p l e x i t i e s o f grammars which gener a t e them. A l t e r n a t i v e l y , one may c l a s s i f y t h e langua g e s a c c o r d i n g t o t h e d e g r e e o f c o m p l e x i t y o f automata which are c a p a b l e o f d i s t i n g u i s h i n g s t r i n g s i n t h e
172
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language from s t r i n g s o u t s i d e of i t ( t h e s e two approac h e s a r e , i n a s e n s e , d u a l one t o a n o t h e r ) . The a n a l y s i s o f s e m a n t i c s c o n s i s t s o f s t u d y i n g t h e p r o p e r t i e s e x p r e s s i b l e i n t e r m s of t h e r e l a t i o n s conn e c t i n g s t r i n g s i n t h e language w i t h e l e m e n t s o f t h e s p a c e o f meanings ( s e e Marcus 1 9 6 8 ) . I n m a t h e m a t i c a l l i n g u i s t i c s t h e r e i s no r e s t r i c t i o n on t h e i n t e r p r e t a t i o n of t h e e l e m e n t s o f t h e a l p h a b e t ; c o n s e q u e n t l y , these e l e m e n t s may a l s o be i n t e r p r e t e d as some u n i t s o f b e h a v i o u r . The s t r i n g s would t h e n c o r r e s p o n d t o s e q u e n c e s of s u c h u n i t s , performed one a f t e r a n o t h e r . T h i s i d e a , c a r r i e d o u t i n Nowakowska 1973, l e a d s t o c r e a t i o n o f t h e t h e o r y of l a n g u a g e s of a c t i o n s . It i s w o r t h w h i l e t o mention here t h a t i n t h e f i n a l p a r t of t h e i r p a p e r , M i l l e r and Chomsky (1963), w r o t e : "It i s p r o b a b l y no a c c i d e n t t h a t a t h e o r y of grammatical s t r u c t u r e s can b e s o r e a d i l y and n a t u r a l l y g e n e r a l i z e d t o a schema for t h e o r i e s of o t h e r ( t h a n l i n g u i s t i c ) k i n d s o f c o m p l i c a t e d human b e h a v i o u r "
.
I n Nowakowska ( 1 9 7 3 ) , n o t o n l y t h e s y n t a x of a c t i o n l a n g u a g e s was a n a l y s e d , b u t a l s o t h e i r s e m a n t i c s , where t h e "meaning" o f a s t r i n g of a c t i o n s was i d e n t i f i e d w i t h t h e outcome o f t h i s s t r i n g . I n psychology , under t h e l i n g u i s t i c i n t e r p r e t a t i o n , t h e p s y c h o l o g i c a l t r a i t s would no l o n g e r be d e f i n e d as f a c t o r s : b e i n g c e r t a i n g e n e r a l p r o p e r t i e s of b e h a v i o u r , t h e y would a c q u i r e a n i n t e r p r e t a t i o n i n t e r m s o f synt a c t i c or s e m a n t i c p r o p e r t i e s of t h e " b e h a v i o u r a l l a n guage" o f a g i v e n p e r s o n . Such a way of d e f i n i n g t r a i t s i s , a t l e a s t a t p r e s e n t , d i f f i c u l t t o apply t o t r a i t s
FACTOR ANALYSIS
173
on h i g h e r o r d e r o f a b s t r a c t i o n . The p r o j e c t i s n o t ent i r e l y u n r e a l i s t i c , t h o u g h , as s u g g e s t e d b y Suppes (1963), who p r o p o s e s t o taxonomize v a r i o u s l e v e l s o f " t r a i t s " s u c h a s grammar a c q u i s i t i o n ( o r : t h e complexi t y o f t a s k s which v a r i o u s s p e c i e s o f a n i m a l s a r e capabl e o f p e r f o r m i n g ) , b y a n a l y s i n g t h e t y p e s o f automata which r e c o g n i z e t h e a p p r o p r i a t e language ( o f a c h i l d , o r of t h e behaviour of animals of a given s p e c i e s ) . One can hope t h a t a p p l i c a t i o n o f m a t h e m a t i c a l l i n g u i s t i c s t o psychology w i l l p r o v i d e n o t o n l y a d i f f e r e n t a p p r o a c h t o o l d problems, b u t ( a s was t h e c a s e w i t h o t h e r m a t h e m a t i c a l models) a l s o a s t i m u l u s l e a d i n g t o new c o n c e p t s o r even new a r e a s o f r e s e a r c h . The l i n g u i s t i c i n t e r p r e t a t i o n o f f a c t o r i a l t h e o r i e s o f
p e r s o n a l i t y may be v a l u a b l e f o r q u a n t i t a t i v e l i n g u i s t i c s , i n p a r t i c u l a r t e x t t h e o r y , and a l s o f o r t h e o r i e s o f v e r b a l and n o n - v e r b a l communication, where t h e r e l a t i o n s between v a r i o u s t y p e s o f d a t a a r e of g r e a t t h e o r e t i c a l and p r a c t i c a l i m p o r t a n c e .
174
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of M a t h e m a t i c a l P s y c h o l o -
1; 54-87.
A Taxonomy of Some P r i n c i p a l Types o f Data a n d
o f M u l t i d i m e n s i o n a l Methods o f T h e i r A n a l y s i s . I n R.N. Shepard e t a l . ( e d s . ) Multidimensional S c a l i n g : Theory a n d A p p l i c a t i o n i n t h e Behavi o r a l Sciences. V o l .
1; 21-47.
N e w York. Semi-
nar.
1974
R e p r e s e n t a t i o n of S t r u c t u r e i n S i m u l a t e d Data: P r o b l e m s a n d P r o s p e c t s . P s y c h o m e t r i k a . 39; 373-
421. SUPPES, P .
1963 S t i m u l u s R e s p o n s e Theory o f F i n i t e Automata. J o u r n a l of M a t h e m a t i c a l P s y c h o l o g y , 6 ; 327-3515.
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SUPPES, P . ,
ZINNES, J . L .
1963 B a s i c Measurement T h e o r y . I n R . D .
Luce, R . R .
Bush and E . G a l a n t e r ( e d s . ) Handbook o f Mathem a t i c a l P s y c h o l o g y . Vol. 1, 1-80.
N e w York.Wi-
ley.
THURSTONE, L . L .
1947 .
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Chicago. Univ. Press.
TORGERSON, W.S.
1952
M u l t i d i m e n s i o n a l S c a l i n g . I. T h e o r y a n d Method. P s y c h o m e t r i k a . 1 7 ; 401-419.
1965 M u l t i d i m e n s i o n a l S c a l i n g of S i m i l a r i t y . Psychom e t r i k a . 30; 379-393. TUCKER, L . R .
1958 An I n t e r - B a t t e r y Method o f F a c t o r A n a l y s i s . P s y c h o m e t r i k a . 2 3 ; 111-136. WEEKS, D . G . ,
BENTLER,
P.M.
1 9 7 9 A Comparison of L i n e a r and Monotone M u l t i d i m e n s i o n a l S c a l i n g Models. P s y c h o l o g i c a l B u l l e t i n .
86; 349-354.
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SOME PSYCHOLOGICAL PROBLEMS OF CONSTRUCTION AND APPLICATION OF Q U E S T I O N N A I R E S
T h i s c h a p t e r w i l l d e a l w i t h two main t o p i c s : (1) t h e p s y c h o l o g i c a l mechanisms u n d e r l y i n g r e s p o n s e s t o i t e m s and s t a t i s t i c a l c o u n t e r p a r t s o f t h e s e mechanisms, ( 2 ) c e r t a i n new i d e a s i n c o n s t r u c t i o n o f t e s t s , where t h e o b s e r v e d s c o r e i s d e f i n e d i n a way o t h e r t h a n t h e t r a d i t i o n a l sum (or l i n e a r c o m b i n a t i o n ) o f s c o r e s i n p a r t i c u l a r i t e m s , and ( 3 ) some a p p l i c a t i o n s o f t e s t s .
1. MODELS OF RESPONSE TO A Q U E S T I O N N A I R E ITEM
We b e g i n w i t h p s y c h o l o g i c a l mechanisms i n v o l v e d i n answering q u e s t i o n s , e s p e c i a l l y t h e t y p e s o f q u e s t i o n s which a p p e a r i n p s y c h o l o g i c a l or s o c i o l o g i c a l q u e s t i o n naires. The t h r e e main o b s e r v a b l e v a r i a b l e s which c o u l d p r o v i d e a n a c c e s s t o p s y c h o l o g i c a l mechanisms i n v o l v e d i n q u e s t i o n a n s w e r i n g a r e (1) t h e answer, ( 2 ) i t s l a t e n c y , and ( 3 ) i t s v a r i a b i l i t y under r e p e t i t i o n . I n case of questionnaire items, latency i s typically n o t o b s e r v e d . Consequently, t h e a n a l y s i s o f t h e p r e s e n t
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s e c t i o n w i l l c o n c e r n o n l y t h e answers and t h e i r v a r i a b i l i t y . The main o b j e c t i v e w i l l be t o a n a l y s e psycholo-
g i c a l mechanisms a c c o u n t i n g f o r v a r i a b i l i t y , s u c h as (among o t h e r s ) a m b i g u i t y and d i f f i c u l t y o f q u e s t i o n s . The concept o f a m b i g u i t y a c q u i r e d c o n s i d e r a b l e i n t e r e s t a f t e r t h e a p p e a r e n c e o f t h e t h e o r y o f f u z z y s e t s o f Zadeh ( 1 9 6 5 ) : t h i s t h e o r y p r o v i d e s formal t o o l s of prec i s e a n a l y s i s o f vague c o n c e p t s ( t h e s e t o p i c s w i l l be t a k e n up i n n e x t c h a p t e r , where t e s t measurement w i l l be suggested f o r c e r t a i n t y p e s of fuzzy c o n c e p t s ) . One of t h e problems d i s c u s s e d i n t h i s s e c t i o n w i l l conc e r n p s y c h o m e t r i c paradox: a n e g a t i v e r e l a t i o n between s t a b i l i t y ( h e n c e a l s o l a c k o f a m b i g u i t y ) o f i t e m s , and t h e i r d i s c r i m i n a t i n g power (which e x p r e s s e s how w e l l t h e i t e m s e r v e s as a measure o f t h e c o n c e p t ) . Next, some e m p i r i c a l r e s u l t s w i l l be p r e s e n t e d concerni n g r e l a t i o n s between v a r i a b i l i t y o f i t e m s and t h e i r perception by respondents. Analysis of t h e s e data l e a d t o a c o n c e p t u a l model o f r e s p o n s e t o an i t e m , and f i n a l l y , t o a f o r m a l i z a t i o n o f t h i s model i n t e r m s o f e x p l i c i t assumptions a l l o w i n g t h e o r e t i c a l a n a l y s i s and s i m u l -
ation. The aim of t h i s s e c t i o n i s n o t s o much t o e x p l i c a t e t h e
s t a t i s t i c a l techniques of item s e l e c t i o n , but rather t o show some p s y c h o l o g i c a l phenomena which a c c o u n t f o r s t a t i s t i c a l f e a t u r e s of item behaviour.
1.1. V a r i a b i l i t y and a m b i g u i t y
I n most o f t h i s s e c t i o n t h e a n a l y s i s w i l l c o n c e r n b i n a r y
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i t e m s , i . e . i t e m s which may be answered "Yes" o r rrNol'. Sometimes we s h a l l a l s o a l l o w t h e i n t e r m e d i a t e answer "I d o n ' t know", a b b r e v i a t e d D K . A s shown by numerous e m p i r i c a l s t u d i e s (most of them
i n t h e 5 0 ' s and 6 0 r s ) , t h e i n t r a - s u b j e c t v a r i a b i l i t y o f answers t o q u e s t i o n n a i r e i t e m s i s c o n s i d e r a b l e , sometimes as h i g h as 35 % ( s e e Goldberg 1963, o r Goldberg and J o n e s 1 9 6 4 f o r a r e v i e w o f s t u d y of v a r i a b i l i t y ) . A l l psychological explanations of v a r i a b i l i t y , includ-
i n g t h e model p r e s e n t e d i n t h i s s e c t i o n , a r e f o r m u l a t e d w i t h i n t h e g e n e r a l c o n c e p t u a l framework of t h e model o f p a i r w i s e d i s c r i m i n a t i o n o f Thurstone ( 1 9 2 7 a , b ) . I n T h u r s t o n e ' s model o f c o m p a r a t i v e judgment, t h e r e s ponse ( o r , more p r e c i s e l y , t h e p r o b a b i l i t y o f a g i v e n t y p e o f r e s p o n s e ) i s d e t e r m i n e d by comparison o f two p r o c e s s e s , r e p r e s e n t i n g " l o c a t i o n s " o f two s t i m u l i on some continuum. I n c a s e o f q u e s t i o n a n s w e r i n g , one proc e s s t y p i c a l l y r e p r e s e n t s t h e p e r c e i v e d v a l u e o f some a t t r i b u t e x of an o b j e c t 0 , while t h e o t h e r process i s t h e "boundary" o f t h e q u e s t i o n , s a y x Typically, the q q u e s t i o n i s r e d u c i b l e (however o r i g i n a l l y p h r a s e d ) t o
.
t h e form "Is 0 such-and-such?". The d e s c r i p t o r "suchand-such" i s r e p r e s e n t a b l e as a s e m i - i n f i n i t e s e t on a continuum, hence i s d e t e r m i n e d by t h e p o i n t x The q answer t o t h e q u e s t i o n s h o u l d be ( s a y ) "Yes" i s x i s t o t h e l e f t of x and "No" o t h e r w i s e . N a t u r a l l y , t h e 9' o b j e c t 0 may be t h e r e s p o n d e n t h i m s e l f , as i n t h e i t e m such as "DO you c o n s i d e r y o u r s e l f s h y ? " , e t c .
.
G e n e r a l l y , b o t h d i f f i c u l t y and a m b i g u i t y o f t h e i t e m a r e i n t e r p r e t e d as t h e p r o p e r t y t h a t x and x a r e c l o s e q
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one t o a n o t h e r ( t h e c o n c e p t s o f d i f f i c u l t y and a m b i g u i t y , though c o n c e p t u a l l y v e r y much d i f f e r e n t , c a n n o t be d i s t i n g u i s h e d w i t h i n t h e model, w i t h o u t r e f e r e n c e t o t h e s p e c i f i c c o n t e n t of t h e i t e m ) . I n s u b s e q u e n t p a r t s o f t h i s s e c t i o n , a model w i l l be p r e s e n t e d which s p e c i f i e s t h e above framework, by i n c l u s i o n o f v a r i o u s b i a s e s . A t t h i s p l a c e , one s h o u l d ment i o n t h e work o f Goldberg (1963), who p o s t u l a t e d s p e c i f i c d i s t r i b u t i o n o f x and x and assumed t h a t t h e r e i s q' a c o n s t a n t A , such t h a t p e r s o n s w i t h / x - x A will q5 change t h e i r answers upon r e p e t i t i o n . Given t h e e m p i r i c a l f r e q u e n c i e s o f answers r r y e s r ' , and t h e f r a c t i o n o f changes under r e p e t i t i o n , one c a n t h e n e s t i m a t e t h e v a l u e o f A ( c a l l e d Ambdex, o r a m b i g u i t y i n dex). One s h o u l d mention h e r e t h e f o l l o w i n g laws and p r e d c i c t i o n s , which c o n n e c t a m b i g u i t y , d i f f i c u l t y and v a r i a b i l i t y of i t ems.
a ) The q u e s t i o n s whose b h u n d a r i e s are p e r c e i v e d by t h e s u b j e c t s as c l o s e t o t h e l o c a t i o n o f t h e o b j e c t ( e . g . themselves, i n self-evaluation items) are: -- o f t e n answered by "I d o n ' t know"; -- w i t h l o n g l a t e n c y ; -- e v a l u a t e d as d i f f i c u l t t o answer. Consequently, t h e c l o s e r i s x t o x
i n the perception of q t h e r e s p o n d e n t , t h e l e s s s t a b l e w i l l be h i s answer.
From t h e p o i n t o f view o f t h e o b s e r v e d answer, we may say that :
--
i t e m s w i t h answer D K w i l l be r e l a t i v e l y l e s s s t a b l e ;
PSYCHOLOGICAL PROBLEMS OF CONSTRUCTION
--
--
187
i t e m s w i t h l o n g e r l a t e n c y w i l l be l e s s s t a b l e ; i t e m s e v a l u a t e d as d i f f i c u l t w i l l b e l e s s s t a b l e .
Next, l e t u s assume t h a t t h e d i s t r i b u t i o n o f v a l u e s o f x i n t h e p o p u l a t i o n i s normal. Hence i f x i s f a r from t h e c e n t r e o f t h e d i s t r i b u t i o n , t h e r e w i l l be few p e r s o n s who w i l l p e r c e i v e t h e v a l u e s o f x c l o s e t o t h e boundary o f t h e q u e s t i o n . C o n s e q u e n t l y , items w i t h h i g h agreement on t h e answer ( s o - c a l l e d unbalanced i t e m s ) w i l l be seldom p e r c e i v e d a s d i f f i c u l t . Thus:
--
balanced items w i l o f t e n t h a n unbalanced -- b a l a n c e d i t e m s w i l -- b a l a n c e d i t e m s a r e t h a n unbalanced o n e s .
l be p l a c e d i n c a t e g o r y DK more
items; l have l o n g e r l a t e n c y ; more o f t e n e v a l u a t e d as d i f f i c u l t
A s r e g a r d s t h e v a l u e s of Ambdex, i t s h o u l d c o r r e l a t e
p o s i t i v e l y w i t h v a r i a b i l i t y of a n s w e r s . To sum up, s t a b i l i t y o f r e s p o n s e i s c o n n e c t e d w i t h
--
p e r c e i v i n g o n e ' s own p o s i t i o n as f a r from t h e boundar y o f t h e q u e s t i o n , and -- u n b a l a n c e d n e s s of t h e i t e m , i . e . p e r c e i v i n g t h e bounda r y o f i t e m a t one o f t h e e n d p o i n t s o f t h e continuum; On t h e o t h e r hand, v a r i a b i l i t y of r e s p o n s e i s c o n n e c t e d with:
-- a m b i g u i t y o f t h e i t e m ; -- d i f f i c u l t y o f t h e i t e m ; -- u n d e c i s e v e n e s s . Now, one of t h e s h o r t c o m i n g s o f G o l d b e r ' s a s s u m p t i o n s i s t h a t p e r s o n s from t h e a m b i g u i t y band change t h e i r answer upon r e p e t i t i o n o f i t e m , w h i l e t h e o t h e r s do n o t .
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It i s e a s y t o s e e why a c c e p t i n g s u c h a r e s t r i c t i v e a s -
sumption would l e a d i n t o l o g i c a l d i f f i c u l t i e s . Imagine namely t h a t t h e q u e s t i o n i s r e p e a t e d n o t t w i c e , b u t t h r e e t i m e s . Then t h e r e s h o u l d b e no p e r s o n s whose answer p a t t e r n would be llYes'r - ''No" - "No", s i n c e t h e f i r s t change means t h a t t h e p e r s o n i s i n t h e a m b i g u i t y band, hence s h o u l d change t h e answer a g a i n . The f o l l o w i n g s i m p l e model shows t h e way i n which t h i s d i f f i c u l t y may b e overcome. Suppose t h a t t h e s u b j e c t s may be d i v i d e d ( w i t h r e s p e c t t o a g i v e n i t e m ) i n t o t h r e e c a t e g o r i e s : t h o s e , who a l w a y s answer "Yes", t h o s e who
a l w a y s answer "No" , and t h o s e u n d e c i d e d . About t h e l a s t c a t e g o r y we assume t h a t t h e y g i v e answers "Yes" o r "No" a t random, i n d e p e n d e n t l y from t r i a l t o t r i a l , w i t h prob a b i l i t y $. L e t p , q and z d e n o t e t h e p r o p o r t i o n s o f s u b j e c t s i n e a c h o f t h e above t h r e e c a t e g o r i e s , s o t h a t p t q t z = 1. C l e a r l y , two o f t h e s e numbers may b e t a ken as p a r a m e t e r s of t h e q u e s t i o n , s a y p and z . T h e obj e c t i s t h e n t o e s t i m a t e t h e s e parameters from t h e s t a t i s t i c a l d a t a on a n s w e r s , c o l l e c t e d from a d o u b l e t r i a l on t h e same q u a t i o n . Thus, t h e d a t a have t h e form of a t r i p l e t ( Y , N , C ) , where Y , N and C d e n o t e t h e numbers o f p e r s o n s i n t h e sample who answered "Yes" on b o t h times, who answered ''No" on b o t h t i m e s , and who changed t h e i r answer. F u r t h e r m o r e , l e t n = Y + N + C , and a s s u me that C
5
min ( 2 N , 2 Y ) .
The l a s t c o n d i t i o n i s j u s t i f i a b l e i n view of t h e assumed model of random answers from c a t e g o r y o f " c h a n g e r s " : on t h e a v e r a g e , t o e v e r y 2 p e r s o n s from t h i s group who answered "Yes" on t h e f i r s t t r i a l , one p e r s o n w i l l
PSYCHOLOGICAL PROBLEMS OF CONSTRUCTION
189
change h i s answer, w h i l e t h e o t h e r w i l l n o t , and t h e
.
same h o l d s f o r a n s w e r ''No" T o p u t i t d i f f e r e n t l y , t h e p r o b a b i l i t y of a c h a n g e of a n s w e r i s 3 , w h i l e t h e p r o b a b i l i t y o f two i d e n t i c a l a n s w e r s ( b y a p e r s o n f r o m t h e " u n d e c i d e d " g r o u p ) i s $ for a n s w e r "Yes" a n d $ for a n swer "No". L e t u s now t r u t o f i n d t h e maximum l i k e l i h o o d e s t l m a t a n d z . The p r o b a b i l i t y o f
ors o f p r o b a b i l i t i e s p , q
a n s w e r "Yes" i s p t { z , p r o b a b i l i t y o f a n s w e r "No" i s q t $z = 1
-
p
-
T3 z , a n d p r o b a b i l i t y o f a c h a n g e o f
answer i s gz. Consequently, t h e l i k e l i h o o d f u n c t i o n i s
which g i v e s
+ c
log z
- c
log 2.
D i f f e r e n t i a t i o n w i t h r e s p e c t t o p and z y i e l d s , a f t e r some s i m p l i f i c a t i o n s , t h e e q u a t i o n s (N
+
Y)p t g ( N
+ 3Y)
= Y,
Cp t $(2N t 3 C ) = C . S o l v i n g , w e o b t a i n f i n a l l y f o r p , q and z p = (2Y
-
C ) / 2 n , q = ( 2 N - C ) / 2 n , z = 2C/n
where n = Y t N
+
C.
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I f one now a d d s some d i s t r i b u t i o n a l a s s u m p t i o n s ,
one may u s e t h e above f o r m u l a s t o g e t i n f o r m a t i o n a b o u t t h e l o c a t i o n o f c e r t a i n p o i n t s on t h e co n tin u u m of t h e t r a i t . Suppose t h a t t h e d i s t r i b u t i o n o f v a l u e s of t h e t r a i t i n t h e p o p u l a t i o n i s n o r m a l , w i t h known mean m and known s t a n d a r d d e v i a t i o n 6. F u r t h e r , s u p p o s e t h a t t h e p e r s o n s who w i l l a n s w e r "Yes" a r e c h a r a c t e r i z e d b y t h e i r v a l u e s x b e i n g l e s s t h a n some t h r e s h o l d a . S i m i l a r l y , t h o s e who w i l l a nsw er "No" a r e c h a r a c t e r i z e d by t h e c o n d i t i o n b c: x , w h i l e t h o s e u n d e c i d e d an d a n s w e r i n g randomly sat i s f y the condition a < x < b. Then t h e v a l u e s a an d b may be d e t e r m i n e d f r o m e q u a t i o n s
where
( I \ ( x) i s t h e s t a n d a r d n o r m a l d i s t r i b u t i o n .
The d i f f e r e n c e b-a may be c a l l e d t h e w i d t h o f t h e un-
c e r t a i n t y area ( e x p r e s s e d i n u n i t s e q u a l :i, w i t h z e r o a t m ) . If w e r e t u r n t o t h e i n t e r p r e t a t i o n a c c o r d i n g t o whic h e a c h p e r s o n (or o b j e c t ) i s c h a r a c t e r i z e d by t h e v a l u e x , t h e n t h e u n c e r t a i n t y area w i l l c o n s i s t o f a l l those x with a < x L b. The f r a c t i o n o f c h a n g e s o f a n s w e r C d e p e n d s n o t o n l y on
t h e w i d t h o f t h e u n c e r t a i n t y a r e a , b u t a l s o on t h e dens i t y o f t h e n o r m a l d i s t r i b u t i o n o v e r t h i s area; t h u s ,
i t d e p e n d s on t h e i n t e g r a l b
One may f o r m u l a t e t h e f o l l o w i n g h y p o t h e s i s r e l a t i n g
191
PSYCHOLOGICAL PROBLEMS OF CONSTRUCTION
t h e w i d t h of t h e u n c e r t a i n t y a r e a b-a and i t s l o c a t i o n on t h e a x i s ( c e n t r a l , i . e . c l o s e t o m y v s . e x t r e m a l , i . e . i n t h e t a i l of t h e d i s t r i b u t i o n ) .
i s small, t h e n t h e v a r i a b i l i t y of answers i s low, r e g a r d l e s s of t h e l o c a t i o n o f t h e u n c e r t a i n t y a r e a . The q u e s t i o n i s b a l a n c e d o r n o t , depending on t h e l o c a t i o n of t h i s a r e a . If t h e w i d t h b-a
On t h e o t h e r hand, i f t h e w i d t h b-a i s l a r g e , t h e v a r i a b i l i t y i s l a r g e i n c a s e o f c e n t r a l l o c a t i o n , and smalle r i n c a s e of extreme l o c a t i o n of t h e u n c e r t a i n t y a r e a .
1 . 2 . A f u z z i f i c a t i o n o f t h e model
The above model n e e d s t o b e f u z z i f i e d , s i n c e i n t h e
p r e s e n t form i t p o s t u l a t e s d i s c o n t i n u i t i e s a t t h e p o i n t s a and b y i n t h e s e n s e t h a t a l l s u b j e c t s w i t h t h e v a l u e x of t h e t r a i t l y i n g t o t h e l e f t from a behave i n a d i f f e r e n t way t h a n t h o s e w i t h a G ' X b y and t h o s e i n t u r n behave d i f f e r e n t l y from s u b j e c t s w i t h x < b .
<
\
A more r e a l i s t i c model would assume t h a t t h e v a l u e s a
and b a r e f u z z y . To c o n s t r u c t t h e membership f u n c t i o n f o r t h e u n c e r t a i n t y a r e a , s a y v ( x ) , choose a number 5 < ( b - a ) / 2 , and assume t h a t v ( x ) = 1 f o r a + ? ' < x c : b - E , v ( x ) = 0 f o r x :a - t and f o r x ,s b+ E , w h i l e on t h e i n t e r v a l s La-?, a+c] and [b-t , b + r ] t h e membership funct i o n v ( x ) i s g i v e n by t h e f o r m u l a
=
4 + 2li- a r c
tan
a - x ( x - a ) 2 - L- 2
f o r a-t: r x / a + E
,
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and v(x) =
i
t
-1 a r c t a n Ti-
(x-b)2
-
for b-C<x< b t i
.
Here t i s a number i n d i c a t i n g , i n a s e n s e , t h e d e g r e e of
f u z z i n e s s : t h e smaller t h e v a l u e i', t h e s t e e p e r i s t h e i n c r e a s e and d e c r e a s e of t h e membership f u n c t i o n v ( x ) i n t h e neighbourhoods of t h e p o i n t s a and b . Another way of a v o i d i n g t h e a s s u m p t i o n o f d i s c o n t i n u i t i e s c o n s i s t s o f assuming t h a t t h e answer "Yes" i s not s p e c i f i e d i n a d e t e r m i n i s t i c way by t h e v a l u e s x and x b u t depends on t h e s e v a l u e s i n a s t o c h a s t i c manner. T o
q'
b e more s p e c i f i c , l e t us assume t h e f o l l o w i n g model o f
answer: t h e s u b j e c t f i r s t v i s u a l i z e s t h e v a l u e s x and x on t h e continuum of t h e t r a i t , and t h e n t h e r u l e o f q a n s w e r i n g i s t h e same as b e f o r e , e x c e p t t h a t x and x 9
a r e random v a r i a b l e s . L e t u s t a k e a s d i s t r i b u t i o n o f x and x
t h e independent 9 random v a r i a b l e s w i t h normal d i s t r i b u t i o n s N ( m , r ) and N(mq,
r q )I.n o r d e r t o c a l c u l a t e t h e d e s i r e d p r o b a b i l i -
t i e s i t i s n e c e s s a r y t o s p e c i f y how t h e p r o b a b i l i t y o f answer "Yes" depends on t h e mutual p o s i t i o n o f p o i n t s x and x . Let u s assume t h a t t h i s p r o b a b i l i t y is g i v e n 9 by some f u n c t i o n f , depending on t h e d i f f e r e n c e x - x . 9' n a t u r a l l y , h i g h p o s i t i v e v a l u e s s h o u l d g i v e low probab i l i t y o f answer " y e s " , w h i l e h i g h n e g a t i v e v a l u e s o f t h i s d i f f e r e n c e s h o u l d g i v e h i g h p r o b a b i l i t y o f answer Y!esl'. Then t h e p r o b a b i l i t y o f "Yes" i s t h e i n t e g r a l
PSYCHOLOGICAL PROBLEMS OF CONSTRUCTION
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T h i s a p p r o a c h , a l t h o u g h p r o b a b l y t h e most a d e q u a t e l y
r e f l e c t i n g t h e c o n d i t i o n s of answering, i n v o l v e s t h e unknown f u n c t i o n f , which must be p o s t u l a t e d , and f o u r c o n s t a n t s m y m ,cr and f The e s t i m a t i o n would r e q u i r e q q complex o p t i m i z a t i o n r o u t i n e s .
.
1.3. Psychometric paradox The o b j e c t o f t h i s s u b s e c t i o n w i l l b e t h e a n a l y s i s o f t h e phenomenon c a l l e d p s y c h o m e t r i c p a r a d o x , which appea r s i n a p p l i c a t i o n o f methods o f i t e m a n a l y s i s . The par a d o x c o n s i s t s o f t h e f a c t t h a t , as a r u l e , items which a r e a "good" measure of t h e c o n t r u c t ( h a v e h i g h d i s c r i m i n a t i n g power) have a l s o h i g h v a r i a b i l i t y , w h i l e s t a b l e i t e m s have t y p i c a l l y low d i s c r i m i n a t i n g power. T h i s phenomenon was o b s e r v e d q u i t e e a r l y : a l r e a d y e . g . E i s e n b e r g (1941) and E i s e n b e r g and Wesnman (1941) had found t h a t i t e m s which d i s c r i m i n a t e w e l l n e u r o t i c s from n o n - n e u r o t i c s have o f t e n h i g h v a r i a b i l i t y , w h i l e s t a b -
l e i t e m s a r e poor d i s c r i m i n a t o r s o f t h e s e two g r o u p s . The t o p i c was t a k e n up by Goldberg
(19631, who seems
t o be t h e f i r s t t o u s e t h e term "psychometric p a r a d o x " . A c t u a l l y , t h e r e s u l t s p r e s e n t e d i n Chapter 1 concerning a n e g a t i v e r e l a t i o n between v a l i d i t y and r e l i a b i l i t y o f t e s t s i n d i c a t e a k i n d o f paradox o f t h e t y p e d i s c u s s e d on t h e l e v e l o f t e s t s ; i n c a s e o f i t e m s , r e l i a b i l i t y (measured by t e s t - r e t e s t c o r r e l a t i o n ) i s r e l a t e d by a s i m p l e f u n c t i o n a l r e l a t i o n t o v a r i a b i l i t y o f an i t e m . Thus, t h e p a r a d o x s h o u l d come as no s u r p r i s e . To e x p l i c a t e t h e paradox, c o n s i d e r a n i t e m which may be
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answered "Yes" or "No" , and l e t u s a n a l y s e i t s p o s s i b l e r e l a t i o n t o a c r i t e r i o n ( e . g . t e s t , or some e x t e r n a l variable).
1.3.1. Discriminativeness.
The n o t i o n o f d i s c r i m i n a t i v e n e s s w i l l b e u n d e r s t o o d g e n e r a l l y as t h e a b i l i t y of a n i t e m to d i s c r i m i n a t e t h e p o p u l a t i o n . One c a n d i s t i n g u i s h h e r e two c o n c e p t s o f d i s c r i m i n a t i v e n e s s , uncond i t i o n a l and c o n d i t i o n a l , depending whether one wants t o d i f f e r e n t i a t e p o p u l a t i n n w i t h r e s p e c t t o some c r i t e r i o n or n o t . We b e g i n w i t h t h e n o t i o n o f u n c o n d i t i o n a l d i s c r i m i n a t i -
v e n e s s . T h i s concept i s n o t r e l a t e d t o any c r i t e r i o n , and depends o n l y on t h e f o r m a l and s t a t i s t i c a l p r o p e r t i e s of t h e measurement t o o l . C o n s i d e r a measurement t o o l X ( t e s t , i t e m , e t c . ) . We say t h a t p e r s o n s a and b are d i s c r i m i n a t e d by X , i f t h e
r e s u l t s o f measurement o f a and b by X y i e l d d i s t i n c t
r e s u l t s . I f t h e p o p u l a t i o n c o n s i s t s o f N members, and X w i l l be a p p l i e d t o e a c h o f them, t h e n some p a i r s w i l l be d i s c r i m i n a t e d and some w i l l n o t . Suppose t h a t X may assume r d i s t i n c t v a l u e s ( t y p i c a l l y , r = 2 o r 3 i n c a s e o f an item); an a p p l i c a t i o n o f X t o the population r e s u l t s therefore i n p a r t i t i o n of t h i s p o p u l a t i o n i n a t most r c l a s s e s ( o f p e r s o n s w i t h t h e
same v a l u e o f X ) . Suppose t h a t t h e s i z e s o f t h e s e o l a s s e s s a r e ml, ,m r N a t u r a l l y , t h e p e r s o n s i n t h e same c l a s s a r e n o t d i s c r i m i n a t e d , w h i l e t h e p e r s o n s from d i s t i n c t c l a s s e s a r e d i s c r i m i n a t e d . Consequently, t h e number o f n o n - d i s c r i m i n a t e d p a i r s w i l l be
...
.
195
PSYCHOLOGICAL PROBLEMS OF CONSTRUCTION
(1.1)
since m.(mi-1)/2 i s t h e number o f p a i r s which may be 1 formed o u t o f mi e l e m e n t s . The number o f a l l p o s s i b l e pairs i s N(N-1)/2, s o t h a t t h e number o f d i s c r i m i n a t e d pairs equals (1.2)
D i v i d i n g by N(N-1)/2 we g e t t h e f r a c t i o n o f d i s c r i m i n a t ed p a i r s t o be r r mi(mi-l) 2 1-_/ z Q = l - L ) _ (1.3) N(N-1) i' r=l r=l
-
\
'V
where z = m . / N i s t h e f r a c t i o n o f p o p u l a t i o n which i s i 1 c l a s s i f i e d i n t o i - t h c l a s s (obtain "score" i)
.
...
t zr = 1, i t i s easy t o s e e t h a t Q a t t a i n s Since z t 1 i t s maximum when a l l z i a r e e q u a l .
+ = l/r, we get t h e f r a c t i o n of In t h i s case, putting zI discriminated pairs equal Q = 1 - l/r, so that f o r bin a r y items we have Q = $.
The r a t i o o f a c t u a l number o f d i s c r i m i n a t e d p a i r s D t o
t h e maximal number of p a i r s which can be d i s c r i m i n a t e d by X i s c a l l e d t h e Ferguson i n d e x ( s e e G u i l f o r d 1 9 5 4 ) ; it measures how w e l l X u t i l i z e s i t s p o t e n t i a l p o s s i b i l i t i e s of discrimination. Now, b y c o n d i t i o n a l d i s c r i m i n a t i v e n e s s we mean t h e r e l a t i o n between t h e c l a s s i f i c a t i o n o f t h e p o p u l a t i o n , as g i v e n by X , and some c r i t e r i o n . The l a t t e r w i l l be i n -
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t e r p r e t e d as a c e r t a i n c l a s s i f i c a t i o n o f p o p u l a t i o n . We s h a l l assume t h a t b o t h X and t h e c r i t e r i o n t a k e s on
numerical v a l u e s , s o t h a t t h e degree of consistency between X and c r i t e r i o n may be measured by c o r r e l a t i o n . I f X i s a t e s t , t h e f o l l o w i n g c a s e s are w o r t h c o n s i d e r i n g : t h e c r i t e r i o n may (1) r e f l e c t t h e c o n s t r u c t which X i s i n t e n d e d t o measure, ( 2 ) i t may be o b t a i n e d b y a n a p p l i c a t i o n of a p a r a l l e l v e r s i o n of X . I f X i s an item, t h e n
t h e c r i t e r i o n may b e ( 3 ) t h e
i n t e n d e d c o n s t r u c t , ( 4 ) a p p l i c a t i o n o f X f o r t h e second t i m e , and ( 5 ) i t may r e s u l t s form an a p p l i c a t i o n o f t h e t e s t which c o n t a i n s i t e m X . Measuring t h e r e l a t i o n between X and t h e c r i t e r i o n by c o r r e l a t i o n c o e f f i c i e n t , one o b t a i n s (1) v a l i d i t y o f t e s t , ( 2 ) r e l i a b i l i t y of t e s t , ( 3 ) v a l i d i t y o f t h e i t e m , ( 4 ) s t a b i l i t y o f t h e i t e m , and (5) i t e m d i s c r i m i n a t i n g power. A s m e n t i o n e d , one of t h e p o s s i b l e measures o f s t a b i l i t y
i s t h e c o r r e l a t i o n between two a p p l i c a t i o n s of an i t e m .
One c a n , however, use a s i m p l e r measure o f s t a b i l i t y , namely t h e f r a c t i o n o f changes o f answer; t h e r e a s o n i s t h a t w e a r e n o t s o much i n t e r e s t e d i n t h e d i r e c t i o n o f change as i n i t s magnitude. A c c o r d i n g l y , we s h a l l consider a p a r t i t i o n o f t h e population i n t o three s e t s : t h e s e t Qy, o f p e r s o n s who answered "Yes" o n b o t h a p p l i c a t i o n s o f t h e t e s t , t h e s e t QNN o f p e r s o n s who answered "No" on b o t h a p p l i c a t i o n s , and t h e s e t Qc o f p e r s o n s who changed t h e i r answer. A s r e g a r d s measures of d i s c r i m i n a t i n g power o f a n i t e m ,
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t h e y are a l l e x p r e s s e d t h r o u g h c o r r e l a t i o n between t h e answer t o t h e i t e m , and t h e r e s u l t s o f t h e t e s t which c o n t a i n s t h i s i t e m . S i n c e we d e a l h e r e w i t h c o r r e l a t i o n between a dichotomous and a c o n t i n u o u s v a r i a b l e , t h e p r o p e r measure i s t h e p o i n t - b i s e r i a l c o r r e l a t i o n . I n p r a c t i c e , one o f t e n t r u n c a t e s t h e lower and upper 2 7 % o f t h e sample, and t a k e s as t h e d i s c r i m i n a t i n g power the coefficient
.
Thus, t h e p r o c e d u r e l e a d i n g t o c a l c u l a t i n g '{ g i v e s a p a r t i t i o n o f t h e p o p u l a t i o n i n t o t h r e e g r o u p s : t h e lowe r group K L' t h e upper group KU, and t h e middle group KM. I n c a l c u l a t i o n s , one d i s r e g a r d s t h e i n t e r n a l o r d e r o f t h e s e groups and s i m p l y compares t h e f r a c t i o n s o f answers n i n t h e lower and upper g r o u p s .
1.3.2. A n a l y s i s o f t h e p a r a d o x .
I n a n a l y s i n g t h e pa-
r a d o x , we s h a l l c o n s i d e r m u t u a l r e l a t i o n s between t h e p a r t i t i o n o f t h e p o p u l a t i o n i n t o g r o u p s Qyy, QNN and and t e s t s c o r e s . It i s w o r t h t o mention t h a t i f &C , c o e f f i c i e n t ;\ i s u s e d , one d i s r e g a r d s i n f o r m a t i o n coming from 46 % o f t h e sample i n e v a l u a t i n g t h e d i s c r i m i n a t i n g power, w h i l e t h e s e 46 % a r e used i n a s s e s s i n g variability. A s r e g a r d s t h e p a r t i t i o n s i n t o s e t s Qyy, QNN and Qc,
t h e f o l l o w i n g t h r e e t y p e s o f i t e m s may be d i s t i n g u i s h e d : Type 1. I t e m s t a b l e and b a l a n c e d : h e r e t h e s e t Qc i s small as compared w i t h Qyy and QNN, w i t h t h e l a t t e r
s e t s b e i n g o f comparable s i z e s ; Type 2 . I t e m s t a b l e and u n b a l a n c e d : h e r e Qc i s small as compared w i t h Qyy and QNN t a k e n j o i n t l y , b u t one
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o f t h e l a t t e r s e t s i s much l a r g e r t h a n t h e o t h e r . Type 3 . V a r i a b l e i t e m . Here Q, t h e s i z e o f Qyy and Q N N .
i s o f s i z e comparable t o
A s r e g a r d s t h e r e l a t i o n between t h e p a r t i t i o n i n t o Qyy,
QNN and Qc, and p a r t i t i o n i n t o K L ? K M and KU ( w i t h r e s p e c t t o t e s t s c o r e s ) , we s h a l l c o n s i d e r o n l y s u c h i t e m s i n which t h e p r o b a b i l i t y o f answer "Yes" i s a monotone f u n c t i o n o f t r u e s c o r e o f t h e t e s t . T h i s means t h a t i f one c o u l d o r d e r t h e p o p u l a t i o n a c c o r d i n g t o t r u e s c o r e s , t h e n t h e f r e q u e n c y o f p e r s o n s from t h e s e t Q, would f i r s t i n c r e a s e and t h e n d e c r e a s e , w h i l e t h e f r e q u e n c i e s of p e r s o n s from s e t s Qyy and QNN would behave i n oppos i t e w a y s , one d e c r e a s i n g and t h e o t h e r i n c r e a s i n g . I n o t h e r words, p e r s o n s from s e t s Qyy and QNN would occupy t h e o p p o s i t e ends o f t h e s c a l e , w i t h p e r s o n s from Q, a p p e a r i n g i n between. Now, t h e same s h o u l d be t r u e i f t h e s u b j e c t s were o r d e r ed a c c o r d i n g t o t h e o b s e r v e d s c o r e s . C o n s e q u e n t l y , we have o n l y t h r e e p o s s i b l e s i t u a t i o n s , corresponding t o t h e t h r e e t y p e s of q u e s t i o n s . I t e m s o f Type 1 ( s t a b l e and b a l a n c e d ) have h i g h d i s c r i m i n a t i n g power ( c o r r e l a t e h i g h l y w i t h t e s t scores). Howe v e r , s u c h i t e m s are r a t h e r h a r d t o f i n d , and i n some classes of psychological constructs they are p r a c t i c a l l y non-existent. I t e m s o f Type 2 ( s t a b l e and u n b a l a n c e d ) have low d i s c r i m i n a t i n g power: t h e r e a s o n i s t h a t w h i l e one end of t h e s c a l e ( s a y , KL> i s o c c u p i e d a l m o s t s o l e l y b y members o f one o f t h e g r o u p s Qyy or QNN, t h e o t h e r end o f t h e
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199
s c a l e w i l l c o n t a i n members of a l l t h r e e ( o r a t l e a s t two) s e t s . C o n s e q u e n t l y , b o t h answers "Yes" and "No" w i l l a p p e a r i n c o n s i d e r a b l e p r o p o r t i o n s i n t h e s e t KU, which w i l l make t h e d i s c r i m i n a t i n g power low. We s h a l l t h e r e f o r e have one k i n d o f p a r a d o x : s t a b l e item, w i t h low d i s c r i m i n a t i n g power. F i n a l l y , Type 3 g i v e s a n o t h e r p i c t u r e . S i n c e t h e s e t Qc i s l a r g e , i t w i l l occupy t h e " c e n t r a l " p o r t i o n o f t h e s c a l e , l e a v i n g t h e ends o f t h e s c a l e r e a s o n a b l y "homogeneous". We s h a l l t h e r e f o r e have h i g h d i s c r i m i n a t i n g power, and hence t h e o t h e r k i n d o f p a r a d o x , o f h i g h b o t h v a r i a b i l i t y and d i s c r i m i n a b i l i t y . For t h e formal c o u n t e r p a r t of these c o n s i d e r a t i o n s , l e t T be t h e t r a i t i n q u e s t i o n , and l e t f ( t ) be i t s d e n s i t y i n t h e p o p u l a t i o n . Without loss o f g e n e r a l i t y we may assume t h a t T i s s t a n d a r d i z e d , s o t h a t E ( T ) = 0 , V a r T = 1.
(1.4)
C o n s i d e r now a n i t e m , and l e t X be t h e r e s p o n s e t o i t , w i t h X = 1 o r 0 depending whether t h e answer i s "Yes" o r "No" L e t p ( t ) be t h e p r o b a b i l i t y t h a t t h e answer w i l l be "Yes", o f a p e r s o n w i t h T = t ( s o t h a t p ( t ) i s t h e i t e m c h a r a c t e r i s t i c c u r v e , as d e f i n e d i n C h a p t e r 1 ) . Moreover, assume t h a t under r e p e t i t i o n , t h e answers a r e independent.
.
Then t h e p r o b a b i l i t y o f a change o f ' a n s w e r by a p e r s o n w i t h T = t i s 2 p ( t ) ( l - p ( t ) ) , s o t h a t p r o b a b i l i t y o f change o f answer ( v a r i a b i l i t y o f an i t e m ) i s v = 2 Jp(t)(l-p(t))f(t)dt.
(1.5)
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On t h e o t h e r hand, w e have E ( X ) = P ( X = 1) = \ p ( t ) f ( t ) d t
(1.6)
and t h e r e f o r e (1.7) Finally, Cov ( X , T )
=
E(XT) = E [ E ( X T i t ) l t ) ] = E[Tf(T)I
= ECTE(X
=
(tp(t)f(t)dt.
(1.8)
We s h a l l now assume t h a t e a c h i t e m i s c h a r a c t e r i z e d b y some p a r a m e t e r m , and t h a t i t s c h a r a c t e r i s t i c f u n c t i o n p m ( t ) i s g i v e n by
where h ( t ) i s some s t r i c t l y i n c r e a s i n g f u n c t i o n , s u c h t h a t h ( - t ) = 1 - h ( t ) and h ( t ) 4 1 as t -39. A s an i l l u s t r a t i o n , some n u m e r i c a l r e s u l t s a r e p r e s e n t -
ed below f o r t h e c a s e when T h a s normal d i s t r i b u t i o n w i t h mean 0 and v a r i a n c e 1, s o t h a t -t2/2
f(t) =
$re
,
and h ( t ) i s t h e l o g i s t i c f u n c t i o n , g i v e n b y
t
t
h(t) = e /(lt e ) .
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The t a b l e below i l l u s t r a t e s t h e r e l a t i o n between f r a c t i o n o f answers "Yes" ( d i f f i c u l t y o f t h e i t e m ) , p r o b a b i l i t y o f change o f answer ( v a r i a b i l i t y , or s t a b i l i t y o f
t h e i t e m ) , and t h e c o r r e l a t i o n between X and T ( d i s c r i m i n a t i n g power of t h e i t e m ) . Probability
o f anwer "Yes"
Probability
of change o f answer
Discriminating power
0.50
0.41
0.404
0.30
0.36 0.23
0.378
0.15 0.07
0.12
0.226
0.03
0.05
0.150
A s may be s e e n ,
0.310
f o r balanced items ( t h e t o p r o w ) , t h e
v a r i a b i l i t y i s h i g h , and s o i s t h e d i s c r i m i n a t i n g power. A s t h e i t e m becomes l e s s b a l a n c e d , i t s s t a b i l i t y i n c r e a -
ses (or: v a r i a b i l i t y d e c r e a s e s ) , and t h e d i s c r i m i n a t i n g power a l s o d e c r e a s e s . 'fhese r e s u l t s a g r e e w i t h t h e phenomenon known as p s y c h o m e t r i c p a r a d o x . The c o n s i d e r a t i o n s above p r o v i d e a n e x p l a n a t i o n of psyc h o m e t r i c p a r a d o x on a s y n t a c t i c ( p s y c h o m e t r i c ) l e v e l . One may a l s o i n t e r p r e t t h i s phenomenon on t h e s e m a n t i c (psychological) level. G e n e r a l l y s p e a k i n g , i t seems t h a t , p h e n o m e n a o f t h e t y pe of p s y c h o m e t r i c p a r a d o x a r e c h a r a c t e r i s t i c o n l y for t h e s o c i a l s c i e n c e s . Indeed, only i n t h e s o c i a l s c i e n c e s one b u i l d s measurement t o o l s which undergo i n t r a and i n t e r i n d i v i d u a l changes and t r a n s f o r m a t i o n , w h i l e
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r e m a i n i n g f o r m a l l y i n v a r i a n t . It may b e a r g u e d t h a t b o t h t h e s e t y p e s of t r a n s f o r m a t i o n s a r e r e s p o n s i b l e f o r psychometric paradox. The int1r.a- and i n t e r i n d i v i d u a l t r a n s f o r m a t i o n s of meani n g may be o b s e r v e d t h r o u g h t h e c o r r e s p o n d i n g ( i n t r a or i n t e r i n d i v i d u a l v a r i a b i l i t y o f answers). A s regards the interindividual v a r i a b i l i t y , it i s a
b a s i s o f c l a s s i f i c a t i o n o f members o f t h e p o p u l a t i o n . T h i s v a r i a b i l i t y has two s o u r c e s : one i s t h e d i f f e r e n -
c es of t h e v a l u e s of t h e t r a i t a t d i f f e r e n t s u b j e c t s ; t h e o t h e r a r e t h e d i f f e r e n c e s i n p e r c e p t i o n of items. N a t u r a l l y , i n c o n s t r u c t i o n o f p s y c h o m e t r i c t o o l s one t r i e s t o r e d u c e t h i s second t y p e o f v a r i a b i l i t y , by t h e so-called adaptations of t e s t s , i . e . including only t h e i t e m s which p r o v i d e i n f o r m a t i o n a b o u t t h e p o p u l a t i o n . T h i s p r o c e d u r e , however, does n o t g u a r a n t e e t h a t 211 i t e m s w i l l be p e r c e i v e d i n t h e same way b y a l l subj e c t s , e s p e c i a l l y because t h e vague items ( i . e . i t e m s which may be i n t e r p r e t e d i n v a r i o u s w a y s ) have h i g h e s t prognostic value. A s reagards intra-individual v a r i a b i l i t y , there a r e
a t l e a s t t h r e e s o u r c e s : v a r i a b i l i t y o f meanings, d i f f i c u l t y i n d e c i s i o n a b o u t t h e r e s p o n s e , and f l u c t u a t i o n s o f t h e v a l u e of t h e t r a i t . The l a t t e r may p e r h a p s be e l i m i n a t e d from c o n s i d e r a t i o n s by s h o r t e n i n g t h e t e s t r e t e s t d i s t a n c e , t h u s reducing t h e sources of v a r i a b i l i t y t o two, namely v a r i a t i o n s i n meaning, and d i f f i culty of decision. The p s y c h o m e t r i c paradox i s d e s c r i b e d by two pararnetrs:
v a l i d i t y ( d i s c r i m i n a t i n g power) and s t a b i l i t y . T o g i v e
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a p s y c h o l o g i c a l i n t e r p r e t a t i o n o f t h e p a r a d o x , one may assume t h a t t h r e e f a c t o r s d e t e r m i n e t h e r e s p o n s e t o a n i t e m : v a l u e o f t h e t r a i t o f t h e s u b j e c t , t h e meaning t h a t he a s s i g n s t o t h e i t e m , and d e g r e e o f d i f f i c u l t y i n making t h e d e c i s i o n a b o u t t h e answer. The s u b j e c t must e a c h time i n t e r p r e t t h e i t e m , and e v a l u a t e h i m s e l f a c c o r d i n g t o t h e g i v e n c a t e g o r i e s of r e s p o n s e . L e t u s d i s t i n g u i s h two t y p e s of and ambiguous o n e s . The l a t t e r , a s u n c l e a r by t h e s u b j e c t s , a r e a c t i n a way s i m i l a r t o s t i m u l i
i t e m s : unambiguous subjectively evaluated r i c h i n meanings, and which c a u s e p r o j e c t i o n s .
The p s y c h o m e t r i c paradox may now be e x p l a i n e d b y t h e f o l l o w i n g hypotheses: I . The p a r a d o x o f h i g h d i s c r i m i n a t i n g power accompanied by high v a r i a b i l i t y :
1. Concerns ambiguous i t e m s , a n d / o r i t e m s w i t h d i f f i -
c u l t d e c i s i o n about t h e r e s p o n s e (most l i k e l y , s u c h i t e m s a r e connected w i t h s e l f - e v a l u a t i o n s and a t t i t u d e s ) ; 2 . Concerns p e r s o n s w i t h moderate ( n o t e x t r e m a l ) va-
l u e s o f t h e measured t r a i t ( s i n c e e x t r e m a l p e r s o n s would t y p i c a l l y have l e s s t r o u b l e i n d e c i s i o n a b o u t t h e a n s w e r ) . 11. The paradox o f s t a b l e items w i t h low v a l i d i t y :
3 . Concerns unambiguous i t e m s , and a l s o i t e m s which a r e " u n i l a t e r a l l y d i a g n o s t i c " , i . e ; s u c h t h a t o n l y one c a t e g o r y o f r e s p o n s e i s d i a g n o s t i c . Such i t e m s w i l l most l i k e l y be q u e s t i o n s c o n c e r n i n g some e v e n t s o r f a c t s . From t h e d i s c u s s i o n o f t y p e s of i t e m s , and a l s o from t h e c o n s i d e r a t i o n s above , i t f o l l o w s t h a t p r a c t i c a l l y a l l
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i t e m s w i l l l e a d t o one o f t h e above two t y p e s o f paradox. It i s c l e a r , t h a t t h e l a r g e r i s t h e p r o p o r t i o n o f i t e m s
which g i v e p a r a d o x of t y p e I1 i n t h e t e s t , t h e more s t a b l e a r e t h e r e s u l t s . A t t h e same t i m e , however, t h e a v e r a g e v a l i d i t y of a n i t e m (mean d i s c r i m i n a t i o n power) decreases. A s a consequence (see Goldberg 1963), v a r i a b i l i t y o f t h e t e s t may be c o n t r o l l e d by c h o i c e o f i t e m s w i t h
appropriate parameters. From t h e p o i n t o f view o f t h e p u r p o s e o f measurement, t o s a t i s f y some p s y c h o m e t r i c c r i t e r i a , however i m p o r t a n t , c a n n o t s e r v e as a means for i t s e l f , s i n c e t h e main issue i s t o c o l l e c t t h e r i c h e s t possible psychological i n f o r m a t i o n about t h e s u b j e c t s . The p s y c h o m e t r i c paradox, e s p e c i a l l ) of t y p e I , imposes some c o n s t r a i n t s on t h e p s y c h o l o g i c a l i n t e r p r e t a t i o n o f t h e r e s u l t s o f t h e t e s t , and one may a n a l y s e some methods o f weakening t h e s e c o n s t r a i n t s . It seems t h a t a t p r e s e n t , t h e s e m e t hods l i e n o t i n improving f o r m a l a s p e c t s o f t h e t o o l s of measurement, b u t i n a t t e m p t s t o b e t t e r u n d e r s t a n d t h e p s y c h o l o g i c a l a c t i o n s of i t e m s . An a p p r o a c h i n which items a r e t r e a t e d as e l e m e n t s o f a c e r t a i n p o p u l a t i o n o f i t e m s , endowed w i t h c e r t a i n p s y c h o l o g i c a l t r a i t s , might a l l o w t o d i s t i n g u i s h , for s u b j e c t s w i t h t h e same v a l u e o f t h e t r a i t , c l a s s e s o f i t e m s w i t h similar meanings or w i t h s i m i l a r t r a i t s , and i n consequence, b u i l d t o o l s w i t h known p s y c h o l o g i a.al (and n o t o n l y p s y c h o m e t r i c ) p r o p e r t i e s ( h e r e a p s y c h o l o g i c a l t r a i t o f a n i t e m i s u n d e r s t o o d as r e f l e c t i o n o f r e g u l a r i t i e s of e v a l u a t i o n s of t h i s i t e m by a
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205
group o f p e o p l e ) . P e r h a p s i n t h i s way one c o u l d d e c r e a s e t h e measurement e r r o r due t o i n t r a - and i n t e r i n d i v i d u a l meaning t r a n s f o r m a t i o n s .
1 . 4 . F a c t o r s d e t e r m i n i n g t h e answer t o a q u e s t i o n n a i r e i t e m and i t s v a r i a b i l i t y It a p p e a r s r e a s o n a b l e t h a t t h e r e s p o n s e t o a q u e s t i o n -
n a i r e i t e m i s d e t e r m i n e d b y two l a r g e c l a s s e s o f v a r i a b l e s : t h o s e r e l a t e d t o t h e c o n t e n t o f t h e i t e m , and t h o s e unrelated t o it ( e . g . p r e f e r e n c e t o a c e r t a i n t y p e o f r e s p o n s e , or group p r e f e r e n c e , which d e t e r m i n e s " s o c i a l l y desirable" responses). A q u a l i t a t i v e a n a l y s i s of t h e p r o c e s s o f a n s w e r i n g t o
a q u e s t i o n n a i r e i t e m s u g g e s t s t h a t t h e answer may depend on s u c h v a r i a b l e s as i n t e l l e c t u a l e v a l u a t i o n o f b o t h t h e item and t h e i n t e n d e d r e s p o n s e , p r e v i o u s e x p e r i e n c e , e m o t i o n a l and m o t i v a t i o n a l a t t i t u d e , and c o n t r o l o v e r t h e p r o c e s s o f a n s w e r i n g . These c o n s i d e r a t i o n s l e d t o t h e c o n s t r u c t i o n o f a q u e s t i o n n a i r e for e v a l u a t i n g t h e questionnaire items. T h i s q u e s t i o n n a i r e c o n s i s t e d o f 17 s c a l e s , o f s e v e n
p o i n t s e a c h . We g i v e below one o f t h e end p o i n t s o f each o f these s c a l e s . 1. T h i s i t e m a p p e a r e d t o y o u t r i v i a l . 2. This item clearly specified the s i t u a t i o n .
3. T h i s i t e m c l e a r l y s p e c i f i e d t h e r e a c t i o n t o t h e s i t u a t i o n presented i n t h e i t e m .
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4 . T h i s i t e m d i d n o t r e q u i r e l o n g r e c o l l e c t i o n of f a c t s a n d / o r emotions.
5. I t was d i f f i c u l t t o u n d e r s t a n d t h e q u e s t i o n b e c a u s e o f t h e c h o i c e of words. 6 . You seldom t h i n k a b o u t t h e problems p r e s e n t e d i n t h e
item. 7 . You would p r e f e r t h a t someone c l o s e t o you d i d not know your answer t o t h e i t e m . 8. It was d i f f i c u l t t o u n d e r s t a n d t h e q u e s t i o n b e c a u s e o f t h e complex s y n t a x . 9. You r a r e l y r e a c t e d i n t h i s way. 1O.You l i k e t o t h i n k a b o u t t h e problems mentioned i n t h e item. 11.You have n e v e r been i n t h e s i t u a t i o n p r e s e n t e d i n t h e item. 1 2 .The q u e s t i o n was u n p l e a s a n t f o r you. l3.Your r e s p o n s e t o t h e i t e m was c o n s i s t e n t w i t h t h e o p i n i o n o f your group. 1 4 . The i t e m caused a n x i e t y . 1 5 . You c o u l d n o t change your r e s p o n s e t o t h e i t e m . 1 6 . The i t e m reminded you o f a s i t u a t i o n of f a i l u r e o r threat. 17. It was n o t h a r d t o answer s i n c e r e l y t o t h e i t e m . A s mentioned, t h e s e a r e o n l y t h e e n d p o i n t s . A t y p i c a l
s c a l e looked as f o l l o w s ( t a k i n g s c a l e 6 as an e x a m p l e ) : 6. You seldom t h i n k about t h e problems p r e s e n t e d i n t h e item
0 0 0 0 0 0 0
You o f t e n t h i n k about t h e problems presented i n t h e item
A p o p u l a t i o n o f 28 i t e m s chosen from t h e 1 6 F a c t o r P e r -
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s o n a l i t y Q u e s t i o n n a i r e of R . B . C a t t e l l was t h e t e s t e d on a s a m p l e of s t u d e n t s (N = 56) o f Psychology Department o f t h e U n i v e r s i t y o f Warsaw. The q u e s t i o n n a i r e was r e p e a t e d a f t e r two weeks. Each s u b j e c t responded f i r s t t o t h e i t e m , and t h e n e v a l u a t e d t h e i t e m and h i s r e s p o n s e o f t h e above 17 s c a l e s .
1.5. Analysis of t h e data 1.5.1. Factor analysis,
T o o b t a i n i n f o r m a t i o n about
t h e mechanisms u n d e r l y i n g p e r c e p t i o n of i t e m s , f a c t o r a n a l y s i s of t h e 17 s c a l e s g i v e n i n t h e p r e c e d i n g s e c t i o n was performed. The f a c t o r s were e x t r a c t e d by method o f p r i n c i p a l components, and r o t a t e d t o s i m p l e s t r u c t u r e by Varimax. The f o l l o w i n g four common and t h r e e s p e c i f i c f a c t o r s
were o b t a i n e d . I . Emotional and m o t i v a t i o n a l a t t i t u d e . T h i s f a c t o r was ...................................
defined by s c a l e s 1 2 (unpleasant vs. n e u t r a l content o f t h e i t e m ) , 1 4 ( a n x i e t y ) and 7 ( p e r c e p t i o n o f t h e s i t u a t i o n o f r e s p o n d i n g as a t h r e a t ) . The f a c t o r r e f l e c t s e m o t i o n a l and m o t i v a t i o n a l a t t i t u d e , a c t i v a t e d by b o t h t h e c o n t e n t of t h e item and by t h e f a c t o f a n s w e r i n g . T h i s a t t i t u d e i s u n d e r s t o o d here as a s s i g n i n g an "emot i o n a l s i g n " t o t h e i t e m , where t h i s s i g n has m o t i v a t i n g f u n c t i o n , and d e t e r m i n e s t h e dynamic f e a t u r e s o f t h e response s i t u a t i o n . 11. ............................ P r e v i o u s s p e c i f i c e x p e r i e n c e . T h i s f a c t o r i s des c r i b e d by s c a l e s 10 ( i n t e r e s t i n t h e problem o f t h e i t e m ) , 11 ( f r e q u e n c y o f s i t u a t i o n s d e s c r i b e d i n t h e i t e m )
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and 9 ( f r e q u e n c y o f r e a c t i o n s d e s c r i b e d i n t h e i t e m ) . Operationally, t h e f a c t o r here i s defined by t h e frequency and a t t r a c t i v e n e s s o f e x p e r i e n c e d r e a c t i o n s and s i t u a t i o n s . I t may be i n t e r p r e t e d as a s p e c i f i c e x p e r i e n c e of t h e s u b j e c t , a l l o w i n g him t o p r e d i c t t h e o u t come o f h i s d e c i s i o n . I n o t h e r words, f a c t o r I1 r e p r e s e n t s memory o f p r e f e r e n c e s , e x p e c t a t i o n s and s t r a t e g i e s , i n form o f some dynamic schemata. The use o f s u c h memory i n a new s i t u a t i o n c o n s i s t s of s u p e r i m p o s i n g one of t h e schemes on t h e problem a t hand; i t i s t h e r e f o r e a n a c t u a l i z a t i o n o f knowledge o f t h e e f f e c t s o f p r e v i o u s d e c i s i o n s and p r o c e d u r e s . I n s h o r t , f a c t o r I1 i s a r e f e r ence scheme f o r new s i t u a t i o n s . 111. ............................................ I n t e l l e c t u a l e v a l u a t i o n o f i t e m and r e s p o n s e . T h i s
f a c t o r i s d e s c r i b e d by s c a l e s 2 and 3 ( c l a r i t y o f i t e m ) , 4 ( l a t e n c y of r e s p o n s e ) , 5 and 8 ( d i f f i c u l t y i n unders t a n d i n g t h e q u e s t i o n ) and 1 7 ( d i f f i c u l t y i n s e l f - e v a l u a t i o n ) . Thus, f a c t o r I11 r e f l e c t s e l e m e n t s o f i n t e l l e c t u a l c o n t r o l o v e r i t e m and r e s p o n s e . A s r e g a r d s t h e i t e m , t h e c o n t r o l i s e x p r e s s e d t h r o u g h t h e way t h e i t e m i s i n t e r p r e t e d ; as r e g a r d s t h e r e s p o n s e , i t c o n c e r n s
i t s s u b j e c t i v e c e r t a i n t y , and e f f e c t s o f d i s c l o s u r e o f t h e answer. O p e r a t i o n a l l y , t h i s f a c t o r may be d e f i n e d a s s u b j e c t i v e p r o b a b i l i t y o f change o f r e s p o n s e upon repetition. I V . Value o f i t e m . T h i s f a c t o r i s d e f i n e d by s c a l e s 1 -------------
( i m p o r t a n c e and a t t r a c t i v e n e s s o f t h e i t e m ) , 6 ( s t r e n g t h o f i t s r e l a t i o n s w i t h r e s p o n d e n t ' s own p r o b l e m s ) , and 1 6 ( s t r e n g t h of r e l a t i o n s w i t h t r a u m a t i c e x p e r i e n c e s ) . F a c t o r I V r e f l e c t s t h o s e mechanisms which a l l o w t h e s u b j e c t to d i s t i n g u i s h t o p i c s t h a t a r e i m p o r t a b t f o r Thus, i t d e t e r m i n e s t h e t y p e o f b e h a v i o u r which
him.
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may b e g e n e r a l l y d e s c r i b e d as v a l u a t i o n of s t i m u l i i n t o s i g n i f i c a n t and i n s i g n i f i c a n t , and c o n s e q u e n t l y , p r e f e r -
e n t i a l b e h a v i o u r t o w a r d s p o s i t i v e s t i m u l i , and avoidance and d e f e n s e r e a c t i o n s w i t h r e s p e c t t o n e g a t i v e ones. V. S o c i a l a p p r o v a l . A s p e c i f i c f a c t o r measured by s c a l e ---------------
13 ( c o n s i s t e n c y o f t h e r e s p o n s e w i t h t h e o p i n i o n of t h e r e f e r e n c e g r o u p ) . It c o r r e s p o n d s to t h e n o t i o n o f s o c i a l d e s i r a b i l i t y , as i n t r o d u c e d by Edwards ( 1 9 5 7 ) , d e f i n e d o p e r a t i o n a l l y by s c a l e v a l u e s e x p r e s s i n g t h e d e g r e e t o which t h e b e h a v i o u r d e s c r i b e d i n t h e i t e m i s s o c i a l l y d e s i r a b l e or u n d e s i r a b l e . F a c t o r V, however, i s n o t i d e n t i c a l w i t h s o c i a l d e s i r a b i l i t y : it r e f l e c t s t h e degree t o which t h e s u b j e c t t h i n k s t h a t he a g r e e s w i t h t h e o p i n i o n o f h i s s o c i a l r e f e r e n c e group. T h i s f a c t o r , as f a c t o r I V , d e t e r m i n e s a g e n e r a l d i r e c t -
i o n o f behaviour c o n s i s t e n t w i t h t h e accepted s e t of s o c i a l norms. It may t h e r e f o r e b e t r e a t e d as a s p e c i f i c component of a more g e n e r a l f a c t o r I V . V I . .......................... S p e c i f i c e m o t i o n a l c o n t e x t . A s p e c i f i c f a c t o r mea-
s u r e d by s c a l e 1 6 ( s t r e n g t h o f r e l a t i o n s w i t h t r a u m a t i c e x p e r i e n c e ) . It r e f l e c t s t h e e x i s t e n c e (or l a c k ) of a n e g a t i v e e m o t i o n a l c o n t e x t connected w i t h t h e c o n t e n t of the item. V I I . Frequency of b e h a v i o u-r s . T h i s f a c t o r i s measured - .................... by s c a l e
9 (frequency of r e a c t i o n s described i n the
i t e m ) . It i s of i n t e r e s t low l o a d i n g on f a c t o r T I as a s p e c i f i c f a c t o r . An ween s c a l e s 9 ( f r e q u e n c y
t h a t t h i s s c a l e , w i t h rather
(previous experience) appears u n e x p e c t e d l y low r e l a t i o n b e t o f r e a c t i o n s ) and 11 ( f r e q u e n c y
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o f s i t u a t i o n s ) may be i n t e r p r e t e d as f o l l o w s : t h e subj e c t s more e a s i l y remember s i t u a t i o n s which t h e y encount e r e d , w h i l e t h e e s t i m a t i o n o f f r e q u e n c y of own behavi o u r i s more d i f f i c u l t .
1 . 5 . 2 . Model o f a n s w e r i n g t o a q u e s t i o n n a i r e i t e m . The
scheme on F i g . 1 . 5 r e p r e s e n t s t h r e e p o s s i b l e "ways" o f a n s w e r i n g t o a n i t e m , r e p r e s e n t e d as p a t h s 1, 2 and 3 , The f i n a l r e s p o n s e e n d i n g w i t h r e s p o n s e s R1, R 2 and R 3' i s o b t a i n e d from R1, R2 and R a c c o r d i n g t o some r u l e 3 ( v a r i o u s r u l e s of combining t r i a l answers i n t o t h e f i n a l answer w i l l be d i s c u s s e d i n n e x t s e c t i o n s ) . The r e s p o n s e R1 i s d e t e r m i n e d by t h e c o n t e n t o f t h e i t e m , w h i l e R 2 i s based on some s o c i a l l y approved schemes, and R i s based on p r e f e r e n c e s t o a c e r t a i n c a t e 3 gory o f r e s p o n s e s , s u c h as p r e f e r e n c e t o s a y i n g " y e s ' : , etc. P a t h s 1 and 2 have a common b e g i n n i n g . I n t h e s e p a t h s
t h e r e s p o n s e s a r e d e t e r m i n e d by f a c t o r I11 ( I n t e l l e c t -
u a l e v a l u a t i o n o f t h e i t e m and a n s w e r ) , i n c o m b i n a t i o n w i t h i n f o r m a t i o n s e l e c t e d from t h e g e n e r a l p r e v i o u s ex-
p e r i e n c e b y f a c t o r I1 ( S p e c i f i c p r e v i o u s e x p e r i e n c e ) . Thus, f a c t o r I1 p r o v i d e s r e a d y schemes o f d e c i s i o n s for some c l a s s e s o f s t i m u l i and s i t u a t i o n s , and a l l o w s t h e p r e d i c t i o n o f t h e outcomes o f d e c i s i o n s . Among t h e schemes o f b e h a v i o u r , a n i m p o r t a n t r o l e i s p l a y e d by s o c i a l l y a c c e p t e d o n e s : t h e d e c i s i o n s based on t h e s e schemes a r e rewarded by t h e i r l a c k o f c o n f l i c t and s o c i a l a p p r o v a l . FacP;or I1 d e c i d e s whether t h e subj e c t w i l l choose p a t h 1 or 2 . I f t h e s t r e n g t h o f r e a d y
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schemes d o m i n a t e s , t h e s u b j e c t chooses p a t h 2 , where t h e r e s p o n s e i s d e t e r m i n e d by f a c t o r
V ( S o c i a l approv-
a l ) . On t h e o t h e r hand, i n c a s e o f a b s e n c e of s u c h sche-
mes, or when t h e i r s t r e n g t h w i l l be small, t h e s u b j e c t w i l l choose p a t h 1, where t h e r e s p o n s e i s d e t e r m i n e d by s u b s e q u e n t f a c t o r s : I V (Value o f t h e i t e m ) and I
( e m o t i o n a l and m o t i v a t i o n a l a t t i t u d e ) . It s h o u l d be n o t e d , t h a t here t h e response i s c o n t r o l l e d by feedback mechanism, r e f l e c t e d i n t h e s u b j e c t i v e p r o b a b i l i t y t h a t t h e answer i s c o r r e c t .
F i n a l l y , i t i s p o s s i b l e t h a t p a t h 3 w i l l be c h o s e n , where t h e v a r i a n t s of answers s e r v e as s t i m u l i , and t h e answer i s d e t e r m i n e d by s t a t i s t i c a l p r e f e r e n c e s t o p a r t i c u l a r c a t e g o r i e s of a n s w e r s . Each o f t h e t h r e e p a t h s (or some o f them) s u p p l y a t r i a l r e s p o n s e ; t h e s e a r e t h e n combined i n t o t h e f i n a l response R , c o n t r o l l e d a g a i n by t h e f e e d b a c k mechanism t o f a c t o r 111 ( o f i n t e l l e c t u a l e v a l u a t i o n ) .
The c o n t r o l o f t h e answer, e x p r e s s e d by a f e e d b a c k t o f a c t o r 111, o p e r a t e s , i n a s e n s e , on two l e v e l s : o f t r u t h and u t i l i t y . On t h e f i r s t l e v e l , o f t r u t h , t h e c o n t r o l r e f l e c t s d e t e r m i n i n g t h e r e s p o n s e R1 ( n o t neces s a r i l y d i s c l o s e d ) , which r e p r e s e n t e s t h e " i n n e r t r u t h " o f t h e s u b j e c t . The c o n t r o l c o n s i s t s on s t a b i l i z i n g t h e u n d e r s t a n d i n g o f t h e i t e m , and t h e n s t a b i l i z i n g t h e r e s p o n s e under g i v e n u n d e r s t a n d i n g . The second l e v e l of c o n t r o l i s t h a t o f u t i l i t y . Here t h e t r i a l f i n a l answer R , d e r i v e d from R1, R2 and R ; t h e 3 l a t t e r r e s p o n s e i s assumed t o i n t e r v e n e o n l y i n t h e c a s e when t h e r e s p o n s e s R1 and R2 are m i s s i n g , o r a r e
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v e r y vague. Now, t h e r e s p o n s e s R1 and R2 may be i n agreement or n o t . I n t h e f i r s t c a s e , t h e f i n a l answer c o i n c i d e s w i t h them. I n t h e second c a s e , t h e f i n a l answer must be i n c o n s i s t e n t w i t h e i t h e r R1 or w i t h R2 ( i f o n l y answers "yes" and "no" a r e a l l o w e d ) . The c o n t r o l o f u t i l i t y of answer c o n s i s t s o f p r e d i c t i o n
o f t h e e f f e c t s o f d i s c l o s i n g t h e r e s p o n s e . The r e s p o n s e i n c o n s i s t e n t w i t h R 1 ( l y i n g ) may l e a d t o punishment i n form o f a lowered s e l f - e s t e e m ( " g u i l t y c o n s c i e n c e " ) , w h i l e a r e s p o n s e i n c o n s i s t e n t w i t h R2 ( n o n - c o n f o r m i s t ) may l e a d t o a d i s a p p r o v a l o f t h e group; i t may a l s o b r i n g some r e w a r d s , e . $ . i n form o f a t t r a c t i n g a t t e n t ion, e t c . I n b o t h c a s e s of control', i t l a s t s u n t i l t h e t r i a l answers s t a b i l i z e . The scheme of a n s w e r i n g s u g g e s t e d i n t h i s s e c t i o n w i l l s e r v e as a b a s i s f o r a s i m u l a t i o n model d i s c u s s e d i n s e c t i o n 1 . 6 . B e f o r e p r e s e n t i n g t h i s model, however, i t i s w o r t h w h i l e t o show c e r t a i n r e s u l t s c o n c e r n i n g change s o f answers.
1.5.3. P e r c e p t i o n o f i t e m s and v a r i a b i l i t y o f a n s w e r s . A s opposed t o t h e a n a l y s i s o f t h e p r e c e d i n g s e c t i o n , which was aimed a t d e t e r m i n i n g p s y c h o l o g i c a l f a c t o r s u n d e r l y i n g t h e p r o c e s s of answering, t h i s s e c t i o n w i l l be d e v o t e d t o p s y c h o l o g i c a l d e t e r m i n a n t s of v a r i a b i l i t y o f answers t o q u e s t i o n n a i r e i t e m s ( s e e Nowakowska
1973 ) .
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The d a t a which p r o v i d e d i n f o r m a t i o n about t h e d e t e r m i n a n t s of v a r i a b i l i t y were t h e same e v a l u a t i o n s on 17 s c a l e s which s e r v e d f o r f a c t o r a n a l y s i s and subseq u e n t c o n s t r u c t i o n o f t h e model o f a n s w e r i n g t o q u e s t ionnaire items. A s already explained i n Section 1 . 4 ,
each subject responded t o a n i t e m , and t h e n e v a l u a t e d t h e i t e m and h i s r e s p o n s e on 17 s e v e n p o i n t s c a l e s . T h i s p r o c e d u r e was r e p e a t e d a f t e r t h e p e r i o d of 1 4 d a y s . Regarding t h e answers t o t h e i t e m s , t h e r e were o n l y 4 p o s s i b i l i t i e s , namely YY, YN, NY and NN ( o r 9 p o s s i b i l i t i e s , i f t h e answer ? was a l l o w e d , namely YY, Y ? , YN, ?Y, ? ? , ?N, NY, N?, NN). If one i s i n t e r e s t e d i n v a r i a b i l i t y o f r e s p o n s e s , and i s w i l l i n g
t o a b s t r a c t from
t h e d i r e c t i o n o f changes, t h e n i n e i t h e r c a s e , t h e r e s p o n s e s t o i t e m s may be c a t e g o r i z e d i n t o + ( a change i n r e s p o n s e ) , and
-
(no change i n r e s p o n s e ) .
The r e l a t i o n between v a r i a b i l i t y o f answers and t h e e v a l u a t i o n o f t h e i t e m on s c a l e s was d e t e r m i n e d b y t h e u s e o f o p t i m a l method o f e s t i m a t i o n o f v a r i a b i l i t y on t h e b a s i s o f e v a l u a t i o n s . The problem was t r e a t e d as one o f d i s c r i m i n a t i o n : t h e v a r i a b l e d i s c r i m i n a t e d i s t o r - (change o f answer o r i t s l a c k ) , and t h e d i s c r i m i n a t o r i s t h e e v a l u a t i o n o f an i t e m on a g i v e n s c a l e ( a c t u a l l y , a p a i r of s u c h e v a l u a t i o n s ) . The p r o b a b i l i t y o f c o r r e c t d i s c r i m i n a t i o n under t h e o p t i m a l p r o c e d u r e e v a l u a t e d from t h e d a t a y i e l d t h e e s t i m a t e s o f t h e corresponding population value. To b e t t e r u n d e r s t a n d t h e problem o f d i s c r i m i n a t i o n o f
v a r i a b i l i t y on t h e b a s i s of e v a l u a t i o n s on a g i v e n sca-
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l e y one may t h i n k m e t a p h o r i c a l l y of t h e e m p i r i c a l d a t a as f o r m i n g two s e t s o f p a i r s o f e v a l u a t i o n s , one from s u b j e c t s o f " t y p e t" ( t h o s e who changed t h e r e s p o n s e t o t h e i t e m ) , and one from s u b j e c t s o f t y p e -, t h e s e s e t s o f d a t a b e i n g p l a c e d i n two u r n s , marked r e s p e c t i v e l y t and A p a i r o f e v a l u a t i o n s i s chosen a t r a n dom from one o f t h e u r n , and t h e o b j e c t i s t o g u e s s from which u r n t h e p a i r was c h o s e n .
-.
B e f o r e p r e s e n t i n g t h e method and t h e c o m p u t a t i o n a l scheme, i t i s w o r t h t o e x p l a i n : (1) why t h e method o f d i s c r i m i n a t i o n f u n c t i o n was c h o s e n , i n s t e a d o f t r a d i t i o n a l method o f s t u d y i n g dependence by c o r r e l a t i o n methods ; ( 2 ) w h y , i n s t e a d of b u i l d i n g one d i s c r i m i n a t i o n f u n c t i o n
based on a l l 17 p a i r s of e v a l u a t i o n s , t h e problem was s p l i t i n t o s e p a r a t e d i s c r i m i n a t i o n problems, one f o r each e v a l u a t i o n s c a l e . The answer t o t h e f i r s t q u e s t i o n i s t h e f o l l o w i n g : t h e v a r i a b l e which s e r v e d as a b a s i s f o r d i s c r i m i n a t i o n ( a p a i r o f e v a l u a t i o n s ) c o u l d n o t be l i n e a r l y o r d e r e d i n a m e a n i n g f u l w a y , which made i t i m p o s s i b l e t o u s e even the rank correlation c o e f f i c i e n t . The answer t o t h e second q u e s t i o n i s t h a t ( a p a r t from t h e o b v i o u s t e c h n i c a l d i f f i c u l t y o f d i s c r i m i n a t i o n on 17 t h e b a s i s o f a v a r i a b l e which may assume ( 7 x 7 ) , i . e . about 5 . 4x102* v a l u e s ) , s e p a r a t e a n a l y s i s o f s c a l e s p r o v i d e d more p s y c h o l o g i c a l i n s i g h t i n t o t h e r e l a t i o n s between changes o f answers and e v a l u a t i o n s . The g e n e r a l problem o f d i s c r i m i n a t i o n o f a dichotomous
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v a r i a b l e may be f o r m u l a t e d as f o l l o w s . We o b s e r v e a c e r t a i n v a r i a b l e x , chosen a c c o r d i n g t o one o f t h e two p r o b a b i l i t y d i s t r i b u t i o n s , s a y p - ( x ) and p t ( x ) , c o r r e s ponding t o t h e v a l u e s - and t of t h e d i s c r i m i n a t e d v a r i a b l e . Knowing x we have t o d e c i d e whether t h e v a l u e o f t h e d i s c r i m i n a t e d v a r i a b l e ( t h e " u r n " from which x was selected) i s + o r Any r u l e which t e l l s u s how t o a s s i g n a v a l u e t o r - t o t h e observed v a l u e x , i s by d e f i n i t i o n , a d i s c r i m i n a t i o n f u n c t i o n . The o p t i m a l d i s c r i m i n a t i o n f u n c t i o n i s t h a t one which l e a d s t o t h e l e a s t number o f e r r o r s .
-.
I n g e n e r a l , a d i s c r i m i n a t i o n f u n c t i o n a s s i g n s t o each o b s e r v e d v a l u e x t h e p r o b a b i l i t y w i t h which one s h o u l d a s s e r t t h a t the value of t h e discriminated variable i s +. Thus, i f f ( x ) i s t h i s p r o b a b i l i t y , t h e n w i t h probab i l i t y l - f ( x ) one s h o u l d (upon o b s e r v i n g x ) a s s e r t t h a t t h e va l ue of t h e d i s c r i m i n a t e d v a r i a b l e i s
-.
< <
I n t h e present case x = (u,v) w i t h 1 < u v 7, with ( u , v ) being a p a i r (ordered so t h a t u $ v ) of evaluati o n s o f a n i t e m on a g i v e n s c a l e . S i n c e no d i r e c t i o n o f change o f answer was t a k e n i n t o a c c o u n t , t h e o r d e r o f e v a l u a t i o n s was d i s r e g a r d e d . A s t h e c r i t e r i o n o f o p t i m a l i t y , t h e minimax r u l e was
chosen: t h e l a r g e r o f t h e two p r o b a b i l i t i e s o f m i s c l a s s i f i c a t i o n ( a s s e r t i n g t when i n f a c t i t i s -, or v i c e v e r s a ) i s minimal. An a l t e r n a t i v e a p p r o a c h might c o n s i s t o f f i n d i n g t h e p r o c e d u r e t h a t minimizes some weighted a v e r a g e o f t h e s e two e r r o r s . T h i s , however, d i d n o t seem a p p r o p r i a t e s i n c e t h e r e were many fewer c a s e s o f changes o f answer cha-
PSYCHOLOGICAL PROBLEMS OF CONSTRUCTION
217
nged, t h a n c a s e s o f no change. Thus, u n l e s s a v e r y l a r g e weight was a s s i g n e d t o one o f t h e e r r o r s , t h e o p t i m a l p r o c e d u r e would a l w a y s t e l l us t o d e c i d e ”no c h a n g e “ , r e g a r d l e s s of e v a l u a t i o n s . We s h a l l s t a r t from d e s c r i b i n g t h e a l g o r i t h m o f const.r u c t i o n o f t h e o p t i m a l minimax d i s c r i m i n a t i o n f u n c t i o n . L e t mUv and nUv d e n o t e t h e numbers of p e r s o n s i n t h e s a m p l e , whose e v a l u a t i o n s on a g i v e n s c a l e were ( u , v ) , and who changed (mu,), o r d i d n o t change (nu,) t h e i r r e s p o n s e s t o t h e i t e m . F u r t h e r , l e t M and N be t h e t o t a l numbers o f p e r s o n s who changed ( r e s p . d i d n o t change) t h e i r responses. D e f i n e p + ( u , v ) = mUv/M, p - ( u , v ) = n uv / N , and l e t u s order t h e points (u,v) according t o the r a t i o s p+(u,v)/ p- ( u , v ) [ o r e q u i v a l e n t l y , according t o t h e r a t i o s muv/nuv]. The o r d e r i n g i s t o b e i n t h e d i r e c t i o n o f i n c r e a s e o f t h e r a t i o s , w i t h t h e convention t h a t i f muv > 0 , nUv = 0 t h e r a t i o i s t a k e n t o be i n f i n i t e , and t h e p o i n t s ( u , v ) w i t h mUv = n = 0 are d i s r e g a r d e d . uv
...,
K, where K L e t t h i s o r d e r i n g be ( u i , v i ) , i = 1,2, i s t h e t o t a l number o f p a i r s ( u , v ) w i t h a t l e a s t one o f n o t z e r o . Thus, we have t h e numbers muv,n uv
D e f i n e now
L e t r be t h e smallest i n d e x s u c h t h a t C r
5 D 1” and
CHAPTER 3
218
Drtl. Such a n i n d e x e x i s t s s i n c e t h e sequence i n c r e a s e s from 0 t o 1, w h i l e t h e sequence D k d e c r e a -
Crtl) C
k
s e s from 1 t o 0 . A f t e r f i n d i n g r , we d e f i n e t h e o p t i m a l d i s c r i m i n a t i o n * n f u n c t i o n f ( u , v ) as f o l l o w s . F i r s t l y , we p u t f (ui,vi) n = 0 f o r i = 1, r . Next, i f C r = D r , w e p u t f (u , v . ) j J = 1 f o r a l l r e m a i n i n g j. I n c a s e o f C r Dr and C r t l ) * Drtl, we p u t f (u , v . ) = 1 f o r j = r t 2 K.
..., j
<
,...,
J
It remains t o d e s c r i b e t h e method o f d e f i n i n g t h e va#
lue f (urtl,vrtl)
i n t h e l a t t e r c a s e . We p u t h e r e
where Q = P + ( u , + ~ > v , + ~ )+
P-(U,+~,V,+~).
n It i s easy t o s e e t h a t t h e v a l u e f ( u r t l , v r t l )
is well
d e f i n e d , s i n c e Q ? 0 , and l i e s between 0 and 1. The p r o b a b i l i t y o f c o r r e c t d i s c r i m i n a t i o n under t h e n optimal discrimination function f equals
*
We s h a l l now p r e s e n t t h e p r o o f t h a t t h e f u n c t i o n f def i n e d above i s i n d e e d o p t i m a l i n t h e minimax s e n s e . For an a r b i t r a r y d i s c r i m i n a t i o n f u n c t i o n f , l e t a ( f ) and b ( f ) be t h e p r o b a b i l i t i e s o f e r r o n e o u s d i s c r i m i n a t i o n , f o r d i s t r i b u t i o n s p - and p,, t h a t i s ,
a ( f ) = p r o b a b i l i t y t h a t t h e d e c i s i o n r e a c h e d on t h e b a s i s o f f i s t when i n f a c t t h e o b s e r v e d v a l u e x was chosen from t h e d i s t r i b u t i o n p - ,
219
PSYCHOLOGICAL PROBLEMS OF CONSTRUCTION
and s i m i l a r l y f o r b ( f ) . THEOREM. The d i s c r i m i n a t i o n f u n c t i o n f
*
i s optimal i n t h e minimax s e n s e , ( f o r loss f u n c t i o n d e f i n e d as t h e p r o b a b i l i t y o f e r r o n e o u s d e c i s i o n ) , t h a t i s , f o r anx d i s c r i m i n a t i o n f u n c t i o n f we have
To p r o v e t h e theorem l e t u s o b s e r v e t h a t ( b y c o n s t r u c t -
*
i o n of f ) we have a ( f
*
*
*
*
= b ( f ) . Let a ( f ) = b ( f ) = c .
We have t o show t h a t f o r any d i s c r i m i n a t i o n f u n c t i o n f we have
i n o t h e r words, we have t o p r o v e t h e i m p l i c a t i o n if a ( f )
4 c , then
b(f)
2
C.
L e t u s c o n s i d e r t h e r e f o r e an a r b i t r a r y d i s c r i m i n a t i o n
f u n c t i o n f , and l e t a ( f ) C c . F o r t h e p r o o f , l e t u s t r e a t t h e d i s c r i m i n a t i o n f u n c t i o n f as a s t a t i s t i c a l t e s t o f t h e h y p o t h e s i s Ho: d i s t r i b u t i o n i s p - ( x ) , a g a i n s t t h e a l t e r n a t i v e H1: d i s t r i b u t i o n i s p + ( x ) . The c o n s i d e r e d d i s c r i m i n a t i o n f u n c t i o n , t r e a t e d a s a s t a t i s t i c a l t e s t , has t h e s i g n i f i c a n c e l e v e l a ( f ) and. power 1 - h(f). F o r s i m p l i c i t y o f n o t a t i o n s , p u t a ( f ) = a , and l e t u s c o n s i d e r t h e most p o w e r f u l t e s t o f hypot h e s i s Ho a g a i n s t t h e a l t e r n a t i v e HI a t t h e s i g n i f i c a n ce l e v e l a . Suppose t h a t s u c h a t e s t i s f a . By t h e Neyman-Pearson
220
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lemma ( s e e Lehmann, 1 9 6 0 ) such a most p o w e r f u l t e s t e x i s t s , and i s based on t h e r a t i o p + ( x ) / p ( x ) i n a manner -* similar t o t h e d i s c r i m i n a t i o n function f constructed above, e x c e p t t h a t t h e c r i t i c a l r e g i o n for t h e t e s t f a i s such t h a t t h e s i g n i f i c a n c e l e v e l i s a (as d i s t i n c t n from t h e c o n s t r u c t i o n of f , where i t was r e q u i r e d t h a t t h e s i g n i f i c a n c e l e v e l e q u a l s t h e p r o b a b i l i t y of t h e e r r o r of t h e second k i n d ) . For t h e most p o w e r f u l t e s t f
a
w e have
The second i n e q u a l i t y f o l l o w s from t h e f a c t t h a t f i s a t h e most p o w e r f u l t e s t a t t h e s i g n i f i c a n c e l e v e l a . It remains t o prove t h e r e f o r e t h a t t h e i n e q u a l i t y a ( f )
<
c i m p l i e s b ( f a ) 3 c . T h i s . however, follows d i r e c t l y from t h e f a c t t h a t t h e most p o w e r f u l t e s t of Ho a g a i n s t H i s s u c h t h a t a d e c r e a s e of s i g n i f i c a n c e l e v e l l e a d s 1 t o a n i n c r e a s e of t h e p r o b a b i l i t y of e r r o r o f t h e second k i n d . T h i s may be s e e n from F i g . 2 .
significance
/
level
p r o b . of e r r o r of second k i n d f o r t h e most powerful t e s t 3 Fig. 2
significance l e v e l o f minimax t e s t
PSYCHOLOGICAL PROBLEMS OF CONSTRUCTION
221
I n o t h e r words, an a r b i t r a r y d i s c r i m i n a t i o n f u n c t i o n f i s c h a r a c t e r i z e d by a p a i r of p r o b a b i l i t i e s , a ( f ' ) , b ( f ) , o f e r r o r s o f t h e f i r s t and second k i n d , and t h e p o i n t [ a ( f ) , b ( f ) ] l i e s i n t h e shaded a r e a of t h e p i c t u r e .
Thus, t h e p o i n t w i t h c o o r d i n a t e s ( a ( f ) , m a x C a ( f ) , b ( f ) l ) l i e s i n t h e doubly shaded a r e a o f t h e p i c t u r e , bounded from below by t h e s e t o f p o i n t s w i t h c o o r d i n a t e s ( a , * max [ a , b ( a ) ] ) . The minimax p r o c e d u r e c o r r e s p o n d s t o t h e l o w e s t p o i n t i n t h e doubly shaded a r e a ; i t i s char a c t e r i z e d by t h e c o n d i t i o n s : (1) t h e p r o b a b i l i t y o f e r r o r o f t h e second k i n d e q u a l s
the significance level;
(2) i t i s most p o w e r f u l . Thus, t h e minimax p r o c e d u r e must have t h e form o f a most p o w e r f u l t e s t , hence, by Neyman-Pearson lemma, must be based on t h e r a t i o s p , ( x ) / p - ( x ) . Among most pow e r f u l t e s t s , one must choose t h e one which s a t i s f i e s * (l), which i s p r e c i s e l y t h e p r o c e d u r e f c o n s t r u c t e d above. A l l 17 d i s c r i m i n a t i o n f u n c t i o n s o b t a i n e d i n t h i s way
were s u b j e c t t o p s y c h o l o g i c a l i n t e r p r e t a t i o n , t a k i n g i n t o account t h e p r o b a b i l i t y PC o f c o r r e c t d i s c r i m i n a t i o n , and f r e q u e n c y d i s t r i b u t i o n s p, and p -
.
The most i m p o r t a n t r e s u l t s o b t a i n e d from such a n a l y s e s
can be f o r m u l a t e d as f o l l o w s . 1. The more d i f f i c u l t i t i s t o answer t h e q u e s t i o n s i n -
c e r e l y , t h e more l i k e l y i s t h e change o f r e s p o n s e ( t h e converse i s n o t t r u e , i . e . h i g h v a r i a b i l i t y does n o t
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i m p l y t h a t i t was d i f f i c u l t t o answer t h e q u e s t i o n s i n -
c e r e l y ) . Here P c = 0 . 6 5 . 2 . The l a r g e r b i a s a g a i n s t t h e problem asked i n t h e i t e m ,
t h e h i g h e r i s t h e p r o b a b i l i t y o f no change of answer ( t h e c o n v e r s e i s n o t t r u e ) . Here P = 0 . 6 5 . C
3 . The more u n p l e a s a n t t h e c o n t e n t of t h e i t e m , t h e h i g h e r i s t h e p r o b a b i l i t y o f s t a b i l i t y of r e s p o n s e ( c o n v e r s e n o t t r u e ) . Here P c = 0 . 6 2 .
4 . T h e c l o s e r t h e c o n n e c t i o n between t h e c o n t e n t of t h e i t e m w i t h t r a u m a t i c e x p e r i e n c e s of t h e s u b j e c t , t h e more l i k e l y t h a t t h e answer w i l l n o t be changed (conv e r s e n o t t r u e ) . Here P c = 0 . 6 1 .
5 . The more t h e p r o c e s s of a n s w e r i n g i s p e r c e i v e d as a t h r e a t , t h e more l i k e l y i s t h e s t a b i l i t y of answer (conv e r s e n o t t r u e ) . Here P c = 0 . 5 9 .
6 . The c l o s e r t h e c o n n e c t i o n between t h e c o n t e n t o f t h e i t e m and t h e s u b j e c t ' s own problems, t h e more l i k e l y i t i s t h a t t h e answer w i l l be s t a b l e . The c o n v e r s e i s a l s o t r u e : t h e l e s s i n t e r e s t i n t h e problem o f t h e i t e m , t h e more l i k e l y t h e change o f answer. Here P c = 0.54.
7 . The s t r o n g e r i s t h e a n x i e t y caused b y t h e i t e m , t h e more l i k e l y i s t h e s t a b i l i t y o f r e s p o n s e ( c o n v e r s e i s not t r u e : s t a b l e items are not n e c e s s a r i l y those c a u s i n g a n x i e t y ) . Here P c = 0 . 5 6 .
8 . The l e s s c l e a r t h e i t e m , t h e more l i k e l y i t i s t h a t t h e answer w i l l b e changed ( c o n v e r s e n o t t r u e ) , Here
Pc = 0 . 5 6 .
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PSYCHOLOGICAL PROBLEMS OF CONSTRUCTION
The q u e s t i o n a r i s e s of t h e s i g n i f i c a n c e of t h e o b t a i n ed v a l u e s o f Pc ( t h e v a l u e P c = 0 . 5 0 r e p r e s e n t i n g l a c k
of r e l a t i o n , s i n c e p u r e l y random g u e s s i n g would g i v e , on t h e a v e r a g e , 5 0 % chance of b e i n g c o r r e c t ) . The following p r o c e d u r e was a p p l i e d h e r e . Given a t a b l e o f raw d a t a , c o n t a i n i n g M s i g n s t and N s i g n s -, t h e M s i g n s t and N s i g n s - were p l a c e d randomly i n t h e t a b l e ( u s i n g t h e t a b l e s o f random numbers) i n s u c h a way t h a t t h e p r o b a b i l i t y of a given s i g n b e i n g placed i n a given c e l l was p r o p o r t i o n a l t o t h e t o t a l number of s i g n s o f a c o r r e s p o n d i n g c e l l i n t h e t a b l e o f raw d a t a . The v a l u e P was t h e n c a l c u l a t e d f o r s u c h a random t a b l e . C The v a l u e of Pc f o r t h e raw d a t a was deemed s i g n i f i c a n t i f f o r f o u r random t a b l e s t h e v a l u e s of P were a l l C lower t h a n t h a t f o r t h e t a b l e i n q u e s t i o n . T h i s g i v e s t h e s i g n i f i c a n c e l e v e l 0 . 2 . A s t h e p r o c e d u r e was r a t h e r cumbersome, i t was performed o n l y f o r two s c a l e s ( 1 5 and 1 4 ) . I n a l l e i g h t c a s e s , t h e v a l u e s of Pc were cons i d e r a b l y lower t h a n t h e ones o b s e r v e d f o r t h e d a t a . I t may now be e a s i l y s e e n t h a t t h e r e s u l t s 1
-
8 above
form a l o g i c a l l y c o n s i s t e n t system: s t a b i l i t y o f answers i s connected w i t h t h e n e g a t i v e emotional r e a c t i o n t o t h e c o n t e n t of i t e m s and w i t h p r e v i o u s n e g a t i v e e x p e r i e n c e , w h i l e v a r i a b i l i t y o f i t e m s i s connected w i t h n e g a t i v e i n t e l l e c t u a l e v a l u a t i o n o f b o t h t h e i t e m and t h e answer t o it. T h i s s u g g e s t s t h e f o l l o w i n g h y p o t h e s i s , which e x p l a i n s
t h e a n a l y s e d phenomenon i n terms o f p s y c h o a n a l y t i c t h e ory. HYPOTHESIS. S t a b i l i t y o f i t e m s may b e a f u n c t i o n of def e n s e mechanisms, p r o v i d e d t h a t t h e p r e v i o u s e x p e r i e n c e
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u n d e r s t o o d h e r e as a s p e c i f i c s y s t e m o f e x p e c t a t i o n s d e t e r m i n i n g t h e a t t i t u d e towards t h e i t e m , a c t i v a t e s some d e f e n s e mechanisms c h a r a c t e r i s t i c f o r a goven subj e c t . If t h e p e r c e p t i o n o f an i t e m i s n o t d i s t u r b e d i n t h e above s e n s e , t h e n t h e b a s i c f a c t o r d e t e r m i n i n g s t a -
b i l i t y or v a r i a b i l i t y of answer i s t h e i n t e l l e c t u a l eval u a t i o n o f b o t h t h e i t e m and t h e answer.
1 . 6 . Q u e s t i o n a n s w e r i n g as a m u l t i p l e - c r i t e r i o n d e c i s -
i o n making: Model M A S I A
When r e s p o n d i n g t o a q u e s t i o n i n an i n t e r v i e w , and t o a l a r g e t e x t e n t a l s o i n responding t o a t e s t item, t h e respondent i s faced with a c e r t a i n m u l t i c r i t e r i a l decisi o n making. T h i s i s t r u e e s p e c i a l l y i n c a s e o f “ d e l i c a t e ” q u e s t i o n s , The c r i t e r i a which e n t e r i n making t h e d e c i s i o n which r e s p o n s e s h o u l d b e g i v e n may be i d e n t i f i e d w i t h v a r i o u s f a c t o r s which o p e r a t e i n d e t e r m i n i n g t h e answer. To mention j u s t some o f t h e s e f a c t o r s , t h e r e s p o n d e n t may wish t o g i v e a n answer which i s “ p l e a s i n g “ t o t h e i n t e r v i e w e r b y s a t i s f y i n g what he p e r c e i v e s
to be t h e i n t e r v i e w e r ’ s e x p e c t a t i o n s ; he may, a t t h e same t i m e have a tendency t o a v o i d t h e answer “ I d o n ’ t know”, s o as t o a p p e a r d e c i s i v e t o t h e i n t e r v i e w e r , and so forth. The model p r e s e n t e d i n t h i s s e c t i o n aims a t c a p t u r i n g t h e above mentioned (and o t h e r ) f a c t o r s which d e t e r m i n e t h e answer. D e s p i t e some s i m p l i f i c a t i o n s , i t a p p e a r s t h a t t h e model d e s c r i b e s w e l l t h e u n d e r l y i n g psycholog i c a l d e c i s i o n p r o c e s s , and a l s o i t a l l o w s us t o r e c o n s t r u c t ( b y means o f s i m u l a t i o n ) t h e d a t a from s o c i o l o g i -
PSYCHOLOGICAL PROBLEMS OF CONSTRUCTION
225
c a l i n t e r v i e w i n which b o t h t h e t r u e and t h e g i v e n answers a r e known. Such t y p e s of d a t a e x i s t f o r t h e soc a l l e d Denver V a l i d i t y S t u d i e s ( s e e van d e r Zouwen, D i j k s t r a and Nowakowska
1 . 6 . 1 . The model. w i l l be r e s t r i c t e d
o f b e t t e r name, w i l y be assumed t h a t x o f some t r a i t T ,
1979 f o r t h e d e t a i l s ) .
I n what f o l l o w s , t h e c o n s i d e r a t i o n s t o q u e s t i o n s which, f o r t h e l a c k l l be termed " l i n e a r " . It w i l l namee a c h r e s p o n d e n t has a c e r t a i n v a l u e and t h a t t h e q u e s t i o n , however
p h r a s e d , may be reduced t o t h e form
"Is your v a l u e o f t r a i t T l e s s t h a n p ? " . The t r a i t T need n o t be o f p s y c h o l o g i c a l n a t u r e : "Are you o v e r 40?" or "DO you e a r n l e s s t h a n $ 3 0 , 0 0 0 p e r y e a r ? " may s e r v e h e r e a s i l l u s t r a t i o n s . One s h o u l d a l s o mention t h a t i n many c a s e s t h e " t h r e s h o l d " i n q u e s t i o n i s f u z z y ( e . g . "DO you c o n s i d e r y o u r s e l f s h y ? " ) .
For a moment we s h a l l d i s r e g a r d t h e f u z z i n e s s o f q u e s t i o n s , and assume t h a t e a c h o f them i s r e p r e s e n t a b l e as a s i n g l e number p . Denoting b y 1 and - l t h e answers " y e s " and "no" r e s p e c t i v e l y , t h e t r u e answer t for a person with t h e value x o f t h e t r a i t T i s 1 i f x < p
t
=
(1.10) -1
if x &p.
Observe t h a t we assumed t h a t t h e t r u e answer i s a l w a y s unambiguous, e i t h e r rryesl' o r "no" N a t u r a l l y , t h e g i v e n
.
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CHAPTER 3
answer need n o t c o i n c i d e w i t h t h e t r u e answer. A s a n example, one may t a k e t h e q u e s t i o n s u c h as “ I s
your p r e s e n t income l e s s t h a n 2 0 , 0 0 0 p e r y e a r ? ” . Note t h a t a n a l y s i s c p n c e r n s q u e s t i o n s which may be answered “ y e s “ or “No”; open ended q u e s t i o n s , s u c h as “What i s y o u r p r e s e n t income?” a r e n o t c o n s i d e r e d . The v a l u e x i n t h e p o p u l a t i o n f o l l o w s some l a w , i . e . i t h a s some u n d e r l y i n g d i s t r i b u t i o n . I n t h e s e q u e l , i t w i l l be assumed t h a t t h i s d i s t r i b u t i o n i s uniform on t h e T h i s does not r e s t r u c t t h e g e n e r a l i t y , i n t e r v a l [O,l]. a t l e a s t i n c a s e o f c o n t i n u o u s d i s t r i b u t i o n s , s i n c e every t r a i t may be s u i t a b l y r e s c a l e d s o as t o o b t a i n t h e uniform d i s t r i b u t i o n on [0,1].
I n d e e d , i f F ( t ) i s t h e d i s t r i b u t i o n o f T i n t h e popul-
<
t ) i s t h e f r a c t i o n of p e r a t i o n , s o t h a t F ( t ) = P(T s o n s w i t h t r a i t T f x, t h e n T I = F ( T ) y i e l d s new v a l u e s T I , and t h e t r a i t T I has uniform d i s t r i b u t i o n on [O,l]. One o f t h e main consequences o f t h i s a s s u m p t i o n i s t h a t t h e p a r a m e t e r p c h a r a c t e r i z i n g t h e q u e s t i o n becomes a t t h e same t i m e a p a r a m e t e r c h a r a c t e r i z i n g t h e p o p u l a t i o n ( w i t h r e s p e c t t o t h e q u e s t i o n ) : p e q u a l s namely t h e p r o b a b i l i t y t h a t t h e t r u e answer o f a randomly chosen p e r s o n w i l l be ” y e s ” . I n o t h e r words, p i s t h e f a c t i o n o f t h e p o p u l a t i o n of r e s p o n d e n t s , f o r whom t h e t r u e val u e of t h e t r a i t T i s l e s s t h a n p . One of t h e p u r p o s e s o f a n i n t e r v i e w ( t e s t ) i s t o e s t i m a t e t h e (unknown) val u e p from t h e g i v e n answers i n a sample o f r e s p o n d e n t s .
1.6.2. The t r i a l a n s w e r s .
A s i m p l e i n t r o s p e c t i o n shows
227
PSYCHOLOGICAL PROBLEMS OF CONSTRUCTION
t h a t b e f o r e g i v i n g t h e f i n a l answer, t h e r e s p o n d e n t f o r -
m u l a t e s a number o f t r i a l answers ( n o t d i s c l o s e d ) , and t h e n b u i l d s t h e f i n a l answers on t h e b a s i s of comparison of t r i a l answers w i t h t h e t r u e answer. T h i s mechanism w i l l be d i s c u s s e d i n d e t a i l i n t h e n e x t s e c t i o n ; i n t h e p r e s e n t s e c t i o n , t h e c o n s i d e r a t i o n s w i l l be d e v o t e d t o t h e a n a l y s i s of a p o s s i b l e n a t u r e o f t r i a l a n s w e r s .
It w i l l be assumed namely t h a t t r i a l a n s w e r s , d e n o t e d by t l , t 2 ,... a r e o b t a i n e d by t h e same b a s i c mechanism as t h e t r u e a n s w e r s : by comparison o f a n a p p r o p r i a t e v a l u e xi w i t h t h e t h r e s h o l d p of t h e q u e s t i o n . I n o t h e r words, t h e i - t h t r i a l answer t i w i l l be d e f i n e d as 1 if
ti
xi ;P
-
-
(1.11) -1
if
xi ? P .
The problem i s t h u s reduced t o d e f i n i n g t h e v a l u e xi f o r t h e i - t h t r i a l answer. T h i s v a l u e w i l l be assumed t o be a " d i s t o r t e d " t r u e v a l u e of t h e t r a i t , s o thzt. xi = x + > r i'
(1.12)
and i t r e m a i n s t o s p e c i f y t h e a s s u m p t i o n a b o u t
fi '
G e n e r a l l y , i t w i l l be assumed t h a t -'i i s governed by three factors:
----
t
t h e tendency t o answer ''yes" ( z i ) ;
-
t h e t e n d e n c y t o answer "no" ( Z i ) ; t h e (un)importance of the question ( u ) .
F o r m a l l y , w e assume t h a t
228
CHAPTER 3
where u , z
+ i
and z- a r e p o s i t i v e q u a n t i t i e s . i
O m i t t i n g f o r s i m p l i c i t y t h e s u b s c r i p t i , we s e e t h a t if
z+ exceeds z - , t h e v a l u e o f
c o n s e q u e n t l y , xi
(
I:
w i l l be n e g a t i v e , and
x . Thus, i f t h e t r u e answer i s
( x < p ) , t h e n t h e t r i a l answer w i l l a l s o be r r y e s r l ,whil e i n some c a s e s t h e t r u e answer "No" w i l l b e changed i n t o t h e t r i a l answer " y e s " .
-
Thus, t h e c o n d i t i o n z + 3 z , i . e . h i g h e r t e n d e n c y t o wards "yes'' t h a n towards "no" w i l l r e s u l t i n some s h i f t towards
i n t h e s e n s e t h a t some o f t h e t r u e a n s -
wers "no" w i l l be changed i n t o " y e s " , w h i l e a l l answers Iryesr1w i l l remain unchanged,
+
I n a s i m i l a r w a y , i f z- .:, z , i . e . i f t h e t e n d e n c y t o ward s Ifno exceeds t h e tendency t o w a r d s " y e s " , t h e n t h e r e w i l l be a s h i f t i n t h e o p p o s i t e d i r e c t i o n : some of t h e " y e s " answers w i l l become changed i n t o "no1', whil e a l l trno" answers w i l l remain unchanged. G e n e r a l l y , t h e magnitude of t h e s h i f t , i n whichever d i -
r e c t i o n , i s l a r g e r f o r l a r g e r d i f f e r e n c e s ( z+ - z-/ , and f o r l a r g e r v a l u e s of u , i . e . when t h e q u e s t i o n i s p e r c e i v e d as unimportank . I n t h e s e q u e l , i t w i l l b e assumed t h a t z + and z- a r e sampled i n d e p e n d e n t l y from t h e uniform d i s t r i b u t i o n on
[0,1], and t h e n t r u n c a t e d from below a t c e r t a i n t h r e s h o l d s a and b . More p r e c i s e l y , l e t
2 29
PSYCHOLOGICAL PROBLEMS OF CONSTRUCTION
/ / y
if
y > r
(1.14)
Then
where v l , v 2 a r e i n d e p e n d e n t random v a r i a b l e s d i s t r i and a , b a r e some p o s i t i v e b u t e d u n i f o r m l y on [O,l], constants, representing the thresholds f o r tendencies towards and towards "no". Alternative
i n t e r p r e t a t i o n h e r e i s as f o l l o w s : t h e
t e n d e n c i e s towards " y e s r r and "no" a r e e q u a l v1 and v 2' b u t s t a r t t o o p e r a t e o n l y i f t h e y exceed t h e i r r e s p e c t i v e t h r e s h o l d s a and b ( a r e " p e r c e i v e d " by t h e respondent)
.
C o n s e q u e n t l y , t h e v a l u e s o f a and b w i l l c a u s e some bias i n responses.
1.6.3. The f i n a l answer. A s a l r e a d y mentioned, t h e f i n a l answer, t o be d e n o t e d by t ' , i s based on t r i a l answers. N a t u r a l l y , depending on t h e q u e s t i o n , t h e r e s p o n d e n t may t a k e up a l o n g t i m e t o answer, r e v i e w i n g i n h i s mind a l l p o s s i b i l i t i e s , and presumably " t e s t i n g " t h e t r i a l answers. I n t h e model o f t h i s s e c t i o n , it w i l l be assumed t h a t t h e f i n a l answer i s based on j u s t two t r i a l a n s w e r s , t l and t 2 ; v e r s i o n s o f t h e model w i t h h i g h e r number of t r i a l answers a r e e a s i l y c o n s t r u c t e d .
230
CHAPTER 3
To be more p r e c i s e , t h e f i n a l answer w i l l depend on t h e v a l u e s x1 = x t
)2
rl
and x2 = x
+
r,,
where
il
and a r e sampled i n d e p e n d e n t l y a c c o r d i n g to t h e r u l e s de-
s c r i b e d i n t h e p r e c e d i n g s e c t i o n . Observe t h a t xI and
x2 are n o t i n d e p e n d e n t , even though
tl
r
are i n d e p e n d e n t , because o f t h e p r e s e n c e o f common component x. and
j2
One c o u l d t h i n k h e r e o f a v a r i e t y o f r u l e s , Some o f them have been t r i e d i n s u c c e s s i v e v e r s i o n s of t h e p r e s e n t model, which i s g e n e r a l l y termed MASIA. The s i m p l e s t v e r s i o n , c a l l e d MASIA 1, was d e f i n e d s o t h a t t h e g i v e n answer t 1 was " y e s " i f b o t h t l and t 2 were " y e s " , t I was rrno" i f b o t h tl and t 2 were "no" , and t 1 i s "I d o n ' t know" ( D K ) , i f t l # t 2 . A l g e b r a i c a l l y , i f t h e answer DK i s d e s i g n a t e d by 0 , MASIA 1 o p e r a t e d a c c o r d i n g t o the rule
t ' = (tI t t p 2 . S u c c e s s i v e models were d e s i g n e d to overcome c e r t a i n s h o r t c o m i n g s o f t h e p r e c e d i n g models. What i s p r e s e n t e d h e r e i s t h e v e r s i o n c a l l e d MASIA 7 . The i d e a i s t o l o o k a t t h e d i f f e r e n c e s p - x1 and p - x2; h i g h p o s i t i v e v a l u e s i n d i c a t e " s t r o n g y e s " , and h i g h n e g a t i v e v a l u e s i n d i c a t e " s t r o n g no". T h i s s u g g e s t s t o d e t e r m i n e t h e f i n a l answer on t h e b a s i s of t h e d i f f e r e n ce between t h e s e v a l u e s , namely
u
= (p
-
XI)
Here U = 0 , i . e . x
- (x*
-
p ) = 2p - (xl t x 2 >
(1.16)
1 t x2 = 2p i n d i c a t e s t h a t x1 and x2 are a t t h e same d i s t a n c e from p , l y i n g on o p p o s i t e s i d e s o f i t , i . e . i f one t r i a l answer i s " y e s " , t h e o t h e r i s
PSYCHOLOGICAL PROBLEMS OF CONSTRUCTION
23 1
, i n t h e same d e g r e e . T h i s s u g g e s t s t h a t i n such a c a s e , t h e answer s h o u l d be D K , i f i t i s t o b e based on j u s t t h e two t r i a l a n s w e r s . llno"
Somewhat more g e n e r a l l y , t h e g i v e n answer i s D K , i f t h e sum x1 t x2 i s c l o s e enough t o 2 p . F o r m a l l y , M A S I A
7 operates according t o the r u l e if
x1
+ x2
t 1 = ? DK
if
2p
-
-1
if
2p t k
/'
\
1
k
$
C_
- k
2p
c x1 x1
t x t
2
2p t k
(1.17)
x2.
Here k i s a p a r a m e t e r r e p r e s e n t i n g t h e ( u n ) i m p o r t a n c e o f a p p e a r i n g d e c i s i v e : i f k i s s m a l l , t h e n x1 t x 2 must l i e v e r y c l o s e t o 2p t o e l i c i t t h e r e s p o n s e D K . G e n e r a l l y , t h e a r e a s "Yes", D K , and "No" are shown on F i g . 1, as s e t s of p o i n t s on t h e ( x l , x 2 ) - p l a n e , which c o r r e s p o n d t o t h e marked r e s p o n s e s . The w i d t h o f t h e DK area r e p r e s e n t s h e r e t h e ( u n ) i m p o r t a n c e of a p p e a r i n g decisive. The a s s u m p t i o n s u n d e r l y i n g t h e model may now be summari z e d a s f o l l o w s . The r e s p o n d e n t i s i n f l u e n c e d , t o a varyi n g d e g r e e , b y s e v e r a l t e n d e n c i e s , s u c h as " y e s - s a y i n g " , "no-saying", which i n t u r n may r e s u l t from o t h e r t e n d e n c i e s , s u c h as "need of a c c e p t a n c e " , "need o f c o n f o r m i t y " , and s o on. These t e n d e n c i e s a f f e c t t h e r e s p o n s e o n l y i f t h e y exceed t h e i r r e s p e c t i v e t h r e s h o l d s . I f these t e n d e n c i e s o p e r a t e , t h e n t h e i r i n t e n s i t y w i l l be h i g h e r when g i v i n g t h e t r u e answer w i l l be s e e n as
less important. I n g i v i n g t h e f i n a l answer, t h e r e s p o n d e n t f o r m u l a t e s
232
CHAPTER 3
f i r s t two t r i a l a n s w e r s , based on t h e t r a i t v a l u e d i s t o r t e d by t h e above two t e n d e n c i e s and by t h e import-
ance o f t h e q u e s t i o n . The f i n a l answer i s t h e n g i v e n
X
2 A
2P
,
1 I 2p-k
"Yes '\,
area \ '
'\\\
1
DK a r e a IINoII
\.,\,
2p-k
2p ,
2 p t, k
>
X
1
Fig. 1
a c c o r d i n g to t h e r u l e (1.17) as i l l u s t r a t e d on t h e F i g u r e above, where k i s t h e p a r a m e t e r measuring t h e ( u n ) i m p o r t a n c e of a p p e a r i n g d e c i s i v e .
1.6.4. Theoretical analysis.
Suppose now t h a t we have
t h e d a t a from some i n t e r v i e w , and we know n o t o n l y t h e answers g i v e n by t h e r e s p o n d e n t s , b u t a l s o t h e i r t r u e a n s w e r s . Such d a t a would have t h e form of a t a b l e w i t h s i x e n t r i e s , as p r e s e n t e d below.
PSYCHOLOGICAL PROBLEMS OF CONSTRUCTION
233
answer given ( t ' )
'-.
answer
(t)
I i
n
I
1
i
n
2
i
n
3
no
T a b l e 1. Form o f t h e d a t a F o r i n s t a n c e ( s e e van d e r Zouwen, Nowakowska and D i j k s t r a 1979) t h e d a t a on q u e n t i o n "Voted i n 1946 Congressi o n a l E l e c t i o n ? " are g i v e n i n T a b l e 2 .
4-1
I
total
I
i
Table 2 The problem i s t h e n t o e s t i m a t e t h e v a l u e s o f t h e param e t e r s a J b J u and k for t h i s q u e s t i o n , p r o v i d i n g t h u s some i n s i g h t i n t o t h e p s y c h o l o g i c a l p r o c e s s e s i n v o l v e d i n t h e answer.
234
CHAPTER 3
A r e l a t i v e l y s i m p l e a p p r o a c h t o t h i s problem i s t o s i -
m u l a t e t h e a n s w e r i n g p r o c e s s f o r any v a l u e s of paramet e r s , and t h e n u s e t h e t r i a l - a n d - e r r o r method t o f i n d t h e p a r t i c u l a r combination o f p a r a m e t e r s which would produce a t a b l e o f data s u f f i c i e n t l y c l o s e t o t h e emp i r i c a l l y g i v e n t a b l e . Such a n a p p r o a c h was used ( f o r a somewhat d i f f e r e n t v e r s i o n o f M A S I A model) i n van d e r Zouwev, Nowakowska and D i j k s t r a ( 1 9 7 9 ) . One c o u l d , however, u s e t h e maximum l i k e l i h o o d method i n t h e following way. Denote by F ( z ) = F
a,b
(z) t h e p r o b a b i l i t y d i s t r i b u t i o n
f u n c t i o n o f t h e random v a r i a b l e w = z - - z
+
= Ib(V1)
- Ia(v2),
(1.18)
where v1 and v 2 a r e i n d e p e n d e n t random v a r i a b l e s uniformly d i s t r i b u t e d on [O,l],
and I r ( x ) i s d e f i n e d b y
t h e formula ( 1 . 1 4 ) . C l e a r l y , f o r a n a p p l i c a t i o n o f maximum l i k e l i h o o d met h o d , we need t o have t h e o r e t i c a l p r o b a b i l i t i e s o f v a r i o u s c o m b i n a t i o n s o f answers ( t , t ' ) as i n T a b l e 1, as f u n c t i o n s of p a r a m e t e r s o f t h e model, t h a t i s , a , b , u and k . Denote t h e s e p r o b a b i l i t i e s by p,, . . . , p 6 , w i t h i n d i c e s c o r r e s p o n d i n g t o T a b l e 1. For i n s t a n c e , p1 = P ( t r u e and g i v e n answer a r e b o t h " y e s " )
235
PSYCHOLOGICAL PROBLEMS OF CONSTRUCTION
so t h a t P p1
-
+
PIWl
,
W2 < ( 2 p
-
k
- 2x)/u]dx
(1.19)
0 w h e r e W1 a n d W2 a r e i n d e p e n d e n t random v a r i a b l e s w i t h
F.
distribution
-
S u b s t i t u t i n g (2p
k
-
2x)/u = y , w e g e t dx =
-
U
dy,
(1.20)
1 t W2 L y ) i s t h e d i s t r i b u t i o n f u n c t i o n o f t h e sum of two i n d e p e n d e n t random v a r i a b l e s ,
where G ( y ) = P(W
each with d i s t r i b u t i o n F. I n a s i m i l a r way we o b t a i n p2 = P ( x
= P[x
< p
<
&
(xl,x2)
a r e i n DK a r e a )
p & ( 2 p - k - 2 ~ ) / <,~ W1+W2
C (2p+k-2~)/~]
(1.22)
(1.23)
236
CHAPTER 3
...
N a t u r a l l y , w e have p1 t p 2 t t p 6 = 1. T h e r e r e m a i n s o n l y t h e problem of d e t e r m i n i n g t h e d i s t r i b u t i o n f u n c t i o n F ( t ) o f t h e random v a r i a b l e W = I ( v ) - I ( v ) ; t h e f u n c t i o n G w i l l t h e n be a convolb 1 a 2 u t i o n of F w i t h i t s e l f . To d e t e r m i n e F , we need t o d i s t i n g u i s h f o u r c a s e s , dep e n d i n g whether a t b 1 or a t b > 1, and whether
<
a
o r a;r b .
Let us c o n s i d e r t h e c a s e a t b ‘i 1, a g b .
(1.26)
The d e t e r m i n a t i o n of F i s r a t h e r cumbersome, b u t i t may be f a c i l i t a t e d w i t h t h e h e l p of g e o m e t r i c i n t e r p r e t a -
t ion. The g e n e r a l scheme w i l l be p r e s e n t e d on F i g . a . The v a r i a b l e s v1 and v2 a r e i n d e p e n d e n t and u n i f o r m l y d i s t r i b u t e d on [O,l], s o t h a t t h e p o i n t ( v , , v 2 ) i s uniformly d i s t r i b u t e d on t h e u n i t s q u a r e . We have I a ( v l ) = 0 i f v1 i s to t h e l e f t o f a , I b ( v 2 ) = 0 i f v 2 i s below b . O t h e r w i s e I a ( v l ) I ( v ) = v 2 . It f o l l o w s t h a t W = Ib - Ia i s 0 b 2 t h e r ( v l , v 2 ) f a l l s to t h e shaded a r e a on F i g .
and = v1 and if eia , on
231
PSYCHOLOGICAL PROBLEMS OF CONSTRUCTION
the diagonal.
V
1
4
The p r o b a b i l i t y o f f a l l i n g on t h e d i a g o n a l i s 0
1
s o that
----------------
b
I’7;”
P(W = 0) = a b .
/’<
/
T o c a l c u l a t e P(W
(1.27)
.t ) i t
w i l l be c o n v e n i e n t t o
a
1
start with positive t , marked on t h e v e r t i c a l axis.
Fig. a
I n e a c h c a s e , P(W
v1
< t)
w i l l be e q u a l t o t h e shaded a r e a
on f i g u r e s below, c o r r e s p o n d i n g t o v a r i o u s r a n g e s o f t . Observe t h a t t h e a r e a below t h e d i a g o n a l , and a l s o below t h e l i n e v 2 = b i s a l w a y s s h a d e d : i n d e e d , below t h e d i a g o n a l we have I a ( v l ) >, I b ( v 2 ) , hence FJ
= I
b
- ‘a
5 0 4, t . On t h e o t h e r hand, below t h e l i n e v 2 = b we have I ( v ) = 0 , hence W = - I a ( v l ) 5 0 < t . b
2
V
f
1 b 1-a b- a
t a Fig, b
1
Fig. c
238
CHAPTER 3
I n case 0 C t
:'
b-a,
t h e s i t u a t i o n i s a s p r e s e n t e d on
F i g . b . The shaded a r e a i s e q u a l t o 1-unshaded a r e a , hence we have
P(WC t ) = 1
-
1-b ~ ( l t b - 2 t )for 0
<
t .' b-a,
(1.28)
s i n c e t h e t r a p e z o i d which r e m a i n s unshaded h a s h e i g h t 1-b and s i d e s 1-t and b - t .
-
a
t
-
a , t h e s i t u a t i o n i s p r e s e n t e d on F i g . c . I n t h i s c a s e we have for t h e shaded a r e a :
For b
P(W
&
5 1
t ) = 1 - a ( 1 - b ) - ( 1-a-t ) 2
, b-a c- t c
1-a. ( 1 . 2 9 )
<
V
V
.g2 1
1
a
;
a
Fig. d The n e x t c a s e i s 1-a
Fig. e
c t
C,
b . Here we have t h e s i t u a -
t i o n a s on F i g . d . I n t h e c o n s i d e r e d r a n g e of t , t h e unshaded a r e a d o e s n o t change w i t h t , s o t h a t w e have
P(W C t )
= 1 - a(1-b)
f o r 1-a 5 t
<
b.
(1.30)
F i n a l l y , i n t h e c a s e when b i t C 1, t h e s i t u a t i o n i s an on F i g . e . We have h e r e
239
PSYCHOLOGICAL PROBLEMS OF CONSTRUCTION
P(W ( t) = 1
-
a ( 1 - t ) for b
<
t
5 1.
(1.31)
S i m i l a r geometric c o n s i d e r a t i o n s l e a d t o determining t h e v a l u e s P ( W c t ) for t 2 0 i n t h e r e m a i n i n g c a s e s , as d e t e r m i n e d by t h e v a l u e s a+b and comparison o f a and b . F o r n e g a t i v e v a l u e s o f t , w e may use t h e r e s u l t s for t 2 0, since
= PCIa(v,)
- Ib(V2)
>
tl
which shows t h a t t h e r e s u l t s for n e g a t i v e t may be obt a i n e d from t h e r e s u l t s f o r p o s i t i v e t b y i n t e r c h a n g i n g t h e r o l e s o f a and b and s u b t r a c t i o n from 1. P u t t i n g t h e above o c o n s i d e r a t i o n s t o g e t h e r , w e have for a t b > 1, a 5 b f o r t 5 -1
0 b(l+t) b(1-a)
= const.
2
b(1-a) t (1-b+t) / 2
F(t) =
f o r -1 <. t /= -a for -ad t C - ( l - b ) f o r - ( 1-b) < t 0
<
mass ab a t t h e p o i n t t = 0 2 f o r 0 i t (i b-a 1 - (l-b)(ltb-2t) /2 1 - a ( 1-b) (1-a-t)2/2 f o r b-a < t C 1-a 1 - a(1-b) = const f o r b < t S l 1 for t > 1.
-
For t h e formulas i n t h e t h r e e remaining c a s e s , s e e Nowakowska ( 1 9 8 1 ) .
240
CHAPTER 3
1 . 6 . 5 . Some r e s u l t s .
As m e n t i o n e d , t h e model was a p p l i e d t o t h e d a t a from i n t e r v i e w s ; t h e s e r e s u l t s , o b t a i n e d by s i m u l a t i n g t h e a n s w e r s a n d t h e n m a t c h i n g t h e res u l t s w i t h t h e d a t a , are presented i n Table 3. N o r m a l l y , s u c h d a t a are n o t a v a i l a b l e : t h e o n l y i n f o r m a t i o n i s t h e g i v e n r e s p o n s e . It may be h o p e d , h o w e v e r , t h a t i t would b e u l t i m a t e l y p o s s i b l e t o r e l a t e t h e v a l u e s o f p a r a m e t e r s a , b , u and k w i t h t h e c o n t e n t of t h e q u e s t i o n . I f t h i s i s s u c c e s s f u l , it w i l l be p o s s i b -
l e t o correct t h e r e s u l t s of sociological interviews
for v a r i o u s b i a s e s . estimated parameters
Q u e s ti o n
estimated probability p
a
b
U
k
P
1. R e g i s t e r e d since l43?
.15
.a5
.45
.OO
.69
2 . Voted i n t h e ' 4 8 Presidential e l e c tion?
.20
.85
.35
-00
.61
.20
.76
.97
.12
.21
.25
.75
.74
.12
.32
.25 1.00
.74
.02
.34
verls licence?
.30
.70
.35
.oo
.46
Do y o u own your house?
.35
.52
.12
.oo
.54
3 . Voted i n ' 4 7 c i t y charter election?
4 . V o t e d i n 146 congressional election?
5. C o n t r i b u t e d t o
Community C h e s t ?
6. Do you own a d r i -
7.
T a b l e 3. E s t i m a t e d parameters
Looking a t T a b l e 3 it a p p e a r s t h a t one might reasonab-
PSYCHOLOGICAL PROBLEMS OF CONSTRUCTION
241
l y e x p e c t some r e l a t i o n between t h e p a r a m e t e r s and t h e
c o n t e n t o f t h e q u e s t i o n . Remembering t h a t a and b a r e t h e t h r e s h o l d s for t e n d e n c i e s f o r ' ' y e s r 1 and "no" t o o p e r a t e , we s e e t h a t t h e r e i s a marked tendency t o s a y ''yes1', even f a l s e l y , i n a l l t h e q u e s t i o n s . T h i s t e n dency i s l e a s t for t h e l a s t q u e s t i o n , about owning t h e house. W i t h tendency t o ''no''
, the situation
i s j u s t t h e oppo-
s i t e . It i s highest i n t h e case o f t h e l a s t q u e s t i o n , and i t i s 0 f o r t h e q u e s t i o n about c o n t r i b u t i o n t o t h e Community C h e s t . A p p a r e n t l y t h e answer " y e s " has h e r e a h i g h s o c i a l a p p r o v a l , and c h a n g i n g t r u e ''yes" t o f a l s e "no" d o e s n o t make much s e n s e . Regarding t h e t h i r d column, i t r e p r e s e n t s t h e (un)img ' i r t a n c e o f t h e q u e s t i o n . Here a g a i n t h e r e s u l t s a g r e e w i t h i n t u i t i o n and common s e n s e , w i t h t h e q u e s t i o n about t h e l o c a l e l e c t i o n s b e i n g l e a s t i m p o r t a n t , and a b o u t owning t h e house - most i m p o r t a n t . F i n a l l y , t h e f o u r t h c o l u m n shows g e n e r a l l y t h a t for t h e r e s p o n d e n t s i t seemed q u i t e i m p o r t a n t t o a p p e a r d e c i s i v e and n o t t o g i v e t h e "I d o n ' t know" answer. From t h e p o i n t o f view o f m u l t i c r i t e r i a l d e c i s i o n maki n g , it i s obvious t h a t answering a q u e s t i o n i s j u s t such a p r o c e s s , w i t h t h e c r i t e r i a c o n s i s t i n g of s a t i s fying various tendencies. The s o l u t i o n s s u g g e s t e d h e r e , namely a p a r t i c u l a r way
of combining t h e o p p o s i n g t e n d e n c i e s i s , o f c o u r s e , li-
m i t e d t o t h e t y p e o f problems c o n s i d e r e d . N e v e r t h e l e s s , t h e way o f combining t h e o p p o s i n g t e n d e n c i e s may have applicability also i n other decision processes.
242
CHAPTER 3
A s mentioned, MASIA i s i n f a c t a f a m i l y o f models, which
may be developed i n v a r i o u s d i r e c t i o n s . One o f them i s
t o l o o k f o r o t h e r p a r a m e t e r s . However, t h e t a b l e o f d a t a has o n l y s i x e n t r i e s , o f which f i v e a r e independ e n t , and t h i s l i m i t s t h e number o f p a r a m e t e r s which may be e s t i m a t e d . O f c o u r s e , one c o u l d a l w a y s t r y a comp l e t e l y new way o f p a r a m e t r i z a t i o n . T h i s , however, i s n o t v e r y s i m p l e , s i n c e one i s c o n s t r a i n e d by t h e
psychological f e a s i b i l i t y of t h e process of i n t e r n a l d e c i s i o n s i n a n s w e r i n g . One o f t h e p o s s i b i l i t i e s i s t o i n c l u d e r i s k f a c t o r s , a l o n g t h e r i s k model p r e s e n t e d i n Chapter 5 . F u z z i f i c a t i o n o f t h e model may l e a d n o t o n l y t o some new i n t e r e s t i n g v e r s i o n s o f i t , b u t may a l s o a p p e a r t o have some i m p o r t a n t consequences f o r t h e f o u n d a t i o n s o f f u z z y r e a s o n i n g , and more g e n e r a l l y , f o r t h e t h e o r y o f judgment and p r e f e r e n c e . Such a n a p p r o a c h might a l l o w t o c o n n e c t t h e p r e s e n t model w i t h t h e e x i s t i n g l o g i c a l and l i n g u i s t i c t h e o r i e s o f q u e s t i o n s .
2 . SOME ALTERNATIVE IDEAS I N TESTING
I n C h a p t e r 1, a p s y c h o l o g i c a l t e s t was d e f i n e d as a t o o l which, when a p p l i e d t o a s u b j e c t , y i e l d s a v a l u e o f a random v a r i a b l e ( t e s t s c o r e ) whose d i s t r i b u t i o n depends on t h e v a l u e o f t h e t r a i t o f t h i s s u b j e c t . Now, t h i s g e n e r a l i d e a may be e x p l o i t e d i n many w a y s , w i t h t e s t s c o r e s d e f i n e d i n v a r i o u s ways, not necessar i l y n u m e r i c a l . Two such a p p r o a c h e s w i l l be o u t l i n e d
PSYCHOLOGICAL PROBLEMS OF CONSTRUCTION
243
i n t h i s s e c t i o n , l e a d i n g t o c o n t e x t u a l t e s t s , and t o dynamic q u e s t i o n n a i r e s . I n t h e f i r s t c a s e , t h e p o i n t i s t o l o c a t e t h e p e r s o n ' s v a l u e o f t h e t r a i t (or s h o r t l y : t o l o c a t e t h e o e r s o n ) on a continuum o f t h e t r a i t , g i v e n t h e i n f o r m a t i o n a b o u t h i s c h o i c e s made i n e s p e c i a l l y d e s i g n e d c o n t e x t s i t u a t i o n s . I n t h i s way, "scor e " has t h e form o t h e r t h a n n u m e r i c a l ( a p r o f i l e and a g r a p h ) , , and t h e t h e o r y o f u n f o l d i n g s c a l e s i s t h e main t h e o r e t i c a l t o o l f o r i n f e r e n c e a b o u t t h e v a l u e o f t h e t r a i t from t h e p e r s o n ' s a n s w e r s . These t e s t s l e a d t o f u z z y c l a s s i f i c a t i o n s o f s u b j e c t s , i n t h e s e n s e t h a t t h e y l e a d t o " s t r e n g t h s o f membership" i n various categories. I n t h e second c a s e , o f dynamic q u e s t i o n n a i r e s , w e r e t u r n t o t h e i d e a of q u e s t i o n n a i r e i t e m s , but w i t h t h e p r o v i s i o n t h a t t h e n e x t q u e s t i o n t o be asked may depend on t h e answers t o t h e p r e c e d i n g q u e s t i o n s . The t e r m i n a t i o n o f t e s t i n g i s governed by a s p e c i a l s t o p p i n g r u l e , s o t h a t t h e t o t a l number o f q u e s t i o n s asked i s a random v a r i a b l e . T h i s a p p r o a c h n o t o n l y a l l o w s a mor e economical w a y s o f t e s t i n g , i n t h e s e n s e o f t i m e and c o s t , b u t a l s o opens up new t h e o r e t i c a l and empir i c a l p o s s i b i l i t i e s i n t e s t t h e o r y . Moreover, i t may prove u s e f u l i n domains s u c h as programmed t e a c h i n g , information storage, e t c .
2.1.
Contextual tests
The f o u n d a t i o n s . We assume t h a t t h e continuum o f t h e t r a i t i s p a r t i t i o n e d i n t o 2n - 1 connected s e t s . From t h e assumption o f c o n n e c t e d n e s s i t f o l l o w s t h a t 2.1.1.
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t h e s e t s a r e i n t e r v a l s , f i n i t e or s e m i - i n f i n i t e , o r d e r ed on t h e continuum. I t w i l l be c o n v e n i e n t t o d e n o t e t h e s u c c e s s i v e s e t s , from l e f t t o r i g h t , by
s o t h a t t h e r e are n s e t s Ai
and n-1 s e t s B i , i + l .
We s h a l l assume t h a t a l l s e t s ( 2 . 1 ) e x c e p t A1 and An a r e of t h e same l e n g t h . The problem w i l l be t o d e t e r m i n e t h e p o s i t i o n o f t h e s u b j e c t on t h e continuum, up t o i t s l o c a t i o n i n one o f the sets (2.1).
The i d e a o f t h e t e s t i s based on t h e u n f o l d i n g p r i n c i p -
l e , and may be d e s c r i b e d as f o l l o w s . Suppose t h a t we choose a s t i m u l u s , r e p r e s e n t a t i v e for e a c h o f t h e s e t s Ai i n ( 2 . 1 ) . The e x a c t n a t u r e o f t h e s e s t i m u l i w i l l be d e s c r i b e d l a t e r ; a t p r e s e n t , we d e n o t e t h e s e s t i m u l i
Sn, s o t h a t Si i s t h e s t i m u l u s c o r r e s p o n d i n g t o t h e s e t A i . These s t i m u l i w i l l be p r e s e n t e d t o the subject i n pairs,
by S l y . . . ,
I n e a c h p a i r , t h e s u b j e c t i s t o choose t h a t s t i m u l u s which he f e e l s i s " c l o s e r " t o h i m , o r d e c i d e t h a t t h e s t i m u l i are " i n e q u i l i b r i u m " . L e t u s f o r a moment assume t h a t t h e c h o i c e s a r e n o t s u b j e c t t o any random e r r o r . Then t h e i n f e r e n c e about t h e v a l u e o f t h e s u b j e c t ' s t r a i t i s based o n t h e f o l l o w i n g h y p o t h e s i s c o n n e c t i n g t h e c h o i c e s from p a i r s of
PSYCHOLOGICAL PROBLEMS OF CONSTRUCTION
245
s t i m u l i and t h e l o c a t i o n o f t h e s u b j e c t on t h e c o n t i nuum. HYPOTHESIS. I n c h o o s i n g from a p a i r o f s t i m u l i ( S i , S . ) , J
t h e s u b j e c t w i l l choose t h a t s t i m u l u s which i s c l o s e r t o h i s own l o c a t i o n on t h e continuum o f t h e t r a i t ( i t i s assumed f o r s i m p l i c i t y t h a t t h e s u b j e c t s and s t i m u l i a r e l o c a t e d a t c e n t r e s of t h e c o r r e s p o n d i n g s e t s ( 2 . 1 ) . In case of equal distances, t h e subject w i l l decide that the stimuli are i n equilibrium.
F o r i n s t a n c e , a s u b j e c t from t h e s e t A2, when c o n f r o n t ed w i t h a c h o i c e between s t i m u l i S l and S4 w i l l choose S1. When c o n f r o n t e d w i t h c h o i c e between s t i m u l i S1 and
S2, he w i l l choose S
2'
etc.
S i m i l a r l y , a s u b j e c t from t h e s e t
B23,
when c h o o s i n g
from t h e p a i r S1, S4 w i l l d e c i d e t h a t t h e s e s t i m u l i a r e i n e q u i l i b r i u m , s i n c e he i s e q u i d i s t a n t from them. One may i n t e r p r e t t h e c h o i c e p r i n c i p l e a s s e r t e d i n t h e H y p o t h e s i s as f o l l o w s : t h e s c a l e (continuum o f t h e t r a i t ) becomes " f o l d e d " a t t h e l o c a t i o n o f t h e s u b j e c t . The two h a l f - l i n e s f a l l t h e n one on a n o t h e r , and t h e s t i mulus which f a l l s c l o s e r t o t h e new o r i g i n i s c h o s e n . T h i s i s i l l u s t r a t e d on f i g u r e s below.
F o l d i n g a t S2
F o l d i n g a t midpoint between S1 and S2
246
CHAPTER 3
Obviously, t h e choice
can be r e p r e s e n t e d as a d i r e c t e d
g r a p h , w i t h a r r o w s i n d i c a t i n g t h e c h o i c e , and l a c k o f a n arrow i n d i c a t i n g e q u i l i b r i u m . F o r i n s t a n c e , assume t h a t n = 4 ( s o t h a t t h e r e a r e s e v e n areas a l t o g e t h e r , f o u r o f them r e p r e s e n t e d by s t i m u l i ) . Suppose t h a t t h e c h o i c e s made a r e as marked w i t h a s t e r i s k s :
Then
t h e g r a p h would b e
Such a g r a p h i s more c o n v e n i e n t l y r e p r e s e n t e d i n form of a v e c t o r , w i t h c o o r d i n a t e s l a b e l l e d i j (correspondi n g t o p a i r ( S i , S . ) o f s t i m u l i ) , and v a l u e s 1,-1 o r 0 J
depending whether t h e f i r s t or second element was chose n , or t h e r e was a n e q u i l i b r i u m . Thus, t h e above g r a p h w i l l have t h e v e c t o r
12 -1
13 0
14 1
23 0
24 1
34 0
Under t h e h y p p t h e s i s f o r m u l a t e d a b o v e , e a c h a r e on t h e continuum has i t s c o r r e s p o n d i n g g r a p h , or e q u i v a l e n t l y ,
i t s c o r r e s p o n d i n g v e c t o r , t o be c a l l e d t h e c h a r a c t e r i s t i c vector. For the case of n =
4 ( h e n c e f o r seven a r e a s ) ,
v e c t o r s a r e g i v e n i n t h e t a b l e below.
these
247
PSYCHOLOGICAL PROBLEMS OF CONSTRUCTION
area
A1
B12
A2
B23
A3
B34 A4
12
13
14
23
24
34
1
1
1
1
1
1
0
1
1
1
1
1
-1
0
1
1
1
1
-1
-1
0
0
1
1
-1
-1
-1
-1
0
1
-1
-1
-1
-1
-1
0
-1
-1
-1
-1
-1
-1
2.1.2. The p r i n c i p l e of e s t i m a t i o n . I n p r a c t i c a l s i t u a t i o n s , t h e c h o i c e i s t o some e x t e n t random, and one cannot expect it t o f ollow e x a c t l y t h e hypothesis of t h e p r e c e d i n g s e c t i o n . It may, however, be r e a s o n a b l y expected t h a t t h e choices w i l l s a t i s f y t h e following stochastic version of the hypothesis. HYPOTHESIS. I n c h o o s i n g between a p a i r of s t i m u l i , t h e s u b j e c t i s more l i k e l y to choose t h a t s t i m u l u s which
i s c l o s e r t o h i s own l o c a t i o n on t h e continuum. L e t u s now t a k e t h e d i s t a n c e between v e c t o r s X and Y as n
i=1 I f X and Y a r e two v e c t o r s r e p r e s e n t i n g g r a p h s , t h e n
i s simply t h e number o f changes n e c e s s a r y t o t r a n s f o r m t h e g r a p h c o r r e s p o n d i n g t o one v e c t o r t o t h e
d(X,Y)
o t h e r g r a p h , i f removing an arrow o r a d d i n g i t i s
248
CHAPTER 3
i s c o u n t e d as one change, and r e v e r s i n g an arrow i s
counted a s two c h a n g e s .
...
L e t now X1,
X12, X 2 , d e n o t e t h e c h a r a c t e r i s t i c vect o r s of t h e s e t s A1, B12, A 2 and l e t Y d e n o t e t h e
,...,
v e c t o r o b t a i n e d from t h e t e s t ( t h e s u b j e c t ' s s c o r e ) . One may now c a l c u l a t e t h e d i s t a n c e s
and t a k e as a n e s t i m a t e o f t h e p o s i t i o n o f t h e s u b j e c t t h a t area which i s c l o s e s t t o Y ( t h e d i s t a n c e i s minim a l ) . If t h e r e a r e two o r more s u c h a r e a s , one may t a k e t h e i r goemetric c e n t e r as an e s t i m a t e . F o r i n s t a n c e , suppose 'chat n = 4 , and t h a t t h e g r a p h corresponding t o vectop Y i s
s o t h a t we have
Y = [ 12
13
14
23
24
1
0
a
1
1
34, -1
We have t h e n ( s e e t h e t a b l e o f c h a r a c t e r i s t i c v e c t o r s ) :
The minimal d i s t a n c e i s 4 t o v e c t o r X1, e s t i m a t e d v a l u e o f t h e t r a i t i s A1.
s o that the
PSYCHOLOGICAL PROBLEMS OF CONSTRUCTION
2 49
2.1.3. C o n s t r u c t i o n o f t h e t e s t . The g e n e r a l p r i n c i p l e p r e s e n t e d i n t h e p r e c e d i n g s e c t i o n must now be implemented by c o n s t r u c t i o n o f t h e s t i m u l i S l y S2,.... One o f t h e p o s s i b i l i t i e s h e r e i s as f o l l o w s .
Suppose t h a t w i t h e a c h area A1, A2,..., A one may asn s o c i a t e a s e t of' m " m i c r o s t i m u l i " , s o t h a t
.. . . . . . . . sn
= [ s (, n ) y . . . ,
sm q .
These m i c r o s t i m u l i may be s e n t e n c e s , a s s e r t i o n s , p i c t u r e s , p h o t o g r a p h s , e t c . F o r t h e sake o f p r e s e n t a t i o n , a r e s t a t e m e n t s which one may l e t us assume t h a t e n d o r s e or n o t . These s t a t e m e n t s a r e c o n s t r u c t e d s o a s t o be r e p r e s e n t a t i v e for v a r i o u s a r e a s , i n t h e s e n s e t h a t a p e r s o n from a r e a Si would be most l i k e l y to end o r s e s t a t e m e n t s c o r r e s p o n d i n g t o h i s a r e a , and l e s s l i k e l y t o e n d o r s e s t a t e m e n t s from o t h e r a r e a s , w i t h t h e p r o b a b i l i t y o f e n d n r s i n g d e c r e a s i n g as t h e two a r e a s i n q u e s t i o n become more d i s t a n t . A p a i r of s t i m u l i ( S i , S . ) w i l l simply be a s e t of 2m J s t a t e m e n t s c o r r e s p o n d i n g t o S and S The s u b j e c t i s i j i n s t r u c t e d t o choose e x a c t l y m o f them, which he f e e l s
.
a r e " c l o s e s t " t o h i m , i . e . s u c h t h a t he f e e l s most a p p r o p r i a t e t o e n d o r s e ( i n t h e g i v e n s e t ) . The s t a t e ments, o f c o u r s e , a r e e i t h e r a r r a n g e d a t random, or matched i n p a i r s , w i t h i n s t r u c t i o n t h a t t h e s u b j e c t i s t o e n d o r s e one s t a t e m e n t from e a c h p a i r .
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Thus, t h e s u b j e c t does n o t know from which s e t t h e s t a t e m e n t s a.re coming, n o r i s he aware t h a t t h e y b e l o n g to different sets. Naturally, each statement appears several times i n t h e t e s t , combined w i t h o t h e r s t a t e m e n t s , and t h e subj e c t i s made aware o f t h i s f a c t . H e s h o u l d r e a l i z e t h a t h i s c h o i c e o f one s t a t e m e n t from a g i v e n s e t (cont e x t ) does n o t mean t h a t he s h o u l d choose t h e same s t a tement i n a n o t h e r c o n t e x t . When a c h o i c e of m s t i m u l i o u t of 2 m i s made, k of them w i l l come from t h e s e t S and m-k w i l l come from t h e i' set S High v a l u e s of k w i l l i n d i c a t e t h e s t i m u l u s S j'
i
was c h o s e n , w h i l e low v a l u e s of k i n d i c a t e t h e c h o i c e of t h e st i mulus S Intermediate values indicate that j' S . and S . are i n e q u i l i b r i u m . The c h o i c e of m , as w e l l 1 J as t h e c h o i c e of c r i t i c a l v a l u e s o f k i n d i c a t i n g v a r i o u s c h o i c e s must be based on a p p r o p r i a t e s t a t i s t i c a l considerations.
A possible selection f o r m i s here m = 6 , so that there
are 1 2 s t a t e m e n t s i n a s e t , 6 o f them from one s t i mulus and 6 from t h e o t h e r . Domination o f S o v e r S i
j
o c c u r s i f 5 o r 6 s t a t e m e n t s a r e from Si, w h i l e domina t i o n o f S o v e r Si o c c u r s i f o n l y one or 0 s t a t e m e n t s j a r e from S The r e m a i n i n g v a l u e s k = 2 , 3 , 4 i n d i c a t e i'
equilibrium. A j u s t i f i c a t i o n h e r e i s as f o l l o w s . F i r s t l y , i t i s
c l e a r t h a t t h e number of s t a t e m e n t s i n a s e t s h o u l d n o t be t o o l a r g e ( u n l e s s t h e y a r e matched i n p a i r s ) , s i n c e t h e s u b j e c t s would t y p i c a l l y e n c o u n t e r d i f f i c u l t i e s i n
a s s e s s i n g a l a r g e s e t o f s t a t e m e n t s and c h o o s i n g e x a c t l y h a l f o f them. I n o t h e r words, t o o l a r g e s e t s o f s t a -
PSYCHOLOGICAL PROBLEMS OF CONSTRUCTION
25 1
t e m e n t s would i n t r o d u c e t o o much randomness i n t h e c h o i c e , beyond what i s n e c e s s a r y and manageable. Thus, t h e number o f s t a t e m e n t s ought t o be s m a l l , s a y 8 , 1 0 o r 1 2 . The v a l u e 1 2 seems b e s t , i n t h e s e n s e t h a t i t a l l o w s t o e l i m i n a t e , on t h e s i g n i f i c a n c e l e v e l a b o u t 0 . 0 5 , t h e f l u c t u a t i o n s which may w i s e i n p u r e l y random c h o i c e . I n d e e d , i f we have 6 s t a t e m e n t s from Si and 6 from S t h e n under p u r e l y random c h o i c e t h e p r o b a b i l i t y j’ t h a t a l l s e l e c t e d s t a t e m e n t s would come from Si e q u a l s 12 1 / ( 6 ) = 1 / 9 2 4 which i s a b o u t 0 . 0 0 1 . T h i s i s due t o t h e f a c t that t h e r e are = 924 ways f o r c h o o s i n g 6 ob-
(F)
j e c t s o u t of 1 2 . S i m i l a r l y , t h e p r o b a b i l i t y o f choosing 5 s t a t e m e n t s under p u r e l y random c h o i c e , from Si and one from S j’ is
6
s i n c e t h e r e e x i s t ( ) = 6 ways o f c h o o s i n g 5 o b j e c t s 6 5 o u t o f 6, and (1) = 6 ways of c h o o s i n g one o b j e c t o u t of t h e remaining 6. C o n s e q u e n t l y , under a p u r e l y random c h o i c e of s i x o u t o f 1 2 s t a t e m e n t s , t h e p r o b a b i l i t y t h a t one s t i m u l u s w i l l b e t a k e n as d o m i n a t i n g o v e r t h e o t h e r ( w h i l e i n 6 6 12 f a c t t h e r e i s no d o m i n a t i o n ) i s [ ( ) ( ) + I]/( 6 ) 2
5
1
The p r o b a b i l i t y of a wrong c o n c l u s i o n about dom i n a t i o n i n e i t h e r d i r e c t i o n i s t w i c e t h a t h i g h , hence equals about 0 . 0 8 . 0.04.
I n s t a t i s t i c a l t e r m i n o l o g y , t h e above c o n s i d e r a t i o n s c o r r e s p o n d t o a two-sided c r i t i c a l r e g i o n f o r dominat-
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o n , w i t h t h e s i g n i f i c a n c e l e v e l about 0 . 0 8 , f o r t h e v e r i f i c a t i o n o f t h e h y p o t h e s i s of complete randomness of c h o i c e . The s i g n i f i c a n c e l e v e l 0 . 0 8
d i f f e r s from t h e t r a d i t i o n -
a l l e v e l s 0 . 0 5 o r 0 . 0 1 . T h i s i s because t h e random v a r i a b l e i n q u e s t i o n assumes o n l y t h e v a l u e s 0,1,...,6 and i t i s n o t p o s s i b l e t o c o n s t r u c t a c r i t i c a l r e g i o n w i t h s i g n i f i c a n c e l e v e l e x a c t l y 0 . 0 5 . The v a l u e 0 . 0 1 i s somewhat t o o s m a l l , s i n c e w i t h s u c h h i g h s i g n i f i c a n c e l e v e l t h e c h o i c e s would seldom show d o m i n a t i o n a t all. Analogous c a l c u l a t i o n s f o r t h e c a s e o f 1 0 s t a t e m e n t s and t h e c h o i c e o f 5 o u t o f 1 0 show t h a t one cannot b u i l d a c r i t i c a l region w i t h significance value closer t o 0.05 than the value 0 . 0 8 obtained f o r 12 statements. I n d e e d , under random c h o i c e of 5 s t a t e m e n t s o u t o f 1 0 t h e p r o b a b i l i t y of g e t t i n g a l l f i v e o u t o f one s e t i s 1/( 10 ) = 1 / 2 5 2 = 0 . 0 0 4 , w h i l e t h e p r o b a b i l i t y o f g e t t i n g 5 5 5 10 4 o u t of one s e t and 1 o u t o f a n o t h e r s e t i s (4)(1)/( 5 = 2 5 / 2 5 2 Y 0.1.
Thus, i f we a c c e p t e d t h a t one area dominates a n o t h e r i f a l l f i v e s t a t e m e n t s a r e from t h i s a r e a , w e would g e t s i g n i f i c a n c e l e v e l about 0 . 0 0 8 , w h i l e i f w e t o o k as dom i n a t i o n c h o i c e o f 5 or 4 s t a t e m e n t s from one s e t , t h e s i g n i f i c a n c e l e v e l would b e about 0 . 2 . S i m i l a r c a l c u l a t i o n s a l l o w u s t o e l i m i n a t e t h e c a s e of choice of 4 statements out of 8.
Some p r e l i m i n a r y s i m u l a t i o n r e s u l t s . I n o r d e r t o check t h e d i s c r i m i n a b i l i t y o f t h e t e s t , some p r e l i 2.1.4.
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253
minary s i m u l a t i o n s t u d i e s were c a r r i e d o u t f o r t h e cas e n = 7 ( s o t h a t t h e r e were 7 a r e a s ) on t h e continuum under c o n s i d e r a t i o n ) . These a r e a s were a s s i g n e d t h e v a l u e s 1;1.5; 2; 2 . 5 ; 3; 3.5 and 4. It was assumed t h a t i n c h o o s i n g from a p a i r o f s t a t e m e n t s from u and v ( w i t h u C v ) , b y a p e r s o n from a r e a x , t h e p r o b a b i l i t y ' of c h o o s i n g a s t a t e m e n t from u ( h e n c e t h e one "more pu to the left") is
PU
=
c + a(v-u) 1-c-a(v-u) (v-x) / ( v - u ) 4 t b(v-U)/(V-X) - b(v-U)/(X-U)
for for for for for
x = u L V u 4 v = x u C X Cv X LU C v u L v C x,
where a , b and c a r e some c o n s t a n t s . For s i m u l a t i o n , t h e v a l u e s were a = 0 . 1 , b = 0.5 and c = 0 . 6 .
For i n s t a n c e , i f a p e r s o n i s from A3 ( i . e . x = 3 ) , t h e n i n c h o o s i n g between a s t a t e m e n t from a r e a A and a s t a 1 tement from area A 2 ( i . e . u = 1, v = 2 ) , t h e p r o b a b i l i t y o f c h o o s i n g a s t a t e m e n t from A1 i s $ - 0 . 5 ( 2 - 1 ) / ( 3 - 1 ) = 0 . 2 5 . I n c h o o s i n g between a s t a t e m e n t from A1 and a s t a t e m e n t from A4 ( i . e . u = 1, v = 4 > , t h e p r o b a b i l i t y o f c h o o s i n g t h e f o r m e r i s (4-3)/(4-1) = 0 . 3 3 , and s o on.
For e a c h of t h e v a l u e s x = 1, 1 . 5 , 2 and 2 . 5 t h e r e were 10 s i m u l a t e d t r i a l s , each c o n s i s t i n g of choosing 6 o u t o f 1 2 s t a t e m e n t s from e a c h o f t h e c o r r e s p o n d i n g s e t s of s t i m u l i .
A t y p i c a l c h o i c e , for x
=
2 . 5 , was
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12 1-5
13 1-5
23
14 3-3
24 4-2
3-3
34
5-1
s o t h a t t h e v e c t o r c o r r e s p o n d i n g t o t h e g r a p h o f domination i s [-1, -1, 0 , 0 , 0 , 1 3 . An easy check shows t h a t t h e d i s t a n c e s from t h e v e c t o r s
c h a r a c t e r i s t i c for t h e s e t s A1, B 1 2 , . . . , A4 a r e 7 , 6, 4 , 1, 2 , 5 s o t h a t t h e e s t i m a t e d v a l u e e q u a l s t o t h e
t r u e value x = 2.5. I n g e n e r a l , t h e r e s u l t s o f s i m u l a t i o n s were as f o l l o w s : t r u e value
estimated value ( # of c a s e s )
1.5
2
5
4.5
0.5
1
7 4
2
1
1 1.5 2
2.5
3 1
2.5
3
3 9
The t r i a l s f o r v a l u e s x = 3 , 3.5 and 4 were n o t r u n because o f t h e symmetry o f t h e problem. The n o n i n t e g e r v a l u e s a p p e a r e d because o f t i e s i n t h e e s t i m a t e d val u e s , i . e . t h e minimum o c c u r r i n g a t more t h a n one v a l u e . T h i s t a b l e shows, f o r i n s t a n c e , t h a t when t h e t r u e va-
l u e was x = 1, t h e e s t i m a t e d v a l u e was a l s o 1 i n 5 o u t o f 10 c a s e s ; i n f o u r c a s e s i t was e q u a l 1 . 5 , and i n one c a s e t h e minimum was a t p o i n t s 1 . 5 and 2.
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255
S i m i l a r l y , when t h e t r u e v a l u e was x = 1 . 5 , t h e e s t i m a t ed v a l u e was a l s o 1.5 i n 7 o u t o f 10 c a s e s . I n one cas e i t was 1, and i n two c a s e s i t was 2 . I t may be s e e n t h a t t h e w o r s t r e s u l t s were o b t a i n e d for
x = 2 : t h e f u z z i n e s s o f t h e e s t i m a t e s was l a r g e s t . S t i l l , i n no c a s e t h e error exceeded 0 . 5 . These r e s u l t s a r e , o f c o u r s e , o n l y p r e l i m i n a r y , and f a r from b e i n g c o n c l u s i v e , b e c a u s e o f t h e small s i z e o f t h e sample, and b e c a u s e one needs a l s o o t h e r v a l u e s o f t h e p a r a m e t e r s . N e v e r t h e l e s s , t h e y i n d i c a t e t h a t t h e method of e s t i m a t i o n by l o o k i n g a t t h e " c l o s e s t graph" may be more p r e c i s e t h a n t h e method of s i m p l e s c o r i n g t h e numb e r o f c h o i c e s from e a c h o f t h e s t i m u l i . A t t h e c o n c l u s i o n , it i s worth t o p o i n t o u t t h a t t h e
c o n t e x t u a l t e s t s are c h a r a c t e r i z e d by t h e f a c t t h a t t h e y make a d e l i b e r a t e u s e n o t o n l y o f t h e c o n t e x t , b u t a l s o o f f u z z i n e s s . I n d e e d , i n t h e u s u a l c h o i c e between s t i m u l i , t h e s u b j e c t ' s d e c i s i o n i s e i t h e r b i n a r y or t e r n a r y , i f t h e t i e s are a l l o w e d . Here, because o f t h e complex n a t u r e o f t h e s t i m u l u s , t h e c h o i c e may be r e g a r d e d as f u z z y , I n d e e d , i n c a s e o f a s p l i t s u c h a s 4 - 2 , one can s a y t h a t t h e c h o i c e was made i n 2/3 f o r one s t i m u l u s and l/3 f o r t h e o t h e r s t i m u l u s . S u r e l y , f o r t h e c o n s t r u c t i o n o f t h e g r a p h , i t was nec e s s a r y t o a d o p t a c o n v e n t i o n which removes f u z z i n e s s , b u t s t i l l t h e c o n t e x t u a l t e s t s show how f u z z i n e s s may be u s e f u l l y employed ( b e s i d e randomness) t o i n c r e a s e t h e p r e c i s i o n o f measurement, and a l s o t o a l l o w r i c h e r psychological i n t e r p r e t a t i o n s of the r e s u l t s .
256
2.2.
CHAPTER 3
Dynamic q u e s t i o n n a i r e
T h i s s e c t i o n w i l l be d e v o t e d t o p r e s e n t a t i o n o f a n o t h e r
measurement t o o l i n p s y c h o l o g y , namely t h e dynamic ques t i o n n a i r e . I t d i f f e r s from t h e t r a d i t i o n a l q u e s t i o n n a i r e by system o f a s k i n g q u e s t i o n s . I n c l a s s i c a l ques t i o n a i r e , t h e r o l e of experimenter i s r a t h e r passive: u s u a l l y he would o n l y d i s t r i b u t e t h e forms, r e a d t h e i n s t r u c t i o n s , and c o l l e c t t h e r e s u l t s , p e r h a p s o c c a s i o n a l l y a n s w e r i n g some q u e r i e s and c l e a r i n g some a m b i g u i t i e s . A s opposed t o t h a t , i n dynamic q u e s t i o n n a i r e s , t h e experimenter’s r o l e i s a c t i v e i n asking questions, o r p r e s e n t i n g i t e m s , i n an o r d e r depending on t h e previ o u s answers o f t h e s u b j e c t . C o n s e q u e n t l y , e v e r y p e r s o n may answer o n l y some o f t h e q u e s t i o n s . To u s e an example, a t r a d i t i o n a l q u e s t i o n n a i r e may b e compared t o a v i s a a p p l i c a t i o n form, which would u s u a l l y cont a i n q u e s t i o n s s u c h as a b o u t maiden name, e t c , whether o r n o t t h e form i s f i l l e d b y males o r f e m a l e s . The dynamic q u e s t i o n n a i r e may be compared t o a c l i n i c a l i n t e r v i e w , where n e x t q u e s t i o n depends l a r g e l y on t h e r e p l i e s t o t h e preceding ones.
The f o r m a l scheme i n g e n e r a l c a s e . The g e n e r a l scheme o f a dynamic q u e s t i o n n a i r e may be d e s c r i b e d b y s p e c i f i c a t i o n o f t h e n e x t d e c i s i o n (which may be e i t h e r t h e n e x t q u e s t i o n t o b e a s k e d , or t h e d e c i s i o n t o s t o p 2.2.1.
q u e s t i o n i n g and t o a c c e p t f i n a l s c o r e , n o t n e c e s s a r i l y one d i m e n s i o n a l . To d e s c i r i b e i t f o r m a l l y , assume f o r s i m p l i c i t y t h a t e a c h q u e s t i o n may be answered o n l y “ y e s ” o r “no” ( t h i s
257
PSYCHOLOGICAL PROBLEMS OF CONSTRUCTION
i s n o t a s e v e r e r e s t r i c t i o n of g e n e r a l i t y ; t h e model may be e a s i l y a d a p t e d t o t h e c a s e o f m u l t i p l e p o s s i b i l i t i e s i n answers.) 4
Suppose t h a t a l l q u e s t i o n s a r e l a b e l e d , and l e t K be t h e s e t o f a l l l a b e l s . G e n e r a l l y , a l a b e l may be a number,
or a v e c t o r ( e . g . i n c a s e t h e q u e s t i o n s a r e p a r t -
i t i o n e d i n t o groups and numbered w i t h i n e a c h g r o u p ; i n t h i s c a s e a l a b e l may be a p a i r ( k , r ) , where k i s t h e number o f t h e group and r i s t h e number o f i t e m i n t h e group, e t c . )
.
L e t p k ( x ) be t h e p r o b a b i l i t y of answer " y e s " t o t h e q u e s t i o n l a b e l e d k b y a p e r s o n w i t h v a l u e x of t h e
trait. A dynami.c q u e s t i o n n a i r e must s p e c i f y
( a ) t h e l a b e l , k l , o f t h e q u e s t i o n t o be asked f i r s t ; ( b ) t h e r u l e , which t o e v e r y p a r t i a l h i s t o r y ( a n s w e r s t o q u e s t i o n s asked t h u s f a c r ) a s s i g n s e i t h e r t h e l a b e l o f t h e n e x t q u e s t i o n , or t h e f i n a l s c o r e .
...
L e t il, i 2 ,
s t a n d f o r answers t o s u c c e s s i v e q u e s t -
i o n s , w i t h in = 1 s i g n i f y i n g t h e answer " y e s " ,
and in
= -1 s i g n i f y i n g t h e answer "no".
Then a " p a r t i a l h i s t o r y " i s a s t r i n g o f t h e form (kl,il; where kl, and i l , i
k2,.
2'"'
k2,i2;
...,
kn > i n )
.,
a r e l a b e l s of c o n s e c u t i v e q u e s t i o n s , a r e answers t o them.
If D stands f o r the set of a l l p o s s i b l e f i n a l scores
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t h e n a dynamic q u e s t i o n n a i r e may be i d e n t i f i e d
with
a s y s t e m o f t h e form
where k l
K i s t h e l a b e l of t h e i n i t i a l q u e s t i o n t o
be a s k e d , and f n : Hn
-+
K
LJ
D
w i t h Hn b e i n g t h e c l a s s of a l l h i s t o r i e s s a t i s f y i n g t h e following conditions:
a . They a r e of l e n g t h n; b.
then f o r every r
I f ( k l Y i l ; , . . . , k ,i-.) Hn,
(which i m p l i e s t h a t ( k l , i l , . . c. I f
(kl,il,...,kn,in)
i and j
.,kr,ir)
n
E Hr);
t h e n ki # k
Hn,
<
j
for all
( t h e same q u e s t i o n i s n e v e r a s k e d t w i c e ) ;
d . w i t h p r o b a b i l i t y one t h e r e w i l l come a moment s u c h that fn(kl,il;...,
kn,in) E
D
( t h e t e s t i n g w i l l sooner o r later terminate). N a t u r a l l y , t h e main problem i s t o c o n s t r u c t t h e s e t D
..
s o t h a t t h e f i n a l s c o r e woulc. p r o v i d e as much i n f o r m a t i o n a b o u t t h e s u b j e c t as p o s s i b le. and t h e r u l e s f l J f 2 , .
PSYCHOLOGICAL PROBLEMS OF CONSTRUCTION
259
N a t u r a l l y , t h e p o s s i b i l i t i e s h e r e depend on t h e probab i l i t i e s p ( x ) of answers. k
The n e x t s e c t i o n w i l l show a s o l u t i o n ( s e e Nowakowska 1 9 6 8 ) which i s based on a random w a l k model, and i s a i m ed a t e s t i m a t i n g t h e v a l u e o f a l i n e a r t r a i t . For ano t h e r s o l u t i o n , s u i t a b l e for t e s t i n g s k i l l s and a b i l i t i e s , see Marshall ( 1 9 8 1 ) .
Random w a l k model. An i m p o r t a n t s p e c i a l c a s e o f a dynamic q u e s t i o n n a i r e i s o b t a i n e d by c o n s t r u c t i o n o f a s p e c i a l random walk. T h i s may be accomplished as follows. 2.2.2.
Assume t h a t q u e s t i o n n a i r e i t e m s a r e p a r t i t i o n e d i n t o g r o u p s , c h a r a c t e r i z e d by t h e same p r o b a b i l i t y of answer " y e s " . Suppose t h a t t h e s e groups a r e numbered fro% -r t o tr, s o t h a t t h e r e a r e 2 r t 1 g r o u p s . F u r t h e r , l e t t h e i t e m s w i t h i n e a c h group be numbered 1,2, ..., and l e t u s t a k e a s t h e l a b e l k t h e p a i r ( k , n ) , where k i s t h e i n d e x o f t h e group, and n i s t h e number o f i t e m i n t h i s group. D e f i n e n o w t h e dynamic q u e s t i o n n a i r e as f o l l o w s . L e t kl = ( O , l ) , i . e .
t h e first q u e s t i o n asked i s always t h e f i r s t q u e s t i o n i n t h e group numbered 0 ( t h e "cent e r " group). a . I f ( k l , i l ,..., k n , in ) 'c H n and kn = ( m , j ) w i t h m f r , mf-r, then fn(kl,il, k ,i ) = (mti,, s), n n where s i s t h e f i r s t i n d e x a v a i l a b l e i n group number-
...,
ed m
+
in.
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T h u s , i f t h e l a s t q u e s t i o n was from any group o t h e r t h a n
t h e e x t r e m a l o n e s , (numbered r and - r ) y t h e n e x t i t e m comes always from t h e group immediately h i g h e r o r i m m e d i a t e l y l o w e r , depending whether t h e answer t o t h e
l a s t q u e s t i o n was "yes1' o r ''no". b . I n c a s e when k = ( r , j ) n a p p l i e s i f in = -1, and i f a p p l i e s i f i = 1, i . e . i f n a t r a n s f e r from group r t o t o group - r t l .
f o r some j , t h e r u l e ( a ) kn = ( - r , j ) , t h e r u l e ( a ) the last question leads t o group r-1, or from group -r
c . I f kn = ( r , j ) or kn = ( - r , j ) f o r some j , and i n ( r e s p . i = -I), t h e n n
= 1
and t h e e x a m i n a t i o n t e r m i n a t e s . T h e f i n a l s c o r e ( n , i n ) shows t h e number o f q u e s t i o n s a s k e d ( n ) , and t h e " e x i t boundary", b e i n g t 1 i f t h e l a s t q u e s t i o n was answered " y e s " , and -1 i f t h e l a s t q u e s t i o n was answered -1 ( s o t h a t t h e l a s t q u e s t i o n i s always from t h e group r o r
-r)
.
Now, t h e s c o r e ( n , i ) would c a r r y t h e i n f o r m a t i o n a b o u t n t h e v a l u e x of t h e t r a i t o n l y i f t h e r e e x i s t a p p r o p r i a t e r e l a t i o n s between t h e p r o b a b i l i t y p k ( x ) o f answer "yes" f o r a p e r s o n w i t h t r a i t l e v e l x , and t h e l a b e l k of t h e item. A p o s s i b l e r e l a t i o n under which one may e x p e c t t o be
a b l e t o e s t i m a t e x on t h e b a s i s o f ( n y i n ) i s s u c h t h a t p k ( x ) , for k = ( m , n ) i s a f u n c t i o n o f t h e group number ( x ) = p,(x), and t h e l a t t e r vam only, so t h a t p (m,n)
PSYCHOLOGICAL PROBLEMS OF CONSTRUCTION
261
l u e i s monotone i n m . Roughly s p e a k i n g , t h e n t h e l e n g t h o f t h e e x a m i n a t i o n would i n d i c a t e w h e t h e r x i s c l o s e r
or more d i s t a n t from t h e c e n t e r of t h e s c a l e , and in would i n f o r m w h e t h e r x i s above or below t h e m i d p o i n t . Naturally
, the
r u l e o f t e r m i n a t i o n o f examining t h e
s u b j e c t given i n ( c ) , connected w i t h t h e e x i t o f t h e random walk t h r o u g h one o f t h e b o u n d a r i e s , i s n o t t h e o n l y o n e p o s s i b l e . One may e a s i l y i m a g i n e o t h e r r u l e s , whcich a r e f e a s i b l e and s h o u l d l e a d t o a s c o r e c o n t a i n i n g i n f o r m a t i o n a b o u t t h e v a l u e x . To g i v e one e x a m p l e ,
a r u l e might be " s t o p when t h e a c c u m u l a t e d number o f a n s w e r s " y e s " r e a c h e s a c e r t a i n l e v e l , and t h e n t a k e a s t h e s c o r e t h e number o f answers "no" , o r i t s oppos i t e , with t h e r o l e s of
and "no" i n t e r c h a n g e d .
The l a t t e r r u l e , or i t s v e r s i o n , i s i n f a c t a p p l i e d ( p e r h a p s on a s u b c o n s c i o u s l e v e l ) i n e x a m i n a t i o n s a i m e d a t t e s t i n g t h e a b i l i t i e s . The e x a m i n e r h a s a number
o f groups of i t e m s , r a n g i n g i n d i f f i c u l t y , w i t h items i n one group b e i n g of t h e same d i f f i c u l t y ( s o t h a t t h e p r o b a b i l i t y o f c o r r e c t r e s p o n s e i s t h e same f o r e a c h i t e m , and e a c h examinee w i t h l e v e l o f a b i l i t i e s x ) . Whenever t h e r e s p o n s e i s c o r r e c t , t h e examiner would c h o o s e t h e n e x t i t e m from t h e group of i t e m s w i t h d i f f i c u l t y h i g h e r by one " u n i t " ( i . e . from t h e group " n e x t t o t h e r i g h t " , and s i m i l a r l y i n c a s e o f a negat i v e r e s p o n s e . The " s t o p p i n g " r u l e may v a r y . T y p i c a l l y , t h e e x a m i n e r would s t o p i f he f i n d s two n e i g h b o u r i n g g r o u p s , s u c h t h a t i n t h e l o w e r o f them t h e number o f c o r r e c t r e s p o n s e s i s h i g h , and i n t h e h i g h e r o f them
it i s low.
26 2
CHAPTER 3
3. APPLICATIONS OF TEST THEORY: METHODOLOGICAL PROBLEMS OF MEASUREMENT OF FUZZY CONCEPTS I n t h i s s e c t i o n w e s h a l l show how t h e i d e a s and r e s u l t s o f t e s t t h e o r y may be u s e f u l i n p r o v i d i n g measurement o f c e r t a i n fuzzy c o n c e p t s ( s e e Nowakowska 1 9 7 9 , 1981). These r e s u l t s y i e l d a l s o an e m p i r i c a l counterpart f o r t h e new a p p r o a c h t o s e m a n t i c s proposed by Zadeh (1981), where he h o l d s t h e view t h a t "almost e v e r y t h i n g t h a t rel a t e s t o n a t u r a l language i s a m a t t e r o f d e g r e e . Thus, i n t e s t - s c o r e s e m a n t i c s , p r e d i c a t e s , p r o p o s i t i o n s and o t h e r t y p e s o f l i n g u i s t i c e n t i t i e s a r e t r e a t e d as c o l l e c t i o n s o f e l a s t i c c o n s t r a i n t s on a s e t o f o b j e c t s o r r e l a t i o n s i n a u n i v e r s e o f d i s c o u r s e " (Zadeh 1 9 8 1 ) . What w i l l be shown i n t h i s s e c t i o n i s t h a t c e r t a i n con-
c e p t s ( u n d e r s t o o d as f u z z y s u b s e t s o f some s e t ) a l l o w
a t e s t measurement o f t h e i r membership f u n c t i o n s , o r more p r e c i s e l y - t h a t i t i s p o s s i b l e t o measure t h e s u i t a b l y d e f i n e d d i s t a n c e between f u z z y c o n c e p t s . Roughly, t h e s e c o n c e p t s w i l l be s u c h which may b e decomposed i n t o a l a r g e number of " e l e m e n t a r y " c o n c e p t s , t h e l a t t e r b e i n g m e a s u r a b l e by some a p p r o p r i a t e i t e m s . The a n a l y s i s w i l l s t a r t from a c e r t a i n t h e o r y o f conc e p t s t r u c t u r e , which w i l l a l l o w d i s t i n g u i s h i n g t h o s e c o n c e p t s which admit t e s t measurement. It i s w o r t h t o mention t h a t t h e s u g g e s t e d t h e o r y o f con-
c e p t s c a n a l s o be c o n s i d e r e d i n i t s dynamic a s p e c t s , where c o n c e p t s a r e r e p r e s e n t e d as g r a p h s which e v o l v e randomly, w i t h e d g e s ( r e p r e s e n t i n g s e m a n t i c r e l a t i o n shops between c o n s t i t u e n t p a r t s o f t h e c o n c e p t ) a p p e a r and d i s a p p e a r a c c o r d i n g t o some random mechanism. T h i s
PSYCHOLOGICAL PROBLEMS OF CONSTRUCTION
263
a p p r o a c h m i g h t l e a d t o a new way o f a n a l y s i n g t h e t o p i c s o f c o n c e p t development and change.
3 . 1 . Fuzzy s e t s and f u z z y i d e n t i t y L e t X be a f i x e d s e t . A f u z z y s u b s e t A of X i s , by d e f i n i t i o n , a function fA:
x
3
[O,ll
(3.1)
( s e e Zadeh 1965). The f u n c t i o n f A i s c a l l e d bhe memb e r s h i p f u n c t i o n ; t h e r e l a t i o n f ( x ) = 1 means t h a t t h e A point x belongs t o A i n f u l l degree, while f A ( x ) = 0 means t h a t x d o e s n o t b e l o n g t o A a t a l l . The i n t e r n e d i a t e v a l u e s r e p r e s e n t " p a r t i a l membership" o f x i n t h e fuzzy s e t A . The s e t t h e o r e t i c a l o p e r a t i o n s on f u z z y s e t s a r e d e f i ned i n t h e u s u a l w a y , s o as t o p r e s e r v e t h e v a l i d i t y of t h e d e f i n i t i o n i n t h e " c r i s p " c a s e s , so t h a t t h e u n i o n and i n t e r s e c t i o n o f f u z z y s e t s A and B a r e s e t s w i t h membership f u n c t i o n s max ( f A , f B ) and min ( f A , f g ) , w h i l e complement o f t h e s e t A has membership f u n c t i o n 1
-
fA(X).
Two f u z z y s e t s A and B a r e e q u a l , i f t h e i r membership f u n c t i o n s c o i n c i d e f o r every x , t h a t i s A = B
iff
fA(x)
=
f B ( x ) f o r every x.
(3.2)
The t h e o r y o f fuzzy s e t s was i n t r o d u c e d i n o r d e r t o c r e a t e p o s s i b i l i t i e s o f a p r e c i s e way o f s p e a k i n g about i n e x a c t , vague o r f u z z y n o t i o n s . The s t a n d a r d examples
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g i v e n i n n e a r l y e v e r y t e x t on f u z z y s e t t h e o r y a r e t h e s e t of a l l v e r y t a l l men, s e t o f numbers a p p r o x i m a t e l y e q u a l 5 , s e t of s i g n i f i c a n t c o n t r i b u t i o n s t o a given domain , and s o o n . The concept of f u z z i n e s s s h o u l d be d i s t i n g u i s h e d from t h a t of s t o c h a s t i c i t y .
I n t h i s s e c t i o n we s h a l l a p p l y f u z z y s e t t h e o r y t o t h e c o n c e p t s i n t h e s o c i a l s c i e n c e s , These c o n c e p t s ( a t l e a s t those w i t h s u f f i c i e n t l y high g e n e r a l i t y ) possess c e r t a i n i n t e r n a l s t r u c t u r e . This s t r u c t u r e w i l l allow u s t o e x p l o r e t h e c o n n e c t i o n between t h e c o n c e p t s and t h e i r membership f u n c t i o n s , and e v e n t u a l l y d e s i g n some methods o f e s t i m a t i n g t h e v a l u e s o f t h e l a t t e r . The a n a l y s i s w i l l concern a f i x e d c l a s s o f c o n c e p t s , a l l p e r t a i n i n g t o t h e same b a s i c s e t X o f o b j e c t s . F o r most p s y c h o l o g i c a l c o n c e p t s , t h e n a t u r a l domain w i l l be t h e s e t X o f p e r s o n s . I f w e a b s t r a c t a t t h e s t a r t from t h e i n t e r n a l s t r u c t u r e of t h e c o n c e p t s , we may i d e n t i f y them w i t h f u z z y s e t s i n X . By d e f i n i t i o n , two c o n c e p t s A and B c o i n c i d e , i f t h e i r membership f u n c t i o n s f and f B a r e i d e n t i c a l . A s l o n g A as t h e d e t e r m i n a t i o n o f membership f u n c t i o n s i s l e f t
open, t h e d e c i s i o n whether or n o t two c o n c e p t s c o i n c i d e i s p u r e l y s u b j e c t i v e . The t h e o r y o f f u z z y s e t s i s i r r e l e v a n t when one makes _ t h i s d e c i s i o n . Moreover, even i f one c o u l d d e t e r m i n e t h e membership f u n c t i o n s a d e q u a t e l y and w i t h o u t e r r o r s , i t i s d o u b t f u l t h a t s u c h a s t r i n g e n t d e f i n i t i o n o f i d e n t i t y i s r e a l l y neede d . Two c o n c e p t s may w e l l b e i d e n t i c a l , i f t h e i r memb e r s h i p f u n c t i o n s a r e s u f f i c i e n t l y c l o s e for most o f t h e p o i n t s . S i n c e t h e t e r m s " s u f f i c i e n t l y c l o s e " and
265
PSYCHOLOGICAL PROBLEMS OF CONSTRUCTION
"most p o i n t s " a r e f u z z y , i t a p p e a r s t h a t one s h o u l d f u z z i f y t h e n o t i o n o f i d e n t i t y o f fuzzy s e t s . T o i l l u s t r a t e t h e t y p e o f a p p l i c a t i o n s , imagine t h a t
two s c i e n t i s t s propose e a c h a c e r t a i n c o n c e p t . The name t h e y use i s t h e same, b u t t h e d e f i n i t i o n s o f t h e c o n c e p t s d i f f e r somewhat. The q u e s t i o n i s : i s i t pos s i b l e t o d e t e r m i n e whether t h e c o n c e p t s d e f i n e d by t h e two s c i e n t i s t s a r e i n d e e d t h e same, a s t h e c h o i c e o f t h e name would s u g g e s t , o r a r e t h e y e s s e n t i a l l y d i f f e r e n t ? Let F d e n o t e t h e c l a s s of f u z z y s e t s under c o n s i d e r a t i o n s , s o t h a t f 6 F means t h a t f i s a f u n c t i o n which maps t h e s e t X i n t o [ O , l l . I f f , g c F , we d e f i n e
(3.3) and
(3.4) where p i s a c e r t a i n p r o b a b i l i t y d i s t r i b u t i o n on t h e s e t X . Thus, d l and d 2 a r e two m e t r i c s on F . P u t t i n g h i ( f , g ) = 1 - d i ( f , g ) , i = 1,2 we e s t a b l i s h a f u n c t i o n from F K F t o [O,l], hence a fuzzy s u b s e t of F .x F , or i n o t h e r words, a f u z z y b i n a r y r e l a t i o n i n F . T h i s r e l a t i o n w i l l be c a l l e d t h e fuzzy i d e n t i t y
o f s u b s e t s of X . It may be a r g u e d t h a t , a t l e a s t i n some c a s e s , d 2 ( a n d hence h 2 ) y i e l d s a n i n t u i t i v e l y more a c c e p t a b l e i d e n t i t y t h a n d l . I n d e e d , t h e r e q u i r e m e n s imposed by d 1
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a r e t h a t f ( x ) and g ( x ) a r e c l o s e one t o a n o t h e r f o r e v e r y x . On t h e o t h e r hand, d 2 i s small when l a r g e d i f f e r e n c e s between f ( x ) and g ( x ) o c c u r seldom, which i s a more r e l a x e d r e q u i r e m e n t . A s w i l l b e s e e n , t h e m e t r i c , and i d e n t i t y imposed b y has a n o t h e r a d v a n t a g e : i t may o c c a s i o n a l l y l e n d i t 2 s e l f t o estimation procedures.
d
me main problem w i t h f u z z y i d e n t i t y , whether based on d l o r on d 2 i s t h a t i t i s n o t t r a n s i t i v e . T h i s i s a phenomenon analogous t o t h a t e n c o u n t e r e d i n psychophys i c s f o r t h e concept of i n d i s t i n g u i s h a b i l i t y : i n a c h a i n o f p a i r w i s e i n d i s t i n g u i s h a b l e s t i m u l i , s u c h as h u e s , t h e extreme members need n o t b e d i s t i n g u i s h a b l e . L e t us w r i t e A = B i f h ( f A , g A )>, 21. a based on e i t h e r d l o r d 2 ) . THEOREM 1.
I f A =a
€3
g& B
=b
( h e r e h may be
c , then
A =a+b- 1 c
.
I n d e e d , l e t f A , f and f c be t h e membership f u n c t i o n s B o f A , B and C . Then we have di(A,B) 1 - a and d i ( B , C ) 5 1 - b . The a s s e r t i o n now f o l l o w s from t h e t r i a n g l e i n e q u a l i t y f o r the metric di.
<
We have t h e r e f o r e t h e phenomenon o f d i s s i p a t i o n o f i d e n t i t y , o r d e c r e a s e o f s i m i l a r i t y , as one moves a l o n g B and B = C t h e chain of similar objects. I f A = 0.90 0.95 t h e n A =0 . 8 5 etc'
';
Next, s i n c e t h e m e t r i c s d l and d 2 a r e i n v a r i a n t under a d d i t i o n of c o n s t a n t s , and s i m u l t a n e o u s change of s i g n , w e have a l s o t h e f o l ' l o w i n g theorem.
267
PSYCHOLOGICAL PROBLEMS OF CONSTRUCTION
THEOREM 2 . If A =a B, t h e n
A' = a B'.
The q u e s t i o n t o w h i c h e x t e n t t h e f u z z y i d e n t i t y i s p r e -
s e r v e d under fuzzy a d d i t i o n and i n t e r s e c t i o n i s p a r t i a l l y a n s w e r e d by t h e f o l l o w i n g t h e o r e m . THEOREM 3 . L e t u s c o n s i d e r t h e i d e n t i t y b a s e d on t h e
metric dl.
If
A =
a C
and
B =b D ,
=rnin(a,b) C U D , A n B
then
=min ( a , b ) C / 7
D
(3.5)
I n many i n s t a n c e s , o n e d e a l s w i t h more t h a n two f u z z y c o n c e p t s w h i c h a r e t o a l a r g e e x t e n t i d e n t i c a l . Because of i n t r a n s i t i v i t y o f t h e i n t r o d u c e d r e l a t i o n =
for a' of t h e s e c o n c e p t s i t i s n o t enough t h a t t h e y may b e combined i n t o a c h a i n o f p a i r w i s e a - i d e n t i c a1 c o n c e p t s . One n e e d s h e r e a s t r o n g e r n o t i o n . the a-identity
DEFINITION. L t F ' C F b e a c l a s s o f f u z z y c o n c e p t s . We s a y t h a t F' f o r m s a n a - i d e n t i t y b u n d l e , i f A = B f o r a a l l A , B 6 F ' . Here a - i d e n t i t y may be b a s e d on e i t h e r d l or d 2 . I n p r a c t i c a l s i t u a t i o n s , a - i d e n t i c a l b u n d l e s , or s i m p l y a - b u n d l e s , a p p e a r i f a c o n c e p t b e a r i n g t h e same name i s e x p l i c a t e d i n a number o f s i m i l a r , b u t n o t i d e n t i c a l w a y s by v a r i o u s r e s e a r c h e r s . To be a b l e t o s a y t h a t t h e s e r e s e a r c h e r s s p e a k o f t h e same c o n c e p t , t h e conc e p t s must form a n a - b u n d l e f o r some. s u f f i c i e n t l y small value of a. The n o t i o n o f a - b u n d l e a l l o w s u s t o s i m p l i f y c h e c k i n g t h a t two c o n c e p t s are s u f f i c i e n t l y d i s t i n c t t o be a c c e p t e d as d i f f e r e n t .
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THEOREM 4. & e J F', F" C F b e , r e s p e c t i v e l y , a n a - b u n d l e C F" and a b - b u n d l e i n F . If t h e r e e x i s t A ' C F' &A" such t h a t A '
a l l B' 6 F ' Proof.
f o r some c ,
ZcA"
and
B"t
#,
A",
If A '
then B'#c-(,t,)
for
B"
F".
> 1-c.
t h e n d(A',A")
Since d(A',B')c
1 - a a n d d ( A " , B " ) 6 1-b f o r a l l B ' c< F ' and B "
>
w e must have d ( B ' , B " ) l-c-(l-a)-(l-b) which i s e q u i v a l e n t t o t h e a s s e r t i o n .
S_
F",
= 1-Cc-(atb)],
We have a l s o
THEOREM b-bundle
5. Let
F'
, F"
be r e p s e c t i v e l y an a-bundle
i n F . I f t h e r e e x i s t s A ' & F' and A"
t h ' a t d(A1,A")
c
F" s u c h c , t h e n F l u F" f o r m s a n ( a t b t c ) - b u n d l e .
I n d e e d , f o r any B '
4
F ' a n d B" C F " we may w r i t e , u s i n g
t h e t r i a n g l e i n e q u a l i t y , d(B' ,B")
+
and
d(A",B")
('
d(B' , A f ) t d(A' ,A")
a t b t c , which p r o v e s t h e a s s e r t i o n .
We may now f o r m u l a t e t h e f o l l o w i n g c o r o l l a r y . COROLLARY.
If t h e a - b u n d l e F 7 a n d b - b u n d l e F" a r e n o t
disjoint, then Ff
F" i s a n ( a t b ) - b u n d l e .
F o r t h e p r o o f i t s u f f i c e s t o n o t e t h a t w e may t h e n t a k e i n t h e preceding theorem c = 0 .
3.2. Structure of concepts L e t now L b e a s e t , whose e l e m e n t s w i l l be i n t e r p r e t e d
as l a b e l s , or names, o f c o n c e p t s . We assume t h a t e l e m e n t s o f L may b e n e g a t e d , a n d j o i n e d w i t h c o n j u n c t i o n
269
PSYCHOLOGICAL PROBLEMS OF CONSTRUCTION
and d i s j u n c t i o n . The s e t L need n o t , however, be c l o s e d under t h e s e o p e r a t i o n s , i . e . t h e c o n d i t i o n s u,v C L do L or u v v L. n o t , i n g e n e r a l , e n t a i l -u 6 L , u A v T h i s means t h a t a c o n c e p t formed as a c o n j u n c t i o n of two c o n c e p t s b e a r i n g a name need n o t i t s e l f b e a conc e p t b e a r i n g a name. Next, l e t T: F +L
(3.6)
where @ L i s a s p e c i a l symbol s i g n i f y i n g “no name”. The r e l a t i o n T ( f ) = @ means t h a t t h e f u z z y s e t w i t h memb e r s h i p f u n c t i o n f has no name, s i m p l e or composite. It w i l l be a s s u j e d t h a t t h e f u n c t i o n T s a t i s f i e s t h e
following conditions.
L2
I f T ( f ) = u and
L3
If T ( f i )
= ui,
t h e n T[min(fl,
L4
If T(fi)
= ui,
-u E L , t h e n T ( 1 - f )
,..., n ,
i = 1
..., f n ) l =
ulJ\
i = l,...,n,
t h e n T[max ( f l , . . . , f n ) l =
and u 1 4
... A
and ulv ulV
... v
= -u,
...
A
u
n C L,
un ’
... v
un C L ,
u . n
C o n d i t i o n L 1 a s s e r t s t h a t e a c h c o n c e p t w i t h l a b e l from L i s r e p r e s e n t a b l e by a t l e a s t one f u z z y s e t i n F , whil e c o n d i t i o n s L2 - L4 a s s e r t a c e r t a i n c o n s i s t e n c y between o p e r a t i o n s on l a b e l s i n L and s e t - t h e o r e t i c a l
270
CHAPTER 3
o p e r a t i o n s on f u z z y s e t s . Given a s e t
AC' F , i t s
d i a m e t e r i s d e f i n e d as
(3.7) We have t h e n THEOREM
6 . Suppose t h a t u, -u L L . Then f o r i
= 1,2
The i d e n t i t y ( 3 . 8 ) f o l l o w s i m m e d i a t e l y from t h e i n v a r i a n c e o f e i t h e r o f t h e m e t r i c s d o r d 2 under n e g a t i o n , 1 as a s s e r t e d i n Theorem 1. If u
/,
v E L, define
Tmin -1 ( U , A V ) =
f
h : h = min ( f , g ) , f 6 T - ' ( u ) , g
and s i m i l a r l y , i f u
-1
TmaX(uv v ) =
1h:
v
C_
CT-'(v)j
L , we put
h = max (f, g )
,
ft
T-'(u)
, g 6 T-l(v)?
From L 3 and L4 i t follows t h a t w e have THEOREM
7.
If u
A
v, u
L.'
v C L,
then
For t h e c o n j u n c t i o n and d i s j u n c t i o n , w e have ( i n c a s e of m e t r i c d l ) , t h e f o l l o w i n g t h e o r e m , c o r r e s p o n d i n g t o
PSYCHOLOGICAL PROBUMS OF CONSTRUCTION
271
Theorem 3 . THEOREM 8 .
Let u ,
v
and
Similarly, i f u v v E L,
u,,
v b e l o n g t o L . Then
then
Proof. We s h a l l prove t h e f i r s t i n e q u a l i t y , t h e p r o o f o f t h e second b e i n g a n a l o g o u s .
Observe f i r s t t h a t -1 ( u n v ) i s w e l l d e f i n e d , i n view of t h e assumption Tmin t h a t u A v (i L . By L 3 we have
Let D [T-'(u)] = a , D,[T-'(v)] = b . Then for a l l f , f ' -1 i n T ( u ) and a l l x we have f ' ( x ) - a g f ( x ) 5 f ' ( x ) t a , and s i m i l a r l y , g ' ( x ) - b e g ( x ) : g ' ( x ) t b f o r 1 a l l x and a l l g , g ' < T- ( v ) . We can t h e r e f o r e w r i t e min [ f ( x ) , g ( x ) ] L min [ f ' ( x ) t a , g ' ( x ) t b J
I n a s i m i l a r way we g e t
which y i e l d s lmin [ f ( x ) , g ( x ) l
-
min [ f ' ( x ) , g ' ( x ) ] ' < m a x ( a , b ) .
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CHAPTER 3
Taking supremum f i r s t w i t h r e s p e c t t o x , and t h e n w i t h r e s p e c t t o f , f t ,g,gl we o b t a i n t h e a s s e r t i o n . L e t now u t L and l e t us c o n s i d e r a l l d e c o m p o s i t i o n s
u = u
fi
u2
1 any k w i t h 2
A
<
. ..
A
u
r’ w i t h u.1 f
.
L, i=l,..,r. I f for
k 2 r and any sequence 1 C i l :i 2-:i
k
r we have u f i u A ... A u 4 L , we s a y t h a t il i2 ik ( u l , ..., u r ) c o n s t i t u t e s a b a s i c d e c o m p o s i t i o n o f u. Nat u r a l l y , b a s i c d e c o m p o s i t i o n s need n o t be u n i q u e .
r
T h e i d e a o f t h i s d e f i n i t i o n i s as f o l l o w s . A decompo-
s i t i o n o f u i s simply a r e p r e s e n t a t i o n o f u as a confrom L , i . e . a decomj u n c t i o n o f e l e m e n t s ul, u 2 , . p o s i t i o n o f u i n t o c o n c e p t s e a c h h a v i n g a name. Now, t h i s d e c o m p o s i t i o n i s b a s i c , i f no c o n j u n c t i o n o f comp o n e n t s , e x c e p t t h e c o n j u n c t i o n o f a l l o f them, i s a c o n c e p t w i t h a name.
..
DEFINITION. The components which a p p e a r i n e v e r y b a s i c d e c o m p o s i t i o n o f u a r e c a l l e d t h e c o r e o f concept u.
-
Decomposing b a s i c components u l , u2 ” * f u r t h e r , one o b t a i n s subsequent decompositions, u l t i m a t e l y r e d u c i b l e t o a t r e e , w i t h atoms a t t h e t o p ( v i s c a l l e d an atom i n L , if i t cannot be r e p r e s e n t e d as a c o n j u n c t i o n o f any c o n c e p t s i n L d i f f e r e n t t h a n i t s e l f ) . F o r u C- L , l e t I(u)
=?XC;-
X:
( Y f G F ) f G T - l ( u ) ==$ f ( x ) = 1:.
T h e s e t I ( u ) w i l l be c a l l e d t h e s e t o f i d e a l o b j e c t s
f o r t h e c o n c e p t u. Similarly, let
PSYCHOLOGICAL PROBLEMS OF CONSTRUCTION
213
The s e t E ( u ) i s c a l l e d t h e s e t of exemplars concept u.
of t h e
Thus, a n o b j e c t x i s a n i d e a l o f t h e concept u , i f whenever t h e f u z z y s e t f b e l o n g s t o T - l ( u )
,
i . e . when-
e v e r T ( f ) = u , t h e n x belongs t o t h e set f i n f u l l d e g r e e . S i m i l a r l y , a n o b j e c t i s a n exemplar of t h e conc e p t u, i f t h e r e e x i s t s a s e t f w i t h T ( f ) = u, s u c h t h a t x belongs t o t h i s s e t i n f u l l degree. We have THEOREM 9 .
the
Let
( ul , . . . , u r )
b a s i c decomposition of
concept u , and l e t x I ( u ) . Then for any i ( i = 1,2,. , r ) and any f , t h e c o n d i t i o n f € T-'(u) implies
that
..
f ( x ) = 1.
P r o o f . Let ( ul , . . . , u r ) be a b a s i c d e c o m p o s i t i o n of u , and l e t f , E T - l ( u ) for i = 1,. , r . By L 3 we have I f = min (fl fr) C T-l(u). If x I ( x ) , then f ( x )
..
,...,
= 1, hence we must have f i ( x ) = 1
for a l l i .
We have a l s o THEOREM 1 0 .
If x
E' E ( u ) &u
t h e n f . ( x ) = 1 g f . C= u, 1 1
b e l o n g s t o t h e c o r e of
-1 T (u).
P r o o f . L e t x 6 E ( u ) , and l e t f be a f u n c t i o n i n T - l ( u )
i s i n t h e c o r e of u , t h e n any d e c o m p o s i t i o n of u i n t o u1 A A un must c o n t a i n u i' By L 3 w e have f = min (fl, fn), hence f i 4 f, and we must have f i ( x ) = 1. w i t h f ( x ) = 1. I f ui
... ...,
2 74
CHAPTER 3
These two theorems a s s e r t t h e r e f o r e t h a t any i d e a l of u must b e l o n g to any s e t i n t h e b a s i c d e c o m p o s i t i o n o f u i n f u l l d e g r e e , and any exemplar of u must b e l o n g i n f u l l d e g r e e to any s e t i n t h e c o r e o f u . We s h a l l s a y t h a t t h e concept u l e a d s to t h e a m b i g u i t y o f t h e f i r s t k i n d , i f t h e r e e x i s t exemplars which are n o t i d e a l s , s o t h a t E ( u ) # I ( u ) . We have t h e n THEOREM 11.
If u l e a d s to a m b i g u i t y o f t h e f i r s t k i n d ,
then there e x i s t u -
and u
ij s i t i o n o f u such t h a t f i ( x )
object x.
a p p e a r i n g i n t h e decompo= 1 and f . ( x ) < 1 f o r some J
Concept which do n o t l e a d t o t h e a m b i g u i t y o f t h e f i r s t k i n d , b u t f o r which t h e s e t s I ( u ) and E ( u ) a r e d i s t i n c t l e a d to a m b i g u i t y o f t h e second k i n d : a l t h o u g h t h e r e e x i s t d i f f e r e n c e s i n e x p l i c a t i o n s of t h e concept, t h e r e i s no d i s a g r e e m e n t as to t h e i d e a l t y p e s .
...,
When a c o n c e p t u i s r e d u c e d t o atomic c o n c e p t s v l , i t i s sometimes ( e . g . i n c a s e o f c e r t a i n psycholoVN ,
g i c a l c o n c e p t s ) p o s s i b l e to o b t a i n c l a s s i f i c a t i o n o f an
...,
vN, o b j e c t x i n t o c l a s s e s d e f i n e d i n t e r m s of v l , and c o n s e q u e n t l y , b u i l d an e s t i m a t o r of f ( x ) f o r f i n T- 1 ( u ) . The d e t a i l s , u s i n g t h e t h e o r y o f p s y c h o l o g i c a l t e s t s from C h a p t e r 1, w i l l be g i v e n i n n e x t s e c t i o n . The g e n e r a l i d e a , however, w i l l be e x p l i c a t e d h e r e , as i t d o e s n o t depend on s p e c i a l a s s u m p t i o n s concerni n g mechanisms of r e s p o n s e s t o i t e m s . By a c l a s s i f i c a t i o n of o b j e c t s x o n e u n d e r s t a n d s a map-
c = Tc1, c 2 , . . . /' , where C i ' s a r e caping t : x t e g o r i e s (defined through vl, v N ) . The f u n c t i o n t 7
...,
215
PSYCHOLOGICAL PROBLEMS OF CONSTRUCTION
may b e p a r t i a l , and i t may change between t h e c l a s s i f y -
i n g s u b j e c t s , and from o c c a s i o n t o o c c a s i o n f o r t h e same s u b j e c t . The r e l a t i o n t ( x ) = C . means t h a t o b j e c t J x was c l a s s i f i e d i n t o c a t e g o r y C ( i n p a r t i c u l a r , t ( x ) j may be an a s s e r t i o n about x u s i n g fuzzy c o n c e p t s , and w i t h a modal frame, s u c h as "I a m c e r t a i n t h a t " , e t c . ) C l a s s i f i c a t i o n s o f t y p e 1 a r e such t h a t i n a d d i t i o n * n one has a f u n c t i o n t : X -+ C , where t ( x ) i s t h e " t r u e c a t e g o r y " o f o b j e c t x . Then a measure o f q u a l i t y o f c l a s s i f i c a t i o n ( w i t h r e s p e c t t o x) i s t h e probabi* l i t y of t h e c o r r e c t c l a s s i f i c a t i o n P [ t ( x ) = t ( x ) ] . I n c l a s s i f i c a t i o n s of type 2, t h e n o t i o n o f t r u e c a t e gory makes no s e n s e ( e . g . i n c a s e o f g r a d i n g t h e exams) I n such a c a s e , t h e q u a l i t y of c l a s s i f i c a t i o n ( w i t h r e s p e c t t o x ) i n expressed by t h e requirement t h a t P [ t ( x ) = t l ( x ) ] shouod be as h i g h as p o s s i b l e and f o r as many x a s p o s s i b l e , where t and t ' a r e two independent c l a s s i f i c a t i o n s . Now, we have P[t(x) = t ' ( x ) ] =
2i
PCT(X) = C i l P C t ' ( x ) =
which i s r e d u c i b l e t o f, P [ t ( x ) = ci] t i c a l l y distributed clahsifications
2
Cil,
i n c a s e of iden-
.
The e m p i r i c a l a c c e s s t o t h e v a l u e s o f membership funct i o n s f ( x ) f o r some c o n c e p t s i s p r o v i d e d , i n p a r t a t l e a s t , by p l a u s i b l e assumptions s t a t i n g t h a t P [ t ( x ) = C . 1 depends on t h e v a l u e s of membership f u n c t i o n s which J This i s e n t e r i n t o t h e d e f i n i t i o n of t h e category C j' t h e i d e a which w i l l b e e x p l o r e d i n t h e n e x t s e c t i o n .
2 76
CHAPTER 3
3 . 3 . Measurement o f c o n c e p t s L e t F ' C F b e a c l a s s of f u z z y c o n c e p t s , t o be c a l l e d * e l e m n t a r y , and l e t F be t h e c l a s s o f c o n c e p t s u w i t h
T(u) # @ ( i . e . c o n c e p t s w i t h n a m e s ) , which a r e r e p r e s e n t a b l e as r e s u l t s o f s e t - t h e o r e t i c a l o p e r a t i o n s on elements of F ' Let f U
T-
1
.
(u)
,-, F * , s o
t h a t f U i s a membership Punc-
t i o n o f c o n c e p t u, which i s e x p l i c a b l e i n t e r m s o f e l e mentary c o n c e p t s from F'. Suppose t h a t ul,
u2,.
..
a r e t h e c o n c e p t s from F' which
...
be e n t e r i n t o t h e d e s c r i p t i o n o f u , and l e t f l , f2, t h e i r membership f u n c t i o n s . Thus, f u i s b u i l t o u t o f f . ' s by o p e r a t i o n s of maxima, minima and complemental
t i o n t o 1. C o n s e q u e n t l y , f o r any x , t h e v a l u e f U ( x ) , i . e . t h e d e g r e e t o which concept u may be a s s i g n e d t o x ( r e g a r d e d now as a p e r s o n ) , i s e q u a l t o some f u n c t ion of t h e values f i ( x ) . The v a l u e s f i ( x ) a r e , o f c o u r s e , unknown, and c a n n o t be e a s i l y e s t i m a t e d . F o r c o n s t r u c t which a r e o f s u f f i c i e n t g e n e r a l i t y , and s u c h t h a t t h e f u n c t i o n f u , i f e x p r e s s ed e x p l i c i t l y , would be a l o n g and i n v o l v e d e x p r e s s i o n o f f l , f 2 , ..., one may b u i l d a measurement p r o c e d u r e f o r f ( x ) based on t h e f o l l o w i n g a s s u m p t i o n s . U
ASSUMPTION 1. I f a s u b j e c t x i s asked about membership
i n any o f t h e s e t s w i t h membership f u n c t i o n s f i , t h e p r o b a b i l i t y o f a p o s i t i v e response i s an i n c r e a s i n g function of t h e value f i ( x ) . F o r m a l l y , we may t h e r e f o r e w r i t e
277
PSYCHOLOGICAL PROBLEMS OF CONSTRUCTION
t h e s u b j e c t x w i l l r e p l y "yes" ' [ t o t h e q u e s t i o n : "DO you b e l o n g ] = t o t h e s e t ui?"
V[fi(X)l
where v i s some i n c r e a s i n g f u n c t i o n from [ 0 , 1 ]
t o [O,l].
N e e d l e s s t o s a y , t h e q u e s t i o n i n q u o t a t i o n marks i s usua l l y f o r m u l a t e d i n some n a t u r a l w a y , e . g . "DO you o f t e n
f e e l t i r e d i n the
m o r n i n g ? " ( = "DO you b e l o n g t o t h e
f u z z y s e t o f p e r s o n s who u s u a l l y f e e l t i r e d i n t h e morning? )
.
ASSUMPTION 2 . The t o t a l number o f r e s p o n s e s "Yesr1 t o
a l l q u e s t i , o n s which e n t e r t h e d e c o m p o s i t i o n o f u, i s monotonically r e l a t e d t o t h e value f U ( x ) . T h i s , e s s e n t i a l l y , i s t h e way i n which t h e p s y c h o l o g i c a l t o o l s o f measurement a r e c o n s t r u c t e d : t h e y a r e
l i s t s of items ( n o t n e c e s s a r i l y q u e s t i o n s ) p l u s a s c o r i n g key f o r computing t h e t e s t s c o r e . N a t u r a l l y , some i t e m s a r e f o r m u l a t e d i n s u c h a way t h a t t h e p o s i t i v e r e s p o n s e i n d i c a t e s membership i n t h e complement o f t h e f u z z y s e t i n q u e s t i o n . T h i s r e q u i r e s reversing
t h e s c o r e f o r a g i v e n i t e m . A l s o , Assumpt-
i o n 2 may be r e a s o n a b l y e x p e c t e d t o be s a t i s f i e d o n l y f o r t h e c o n c e p t s u which a r e c o h e r e n t , i n t h e s e n s e o f t h e following d e f i n i t i o n . DEFINITION. A c o n c e p t u i s c o h e r e n t ,
if f o r any f
such t h a t T ( f ) = u , a r e p r e s e n t a t i o n of f i n terms of
..
u2,. u1, e i t h e r ui,
from F 1 s a t i s f i e s t h e f o l l o w i n g p r o p e r t y : o r -u
i'
but not both, e n t e r i n t o t h e repre-
sentation of f . The i n t u i t i v e j u s t i f i c a t i o n o f Assumption 2 i s t h a t i f
278
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a c o n c e p t i s c o h e r e n t and i n v o l v e s a l a r g e number o f e l e m e n t s from F', some s o r t of a v e r a g i n g , o r l a r g e number e f f e c t , t a k e s p l a c e .
If u i s a c o n c e p t , l e t t u ( x ) d e n o t e s t h e t e s t s c o r e f o r s u b j e c t x , i n t h e s e n s e d e f i n e d above ( a s t h e numb e r of answers "yes" to i t e m s ul, u*,... .The r e l a t i o n between t U ( x ) and f u ( x ) i s , of c o u r s e , unknown. The b e s t one can do i s t o r e s c a l e t h e v a l u e s of t Us o t h a t t h e y f a l l i n t o [ O , l l , and impose some assumptions under which t h e t e s t s c o r e may s e r v e as a n e s t i m a t e o f f U ( x ) . Without loss of g e n e r a l i t y , assume t h a t t h e t e s t s c o r e s
<
1. The r e l i a are already resealed, so t h a t 0 5 t ( x ) U b i l i t y of t h e t e s t , d e f i n e d as t h e r a t i o n of t h e v a r -
i a n c e , averaged o u t o v e r a l l s u b j e c t s , of t h e expecta t i o n of t h e s c o r e ( t h e t r u e s c o r e s ) , t o t h e v a r i a n c e o f t h e t e s t i n p o p u l a t i o n , may s e r v e as a measure o f how e x a c t i s t h e a s s e s s m e n t o f t h e t r u e v a l u e . To b e more p r e c i s e , assume t h a t t h e e x p e c t a t i o n of
t U ( x ) f o r any p e r s o n x e q u a l s t o t h e v a l u e of t h e membership function f U ( x ) , so that
L e t us w r i t e ( s e e C h a p t e r 1 f o r t h e d e t a i l s o f t h e
c o n s t r u c t i o n i n g e n e r a l c a s e , and t h e p r o o f s of v a r i o u s p r o p e r t i e s of t h e concepts i n t r o d u c e d ) :
where e ( x ) i s a random v a r i a b l e , t h e e r r o r s c o r e . By 2 c o n s t r u c t i o n we have E e ( x ) = 0 . L e t Var e ( x ) = d ( x ) .
PSYCHOLOGICAL PROBLEMS OF CONSTRUCTION
2 19
I f t h e p e r s o n x i s sampled randomly, t h e n f U ( x ) may be
r e g a r d e d a s a v a l u e o f a random v a r i a b l e , s a y f U , w i t h One can t h e n w r i t e , for t h e t e s t t h e v a r i a n c e 2f U s c o r e o f a randomly s e l e c t e d p e r s o n
.
t U=
f U -t e ,
and one can p r o v e ( s e e C h a p t e r 1) t h a t e and f a r e U uncorrelated. The r e l i a b i l i t y o f t h e t e s t i s d e f i n e d as t h e r a t i o 2 2 2 2 r t = Q f u / b t, = G f u / ( O f U
f
2
6 e).
where t and t ' are One may show t h a t r t = p ( t , , t A ) , two p a r a l l e l a p p l i c a t i o n s of t h e t e s t t U . C o n s e q u e n t l y , t h e e s t i m a t e o f t h e e r r o r v a r i a n c e i s o b t a i n e d from t h e f o rmul s
L e t now u ' and u" b e two c o n c e p t s , and l e t
be t h e d i s t a n c e between them ( h e r e p ( x ) i s t h e sampling distribution). One can show t h a t i f t U , and t U l la r e two t e s t s measuri n g c o n c e p t s u ' and u " , and t h e r e l i a b i l i t i e s of these t e s t s a r e r ' and r " , t h e n t h e c o r r e l a t i o n between t h e t r u e v a l u e s i s g i v e n by t h e a t t e n u a t i o n formula
2 80
CHAPTER 3
S i n c e t h e q u a n t i t i e s on t h e r i g h t hand s i d e of t h e l a s t f o r m u l a a r e e m p i r i c a l l y a c c e s s i b l e , one may e s t i m a t e t h e l e f t hand s i d e , e q u a l t h e c o r r e l a t i o n between f U , and f u l l . L e t us now w r i t e
d 2( u ' , u " )
=
E[fU,
-
2
fUI,] =
Adding and s u b t r a c t i n g t h e terms [ E f U , 1 2 and CEfU,,I
2
as w e l l as 2 E f u r E f U l , ,t h e l a s t e x p r e s s i o n i s e a s i l y transformed t o
+
LEYUl - E f U , , I
2
.
2 2 2 We have h e r e G f U , = r ' G t U ,- r ( t U , , t i , , ) G t U , 2 and s i m i l a r l y for $ t U I land , also
where i n
t h e l a s t l i n e t h e a t t e n u a t i o n f o r m u l a was
u s e d . F i n a l l y , E ( f U , ) = E ( t U , ) , E ( f U , , ) = E ( t u , , ) , and we o b t a i n
,
PSYCHOLOGICAL PROBLEMS OF CONSTRUCTION
281
and a l l q u a n t i t i e s on t h e r i g h t hand s i d e a r e e m p i r i c a l l y a c c e s s i b l e , t h u s l e a d i n g t o an e s t i m a t o r o f t h e d i s t a n c e d 2 ( u f , u I I ) between two c o n c e p t s . One s h o u l d s t r e s s once more t h a t t h e d i s t a n c e between fuzzy concepts analysed her e i s e m p i r i c a l l y accessibl e o n l y f o r t h e c l a s s o f c o n c e p t s whose m o d a l i t y may b e e s t i m a t e d t h r o u g h measurement t o o l s d e s c r i b e d i n t h i s s e c t i o n ( f o r a d e t a i l e d d i s c u s s i o n , s e e Nowakows k a 1 9 7 9 ) . T h i s c o n n e c t i o n , between t h e v e r y s u g g e s t i v e t h e o r y o f Zadeh, and t h e use o f t e s t t h e o r y , shows a p o s s i b l e p a t h o f development o f b o t h measurement problems and t e s t t h e o r y (where many p s y c h o l o g i s t s would want t o s e e i n t r o d u c t i o n o f f u z z y s e t t h e o r y , as a r e f r e s h m e n t o f a n o l d t o p i c ) . One s h o u l d n o t , however, o v e r e s t i m a t e t h i s p o s s i b i l i t y , and look r a t h e r f o r s o l u t i o n s based on models u s i n g new p s y c h o l o g i c a l i n t u i t i o n s . It seems t h a t f u t u r e development o f t e s t t h e o r y can go t o w a r d s dynamic a n a l y s i s o f development and changes o f concept s t r u c t u r e . Such a t h e o r y s h o u l d underlie the future theory of tests, w i t h t h e s t r e s s p u t on dynamic a s p e c t s , i n p a r t i c u l a r , on a n a l y s i s o f development o f knowledge s t r u c t u r e s (which a r e o f fundamental importance f o r computer s c i e n c e s and a r t i f i c i a l i n t e l l i g e n c e ) . One would p o s t u l a t e t h a t t r a i t s imply some way o f v i e w i n g t h e w o r l d , o r s t a t e o f knowl e d g e a b o u t t h e w o r l d . The problem would l i e i n t r a c i n g t h e development of v a r i o u s t y p e s o f r e p r e s e n t a t ion, t h e i r interactions, s u b s t i t u t a b i l i t y , distortions,
282
CHAPTER 3
Some i n t u i t i o n s h e r e w i l l be s u p p l i e d b y t h e n e x t c h a p t e r , d e a l i n g w i t h problems o f s e m i o t i c s and (among o t h e r s ) s t u d y o f knowledge s t r u c t u r e s .
283
PSYCHOLOGICAL PROBLEMS OF CONSTRUCTION
BIBLIOGRAPHY TO CHAPTER THREE
EDWARDS, A .
1957
S o c i a l D e s i r a b i l i t y i n P e r s o n a l i t y Assessmentand R e s e a r c h . N e w York. Dryden.
EISENBERG, P
.
1 9 4 1 I n d i v i d u a l I n t e r p r e t a t i o n of P s y c h o n e u r o t i c I n ventory Items. J o u r n a l o f General Psychology.
25. 1 9 - 4 0 . EISENBERG, P . , WESNMAN, A . G .
1 9 4 1 C o n s i s t e n c y of R e s p o n s e a n d L o g i c a l I n t e r p r e t a t i o n o f Psychoneurotic I n v e n t o r y Items. n a l o f E d u c a t i o n a l P s y c h o l o g y . 1 0 . 321-338.
*-
GOLDBERG. L . R .
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Model of Item Ambiguity i n P e r s o n a l i t y Assessment. E d u c a t i o n a l and P s y c h o l o g i c a l Measurement.
2 3 . 467-500. GUILFORD, J . P .
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P s y c h o m e t r i c M e t h o d s , N e w York. McGraw-Hill. GOLDBERG, L . R .
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Parameters. I t e m Response S t a b i l i t y a n d S c a l e Reliability. E d u c a t i o n a l and P s y c h o l o g i c a l Measurement. 2 7 .
. .
LEHMANN , E L
1959 T e s t i n g S t a t i s t i c a l H y p o t h e s e s . New Y o r k . W i l e y .
. .
MARSHALL , S P
1 9 8 1 S e q u e n t i a l Item S e l e c t i o n : O p t i m a l and H e u r i s t i c P o l i c i e s . J o u r n a l o f M a t h e m a t i c a l Psycho-
logy. 2 3 . 134-152.
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NOWAKOWSKA, M .
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K w e s t i o n a r i u s z dynamiczny ( i n P o l i s h ) . Psycho-
85-93.
l o g i a Wychowawcza. X I .
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P e r c e p t i o n of I t e m s and V a r i a b i l i t y o f Answers.
1979
Fuzzy C o n c e p t s : T h e i r S t r u c t u r e a n d P r o b l e m s
B e h a v i o r a l S c i e n c e . 1 8 . 99-108. o f Measurement. and R . R .
I n M.M.
Gupta, R . K .
Ragade
Yager ( e d s . ) Advances i n Fuzzy S e t
T h e o r y a n d A p p l i c a t i o n . Amsterdam: N o r t h Holl a n d . p p . 361-387. THURSTONE, L . L .
1927a
P s y c h o p h y s i c a l S c a l i n g . American J o u r n a l o f P s y c h o l o g y . 3 8 . 368-369.
1927b
A Law o f C o m p a r a t i v e J u d g m e n t . P s y c h o l o g i c a l
Review. 34. 273-286. VAN ZOUWEN H . ,
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NOWAKOWSKA, M . ,
DIJKSTRA, W .
S i m u l a t i o n Model f o r Answering Q u e s t i o n n a i r e
Items. P r o c . C o n f . on S y s t e m M o d e l i n g a n d C o n t r o l . Zakopane 1 9 7 9 . ZADEH, L . A .
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F u z z y S e t s . I n f o r m a t i o n a n d Contro1.8.338-353.
285
CHAPTER FORMAL SEMIOTICS:
4 REPRESENTATIONS,
OBSERVABILITY AND PERCEPTION
F o r t h e o r y o f t e s t s , and as i t w i l l a p p e a r l a t e r , f o r measurement t h e o r y , t h e most i m p o r t a n t problems a r e t h o s e o f c o n s t r u c t i o n o f a s p e c i a l c l a s s o f knowledge r e p r e s e n t a t i o n s , and d e t e r m i n i n g t h e c o n d i t i o n s o f c o r r e c t n e s s and adequacy o f t h e s e r e p r e s e n t a t i o n s . I n t h e present chapter (using the notion of semiotic syst e m s ) some g e n e r a l problems o f knowledge r e p r e s e n t a t i o n w i l l be shown; t h e y w i l l a l l o w a s y n t h e s i s of c o g n i t i v e t o p i c s , useful i n p a r t i c u l a r f o r those psychologists, who succumb t o s t r o n g e r s p e c i a l i z a t i o n p r e s s u r e s , and l o s e c e r t a i n g e n e r a l p o i n t o f view, even i n t h e a r e a of psychological sciences. T h i s l e a d s , among o t h e r s , t o phenomena s u c h as i n s u f f i -
c i e n t usage of methods and c o n c e p t s i n t r o d u c e d i n o t h e r subdomains. A good example h e r e may be almost complete l a c k o f s u c h p s y c h o m e t r i c p a r a m e t e r s as r e l i a b i l i t y , i n t e r n a l consistency , external v a l i d i t y , e t c . , i n s t u d i e s on s u b j e c t i v e p r o b a b i l i t y , which c a u s e s some d e c r e a s e o f q u a l i t y o f t h e s e s t u d i e s . Another example may be a n e x t e n s i v e development o f t h e o r e t i c a l c o n s t r u c t i o n s i n v o l v i n g s u b j e c t i v e j u d g m e n t s , as i n fuzzy s e t t h e o r y , w i t h o u t e n t e r i n g t h e problems of p s y c h o m e t r i c and p s y c h o l o g i c a l a s p e c t s o f c o n s t r u c t i o n o f s u c h judgments ( s e e e . g . Nowakowska 1 9 7 9 ) .
286
CHAPTER 4
S e g m e n t a t i o n o f domains c a u s e s a l s o t h e need o f new attempts of synthesis, especially i n t h e theoretical-m e t h o d o l o g i c a l and p h i l o s o p h i c a l -- l e v e l s , which c o u l d p o t e n t i a l l y p r o v i d e new i n t u i t i o n s and frameworks f o r s t u d i e s . The r e f l e x i o n on s e m i o t i c systems i s t o s e r v e such g e n e r a l g o a l s . The main n o t i o n s o f f o r m a l s e m i o t i c s a r e t h o s e o f o b s e r v a b i l i t y and p e r c e p t i o n , o b j e c t and i t s r e p r e s e n t a t i o n , as w e l l as o p e r a t i o n s on t h e l a t t e r . The i n t e r p r e t a t i o n of o b j e c t i s f l e x i b l e ; f o r p s y c h o l o g i s t s o f s p e c i a l i n t e r e s t may be t h e c a s e when t h e o b j e c t i s u n d e r s t o o d a s a m u l t i d i m e n s i o n a l u n i t o f communication, t h a t i s , meaning-carrying composite s t i m u l u s . T h i s allows t o analyse t h e r e l a t i o n a l s t r u c t u r e of t h i s s t i m u l u s ( o b j e c t ) , which i n many c a s e s i s l e f t u n s p e c i f i e d i n p s y c h o l o g i c a l r e s e a r c h , and c o n n e c t t h i s s t r u c t u r e w i t h some c h a r a c t e r i s t i c o f o b s e r v e d a c t i o n s and explanatory notions, The l a t t e r r e p r e s e n t t h e a c q u i r e d knowledge c o n c e r n i n g
the relations analysed.Including, or identifying
such a n o t i o n i n a l a r g e r network c o r r e s p o n d s t o some o p e r a t i o n s on r e p r e s e n t a t i o n s .
The n a t u r e o f t h e s e r e p r e s e n t a t i o n s i s l e f t u n s p e c i f i e d i n f o r m a l s e m i o t i c s , and i s open t o v a r i o u s i n t e r p r e t a t i o n s . One may also c o n s i d e r r e p r e s e n t a t i o n s o f representations, i . e . hierarchical representation struct u r e s , which c o r r e s p o n d t o more and more g e n e r a l relational structures. Multimodal and i n t e r a c t i o n a l u n d e r s t a n d i n g o f a s t i m u l -
u s (communication u n i t ) a l l o w s f o r a u s u a l l y n e g l e c t ed c o g n i t i v e u n i f i c a t i o n o f d i f f e r e n t a s p e c t s o f p e r -
F O R M A L SEMIOTICS
287
c e p t i o n on d i f f e r e n t media, such a s v e r b a l , v i s u a l , motions, e t c .
1. I N T R O D U C T I O N
Formal s e m i o t i c s was i n t r o d u c e d i n Nowakowska (1977; s e e a l s o 1 9 8 1 ) . I n t h i s t h e o r y , one d i s t i n g u i s h e s t h e b a s e , or l o c a l s e m i o t i c s y s t e m s , and g l o b a l , or macrol e v e l s e m i o t i c s y s t e m s , composed o u t o f t h e l o c a l o n e s . The t h e o r y o f l o c a l s e m i o t i c systems w i l l a l t e r n a t i v e l y be c a l l e d f o r m a l s e m i o t i c s Besides t h e a l r e a d y mentioned p o s s i b i l i t y o f u n i f i c a t -
i o n of v a r i o u s a p p r o a c h e s i n c o g n i t i v e p r o c e s s e s , forma l s e m i o t i c s a l l o w s a l s o t o re-examine such n o t i o n s as v a g u e n e s s , i m p r e c i s i o n and f u z z i n e s s . On m a c r o - l e v e l s , one c o n s i d e r s c o n d i t i o n s for c r e a t i o n o f more or l e s s i n t e g r a t e d g l o b a l s e m i o t i c s y s t e m s , where t h e b a s e s y s t e m s c o o p e r a t e i n f o r m i n g t h e o v e r a l l s e m a n t i c s of t h e g l o b a l s y s t e m s . Formal s e m i o t i c s , i n i t s s t r u c t u r e and i n f e r e n c e , goes f a r beyond t h e well-known c l a s s i f i c a t i o n s and d e f i n i t i o n s of signs by Peirce. Out o f v a r i o u s media on which a n o b j e c t may be r e p r e s e n t e d on d i f f e r e n t l e v e l s of a b s t r a c t i o n , t h e most i m p o r t a n t i s v e r b a l medium. I n o t h e r words, f o r v a r i ous " c o p i e s " -- i c o n i c , s y m b o l i c , e t c . -- o f an o b j e c t , t h e most i m p o r t a n t i s a v e r b a l copy, or d e s c r i p t i o n . A s p e c i a l c a s e o f such v e r b a l c o p i e s a r e l i n g u i s t i c measurements o f o b j e c t s ( s e e C h a p t e r 5 ) , where one a s s i g n s t o a n o b j e c t o r i t s f e a t u r e s , some e x p r e s s i o n s
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from c e r t a i n s p e c i f i c l i n g u i s t i c s c a l e s . O b j e c t s may a l s o be r e p r e s e n t e d n u m e r i c a l l y , when som e f e a t u r e s a r e mapped i n t o a n u m e r i c a l s c a l e o f a g i v e n t y p e . I n t h e c o n s i d e r a t i o n s below, a l l r e p r e s e n t a t i o n s o f o b j e c t s w i l l f o r s i m p l i c i t y be c a l l e d s i g n s . The f o r m a l t h e o r y o f s e m i o t i c s d i s t i n g u i s h e s t h r e e domains: (1) t h e domain o f o b j e c t s , ( 2 ) t h e domain o f s i g n s , and ( 3 ) t h e domain o f meanings. Each o f t h e s e domains has i t s own c h a r a c t e r i s t i c s t r u c t u r e , w i t h p r i m i t i v e n o t i o n s and schemes o f i n t e r p r e t a t i o n . The most b a s i c c o n c e p t s o f t h e t h e o r y a r e t h o s e o f t h e s e t o f s i g n s , s e t o f meanings which may be a t t a c h e d t o t h e s e s i g n s , and t h e s e t o f p e r s o n s -- o b s e r v e r s , who p e r c e i v e and i n t e r p r e t t h e s i g n s , i . e . a t t a c h meanings t o them. The b a s i c p r i m i t i v e n o t i o n , used f o r e x p r e s s i n g o t h e r n o t i o n s o f t h e t h e o r y , i s t h e f a c t t h a t "pers o n p a t t a c h e s meaning z to s i g n s " . I n a "non-fuzzy'' w o r l d , i t would b e s u f f i c i e n t t o i n t r o d u c e a n appropr i a t e symbol f o r t h i s f a c t , and p r o c e e d f u r t h e r . Howe v e r , i t may be t h a t e i t h e r meanings a r e f u z z y , or
a p e r s o n may n o t be s u r e about t h e meanings, s o t h a t it i s more c o n v e n i e n t t o u s e " f u z z i f i c a t i o n " o f t h e above f a c t , and i n t r o d u c e a n a p p r o p r i a t e f u n c t i o n , w i t h v a l u e s between 0 and 1, which r e p r e s e n t s t h e deg r e e t o which a p e r s o n a t t a c h e s a g i v e n meaning t o a g i v e n s i g n . Two s i g n s may t h e n be d e f i n e d as synonymous f o r a g i v e n p e r s o n , i f he a t t a c h e s t o them t h e same meanings t o t h e same d e g r e e . I f w e r e q u i r e t h i s t o h o l d f o r a l l p e r s o n s , we o b t a i n u n i v e r s a l synonymy of signs. Next, w i t h e a c h s i g n and p e r s o n , it i s n a t u r a l t o d i s -
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t i n g u i s h two s e t s , namely t h e c a r r i e r and k e r n e l o f t h i s s i g n . The f i r s t i s t h e s e t of a l l meanings which t h i s s i g n e x p r e s s e s i n some p o s i t i v e d e g r e e , w h i l e k e r n e l i s t h e s e t o f a l l t h o s e meanings, which a r e e x p r e s s e d i n f u l l d e g r e e ( s o t h a t any meaning which belongs t o t h e kernel a l s o belongs t o the c a r r i e r ) . These n o t i o n s , i n t u r n , a l l o w u s t o d e f i n e l e s s s t r i n g e n t n o t i o n s t h a n full synonymity, namely c a r r i e r synonymity and k e r n e l synonymity. Thus, c a r r i e r synonymity ( i d e n t i t y o f c a r r i e r s ) means t h a t any meaning which i s n o t e x p r e s s e d b y one s i g n a t a l l , i s n o t e x p r e s s e d b y t h e second s i g n e i t h e r . On t h e o t h e r hand, k e r n e l synonymity means t h a t whenever a meailing i s e x p r e s s e d by t h e f i r s t s i g n i n f u l l d e g r e e , t h e same h o l d s for t h e second s i g n , and c o n v e r s e l y . I n a s i m i l a r way one may d e f i n e c a r r i e r and k e r n e l for a meaning -- as opposed t o t h o s e of a s i g n . Thus, c a r r i e r o f a meaning i s t h e s e t o f a l l s i g n s ( o f a g i v e n c a t e g o r y under c o n s i d e r a t i o n ) which e x p r e s s t h e meaning i n some p o s i t i v e d e g r e e . K e r n e l o f a meaning i s t h e c l a s s o f a l l s i g n s which e x p r e s s t h i s meaning i n d e g r e e one. Given t h e s e n o t i o n s , one may i n t u r n d e f i n e t h e most i m p o r t a n t n o t i o n -- t h a t of e x p r e s s i b i l i t y : a meaning i s e x p r e s s i b l e , if i t s k e r n e l i s n o t empty ( i . e . i f t h e r e e x i s t s a s i g n which e x p r e s s e s t h i s meaning i n f u l l d e g r e e ) . The f i r s t enrichment of t h e s y s t e m , i n s p i r e d by t h e r e a l i t y which one may want t o d e s c r i b e , i s t h a t o f p a r t i t i o n o f t h e s e t of s i g n s i n t o c l a s s e s c o r r e s p o n d i n t o v a r i o u s media of e x p r e s s i o n . Thus, t h e f i r s t cat e g o r y may c o n t a i n v e r b a l s i g n s -- u t t e r e d words or s e n t e n c e s , second may be t h e c a t e g o r y o f p i c t o r i a l
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s i g n s , t h e t h i r d -- f a c i a l e x p r e s s i o n s , and s o on. The a c t u a l c h o i c e o f c a t e g o r i e s depends i n e a c h c a s e o f t h e g o a l of a n a l y s i s . The p a r t i t i o n i n t o c a t e g o r i e s l e a d s i n a n a t u r a l way t o c o n s i d e r i n g s i g n s which a r e m u l t i d i m e n s i o n a l , or m u l t i m e d i a l , b e i n g v e c t o r s of s i m u l t a n e o u s s i g n s , e a c h on i t s medium o f e x p r e s s i o n . T h i s a l l o w s u s t o e x p l i c a t e several very c r u c i a l notions concerning semantics of s i g n s , namely t h e r o l e s p l a y e d by v a r i o u s s i g n s on s p e c i f i c media i n t h e i r c o n t r i b u t i o n t o t h e o v e r a l l meaning. T h i s l e a d s t o c o n s t r u c t i o n of dynamic s e m a n t i c s , o f c o n s i d e r a b l e i m p o r t a n c e and i n t e r e s t i n a n a l y s i s o f m u l t i d i m e n s i o n a l language of communication. T h e main and most i n t e r e s t i n g problem h e r e i s t h a t o f exp r e s s i b i l i t y i n a maximal d e g r e e ( p o s s i b l y l e s s t h a n 1) i n c a s e when some media a r e r e s t r i c t e d i n t h e i r v o c a b u l a r i e s , or even t o t a l l y e l i m i n a t e d . A s examples o f s i t u a t i o n s when some media a r e banned one may t a k e communication between deaf and mute, b a l l e t , e t c . F o r a n a l y s i s o f t h e r e l a t i o n s h i p between o b j e c t s (which may mean h e r e p h y s i c a l o b j e c t s , r e a l or imagined, a c t -
i o n s , or even a b s t r a c t i d e a s or t h e o r i e s , sometimes r e f e r r e d t o as s i t u a t i o n s ) , t h e main problem i s t o e x p l i c a t e how t h e s t r u c t u r e o f t h e o b j e c t i s r e p r e s e n t ed by t h e s i g n . F o r t h a t p u r p o s e , i t i s n e c e s s a r y t o i n t r o d u c e some f o r m a l s y s t e m f o r a n a l y s i n g t h e s t r u c t u r a l a s p e c t s o f o b j e c t s . The systems used h e r e w i l l depend, o f c o u r s e , on t h e g o a l f o r which t h e f o r m a l i z a t i o n i s t o be u s e d . T h e r e a r e two main t y p e s o f e x p l i c a t i o n o f t h e s t r u c -
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t u r e o f o b j e c t s , which w i l l be c o n v e n i e n t t o c a l l cons t r a i n e d and u n c o n s t r a i n e d , or f r e e . B a s i c a l l y , t h e f i r s t c a s e -- o f c o n s t r a i n e d s t r u c t u r es -- a p p l i e s when w e d e a l w i t h a c l a s s o f o b j e c t s which a r e a l l s u f f i c i e n t l y s i m i l a r s t r u c t u r a l l y , s o t h a t one may d i s t i n g u i s h a number o f w e l l d e f i n e d a t t r i b u t e s on which t h e s e o b j e c t s may d i f f e r . The second c a s e -- or u n c o n s t r a i n e d s t r u c t u r e s
--
appli e s when we u s e t h e f o r m a l s y s t e m f o r a n a l y s i s o f j u s t one o b j e c t , s o t h a t t h e n e x t a p p l i c a t i o n has v e r y l i t t l e i n common w i t h t h e p r e c e d i n g one. The t y p e s o f q u e s t i o n s one asks i n t h e f i r s t c a s e a r e u s u a l l y d i f f e r e n t from t h o s e which one asks i n t h e second c a s e . I n t h e f i r s t c a s e , t h e main i s s u e i s t h a t o f i d e n t i f i c a t i o n of t h e o b j e c t , g i v e n i t s s i g n . T h e s i t u a t i o n i s w e l l i l l u s t r a t e d by examples s u c h a s books i n a library catalogue, hospital records of p a t i e n t s, and so on. I n t h e second c a s e , one may a s k v a r i o u s q u e s t i o n s reg a r d i n g t h e r e l a t i o n between t h e o b j e c t and i t s s i g n , f o r i n s t a n c e i n t h e form o f v e r b a l copy, i n o r d e r t o g a i n some i n s i g h t i n t o t h e p r o c e s s o f p e r c e p t i o n . I n c a s e o f c o n s t r a i n e d s t r u c t u r e s , i t w i l l b e conveni e n t t o u s e t h e example o f l i b r a r y c a t a l o g u e as a g u i de f o r i n t u i t i o n . We have here a c l a s s o f o b j e c t s -l i b r a r y books -- which a r e a l l " a l i k e " , e x c e p t t h a t t h e y may d i f f e r on some a t t r i b u t e s , s u c h as " a u t h o r ' s name", " p u b l i s h e r " , " t i t l e " , and s o o n . One c a n v i s u a l i z e t h e s e a t t r i b u t e s as h e a d i n g s o f e n t r i e s i n t h e c a t a l o g u e , and t h e e n t r i e s t h e m s e l v e s -- as t h e v a l u e s
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of a t t r i b u t e s . Now, i t s h o u l d be c l e a r t h a t t h e c h o i c e o f a t t r i b u t e s and o f t h e s e t s o f t h e i r v a l u e s depends v e r y s t r o n g l y on t h e g o a l , o r c l a s s o f g o a l s , f o r which t h e system w i l l b e u s e d . However, t h e c l a s s o f a t t r i b u t e s may
a l w a y s be assumed s u f f i c i e n t l y r i c h , s o t h a t s p e c i f i cation of a l l values of a t t r i b u t e s provides t h e requir e d i n f o r m a t i o n about t h e o b j e c t ( s o t h a t i f some a t t r i b u t e v a l u e s a r e n o t needed i n a p a r t i c i l a r i n s t a n c e , t h e y may simply be d i s r e g a r d e d ) . The second p o i n t c o n c e r n s t h e n a t u r e o f a t t r i b u t e s . I n many c a s e s , t h e assignment o f t h e v a l u e o f an a t t r i b u t e t o a n o b j e c t i s non-fuzzy ( f o r i n s t a n c e , i n c a s e of books, t h e y e a r o f p u b l i c a t i o n i s such a n a t t r i b u t e ) . There a r e , however, a l s o fuzzy a t t r i b u t e s , and t h i s may happen even i n c a s e o f l i b r a r y books; f o r i n s t a n c e , i n s u b j e c t c l a s s i f i c a t i o n , one o f t e n f i n d s s e v e r a l e n t r i e s i f t h e book b e l o n g s t o s e v e r a l c a t e gories). Thus, i n t h e f o r m a l s y s t e m , i t w i l l be c o n v e n i e n t t o a l l o w t h e a t t r i b u t e v a l u e s t o be f u z z y . The problem c o n c e r n s t h e n t h e s i g n systems f o r t h e d e s c r i p t i o n o f t h e o b j e c t s ( e . g . code numbers i n l i b r a r y c l a s s i f i c a t i o n s ) . It i s worth t o mention t h a t t h e s e t s o f v a l u e s o f a t t r i b u t e s may be o f d i f f e r e n t char a c t e r . They may be n u m e r i c a l o r n o t , and need n o t be d i s j o i n t . Thus, " a u t h o r ' s name" has v a l u e s b e i n g names, y e a r o f p u b l i c a t i o n " and "number o f pages" a r e both n u m e r i c a l , and s o on. To d e s c r i b e t h e s y s t e m , one must a s s i g n t o e a c h o b j e c t
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t h e s e t ( p o s s i b l y f u z z y ) o f i t s v a l u e s on e a c h a t t r i b u t e . Thus, t o e a c h o b j e c t t h e r e i s a s s i g n e d a v e c t o r of fuzzy s e t s , t h e i - t h set i n t h e v e c t o r being t h e s e t o f v a l u e s of i - t h a t t r i b u t e . 3
On t h e o t h e r hand, o b j e c t s a r e a l s o r e p r e s e n t e d b y s i g n s , so t h a t we must have a n o t h e r s e t , o f s i g n s , s i m p l e or c o m p o s i t e , and t h e r e l a t i o n o f r e p r e s e n t a t i o n , s i m i l i a r t o t h a t between s i g n s and t h e i r meanings. I n t h e s i m p l e s t c a s e , t h e r e may be j u s t one s i g n a s s o c i a t e d w i t h e a c h o b j e c t ; i n more complex c a s e s , t h e r e may b e o b j e c t s r w r e s e n t e d by s e v e r a l s i g n s , and moreo v e r , t h e r e p r e s e n t a t i o n may be f u z z y . To o u t l i n e t h e b a s i c n o t i o n s , we have h e r e two s e t s o f e q u i v a l e n c e s on t h e s e t o f o b j e c t s . One s e t c o m p r i s e s t h e r e l a t i o n s which r e p r e s e n t s o r t o f e x c h a n g e a b i l i t y . These r e l a t i o n s concern t h e g o a l s , or t h e i n f o r m a t i o n which one s e e k s . 46
The second c l a s s o f r e l a t i o n s c o n c e r n s means -- t h a t i s , t h e i n f o r m a t i o n p r o v i d e d b y t h e s i g n s . Such r e l a t i o n s may b e c a l l e d e q u i r e p r e s e n t a t i o n s . These two s e t s o f r e l a t i o n s need n o t be c o n n e c t e d i n any way ( a l t h o u g h i t i s d e s i r a b l e t o have them c o n n e c t e d ) . The most d e s i r a b l e p r o p e r t y i s t h a t o f i n c l u s i o n of c l a s s e s o f equir epr esented o b j e c t s i n c l a s s e s of e x c h a n g e a b i l i t y $ e l a t i o n s . Such p r o p e r t y i s r e f e r r e d t o as i d e n t i f i a b i l i t y . More s p e c i f i c a l l y , e x c h a n g e a b i l i t y r e l a t i o n s a r e o b t a i n ed b y t r e a t i n g as e q u i v a l e n t ( e x c h a n g e a b l e ) two o b j e c t s t h a t have t h e same v a l u e s on a l l a t t r i b u t e s (more ge-
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n e r a l l y , two o b j e c t s a r e G-exchangeable,
i f t h e y have
t h e same a t t r i b u t e s from t h e s e t G ) . Another u s e f u l c o n c e p t of e x c h a n g e a b i l i t y (weaker t h a n t h e above) i s t o r e q u i r e t h e i d e n t i t y o f k e r n e l s or c a r r i e r s . The g o a l d e t e r m i n e s which e x c h a n g e a b i l i t y r e l a t i o n s a r e a d m i s s i b l e f o r t h e g o a l and which a r e n o t ; i n o t h e r words, which o b j e c t s may and which may n o t , be t r e a t e d as " i d e n t i c a l " from t h e p o i n t o f view of t h e g o a l . On t h e o t h e r hand, e q u i r e p r e s e n t a t i o n s a r e c o n n e c t e d w i t h t h e chosen s i g n s y s t e m s , These r e l a t i o n s h o l d between o b j e c t s , i f t h e y have i d e n t i c a l s i g n s . Again, w e may r e q u i r e o n l y i d e n t i t y o f some a s p e c t s o f s i g n s , and so on. G e n e r a l l y , s i g n s r e p r e s e n t t h e i n f o r m a t i o n a.bout t h e o b j e c t -- t h e y c h a r a c t e r i z e t h e o b j e c t o n l y up t o a n e q u i v a l e n c e c l a s s o f e q u i r e p r e s e n t e d s i g n s . I d e n t i f i a b i l i t y means now t h a t a c l a s s o f a b s t r a c t i o n of equirepresentation r e l a t i o n i s contained i n a c l a s s o f a b s . t r a c t i o n o f e x c h a n g e a b i l i t y r e l a t i o n . The g e n e r a l idea i s c l e a r : t h e information contained i n a sign s h o u l d a l l o w u s t o i d e n t i f y t h e o b j e c t up t o t h e c l a s s of exchangeable o b j e c t s , i . e . should i d e n t i f y t h e e x c h a n g e a b i l i t y c l a s s t o which t h e o b j e c t b e l o n g s . These c o n s i d e r a t i o n s may a l s o p r o v i d e f o u n d a t i o n s f o r t h e f u z z y s e t t h e o r y . The p o i n t i s t h a t t h e i n f o r m a t i o n c o n t a i n e d i n t h e s i g n -- a b o u t t h e e q u i r e p r e s e n t a t i o n c l a s s t o which t h e o b j e c t b e l o n g s -- o f t e n does n o t p r o v i d e i n f o r m a t i o n about t h e e x c h a n g e a b i l i t y c l a s s . I n o r h e r words, t h e r e may b e no i d e n t i f i a b i l i t y . T h i s , of c o u r s e , d o e s n o t mean t h a t t h e i n f o r m a t i o n i s u s e l e s s . Even i f i t does n o t a l l o w l o c a t i o n of t h e
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o b j e c t i n any o f t h e e x c h a n g e a b i l i t y c l a s s e s , i t may allow determining the p o s s i b i l i t y d i s t r i b u t i o n , i . e . d e g r e e o f p o s s i b i l i t y t h a t t h e o b j e c t i s i n any o f t h e c l a s s e s . Such a p o s s i b i l i t y d i s t r i b u t i o n -- i n e f f e c t a d e s c r i p t i o n o f s e m a n t i c s of s i g n s -- i s a f o u n d a t i o n of fuzzy set t h e o r y . One can a l s o p r o c e e d i n a d i f f e r e n t d i r e c t i o n : i n s t e a d o f p r o c e e d i n g from s i g n s t o o b j e c t s , one can p r o c e e d from o b j e c t s t o ' - s i g n s , t h u s o b t a i n i n g p o t e n t i a l i t y d i s t r i b u t i o n , i . e . c l a s s e s o f a l l s i g n s which r e p r e s e n t a g i v e n o b j e c t i n some p o s i t i v e d e g r e e . A s a l r e a d y mentioned, s i g n s a r e a l s o r e g a r d e d as r e l a t -
i o n a l s t r u c t u r e s and one can a s k how t h i s s t r u c t u r e , w i t h i t s complexity and r i c h n e s s , d e t e r m i n e s t h e p r e c i s i o n and p o s s i b i l i t y o f i d e n t i f i c a t i o n , i . e . t h a t p a r t i c u l a r c l a s s of e x c h a n g e a b i l i t y t o which a g i v e n o b j e c t b e l o n g s . Given t h e s i g n and i t s s t r u c t u r e , one can t h e n d e t e r m i n e t h e c l a s s o f e q u i r e p r e s e n t a t i o n r e l a t i o n t o which i t b e l o n g s . I f t h i s c l a s s i s completel y c o n t a i n e d i n some c l a s s o f ( a d m i s s i b l e ) exchangeab i l i t y r e l a t i o n , we have t h e c a s e o f complete i d e n t i f i a b i l i t y , i . e . s u f f i c i e n t information f o r i d e n t i f i c a t i o n . I f t h i s c l a s s i n t e r s e c t s w i t h some o t h e r c l a s s e s o f e x c h a n g e a b i l i t y r e l a t i o n , we have u n c e r t a i n i n f o r m a t i o n , t h a t may or may n o t b e accompanied by i n f o r m a t i o n about t h e p o s s i b i l i t y d i s t r i b u t i o n . Such i n f o r m a t i o n i s n o t s u f f i c i e n t for i d e n t i f i c a t i o n ; however, l a c k o f s u f f i c i e n c y may be g r a d e d , i n t h e s e n s e t h a t s i g n s may p r o v i d e v e r y c o n f u s i n g i n f o r m a t i o n , or i n f o r m a t i o n which i s a l m o s t c e r t a i n . I f one has s u f f i c i e n t i n f o r m a t i o n , i t might be p o s s i b l e
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t h a t l e s s i n f o r m a t i o n would a l s o be s u f f i c i e n t ; i n
o t h e r words, some i n f o r m a t i o n c o n t a i n e d i n t h e s t r u c t u r e o f s i g n i s not necessary. This allows us t o i n t r o duce t h e c o n c e p t o f atom o f r e p r e s e n t a t i o n . A special class of signs are verbal copies, i . e .
signs e x p r e s s e d c o m p l e t e l y on v e r b a l medium ( d e s c r i p t i o n s ) .
A f t e r p o s s i b l e r e a r r a n g e m e n t s , such a v e r b a l copy may be r e d u c e d t o i t s normal form, t h a t i s , a v e c t o r o f fuzzy s e t s d e s c r i b i n g t h e v a l u e s of c e r t a i n a t t r i b u t e s of t h e o b j e c t . V e r b a l c o p i e s may be combined, by termwise i n t e r s e c t i o n s o r d i s j u n c t i o n s o f t h e s e t s which s e r v e as comp o n e n t s o f t h e v e c t o r s . On t h e o t h e r hand, t h e o b j e c t i s a l s o c h a r a c t e r i z e d by i t s t r u e s t a t e ( f u z z y o r n o t )
and one may s a y t h a t a v e r b a l copy i s f a i t h f u l , i f t h e components o f t h e d e s c r i p t i o n p r o v i d e d c o v e r t h e
t r u e s t a t e . If they a r e exactly equal t o the t r u e s t a t e s , t h e copy i s s a i d t o be e x a c t . The main problem w i t h v e r b a l c o p i e s i s t h a t one i s
c o n s t r a i n e d i n t h e d e s c r i p t i o n s -- simply b e c a u s e o n l y c e r t a i n s e t s o f a t t r i b u t e s may be e x p r e s s e d v e r b a l l y . T h i s l i m i t a t i o n o f t h e v o c a b u l a r y imposes t h e c o r r e s ponding l i m i t a t i o n s on t h e p o s s i b i l i t i e s o f v e r b a l c o p i e s t o be e x a c t . F o r m a l l y , t o e a c h a t t r i b u t e t h e r e c o r r e s p o n d s a vocab u l a r y , and a n assignment o f a s e t ( f u z z y or n o t ) t o e v e r y element o f t h e v o c a b u l a r y . T h i s s e t r e p r e s e n t s a c o n c e p t w i t h a name from t h e v o c a b u l a r y . The v o c a b u l a r i e s a r e assumed t o be c l o s e d under n e g a t i o n , s o t h a t a complement o f a s e t w i t h a name i s a l s o
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a s e t w i t h a name. To i l l u s t r a t e t h e o p e r a t i o n o f t h e s e c o n s t r a i n t s on verbal copies i n p r a c t i c a l s i t u a t i o n s , consider t h e f o l l o w i n g example. Suppose t h a t one o f t h e a t t r i b u t e s o f t h e o b j e c t s d e s c r i b e d i s l e n g t h , which may t a k e - on any n u m e r i c a l v a l u e . I n d e s c r i b i n g t h e o b j e c t , we a r e c o n s t r a i n e d t o a " v o c a b u l a r y " which has e x p r e s s i o n s o f t h e form " l o n g e r t h a n A i l ' , t h e i r n e g a t i o n s , and c o n j u n c t i o n s and d i s j u n c t i o n s o f t h e above c a t e g o r y o f e x p r e s s i o n s . The o b j e c t s which a r e allowed t o u s e as means o f comparison are f i x e d . Then t h e p r e c i s i o n w i t h which one can d e s c r i b e t h e l e n g t h o f an o b j e c t depends on t h e r i c h n e s s o f t h e c l a s s . S i m i l a r c o n s t r a i n t o c c u r i n everyday l i f e , n o t nece-
s s a r i l y for v e r b a l communication. For i n s t a n c e , g i v e n
a s c a l e and a s e t o f w e i g h t , one c a n d e t e r m i n e t h e weight o f a c e r t a i n o b j e c t o n l y w i t h some a c c u r a c y , depending on t h e weight a t o u r d i s p o s a l . A s a n o t h e r example, most TV s t a t i o n s have some symbols f o r w e a t h e r f o r e c a s t , l i k e p i c t u r e s o f r a i n , wind, e t c . With s u c h
a " v o c a b u l a r y " , t h e w e a t h e r may be d e s c r i b e d o n l y w i t h a c e r t a i n degree of accuracy. These examples i l l u s t r a t e t h e bounds imposed by t h e a c c e p t e d language o f d e s c r i p t i o n s , as w e l l a s unavoida b i l i t y o f s e m a n t i c l i m i t a t i o n s on t h e o r i e s , t h a t may sometimes l e a d t o i n c o m p a r a b i l i t y o f d e s c r i p t i o n s o f t h e same phenomenon. I n o t h e r words, i t may be imposs i b l e t o have a n a d e q u a t e t r a n s l a t i o n o f one d e s c r i p t i o n t o a n o t h e r , even i n t h e c a s e o f d e s c r i p t i o n o f t h e same phenomenon. A s r e g a r d s t h e p a r a m e t e r s d e s c r i b i n g v e r b a l c o p i e s and
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t h e i r q u a l i t y , i t i s s e e n t h a t t h e r e may e x i s t u s e f u l
n o t i o n s d e r i v e d from t h a t o f t r u t h , namely f a i t h f u l n e s s and e x a c t n e s s . These n o t i o n s c o n c e r n v e r b a l copy as a whole. N a t u r a l l y , t h e y depend on t h e p a r t i c u l a r conc e p t o f t r u t h one a d o p t s f o r c h e c k i n g t h e t r u t h about the object. It s h o u l d b e mentioned t h a t i n t h e above mentioned c a s e o f l e n g t h , t h e l i m i t a t i o n s would d i s a p p e a r , i f one were a l l o w e d t o u s e t h e e x p r e s s i o n s s u c h as "longe r t h a n o b j e c t A t a k e n t w i c e and o b j e c t A2 t a k e n 1 t h r e e t i m e s , p u t end t o e n d " , e t c . T h i s i s b e c a u s e o f the p o s s i b i l i t y of empirical operation of a d d i t i o n f o r l e n g t h . However, s u c h l i m i t a t i o n s would remain i n f o r c e i n t h e c a s e of e x p r e s s i n g a t t i t u d e s ( s a y ) . I n r e d u c i n g a v e r b a l copy t o i t s normal form, one may a l s o include i n the l a t t e r representations of sentenc e s t h a t were n o t a c t u a l l y u t t e r e d , b u t t h a t were i m p l i e d by t h e u t t e r e d s e n t e n c e s . Such a r e d u c t i o n r e q u i r e s i n t r o d u c i n g a p r i n c i p l e o f making i n f e r e n c e from u t t e r e d s e n t e n c e s , which goes beyond t h e r u l e s o f i n f e r e n c e o f f o r m a l l o g i c . The r u l e s o f s u c h i n f e r e n c e , based on t h e n o t i o n of s e m a n t i c i m p l i c a t i o n , were i n t r o d u c e d i n Nowakowska (1973); t h e y are based on t h e n o t i o n o f a d m i s s i b i l i t y and i n a d m i s s i b i l i t y o f an u t t e r a n c e . The above concerned c o n s t r a i n e d s t r u c t u r e s . I n t h e
c a s e o f f r e e s t r u c t u r e s , w e a g a i n have a n o b j e c t i n t h e form o f a r e l a t i o n a l s t r u c t u r e ( c o n s i s t i n g o f i t s c o n s t i t u e n t p a r t s , t h e i r a t t r i b u t e s , and r e l a t i o n s ) . One o f t h e r e l a t i o n s would t y p i c a l l y b e a p a r t i a l o r d e r corresponding t o the r e l a t i o n "is a p a r t of", so t h a t
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t h e o b j e c t i s r e p r e s e n t a b l e as a t r e e , w i t h t h e f l o o r s c o r r e s p o n d i n g t o l e v e l s o f g e n e r a l i t y , and t h e t o p f l o o r r e p r e s e n t i n g t h e meaning. The d i f f e r e n c e between c o n s t r a i n e d and f r e e d e s c r i p t -
i o n i s t h a t one i s n o t bounded i n t h e c h o i c e o f a t t r i b u t e s , s i n c e t h e r e i s no need t o compare i t w i t h o t h e r obje c ts
.
Now, t h e s i g n i s a l s o a r e l a t i o n a l s t r u c t u r e , and i n t h e most i n t e r e s t i n g c a s e s o f i c o n i c s i g n s i t p o s s e s s e s c e r t a i n p r o p e r t i e s r e q u i r e d for a model, namely t h e e x i s t e n c e o f a homomorphism c o n n e c t i n g t h e two r e l a t i o n a l s t r u c t u r e s . S u c h a homomorphism r e q u i r e s t h a t ( a t l e a s t some) e l e m e n t s o f t h e s i g n remain i n t h e same r e l a t i o n s as t h e c o r r e s p o n d i n g e l e m e n t s o f t h e object. One can a l s o c o n s i d e r o t h e r s i g n s , b e i n g models o f t h e f i r s t , s o t h a t i n s u c h c a s e s , t h e s i g n o f t h e "second f l o o r " , b y t r a n s i t i v i t y o f a homomorphism, i s a l s o a model o f t h e o r i g i n a l o b j e c t . L e t u s now a n a l y s e b r i e f l y t h e problems c o n n e c t e d w i t h changes o f t h e o b j e c t s and t h e i r o b s e r v a b i l i t y . An obj e c t c h a n g i n g i n t i m e may b e t r e a t e d s i m p l y as a sequence of o b j e c t s , corresponding t o observations a t dif f e r e n t moments
.
I n t h i s c a s e , i t i s c o n v e n i e n t t o p u t more s t r u c t u r a l c o n s t r a i n t s on o b j e c t s . We a g a i n c o n s i d e r o b j e c t s r e p r e s e n t a b l e as v e c t o r s o f a t t r i b u t e v a l u e s , w i t h p o s s i b l e "empty" v a l u e s # , i f a g i v e n o b j e c t i s n o t r e p r e s e n t e d . However, we remove
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now t h e f u z z i n e s s , and assume t h a t a t any g i v e n t i m e , t h e o b j e c t has a t most one v a l u e a t e a c h a t t r i b u t e . C o n s e q u e n t l y , t h e s t a t e of t h e o b j e c t a t any t i m e i s c h a r a c t e r i z e d by a v e c t o r , whose components a r e e i t h e r a t t r i b u t e v a l u e s , or # , i f t h i s a t t r i b u t e i s n o t approp r i a t e for t h e o b j e c t . The s e t o f a l l a t t r i b u t e s t h a t a r e represented i s called t h e base. Now, t o d e s c r i b e t h e c h a n g e s , one may d i s t i n g u i s h two c a t e g o r i e s o f t h e l a t t e r : one c a t e g o r y c o m p r i s e s c a s e s when a n a t t r i b u t e v a l u e s change, but t h e base d o e s n o t , w h i l e t h e second c a t e g o r y i n v o l v e s change of t h e b a s e , i . e . removal or a d d i t i o n o f a n a t t r i b u t e . Such c a t e g o r i z a t i o n i s u s e f u l i n d e s c r i b i n g changes o f o b j e c t s a n d / o r s i g n s . Moreover, one c a n o b t a i n a c e r t a i n char a c t e r i z a t i o n o f a c e r t a i n s p e c i a l c l a s s of changes.
For a meaning m , l e t us namely c o n s i d e r t h e d e g r e e t o which a t t i m e t , t h e s i g n r e p r e s e n t s t h i s meaning. We may t h e n s a y t h a t t h e o b j e c t e v o l v e s n o r m a l l y , i f t h i s d e g r e e ( a s a f u n c t i o n o f t ) may have a t most one p e a k : if i t e v e r b e g i n s t o d e c r e a s e , i t w i l l n e v e r i n c r e a s e a g a i n . One can t h e n p r o v e t h a t t h e c l a s s o f a l l r e g u l a r meanings i s c l o s e d under t h e o p e r a t i o n of conjunction. I
L e t u s now c o n s i d e r t h e problems o f o b s e r v a b i l i t y . The main i s s u e h e r e i s t h a t t h e o b s e r v a t i o n of some a t t r i b u t e s may make i t i m p o s s i b l e t o o b s e r v e a t t h e same t i m e some o t h e r a t t r i b u t e s . I n o t h e r words, n o t a l l s e t s of a t t r i b u t e s a r e j o i n t l y o b s e r v a b l e . I f a s e t o f a t t r i b u t e s which may b e j o i n t l y o b s e r v e d
i s c a l l e d a mask, t h e n t h e problem o f d e t e r m i n i n g t h e
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c l a s s o f a l l masks, and t h e c h o i c e o f one i n a c o n c r e t e s i t u a t i o n , i s o f extreme i m p o r t a n c e , n o t o n l y i n t h e b i o l o g i c a l sciences, but a l s o i n t h e s o c i a l sciences. The p o i n t i s simply t h a t o b s e r v i n g a phenomenon i n v o l v e s a d e c i s i o n about t h e p a r t i c u l a r f e a t u r e s which one wants t o o b s e r v e , and t h i s d e c i s i o n may r u l e o u t t h e p o s s i b i l i t y of observing other features. F o r i n s t a n c e , it may happen t h a t i n any e x p e r i m e n t a l
a n i m a l one may o b s e r v e X o r Y , b u t n o t b o t h . Such a s i t u a t i o n o c c u r s i f o b s e r v i n g X would r e q u i r e k i l l i n g t h e a n i m a l b e f o r e t h e t i m e when Y c o u l d be o b s e r v e d . F o r m a l l y , t h e s e t s c o n s i s t i n g o f X a l o n e , or o f Y a l o n e a r e masks, b u t t h e s e t c o n s i s t i n g o f X and Y i s n o t a mask. F o r a n example i n t h e s o c i a l s c i e n c e s , suppose t h a t t h e e x p e r i m e n t e r f a c e s a group o f , s a y , 8 p e r s o n s . H e can measure t h e r e s i s t a n c e t o s o c i a l p r e s s u r e o f any o f t h e s e 8 p e r s o n s , by c o n d u c t i n g A s c h ' s e x p e r i m e n t . Howe v e r , he can measure t h i s v a r i a b l e f o r one p e r s o n o n l y , s i n c e he would have t o l e t t h e r e m a i n i n g seven i n on the s e c r e t of experiment. The f o r m a l c o u n t e r p a r t o f s u c h a s i t u a t i o n i s t h a t o f
a m a s k , i . e . a s e t o f a t t r i b u t e s which a r e j o i n t l y o b s e r v a b l e . T h i s n o t i o n l e a d s i n a n a t u r a l way t o t h e c o n c e p t o f maximal mask, i . e . s u c h mask t h a t any l a r g e r
s e t i s a l r e a d y n o t a mask. Maximal masks need n o t b e unique (e.'g. i n t h e c a s e o f t h r e e v a r i a b l e s , X , Y , Z of which e v e r y two are j o i n t l y o b s e r v a b l e , b u t a l l t h r e e a r e n o t , m a x i m a l masks a r e t h o s e c o n t a i n i n g e x a c t l y two e l e m e n t s , , ( X , Y ) , ( X , Z ) and ( Y , Z > ) .
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One c a n p r o v e t h a t t h e c l a s s o f a l l maximal masks det e r m i n e s u n i q u e l y t h e c l a s s o f a l l masks ( f o r t h e p r o o f , s e e Nowakowska 1 9 7 3 ) . The c h o i c e o f a mask i n a c t u a l s i t u a t i o n d e t e r m i n e s t o some e x t e n t t h e " l e v e l o f t r u t h " t h a t may be approa c h e d , and t h e r e f o r e a mask a l s o imposes some c o n s t r a i n t s on t h e d e r i v e d c o n c e p t s of f a i t h f u l n e s s and e x a c t n e s s . T h i s t y p e o f i n t e r r e l a t i o n o f c o n c e p t s prov i d e s d e e p e r i n s i g h t i n t o t h e f o u n d a t i o n o f knowledge f o r m a t i o n and b a s i c f u n c t i o n s o f s c i e n c e . A t t h e end, i t i s w o r t h w h i l e t o c o n s i d e r s t i l l a n o t h e r
c l a s s of s i g n s , i n t h e c l a s s i f i c a t i o n o f P e i r c e c a l l e d i n d e x e s . Here t h e i s s u e c o n c e r n s t h e c a u s a l r e l a t i o n between o b j e c t s . When o b j e c t A i s a c a u s e o f o b j e c t B , t h e n B i s a s i g n o f A . From t h e f o r m a l p o i n t o f v i e w , t o some p a i r s o f o b j e c t s c o n n e c t e d by c a u s a l laws, there a r e assigned explanatory sentences. Thus, for e a c h e x p l a n a t o r y s e n t e n c e ( a l a w ) , one may s p e a k o f a s e t o f o b j e c t s which s a t i s f y t h i s l a w . T h i s g i v e s t h e domain o f a l a w , and a l s o i t s i n v e r s e domain, and t h e i n t e r s e c t i o n o f t h e l a t t e r c o r r e s p o n d s t o a c o n f i g u r a t i o n o f o b j e c t s . I n o t h e r words, a n o b j e c t may be d e s c r i b e d by many laws; more p r e c i s e l y , b y a l l t h o s e laws, t o whose domains i t b e l o n g s . Coherent s e t s o f laws form t h e o r i e s o f some s e t s o f o b j e c t s , s u c h as s i t u a t i o n s or phenomena. The d e c i s i o n whether or n o t an o b j e c t b e l o n g s t o a
domain o f some law depends i n many c a s e s on t h e o b s e r v a b i l i t y r e s t r i c t i o n s , which were d i s c u s s e d above. O b s e r v a t i o n of an o b j e c t i s r e l a t i v e t o a s i t u a t i o n ,
FORMAL SEMIOTICS
3 03
t o measurement p r o c e d u r e , and mappings i n t h e n u m e r i c a l and l i n g u i s t i c domains, t h a t a r e t h e p r e r e q u i s i t e f o r f o r m u l a t i n g t h e o b s e r v a t i o n a l judgments. The l a t t e r mappings a l l o w u s t o r e p l a c e t h e o p e r a t i o n s on o b j e c t s b y o p e r a t i o n s on symbols -- e i t h e r i n m a t h e m a t i c a l domain, o r i n l i n g u i s t i c s p a c e o f c l a s s i f i c a t i o n s o r d e s c r i p t i o n s . T h i s mapping -- and i t s c h a r a c t e r -i s d e t e r m i n e d b y t h e c l a s s o f t r a n s f o r m a t i o n r u l e s of o b s e r v a t i o n a l s y s t e m s , t h a t impose c o n s t r a i n t s on i t s adequacy and i n v a r i a n c e . I n o t h e r words, one may d e a l w i t h t h e whole c l a s s o f d e s c r i p t i o n s o r n u m e r i c a l a s s i g n m e n t s , from which one chooses onl y t h o s e t h a t are c h a r a c t e r i z e d by s u f f i c i e n t l e v e l of i n v a r i a n c e . An i m p o r t a n t problem here i s t h e c h o i c e o f a d e q u a t e r e p r e s e n t a t i o n o f o b s e r v a t i o n s of o b j e c t s a r e r e l a t i o n s between t h e m . Both f o r t h e r e p r e s e n t a t i o n i n n u m e r i c a l s p a c e and i n l i n g u i s t i c s p a c e , t h e c r i t e r i o n o f adequacy i s v a l i i d i t y , and a l s o parsimony: s i m p l i c i t y , e f f i c i e n c y , e t c . The s y s t e m o f c a u s a l r e l a t i o n s between o b j e c t s , i n t r o -
duced p r i m a r i l y f o r o b j e c t s , must b e e x t e n d e d t o s e t s of o b j e c t s t h a t c o n s t i t u t e s i t u a t i o n s , t h e c a u s a l rel a t i o n s between networks of o b j e c t s , s e t s o f s e n t e n c e s d e s c r i b i n g f a c t s o r laws, c l a s s e s o f p r o c e d u r e s , num e r i c a l and l i n g u i s t i c mappings, and c o n d i t i o n s f o r invariance. The s y s t e m s o g e n e r a l i z e d a l l o w s US t o i n t r o d u c e t h e
g e n e r a l i z e d d e s c r i p t i o n s o f o b j e c t s , i n form of laws or theories. T h i s i s a dynamical s y s t e m , n o t o n l y i n t h e s e n s e of
304
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changes o f o b j e c t s and t h e c o r r e s p o n d i n g changes of d e s c r i p t i o n s , p o s s i b i l i t i e s of continuous c o l l e c t i o n o f i n f o r m a t i o n , b u t a l s o b e c a u s e of some m o d i f i c a t i o n s o f t h e o b s e r v a t i o n a l system -- f o r i n s t a n c e due t o i n t r o d u c t i o n o f new measurement tools, or improvement o f t h e r e p r e s e n t a t i o n s ( e . g . by i n t r o d u c i n g a new model) t h a t i n d u c e t h e e n r i c h e d and improved d e s c r i p t i o n s o f r e l a t i o n s between o b j e c t s . I n consequence, i t l e a d s t o e l i m i n a t i o n o f e r r o r s , e x t e n s i o n o f o l d and i n t r o d u c t i o n o f new t h e o r y , f o r t h e e x t e n d e d s e t of observational sentences. To sum up, t h e dynamics of r e l a t i o n s between o b s e r v a t i o n s and c o l l e c t e d knowledge o f o b j e c t s on t h e one hand, and p r o d u c t i o n o f t h e i r c o p i e s on a g i v e n l e v e l o f r e p r e s e n t a t i o n , shows t h e m e t h o d o l o g i c a l and i n t e g r a t i v e r o l e of semiotic conceptions of t h e foundation of cognition.
2. MULTIMEDIAL C O M M U N I C A T I O N I n t h i s s e c t i o n , t h e a n a l y s i s w i l l c o n c e r n t h e problems of communication c a r r i e d o u t s i m u l t a n e o u s l y on s e v e r a l media, s u c h as v e r b a l medium, medium o f f a c i a l e x p r e s s i o n s , medium o f g e s t u r e s , and s o f o r t h . These t o p i c s were f i r s t f o r m a l l y a n a l y s e d i n Nowakowska (1977, 1981, 1983). Here an e x t e n d e d and e n r i c h e d model
1978,
o f s i t u a t i o n w i l l be shown, c a p t u r i n g t h e i n h e r e n t f u z z i n e s s o f b o t h t h e communication u n i t s , as w e l l as t h e meanings of t h e s e u n i t s .
305
FORMAL SEMIOTICS
2.1.
Formal a p p r o a c h t o s y n t a x o f m u l t i m e d i a 1 communic-
a t i o n languages We s h a l l c o n s i d e r communication c a r r i e d o u t s i m u l t a n e o u s l y on a number of media, l a b e l e d m l , m 2 , ,mr. The c h o i c e o f media f o r a n a l y s i s depends on t h e g o a l
...
o f s t u d y . For i n s t a n c e , one may r e g a r d t h e arm movements as a s e p a r a t e medium; however, f o r a d e s c r i p t i o n o f I n d i a n d a n c e s , s a y , one s h o u l d c o n s i d e r movements o f e a c h f i n g e r s e p a r a t e l y as c a r r i e d on some medium. With e a c h medium we a s s o c i a t e s e t o f s t a t e s ( o f t h e r e l e v a n t p h y s i c a l s y s t e m , s u c h as arm, f a c e , e t c . ) L e t S . be t h e s e t o f s t a t e s c h a r a c t e r i s t i c f o r medium mi. 1
We s h a l l assume t h a t t h e s t a t e s a r e d e f i n e d i n a d i s j u n c t i v e and e x h a u s t i v e w a y , s o t h a t a t e a c h t i m e e x a c t l y one o f t h e s t a t e s t a k e s p l a c e . The s t a t e s on i - t h medium mi w i l l g e n e r a l l y be d e n o t e d b y s i , or s i ( t ) , i f w e need t o s p e c i f y t h e s t a t e a t t i m e t . F u r t h e r , w i t h e a c h medium m we s h a l l a s s o c i a t e i t s i (fuzzy) vocabulary V i n form o f a f a m i l y of f u z z y i’ s u b s e t s of Si. The e l e m e n t s o f V . w i l l be d e n o t e d by 1 v i1, v i2, and t h e same symbol w i l l be udes t o d e n o t e
...,
k v i ( s ) , s C Si, i s k t h e d e g r e e t o which t h e s t a t e s b e l o n g s t o t h e s e t vi.
t h e membership f u n c t i o n , s o t h a t
L e t u s now f i x some q w i t h $ 4 q 5 1. Given si
let
and l e t
S
i’
3 06
CHAPTER 4
n
i ,q
( s i ) = # o f elements i n B
i ,q
(si)
.
(2.2)
Thus, B ( s . ) i s t h e s e t of a l l e l e m e n t s o f t h e vocai,q 1 b u l a r y Vi o f medium m which a r e e x p r e s s e d by s i i n i degree at l e a s t q. k If max v . ( s . )
4
remains n e u t $ a l (on l e v e l q ) , and t h a t i t t h e n e x p r e s s e s t h e 1
p a u s e It
1
q , we s h a l l s a y t h a t medium m
i
#i.
We have h e r e t h r e e p o s s i b l e c a s e s : t h e s e t Bi (si) ,q may be e m p t y ; i t may c o n t a i n j u s t one e l e m e n t ; and i t may c o n t a i n more t h a n one e l e m e n t . I n t h e f i r s t c a s e , i n t h e second c a s e , w e s h a l l medium m e x p r e s s e s # i i’ s a y t h a t it i s unequivocal (on l e v e l q ) , w h i l e i n t h e t h i r d c a s e , i t i s ambiguous ( o n l e v e l 9 ) .
-
L e t u s now c o n s i d e r t h e v e c t o r 2 = ( s l,.
..,sr)
of s t a t e s o f c o n s e c u t i v e media, o c c u r r i n g a t some t i m e t. Some o f t h e s e media w i l l e x p r e s s # i’ w h i l e some o t h e r s w i l l e x p r e s s c e r t a i n elements o f t h e i r r e s p e c t i v e vocabulark i e s . L e t u s w r i t e B i ,q ( s i ) = i f max v i ( s i ) < q , 1 k and l e t u s a s s i g n t o 2 t h e s e t
F#.l
The number o f e l e m e n t s i n Q(g) i s t h e n t h e p r o d u c t o f t h e s i z e s of components, i . e . we have
If n
(5;)
g -
= 1, t h e s t a t e
2 i s u n e q u i v o c a l o r unambiguous,
FORMAL SEMIOTICS
307
( a t t h e l e v e l q ) , s i n c e e a c h medium must e i t h e r e x p r e s s #. or e x p r e s s some unique element o f t h e v o c a b u l a r y V i ' 1 On t h e o t h e r hand, i f n (s) > 1, t h e n a t l e a s t one q -
medium e x p r e s s e s more t h a n one element o f i t s vocabulary.
The number o f e l e m e n t s i n t h e s e t Q ( s ) i n c r e a s e s r a p i d q l y when more and more media become ambiguous. T h i s phenomenon, a t l e a s t i n t h e o r y , would make communication i m p o s s i b l e i n c a s e o f even moderate a m b i g u i t y on some o f t h e media. However, w e s h a l l show t h a t s e m a n t i c s e l i m i n a t e s most o f t h e a m b i g u i t y .
D e f i n i t i o n of m u l t i m e d i a 1 l a n g u a g e . We may now i n t r o d u c e f o r m a l l y t h e system which w i l l a l l o w u s t o d e f i n e f u r t h e r n o t i o n s . Let 2.1.1.
be t h e c l a s s o f a l l s t a t e - v e c t o r s .
By
a s y n t a c t i c s t r u c t u r e o f m u l t i m e d i a 1 communication,
w e s h a l l mean a system
h
where S i s a c l a s s o f s t r i n g s 2 ( 1 ) , 2 ( 2 ) , . . . , 2 ( T ) , T = 1 , 2 , ..., and g ( n ) f5S f o r n = 1,..., T.
*
with
I n t u i t i v e l y , e l e m e n t s o f S w i l l be t h e a d m i s s i b l e s t r i n g s o f s t a t e s , as o b s e r v e d a t some f i x e d t i m e s , s a y t = 1,2,...,T f o r some T .
3 08
CHAPTER 4
Now, w i t h e a c h q and e a c h element o f S
*
we may a s s o c i a -
t e a c l a s s o f s t r i n g s o f v e c t o r s o f elements o f t h e v o c a b u l a r i e s o f media. F o r m a l l y , l e t V; = Vi
V
=
V'1 n
...
s o t h a t e l e m e n t s or where vi:
€
u
and l e t
V;
v
(2.7) k
,. . . ,vpr)
a r e v e c t o r s v_ = ( v k l l
Vl.
n Given q w i t h $ 4 q f 1 and a s t r i n g s = ( ~ ( l. ) . . ,, g (T)) w e a s s i g n t o i t a s e t o f s t r i n g s o f t h e form
where
(2.9) with
viki(n) 6 Bi
,q
(s.(n)). -1
The c l a s s o f a l l s t r i n g s o b t a i n e d i n t h i s way w i l l b e
c a l l e d t h e m d t i m e d i a l l a n g u a g e o f communication a c t ions. To b e t t e r v i s u a l i z e t h e c o n s t r u c t i o n , one may p r o c e e d
*
as f o l l o w s . The s t r i n g s = ( g ( l ) ,. . . , g (T)) h a s v e c t o r s of s t a t e s on p a r t i c u l a r media as i t s e l e m e n t s . It may t h e r e f o r e b e r e p r e s e n t e d as a m a t r i x
3 09
FORMAL SEMIO’ITCS
I .
...
(2.10)
where s . ( n ) i s t h e s t a t e on medium m 1
i
at t i m e t = n.
If now q i s f i x e d , t h e n w i t h t h e s t a t e s i ( n ) on i - t h
medium t h e r e i s a s s o c i a t e d one or more e l e m e n t s o f t h e v o c a b u l a r y V’ i . e . t h e o r i g i n a l v o c a b u l a r y , e n r i c h e d i’ by the pause #i. Thus, e a c h element o f m a t r i x ( 2 . 1 0 ) may b e r e p l a c e d by t h e a p p r o p r i a t e element o f V 1 * t h i s i’ yields a matrix
...
vk 1l ( T )
...
vk 2 2( T )
(2.11)
Whenever t h e r e i s a m b i g u i t y for some s t a t e s . ( n ) , t h e 1
c o r r e s p o n d i n g element may b e f i l l e d w i t h any o f t h e * e l e m e n t s of t h e s e t B ( s i ( n ) ) . Thus, a s t r i n g s iq w i l l t y p i c a l l y have s e v e r a l m a t r i c e s o f t h e form ( 2 . 1 1 ) assigned t o it. Any column which may a p p e a r i n t h e m a t r i x ( 2 . 1 1 ) w i l l be c a l l e d a f e a s i b l e u s i t o f m u l t i m e d i a 1 communication a c t i o n s , and t h e c l a s s of a l l p o s s i b l e i - t h rows i n m a t r i c e s ( 2 . 1 1 ) w i l l be c a l l e d t h e language Li of i - t h medium.
CHAPTER 4
310
A s a n example of a u n i t , c o n s i d e r " G r e e t i n g " , performed
s i m u l t a n e o u s l y on v e r b a l medium, medium of f a c i a l expr e s s i o n s , body movements, and s o on. N e x t t o i t , w e g i v e a n example o f an i n a d m i s s i b l e u n i t ( n o t b e l o n g i n g t o t h e language of a c t i o n s on communication m e d i a ) . medium
I'
verbal
s a y i n g "Hello"
Say i n g " H e 1l o
body movements
bow
Say i n g " H e 11o I'
gestures
t i p p i n g t h e hat
tipping the hat
facial e x p r e s si o n s
smile
tipping the hat
......
Greet i ng
....
inadmissible unit
....
Here t h e i n a d m i s s i b l e u n i t has simply i m p o s s i b l e a c t i o n s on v a r i o u s media, i . e . a c t i o n s a s s o c i a t e d w i t h d i f f e r e n t media. The i n t u i t i o n s of language of communication a c t i o n s
c o n c e r n s a d m i s s i b i l i t y n o t of u n i t s , but o f s t r i n g s o f u n i t s , even if e a c h s t r i n g s e p a r a t e l y i s a d m i s s i b l e . A s t r i n g ( m a t r i x of a c t i o n s , or a l t e r n a t i v e l y , a bunch o f s i m u l t a n e o u s s t r i n g s on v a r i o u s media) i s a d m i s s i b l e , i f i t i s p h y s i c a l l y p o s s i b l e t o p e r f o r m , and a l s o i f i t r e p r e s e n t s a fragment o f b e h a v i o u r which i s meani n g f u l , s o c i a l l y a c c e p t a b l e , and s o o n . I t i s n o t n e c e s s a r y t o g i v e h e r e more e x a c t i n t e r -
FORMAL SEMIOTICS
31 1
p r e t a t i o n , mainly because f l e x i b i l i t y of t h i s i n t e r p r e t a t i o n g i v e s t h e model more a p p l o c a b i l i t y . F o r i n s t a n c e , i f one w a n t s t o d e s c r i b e and a n a l y s e some communication a c t i o n s of s p e c i a l t y p e , s u c h as r o y a l c o u r t p r o t o c o l , J a p a n e s e t e a ceremony, e t c . , t h e s e t o f a l l a d m i s s i b l e a c t i o n s , a nd h en ce a l s o t h e s e t o f s t r i n g s a d m i s s i b l e on p a r t i c u l a r media ( l a n g u a g e s L i ) h a v e t o be a p p r o p r i a t e l y d e f i n e d i n each context (as conforming t o c o u r t protocol, etc.). The f a c t t h a t n o t a l l m a t r i c e s ( 2 . 1 1 )
are o b t a i n a b l e makes i t p o s s i b l e t o d e f i n e v a r i o u s from s t r i n g s t y p e s o f c o n s t r a i n t s b etween e l e m e n t s ( a c t i o n s on t h e same o r v a r i o u s m e d i a ) . Th u s, two s u c h r e l a t i o n s , o f e n f o r c i n g a nd e x c l u s i o n , may be d e f i n e d a s f o l l o w s .
* s
F i r s t l y , w he nev er we h av e v ki( n ) = x i
=+
k.
v J(n) j
z
y
for a l l n,
t h e n a c t i o n x on medium mi e x c l u d e s s i m u l t a n e o u s a c t i o n y on medium m Similarly, i f j' k.
ki
vi
v J ( n ) = y f o r some n, j
(n) = x
t h e n a c t i o n x on medium mi e n f o r c e s s i m u l t a n e o u s a c t i o n y o n medium m j'
These r e l a t i o n s c o n c e r n t h e same t i m e , i . e . t h e y r e f e r
t o t h e same u n i t . I n a s i m i l a r way, o n e may s a y t h a t the condition v ki( n ) = x i
+
k. v J ( n t c ) = y f o r some n j
312
CHAPmR 4
means t h a t a c t i o n x on medium m i e n f o r c e s a c t i o n y on medium m a t c u n i t s o f t i m e l a t e r . S i m i l a r c o n d i t i o n j h o l d s f o r e x c l u s i o n on t h e same medium a c r o s s t i m e . F i n a l l y , one may have e n f o r c i n g or e x c l u s i o n a c r o s s b o t h media and t i m e , e . g .
means t h a t a c t i o n x on medium m medium m
j
i
e x c l u d e s a c t i o n y on
a t c u n i t s of time later.
It i s c l e a r t h a t t h e r e l a t i o n o f e n f o r c i n g i s t r a n s i -
t i v e , b u t n o t n e c e s s a r i f y symmetric, w h i l e t h e r e l a t i o n o f e x c l u s i o n i s symmetric, b u t n o t n e c e s s a r i l y transitive. Somewhat more g e n e r a l l y , w e may c o n s i d e r t h e c o n c e p t o f p a r a s i t i c i t y , b o t h w i t h i n a u n i t o f communication a c t i o n s , and w i t h i n a s t r i n g of s u c h u n i t s . I n c a s e o f u n i t , a p a r t i c u l a r combination o f a c t i o n s on media i s p a r a s i t i c , i f t h e unit i s not admissible regardless of a c t i o n s on t h e r e m a i n i n g media. O b v i o u s l y , i f two a c t i o n s e x c l u d e one a n o t h e r , t h e n t h e y b o t h j o i n t l y are parasitic S i m i l a r l y , p a r a s i t i c i t y may be d e f i n e d f o r s t r i n g s o f
u n i t s : a s t r i n g u i s p a r a s i t i c , i f vuw does n o t b e l o n g t o t h e language ( i . e . i s n o t a d m i s s i b l e ) f o r any s t r i n g s v and w ( t h e s t r u c t u r e o f t h i s d e f i n i t i o n i s t h e same f o r t h e c a s e o f m u l t i m e d i a 1 and u n i m e d i a l strings). P a r a s i t i c i t y p r o v i d e s u s e f u l i n f o r m a t i o n about t h e
313
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s y n t a c t i c c o n s t r a i n t s i n t h e language; i n t u i t i v e l y , t h e more p a r a s i t i c s t r i n g s , t h e " r i c h e r " i s t h e s t r u c t u r e , i . e . t h e more s t r i n g s a r e i n a d m i s s i b l e . P r o b a b i l i s t i c i n t e r p r e t a t i o n . Thus f a r , we used * t h e c o n c e p t of t h e s e t S of s t r i n g s o f s t a t e v e c t o r s * * t o d e f i n e a mapping which t o e a c h s t r i n g s i n S a s s i g n s a s e t o f m a t r i c e s o f t h e form ( 2 . 1 1 ) . 2.1.2.
*
I n f a c t , v a r i o u s e l e m e n t s o f S o c c u r w i t h v a r i o u s prob a b i l i t i e s . T h i s means t h a t some s t r i n g s o f m u l t i m e d i a 1 * a c t i o n s o c c u r more f r e q u e n t l y t h a n o t h e r s . L e t P ( s ) * be t h e p r o b a b i l i t y o f o c c u r r e n c e o f t h e s t r i n g s
.
Denote g e n e r a l l y by f t h e mapping which which a s s i g n s
*.
t h e s e t of m a t r i c e s o f t h e form ( 2 . 1 1 ) t o s t r i n g s s
Then t o e v e r y c l a s s M of m a t r i c e s ( 2 . 1 1 ) one may assign the probability (2.12)
that i s , (2.13)
P(M) = P [ f - ' ( M ) ] .
T y p i c a l l y , t h e s e t M of i n t e r e s t w i l l be c l a s s e s o f m a t r i c e s ( 2 . 1 1 ) which s a t i s f y some c o n d i t i o n s , e . g . have some s p e c i f i c c o m b i n a t i o n s o f a c t i o n s on s p e c i f i c media. Moreover, t h e s t r i n g s o f a c t i o n s on p a r t i c u l a r media, i . e . e l e m e n t s o f l a n g u a g e s L i , may b e r e g a r d e d
a s s t o c h a s t i c p r o c e s s e s of some s o r t , w i t h p r o b a b i l i t y d i s t r i b u t i o n s indhced b y t h e p r o b a b i l i t y d i s t r i b u t i o n * P ( s ) . T h u s , a s t r i n g v i n L w i l l have t h e p r o b a b i -i
lity
i
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3 14
P ( ~ . )=
?(M:
i - t h row of M i s
1
Xi),
e t c . N a t u r a l l y , t h e p r o c e s s e s o f d i f f e r e n t media a r e strongly interrelated. I n f a c t , these interrelations, t h a t i s , a t e n d e n c y o f j o i n t a p p e a r e n c e o f some comb i n a t i o n s o f a c t i o n s on v a r i o u s media, a r e o f t e n s o s t r o n g t h a t t h e y may s e r v e as i n d i v i d u a l c h a r a c t e r i s t i c s o f a p e r s o n . T h i s s e r v e s as a b a s i s f o r i m i t a t i o n s , and a c c o u n t f o r t h e f a c t t h a t i t i s g e n e r a l l y p o s s i b l e t o r e c o g n i z e who i s b e i n g i m i t a t e d . From t h e d e s c r i p t i o n o f u n i t s g i v e n t h u s f a r i t f o l l o w s t h a t t h e y are some s o r t o f " m u l t i - w o r d s " , which form f u n c t i o n a l e n t i t i e s . The a d m i s s i b i l i t y i n a g i v e n l a n guage a l l o w s us t o b u i l d grammatical s t r i n g s o f u n i t s . One can c o n s i d e r , i n a d d i t i o n to t h e i n h e r e n t s t r u c t u r e of s u c h m u l t i - w o r d s , some p r o p e r t i e s which may be obs e r v e d i n a s t r i n g . Thus, i n s i t u a t i o n s o f f o r m i n g mult i - e x p r e s s i o n s ( s t r i n g s ) some a c t i o n s may a p p e a r c y c l i c a l l y , w h i l e o t h e r s may r e p r e s e n t p e r s e v e r a n c e o f a c t i o n s o f some t y p e s on c e r t a i n media. T h i s may o c c u r , i f some a c t i o n i s k e p t c o n s t a n t t h r o u g h s e v e r a l u n i t s , w h i l e a c t i o n s on o t h e r media change. T h i s means some c o u p l i n g o f u n i t s . F o r m a l l y , two n e i g h b o u r i n g u n i t s
aEe c o u p l e g , i f t h e r e e x i s t s i n d e x i such t h a t we have v i ( n ) = v i ( n t l ) for soge n . For c y k l i c a l a p p e a r e n c e , i i t h e c o n d i t i o n would be v i(n+km) = v i i ( n ) f o r k = 1,2, i . . and some rn and n .
.
The p r o c e s s o f c o n t r o l o f
performance c o n s i s t s o f v a r i o u s o p e r a t i o n s on u n i t s of "conventional" c h a r a c t e r , s u c h a s o m i s s i o n o r i n s e r t i o n o f some a c t i o n s , i n t e r c h a n g e of a c t i o n s , e t c .
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S i n c e l a n g u a g e s o f communication have o f t e n l o c a l char a c t e r , r e s t r i c t e d t o some s i t u a t i o n , and g i v e a desc r i p t i o n o f r e g u l a r i t i e s of a c t i o n s t r u c t u r e i n such s i t u a t i o n s , f o r an a c t o r i t i s i m p o r t a n t t o know how t o p a s s from one language t o a n o t h e r , t h u s combining s t r i n g s from v a r i o u s a c t i o n l a n g u a g e s . Such c o m b i n a t i o n s may i n v o l v e embedding ( s i m u l t a n e o u s performance o f s t r i n g s from two or more l a n g u a g e s ) , and c o n c a t e n a t i o n s , i . e . s t a r t i n g new s t r i n g o n l y a f t e r c o m p l e t i n g t h e p r e c e d i n g one. C o n c a t e n a t i o n s o f l a n g u a g e s , s a y L1 and L2, a r e d e f i n e d f o r m a l l y as
L1L2 = f u u ' : u c L1, u '
f
LJ.
grammar, i . e . s y n t a x and l o g i c o f s u c c e s s i o n o f s t r i n g s from d i f f e r e n t l a n g u a g e s of a c t i o n s , one c o u l d e x t e n d t h e p r o b a b i l i s t i c i n t e r p r e t a t i o n t o algebra of languages, where t h e l a t t e r would p l a y t h e r o l e o f words i n t h e v o c a b u l a r y . One c o u l d t h e n form a monoid b u i l t o u t o f l a n g u a g e s . The a d m i s s i b l e s t r i n g s would form here a "super-language" of communication a c t i o n s , t h a t c o u l d be i d e n t i f i e d w i t h s e t o f h i s t o r i e s o f a c t i o n s , or dual l y , h i s t o r i e s o f e v o l v i n g meanings g e n e r a t e d by p e r forming s t r i n g s of a c t i o n s . F o r f o r m a l d e s c r i p t i o n o f some i n t e r - s t r u c t u r a l
2.2.
Semantics o f m u l t i m e d i a 1 l a n g u a g e s
Let u s now e n r i c h t h e system of t h e p r e c e d i n g s e c t i o n by i n c l u s i n g i t s s e m a n t i c s . We s h a l l d e n o t e simply by
3 16
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L t h e c l a s s o f a l l m a t r i c e s o f t h e form ( 2 . 1 1 ) which
*
a r e o b t a i n a b l e from e l e m e n t s o f S , and by P t h e probab i l i t y measure on t h e c l a s s o f t h e s e m a t r i c e s , as indu* ced by t h e measure P on S Moreover, i t w i l l be conv e n i e n t t o d e n o t e by C t h e c l a s s o f a l l p o s s i b l e columns i n m a t r i c e s ( 2 . 1 1 ) , s o t h a t e a c h element o f L i s a s t r i n g c1c2.. cT of columns.
.
.
We s h a l l now c o n s i d e r a system
where C , L and p r o b a b i l i t y measure P have a l r e a d y been d e f i n e d , Z i s t h e s e t o f a l l p o s s i b l e meanings, and g: L x Z 4C O , l l
(2.15)
i s t h e f u n c t i o n d e s c r i b i n g fuzzy semantics, i . e . a
f u n c t i o n which to e v e r y meaning z and s t r i n g v i n L a s s i g n s t h e v a l u e g ( v , z ) , r e p r e s e n t i n g t h e d e g r e e , to which t h e meaning of s t r i n g v i s z . S i n c e t h e language L o f communication a c t i o n s c o n t a i n s , among o t h e r s , u t t e r a n c e s , t h e s e t Z of a l l meanings must c o n t a i n a l l p o s s i b l e meanings o f u t t e r a n c e s . The main o b j e c t o f i n t e r e s t h e r e w i l l b e , however, t h o s e meanings which a r e c a r r i e d m o s t l y b y n o n - v e r b a l media; examples o f s u c h meanings may b e " F r i e d l i n e s s " , "Annoyance", "Sco.rn" , " D i s l i k e " , and s o on. Most o f t h e s e meanings have t h e p r o p e r t y t h a t t h e y may appear i n m o d a l i t i e s ; t h u s t h e set Z w i l l c o n t a i n meanings s u c h as " M i l d annoyance", "Extreme annoyance", and so on.
FORMAL SEMIOTICS
317
A group o f meanings (sememes) r e p r e s e n t i n g v a r i o u s
d e g r e e s o f a meaning z w i l l be c o n s i d e r e d a s f o r m i n g a l i n e a r s c a l e , i . e . t h e y w i l l be assumed t o be o r d e r e d a l o n g a c e r t a i n continuum s p e c i f i c f o r a g i v e n meaning. G e n e r a l l y , e a c h s t r i n g v may have more t h a n one meaning associated with it; thus,
i s t h e s e t o f a l l meanings z which v e x p r e s s e s i t t h e d e g r e e a t l e a s t h. S i m i l a r l y , t h e same meaning z may be o f t e n e x p r e s s e d b y more t h a n one s t r i n g of communication actions, so that
i s t h e s e t o f a l l s t r i n g s i n L which e x p r e s s z i n t h e degree a t l e a s t h.
To b e t t e r a p p r e c i a t e t h e i n t e r p r e t a t i o n o f g , assume t h a t i t a s s i g n s meanings n o t o n l y t o s t r i n g s , b u t a l s o t o u n i t s . T h i s means, t e c h n i c a l l y s p e a k i n g , t h a t we assume t h a t columns i n m a t r i c e s ( 2 . 1 1 ) a r e a l s o e l e ments of L . The assumption o f e x i s t e n c e o f f u z z y membership f u n c t i o n g f o r s i n g l e u n i t s (and n o t o n l y for s t r i n g s ) r e q u i r e s t h e c l a r i f i c a t i o n of two p o i n t s : (1) t o which e x t e n t , and how, i t narrows down t h e po-
s s i b i l i t y of choice of u n i t s ?
( 2 ) how t o i n t e r p r e t t h e c o n c e p t o f f u n c t i o n g i n t h e
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co nt ext of p r e s e n t c o n s i d e r a t i o n s ? Regarding t h e f i r s t q u e s t i o n , t h e answer i s t h a t i n o r d e r t o g e t t h e p r o p e r i n t e r p r e t a t i o n , one cannot d i s t i n g u i s h u n i t s which a r e t o o small: e a c h u n i t must b e l a r g e enough t o c a r r y some meaning. Now, t h e r e are s e v e r a l ways i n which one may i n t e r p r e t t h e a s s e r t i o n t h a t g ( c , z ) = p , where c i s some m u l t i medial communicat,ion u n i t , and z i s some meaning. F i r s t l y , i t may mean t h a t p i s t h e f r a c t i o n o f p e r s o n s
who -- when c o n f r o n t e d w i t h u n i t c -- w i l l c l a i m t h a t i t s meaning i s z . S e c o n d l y , z may be one of p o s s i b l e meanings o f c , and p i s s i m p l y t h e f r a c t i o n o f o c c a s i o n s , when i t i s used t o denote z. T h i r d l y , p may be t h e d e g r e e o f c e r t a i n t y ( o f a s p e c i -
f i c p e r s o n ) t h a t when u n i t c i s used on p a r t i c u l a r o c c a s i o n , i t s meaning was z . Finally, there i s also possible a fourth interpretation, connected w i t h t h e f o l l o w i n g c o n c e p t u a l i z a t i o n o f u n i t c . G e n e r a l l y , c c o n s i s t s o f some u t t e r a n c e s , accompani e d b y a c t i o n s on some o t h e r media, such as g e s t u r e s , f a c i a l e x p r e s s i o n s , s u c h as s m i l e , and s o on. T y p i c a l l y t h e r e i s some d e g r e e o f freedom o f p e r f o r m i n g t h e s e a c t i o n s , w i t h i n t h e c o n s t r a i n t s imposed by human body, o c c a s i o n , and s o f o r t h . We may t h e n t h i n k o f u n i t c as b e i n g i n f a c t one o f a f a m i l y o f u n i t s , s a y ex, where x i s some ( p o s s i b l y m u l t i d i m e n s i o n a l ) p a r a m e t e r . Now, d i f f e r e n t c x d i f f e r t o t h e d e g r e e w i t h which t h e y
3 19
FORMAL SEMIOTICS
e x p r e s s meaning z . I n t h i s i n t e r p r e t a t i o n , g ( c , z ) would be i n t e r p r e t e d as f r a c t i o n of u n i t s c which X c a r r y t h e meaning z . R e g a r d l e s s o f t h e i n t e r p r e t a t i o n of f u n c t i o n g , i t may now b e used t o e x p r e s s t h e c o n t r i b u t i o n t o some meaning o f a s p e c i f i c component o f t h e u n i t . Assume t h a t c and c ' a r e two u n i t s , which have a l l a c t i o n s i d e n t i c a l e x c e p t t h a t on medium m i'* l e t vi be t h e a c t i o n on medium m i n u n i t c , w h i l e l e t u n i t c l i have " n e u t r a l " a c t i o n # i on medium m Then, we s a y i' that ( i n t h e context of other a c t i o n s , i d e n t i c a l i n c and c ' ) ,
--
v
i
s u p p o r t s meaning z , i f
and i n p a r t i c u l a r , vi g e n e r a t e s meaning z , if
--
v
i
i n h i b i t s meaning z , i f
and i n p a r t i c u l a r , v i c a n c e l s meaning z,
-- vi
if
i s n e u t r a l w i t h r e s p e c t t o meaning z , if
g(c,z) = g(c',z).
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These n o t i o n s , which may be e a s i l y e x t e n d e d t o t h e c a s e o f s e v e r a l a c t i o n s ( i . e . combinations o f a c t i o n s on s e v e r a l m e d i a ) , d e s c r i b e b a s i c t y p e s o f i n t e r a c t i o n s between media of communication. Thus, s u p p o r t o f a meaning o c c u r s when a c t i o n v
on i medium m i n c r e a s e s t h e l e v e l o f e x p r e s s i o n o f meaning i z above t h e l e v e l which would have o c c u r r e d i f t h e n e u t r a l a c t i o n # were performed. I n p a r t i c u l a r , i f i n i t h e l a t t e r c a s e z would n o t have been e x p r e s s e d a t a l l , we have g e n e r a t i o n o f meaning. O t h e r d e f i n i t i o n s a r e b u i l t on t h e same p r i n c i p l e : f o r i n s t a n c e i n h i b i t i o n occurs i f performing v
i
causes
l e s s e r d e g r e e of e x p r e s s i o n o f z t h a t would have sappeared i n c a s e of n e u t r a l a c t i o n #i. A t y p i c a l example f o r s u p p o r t i n g a meaning s u c h as
"Friendliness" i n t h e appropriate context i s t h e smile which accompanies o t h e r a c t i o n s . To have a n example o f g e n e r a t i o n o f a meaning, i t s u f f i c e s t o c o n s i d e r r e l i g i o u s c e r e m o n i e s , or magic, where t h e i n t e n d e d meaning i s t o t a l l y a b s e n t , u n l e s s one makes t h e a p p r o p r i a t e g e s t u r e , u t t e r s some words, e t c . Such "magical" components a p p e a r a l s o o u t s i d e o f magic o r r e l i g i o n ; f o r i n s t a n c e , i n many c o u n t r i e s , a p e r son may be h a n d c u f f e d , t a k e n t o t h e p o l i c e s t a t i o n , e t c . , b u t t h e s e a c t i o n s do n o t c o n s t i t i t e " l a w f u l
a r r e s t " u n l e s s f o r s p e c i f i c f o r m u l a i s u t t e r e d by t h e officer
.
I n c a s e of mutual i n t e r r e l a t i o n s between more a c t i o n s , t h e s i t u a t i o n i s , n a t u r a l l y , more complex, b u t t h e b a s i c i d e a s remain t h e same. A s a n example, we may
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321
.
two media, s a y m and m L e t c be a u n i t o f a c t i o n s , i j and l e t c i , c and c b e t h e u n i t s o b t a i n e d from c j ij by r e p l a c i n g r e s p e c t i v e l y i t s a c t i o n o f medium mi b y # i , i t s a c t i o n on medium m b y # j , and b o t h a c t i o n s on j media m and m by #i and # We may t h e n s a y t h a t ( i n i j j' t h e c o n t e x t o f a c t i o n s o f c ) , a c t i o n s v and v on i 5 media m and m a r e p o s i t i v e l y a s s o c i a t e d f o r meaning i 5 z , if g ( c i j , z ) ( m i n [ g ( c i , z ) , g ( c j , z ) l and a l s o max [ g ( c i , z ) , g ( c j , z ) l 4 g ( c , z ) . On t h e o t h e r hand, v catalyses v if 0 = g ( c i j , z ) = g ( c j , z ) < g ( c i , z ) . i j' These two d e f i n i t i o n s do not e x h a u s t a l l p o s s i b i l i t i e s , and one can have a l s o p o s i t i v e a s s o c i a t i o n i n i n h i b i t i n g meaning z , o r n e g a t i v e a s s o c i a t i o n f o r o r a g a i n s t t h i s meaning , e t c . P o s i t i v e a s s o c i a t i o n o c c u r s when e a c h o f t h e a c t i o n s s u p p o r t s t h e meaning, but i n c o n j u c t i o n t h e y s u p p o r t t h i s meaning s t r o n g e r t h a n e a c h o f them s e p a r a t e l y . On t h e o t h e r hand, c a t a l y s i n g means t h a t one a c t i o n has no i n f l u e n c e on t h e meaning, u n l e s s i t i s accompanied by o t h e r a c t i o n ( h e r e a g a i n r e l i g o i u s ceremonies may s e r v e as an e x a m p l e ) . The n o t i o n s which d e s c r i b e t h e r o l e o f a n a c t i o n on a medium i n g e n e r a t i n g , s u p p o r t i n g , i n h i b i t i n g , o r canc e l l i n g a meaning, may b e e x t e n d e d t o t h e c a s e of t h e r o l e s which u n i t s p l a y i n a s t r i n g o f u n i t s . However, i t i s n e c e s s a r y t o i n t r o d u c e t h e concept which i n t h i s c a s e would c o r r e s p o n d t o t h e " n e u t r a l " a c t i o n . If we have s u c h a c o n c e p t , and # s t a n d s f o r t h e " n e u t r a l " u n i t ( f o r a g i v e n meaning z ) , t h e n t h e n o t i o n s o f i n h i b i t i o n , s u p p o r t , e t c . can be d i r e c t l y a p p l i e d , b y comparison o f g ( 2 , z ) and g ( c . , z ) , where -1 c and c d i f f e r o n l y by r e p l a c e m e n t o f i - t h u n i t by -i
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t h e n e u t r a l u n i t # . We s h a l l n o t repeat t h e d e f i n i t i o n s , b u t e x p l a i n i n s t e a d how one may d e f i n e t h e n e u t r a l unit #. For d e f i n i t i o n , one c o n s i d e r s a g i v e n meaning t o g e t h e r w i t h i t s m o d i f i c a t i o n s , b o t h i n t h e p o s i t i v e and neg a t i v e d i r e c t i o n . One can v i s u a l i z a i t as a s c a l e , of t h e form
"
not-z
"
rather not-z
"
somewhat z
"
"
z
very z
"
extremely z
F o r i n s t a n c e , f o r z = F r i e n d l i n e s s , we s h a l l have meanings s u c h a s " V e r y f r i e n d l y " , "Extremely f r i e n d l y " , a s w e l l as " U n f r i e n d l y " , and even "Rude". A s t a n d a r d u n i t for e x p r e s s i n g z w i l l be d e f i n e d as
any u n i t m e e t i n g two c o n d i t i o n s : f i r s t l y , i t d o e s n o t e x p r e s s any n e g a t i o n o f z i n any d e g r e e , i . e . g ( c , - z ) = 0 , and s e c o n d l y , among u n i t s which meet t h e f i r s t c o n d i t i o n , i t e x p r e s s e s z i n h i g h e s t d e g r e e , i . e . c must s a t i s f y g ( c , z ) = max t g ( c 1 , z ) : g ( c ' , - z ) =
01.
It i s w o r t h t o
mention t h a t s t a n d a r d u n i t d o e s n o t e x p r e s s t h e h i g h e s t d e g r e e of z , i . e . "Extremely z " , b u t e x p r e s s e s z i n t h e h i g h e s t d e g r e e . For i n s t a n c e , f o r meaning s u c h as " P o l i t e n e s s " , a s t a n d a r d u n i t w i l l n o t be such b e h a v i o u r which i s " E x t r e m e l y p o l i t e " , b u t a b e h a v i o u r which i s " P o l i t e " i n t h e h i g h e s t p o s s i b l e degree ( i . e . without exaggeration, but a l s o without t r a c e s of i m p o l i t e n e s s ) . It i s c l e a r t h a t t h e n o t i o n o f s t a n d a r d u n i t i s f u z z y , and moreover, a s t a n d a r d u n i t i s n o t unique f o r a
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g i v e n meaning; , , t h e r e may e x i s t s e v e r a l s t a n d a r d u n i t s .
2.3. Pragmatic semantics A s mentioned i n t h e I n t r o d u c t i o n , one o f t h e most i n -
t e r e s t i n g problems c o n n e c t e d w i t h s e m a n t i c s o f m u l t i m e d i a l l a n g u a g e s o f communication i s t h a t o f e x p r e s s i b i l i t y o f meanings when t h e v o c a b u l a r i e s on one o r more media a r e c o n s t r a i n e d ( i n extreme c a s e s , some media may b e t o t a l l y banned from u s e , e . g . v e r b a l medium i n b a l l e t , o r communication between d e a f and m u t e ) . G e n e r a l l y , l e t u s d e n o t e b y R t h e r e s t r i c t i o n s under c o n s i d e r a t i o n s , so t h a t R may c o n c e r n banning one medium, o r r e s t r i c t i n g t h e v o c a b u l a r i e s o f some media, e t c . Such a r e s t r i c t i o n r e s u l t s i n s e l e c t i n g a s u b s e t L ( R ) o f t h e l a n g u a g e L , w i t h t h e c o n d i t i o n t h a t one may u s e o n l y t h e s t r i n g s v from L(R). The e f f e c t o f r e s t r i c t i o n o f u s e o f t h e s t r i n g s i n t h e l a n g u a g e o f communication a c t i o n s t o L(R) l i e s i n l i m i t i n g t h e p o s s i b i l i t y o f e x p r e s s i n g some meanings, and a l s o i n t h e p o s s i b i l i t y o f s e p a r a t i n g two meanings.
<
Let z Z be a f i x e d meaning, and l e t u s d e f i n e t h e q u a n t i t y q(z,R) as
C l e a r l y , t h e value q(z,R) i s always nonnegative, since t h e second supremum i s t a k e n o v e r a s m a l l e r s e t .
The v a l u e q ( z , R ) w i l l b e c a l l e d e x p r e s s i b i l i t y l o s s for
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meaning z , d u e t o r e s t r i c t i o n s R . Thus, i f q ( z , R ) = 0 , t h e n sup g ( v , z ) = sup g ( v , z ) ve L vbL(R) which means t h a t t h e meaning z may b e e x p r e s s e d t o t h e same d e g r e e w i t h o r w i t h o u t r e s t r i c t i o n s R . I n o t h e r words, r e s t r i c t i o n t o R has no e f f e c t on t h e p o s s i b i l i t y o f e x p r e s s i n g z , and t h e r e i s no loss. The m a x i m a l v a l u e of q ( z , R ) i s equality
sup g ( v , z ) , and t h e v LL
implies that
T h i s means t h a t we have
sup
g(v,z)
= 0.
VGL(R)
t o t a l l o s s o f e x p r e s s i b i l i t y of z : meaning z s i m p l y cannot be e x p r e s s e d i n any p o s i t i v e d e g r e e , i f one i s t o o p e r a t e under c o n s t r a i n t s R . On t h e o t h e r hand, i t i s p o s s i b l e t o have no loss of e x p r e s s i b i l i t y , s o t h a t q ( z , R ) = 0 , b u t i t may b e i m p o s s i b l e t o s e p a r a t e t h e meaning z from some o t h e r meaning z' ( o r a t l e a s t , s u c h a p o s s i b i l i t y may be limited). To d e f i n e t h e s e c o n c e p t s , l e t us c o n s i d e r two meanings,
*
z and z ' , and l e t z = z & - z ' . Suppose t h a t q ( z , R ) = 0 ; * i f z i s e x p r e s s i b l e i n full d e g r e e , t h e n z and z ' a r e * c o m p l e t e l y s e p a r a b l e , and i n g e n e r a l , q ( z ,R) r e f l e c t s t h e loss of p o s s i b i l i t y of s e p a r a t i n g meanings z and z f . A s a n example, c o n s i d e r meanings such as " s u r p r i s e " ,
" f e a r " , r t r a g e " , " j o y " , e t c . I f v e r b a l medium i s a l l o w e d , t h e y c a n b e f u l l y e x p r e s s e d , w i t h no c o n f u s i o n of t h e s e meanings. However, i f v e r b a l medium i s n o t allow-
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e d , and one may u s e o n l y f a c i a l e x p r e s s i o n s ( e . g . i n
s i l e n t f i l m s ) , t h e n some o f t h e above meanings may be d i f f i c u l t , i f n o t i m p o s s i b l e , t o s e p a r a t e i n f u l l degree. F i n a l l y , i t i s p o s s i b l e t h a t meaning z i s e x p r e s s i b l e i n f u l l under r e s t r i c t i o n s R ( h e n c e a l s o i n language L ) , b u t i n t h e f o r m e r c a s e i t r e q u i r e s more u n i t s t o e x p r e s s t h e meaning t h a n i n t h e l a t t e r . F o r example, what may d e s c r i b e d i n j u s t few words, may r e q u i r e d much l o n g e r i f o n l y g e s t u r e s a r e a l l o w e d , e t c . To d e s c r i b e f o r m a l l y s u c h " d i l u t i o n " o f r e q u i r e d t i m e ,
l e t Ln and L n ( R ) be s e t s o f s t r i n g s i n L and i n L ( R ) ,
o f l e a n g t h a t most n . We have t h e n
and L = uLn,
n
L(R) =
0 Ln(R). n
We may now d e f i n e
and
Then t h e r a t i o N(z,R)/N(z) d e s c r i b e s how much l o n g e r i t takes t o e x p r e s s z under c o n s t r a i n t s R t h a n w i t h o u t these constraints.
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To d e s c r i b e o t h e r n o t i o n s , l e t u s d e f i n e
so t h a t Ua(z) i s t h e c l a s s of a l l units z i n degree a t least a . Define a l s o
which e x p r e s s
s o t h a t V ( z ) i s t h e complement o f t h e c a r r i e r o f z , and U , ( z ) i s t h e k e r n e l o f z .
These c o n c e p t s a r e r e l a t e d s p e c i f i c a l l y t o f u z z y meani n g s , and have no d i r e c t c o u n t e r p a r t i n u s u a l s e m a n t i c s . They a r e e s p e c i a l l y c o n n e c t e d w i t h two f a c t s : -- t h a t meanings may b e e x p r e s s e d i n v a r y i n g d e g r e e s (which i s n o t , a s s t r e e s s e d , t h e same as e x p r e s s i o n o f h i g h e r d e g r e e of t h e meaning);
--
t h a t an e x p r e s s i o n - u n i t o r s t r i n g o f communication a c t i o n s -- may e x p r e s s s e v e r a l meanings a t o n c e .
We may s a y t h a t meaning z a - s u p p o r t s meaning z ' , i f f o r some a we have 0
for all a
2
a o . T h i s concept d e s c r i b e s t h e s i t u a t i o n when a n e x p r e s s i o n o f a g i v e n meaning, i f i t i s enough c l e a r , o r unambiguous ( i . e . e x p r e s s e d i n s u f f i c i e n t l y h i g h d e g r e e ) m u l s t i n v o l v e t h e e x p r e s s i o n o f some o t h e r meaning a l s o . S u p p o r t i n g of meaning i s q u i t e a n i n t e r e s t i n g phenomenon: i t shows t h e i n t e r r e l a t i o n s between meanings
FORMAL SEMIOTICS
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imposed b y t h e r e s t r i c t i o n s due to c e r t a i n media. The meaning a s such may be c o n c e p t u a l l y d i s t i n c t from ano t h e r , b u t b e c a u s e o f l i m i t a t i o n s o f t h e media, one cannot be e x p r e s s e d w i t h o u t t h e o t h e r . T h i s s e t o f r e l a t i o n s between meanings shows t h e r e f o r e t h e e x t e n t o f " b l u r r i n g " due to v a r i o u s media. An example was g i v e n above: t h e s i l e n t movies, where t h e a c t o r had t o convey meanings by f a c i a l e x p r e s s i o n s o n l y , and had t o d i f f e r e n t i a t e between, s a y , s u r p r i s e and f e a r , o r h a t e , j e a l o u s y and a n g e r , e t c . F i n a l l y , l e t u s o b s e r v e , t h a t t h e c o n s i d e r a t i o n s conc e r n i n g " d i l u t i o n " o f l e n g t h of s t r i n g s n e c e s s a r y t o e x p r e s s a g i v e n meaning may a l s o be c a r r i e d o v e r to t h e c a s e of e x c h a n g e a b i l i t y of media, i . e . d e t e r m i n i n g t h e r e l a t i v e r a t i o s o f l e n g t h o f s t r i n g s needed t o e x p r e s s a g i v e n meaning ( i f e x p r e s s i b l e a t a l l ) w i t h t h e u s e o f one medium o n l y ( o r a s e l e c t e d s e t o f media). To u s e a n example, a d i a l o g u e i n which a man s a y s " L e t ' s go f o r a w a l k " and t h e woman r e p l i e s , "Oh, l e a v e
me a l o n e ! " q a y t a k e s e v e r a l seconds on v e r b a l medium. The same meaning e x p r e s s e d i n a b a l l e t s c e n e may t a k e some m i n u t e s , and i n medium o f f a c i a l e x p r e s s i o n s o n l y i t i s n o t e x p r e s s i b l e a t a l l , at l e a s t t h e man's p a r t , s i n c e no f a c i a l e x p r e s s i o n can convey w i t h any r e a s o n a b l e l a c k o f a m b i g u i t y a s u g g e s t i o n o f a walk. Given two meanings, z and z ' , one can t h e n s a y t h a t z i s c o n t a i n e d i n z ' , i f U ( z ) C U a ( z ' ) , hence a l s o a V ( z ' ) C V ( z ) . Moreover, t h e s e meanings a r e i n s e p a r a b l e , i f V ( z ) = V ( z l ) , and i n c o m p a t i b l e , i f V ( z ) n V ( z ' )
328
=
m.
CHAPTER 4
F i n a l l y , they a r e orthogonal, i f n e i t h e r of t h e
s e t s V ( z ) and V ( z l ) i s c o n t a i n e d i n t h e o t h e r . A s mentioned i n t h e I n t r o d u c t i o n t o t h i s c h a p t e r , one may a l s o d e f i n e v a r i o u s t y p e s of synonymity: t h e s t r o n g -
e s t requirement here i s t h a t Ua(z) = U a ( z l ) f o r a l l a. On t h e o t h e r hand, f o r i n s t a n c e k e r n e l synonymity i s d e f i n e d through t h e requirement t h a t U,(z) = U , ( z t ) , and s o on. The f i r s t c o n c e p t o f i n c l u s i o n ( o r : embedding) c o r r e s ponds t o t h e u s u a l c o n c e p t o f i n c l u s i o n of meaning i n s e m a n t i c s , Here, however, t h e i n c l u s i o n i s n o t due t o t h e c o n t e n t o f z and z ' , b u t r a t h e r t o t h e l i m i t a t i o n s imposed b y t h e n e c e s s i t y of u s i n g o n l y c e r t a i n media, but not t h e o t h e r s . A s r e g a r d s t h e concept of i n s e p a r a b i l i t y , it c o r r e s -
ponds i n s e m a n t i c s t o t h e c o n c e p t o f p a r t i a l synonymy: when t h e meanings o v e r l a p , e x p r e s s i n g one of t h e m means automatically expressing the other, at l e a s t p a r t i a l l y . The c o n c e p t of i n c o m p a t i b i l i t y i s p e r h a p s c l o s e s t t o t h e c o n c e p t of n e g a t i o n , o r
-- more p r e c i s e l y -- e n t a i l -
ment of n e g a t i o n of one meaning by t h e o t h e r . F i n a l l y , t h e last concept, of ortogonality, corresponds t o l o g i c a l independence of meanings. Those c o n c e p t s u t i l i z e d o n l y p a r t o f t h e i n f o r m a t i o n c o n t a i n e d i n t h e f u n c t i o n g , namely t h e f a c t t h a t t h e v a l u e i s 0 o r n o t ; t h u s , t h e y may b e termed " P r e sence-Absence" c o n c e p t s . A s opposed t o t h a t , t h e conc e p t o f synonymy, o r
a-support,utilize
f u l l inforrnat-
i o n c o n t a i n e d i n t h e f u n c t i o n g . They may t h e r e f o r e be
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c a l l e d "Degree of e x p r e s s i o n " c o n c e p t s . It f o l l o w s from t h e above c o n s i d e r a t i o n s , t h a t t h e c o n c e p t u a l scheme o u t l i n e d i n t h e s e c t i o n s above i s s u f f i c i e n t l y r i c h t o allow defining various notions i n terms o f some p r i m i t i v e c o n c e p t s . The q u e s t i o n a r i s e s a b o u t t h e r o l e which s u c h a s y s t e m may p l a y i n a n a l y s i s of r e l a t i o n s between v e r b a l and n o n v e r b a l a c t i o n s . It seems t h a t t h e answer h e r e may be g i v e n i n form o f t h e f o l l o w i n g p o i n t s . 1. The system a l l o w s t o u n i f y t h e e m p i r i c a l
material and c l a s s i f i c a t i o n o f i t a c c o r d i n g t o a w e l l d e f i n e d scheme. T h i s c r e a t e s t h e p o s s i b i l i t y o f c o m p a r a t i v e a n a l y s i s o f communication s y s t e m s . 2 . The s y s t e m a l l o w s a l s o f o r t h e c o o r d i n a t i o n o f
f u t u r e r e s e a r c h , d i r e c t i n g t h e a t t e n t i o n a t t h o s e asp e c t s o f communication which may p r o v i d e b e s t d a t a f o r t e s t i n g v a r i o u s h y p o t h e s e s , f o r m u l a t e d i n terms o f t h e model.
3. The s y s t e m a l l o w s a l s o t o f o r m u l a t e a c e r t a i n numb e r o f t e n t a t i v e h y p o t h e s e s , o f which some a r e as follows. H y p o t h e s i s 1. The r e l a t i o n o f e n f o r c i n g i s much l e s s f r e q u e n t t h a t t h e r e l a t i o n of e x c l u s i o n . The i n t u i t i v e j u s t i f i c a t i o n o f t h i s h y p o t h e s i s i s t h a t i f an a c t i o n on a g i v e n medium a l l o w s f o r a number o f a c t i o n s on a n o t h e r medium, and e x c l u d e s some a c t i o n s on t h i s medium, w e have have e x c l u s i o n , b u t n o t enf o r c i n g , s i n c e t h e r e i s more t h a n one a c t i o n p o s s i b l e . Hypothesis 2 .
The r a t i o of t h e number of e x c l u s i o n s
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t o t h e number of e n f o r c e m e n t s depends on t h e p a i r o f media which i n e c o n s i d e r s . T h i s r a t i o i s s m a l l e s t , i . e . we have r e l a t i v e l y many e n f o r c e m e n t s , f o r media connected w i t h body movements. H y p o t h e s i s 3. There are more a c t i o n s which g e n e r a t e a meaning t h a n t h o s e which i n h i b i t i t . Probably o n l y few a c t i o n s have t h e l a t t e r p r o p e r t y . H y p o t h e s i s 4 . The movements o f h e a d , body and arms a r e p r o b a b l y more o f t e n used f o r s u p p o r t i n g t h e meanings which a r e e x p r e s s e d v e r b a l l y t h a n for g e n e r a t i o n o f new meanings. H y p o t h e s i s 5 . To g e n e r a t e meanings, one most o f t e n u s e s t h e medium of f a c i a l e x p r e s s i o n s . H y p o t h e s i s 6 . Media c o n n e c t e d w i t h i n t o n a t i o n a r e used most o f t e n t o e x p r e s s a meaning which i s a n e g a t i o n o f t h e meaning e x p r e s s e d v e r b a l l y . The t h e o r y developed i n t h i s s e c t i o n i s n o t c l o s e d ,
and may be developed i n many d i r e c t i o n s , s u c h a s a c o n s t r u c t i o n o f g e n e r a l s e m i o t i c c a l c u l u s , b a s e d on some laws o f p e r c e p t i o n o f meanings, o r t o w a r d s t h e i n t e r a c t i o n a l a s p e c t s o f communication, hence i n p a r t i c u l a r , towards a multidimensional t h e o r y o f d i a l o g u e s . F i n a l l y , t h e t h e o r y may be a p p l i e d t o t h e n a t u r a l l a n g a u a g e s , by r e s t r i c t i n g t h e a n a l y s i s t o v e r b a l medi a only ( p i t c h , intonation, accentuation, p a r t i t i o n int o s y l l a b e s , e t c . ) . We would have h e r e a d e s c r i p t i o n o f n a t u r a l language i n terms of m u l t i d i m e n s i o n a l u n i t s , p l a y i n g d e f i n i t e r o l e s i n e x p r e s s i n g c e r t a i n meanings. One c o u l d a l s o c o n s i d e r a n a n a l y s i s o f w r i t t e n l a n g u a g e , where t h e shape o f t y p e , s i z e , u s e o f i t a l i c s , e t c .
33 1
FORMAZ, SEMIOTICS
p l a y t h e r o l e o f media. F o r s u c h t y p e o f a n a l y s e s , i t
would b e v a l u a b l e t o a n a l y s e t h e o r e t i c a l l y and empir i c a l l y t h e t o p o l o g i c a l s t r u c t u r e of t h e s p a c e o f a l l meanings. It s h o u l d
a l s o b e s t r e s s e d t h a t t h i s model c o u l d a l s o be u s e f u l f o r f o u n d a t i o n of i n f o r m a t i o n s c i e n c e s , where t h u s f a r t h e r e were no a t t e m p t s t o d e s c r i b e l a n g u a g e s i n terms of multidimensional u n i t s ( l e t t e r s ) , w i t h a c e r t a i n i n t e r n a l s t r u c t u r e , allowing t h e simultaneous o p e r a t i o n s and new t y p e s o f s e m a n t i c s . F i n a l l y , t h e model s u g g e s t e d h e r e , w i t h some m o d i f i c a t i o n s , l e a d s t o new c o n c e p t s i n s y s t e m s t h e o r y , namel y t o t o p i c s o f system s y n t h e s i s . These problems w i l l be d i s c u s s e d i n Chapter 6 .
2.4.
Semantic l a n g u a g e s
Let u s observe t h a t t h e approach t o semantics i n t r o duced h e r e a l l o w s u s t o d e f i n e t h e f o l l o w i n g c o n s t r u c t i o n , which i s of i n d e p e n d e n t i n t e r e s t . T h i s c o n s t r u c t i o n i s a p p l i c a b l e t o any l a n g u a g e s , n o t n e c e s s a r i l y m u l t i d o m e n s i o n a l , and t h e r e f o r e we s h a l l p r e s e n t i t here i n t h e general case. We assume t h a t we have a c e r t a i n l a n g i a g e L , i . e .
*
a
s u b s e t of t h e monoid V o v e r a c e r t a i n v o c a b u l a r y . The e l e m e n t s o f V may be l e t t e r s ( i n which c a s e e l e m e n t s o f L w i l l be w o r d s ) , o r e l e m e n t s o f V may be words, i n which c a s e e l e m e n t s o f L w i l l be s e n t e n c e s . I n some o t h e r c a s e s , e l e m e n t s of V may be m u l t i m e d i a 1 u n i t s , symbols i n some l o g i c a l c a l c u l u s , e t c .
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Elements of L are t h e r e f o r e a d m i s s i b l e s t r i n g s of e l e m e n t s of t h e v o c a b u l a r y , and w i l l be d e n o t e d by vl...v where v C V f o r i = 1 n , and n = 1 , 2 n’ i
,...,
,... .
L e t now Z be t h e s e t of meanings under c o n s i d e r a t i o n , and as b e f o r e , l e t g ( v , z ) be t h e d e g r e e t o which t h e s t r i n g v e x p r e s s e s meaning z . G e n e r a l l y , e l e m e n t s of Z a r e some complex s t r u c t u r e s b u i l t o u t o f semems, and o f a u x i l i a r y symbols, s u c h as p a r e n t h e s e s , e t c . Such s t r u c t u r e s may be assumed t o be l i n e a r ( . i . e . even i n c a s e of complex s t r u c t u r e s , i n forms of t r e e s , e t c . , one can d e s i g n a method o f l i n e a r i z a t i o n , by e n r i c h i n g , i f n e c e s s a r y , t h e s e t o f symbols). Thus, we s h a l l assume now t h a t t h e r e e x i s t s a n o t h e r v o c a b u l a r y , s a y H , comprisong e l e m e n t a r y semems and some o t h e r symbols, and a mapping b : Z
- 2
H*
,
which t o e v e r y meaning z E A a s s i g n s a s e t o f s t r i n g s of e l e m e n t s o f H . T h i s s e t , d e n o t e d b y b ( z ) , i s t h e r e f o r e t h e c l a s s of a l l semantic r e p r e s e n t a t i o n s o f z . Thus, t h e s e m a n t i c system under c o n s i d e r a t i o n i s now
C o n s i d e r now a s t r i n g v E L and (for f i x e d l e v e l a ) t h e s e t o f meanings
FORMAL SEMIOZTCS
333
t zvla>
we may a s s i g n t o i t t h e s e t b ( z ) , whose e l e m e n t s w i l l be c a l l e d s e m a n t i c markers o f z.
If z
A s a consequence, we have a f a m i l y o f mappings f a :
+ 2H*, which
t o every s t r i n g v i n L a s s i g n s t h e s e t f ( v ) o f s t r i n g s o f s e m a n t i c m a r k e r s , t o be c a l l e d a s e m a n t i c t r a c e of v .
L
Semantic t r a c e s can be c o n c a t e n a t e d , i n t h e s e n s e t h a t fa(u)fa(v) = f x : x = YZ, Y E fa(& Y € fa(V)). By s e m a n t i c language ( l a n g u a g e o f t r a c e s ) o v e r L we s h a l l now mean t h e s e t
sa ( L )
=
u V(
fa(v).
L
Moreover, i f one has an e q u i v a l e n c e r e l a t i o n on t h e s e t o f s e m a n t i c m a r k e r s , one can c o n s i d e r c l a s s e s o f a b s t r a c t i o n o f t h i s r e l a t i o n , and c o n s e q u e n t l y , langua g e s of s e m a n t i c markers of h i g h e r o r d e r , where t h e markers would be l a b e l s o f e q u i v a l e n c e c l a s s e s . Now, t h e most i m p o r t a n t c a s e s o c c u r when f a ( v ) = 0 and when f ( v ) c o n s i s t s o f a s i n g l e e l e m e n t . I n t h e a f i r s t c a s e , we s a y t h a t s t r i n g v i s m e a n i n g l e s s : t h e r e i s no s t r i n g o f s e m a n t i c markers a s s i g n e d t o i t . I f * f ( v ) c o n s i s t s of a s i n g l e s t r i n g i n H o n l y , t h e n v a has a unique meaning ( a t l e v e l a ) , or i s a-unambiguous F i n a l l y , i f f a ( v ) c o n s i s t s o f more t h a n one e l e m e n t , w e have a m b i g u i t y o f s t r i n g v . I n p a r t i c u l a r , we have language o f t r a c e s o f s e m a n t i c markers Sa(L) o v e r t h e m u l t i m e d i a 1 l a n g u a g e o f communication L. Strings of actions generate s t r i n g s of s t a t e s , which form t h e r e f o r e a s t a t e - l a n g u a g e g e n e r a t -
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ed by c o n s e c u t i v e u n i t s ; l e t t h i s language b e S
*
(this
language a g a i n may be c o n s i d e r e d on v a r i o u s l e v e l s o f a b s t r a c t i o n , induced by e q u i v a l e n c e c l a s s e s on s t a t e s ) . * One can now c o n s i d e r p o s s i b l e homomorphisms between S and s e m a n t i c l a n g u a g e . N a t u r a l l y , such homomorphisms need n o t e x i s t s i n c e t h e r e i s no a p r i o r i r e a s o n why t h e s t r u c t u r a l p r o p e r t i e s imposing c o n s t r a i n t s o n , s a y , arm movements, s h o u l d r e s e m b l e i n any way t h e s t r u c t u r -
a l p r o p e r t i e s o f e v e n t s d e s c r i b e d by words and g e s t u r e s . However, one c o u l d e x p e c t some r e l a t i o n s h i p s t o e x i s t between t h e p h y s i c a l s t r u c t u r e of s i g n s and t h e o b j e c t s d e s c r i b e d (some t r a c e s o f such r e l a t i o n s h i p s , on a low l e v e l , e x i s t i n form o f onomatopeic words, g e s t u r e s f o r " l a r g e " and " s m a l l " , e t c . ) . Such homomorphisms c o u l d be s t u d i e d for some s i m p l i f i e d r e g u l a r forms o f m u l t i m e d i a 1 l a n g u a g e s and t h e t r a c e l a n g u a g e s g e n e r a t e d i t . T h i s would l e a d t o s e a r c h f o r e q u i v a l e n c e o f r e p r e s e n t a t i o n o f m u l t i m e d i a l commun i c a t i o n language i n s e m a n t i c domain, as w e l l as i n t h e domain o f s t a t e l a n g u a g e , r e p r e s e n t e d b y i n s t r u c t i o n l a n g u a g e s . Such problems c o u l d be a n a l y s e d a l s o f o r l a n g u a g e s of h i g h e r o r d e r s . Some hope h e r e c o u l d be p r o v i d e d by s i m u l a t i o n s t u d i e s of such l a n g u a g e s . The most i m p o r t a n t would be f i n d i n g r e p r e s e n t a t i o n s o f v e c t o r s ( m u l t i m e d i a l u n i t s ) as n e t works. ,Next s e c t i o n w i l l show d e f i n i t i o n s o f multimedi a l u n i t s and l a n g u a g e s i n terms o f p r o b a b i l i s t i c networks. 2.5.
Graph r e p r e s e n t a t i o n o f a u n i t
I n t h e a p p r o a c h p r e s e n t e d a b o v e , u n i t s are t r e a t e d as
FORMAL SEMIOTICS
335
random o b j e c t s (random s i g n s ) , composed out of e l e m e n t a r y o b j e c t s . We have namely some s p a c e , s a y Q , of c e r -
t a i n o b j e c t s (which may be a c t i o n s , p h y s i c a l o b j e c t s , e v e n t s , e t c . ) ; on Q we have a b i n a r y r e l a t i o n , i n t e r p r e t e d as " b e i n g s e m a n t i c a l l y connected to". Thus, a u n i t may b e i d e n t i f i e d w i t h a g r a p h of such a r e l a t ion. The u s a g e of a g i v e n u n i t may b e , o f a t r a n s i e n t n a t u r e (examples may be happenings i n a r t , some non-conventi o n a l s i g n s i t u a t i o n s a r r a n g e d ad hoe, m e a n i n g f u l o n l y i n a g i v e n c o n t e x t , e t c . ) . One can a l s o d i s t i n g u i s h more s t a b l e o b j e c t s , t h a t e v o l v e i n time s l o w l y . T h e e v o l u t i o n h e r e i s e x p r e s s e d t h r o u g h loss o r g a i n of some nodes o f t h e r e l a t i o n , and g a i n or loss o f some e d g e s . T o d e s c r i b e s u c h s i t u a t i o n s , one may u s e a new appro.ach t o random g r a p h s ( s e e Nowakowkska 1 9 7 9 a , 1 9 8 3 ) , which u t i l i z e s t h e p o s s i b i l i t y of random a p p e a r e n c e s and random d i s a p p e a r e n c e s o f nodes and e d g e s . Such a t h e o r y w i l l be o u t l i n e d below i n t h e p r e s e n t s e c t i o n . The f o r m a l a p p r o a c h w i l l be somewhat d i f f e r ent than i n t h e c i t e d papers. F o r m a l l y , a g r a p h i s a p a i r (S,C) , where S i s t h e s e t of elememts c a l l e d n o d e s , and C i s a b i n a r y r e l a t i o n i n S , i . e . a s u b s e t C C S % S . If xCy, we s a y t h a t e l e m e n t s x and y a r e c o n n e c t e d w i t h a n edge. The e d g e s a r e d i r e c t e d , t h a t i s , one may have xCy, b u t n o t yCx. T y p i c a l l y , random g r a p h s a r e d e s c r i b e d by assuming t h a t t h e s e t of nodes i s f i x e d , and t h e edges a r e c h o s e n a t random a c c o r d i n g t o some s a m p l i n g scheme. I n t h i s s e c t i o n , a model w i l l be s u g g e s t e d which w i l l
336
CHAFER 4
account f o r t h e p o s s i b i l i t y o f randomness a l s o i n t h e s e t S o f nodes. A c c o r d i n g l y , we s h a l l d e n o t e by S ( t ) and C ( t ) t h e s e t o f nodes p r e s e n t a t t , and t h e s e t of e d g e s p r e s e n t
a t t . N a t u r a l l y , f o r e v e r y t w e have
(2.17) To e n s u r e c o n d i t i o n (2.17) one must assume t h a t as soon
as a node d i s a p p e a r s , a l l edges l e a d i n g t o and from t h i s node a l s o d i s a p p e a r . T h i s i s a n a t u r a l r e q u i r e m e n t . The e v o l u t i o n of a random g r a p h ( S ( t ) , C ( t ) ) may be d e s c r i b e d i n terms of' two p r o c e s s e s : moments o f chang e s o f t h e s e t S ( t ) o f n o d e s , and changes of C ( t ) o c c u r i n g between changes o f S ( t ) . A c c o r d i n g l y , we s h a l l need two k i n d s o f a s s u m p t i o n s : (1) t h o s e d e s c r i b i n g t h e changes i n S ( t ) , and ( 2 ) t h o s e d e s c r i b i n g changes i n C ( t ) which o c c u r i n
between changes of S ( t ) , hence when t h e number o f nodes i s c o n s t a n t . A s r e g a r d s t h e f i r s t assumptions, i t i s n a t u r a l t o
r e q u i r e t h a t t h e s e t S ( t ) and a l s o C ( t ) form Markov p r o c e s s e s . S p e c i f i c a l l y , we assume t h a t S ( t ) s a t i s f i e s ASSUMPTION 1. Suppose t h a t t h e s e t S ( t ) has I S ( t ) (
=
n
e l e m e n t s . Then t h e p r o b a b i l i t y of a new node a p p e a r i n g between t and t t A t e q u a l s c(n)At
+
O b t ) .
(2.18)
Here c ( n ) i s some f u n c t i o n o f t h e g r a p h s i z e n . I n t h e
331
FORMAL SEMIOTICS
p a r t i c u l a r c a s e when c ( n ) = 1 f o r a l l n , we have a r r i v a l s according t o Poisson process w i t h i n t e n s i t y Another i m p o r t a n t s p e c i a l c a s e i s c ( n ) = n for a l l n , i n which c a s e t h e a r r i v a l i n t e n s i t y i s p r o p o r t i o n a l t o t h e a c t u a l s i z e o f t h e graph. A s r e g a r d s d i s a p p e a r e n c e s of nodes o f t h e g r a p h , t h e
g e n e r a l assumption may be as f o l l o w s . ASSUMPTION 2 . L e t x b e a node i n S(t), and l e t p, and q x be t h e numbers o f e d g e s l e a d i n g fram x and t o x. Then t h e p r o b a b i l i t y o f d i s a p p e a r e n c e o f t h e node x between t and t t A t e q u a l s
Moreover, t h e dwents d e s c r i b e d i n t h i s a s s u m p t i o n f o r d i f f e r e n t nodes x , and t h e e v e n t s i n Assumption 1 a r e m u t u a l l y independent. Here r ( p x , q x ) i s some f u n c t i o n , which d e s c r i b e s " d e a t h r a t e " o f n o d e s , depending on t h e " t i e s " o f t h i s node w i t h t h e r e s t , o f t h e graph. I n p a r t i c u l a r c a s e s , w e may have r ( p x , q x ) = 1, which means t h a t p r o b a b i l i t y o f d i s a p p e a r e n c e o f a node i s c o n s t a n t , and d o e s n o t depend on t h e number o f e d g e s l e a d i n g t o o r from t h i s node. I n s u c h a c a s e , t h e p r o b a b i l i t y t h a t one node w i l l disappear i n ( t , t t A t ) equals if is the t o t a l number o f nodes a t t i m e t .
rn,
Now, as r e g a r d s t h e a s s u m p t i o n s about t h e changes i n t h e s e t o f e d g e s C ( t ) , one can assume t h e f o l l o w i n g . L e t u s f i x some t i m e t , and c o n s i d e r two n o d e s , x and
CHAPTER 4
338
y , i n S ( t ) . These nodes may be c o n n e c t e d by an e d g e
o r not; denote
a
= 1 if
o
xCy,
i f n o t xcy
(and s i m i l a r l y f o r a
). YX
ASSUMPTION 3. Suppose t h a t x , y
S ( t ) , and l e t a
= 0 XY
( s o t h a t t h e r e i s no edge l e a d i n g from x t o y ) . Then t h e p r o b a b i l i t y of a p p e a r e n c e o f a new edge from x t o y i n ( t , tt A t ) equals
On t h e o t h e r h a n d , i f a
XY
= 1 ( t h e r e i s an edge l e a d -
i n g from x t o y ) , t h e n t h e p r o b a b i l i t y t h a t t h i s edge w i l l disappear i n ( t , tt A t ) i s
Moreover, t h e e v e n t s d e s c r i b e d i n t h i s a s s u m p t i o n a r e independent f o r d i f f e r e n t edges. The form of f u n c t i o n s f and g which a p p e a r i n t h i s a s s u m p t i o n depends on t h e p a r t i c u l a r phenomenon which o n e w a n t s t o d e s c r i b e . I n p a r t i c u l a r , t h e r e may be a t e n d e n c y t o t h e a p p e a r e n c e o r d i s a p p e a r e n c e of an e d g e , d e p e n d i n g on how " p o p u l a r " a r e t h e two n o d e s , Such a t e n d e n Y' cy w i l l be g e n e a r a l l y d e s c r i b e d by a p p r o p r i a t e c h o i c e i.e.
on t h e numbers p,,
of f u n c t i o n s f and g .
q x , py and q
FORMAL SEMIOTICS
339
To c o n s i d e r some s p e c i a l c a s e s , assume t h a t t h e funct i o n s c ( n ) and r ( n ) i n Assumptions 1 and 2 a r e c o n s t a n t s e q u a l 1. L e t P n ( t ) d e n o t e t h e p r o b a b i l i t y t h a t [S(t)\
=
n . Then ( s e e F e l l e r , 1957), we have t h e s y s -
tem o f e q u a t i o n s for P ( t ) : n
for n = 1,2,...
.
S i m i l a r e q u a t i o n s may be o b t a i n e d
a l s o for o t h e r c a s e s of f u n c t i o n s c ( n ) and r(n). I n t h e c a s e under c o n s i d e r a t i o n ( a s l o n g as \ S ( t ) l > 0 ) t h e t i m e X t u n t i l t h e n e x t ( a f t e r t ) a p p e a r e n c e or d i s a p p e a r e n c e o f a node, has e x p o n e n t i a l d i s t r i b u t i o n
so that
I n c a s e when \ S ( t ) ( = 0 , t h e o n l y p o s s i b i l i t y i s t h a t a new node w i l l a p p e a r , s o t h a t
Moreover, i f ‘ S ( t ) l 7 0 , t h e p r o b a b i l i t y t h a t t h e n e x t e v e n t w i l l be a n a p p e a r e n c e o f a new node i s b / ( h t p ) , w h i l e t h e p r o b a b i l i t y t h a t i t w i l l be a d i s a p p e a r e n c e o f a node i s r/(h+p).
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Now, as r e g a r d s a p p e a r e n c e and d i s a p p e a r e n c e o f e d g e s , t h e p o s s i b i l i t i e s of modeling v a r i o u s e f f e c t s ( a s a l r e a d y m e n t i o n e d ) l i e s i n t h e p r o p e r c h o i c e of f u n c t i o n s f and g . I n t h e s i m p l e s t c a s e , assume t h a t f and g a r e c o n s t a n t s , s a y e q u a l a and b . Then t h e a p p e a r e n c e and d i s a p p e a r e n c e o f e d g e s a r e a g a i n P o i s s o n p r o c e s s e s , and moreo v e r , each o f t h e edges t h a t e x i s t a t a given time h a s t h e same chance o f d i s a p p e a r i n g , and e a c h o f t h e p o s s i b l e " m i s s i n g " e d g e s h a s t h e same chance o f a p p e a r i n g . T h i s c o r r e s p o n d s t o random s a m p l i n g of e d g e s from t h e e x i s t i n g and n o n e x i s t i n g o n e s . The w a i t i n g t i m e f o r n e x t e v e n t ( a p p e a r e n c e o r d i s a p p e a r e n c e o f an e d g e ) h a s e x p o n e n t i a l d i s t r i b u t i o n , w i t h p a r a m e t e r s a and b . One can t h e r e f o r e compute t h e e x p e c t e d change i n t h e g r a p h between c o n s e c u t i v e appearences o r d i s a p p e a r e n c e s o f nodes. I n p a r t i c u l a r , i f a t some t i m e t w e have I S ( t ) l = n and I C ( t ) l
= k ( g r a p h h a s n nodes and k e d g e s ) , and
a t t one node d i s a p p e a r s , t h e n t h e e x p e c t e d number o f e d g e s which must a l s o d i s a p p e a r i s k/n ( s i n c e t h e e x p e c t e d number o f e d g e s l e a d i n g from a g i v e n node i s k / 2 n , and e x p e c t e d number o f e d g e s which l e a d t o a g i v e n node i s a l s o k / 2 n ) . A s a n example o f g r a p h ( n e t w o r k ) r e p r e s e n t a t i o n o f u n i t s o f m u l t i m e d i a 1 l a n g u a g e s , one may t a k e changes of ways o f a r t i s t i c e x p r e s s i o n i n t i m e , e . g . e v o l u t i o n
o f G o t h i c s c u l p t u r e , where some means o f e x p r e s s i o n a p p e a r and some o t h e r d i s a p p e a r . A s a n o t h e r example, one may t a k e p l a y s a d a p t e d f o r t h e a t r e ; h e r e removing
FORMAL SEMIOTICS
341
a r o l e ( n o d e ) i m p l i e s removing a l l r e f e r e n c e s t o i t , i . e . a l l e d g e s l e a d i n g t o and from t h i s node. As s t i l l a n o t h e r example, one may c o n s i d e r a p a i n t e r who works on some p a i n t i n g . When he d e c i d e s t o e r a s e some f i g u r e i n t h e background ( s a y ) , he may have t o r e p l a c e i t by some o t h e r f i g u r e o r o b j e c t ( n o d e ) , i n o r d e r t o p r e s e r v e c o m p o s i t i o n a l e q u i l i b r i u m . I n o t h e r words, w e may have h e r e n e c e s s a r y s u b s t i t u t i o n s . S i m i l a r l y , i n t h e a t r e , one may remove a fragment of m u s i c a l background, and i f i t i s c o m p o s i t i o n a l l y n e c e s s a r y , r e p l a c e i t by some o t h e r m u s i c a l f r a g m e n t . The a s s u m p t i o n s c o n c e r n i n g changes o f edges ( a s opposed t o changes of n o d e s ) c o r r e s p o n d t o laws o f changea b i l i t y o f v a r i o u s forms of composite communication u n i t s ; t h e s e laws may be d i f f e r e n t f o r d i f f e r e n t forms o f a c t i v i t y . The r e l a t i o n s h i p between t h e p r o b a b i l i t y of a p p e a r e n c e o r d i s a p p e a r e n c e of an edge and t h e c h a r a c t e r i s t i c s of t h e n o d e s , c o r r e s p o n d t o t h e p r i n c i p l e of c o n t e x t u a l i t y of change. Here we have a law o f meaning p r e s e r v a t i o n i n c a s e o f a p p e a r e n c e of an e d g e , and s i m i l a r l a w s for t h e p r o c e s s of d i s a p p e a r e n c e . Also, some e d g e s may have s m a l l p r o b a b i l i t y o f d i s a p p e a r e n c e s ( o r even z e r o ) ; i n t e r m s of m u l t i m e d i a l u n i t s , t h i s corresponds t o multimedia1 f u n c t i o n a l u n i t s , where some components a r e s i g n - n e c e s s a r y , i . e . must a p p e a r i n any s t a g e of e v o l u t i o n of s i g n . F o r a n a l y s i n g t h e e f f e c t of s u p p o r t i n g o r i n h i b i t i n g a meaning by s e m a n t i c c o n n e c t i o n w i t h a new node ( r e -
p r e s e n t i n g a fragment of random o b j e c t ) , one can i n t r o duce t h e n o t i o n o f a f i l t e r , i . e . a f u n c t i o n which changes t h e meaning of a node, and hence i n f l u e n c e s t h e meaning o f u n i t as a whole.
342
CHAPmR 4
H y p o t h e t i c a l l y , t h i s p r o c e s s may be i n t e r p r e t e d i n two w a y s . F i r s t l y , from r e c e i v e r ' s p o i n t o f v i e w , f i l t e r may be o m i s s i o n o f some p a r t s o f random o b j e c t , due t o
s e l e c t i v i t y o f p e r c e p t i o n . From t h e p o i n t o f view o f t h e c o n s t r u c t o r o f a t r a n s i e n t m u l t i m e d i a 1 u n i t (random o b j e c t ) , f i l t e r s may be i n t e r p r e t e d as c o n s c i o u s oper a t i o n s on d e g r e e s o f meaning c a r r i e d b y n o d e s , and c h o i c e o f s u c h c o m p o s i t i o n o f s u b u n i t s which would g i v e t h e r e q u i r e d l e v e l o f i n t e n d e d o v e r a l l meaning, f o l l o w i n g some p r i n c i p l e s o f s e m a n t i c c a l c u l u s .
For more c o n v e n i e n t i n t e r p r e t a t i o n of t h e p r o c e s s o f f i l t e r i n g and c o n t r o l , i t w i l l be b e t t e r t o t r e a t t h e r e l a t i o n o f s e m a n t i c c o n n e c t e d n e s s i n t e r m s o f amount o f i n f o r m a t i o n about meaning c a r r i e d by one s u b u n i t ( n o d e ) about a n o t h e r . The flow o f i n f o r m a t i o n about meaning i n t h e random o b j e c t c o r r e s p o n d s t o e i t h e r a chosen p a t h of t h e eye o v e r p a r t s o f t h e o b j e c t , o r i n c a s e o f t h e s e n d e r -- c h o i c e o f such s t r u c t u r e o f edges and nodes which imposes some t y p e o f f l o w o f i n f o r m a t i o n i n r e c e i v e r . The p r o c e s s o f c o n t r o l can
--
be i d e n t i f i e d i n b o t h c a s e s w i t h a s t r i n g o f i n s t r u c t i o n s : f o r r e c e i v e r t h e s e w i l l be t h e i n s t r u c t i o n s a b o u t t h e nodes t o o m i t , t o v i s i t , t o a s s i g n g r e a t e r importance, e t c . For t h e s e n d e r , i n s t r u c t i o n s concern t h e d e s i g n o f t h e u n i t w i t h i n t e n d e d meaning and i t s
degree. Of c o u r s e , b o t h s e n d e r and r e c e i v e r have knowledge c o n c e r n i n g s t r u c t u r e o f more c o n v e n t i o n a l communicati o n u n i t s , and knowledge about t h e w a y s o f c o n t r o l l i n g t h e p r o c e s s o f communication t h r o u g h such u n i t s . I n o t h e r words, t h e y have knowledge about some random v a r i a b l e , and p r o c e s s o f p e r c e p t i o n o r p r o c e s s o f
FORMAL SEMIOTICS
343
d e s i g n i n g and p e r f o r m i n g a u n i t may be i d e n t i f i e d w i t h r e a l i z a t i o n o f s u c h random v a r i a b l e . We s h a l l show l a t e r t h e use o f d e f i n i t i o n o f random o b j e c t f o r t h e t h e o r y o f o b s e r v a b i l i t y . Now we s h a l l show two i m p o r t a n t mechanisms o f p e r c e p t i o n , whose i n t e r a c t i o n d e t e r m i n e s t h e p a t h o f t h e eye o v e r t h e structure of t h e object.
3. CONTROL PROCESSES I N PERCEPTION The p e r c e p t u a l p r o c e s s e s i n t h e p r e s e n t e d below new
dynamic models ( s e e Nowakowska 1983) w i l l be assumed’ to b e p a r t i a l l y c o n t r o l l a b l e . The o b s e r v a b l e v a r i a b l e s w i l l be t h e i n s p e c t i o n p r o c e s s t h r o u g h c o n s e c u t i v e g l a n c e s , governed by two b a s i c mechanisms -- o f spont a n e o u s i n s p e c t i o n , and o f p u r p o s e f u l i n s p e c t i o n , t o be d e s c r i b e d b y t h e two models below. The o t h e r two models w i l l c o n c e r n t h e t o p i c o f d a t a g e n e r a t i o n t h r o u g h c o m p a r i s o n s , and f o r m i n g e v e n t r e p r e s e n t a t i o n s o f o b j e c t s i n memory.
3.1.
P r o c e s s of s p o n t a n e o u s i n s p e c t i o n s
A s r e g a r d s t h e p r o c e s s of c o n s e c u t i v e g l a n c e s , it i s
aimed a t e x p l a i n i n g t h e mechanisms which d e t e r m i n e how
l o n g a g i v e n domain ( f r a g m e n t o f t h e o b j e c t , o r a c t i o n o f a m u l t i m e d i a 1 u n i t ) w i l l be i n s p e c t e d , and which w i l l be t h e domain i n s p e c t e d as n e x t . I f , i n a d d i t i o n , one assumes p a r t i a l c o n t r o l l a b i l i t y
of t h e p e r c e p t i o n p r o c e s s , u n d e r s t o o d as a c o n s c i o u s
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component i n placement o f g l a n c e s , t h e n t h e jumping eye movement a l l o w s f o r economy of p e r c e p t u a l work. T h i s economy i s a t t a i n e d under some a d d t i o n a l c o n s t r a i n t s , namely under t h e c o n d i t i o n o f s e m a n t i c c o n t i n u i t y . i . e . supplementary knowledge of t h e s i t u a t i o n , which a l l o w s f o r smoothing t h e d i s c r e t e d a t a s u p p l i e d by t h e e y e , and a c c o u n t s f o r u n d e r s t a n d i n g t h e s e d a t a . To e x p l a i n t h e jumpirlg
e x p l o r a t o r y eye movement, one introduces here t h e notion of p u l l i n g force. T h i s force
r e p r e s e n t s t h e s t r e n g t h w i t h which a g i v e n domain ( o f
a f o c a l p o i n t o f a given fragment) " a t t r a c t s " t h e e y e . The p u l l i n g f o r c e d e t e r m i n e s t h e t i m e o f s t a y i n a domain, and t h e p r o b a b i l i t y o f t r a n s i t i o n t o a g i v e n o t h e r domain. G e n e r a l l y , i t w i l l be assumed t h a t t h e p u l l i n g f o r c e d e c r e a s e s when t h e domain i s i n s p e c t e d , and i n c r e a s e s when some o t h e r domain i s i n s p e c t e d . The mechanisms o f i n c r e a s e and d e c r e a s e may be chosen i n v a r i o u s ways. One o f them i s t o assume t h a t t h e p u l l i n g f o r c e a s s o c i a t e d w i t h a domain d e c r e a s e s exp o n e n t i a l l y when t h i s domain i s i n s p e c t e d , and i n c r e a ses l i n e a r l y d u r i n g i n s p e c t i o n o f some o t h e r domain ( s i m i l a r a s s u m p t i o n s u n d e r l i e t h e model o f change o f a c t i v i t y o f Atkinson and B i r c h (1972); t h e p r e s e n t model, however, d i f f e r s i n v e r y e s s e n t i a l a s p e c t s from t h a t o f Atkinson and B i r c h ) . The main i s s u e o f t h e model l i e s i n c o n n e c t i o n between t h e unobsevable p u l l i n g f o r c e s and t h e o b s e r v e d behavi o u r -- i n t h i s c a s e , c o n s e c u t i v e g l a n c e s . The model p o s t u l a t e s t h a t t h e d e c i s i o n t o t e r m i n a t e a
g l a n c e a t some domain and move t h e eye t o some o t h e r
345
FORMAL SEMIOl7CS
domain r e s u l t s from r e s o l v i n g a " c o n f l i c t " between p u l l s by v a r i o u s domains. I n a s e n s e , v a r i o u s domains "compete" f o r t h e g l a n c e , w h i l e t h e p u l l k e e p i n g t h e g l a n c e a t a g i v e n i n s p e c t e d domain d e c r e a s e s . The eye i s t h e r e f o r e " p u l l e d " i n v a r i o u s d i r e c t i o n s . The mechanisms o f t h e . c h o i c e i s assumed t o b e s t o c h a s t i c , so t h a t t h e model a l l o w s t o o b t a i n t h e p r o b a b i l i t y d i s t r i b u t i o n o f t h e d u r a t i o n o f s t a y i n a g i v e n domain and t h e t r a n s i t i o n p r o b a b i l i t i e s t o o t h e r domains. To p r o c e e d f o r m a l l y , w e assume t h a t t h e o b j e c t i n s p e c t -
ed has a number o f f o c a l p o i n t s and t h e i r domains; t h e f o c a l p o i n t s w i l l b e d e n o t e d by p l , p 2 , and The n o t i o n o f a domain t h e i r domains by D1, D2,
... .
...
here may be f u z z y , i . e . some p o i n t s may b e l o n g i n
v a r y i n g d e g r e e s t o v a r i o u s domains. However, i t w i l l be assumed t h a t a t e a c h moment t h e eye i s f o c u s e d a t some w e l l d e f i n e d domain. Assumption 1. With e a c h f o c a l p o i n t pi t h e r e i s a s s o c i a t e d i t s p u l l i n g f o r c e f i ( t ) , which may change i n t i m e . It i s assumed t h a t f . ( t ) d e c r e a s e s exponent1 i a l l y when domain o f pi i s i n s p e c t e d , and i n c r e a s e s l i n e a r l y when some o t h e r domain i s i n s p e c t e d . S p e c i f i c a l l y , i t i s assumed t h a t i f t h e domain Di
is
i n s p e c t e d between t o and t , t h e n
(3.1) if a domain p
and t
,
then
j'
w i t h j $ i , i s i n s p e c t e d between t
0
CHAPTER 4
346
Assumption 2 .
Suppose t h a t at time t t h e domain D
i s inspected,
and l e t f l y f 2 y . . . be t h e p u l l i n g f o r c e s
a t t h a t t i m e . Then t h e p r o b a b i l i t y t h a t t h e r e w i l l be a change o f domain i n s p e c t e d ( j u m p ) between t and t + h , i s f o r domains o f p l y p 2 , . . .
2
f. h
fi j # i
+
o(h).
(3.3)
J
If a jump o c c u r s , t h e n t h e p r o b a b i l i t y t h a t i t w i l l
be t o t h e domain D k ( k # i ) e q u a l s fk/
z
fj
.
j#i
One can t h e n p r o v e t h e f o l l o w i n g t h e o r e m ( s e e Nowakows k a 1983): THEOREM. & e J f l y f 2 ) . . . be t h e p u l l i n g f o r c e s a t t i m e
t , o f b e g i n n i n g o f i n s p e c t i o n o f t h e domain D i . Then, d e n o t i n g by Ui t h e time o f u n i n t e r r u p t e d i n s p e c t i o n o f D i , w e have
= exp(i-
1 [Bite fiCi
B.
cit t
cit
(2 - ~ . ) ( e c
i
1
-
1)lf
(3.5)
where =
Bi
z
bik,
Fi =
k#i
2 fk.
(3.6)
kfi
The d i s t r i b u t i o n o f t h e d u r a t i o n o f t h e v i s i t i s n o t e x p o n e n t i a l , and t h e r e f o r e t h e p r o c e s s o f v i s i t s i s n o t a Markov p r o c e s s .
FORMAL SEMIOTICS
347
To p r o v e t h e theorem, o b s e r v e t h a t
t P(Ui>t)
=
exp
[--f
u(s)ds]
(3.7)
0
where
u ( s ) i s t h e jump i n t e n s i t y a t t i m e s . We have
here u(s) =
=
rz k#i C
%
fk(S)l/fi(S)
-c. s (f,
+ biks)l/fie
1
k#i Fi c . s
- -
Bi
e ' t - s e f:
cis
.
I n t e g r a t i n g t h e i n t e n s i t y we o b t a i n t h e a s s e r t i o n o f t h e theorem. Q u a l i t a t i v e l y s p e a k i n g , a v i s i t t o a domain i s l i k e l y t o t e r m i n a t e when t h e t o t a l p u l l t o o t h e r domains exc e e d s t h e p u l l t o t h e domain i n s p e c t e d . T h i s t o t a l
pull i n c r e a s e s l i n e a r l y , w h i l e t h e p u l l t o t h e domain i n s p e c t e d d e c r e a s e s e x p o n e n t i a l l y , s o t h a t t h e jump i n t e n s i t y i n c r e a s e s t o i n f i n i t y . Also, i f a t r a n s i t i o n occurs, t h e p r o b a b i l i t y o f t h e n e x t domain depends -a g a i n , i n a s t o c h a s t i c way -- on t h e r e l a t i v e s t r e n g t h s o f p u l l i n g f o r c e s , s o t h a t a domain w i t h s t r o n g e r pull i s more 1.iitely t o a t t r a c t t h e eye for n e x t i n s p e c t i o n . It s h o u l d be p o i n t e d o u t , however, t h a t -- a s opposed t o model o f Atkinson and B i r c h ( 1 9 7 2 ) -- i t i s n o t
g e n e r a l l y t r u e t h a t t h e t r a n s i t i o n w i l l always be t o t h e domain w i t h s t r o n g e s t p u l l .
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The main p o i n t o f t h e theorem i s t h a t t h e d u r a t i o n s o f v i i s i t s do n o t have e x p o n e n t i a l d i s t r i b u t i o n , hence t h e p r o c e s s of v i s i t s i s n o t Markovian. I n t u i t i v e l y , t h e r e a s o n f o r non-Markovian c h a r a c t e r o f t h i s p r o c e s s i s c l e a r : f i r s t l y , t h e l o n g e r a g i v e n domain was i n s p e c t e d , t h e more l i k e l y i t i s t h a t t h e v i s i t w i l l terminate, since i t s p u l l diminishes, while other p u l l s i n c r e a s e . T h i s i m p l i e s t h a t t h e d u r a t i o n s do n o t have e x p o n e n t i a l d i s t r i b u t i o n , s i n c e t h e jump i n t e n s i t y i s not constant.
S e c o n d l y , t h e l o n g e r a domain h a s n o t been v i s i t e d , t h e more l i k e l y i t i s t h a t i t s p u l l w i l l dominate o t h e r p u l l s -- s o t h a t a v i s i t t o t h i s domain becomes more and more l i k e l y . Again , t h i s shows a non-Markovian c h a r a c t e r o f t h e p r o c e s s of v i s i t s . Actually, t h e p u l l i n g force i s not only unobservable, b u t t h e n o t i o n as such i s f u z z y , p a r t i a l l y b e c a u s e o f f u z z i n e s s of t h e n o t i o n o f t h e domain D One can exi' p l i c a t e t h i s f u z z i n e s s as f o l l o w s . Around t h e "nominal" f o c a l p o i n t pi, one can l o c a t e a number o f o t h e r p o i n t s o f i n t e r e s t , say pil,
....
pi2, A s an example imagine a p a i n t i n g , w i t h one o f t h e domains b e i n g t h e f a c e o f t h e p e r s o n on t h e p a i n t i n g . A s t h e f o c a l p o i n t from h i s domain one may t a k e , s a y , t h e t i p o f t h e n o s e , but t h e r e a r e a l s o o t h e r p o i n t s , s u c h as e y e s , mouth, e t c . i n t h e same domain. L e t hij
be t h e d e g r e e t o which p o i n t p
t h e domain Di.
belongs t o One can t h e n assume t h a t w i t h e a c h o f ij
the points there i s associated a pulling force
and t h a t t h e t o t a l p u l l t o w a r d s t h e domain Di by
gij(t) i s given
FORMAL SEMIOlTCS
f . ( t ) = max 1 j
min Chij,
349
gij(t)l.
Here t h e s p e c i f i c a s s u m p t i o n s about changes of g ( t ) ij depend m o s t l y on t h e mutual l o c a t i o n s and d i s t a n c e s may o r may n o t between p o i n t s p i j : i n s p e c t i o n of p ij p r o v i d e i n f o r m a t i o n about pik (and hence d e c r e a s e t h e p u l l i n g f o r c e t o w a r d s t h i s p o i n t ) , depending on are at a d i s t a n c e allowing seeing w h e t h e r p i j and p ik both points o r not.
3.2. Purposeful inspection A s opposed t o t h e model of t h e p r e c e d i n g s e c t i o n , i n which t h e p r o c e s s of i n s p e c t i o n was assumed t o be p u r e l y s t o c h a s t i c , and governed by v a r i o u s p u l l i n g f o r c e s , i n t h i s s e c t i o n we s h a l l s t a r t from a n o t h e r " e x t r e m a l " p o i n t o f view: t h a t t h e p e r s o n i n s p e c t s t h e domains of f o c a l p o i n t s i n a p u r p o s e f u l way, s o as t o a t t a i n t h e g o a l of r e c o g n i t i o n of t h e meaning.
Thus, we c o n s i d e r h e r e a problem somewhat d i f f e r e n t t h a n i n t h e p r e c e d i n g s e c t i o n , where r e c o g n i t i o n may o r may n o t have been t h e g o a l o f i n s p e c t i o n . F o r t h e model, we assume t h a t w i t h e a c h domain ( o r a l t e r n a t i v e l y , w i t h each f o c a l p o i n t ) , t h e r e i s a s s o c i a t e d some s e t of f e a t u r e s . L e t Fi be t h e s e t o f f e a t u r e s a s s o c i a t e d w i t h domain D:; we assume t h a t I t h e s e s e t s a r e d i s j o i n t , so t h a t F i / ) F = @ for i # j , j and l e t
3 50
CHAPTER 4
be t h e s e t o f a l l f e a t u r e s of t h e f i g u r e t o be r e c o g n i -
zed ( i d e n t i f i e d ) . A s a n example t o g u i d e t h e i n t u i t i o n , c o n s i d e r t h e
problem o f i d e n t i f y i n g t h e make and model o f a c a r from i t s p h o t o g r a p h . Then one c a n d i s t i n g u i s h v a r i o u s 'ldomains'l, more o r l e s s f u z z y , such as w h e e l s , f e n d e r s , e t c . , and t h e i r f e a t u r e s ( e . g . s h a p e s , e t c . ) . The two b a s i c a s s u m p t i o n s , s t a t e d i n a q u a l i t a t i v e w a y , a r e h e r e as f o l l o w s :
-- t h e f i g u r e may be r e c o g n i z e d b e f o r e a l l f e a t u r e s a r e ob s e r v e d ;
-- a f e a t u r e o f a domain may
b e n o t i c e d o r missed upon
i n s p e c t i o n of t h i s domain. We s h a l l now p r e s e n t f o r m a l c o u n t e r p a r t s o f t h e s e a s s u m p t i o n s , f i r s t i n a non-fuzzy c a s e , and t h e n i n d i c a t e p o s s i b l e ways o f f u z z i f y i n g them. Thus, t h e f i r s t a s s u m p t i o n may b e e x p r e s s e d as a c l a s s of a l l s u b s e t s o f F which a r e s u f f i c i e n t f o r r e c o g n i t i o n ( t o be c a l l e d r e c o g n i t i o n s e t s ) .
Let
be s u c h a c l a s s .
*
One can t h e n d e f i n e t h e c l a s s
of minimal r e c o g n i t -
i o n s e t s as f o l l o w s : G E
y*
iff G C
18
and whenever H i s a p r o p e r s u b s e t
of G , t h e n H
L
y.
351
FORMAL SEMIOTICS
I n p r a c t i c a l s i t u a t i o n s , t h e n o t i o n o f r e c o g n i t i o n set i s f u z z y , m o s t l y b e c a u s e t h e same s e t o f f e a t u r e s may o r may n o t be s u f f i c i e n t f o r r e c o g n i t i o n , depending on t h e p e r s o n and o c c a s i o n . Thus, i s i n f a c t a fuzzy c l a s s of s u b s e t s of F , s o t h a t w i t h each (nonfuzzy) s u b s e t G C F one may a s s o c i a t e t h e v a l u e r ( G ) , which c h a r a c t e r i z e s t h e d e g r e e t o which G i s a r e c o g n i t i o n
$
set. N e x t , a s r e g a r d s t h e second a s s u m p t i o n , we s h a l l t r e a t t i m e as d i s c r e t e , w i t h p o s s i b l e changes o f t h e domains of inspection being t = 1 , 2 , . . . We assume t h a t i n e a c h u n i t o f t i m e a t most one f e a t u r e may be d e t e c t e d . Moreover, i f a t t i m e t t h e domain D i i s i n s p e c t e d , and t h e s e t o f f e a t u r e s of D i which a r e a l r e a d y d e t e c t t h e n t h e p r o b a b i l i t y o f d e t e c t i o n of ed i s F ' C F i ,
.
i
t h e f e a t u r e x k Fi
-
F! between t and t t l i s 1
We may have h e r e
xGFi-F: The i n e q u a l i t y a c c o u n t s f o r t h e f a c t
t h a t i t i s po-
s s i b l e t h a t no f e a t u r e w i l l be d e t e c t e d . F o r m a l l y t h e r e f o r e , f o r e v e r y F;
c
Fi ( i n c l u d i n g t h e
c a s e F ' = 0 ) , we have a p r o b a b i l i t y d i s t r i b u t i o n on i where N s t a n d s f o r "no d e t e c t i o n t h e s e t Fi - F;g{Nl, o f any f e a t u r e " . One may now f o r m u l a t e t h e problem o f o p t i m a l p a t h o f
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i n s p e c t i o n , t h e o p t i m a l i t y b e i n g c h a r a c t e r i z e d by t h e minimality of r e c o g n i t i o n t i m e . A t any g i v e n t i m e t , t h e s t a t e of i n s p e c t i o n i s d e s c -
r i b e d by t h e v e c t o r
...
F i , F$,
o f f e a t u r e s of domains D1,
D2,
. . . which
have a l r e a d y
been d e t e c t e d . L e t C(Fi,F;,...)
be t h e r e s i d u a l t i m e t i l l r e c o g n i t i o n
under o p t i m a l s t r a t e g y , s t a r t i n g from t h e s t a t e F i , F ; ,
...
o f a l r e a d y d e t e c t e d f e a t u r e s . We have t h e n t h e
Bellman e q u a t i o n
.
(3.8)
C(Fi,F$,. .)
This condition expresses the f a c t t h a t i n t h e state d e s c r i b e d by F i , F;,
... we
must make t h e d e c i s i o n
a b o u t t h e domain t o be i n s p e c t e d as n e x t . I f w e d e c i d e t o i n s p e c t t h e domain Diy
then t h e expected time till
recognition equals 1 plus t h e average of recognition
times under c o n d i t i o n o f d e t e c t i n g v a r i o u s p o s s i b l e f e a t u r e s , o r not d e t e c t i n g any. The boundary c o n d i t i o n s h e r e a r e d e s c r i b e d by t h e i n c l u s i o n o f t h e s t a t e and one o f t h e r e c o g n i t i o n s e t s :
353
FORMAL SEMIOTICS
C ( F i , F:,
...) = 0 i f
(3.9) The i n i t i a l c o n d i t i o n i s F ’ = @ f o r a l l i . i The o p t i m a l c h o i c e o f t h e domain t o i n s p e c t as n e x t
i s o b t a i n e d by s o l v i n g t h e e q u a t i o n ( 3 . 8 ) , and t a k i n g a s t h e domain t o be i n s p e c t d n e x t ( i n a g i v e n s t a t e ) t h e domain w i t h i n d e x i , which g i v e s t h e minimum on t h e r i g h t hand s i d e i n ( 3 . 8 ) . The n o t i o n o f r e c o g n i t i o n s e t s a l l o w s u s t o i n t r o d u c e t h e n o t i o n of r e c o g n i t i o n weights. I n t u i t i v e l y , t h e s e a r e numbers r e p r e s e n t i n g t h e i m p o r t a n c e o f v a r i o u s features f o r the overall recognition. To d e f i n e t h e w e i g h t s f o r m a l l y , assume t h a t t h e s e t F o f a l l f e a t u r e s i s f i n i t e , and c o n s i s t s of e l e m e n t s
F = I f l yf 2 ,
...,
fN3
‘
L e t u s now c o n s i d e r a p e r m u t a t i o n
o f e l e m e n t s o f F.
fil,
fi2,.
..,f
iN
We may now a s s o c i a t e w i t h e a c h
permutation a p i v o t , according t o t h e following definition. DEFINITION. ‘il,
f
The e l e m e n t f .
. .. , f i ,. .. , f
i2’
n longs t o t h e c l a s s
f
1..
il
,..., f . 3 ’k
,..
3
, i f t h e s e t (fi .,f. beiN 1 In o f minimal r e c o n g i t i o n s e t s ,
while f o r k = l Y 2 , . . . , n - l form {f
i s a pivot i n permutation
ri
none o f t h e s e t s o f t h e
has t h i s property.
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Thus, a p i v o t i s t h e l a s t e l e m e n t i n t h e g i v e n permuta t i o n with t h e property t h a t , together with a l l elements preceding i t , t h e s e t of f e a t u r e s thus obtained i s sufficient f o r recognition. DEFINITION. The r e c o g n i t i o n w e i g h t o f t h e f r a g m e n t f i n F i s defi.ned as
# o f p e r m u t a t i o n s i n which f i s t h e p i v o t
w(f)
=
N! T h i s d e f i n i t i o n o f r e c o g n i t i o n w e i g h t s h a s been i n t r o duced by Nowakowska ( 1 9 6 7 ) ; i t i s b a s e d on t h e n o t i o n
o f v o t i n g power and minimal w i n n i n g c o a l i t i o n s o f S h a p l e y and S h u b i k ( 1 9 5 4 ) . A s an example, i m a g i n e t h a t t h e o b j e c t t o be r e c o g n i z e d
i s a word, which i s w r i t t e n i n a random o r d e r ( i n t h e s e n s e o f a p p e a r e n c e o f l e t t e r s i n a random w a y , e a c h o c c u p y i n g i t s p l a c e i n t h e w o r d ) . Assume t h a t t h e word
i s WATER, and t h a t t h e l e t t e r s a r e a p p e a r i n g i n permuta t i o n A , R, W,
T , E.
(so that the features are f l = W,
f 2 = A , and s o o n , and for t h e p e r m u t a t i o n i n q u e s t i o n , i
1
= 2, etc.)
The c o n s e c u t i v e s t a g e s a r e h e r e as f o l l o w s - A - - -
- A - - R
W A - -
R
W A T - R
C l e a r l y , t h e f i r s t t h r e e s t a g e s a r e n o t s u f f i c i e n t for r e c o g n i z i n g t h e words as WATER, w h i l e i n t h e f o u r t h s t a g e , d e s p i t e one l e t t e r s t i l l m i s s i n g , t h e word
FORMAL SEMIOTICS
355
w i l l b e r e c o g n i z e d . T h i s means t h a t i n t h e c o n s i d e r e d permutation, the pivot i s T. By a n a l y s i n g a l l p e r m u t a t i o n s , and i d e n t i f y i n g t h e p i v o t i n e a c h c a s e , one can a s s i g n w e i g h t s t o a l l f i v e l e t t e r s , t h u s e x p r e s s i n g t h e importance o f t h e s e l e t t e r s i n r e c o g n i z i n g t h e p a r t i c u l a r word. Human e x p e r i e n c e a l l o w s a l s o t o e v a l u a t e s u c h w e i g h t s s u b j e c t i v e l y , f o r v a r i o u s o c c a s i o n s and g o a l s , and determine t h e invariance of weights f o r c l a s s e s of o b j e c t s and g o a l s , and f o r c l a s s e s o f t r a n s f o r m a t i o n s . I n t r o d u c t i o n of t h e n o t i o n o f w e i g h t s a l l o w s a l s o f o r s t u d y i n g t h e s t r a t e g i e s of t h e p a t h of t h e eye o v e r t h e o b j e c t , which a r e g e n e r a t e d by s t r u c t u r a l p r o p e r t i e s o f t h i s o b j e c t and p a r t i a l l y c o n t r o l l e d b y t h e s u b j e c t . V a r i o u s models o f d e c i s i o n s about "walking o v e r t h e tree of the object'' ( p i c t u r e ) , r e l a t i o n s with Gestalt t h e o r y , as w e l l as a t t e m p t s o f q u a n t i f i c a t i o n o f some n o t i o n s o f t h i s t h e o r y ( e . g . t h a t o f good f i g u r e ) may be found i n Nowakowska (1967). T h i s p a p e r a l s o shows a p p l i c a t i o n of c o n d i t i o n a l weights t o s e q u e n t i a l a n a l y s i s of recognition. The u s e o f dynamic programming t o t h e s t u d y of p u r p o s e -
f u l i n s p e c t i o n seems t o be o f i m p o r t a n t m e t h o d o l o g i c a l v a l u e , a l s o i n e n g i n e e r i n g and a r t i f i c i a l i n t e l l i g e n c e , for modeling p a t t e r n r e c o g n i t i o n . I n n a t u r a l s i t u a t i o n s , w e d e a l o f c o u r s e w i t h a m i x t u r e o f two proc e s s e s , o f f r e e and p u r p o s e f u l i n s p e c t i o n , and v a r i o u s c o n t r i b u t i o n s o f t h e s e p r o c e s s e s are p r o b a b l y induced by b o t h t h e g o a l o f p e r c e p t i o n and t h e s t r u c t u r e o f t h e p e r c e i v e d o b j e c t . I n o t h e r words, t h e a c t u a l g l a n c e
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p a t h may b e r e g a r d e d as a m i x t u r e o f o p t i m a l g l a n c e p a t h a n d p u r e l y random g l a n c e p r o c e s s , g o v e r n e d by pulling forces. The g l a n c e p r o c e s s may b e assumed t o i n d u c e t h e p r o c e s s o f t r a n s p o r t a t i o n and c o m p a r i s o n s . Such
comparisons,
i n t u r n , may b e r e g a r d e d a s p r e - e v e n t s , t h a t i s , t h e c a n d i d a t e s f o r e v e n t s t o b e remembered. It c a n b e p o s tulated that recognition, identification, e t c . , requir e c o l l e c t i o n o f "raw d a t a " a b o u t t h e r e l a t i o n a l s t r u c t u r e o f t h e p e r c e i v e d o b j e c t . Such raw d a t a must i n v o l v e , among o t h e r s , t h e r e s u l t s o f c o m p a r i s o n s o f values of a t t r i b u t e s of fragments, t o d e t e c t d i f f e r e n ces , symmetries , e t c. The c o m p a r i s o n o f two f r a g m e n t s
--
i f t h e y are located
s u f f i c i e n t l y f a r a p a r t -- r e q u i r e t h a t a g l a n c e a t one p o i n t o f t h e o b j e c t i n v o l v e s a l s o a " t r a n s p o r t a t i o n " o f t h e f r a g m e n t t o b e compared t o t h e l o c a t i o n o f t h e o t h e r f r a g m e n t , w i t h which t h e comparison i s t o b e made. N a t u r a l l y , s u c h a t r a n s p o r t a t i o n l e a d s t o a f u z z i f i c a t i o n , a n d a l s o t o a loss o f p r e c i s i o n i n comparison. Under a p p r o p r i a t e a s s u m p t i o n s , o n e may d e r i v e f o r m u l a s f o r the p r o b a b i l i t y of e r r o r s , t h u s o b t a i n i n g a c l u e t o e s t i m a t i o n o f p a r a m e t e r s o f t h e m o d e l . Also -- w h i c h i s more i m p o r t a n t -- t h e s e f o r m u l a s p r o v i d e means o f t e s t i n g the hypothesis of p a r a l l e l v e r s u s serial proc e s s i n g , i n case of comparisons i n v o l v i n g s e v e r a l a t t ributes. The mechanisms o f t r a n s p o r t a t i o n w i l l b e modeled i n S e c t i o n 3 . 4 , and t h o s e of changing p r e - e v e h ts
into
FORMAL SEMIOTICS
357
e v e n t s , t o b e s t o r e d i n memory -- i n S e c t i o n 3 . 5 . We s h a l l s t a r t , however, w i t h t h e problems o f g e n e r a t i o n o f meaning o f a n o b j e c t ; t h e model s u g g e s t e d w i l l be aimed a t e x p l a i n i n g t h e f a c t t h a t t h e meaning a s s i g n ed may sometimes depend on t h e o r d e r o f i n s p e c t i o n .
3 . 3 . G e n e r a t i o n o f meaning under i n s p e c t i o n The a n a l y s i s of t h e p r e c e d i n g s e c t i o n was c a r r i e d o u t under t h e t a c i t a s s u m p t i o n t h a t t h e o b j e c t i n s p e c t e d has o n l y one meaning. I n many c a s e s i t may happen t h a t t h e o b j e c t i n s p e c t e d has s e v e r a l a l t e r n a t i v e meanings, and t h e problem i s t o a s s i g n one o f t h e m t o t h e o b j e c t . I n such c a s e s , t h e f i n a l d e c i s i o n may depend on t h e o r d e r o f i n s p e c t i o n ( r e g a r d l e s s whether t h e i n s p e c t i o n i s random or p u r p o s e f u l ) . I n t h i s s e c t i o n , we s h a l l p r e s e n t a simple model which a c c o u n t s f o r t h i s dependence.
From t h e knowledge o f p s y c h o p h y s i c a l p r o c e s s e s which accompany t h e i n s p e c t i o n o f f r a g m e n t s (domains) o f a f i g u r e , one may e x p e c t t h a t t h e o b s e r v e r behaves a s i f t e s t i n g t h e h y p o t h e s i s t h a t t h e meaning i s z : s u c c e s s i v e i n s p e c t e d domains p r o v i d e p r e m i s e s f o r o r a g a i n s t t h i s h y p o t h e s i s . He f i n a l l y r e a c h e s e i t h e r t h e d e c i s i o n " f o r ' ' meaning z , i n which c a s e t h e p r o c e s s of d e c i s i o n (though not n e c e s s a r i l y of i n s p e c t i o n ) t e r m i n a t e s , o r t h e d e c i s i o n " a g a i n s t " meaning z , i n which c a s e he c o n t i n u e s t e s t i n g f o r some o t h e r meaning. A t e a c h s t a g e of i n s p e c t i o n , t h e f r a g e m e n t s i n s p e c t e d
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t h u s f a r g i v e a c e r t a i n p r e m i s e f o r t h e meaning z ; t h e j o i n t e f f e c t h e r e may be r e p r e s e n t e d as a d e g r e e o f c e r t a i n t y t h a t t h e meaning i s z . T h i s d e g r e e may i n c r e a s e o r d e c r e a s e , depending on t h e fragment i n s p e c t ed as l a s t . L e t u s d e n o t e t h i s l e v e l of c e r t a i n t y a f t e r i n s p e c t i n g t h e f i r s t k f r a g m e n t s by xk ( x k w i l l , i n g e n e r a l , depend on z , b u t f o r s i m p l i c i t y , w e c o n s i d e r here t e s t i n g f o r a f i x e d meaning z ) . The problem l i e s i n def i n i n g x i n s u c h a way t h a t i t i s n o t i n v a r i a n t w i t h k respect t o the order of inspection. One o f p o s s i b l e a s s u m p t i o n s i s t h a t i f x i s s u f f i c i e n t k ly h i g h , t h e n n e x t p r e m i s e s " f o r " do n o t a f f e c t xk. S i m i l a r l y , when xk i s s u f f i c i e n t l y low, t h e p r e m i s e s " a g a i n s t " do n o t a f f e c t x k .
, fi2 ,... d e n o t e t h e 1 s u c c e s s i v e f r a g m e n t s i n s p e c t e d , and l e t q k ( z ) d e n o t e t h e s t r e n g t h of premise f o r o r a g a i n s t z , r e s u l t i n g from i n s p e c t i o n o f fragment f
For formal d e f i n i t i o n , l e t f i
ik
.
Put xo = 0 and d e f i n e xk a s
where A
B a r e some t h r e s h o l d s .
To s e e t h e consequences o f t h i s f o r m u l a , l e t u s u s e t h e f i g u r e 3 . 1 . below. T h i s f i g u r e i l l u s t r a t e s why t h e f i n a l r e s u l t may depend on t h e o r d e r o f i n s p e c t i o n .
FORMAL SEMIOTICS
X
359
k
Fig. 3.1
When xk e q u a l s t h e upper boundary B , t h e n t h e s u c c e s i v e p r e m i s e s " f o r " do n o t c a u s e any i n c r e a s e o f x k ' It i s s e e n t h a t t h e f i n a l v a l u e o f xk would be h i g h e r i f t h e p o s i t i v e v a l u e s of q k ( z ) o c c u r r e d a t some o t h e r p l a c e s , e . g . t o w a r d s t h e end o f i n s p e c t i o n . T h i s model i s , n a t u r a l l y , o n l y one o f t h e p o s s i b l e models o f i n f l u e n c e of o r d e r on t h e d e c i s i o n .
It would be o f c o n s i d e r a b l e i n t e r e s t t o d e t e r m i n e t h e p r o b a b i l i t y d i s t r i b u t i o n o f xk under random o r d e r o f i n s p e c t i o n . However, f o r s u c h a n answer t o be t h e o r e t i c a l l y s i g n i f i c a n t , one s h o u l d impose some c o n s t r a i n t s on t h e o r d e r o f i n s p e c t i o n , s i n c e n o t a l l o r d e r s a r e e q u a l l y l i k e l y , and c a l c u l a t i o n s under t h e a s s u m p t i o n o f e q u i p r o b a b l e o r d e r s a r e u n r e a l i s t i c . The problem o f v a r i o u s p r o b a b i l i t i e s of o r d e r s i s connected c l o s e l y w i t h weights, discussed i n t h e preceding section.
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It i s w o r t h t o mention h e r e a c e r t a i n c l a s s o f o b j e c t s
f o r which t h e f i n a l d e c i s i o n does n o t depend on t h e o r d e r o f i n s p e c t i o n : t h e s e a r e o b j e c t s such t h a t every fragment " s u p p o r t s " t h e meaning, s o t h a t q k ( z ) i s pos i t i v e . Then xk does n o t depend on t h e o r d e r o f i n s pection. F o r f i g u r e s w i t h some q k ( z ) b e i n g p o s i t i v e and some n e g a t i v e , t h e o r d e r o f i n s p e c t i o n p l a y s an i m p o r t a n t role.
3 . 4 . Mechanism o f t r a n s p o r t a t i o n i n comparisons G e n e r a l l y , t h i s p r o c e s s o c c u r s i n c a s e o f comparison o f v a r i o u s p a r t s of t h e p e r c e i v e d o b j e c t s , which are l o c a t e d a t d i f f e r e n t p l a c e s . One can a r g u e t h a t s u c h comparisons are i n r e a l i t y p r e - e v e n t s , t o be d i s c u s s e d i n t h e n e x t s e c t i o n . More p r e c i s e l y , some o f t h e r e s u l t s o f s u c h comparisons e n t e r t h e memory and form t h e "raw data" which l a t e r s e r v e for t h e p u r p o s e o f identification, recognition, etc. One may namely assume t h a t when a new f e a t u r e i s p e r c e i v e d i n some domain, i t i s compared w i t h some a n a l o gous f e a t u r e s i n o t h e r domains, t o form e v e n t u a l l y , t h e knowledge about t h e o b j e c t , a s a s e r i e s o f a s s e r t ions s u c h as "x i n domain D i s b r i g h t e r t h a n y i n i domain D ' I , o r " X i n domain D i s l o n g e r t h a n y i n j i domain D ", e t c . Such knowledge i s e s s e n t i a l b o t h f o r j r e c o g n i t i o n and a l s o f o r remembering t h e o b j e c t some t i m e afterwards. To show how t h e model works, and f o r e s t i m a t i o n o f
FORMAL SEMIOTICS
36 1
t h e p a r a m e t e r s o f t h e model, w e use a s i m p l e example
o f p e r c e p t i o n o f e l e m e n t a r y forms s u c h as l i n e s , which a r e compared-with r e s p e c t t o t h e i r l e n g t h i n s p e c i a l conditions. G e n e r a l l y , when one i s t o compare t h e f e a t u r e s o f two fragments t h a t a r e s u f f i c i e n t l y f a r a p a r t , so t h a t it i s i m p o s s i b l e t o i n s p e c t them b o t h s i m u l t a n e o u s l y , t h e a c t u a l comparison c o n c e r n s t h e memory t r a c e s of t h e f r a g m e n t s . Thus, t h e model below w i l l be based on some a s s u m p t i o n s about t h e e v o l u t i o n of memory t r a c e s a f t e r t h e stimulus ceases. I n o u r example, imagine t h a t t h e s u b j e c t i n s p e c t s some fragment ( s t i m u l u s S1), f o r m i n g t h e t r a c e o f t h i s s t i m u l u s i n memory. Then h i s e y e i n s p e c t s f o r some t i m e o t h e r f r a g m e n t s ( a n d t h e t r a c e o f S1 becomes g r a d u a l l y more f u z z y ) , f i n a l l y a r r i v i n g a t a p a r t S2, which i s t o b e compared i n l e n g t h w i t h S1. S t i m u l u s S2 i s i n s p e c t e d f o r a c e r t a i n p e r i o d of t i m e , and i t s t r a c e i s formed i n memory. The t r a c e s o f S and S2 1 a r e t h e n compared and d e c i s i o n i s r e a c h e d whether S1 i s o f t h e same l e n g t h as S2 o r n o t . I n e x p e r i m e n t a l s i t u a t i o n s , t h e l e n g t h s of s t i m u l i , d u r a t i o n o f t h e i r e x p o s i t i o n and o f t h e p a u s e between t h e e x p o s i t i o n s , a s w e l l as t h e l o c a t i o n s o f s t i m u l i , would be under t h e c o n t r o l o f t h e e x p e r i m e n t e r , t h u s making i t p o s s i b l e t o employ t h e t e c h n i q u e s o f e s t i mation of parameters. I n p r a c t i c a l s i t u a t i o n s , t h e t i m e s of e x p o s i t i o n s , p a u s e , and l o c a t i o n s o f s t i m u l i depend on t h e o r d e r and dynamics o f i n s p e c t i o n . To f i x t h e i d e a s , t h e model w i l l be p r e s e n t e d on an example o f a h y p o t h e t i c a l e x p e r i m e n t , i n which t h e
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s t i m u l i S1 and S2 a r e d i s p l a y e d on a s c r e e n , and t h e s u b j e c t i s t o d e c i d e whether t h e y a r e o f t h e same l e n g t h , and i f n o t , which o f them is l o n g e r . Assume now t h a t t h e s u b j e c t i s p r e - t r a i n e d ,
and he knows
t h e f u t u r e l o c a t i o n o f t h e s t i m u l u s on t h e s c r e e n , as w e l l as what i s e x p e c t e d o f him. There a r e f o u r s t a g e s , e a c h i n v o l v i n g a d i f f e r e n t proc e s s . These s t a g e s a r e p r e s e n t e d on t h e f i g u r e below.
Stage 1
Stage 2
Stage 3
Stage 4
Fig. 3.2.
During s t a g e 1, s t i m u l u s S1 i s d i s p l a y e d , and i t s t r a c e i s formed i n memory.
During s t a g e 2 , t h e ' s c r e e n i s b l a n k , a n d t h e t r a c e o f
S1 i s "moved" t o t h e f u t u r e l o c a t i o n o f s t i m u l u s S2. During t h i s p r o c e s s , t h e t r a c e undergoes some d i s tortions. During s t a g e 3 , t h e s t i m u l u s S2 i s d i s p l a y e d , and i t s t r a c e i s formed i n memory. A t t h e same t i m e , t h e t r a c e o f S1 undergoes s t i l l more d i s t o r t i o n s . F i n a l l y , i n s t a g e 4 , t h e s c r e e n i s b l a n k a g a i n . Both
363
FORMAL SEMIO'IICS
t r a c e s undergo d i s t o r t i o n ( f u z z i f i c a t i o n ) d u r i n g t h a t stage. I n t h e s e q u e l , we s h a l l d e n o t e by x l ( t ) and x 2 ( t ) t h e s i z e s o f t h e t r a c e s of s t i m u l i S1 and S2 i n memory a t t i m e t , and make t h e f o l l o w i n g a s s u m p t i o n s :
ASSUMPTION 1. The d e c i s i o n t h a t S1 = S 2 i s r e a c h e d , i f a t t i m e t o f comparison, w e have (3.10) where q i s some p o s i t i v e t h r e s h o l d .
ASSUMPTION 2 . Each t r a c e x i ( t ) , i = 1 , 2 i s a Wiener process with
(3.11) where mi
i s t h e t r u e s i z e o f s t i m u l u s Si.
The s u b s e q u e n t a s s u m p t i o n s c o n c e r n t h e v a r i a n c e s o f t h e two Wiener p r o c e s s e s . One c o u l d e x p e c t t h a t t h e f u z z i f i c a t i o n o f t h e t r a c e x l ( t ) i s f a s t e r when t h e a t t e n t i o n i s c o n c e n t r a t e d a t s t i m u l u s S2, i . e . d u r i n g s t a g e 3. Moreover, i t i s n e c e s s a r y t o t a k e i n t o a c c o u n t t h e f a c t t h a t t h e first process, corresponding t o s t i m u l u s S1, l a s t s l o n g e r t h a n t h e p r o c e s s x 2 ( t ) ( i n d e e d , x 1 ( t ) b e g i n s a t t h e end o f s t a g e 1, and l a s t s u n t i l t h e end o f s t a g e 4 , w h i l e t h e ' s e c o n d p r o c e s s b e g i n s only a t t h e end o f s t a g e 3 ) . We now make t h e f o l l o w i n g a s s u m p t i o n s , which d e s c r i b e
t h e v a r i a n c e s o f t h e two p r o c e s s e s .
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3 64
ASSUMPTION 3. D u r i n g s t a g e 2, i . e . p a u s e between t h e e x p o s i t i o n of t h e s t i m u l i , t h e v a r i a n c e of x ( t ) in1 creases proportionally t o a2: Var x , ( t )
= ECxl(t)
- m112
= a 2t .
ASSUMPTION 4 . During t h e e x p o s u r e o f t h e second s t i m u l u s , the variance of x ( t ) increases proportioanally 1 t o b2:
ASSUMPTION 5 . During t h e p r o c e s s o f comparison, t h e v a r i a n c e s of x ( t ) and x 2 ( t ) i n c r e a s e p r o p o r t i o n a l l y 1 t o c2:
ASSUMPTION 6 . The p r o c e s s e s x , ( t ) pendent.
and x 2 ( t ) a r e i n d e -
2
Here a 2 , b and c 2 a r e some c o n s t a n t s . I n p a r t i c u l a r , t h e c o n s t a n t c 2 may depend on t h e mutual d i s t a n c e and l o c a t i o n o f t h e s t i m u l i . T h i s dependence may be assumed t o b e t h e f o l l o w i n g ( s e e F i g . 3 . 3 ) . It may namely be c o n j e c t u r e d , t h a t t h e l e v e l o f d i f f i -
c u l t y o f comparison, as measured by t h e v a r i a n c e , i s l a r g e r when t h e a n g l e 'Q i s l a r g e r ( i . e . i t i s e a s i e s t t o compare t h e s t i m u l i i f t h e y are " l i n e d u p " ) . T h i s c o n j e c t u r e may b e e x p r e s s e d as 2
c 2 = k2R(1 t & sinf'), R2 = u2 t v where k and E a r e some c o n s t a n t s .
2
.
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F i g . 3.3
Thus, v a r i a n c e depends on R , i . e . on t h e d i s t a n c e by which one s t i m u l u s h a s t o be " t r a n s p o r t e d " , and on t h e angle of i n c l i n a t i o n of t h i s t r a n s p o r t .
We have t h e n t h e f o l l o w i n g theorem. THEOREM. The p r o b a b i l i t y t h a t t h e s t i m u l i S1
and
w i l l be p e r c e i v e d as i d e n t i c a l e q u a l s
where
A = m
1
-
m2 = t r u e d i f f e r e n c e between s t i m u l i
S2
3 66
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where * t
-
d u r a t i o n o f p a u s e between s t i m u l i , T I - t i m e o f exposure o f s t i m u l u s S 2 , T c - t i m e between t h e end o f e x p o s u r e o f S and 2t h e end o f s t a g e o f comparison, R - d i s t a n c e between S, and S2,
\4 -
I
a n g l e o f i n c l i n a t i o n between p o s i t i o n s o f s t i m u l i S, and S 2 . I
I n t h i s theorem ion function.
6 stands
f o r s t a n d a r d normal d i s t r i b u t -
From t h e e x p e r i m e n t a l v i e w p o i n t , t h e above theorem p r o v i d e s u s w i t h t h e p o s s i b i l i t y o f t e s t i n g t h e model o f t r a n s p o r t a t i o n . I n d e e d , t h e unknown p a r a m e t e r s a r e h e r e a , b , c and q (or i n s t e a d o f c , w e may t a k e k and t). The c o n t r o l l e d v a r i a b l e s a r e t h e e x p o s i t i o n t i m e s and l e n g t h o f p a u s e , as w e l l as t h e d i f f e r e n c e between * The t h e t r u e l e n g t h s o f s t i m u l i , t h a t i s , t , TI and A v a r i a b l e s o b s e r v e d a r e t h e d e c i s i o n ( " e q u a l " v s . "une q u a l " ) , and t h e d e c i s i o n t i m e T
.
C
.
By s e t t i n g v a r i o u s v a l u e s o f t h e c o n t r o l v a r i a b l e s , i t
i s possible t o observe t h e frequencies of various types o f d e c i s i o n s , and t h e d e c i s i o n t i m e s , and i n t h i s w a y , o b t a i n a n a c c e s s t o e s t i m a t i o n o f t h e unknown parameters. F o r t h e p r o o f o f t h e t h e o r e m , l e t X and Y d e n o t e t h e s i z e s of t r a c e s o f S,I and S, a t t h e t i m e o f comparison. We o b t a i n t h e r e f o r e f o r t h e d i s t r i b u t i o n s o f X and Y : L
Y-
N(m2,
2
c Tc) = N(m2,
B2 ) .
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FORMAL SEMIOTICS
We may t h e r e f o r e w r i t e
P(S1 j u d g e d i d e n t i c a l t o S 2 ) =
P ( \ X - Y \ < q ) = P(Y
-
q
c: x c
Y
+
q)
-(u-m2) 2 / 2 B 2 du.
Substituting u = m 2 t h e theorem.
+
Bz
we o b t a i n t h e a s s e r t i o n o f
L e t u s o b s e r v e t h a t t h i s theorem might p r o v i d e some
means o f t e s t i n g t h e h y p o t h e s i s o f p a r a l l e l v e r s u s s e r i a l p r o c e s s i n g . To i l l u s t r a t e t h i s p o s s i b i l i t y , imagine now t h a t s t i m u l i S1 and S2 a r e c h a r a c t e r i z e d by s e v e r a l a t t r i b u t e s , and t h a t t h e d e c i s i o n t h a t t h e y a r e i d e n t i c a l occurs i f t h e perceived values of t h e c o r r e s p o n d i n g a t t r i b u t e s a r e w i t h i n some t h r e s h o l d d i s t a n c e ( i n t h e s i m p l e s t c a s e , each o f t h e s t i m u l i may b e o f t h e shape o f l e t t e r L , s o t h a t t h e two a t t r i b u t e s t o b e compared a r e t h e l e n g t h s o f v e r t i c a l and horizontal p a r t s ) . I f t h e s e comparisons o c c u r s i m u l t a n e o u s l y , t h e probab i l i t y o f t h e d e c i s i o n o f i d e n t i t y may be o b t a i n e d by
t h e same method a s f o r t h e one a t t r i b u t e c a s e .
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I n c a s e o f s e r i a l c o m p a r i s o n s , t h e t r a c e o f one a t t r i b u t e undergoes f u r t h e r changes d u r i n g t h e t i m e when t h e o t h e r a t t r i b u t e i s p r o c e s s e d . Again, u s i n g t h e same t e c h n i q u e s , one c o u l d d e r i v e t h e f o r m u l a f o r t h e p r o b a b i l i t y o f a g i v e n d e c i s i o n . Given t h e e s t i m a t e s of t h e r e l e v a n t p a r a m e t e r s o b t a i n e d from o n e - a t t r i b u t e c a s e , one c o u l d t e s t t h e h y p o t h e s i s o f p a r a l l e l v e r s u s s e r i a l processing
.
It i s w o r t h w h i l e t o p o i n t o u t h e r e t h a t i n t h e system
o f models o f p e r c e p t i o n shown i n t h i s s e c t i o n , f o r t h e f i r s t t i m e i n t h e l i t e r a t u r e t h e r e were i n t r o d u c e d t o t h e model ( v i a mechanisms o f t r a n s p o r t a t i o n ) t h e memor y p a r a m e t e r s . Also, a s i g n i f i c a n t n o v e l t y i s t h e t h e o r e t i c a l a n a l y s i s of p e r c e p t i o n , open t o e x p e r i m e n t a t i o n v i a i n t e r a c t i o n a l s y s t e m o f models. The new t h e o r y a l s o p a s s e s t h e t e s t , c r u c i a l f o r a l l p e r c e p t i o n mod e l s , namely e x p l a n a t i o n o f v i s u a l i l l u s i o n s ( s e e Nowakowska 1 9 8 3 a , b ) . I n a d d i t i o n t o g l a n c e s , one u s e s t h e phenomenon o f f i n e eye movements, which p l a y e s s e n t i a l r o l e i n s h a r p e n i n g and memorizing t h e p e r c e i v e d image. Now, d u r i n g t h e f i n e eye movements, c e n t e r e d a t some p o i n t , t h e image of t h e f i g u r e on t h e r e t i n a p e r f o r m s a movement, c o n s e c u t i v e l y c o v e r i n g and u n c o v e r i n g any p o i n t c l o s e t o t h e boundary o f t h e f i g u r e . I n o t h e r words, t h e f i g u r e " r e t u r n s " t o t h i s p o i n t from t i m e t o t i m e . The f r e q u e n c y of t h i s r e t u r n p r o c e s s depends on t h e l o c a t i o n o f o f t h e p o i n t r e l a t i v e t o t h e f i g u r e . I n f a c t , we have h e r e many i n t e r r e l a t e d p r o c e s s e s , one f o r e v e r y p o i n t c l o s e t o t h e boundary o f t h e f i g u r e . It t u r n s o u t t h a t t h e p r o p e r t i e s o f t h e s e p r o c e s s e s
FORMAL SEMIOTICS
3 69
o f f e r a p o s s i b l e e x p l a n a t i o n o f t h e MUller-Lyer i l l u s i o n ( f o r a r e v i e w o f d i f f e r e n t a p p r o a c h e s and e x p l a n a t i o n s o f v i s u a l i l l u s i o n s , s e e e . g , Eijkman e t a l . (1981), Lee (19811, C a e l l i ( 1 9 8 1 ) o r Ullmann ( 1 9 7 9 ) ) . An a l t e r n a t i v e e x p l a n a t i o n o f v i s u a l i l l u s i o n s based on f i n e eye movements i s g i v e n i n Nowakowska (198313). The i l l u s i o n p r o c e s s e s are proposed t h e r e as s p e c i f i c modeling o c c u r r i n g a t a r e l a t i v e l y low l e v e l o f cognit i o n . The i l l u s i o n i t s e l f i s e x p l a i n e d t h r o u g h t h e n o t i o n o f competing p e r c e p t u a l s y s t e m s -- b a s i c and c o n t e x t u a l -- which y i e l d d i f f e r e n t v a l u e s o f t h e observed a t t r i b u t e s . The MUller-Lyer i l l u s i o n c o n s i s t s o f m i s p e r c e p t i o n of
l e n g t h s o f arrows on F i g . 3 . 4 .
>
/
F i g . 3.4 C o n s i d e r a n enlargement of t h e e n d p o i n t s o f t h e a r r o w s ( s e e F i g . 3 . 5 ) . The movements of t h e eye a r e e q u i v a l e n t
F i g . 3.5
t o movements o f t h e images of t h e s e a r r o w s . Consequentl y , any f i x e d p o i n t on t h e r e t i n a w i l l be c o n s e c u t i v e l y c o v e r e d and n o t c o v e r e d b y t h e image.
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N a t u r a l l y , s u c h changes w i l l be more f r e q u e n t f o r p o i n t s which a r e c l o s e t o t h e edge o f t h e a r r o w , from o u t s i d e or from i n s i d e . On t h e o t h e r hand, f o r p o i n t s f a r away from t h e e d g e , l y i n g o u t s i d e , t h e p e r i o d s o f b e i n g uncovered w i l l t e n d t o be l o n g e r , and conv e r s e l y f o r p o i n t s l y i n g i n s i d e t h e arrow. Each p e r i o d when a p o i n t i s n o t c o v e r e d c o n s i s t s o f two p a r t s : d u r i n g t h e f i r s t p a r t , t h e t r a c e o f t h e stimul u s s t i l l p e r s i s t s i n memory, w h i l e i n t h e second p a r t (which may be z e r o ) , t h e s t i m u l u s has a l r e a d y faded
.
Q u a l i t a t i v e l y s p e a k i n g , t h i s meachanism e x p l a i n s t h e i l l u s i o n . To s e e i t , c o n s i d e r two p o i n t s , a t t h e same d i s t a n c e from t h e end o f t h e a r r o w , l i k e p and q on Fig. 3.5. I n c a s e of outward a r r o w , t h e p o i n t g w i l l be c o v e r e d r a t h e r o f t e n , and f o r l o n g e r p e r i o d s o f t i m e , simply be v e r t i c a l movements o f t h e arrow ( i n a d d i t i o n t o h o r i z o n t a l movements). On t h e o t h e r hand, t h e p o i n t p on t h e inward arrow w i l l b e covered r a t h e r seldom, and f o r s h o r t e r p e r i o d s o f time. C o n s e q u e n t l y , f o r t h e p o i n t q t h e f r a c t i o n o f times when i t i s c o v e r e d , p l u s t h e times when i t i s n o t covered, but t h e t r a c e of t h e s t i m u l u s s t i l l p e r s i s t s , i s q u i t e h i g h . A s opposed t o t h a t , f o r p o i n t p t h e p e r i o d s when t h e r e i s a t r a c e o f t h e s t i m u l u s i n memor y w i l l t e n d t o be r a r e and s h o r t . I n e f f e c t , two p o i n t s l o c a t e d a t t h e same d i s t a n c e from t h e end o f t h e a r r o w w i l l g i v e d i f f e r e n t f r a c t i o n s o f t i m e when t h e s t i m u l u s p e r s i s t s .
FORMAL SEMIOTICS
371
The e x a c t e x p r e s s i o n for t h e f r a c t i o n o f t h e t i m e when t h e image p e r s i s t s depends on t h e a s s u m p t i o n s about t h e f a d i n g p r o c e s s . I n p a r t i c u l a r , if one assumes t h a t t h e f a d i n g p r o c e s s i s s u c h t h a t t h e t r a c e of t h e s t i mulus d i s a p p e a r s a f t e r e x p o n e n t i a l l l y d i s t r i b u t e d t i m e , one may o b t a i n t h e e x p r e s s i o n f o r t h e e x p e c t d d u r a t i o n o f t h e t i m e Y o f p e r s i s t e n c e of t r a c e i n memory. We have namely t h e f o l l o w i n g theorem
If c i s t h e decay r a t e of t h e t r a c e o f t h e s t i m u l u s i n memory, t h e n
THEOREM.
t
where F i s t h e L a p l a c e t r a n s f o r m o f t h e d i s t r i b u t i o n P F o f t h e "pause" f o r p o i n t p ( i . e . d i s t r i b u t i o n o f P t h e l e n g t h s o f t i m e when p &s n o t c o v e r e d ) . F o r t h e p r o o f , s e e Nowakowska ( 1 9 8 3 a ) A s a consequence, t h e a v e r a g e d u r a t i o n of t h e t r a c e of
t h e s t i m u l u s " p e r c y c l e " i s ( d e n o t i n g by U t h e l e n g t h pf p e r i o d o f b e i n g covered)
When c t e n d s t o 0 ( t r a c e p e r s i s t s f o r a l o n g t i m e ) , t h i s r a t i o t e n d s t o 1. On t h e o t h e r hand, when c t e n d s t o i n f i n i t e ( t r a c e fades f a s t ) , t h i s r a t i o tends t o the proportion E(U)/[E(U) t E(Z)].
3 12
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3 . 5 . Events r e p r e s e n t a t i o n o f a n o b j e c t : p r e - e v e n t s and memory T h e model here i s based on t h e f o l l o w i n g i n t u i t i o n s . During t h e p r o c e s s of i n s p e c t i o n , t h e eye s e n d s a cont i n u o u s s t r e a m of i n f o r m a t i o n . Somehow o n l y a p a r t of i t i s s e l e c t e d for remembering, and t h e q u e s t i o n i s t o f i n d p l a u s i b l e mechanisms which e x p l a i n how such
a s e l e c t i o n i s accomplished. Such t y p e o f s e l e c t i o n o c c u r s , i n f a c t , on two l e v e l s . On t h e one hand, i n t h e p r o c e s s o f r e c o g n i t i o n and i d e n t i f i c a t i o n o f some o b j e c t , t h e f e a t u r e s a r e anal y s e d , matched, e t c . and e v e n t u a l l y t h e d e c i s i o n about t h e meaning i s r e a c h e d . A s a r u l e , a f t e r some t i m e , t h e p e r s o n remembers more t h a n j u s t t h e meaning o f t h e o b j e c t ; he a l s o remembers some d e t a i l s . To i l l u s t r a t e i t , suppose t h a t we a s k a p e r s o n to
i d e n t i f y t h e make o f a c a r from a photograph. Some t i m e a f t e r t h e t a s k i s a c c o m p l i s h e d , t h e p e r s o n may be a b l e t o t e l l t h e c o l o u r o f t h e c a r , even though t h e c o l o u r p l a y e d no r o l e i n i d e n t i f i c a t i o n of t h e make. A s i m i l a r p r o c e s s o c c u r s a t a somewhat h i g h e r l e v e l , w i t h a n i n t e r v e n t i o n o f c o n s c i o u s components. F o r i n -
stance, certain are the appears
a p e r s o n i n s p e c t s a p a i n t i n g and " n o t i c e s " d e t a i l s : t h e two s h a p e s a t d i f f e r e n t c o r n e r s same, some e l e m e n t s a r e symmetric, l i g h t a t some p l a c e s i n a s p e c i a l way, e t c .
It may be t h o u g h t t h a t t h e b a s i c mechanisms a r e t h e same f o r b o t h c a s e s , and one may t r y t o d e s i g n a model
373
FORMAL SEMIOTICS
which would c o v e r b o t h c a s e s
--
o f c o n s c i o u s and sub-
conscious analysis. When a s t i m u l u s a r r i v e s , i t i s e i t h e r d i s r e g a r d e d , o r s t o r e d i n memory f o r l a t e r c o m p a r i s o n , a n a l y s i s , e t c . The c o n c e p t u a l framework h e r e i s as f o l l o w s . The s t r e a m o f i n f o r m a t i o n i s t r e a t e d as d i s c r e t e , f o r m i n g a s t r e a m
of pre-events,
occurring a t times t = 0 , 1 , 2 ,
... .
Each
pre-event' b e l o n g s t o some t y p e . The c o n c e p t of t y p e i s
a p r i m i t i v e o n e , and n e e d s t o be e x p l i c a t e d i n e a c h c a s e o f a p p l i c a t i o n . I n s p e c i a l c a s e s , i t may be s h a p e , c o l o u r , a b i l i t y t o evoke a s s o c i a t i o n s , e t c . A t any g i v e n t i m e , t h e p e r s o n i s r e c e p t i v e t o some
t y p e s , and n o t r e c e p t i v e t o o t h e r t y p e s , He w i l l s t o r e i n h i s memory o n l y t h o s e p r e - e v e n t s which have t h e t y p e t o which he i s r e c e p t i v e a t t h i s t i m e . F i n a l l y , s t a t e o f r e c e p t i v e n e s s c h a n g e s , and i n p a r t i c u l a r , i t may be m o d i f i e d by p r e - e v e n t s . The main q u e s t i o n which w i l l b e a s k e d i s t o d e t e r m i n e t h e p r o b a b i l i t y t h a t a given pre-event
w i l l become
s t o r e d i n memory, t h a t i s , t h a t i t w i l l become as e v e n t . The
frequency of such c a s e s w i l l t h e r e f o r e
d e t e r m i n e t h e o v e r a l l memory l o a d r e s u l t i n g from t h e inspection. Let e l ,
e 2 , . . . b e t h e p r e - e v e n t s which o c c u r a t t i m e s
t = 1,2,.,.,
and l e t M d e n o t e t h e s e t o f t y p e s , w i t h
g ( e . ) b e i n g t h e type o f pre-event 1
e
i'
L e t p x , x 6 M be t h e p r o b a b i l i t y t h a t a pre-event w i l l be o f t y p e x ; t h e t y p e s a r e assumed t o b e sampled i n d e p e n d e n t l y w i t h t h e same d i s t r i b u t i o n .
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N e x t , we make t h e f o l l o w i n g a s s u m p t i o n a b o u t r e c e p t i v i t y : a t any g i v e n t i m e , t h e s t a t e o f r e c e p t i v i t y i s
d e s c r i b e d b y a p a i r ( Q , f t ) , where
-- Q C M i s t h e s e t o f t y p e s o f p r e - e v e n t s t o which t h e person i s receptive at t i m e t ;
--
f t ( x ) , x G &, i s t h e " r e s i d u a l " t i m e o f r e c e p t i v i t y
f o r t y p e x a t time t ( s o t h a t , i n absence o f subsequent changes, t h e person w i l l remain r e c e p t i v e t o t y p e x u n t i l t h e time t t f t ( x ) ) ;
--
w i t h e a c h t y p e t h e r e i s a s s o c i a t e d a s e t B ( x ) of
t y p e s which i t " a c t i v a t e s " ;
-- if y i s s u c h a t y p e , t h e n t h e time o f r e c e p t i v i t y t o y i s Tx P(Tx
,Y
,Y
>
, where t ) = e- k ( X , Y ) t
L e t us d e n o t e now
s o t h a t C(x) i s t h e c l a s s o f a l l t y p e s y which may modify t h e r e c e p t i v i t y f o r t y p e x. F u r t h e r , l e t
be t h e p r o b a b i l i t y t h a t a p r e - e v e n t t h e receptivity t o type x. We have t h e n t h e f o l l o w i n g t h e o r e m .
w i l l n o t cause
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THEOREM. P r o b a b i l i t y P t h a t a p r e - e v e n t
s t o r e d i n memory ( w i l l
w i l l become
become an e v e n t ) e q u a l s
F o r t h e p r o o f , s e e Nowakowska (1983a) To a p p r e c i a t e t h e consequences o f t h i s theorem, i t i s w o r t h w h i l e t o s i m p l i f y t h e s i t u a t i o n , and r e d u c e t h e number o f p a r a m e t e r s . Thus, assume t h a t
--
t h e r e i s o n l y a f i n i t e number m o f t y p e s , a l l w i t h t h e same p r o b a b i l i t y , so t h a t px = l / m . T h i s means t h a t t y p e s a r e sampled i n d e p e n d e n t l y from t h e uniform distribution;
--
t h e a v e r a g e t i m e o f r e c e p t i v i t y i s t h e same f o r
any t y p e , r e g a r d l e s s o f t h e t y p e o f p r e - e v e n t which c a u s e d t h e r e c e p t i v i t y . T h i s means t h a t r ( x , y ) = f o r a l l x,y;
--
e a c h pre-event a c t i v a t e s t h e same number r o f t y p e s , s o t h a t e a c h s e t C(x) c o n s i s t s o f r e l e m e n t s .
Under t h e s e a s s u m p t i o n s , t h e p r o b a b i l i t y t h a t a p r e e v e n t w i l l become s t o r e d i n memory e q u a l s
1
P = 1
+
h m(e r
-
1)
s o t h a t t h e a v e r a g e time between two c o n s e c u t i v e memo-
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r y storages i s
The e s s e n c e o f t h i s model may be summarized as f o l l o w s :
a p r e - e v e n t ( a c a n d i d a t e for an e v e n t ) becomes s t o r e d i n memory, i f i t i s p r e c e d e d s u f f i c i e n t l y r e c e n t l y by a n o t h e r p r e - e v e n t of an a p p r o p r i a t e t y p e . The probl e m l i e s i n b u i l d i n g a model which would d e s c r i b e t h e frequency of e v e n t s , t h e i r t y p e s , e t c . , i . e . d e s c r i b e t h e s t a t i s t i c a l f e a t u r e s of what i s b e i n g perceived, i n t h e sense of storage of t h e m a t e r i a l t o be p r o c e s s e d l a t e r . The theorem g i v e s t h e p r o b a b i l i t y d i s t r i b u t i o n o f
s p a c i n g s between e v e n t s . T h i s d i s t r i b u t i o n i s approx i m a t e l y e x p o n e n t i a l , s o t h a t s t o r a g e moments c o n s t i t u t e a Poisson process. The main g o a l of t h e s y s t e m o f p e r c e p t i o n p r e s e n t e d h e r e was t o e x p l i c a t e t h e phenomenon of f i l t e r i n g , or p a r t i a l c o n t r o l , i n t h e p r o c e s s of p e r c e p t i o n o f obj e c t s . It was shown how one c a n , i n a c o n s i s t e n t , psyc h o l o g i c a l l y p l a u s i b l e , and e x p e r i m e n t a l l y t e s t a b l e way, s p e a k o f t h e p r o c e s s of f i l t e r i n g ( c o n t r o l ) i n
p e r c e p t i o n . Moreover, t h e o r i z i n g shown h e r e d e p a r t s s u b s t a n t i a l l y from t h e known u n t i l now m a t h e m a t i c a l a p p r o a c h e s t o p a t t e r n r e c o g n i t i o n i n t h e domain o f m a t h e m a t i c a l psychology and a r t i f i c i a l i n t e l l i g e n c e . One may hope t h a t i t w i l l i n f l u e n c e t h e new s t y l e of r e s e a r c h .
FORMAL SEMIOTICS
377
4 . OBSERVABILITY AND CHANGE
I n t h i s s e c t i o n w e s h a l l c o n s i d e r some problems conn e c t e d w i t h t a k i n g o b s e r v a t i o n s , d e s c r i p t i o n s and i d e n t i f i c a t i o n of o b j e c t s .
4.1. Joint observability We s t a r t from a g e n e r a l framework t o e x p r e s s t h e problems o f s p e c i a l importance f o r s c i e n t i f i c r e s e a r c h , namely j o i n t o b s e r v a b i l i t y (or: m u l t i o b s e r v a b i l i t y ) . Generally, a researcher i s interested i n a c e r t a i n phenomenon. The l a t t e r c o n c e p t i s a p r i m i t i v e o n e , i n t h e s e n s e t h a t we s h a l l n o t g i v e any f o r m a l means of d i s t i n g u i s h i n g a phenomenon from non-phenomenon; i n s t e a d , we s h a l l g i v e a f o r m a l scheme t o d e s c r i b e phenomena. F o r m a l l y , a c o n c e p t u a l framework i s a f a m i l y
(4.1) where e a c h X i s a random ( i n extreme c a s e : d e t e r g,s m i n i s t i c ) v a r i a b l e , depending on t h e i n d i c e s from G and S , and C s i s a c l a s s o f s u b s e t s of G , t o be i n t e r preted l a t e r . The i n t e n d e d i n t e r p r e t a t i o n i s s u c h t h a t e l e m e n t s o f G
a r e l a b e l s o f v a r i a b l e s ( t h e i r names), w h i l e e l e m e n t s o f S a r e l a b e l s o f " c o n t r o l " p a r a m e t e r s . I n extreme c a s e w e may have S c o n s i s t i n g o f one e l e m e n t o n l y , s o
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t h a t t h e phenomenon i s n o t c o n t r o l l a b l e .
The phenomenon i s now i n t e r p r e t e d as some s o r t o f t h e o r e t i c a l r e l a t i o n s h i p between v a r i a b l e s i i n t h e s y s t e m ( 4 . l ) , t o which one may a s s i g n a name. Such phenomenon may o r may n o t o c c u r i n a p a r t i c u l a r i n s t a n c e , depending on t h e a c t u a l v a l u e s o f t h e v a r i a b l e s . To f i x t h e i d e a s and g u i d e somewhat t h e i n t u i t i o n , i t i s n e c e s s a r y t o g i v e some examples.
Example 1. I n many problems of s c i e n t i f i c r e s e a r c h one i s i n t e r e s t e d i n a r e l a t i o n s h i p between two v a r i a b l e s , s a y x and y . T y p i c a l l y , o b s e r v a t i o n s have t h e form o f pairs (xl,yl), ( x n , y n ) of t h e v a l u e s o f t h e depen-
...,
d e n t v a r i a b l e y a s s o c i a t e d w i t h v a l u e s of t h e independ e n t v a r i a b l e x. One may t h e n t r y t o f i n d t h i s r e l a t i o n s h i p be u s i n g r e g r e s s i o n methods, e t c . I n p r a c t i c e , w e have h e r e two c o n c e p t u a l l y d i s t i n c t s i t u a t i o n s : ( a ) t h e v a l u e s of independent v a r i a b l e x can be c o n t r o l l e d by t h e e x p e r i m e n t e r ; ( b ) t h e v a l u e s of x a r e n o t c o n t r o l l a b l e . The f i r s t c a s e may be exempl i f i e d by t a k i n g x as a d o s e , and y as a r e s p o n s e ; t h e second by t a k i n g a s x t h e number o f c a r a c c i d e n t s , and by y t h e number o f d e a t h s caused by them. The f i r s t c a s e w i l l be r e p r e s e n t e d by t a k i n g p o s s i b l e v a l u e s o f x as l a b e l s s o f c o n t o l l a b l e v a r i a b l e s , and t a k i n g a s G a s i n g l e t o n ( s i n c e o n l y one random v a r i a b l e , namely y , i s o b s e r v e d ) . I n t h e second c a s e , we t a k e S as a s i n g l e t o n ( s i n c e t h e r e i s no c o n t r o l ) , and G = {'1,23 f o r v a r i a b l e x , and 2
--
, where
for variable y.
1 stands
FORMAL SEMIOTICS
Example 2 .
3 79
For a phenomenon s u c h a s " c o n t a g i o n " , t h e
s e t o f v a r i a b l e s which may p o t e n t i a l l y b e o b s e r v e d i s e x t r e m e l y l a r g e . It may c o n t a i n v a r i o u s r e s p o n s e s o f human and a n i m a l organisms t o d i f f e r e n t t y p e s o f cont a c t s w i t h i n f e c t i o n s m a t e r i a l . T h i s s e t may a l s o cont a i n v a r i a b l e s p e r t a i n i n g t o o b s e r v a t i o n o f immunity, e p i d e m i c a l s p r e a d , e t c . The r e s e a r c h e r s t u d y i n g "cont a g i o n " would have t o narrow down t h e scope o f h i s s t u d y t o some s u b s e t s o f S and G . N O W , w i t h o u t any loss of g e n e r a l i t y , we may assume t h a t
e l e m e n t s o f S e x c l u d e one a n o t h e r , s o t h a t i n e a c h i n s t a n c e e x a c t l y one element w i l l be s e l e c t e d ( t h e c a s e o f no c o n t r o l may b e i n t e r p r e t e d a s h a v i n g S cons i s t i n g o f j u s t one e l e m e n t ) . I n c a s e when one may c o n t r o l a number o f p a r a n e t e r s , e l e m e n t s o f S w i l l be v e c t o r s , e a c h r e p r e s e n t i n g a p o s s i b l e c o n f i g u r a t i o n of parameters. A s r e g a r d s C s J i t s e l e m e n t s w i l l be i n t e r p r e t e d as
l a b e l s o f v a r i a b l e s which may be o b s e r v e d j o i n t l y i n t h e same i n s t a n c e ( " r e a l i z a t i o n " o f t h e s e t o f random
v a r i a b l e s ) , when t h e c o n t r o l p a r a m e t e r i s s . The c l a s s e s Cs a r e assumed t o s a t i s f y t h e f o l l o w i n g HYPOTHESIS. F o r e v e r y s d S , t h e c l a s s Cs i s nonempty, and moreover, whenever A C Cs and B C A , t h e n B G Cs. We have t h e n
a l a t t i c e under o p e r a t i o n o f I n t e r s e c t i o n , i . e . whenever A , B E Cs, then A 0 B C Cs.
THEOREM. The c l a s s
If A =
{'g
Cs i s
.,gn3(Cs,
then a l l variables
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may be o b s e r v e d i n t h e same i n s t a n c e . We s h a l l a l s o s a y t h a t i n t h i s c a s e , t h e s e t A a l l o w s multi-observ-
a t i o n under s ( t h i s i s n o t to be c o n f u s e d w i t h r e p l i c a t i o n o f o b s e r v a t i o n s o f t h e same v a r i a b l e ) . We may now i n t r o d u c e some c o n c e p t s which a r e c o n n e c t ed w i t h t h e above s e t u p i n a n a t u r a l w a y . F i r s t l y , l e t u s s a y t h a t a s e t A C G i s s-maximally m u l t i - o b s e r v a b l e , i f A & Cs and moreover, i f A C B and B e C
*
S'
then A = B.
L e t Cs d e n o t e t h e c l a s s o f a l l s-maximally multi-observable subsets of G . Denote now
R(s) =
UA
(4.2)
A €Cs We s h a l l c a l l R(s) t h e o b s e r v a b i l i t y domain under cont r o l s , and i n c a s e when R(s) = G , w e s a y t h a t s a l l o w s complete i n d i v i d u a l o b s e r v a b i l i t y .
If G E Cs, we s a y t h a t s a l l o w s f u l l m u l t i - o b s e r v a b i l i - , . t y . I n d e e d , from Hypotheses a b o u t Cs i t f o l l o w s t h a t we have
If
then
s a l l o w s complete i n d i v i d u a l = {G\, s o t h a t t h e o n l y maximo b s e r v a b i l i t y , and a l l y multi-observable s e t i s G .
THEOREM.
G € Cs,
* Cs
I f R(s) # G , t h e n we s a y t h a t s p r e v e n t s o b s e r v a b i l i t y
381
FORMAL SEMIOTlCS
of any X
g,s
with g
t'
G
-
R(s).
A s a l r e a d y m e n t i o n e d , a phenomenon i s i d e n t i f i e d w i t h
some t h e o r e t i c a l p r o p e r t y o f c e r t a i n v a r i a b l e s . Its r e a l i z a t i o n may t h e r e f o r e be i d e n t i f i e d w i t h :
-- a s e t o f l a b e l s , gl, g 2 , . . . , gn o f v a r i a b l e s , which d e t e r m i n e whether or n o t t h e phenomenon o c c u r r e d i n a given instance;
--
a c e r t a i n s u b s e t Q o f n-dimensional s p a c e , which n c o r r e s p o n d s t o r e a l i z a t i o n s o f t h e phenomenon.
We s a y namely t h a t t h e phenomenon, s a y Ti , o c c u r r e d under s, or o c c u r r e d i n a g i v e n r e a l i z a t i o n under c o n t r o l s, i f
DEFINITION. We s a y t h a t
i s m u l t i - o b s e r v a b l e under
s, i f It i s q u i t e p o s s i b l e t h a t a g i v e n phenomenon i s m u l t i -
o b s e r v a b l e under some c o n t r o l p a r a m e t e r s, and n o t o b s e r v a b l e under some o t h e r c o n t r o l s s'. I f t h e phenomenon
(A,Qn)
i s c h a r a c t e r i z e d by t h e p a i r
w i t h A = {glJ...,gn?
a s d e s c r i b e d above, t h e n
i s t h e s e t o f a l l c o n t r o l v a l u e s s under which multi-observable.
If s k Q ( R ) , t h e n
li i s
Ti i s n o t m u l t i - o b s e r v a b l e under s,
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and one h a s t o r e s t r i c t t h e o b s e r v a t i o n s t o
some
s u b s e t B C A , w i t h B E Cs. Now, observations of X
f o r g E B u s u a l l y do n o t g,s p r o v i d e complete i n f o r m a t i o n whether i s realized o r not i n a p a r t i c u l a r instance. I f B = j gl,...,gml, then the vector
p r o v i d e s o n l y p a r t i a l i n f o r m a t i o n about t h e r e a l i z a t i o n o f f i . L e t f g m + l , . . . , g n l = A - B be t h e unobserva b l e v a r i a b l e s ( i f one d e c i d e s t o o b s e r v e v a r i a b l e s i n B y i . e . v a r i a b l e s unobservable j o i n t l y w i t h t h o s e from B), and l e t
By d e f i n i t i o n , t h e phenomenon
occurred i f
and d i d n o t o c c u r o t h e r w i s e . Given t h e o b s e r v e d v a l u e s can t h e n d e t e r m i n e t h e p r o b a b i l i t y , or p o s s i -
zB, one
b i l i t y , that
i s realized, that i s
or
The first q u a n t i t y r e q u i r e s t h e knowledge of j o i n t
p r o b a b i l i t y d i s t r i b u t i o n of
ZB
and &A-B,
which may
383
FORMAL SEMIOTICS
be a v a i l a b l e o r n o t . The second depends on t h e d e g r e e with additional o f p o s s i b i l i t y of "complementing" c o o r d i n a t e s , s o t h a t t h e r e s u l t i n g j o i n t v e c t o r would be i n Qn.
zB
If t h e j o i n t d i s t r i b u t i o n i s a v a i l a b l e f o r any s e t B ,
one can p o s e t h e problem o f o p t i m a l , o r most informat i v e , o b s e r v a b i l i t y . If I A ( B ) s t a n d s for t h e amount of i n f o r m a t i o n about -A-B X contained i n X ( i . e . I A ( B ) -B i s t h e conditional entropy of X given X B ) , t h e n A-B t h e o p t i m a l s e t B may be chosen as s u c h t h a t o p t i m i zes I A ( B ) . A s a n example, one can t h i n k h e r e o f t h e problem of
p r e d i c t i o n o f some phenomenon. To d e t e r m i n e whether o c c u r r e d or n o t , one n e e d s t h e o b s e r v a t i o n s of v a r i a b l e s
Suppose t h a t l a b e l gn c o n c e r n s t h e v a r i a b l e i n some f u t u r e , hence not o b s e r v a b l e a t p r e s e n t . Other v a r i a b l e s a r e o b s e r v a b l e a t p r e s e n t , though p e r h a p s n o t all ( d u e t o , s a y , c o s t o f o b s e r v a t i o n s , which may a l l o w t o o b s e r v e o n l y a c e r t a i n number, s a y r , o u t o f n-1 v a r i a b l e s ) . The q u e s t i o n i s t h e n o f s e l e c t i n g t h e most informative set of r v a r i a b l e s out of X , s > * * * Ygx l's i . e . v a r i a b l e s which would p r o v i d e as muih informaf?ion
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384
4.2.
Masks and f i l t e r s
I n t h e p r e c e d i n g a n a l y s i s , no s p e c i f i c a s s u m p t i o n s were made about t h e n a t u r e of e l e m e n t s o f t h e s e t of l a b e l s G. I n p r a c t i c e , l a b e l s g 6 G a r e u s u a l l y m u l t i d i m e n s i o n a l , w i t h one c o o r d i n a t e r e p r e s e n t i n g t i m e . Thus, we may w r i t e G as a C a r t e s i a n p r o d u c t G = H x T
where T i s t h e t i m e a x i s , and H i s a s e t o f l a b e l s which d e s c r i b e t h e n a t u r e o f t h e v a r i a b l e w i t h o u t spec i f y i n g t i m e . I n o t h e r words, g = ( h , t ) i s t o be i n t e r p r e t e d as o b s e r v a t i o n of v a r i a b l e l a b e l e d h a t t i m e t ( e . g . h may b e "blood p r e s s u r e " , t o be o b s e r v e d
a t some f i x e d moments t l , t 2 , . . . a f t e r a n a d m i n i s t r a t i o n of a drug, e t c . ) . Instead of X i t w i l l now be c o n v e n i e n t t o w r i t e g,s X ( t ) f o r t h e v a l u e o f t h e v a r i a b l e l a b e l e d h , under h,s c o n t r o l s , a t t i m e t ( i n t h e example above, s m a y s t a n d f o r t h e dose o f t h e drug, e t c . ) . The c l a s s Cs o f j o i n t l y o b s e r v a b l e v a r i a b l e s a c q u i r e s now a s t r u c t u r e , due t o t h e s t r u c t u r e imposed on G. Thus, a s e t A
Cs i s now a s e t o f p a i r s (4.3)
N a t u r a l l y , some o f t h e v a l u e s h i , as w e l l as some o f t h e v a l u e s t i , may r e p e a t . Let u s t h e r e f o r e d e f i n e , for a fixed h E H
and s e t A g i v e n b y ( 4 . 3 )
385
FORMAL SEMIOTICS
t o be c a l l e d t e m p o r a l t r a c e o f h under A , or h-mask of A . The s e t T A ( h ) i s simply t h e s e t o f a l l moments
a t which t h e v a r i a b l e l a b e l e d h i s t o be o b s e r v e d . I f T A ( h ) = t l , . , . , t , t h e n w e observe (under c o n t r o l k s ) the values X ( t 1 ) , . . . ,X h Y s ( t k ) . h,s
On t h e o t h e r hand, one can t a k e a f i x e d moment t , and consider the s e t MA(t)
= { h C H:
( h , t ) 15 A )
o f a l l v a r i a b l e s which a r e t o be o b s e r v e d a t t i m e t . The s e t M A ( t ) w i l l be c a l l e d a t-mask o f o b s e r v a t i o n , and t h e c l a s s o f a l l t-masks w i l l b e c a l l e d s h o r t l y a mask. T h i s n o t i o n g e n e r a l i z e s t h e c o n c e p t o f mask i n t r o d u c e d b y K l i r (1972). Given t h e above n o t i o n s , one can e x p r e s s v a r i o u s cons t r a i n t s on j o i n t o b s e r v a b i l i t y
.
Thus, f o r g i v e n T A ( h ) , d e f i n e AA(h) =
min
Ilt-t'I
: t,t'
T A ( h ) , t # t f f.(4.4)
T h i s v a l u e r e p r e s e n t s t h e minimal t e m p o r a l s p a c i n g
between o b s e r v a t i o n o f v a r i a b l e h, I f we p u t A(h) =
inf
AA(h)
(4.5)
A 6 Cs
then
A ( h ) d e s c r i b e s t h e minimal t e m p o r a l s p a c i n g
a d m i s s i b l e f o r v a r i a b l e h . I n o t h e r words, any h-mask
i n v o l v i n g two moments c l o s e r t h a n A ( h ) i s n o t admis s i b l e ; t h e r e i s no s e t A o f j o i n t l y o b s e r v a b l e v a r i a b l e s which would a l l o w o b s e r v i n g X ( t ) for t c l o s h,s er than A(h>.
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386
Now, masks d e s c r i b e t h e t e m p o r a l
p a t t e r n s of variab-
l e s t o be o b s e r v e d , and a l s o t h e s e t s o f v a r i a b l e s t o b e o b s e r v e d a t any g i v e n t i m e . I n t h i s s e n s e , t h e y describe the observer's decision (within t h e constrai n t s o f j o i n t o b s e r v a b i l i t y ) about t h e v a r i a b l e s which a r e t o be r e c o r d e d i n a database.
given instance, t o e n t e r t h e
However, even i f t h e o b s e r v e r d e c i d e s t o r e c o r d some v a l u e s , i t may happen t h a t t h e s e v a r i a b l e s w i l l n o t be a v a i l a b l e ; t h e y may be c e n s o r e d , o r f i l t e r e d . T h i s means i n p r a c t i c e t h a t some v a l u e s , i n a g i v e n i n s t a n c e o f t h e phenomenon, may n o t " o c c u r " , o r t h e i r obs e r v a t i o n may be o b s c u r e d i n some o t h e r w a y . A t y p i c a l case occurs i n gathering d a t a f o r medical
r e s e a r c h , where t h e d a t a i n a follow-up s t u d y o f a group o f p a t i e n t s may be u n a v a i l a b l e simply b e c a u s e some o f t h e p a t i e n t s d i e d b e f o r e t h e t i m e o f o b s e r v a tion, left the city, etc. G e n e r a l l y , suppose t h a t a p e r s o n d e c i d e s t o o b s e r v e t h e v a r i a b l e s from some s e t A C where A h a s t h e S'
form ( 4 . 3 ) . T h i s r e q u i r e s o b s e r v i n g t h e v a r i a b l e s
By a f i l t e r w e s h a l l mean a random v a r i a b l e , whose
v a l u e s are s u b s e t s o f A , o r simply -- a random s u b s e t o f A . Thus, a f i l t e r i s d e s c r i b e d b y i t s d i s t r i b u t i o n , i . e . a function
f:
2A
+[O,ll
s u c h t h a t t h e sum o f a l l p r o b a b i l i t i e s e q u a l s o n e , that is
FORMAL SEMIOTICS
z
387
f ( B ) = 1.
BC A
The o p e r a t i o n of a f i l t e r i s t o be i n t e r p r e t e d s o t h a t w i t h p r o b a b i l i t y f ( B ) only t h e v a r i a b l e s
w i l l be o b s e r v a b l e , w h i l e t h e v a r i a b l e s w i t h i n d i c e s (h,t)
k A - B w i l l n o t be o b s e r v a b l e ( i n t h e g i v e n
instance). A f i l t e r with f ( 0 )
> 0 may
be c a l l e d a f i l t e r w i t h
b l a c k o u t p o s s i b i l i t y : i t may happen t h a t none of t h e v a r i a b l e s w i l l be o b s e r v a b l e . On t h e o t h e r e x t r e m e , i f f ( A ) = 1, t h e f i l t e r w i l l n o t d i s t u r b any o b s e r v a t i o n s . V a r i o u s t y p e s of f i l t e r s may now be d e f i n e d t h r o u g h t h e p r o p e r t i e s of p r o b a b i l i t y d i s t r i b u t i o n f(B). I n p a r t i c u l a r , a f i l t e r w i l l be c a l l e d s i m p l e , i f there i s a function
such that
T h i s c o r r e s p o n d s t o t h e c a s e when e a c h v a r i a b l e i s
“ f i l t e r e d “ or n o t f i l t e r e d i n d e p e n d e n t l y , w i t h a p p r o p r i a t e p r o b a b i l i t y . Such f i l t e r s , for t h e c a s e of P o i s s o n p r o c e s s e s , were c o n s i d e r e d i n Nowakowska ( 1 9 8 3 a ) . I n p a r t i c u l a r , if p ( h , t ) = p for a l l ( h , t ) & A , and A c o n t a i n s n e l e m e n t s , t h e n t h e p r o b a b i l i t y of a f i l t e r B w i t h k e l e m e n t s i s g i v e n by t h e b i n o m i a l formula
388
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(:)pk(l-p)n-k,
k = O,l,...,n,
so t h a t t h e p r o b a b i l i t y of a blackout ( a l l v a r i a b l e s u n o b s e r v a b l e ) i s (1-p) n
.
The main problem l i e s i n d e t e r m i n i n g t o which e x t e n t t h e e x i s t e n c e of a f i l t e r d i s t o r t s t h e p o s s i b i l i t y of s t a t i s t i c a l i n f e r e n c e about t h e phenomenon. T h i s ext e n t may v a r y ; i n t h e c a s e o f P o i s s o n p r o c e s s e s c i t e d above, f i l t e r i n g m e r e l y slowed down t h e p r o c e s s o f e s t i m a t i o n , w i t h o u t a f f e c t i n g i t i n any o t h e r way. On t h e o t h e r hand, i n c a s e of c e n s o r i n g d a t a e . g . i n med i c a l r e s e a r c h , t h e e f f e c t may be s u b s t a n t i a l , and requiring special s t a t i s t i c a l techniques.
4.3. O p e r a t i o n s on masks and f i l t e r s O f t e n masks, i . e . p a r t i c u l a r c o n s t r a i n t s on j o i n t obs e r v a b i l i t y (and a t t h e same t i m e , p l a n s o f c o l l e c t i n g d a t a , m e e t i n g t h e s e c o n s t r a i n t s ) a r e superimposed on one a n o t h e r . T h i s may happen when t h e c o n s t r a i n t s a r e due t o v a r i o u s f a c t o r s ; f o r i n s t a n c e , one c o n s t r a i n t may be due t o c o s t s o f t a k i n g o b s e r v a t i o n s , o t h e r s may be due t o a v a i l a b i l i t y o f equipment, w h i l e s t i l l
o t h e r s -- t o p h y s i c a l l i m i t a t i o n s . A s a r e s u l t , t h e o b s e r v a t i o n s a r e t a k e n a c c o r d i n g t o a mask which i s an i n t e r s e c t i o n , t o be r e f e r r e d t o as multi-mask.
...,
C s a r e masks, t h e n t h e AN( F o r m a l l y , i f A1, A2, multi-mask g e n e r a t e d by t h e m i s A = A A fi, AN. 1 The theorem o f t h e p r e c e d i n g s e c t i o n e n s u r e s t h a t
.. .
we have A g C s ( i . e . t h e c l a s s Cs i s c l o s e d under t a k ing intersections).
FORMAL SEMIOTICS
389
...
Another problem a r i s e s when a mask A = { ( h l , t l ) , , ( h n , t n ) ] i s t r a n s l a t e d by some amount. Such a t r a n s l a t i o n i s d e f i n e d by
provided t h a t At
Cs.
G e n e r a l l y , we may d e f i n e
s o t h a t t h e s e two t i m e s d e t e r m i n e t h e e a r l i e s t and
l a t e s t t i m e of o b s e r v a t i o n under mask A . These times may be c a l l e d anchor and h o r i z o n of t h e m a s k , r e s p e c t -
ively.
N a t u r a l l y , we have
tmin= t + t min A At
, tmax = t + tmax A, At
i . e . t h e a n c h o r and h o r i z o n o f a t r a n s l a t e d mask a r e a l s o s h i f t e d b y t h e same amount. V a r i o u s r e l a t i o n s between masks may be d e f i n e d by considering t h e temporal equivalences (e.g. i d e n t i t y o f t e m p o r a l t r a c e s up t o a t r a n s l a t i o n , e t c . ) , and i d e n t i t y of v a r i a b l e s o b s e r v e d , r e g a r d l e s s of t h e t i m e s , i . e . i d e n t i t y of t-masks and h-masks a s s o c i a t e d w i t h two masks.
...,
Suppose now t h a t we have t h e masks A1, AN, each accompanied b y a f i l t e r c h a r a c t e r i s t i c f o r t h i s mask. These f i l t e r s g e n e r a t e random s e t s B1,. : ,BN, w i t h Then t h e opaque s u p e r p o s i t i o n of t h e s e Bi C. A i . f i l t e r s i s d e f i n e d as
.
3 90
CHAPTER 4
B
=nB , ,
w h i l e t h e i r t r a n s p a r e n t s u p e r p o s i t i o n i s d e f i n e d as
Thus, an opaque s u p e r p o s i t i o n a l l o w s f o r o b s e r v a t i o n of a v a r i a b l e only i f it i s f i l t e r e d through a l l of t h e f i l t e r s Bi,
while a transparent superposition
allows t h e o b s e r v a t i o n of a v a r i a b l e , i f it i s f i l t e r ed by a t l e a s t one o f t h e f i l t e r s B
i'
Suppose now t h a t t h e f i l t e r s a r e g e n e r a t e d i n d e p e n d e n t -
l y . One c a n t h e n a s k f o r t h e d i s t r i b u t i o n o f t h e r e s u l t i n g opaque and t r a n s p a r e n t s u p e r p o s i t i o n o f t h e filters. T h i s problem may be s o l v e d f o r t h e c a s e o f t h e s i m p l e f i l t e r s , w i t h i d e n t i c a l p r o b a b i l i t i e s . Suppose namely t h a t i n f i l t e r Bi
e a c h o f t h e v a r i a b l e s i n Ai
Bi w i t h p r o b a b i l i t y p i ,
is in
independently of other variab-
l e s . We have t h e n THEOREM.
&&
m be t h e s i z e o f t h e s e t
n
A i . Then th e p r o b a b i l i t y t h a t t h e opaque s u p e r p o s h i o n o f f i l t e r s B1,
...,
BN w i l l c o n s i s t o f e x a c t l y k e l e m e n t s
equals Qqk(l
-
where q = plp 2...pN.
q)m-k , k = 0 , 1 ,
...,m ,
(4.6)
I n t h e c a s e o f t r a n s p a r e n t super-
p o s i t i o n , formula ( 4 . 6 ) z l s o h o l d s , but w i t h
FORMAL SEMIOTICS
391
4.4. O b s e r v a b i l i t y o f change I n t h e p r e c e d i n g s e c t i o n s , t h e a n a l y s i s concerned obs e r v a b i l i t y o f v a l u e s o f some v a r i a b l e s . I f a v a r i a b l e i s o b s e r v e d a t t i m e t l and t 2 , and i t s v a l u e s a t t h o s e t i m e s a r e X ( t ) = x and X ( t , ) = y w i t h x # y , t h e n 1 w e know t h a t a change h a s o c c u r r e d between t l and t 2 . For t h e moment, l e t u s d i s r e g a r d t h e d i s t i n c t i o n b e t ween " e s s e n t i a l " or " s i g n i f i c a n t " c h a n g e s , and " i n e s s e n t i a l " or " i n s i g n i f i c a n t f 1 o n e s , and q u a l i f y any inequa l i t y x # y as a change. Two e s s e n t i a l f a c t s about changes a r e w o r t h s t r e s s i n g here. F i r s t l y , i t may happen t h a t t h e i n f o r m a t i o n whether x = y o r x # y may be o b t a i n e d w i t h o u t a c t u a l l y o b s e r v i n g x and y s e p a r a t e l y . T h i s means t h a t i n o r d e r t o d e t e r m i n e whether or n o t a change has o c c u r r e d w e may n o t r e q u i r e t h e knowledge o f b o t h x and y ( t h e s i m p l e s t example i s i n c a s e when x and y a r e i n t e g e r v a l u e d
s i z e s o f some s e t s : one can t h e n d e t e r m i n e whether
x
x = y or x
>y
by p r o c e s s o f matching p a i r s , w i t h o u t l e a r n i n g t h e v a l u e s x and y ) . S e c o n d l y , t h e i n f e r e n c e about change works o n l y i n one d i r e c t i o n : i f x # y , t h e n a change h a s o c c u r r e d ; i f x = y , t h e n a l l we know i s t h a t t h e v a l u e s a t t h e endp o i n t s o f an i n t e r v a l a r e t h e same. I f t h e i n t e r m e d i a t e v a l u e s a r e n o t o b s e r v e d , i t i s p o s s i b l e t h a t a ser i e s o f changes o c c u r r e d i n between, e v e n t u a l l y c a n c e l l i n g o u t t o t h e same v a l u e . The p o i n t h e r e i s t h a t i n o r d e r t o i n f e r from t h e ob-
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s e r v e d f a c t t h a t x = y t h a t t h e change d i d n o t o c c u r , one n e e d s some a d d i t i o n a l knowledge about t h e n a t u r e o f t h e phenomenon a n a l y s e d , and no f o r m a l t h e o r y can r e p l a c e t h i s knowledge. To g i v e a s i m p l e example, suppose t h a t t l and t 2 a r e 1 0 m i n u t e s a p a r t ( s a y ) , and we know t h a t X ( t l )
equals
X(t2).
i s t h e number o f b i r t h s i n a g i v e n h o s p i t a l , c o u n t i n g from some f i x e d moment ( m i d n i g h t , s a y ) u n t i l t , t h e n X ( t l ) = X ( t , ) means t h a t no b i r t h s o c c u r r e d between t l and t 2 . Here we u s e t h e f a c t t h a t , b y deIf X ( t )
f i n i t i o n , t h e f u n c t i o n X ( t ) may o n l y i n c r e a s e . On t h e o t h e r hand, i f X ( t ) i s t h e number o f p e r s o n s i n a room a t t i m e t , and X ( t , ) = X ( t ) , we c a n n o t i n f e r I 2 t h a t no change o c c u r r e d i n between: some p e r s o n s may have e n t e r e d t h e room, and some o t h e r s may have l e f t it. We s h a l l now g i v e a f o r m a l d e s c r i p t i o n o f s p e c i a l t y p e o f masks, t o be c a l l e d cd-masks ( f o r : change d e t e c t i o n masks). A cd mask i s a s e t o f p a i r s ( h , t ) o f t h e form
(4.7) where t i
ti
f o r i = 1,2,
...,n .
N a t u r a l l y , one c o n s i d e r s o n l y masks which a r e i n C S' i . e . a l l o w j o i n t o b s e r v a b i l i t y of t h e v a r i a b l e s i n q u e s t i o n a t s p e c i f i e d times.
n L e t C s b e t h e c l a s s o f a l l masks o f t h e form ( 4 . 7 )
FORMAL SEMIOTlCS
393
with A & Cs.
S ince i n each consecutive p a i r t h e f i r s t e l e m e n t s c o i n c i d e , i t i s more c o n v e n i e n t t o r e p r e s e n t t h e mask ( 4 . 7 ) i n t h e form
t o be i n t e r p r e t e d : o b s e r v e t h e v a r i a b l e l a b e l e d hi a t t i m e s t i and and d e t e r m i n e whether a change has o c c u r r e d between t h e s e moments. N a t u r a l l y , some o f t h e v a l u e s hi may c o i n c i d e , if a v a r i a b l e i s o b s e r v e d more t h a n two t i m e s .
ti
= 0 i f a change o f v a r i a b l e l a b e l e d hi 1 1 1 o c c u r r e d between t i and ti, and Q ( h i , t i , t i ) = 1 o t h e r wise. One can t h e n d e f i n e t h e c h a n g e - d e t e c t i o n f u n c t i o n a s s o c i a t e d w i t h mask ( 4 . 8 ) as
Let Q(h.,t.,t!)
...
(4.9) We have h e r e G ( A ) = 0 whenever a change o c c u r s i n any o f t h e v a r i a b l e s and between t h e t i m e s s p e c i f i e d b y t h e m ask A , and G ( A ) = 1 i f no change was d e t e c t e d ( w h i c h , as a l r e a d y e x p l a i n e d , d o e s n o t i m p l y t h a t a change d i d n o t o c c u r ) . The s e t
H(A)
= lh:
(h,t,t')
A for some t , t f 3
(4.10)
i s c a l l e d t h e b a s e o f t h e cd-mask, C l e a r l y , mask A i s
c a p a b l e of d e t e c t i n g a change o n l y i f i t c o n c e r n s t h e v a r i a b l e s i n a base; o t h e r changes are u n d e t e c t a b l e under mask A . On t h e o t h e r hand, l e t u s d e f i n e t h e set
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Then t h e s e t T ( A ) i s t h e s e l e c t i o n s e t , i n t h e s e n s e t h a t t h e changes w i l l be r e c o r d e d o n l y i f t h e y o c c u r i n one o f t h e i n t e r v a l s o f t h e s e l e c t i o n s e t . Masks may be o r d e r e d a c c o r d i n g t o t h e r e l a t i o n o f r e -
f i n e m e n t ; g e n e r a l l y , a r e f i n e m e n t o f a cd mask i s a mask c a p a b l e o f d e t e c t i n g m u l t i p l e c h a n g e s , o c c u r r i n g w i t h i n some i n t e r v a l s c o v e r e d by t h e o r i g i n a l m a s k , or changes o f o t h e r v a r i a b l e s ( e x t e n s i o n of a b a s e ) . F o r m a l l y , we s a y t h a t a mask A ' i s a r e f i n e m e n t o f mask A , i f H(A) C H ( A ' ) , and moreover, whenever a t r i p l e (h,t,t')
6 A , t h e n t h e r e e x i s t s a sequence
with (h,t,t('))
1,2
,..., m - 1 ,
t' A ' ,
(h,t (i),t(itl)) E
and ( h , t ( m ) , t ' ) E A ' .
A 1
for i =
Thus, w h i l e mask
A i s " s e n s i t i v e " t o a change i n t h e i n t e r v a l ( t , t ' )
( o f v a r i a b l e l a b e l e d h ) , mask A ' w i l l r e c o r d a change of t h i s v a r i a b l e i f i t o c c u r s anywhere i n one o f t h e i n t e r v a l s of t h e p a r t i t i o n ( 4 . 1 2 ) . O b v i o u s l y , u n d e r r e l a t i o n of r e f i n e m e n t , masks form a l a t t i c e ; however, t h e r e i s no "most r e f i n e d " mask, s i n c e e v e r y i n t e r v a l may be f u r t h e r s u b d i v i d e d , leadi n g t o a more r e f i n e d mask. We have h e r e t h e f o l l o w i n g theorem.
THEOREM. If A'
is
a refinement of A , t h e n G ( A t ) = 1 i m p l i e s G ( A ) = 1, i . e . whenever a r e f i n e d mask d e t e c t s no change, t h e n t h e o r i g i n a l mask d e t e c t s no change either.
FORMAL SEMIOlTCS
Finally,
395
l e t u s o b s e r v e t h a t a mask may be c r e a t e d
sequentially, i n the sense of observations being taken c o n d i t i o n a l l y on p r e v i o u s o b s e r v a t i o n s ( t h i s n o t i o n o f c o n d i t i o n a l mask a p p l i e s t o cd-masks as w e l l as t o t h e u s u a l m a s k s ) . Thus, a c o n d i t i o n a l mask i s r a n dom, and i s formed as f o l l o w s ( f o r s i m p l i c i t y , t h e d e s c r i p t i o n w i l l be g i v e n f o r one v a r i a b l e o n l y ; a g e n e r a l i z a t i o n for t h e c a s e o f more t h a n one v a r i a b l e i s immediate). F i r s t l y , we have a p a i r ( h , t ) , c a l l e d t h e s t a r t e r o f
random mask. Then, s u b s e q u e n t o b s e r v a t i o n s depend on ( t ) . The d e c i s i o n i s g i v e n i n t h e observed v a l u e X h,s form of a f u n c t i o n f , mapping t h e s e t R o f v a l u e s of X ( t ) i n t o t h e set h,s
t h e symbol * s i g n i f i e s s t o p p i n g t h e p r o c e s s o f t a k i n g o b s e r v a t i o n s ; t h u s , w e s t o p t h e p r o c e s s , i f w e have the condition f [ X h , s ( t ) ] = *. O t h e r w i s e , w e have f[XhJs(t)] = t t w i t h t' '> t , and n e x t o b s e r v a t i o n i s t a k e n a t t i m e t t . The p r o c e s s t h e n c o n t i n u e s i n t h i s fashion u n t i l the value * i s obtained, I n t h i s way, t h e moments t l , t 2 , a t which t h e o b s e r v a t i o n s a r e made ( o r , i n general case, the pairs (hi,ti) of variables t o b e o b s e r v e d and t i m e s o f t h e o b s e r v a t i o n s ) depend i n a random way on t h e p r e c e d i n g o b s e r v a t i o n s .
...
4 . 5 . Semantic a s p e c t s o f r e p r e s e n t a t i o n s We s h a l l now c o n s i d e r t h e problems i n t r o d u c t o r y t o
f o r m a l s e m i o t i c s , t o be c o n s i d e r e d i n n e x t s e c t i o n s ,
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396
namely t h e g e n e r a l scheme of r e p r e s e n t a t i o n o f o b j e c t s . We b e g i n w i t h s y s t e m s r e f e r r e d t o i n t h e i n t r o d u c t i o n as " c o n s t r a i n e d " , where we d e a l w i t h a c l a s s o f o b j e c t s which are a l i k e i n many r e s p e c t s , and d i f f e r o n l y by v a l u e s o f some a t t r i b u t e s . T h e s y s t e m h e r e w i l l have t h e form
(4.13) with t h e following intended interpretation. F i r s t l y , X i s a c l a s s o f o b j e c t s under c o n s i d e r a t i o n , and i s a n e q u i v a l e n c e on X . Thus, t h e s e t X i s p a r t i t i o n e d i n t o equivalence c l a s s e s of r e l a t i o n A , , The r e l a t i o n
l~ i s c o n n e c t e d w i t h t h e g o a l f o r which t h e s y s t e m i s u s e d : t h e symbol x m y means t h a t x and y and i n d i s t i n g u i s h a b l e from t h e p o i n t of view of t h e
goal of d e s c r i p t i o n , o r a r e "for p r a c t i c a l purposes will i d e n t i c a l " . Equivalence c l a s s e s of r e l a t i o n be c a l l e d t y p e s , and e l e m e n t s o f t h e same e q u i v a l e n c e
c l a s s w i l l be c a l l e d exemplars of a t y p e . The t y p e s w i l l b e d e n o t e d by T1, T 2 , so t h a t
...,
X = T
1
w T
2
u
...
(4.14)
w i t h XP- y i f f x , y 6 Ti for some i . I n t h i s c a s e , x and y a r e b o t h o f t y p e T or a r e exemplars o f t h a t i'
t4pe. Next, A i s t h e s e t o f a l l a t t r i b u t e v a l u e s , and 2 i s a n e q u i v a l e n c e i n A . Thus, s i m i l a r l y as w i t h t h e s e t X , t h e set A i s p a r t i t i o n e d i n t o equivalence c l a s s e s
of r e l a t i o n
, so
that
397
FORMAL SEMIOTICS
Each s e t A i w i l l be i n t e r p r e t e d as a s e t of a l l p o s s i b l e v a l u e s of some a t t r i b u t e . F o r i n s t a n c e , i f t h e f i r s t a t t r i b u t e s t a n d s for c o l o u r , t h e n e l e m e n t s o f A 1 w i l l be " r e d " , "green", e t c . From t h e c o n s t r u c t i o n of t h e s e t s Ai
a s b iff a,b f A
i
it follows that
for some i .
F i n a l l y , as r e g a r d s r e l a t i o n R , i t w i l l b e i n t e r p r e t e d as a r e l a t i o n o f " h a v i n g a v a l u e on a n a t t r i b u t e " . We p o s t u l a t e t h e r e f o r e ( b e g i n n i n g w i t h nonfuzzy c a s e ) t h a t R i s a binary r e l a t i o n i n X x A s a t i s f y i n g t h e f o l l o w i n g two c o n d i t i o n s :
(vb
c Ai)[xRb + a
=
blf
.
The f i r s t c o n d i t i o n means s i m p l y t h a t e a c h o b j e c t has
a v a l u e on a t l e a s t one a t t r i b u t e . T h e second c o n d i t i o n a s s e r t s t h a t whenever o b j e c t x
has a v a l u e on an a t t r i b u t e , t h i s v a l u e i s u n i q u e . I n
o t h e r words, e i t h e r a n a t t r i b u t e i s i n a p p l i c a b l e f o r a n o b j e c t , or it has e x a c t l y one v a l u e on t h i s a t t r i bute. Consequently, i t i s
I(x)
=
ti:
n a t u r a l t o denote
l a E Ai w i t h xRa]
(4.16)
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398
and from a s s u m p t i o n ( a ) i t f o l l o w s t h a t I ( x ) # 0 f o r e v e r y x. We may now d e f i n e v a r i o u s k i n d s o f e q u i v a l e n c e s on X , e x p r e s s i n g them t h r o u g h a t t r i b u t e s . F i r s t l y , we s h a l l d e f i n e
x
%
A
y i f (+a
E A) xRa i f f Y R ~ .
Thus, two o b j e c t s a r e i n r e l a t i o n -A,
i f t h e y have
t h e same v a l u e s o f a l l a t t r i b u t e s . S e c o n d l y , one may d e f i n e x -I
x
y, i f I(x) = I(y). If
y , t h e o b j e c t s w i l l be s a i d t o belong t o t h e
same a t t r i b u t e t y p e s . I n p r a c t i c e , i t means t h a t t h e s e t s o f a t t r i b u t e s a p p l i c a b l e t o them c o i n c i d e , a l t h o u g h t h e a t t r i b u t e v a l u e s may be d i f f e r e n t . L e t U1,
U2, ... b e t h e e q u i v a l e n c e c l a s s e s o f
-
l e t V1,
V2,
-I.
c e x A,,,
...
be t h e e q u i v a l e n c e c l a s s e s o f
y i m p l i e s t h a t x -I
u n i o n o f some U ' s . j
y , each
Thus, t h e p a r t i t i o n
finement o f p a r t i t i o n
...,
Sin-
Vi must b e t h e
tUi
i s a re-
yVij.
S i n c e our o b j e c t i v e i s t o d i s t i n g u i s h o b j e c t s t h e i r equivalence
A' and
2 , i.e.
up t o
up t o t h e i r t y p e s T1, T 2 ,
t h e f o l l o w i n g d e f i n i t i o n seems a p p r o p r i a t e .
D E F I N I T I O N . The s y s t e m ( 4 . 1 3 ) i s s a i d t o be i n f o r m a -
t i o n a l l y c o m p l e t e , i f f o r e v e r y Ui
there exists a type
.
T . w i t h Ui c T Otherwise, t h e system i s i n f o r m a t i o n J j a l l y incomplete. T h i s means t h a t t h e s y s t e m i s i n f o r m a t i o n a l l y c o m p l e t e ,
399
FORMAL SEMIOZTCS
i f t h e knowledge o f a l l a t t r i b u t e s o f t h e o b j e c t
a l l o w s t o i d e n t i f y i t , up t o t h e c l a s s o f e q u i v a l e n c e i n t h e degree s u f f i c i e n t f o r t h e g o a l . On t h e r o t h e r hand, i f t h e system i s n o t complete i n f o r m a t i o n a l l y , t h e n t h e r e e x i s t o b j e c t s which have t h e same v a l u e s o f a t t r i b u t e s ( h e n c e t h e y cannot be d i s t i n g u i s h e d u s i n g t h e a t t r i b u t e s from A ) , b u t a r e of d i f f e r e n t t y p e s , h e n c e p l a y v a r i o u s r o l e s from t h e p o i n t o f view o f t h e g o a l . of t h e r e l a t i o n v, i . e .
4.5.1. Algebra o f d e s c r i p t i o n s .
For e v e r y x , t h e s e t
I ( x ) describes t h e c l a s s of attributes applicable t o x , and for e v e r y i < I ( x ) t h e r e i s e x a c t l y one element a i ( x ) k A i such t h a t x R a i ( x ) . T h i s means t h a t f o r e v e r y x we have a f u n c t i o n , w i t h domain I ( x ) and values i n s e t s Ai, i & I ( x ) . This function provides a s i m p l e s t d e s c r i p t i o n of o b j e c t x. To d e v e l o p a conveni e n t f o r m a l i s m for i n t r o d u c i n g a n a l g e b r a o f d e s c r i p = {1,2,..,,NI be t h e s e t o f a l l a t t r i b u t e s ions, l e t ( o r , more p r e c i s e l y , a t t r i b u t e names). We s h a l l now d e f i n e a d e s c r i p t i o n as a v e c t o r
(w,
B ~ i, 6
W)
(4.17)
c o n s i s t i n g of a s e t W C i s a nonempty s e t o f a t t r i b u t e s , and Bi C A i i s a s u b s e t o f t h e s e t Ai o f v a l u e s of a t t r i b u t e i . Obviously, s p e c i f i c a t i o n of t h e v a l u e s of a t t r i b u t e s o f a n o b j e c t i s a d e s c r i p t i o n of t h i s o b j e c t i n t h e s e n s e o f d e f i n i t i o n ( 4 . 1 7 ) , w i t h I ( x ) = W , and e a c h Bi b e i n g a s i n g l e t o n Bi = 1 ai ( x ) ? . However, one c a n
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400
t h i n k a l s o o f d e s c r i p t i o n s which do n o t s p e c i f y e x a c t l y t h e v a l u e s o f a t t r i b u t e s , b u t g i v e o n l y some s e t s o f v a l u e s , presumed t o " c o v e r " t h e v a l u e s a i ( x ) , One can now d e f i n e some r e l a t i o n s and o p e r a t i o n s on d e s c r i p t i o n s o f t h e form ( 4 . 1 7 ) . F i r s t l y , i f
a r e two d e s c r i p t i o n s , we s a y t h a t do i s c o n t a i n e d i n d ' , o r : d ' i s an e x t e n s i o n o f d , i f
Bi
(b)
B;
f o r each i
C
W.
I n t h i s case w e shall write d C d ' .
Thus, an e x t e n s i o n c o n t a i n s a t l e a s t a3 many a t t r i b u t e s , and whenever a n a t t r i b u t e i s used i n b o t h d e s c r i p t i o n s , then the set o f values i n extension d l i s a t l e a s t as l a r g e as t h e s e t o f v a l u e s i n d . N e x t , l e t d and d 1 b e two d e s c r i p t i o n s w i t h i d e n t i c a l bases, W = W'. Then t h e i r i n t e r s e c t i o n d n d ' i s def i n e d as t h e d e s c r i p t i o n
while t h e i r disjunction d
(w,
B
~
i~
d ' i s defined as
B U' i C W). iy
F i n a l l y , a n o t h e r o p e r a t i o n w i l l be t h e j o i n o f d e s c r i p t ions, applicable t o descriptions d,dl with W D W f = @
.
401
FORMAL SEMIOTICS
Then t h e j o i n (W
\I
d
W',
+
d ' i s d e f i n e d as
Bi,
i
E W , B;, i < W').
It s h o u l d be o b s e r v e d t h a t w h i l e i n t e r s e c t i o n and d i s j u n c t i o n c o n c e r n t h e same s e t s o f a t t r i b u t e s , t h e o p e r a t i o n o f j o i n a l l o w s u s t o add new a t t r i b u t e s . A description with singleton base, i . e . with W =
CiT,
c o n s i s t i n g t h e r e f o r e o f a s i n g l e s e t Bi w i l l be c a l l e d e l e m e n t a r y . I n a d d i t i o n , i f Bi = ? a i l , w i t h a i E Ai, t h e d e s c r i p t i o n w i l l be c a l l e d simple. It f o l l o w s t h a t e v e r y d e s c r i p t i o n d = (W, Bi, i E W) of t h e form ( 4 . 1 7 ) i s a j o i n o f e l e m e n t a r y d e s c r i p t o n s d(Bi)
= ( f i J Bi), ,
so that
d(Bi).
d =
i E W T h i s form w i l l be c a l l e d a c a n o n i c a l r e p r e s e n t a t i o n o f
a description. Moreover, e v e r y e l e m e n t a r y d e s c r i p t i o n i s a d i s j u n c t i o n o f s i m p l e d e s c r i p t i o n s , namely d(Bi)
=
u
d(ia)).
a e Bi
C o n s e q u e n t l y , e v e r y d e s c r i p t i o n d has a c a n o n i c a l rep r e s e n t a t i o n as a j o i n o f d i s j u n c t i o n s o f s i m p l e descriptions d = i G W
u
d({aJ).
a c Bi
The o p e r a t i o n s d e f i n e d above impose a s t r u c t u r e on
t h e s e t o f a l l d e s c r i p t i o n s . A s w i l l b e shown i n n e x t
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402
s e c t i o n t h i s s t r u c t u r e imposes s t r u c t u r a l c o n s t r a i n t s on s e m a n t i c s o f d e s c r i p t i o n s .
4.5.2. A s s e r t i o n s and q u e r i e s . L e t now D d e n o t e t h e c l a s s o f a l l d e s c r i p t i o n s ( 4 . 1 7 ) . Such d e s c r i p t i o n s may be used i n two w a y s . One way may be d e s c r i b e d as an a s s e r t i o n " o b j e c t x i s d" ( i . e . " d e s c r i p t i o n d f i t s o b j e c t x " , o r " c o v e r s " o b j e c t x , e t c . ) . Here t h e s t a t e m e n t s p e c i f i e s a p a i r ( x , d ) , where x i s a n o b j e c t and d i s a d e s c r i p t i o n , presumed t o f i t t o t h i s o b j e c t . Such an a s s e r t i o n w i l l be d e n o t e d by x!d. I n a n o t h e r way, one may a s k f o r o b j e c t o r o b j e c t s which s a t i s f y a given d e s c r i p t i o n d . I n t h i s c a s e , only d i s g i v e n ; we s h a l l i n t e r p r e t i t as a q u e r y , t o b e d e n o t ed by ? d . I n b o t h c a s e s , w e s h a l l need a r e l a t i o n c o n n e c t i n g objects with descriptions , representing the t r u t h about t h e s e o b j e c t s . We s a y namely t h a t x i s d e s c r i b e d by d = ( W ,
i t W)
Bi,
( o r : d i s a t r u t h about x , d i s
t r u e f o r x ) , if I ( x ) = W,
ai(x)
$r
Bi
for all i t I(x).
N O W , as r e g a r d s a s s e r t i o n s x ! d , we have
x!d i s t r u e i f f
d i s a t r u t h f o r x.
2
Next,let d = d ( B . ) be t h e c a n o n i c a l r e p r e s e n t a t i o n 1 * 4 W of d as a j o i h o f e l e m e n t a r y d e s c r i p t i o n s . We have t h e n THEOREM.
x!d i s t r u e i f f d(Bi) i s t r u e f o r x f o r a l l i .
403
FORMAL SEMIOTlCS
Now, as r e g a r d s q u e r i e s , t h e s i t u a t i o n i s a s f o l l o w s . F i r s t l y , for e a c h d = (W, Bi, i 6 W) d e n o t e S(d) = q x € X : I(x) = W , a i ( x ) t Bi,
i t
W]
(4.18)
t h a t i s , t h e s e t of a l l o b j e c t s f o r which t h e d e s c r i p -
tion d i s true. The o p e r a t i o n s o f d e s c r i p t i o n s may now be a s s o c i a t e d w i t h o p e r a t i o n s ' o n s e t s S ( d ) as f o l l o w s . L e t d and d ' be d e s c r i p t i o n s w i t h t h e same b a s e W ,
so
B;, i € W ) , and l e t S ( d ) and S ( d ' ) be t h e c o r r e s p o n d i n g s e t s o f o b j e c t s f o r which d and d ' a r e t r u e . We have t h e n
that d = (W,
Bi,
i € W ) and d' = ( W ,
S(d u d ' ) =
and S(d n d ' ) = S(d)
#A
(4.20)
S(d').
A s r e g a r d s t h e o p e r a t i o n of j o i n , l e t d = W , Bi,
and d ' = ( W ' , S(d
+
BI, i € W 1 ) w i t h W d ' ) = S(d)
r\
S(d').
fi
W1 =
i € W)
0. Then (4.21)
C o n s e q u e n t l y , f o r m u l a s ( 4 . 1 9 ) - ( 4 . 2 1 ) e s t a b l i s h an isomorphism between t h e s t r u c t u r e o f t h e c l a s s of s e t s S ( d ) under o p e r a t i o n s of u n i o n s and i n t e r s e c t i o n s , and t h e c l a s s o f a l l d e s c r i p t i o n s , w i t h o p e r a t i o n s of i n t e r s e c t i o n , d i s j u n c t i o n and j o i n . We s h a l l now i n t e r p r e t S ( d ) as a f u l l r e p l y t o t h e query ?d. If S ( d ) = 0, i.e. t h e r e a r e no o b j e c t s x
404
CHAPTER 4
f o r w h i c h x ! d , t h e n t h e q u e r y ?d i s a n s w e r e d n e g a t i v e l y . If S ( d ) # @,
t h e answer i s p o s i t i v e . I f , i n a d d i t -
i o n , there e x i s t s an index j such t h a t S ( d ) C T
j’
where j i s a c l a s s of a b s t r a c t i o n of t h e e q u i v a l e n c e yv , w e s a y t h a t t h e q u e r y ? d h a s s a t i s f a c t o r y a n s w e r , o r t h a t d e s c r i p t i o n d i s s a t i s f a c t o r y . The j u s t i f i c a t i o n of t h e t e r m l i e s h e r e i n t h e f a c t t h a t t h e s e t S ( d ) c o n t a i n s o b j e c t s which are i d e n t i c a l from t h e p o i n t o f view o f t h e g o a l , i . e . i f x,y 6 S ( d ) , t h e n X
N
y.
We may now g i v e
a n i n t e r p r e t a t i o n o f t h e a b o v e con-
s i d e r a t i o n s w h i c h w i l l b e u s e f u l i n t h e s e q u e l , when we p a s s t o fuzzy a t t r i b u t e s , l i n g u i s t i c d e s c r i p t i o n s i n form o f v e r b a l c o p i e s , and g e n e r a l l y , l i n g u i s t i c measurements
and s e m i o t i c s .
E a c h d e s c r i p t i o n may b e r e g a r d e d a s a scheme b u i l t o u t o f s e t s o f a t t r i b u t e v a l u e s by l o g i c a l o p e r a t i o n s o f d i s j u n c t i o n , c o n j u n c t i o n and j o i n ( r e d u c i b l e a l w a y s t o c a n o n i c a l f o r m ) . To e a c h scheme t h e r e c o r r e s p o n d s a s e t o f o b j e c t s , whose s t r u c t u r e i s i s o m o r p h i c t o t h e l o g i c a l s t r u c t u r e o f t h e scheme of d e s c r i p t i o n .
4 . 5 . 3 . F u z z y a t t r i b u t e s a n d v e r b a l c o p i e s . L e t us now e x t e n d t h e t h e o r y o f t h e p r e c e d i n g s e c t i o n s s o as t o i n c l u d e t h e p o s s i b i l i t y o f some d e s c r i p t o r s b e i n g fuzzy. T h i s s i t u a t i o n i s o f s p e c i a l importance i n t h e c a s e when w e d e a l w i t h v e r b a l d e s c r i p t i o n s . The m o d i f i c a t i o n w i l l c o n c e r n t h e s e t s B
allowed i n t h e d e s c r i p t i o n s d = (W, Bi, i 6 W ) ; t h e y may now b e f u z z y ; t h e s e t W , h o w e v e r , w i l l n o t be assumed f u z z y . i
405
FORMAL SEMIOnCS
T e c h n i c a l l y s p e a k i n g , B . i s now a fuzzy s u b s e t o f A i , 1 hence c h a r a c t e r i z e d by membership f u n c t i o n mg ( a ) ,
a G A i . Moreover, we s h a l l r e s t r i c t t h e d e s c r i p t i o n s t o f u z z y s e t s o f some s p e c i a l form o n l y ; i n t u i t i v e l y s p e a k i n g , o n l y t o t h o s e fuzzy s e t s which may be "named". T o e x p r e s s t h e s e r e s t r i c t i o n s , we i n t r o d u c e two a d d i -
t i o n a l n o t i o n s t o t h e system (4.13), namely V and m , s o t h a t now we d e a l w i t h t h e s y s t e m
(x,
cu
, A,
3
,
R,
V, m )
(4.22)
where t h e f i r s t f i v e symbols have t h e same i n t e r p r e t a t i o n as b e f o r e , w h i l e V and m a r e i n t e r p r e t e d as follows. F i r s t l y , V i s a s e t o f a d m i s s i b l e names f o r s e t s o f
a t t r i b u t e v a l u e s . We assume h e r e t h a t t h e r e l a t i o n i s d e f i n e d a l s o on V , and t h a t V i s p a r t i t i o n e d i n t o
t h e same number o f d i s j o i n t c l a s s e s as A , i . e .
v = v1 u v*
u
...
LJ
vN'
The e l e m e n t s o f Vi w i l l be i n t e r p r e t e d as a d m i s s i b l e names o f v a l u e s o f a t t r i b u t e s i n A i . S e c o n d l y , m i s a f u n c t i o n which maps V r A i n t o t h e i n t e r v a l 0,1], i . e .
m: V x A 4[0,11, i t s v a l u e s t o be d e n o t e d by m v ( a ) . It w i l l be assumed t h a t the function m satisfies t h e following condition: if v c Vi, t h e n m v ( a > = 0 f o r a l l a 9? A i .
CHAPTER 4
406
The meaning of t h e above c o n d i t i o n i s as f o l l o w s . For
e a c h v t V, t h e f u n c t i o n m v ( a ) , t r e a t e d a s a f u n c t i o n of a , d e t e r m i n e s a f u z z y s u b s e t o f A . I f v V then i' t h e c o n d i t i o n s t a t e s t h a t t h e s u b s e t c h a r a c t e r i z e d by
( a ) has s u p p o r t i n A i J i . e . it i s a fuzzy s u b s e t of
m
vi A . only. 1
O o n s i d e r now t h e c l a s s o f a l l d e s c r i p t i o n s o f t h e form d =
(W, B i ,
i € W)
such t h a t t h e f o l l o w i n g c o n d i t i o n
ho I d s : f o r e a c h i E W t h e r e e x i s t vil, such t h a t
Bi
vi2,.
. . ,v i n
i
CVi
has membership f u n c t i o n
.
mvilr\ vi2 n . . n v i n i
( a ) = min j
mV
(a). i j
Such d e s c r i p t i o n s w i l l b e simply d e n o t e d b y d = (vij,
i
,..., n i ) .
W, j = 1
Now, a s b e f o r e , a d e s c r i p t i o n d may be used i n two w a y s . If d i s used i n c o n j u n c t i o n w i t h some o b j e c t x , as an a s s e r t i o n "x i s d " , i t w i l l be c a l l e d a v e r b a l copy o f x . I f used by i t s e l f , i t c o n s t i t u t e s e i t h e r a q u e r y ? d , as i n t h e p r e c e d i n g s e c t i o n , o r as a g e n e r a t i n g p r o j e c t o r of an imaginary o b j e c t x . To s e e t h e d i f f e r ence between v e r b a l copy and g e n e r a t i n g p r o j e c t o r , it i s enough t o n o t e a d i f f e r e n c e between d e s c r i p t i o n o f , s a y , White House, and a d e s c r i p t i o n o f t h e house of a f i c t i t i o u s c h a r a c t e r i n a novel. I n t h e f i r s t c a s e , i t i s p o s s i b l e t o d e t e r m i n e how w e l l t h e v e r b a l copy d e s c r i b e s t h e o b j e c t , whether some d e t a i l s a r e
407
FORMAL SEMIOTICS
o m i t t e d or m i s r e p r e s e n t e d , e t c ( t e c h n i c a l l y , one can d e t e r m i n e how f a i t h f u l and how e x a c t i s t h e v e r b a l c o p y ) . I n t h e second c a s e , when d i s used as a genera t i n g p r o j e c t o r , i t makes no s e n s e t o c o n s i d e r t h e “ t r u t h ” of t h e d e s c r i p t i o n , hence n e i t h e r i t s f a i t h fulness o r exactness. I n C h a p t e r 6 , t h e problem of v e r b a l c o p i e s w i l l be considered again, i n connection w i t h s c i e n t i f i c descripti o n s and t h e o r y o f knowledge ( h e n c e t h e y w i l l be used i n t h e s e n s e o f d e s c r i p t i o n s o f an o b j e c t x , p e r h a p s of a consi der able complexity) . I n connection with t h e form “x i s d ” , it i s necessary t o analogous t o t h o s e
verbal copies, i . e . assertions of w i t h d being a fuzzy d e s c r i p t i o n , determine t h e t r u t h conditions, i n t h e nonfuzzy c a s e .
N a t u r a l l y , t h e t r u t h i n t h i s c a s e w i l l b e of f u z z y c h a r a c t e r . D e n o t i n g by mg ( a ) t h e membership f u n c t i o n ( a ) , cbrresponding t o a l l fuzzy m V i l 4 Vi2O.. ’ini d e s c r i p t o r s concerning i - t h a t t r i b u t e , t h e t r u t h value a s s o c i a t e d w i t h i - t h a t t r i b u t e w i l l be d e f i n e d as
.
(4.23) where a s b e f o r e , a 1. ( x ) i s t h e unique t r u e v a l u e of t h e a t t r i b u t e i o f o b j e c t x. The s e t pi = f a 6 A ~ :m
(a) Bi
>
0’)
(4.24)
w i l l be c a l l e d t h e e x a c t n e s s s e t . Now, f a i t h f u l n e s s
l e v e l o f t h e v e r b a l copy o f x may be d e f i n e d a s t h e v e c t o r of numbers
408
CHAPTER 4
and i t s e x a c t n e s s i s c h a r a c t e r i z e d by t h e v e c t o r o f sets
(4.26) I n t u i t i v e l y , the highest l e v e l of f a i t h f u l n e s s occurs i f a l l c o o r d i n a t e s o f v e c t o r ( 4 . 2 5 ) are e q u a l 1, w h i l e exactness increases with the decrease of s e t s F so i' t h a t t h e most e x a c t copy has e v e r y F b e i n g a s i n g l e i
ton. I n t h e nonfuzzy c a s e , when e a c h of t h e s e t s B
i
is
" c r i s p " , a f a i t h f u l v e r b a l copy o f x i s a d e s c r i p t i o n
of (some o r a l l ) a t t r i b u t e s of x , which happens t o be t r u e . The t r u t h may be broad o r q u i t e s p e c i f i c ; e . g . i n d e s c r i b i n g an a t t r i b u t e s u c h as h e i g h t o f a p e r s o n , a f a i t h f u l v e r b a l copy might g i v e t h e s e t B o f v a l u e s i o f a t t r i b u t e " h e i g h t " such as "more t h a n one f o o t , and l e s s t h a n 8 f e e t " , o r i t may g i v e more narrow and i n f o r m a t i v e bounds. I n t h e c a s e o f f u z z y a t t r i b u t e s , t h e problem o f t r u t h i s more c o m p l i c a t e d . T o u s e examples t y p i c a l i n f u z z y s e t t h e o r y , i t may be h a r d t o d e t e r m i n e t h e t r u t h o f d e s c r i p t i o n o f someone as " r a t h e r young" and " f a i r l y t a l l " . Here t h e c o n c e p t o f f a i t h f u l n e s s r e p l a c e s t h a t o f t r u t h , and i t s l e v e l i s measured by t h e members h i p v a l u e s of t h e p e r s o n ' s t r u e age and h e i g h t i n t h e f u z z y s e t s " r a t h e r young" and " f a i r l y t a l l " . Similar s i t u a t i o n e x i s t s with t h e notion o f exactness, which i n t h e n i n f u z z y c a s e c o r r e s p o n d s t o p r e c i s i o n .
FORMAL SEMIOTICS
409
A d e s c r i p t i o n o f someone's h e i g h t as 5'11'' i s p r e c i s e up t o t h e n e a r e s t i n c h ; a d e s c r i p t i o n "Over 6 f e e t t a l l " i s much l e s s p r e c i s e ( t h o u g h b o t h may be f a l s e ) I n a s i m i l a r w a y , i n t h e f u z z y c a s e , one may have more e x a c t and l e s s e x a c t d e s c r i p t i o n s , depending on t h e s u p p o r t s of t h e fuzzy s e t s g i v e n as d e s c r i p t o r s . The e s s e n t i a l f a c t here i s t h a t b o t h f a i t h f u l n e s s and
e x a c t n e s s a r e m u l t i d i m e n s i o n a l c o n c e p t s : a v e r b a l copy may b e f a i t h f u l w i t h r e g a r d s t o some a t t r i b u t e s
o n l y , and i t s e x a c t n e s s may a l s o v a r y from a t t r i b u t e to attribute. To sum u p , t h e c o n c e p t o f f a i t h f u l n e s s , though based on t h e underlying concept of t r u t h , i s not e q u i v a l e n t t o
i t . I t r e p r e s e n t s a m u l t i d i m e n s i o n a l composite t r u t h and i s c l o s e r t o t h e n o t i o n o f v a l i d i t y o f d e s c r i p t i o n . I n t h i s s e n s e , t h e n o t i o n o f f a i t h f u l n e s s may be o f i n t e r e s t not only f o r psychologists, but also t o logici a n s , p h i l o s o p h e r s and s p e c i a l i s t s i n a r t i f i c i a l i n t e l l i g e n c e , as a n example o f forming composite t r u t h valuations. A d d i t i o n a l l y , an i m p o r t a n t consequence i s o b t a i n e d t h r o u g h a n a l y s i s o f v a r i a b i l i t y o f s u c h judgments i n t i m e . A s shown i n C h a p t e r 3, i f one s t u d i e s s u c h v a r i a b i l i t y a c r o s s p o p u l a t i o n , one o b t a i n s i n f o r m a t i o n about a m b i g u i t y and u n c e r t a i n t y / v a g u e n e s s o f t r u t h or f a i t h f u l n e s s . This a d d i t i o n a l information, concerning s t a b i l i t y o f j u d g m e n t s , leads t o some p a r a d o x i c a l c o n c l u s i o n s about some a r e a s o f vagueness o f t r u t h -p a r a d o x i c a l b e c a u s e t r u t h i s u s u a l l y e x p e c t e d t o be accompanied by c l a r i t y . The most i m p o r t a n t , however, i s t h e f a c t t h a t vagueness o f t r u t h t e n d s t o i n c r e e a s e
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t h e v a l i d i t y of vague s t a t e m e n t s . I n d e e d , t h i s was t h e r e a s o n why P y t h i a s i n a n c i e n t Greece were s o successful i n t h e i r predictions. The above c o n s i d e r a t i o n s show a l s o some d i f f i c u l t i e s i n t h e f o u n d a t i o n s of f u z z y s e t t h e o r y of Zadeh. T h i s t h e o r y was i n t r o d u c e d i n o r d e r t o s p e a k p r e c i s e l y about i m p r e c i s i o n , but i t i s e a s y t o show t h a t even more s o p h i s t i c a t e d and c l o s e r t o i n t u i t i o n m a t h e m a t i c a l t o o l s d e s i g n e d f o r d e a l i n g w i t h vagueness c a n n o t a v o i d t h e i r own u n c e r t a i n t y and v a g u e n e s s a r e a , due t o spec i fi c psycho l o g i c a1 mechani sms o f j udgment format i o n and change. T h i s has some p h i l o s o p h i c a l consequences c o n c e r n i n g c o g n i t i v e bounds i n removing u n c e r t a i n t y and v a g u e n e s s from judgments. The l a t t e r phenomenon was h a r d l y n o t i c e d t h u s f a r , s i n c e l i t t l e i n t e r e s t was d i r e c t e d a t s t a b i l i t y and v a r i a b i l i t y o f a s s i g n m e n t s o f v a l u e s o f membership f u n c t i o n s ( s e e , however, Zimmermann and Zysno, 1 9 8 0 ) . The r e s u l t s p r e s e n t e d h e r e p o i n t o u t t o t h e n e c e s s i t y o f c o n c e n t r a t i n g t h e r e s e a r c h n o t o n l y on d e v e l o p i n g fuzzy s e t t h e o r y a s a mathematical d i s c i p l i n e , but a l s o on i t s p s y c h o l o g i c a l f o u n d a t i o n s , and on some immanent p r o p e r t i e s of t h e l a t t e r . O m i t t i n g t h e s e a s p e c t s may i n consequence lower t h e v a l u e o f t h e r e s u l t s o b t a i n e d t h u s f a r . Theory o f f u z z y s e t s i s h e r e a good example o f i n t e r v e n t i o n of p s y c h o l o g i c a l p r o c e s s e s i n t o mathem a t i c a l s t r u c t u r e s b u i l t on some a s s u m p t i o n s about these processes. Now, g i v e n t h e above n o t i o n s o f f a i t h f u l n e s s and e x a c t ness,
i t i s e a s y t o e x p l a i n why t h e vagueness o f
P y t h i a ' s predictions increased t h e i r v a l i d i t y . F i r s t
41 1
FORMAL SEMIOTICS
o f a l l , by g i v i n g t o o f e w , d i v e r g e n t , and f u z z y a t t r i b u t e s , i t s was p o s s i b l e t o f i n d many o b j e c t s t h a t somehow s a t i s f i e d t h e p r e d i c t i o n s . The a d d i t i o n a l e f f e c t o f i n c r e a s e o f v a l i d i t y was due t o vagueness By g i v i n g t o o f e w ( o r none) f u z z y v a l u e s o f a t t r i b u t e s , i t
.
was p o s s i b l e t o i n c r e a s e i t s t i l l f u r t h e r . The f u n c t i o n s o f f a i t h f u l n e s s and e x a c t n e s s a l l o w a l s o a d e f o r m a t i o n o f v e r b a l copy o f t h e o b j e c t , o r a l t e r n a t i v e l y -- some imaginary deforming o p e r a t i o n s on objects. Being e x a c t on some some o t h e r open f o r or c o n v e n t i o n , i s a res (deformation of
a t t r i b u t e s , and l e a v i n g v a l u e s o f complementation due t o knowledge f o u n d a t i o n f o r comics o r c a r i c a t u objects).
A copy may be e x a c t b u t n o t f a i t h f u l , i . e . i t may be
"shifted" with respect t o the f u l s e t o f a t t r i b u t e s does n o t s e t ) . T h i s allows t o introduce t h e s o u r c e o f comic e f f e c t s i n
t r u e state ( i t s faithintersect the basic d e f o r m a t i o n s which a r e p l a y s or n o v e l s .
Another d i s t o r t i o n may be due t o o v e r l y e x t e n s i o n o f f a i t h f u l n e s s s e t , beyond t h e p a r s i m o n i o u s optimum for t h e t r u e s t a t e of a given o b j e c t . T h i s c o n c e n t r a t i o n o f f a i t h f u l n e s s f a c t o r may c a u s e p a r t i a l loss o f e x a c t ness. The i m p o r t a n c e o f e x a c t n e s s l i e s i n t h e f a c t t h a t i t
a l l o w s t o i d e n t i f y ( p o s s i b l y w i t h some e r r o r s ) t h e objects. One c a n speak o f i n e x a c t n e s s o f e x a c t n e s s , i n t h e s e n s e o f v a r i a b i l i t y o f judgments i n t i m e ( u n d e r a s s u m p t i o n of i n v a r i a n c e o f t h e t r u e s t a t e of t h e o b j e c t ) .
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F a i t h f u l n e s s has a l s o i t s v a g u e n e s s , s i n c e f o r a g i v e n f i x e d t r u e s t a t e , t h e f a i t h f u l n e s s s e t may b e t o o
small o r t o o l a r g e w i t h r e s p e c t t o i t s o p t i m a l s i z e , e . g . b e c a u s e o f changes i n l o c a t i o n o f t h e s e s e t s , due t o change i n judgment from o c c a s i o n t o o c c a s i o n . T h i s may l e a d t o a d e c r e a s e o f t h e p o s s i b i l i t y of i d e n t i f i c a t i o n of the object. The n o t i o n o f v e r b a l copy shows how c o m p l i c a t e d c o g n i t i v e l y may be even t h e s i m p l e s t r e l a t i o n s between o b j e c t
and i t s r e p r e s e n t a t i o n , and i n d i c a t e s t h e n e c e s s i t y o f t a k i n g i n t o a c c o u n t ( i n a model o f a g i v e n phenomenon) t h e c o g n i t i v e bounds i n h e r e n t f o r t h i s phenomenon.
5 . FORMAL SEMIOTIC SYSTEMS
5.1. Eepresentations A t t h e e n d , w e s h a l l c o n s i d e r t h e problem o f r e p r e s e n t -
a t i o n of o b j e c t s i n t h e g e n e r a l c a s e , n o t n e c e s s a r i l y b y v e r b a l u t t e r a n c e s ; t h e r e p r e s e n t a t i o n s w i l l be
g e n e r a l l y r e f e r r e d t o as s i g n s . A s opposed t o t h e a n a l y s i s of t h e p r e c e d i n g s e c t i o n s ,
we s h a l l d e a l h e r e w i t h " f r e e " or " u n c o n s t r a i n e d " des c r i p t i o n s . I n o t h e r words, we no l o n g e r have a s i t u a t i o n o f a s e t o f o b j e c t s which a r e a l l " a l i k e " , exc e p t t h a t t h e y may d i f f e r o n l y on some s e t o f w e l l d e f i n e d a t t r i b u t e s . C o n s e q u e n t l y , w e do n o t assume any s p e c i f i c s i m i l a r i t i e s between o b j e c t s which may be described o r represented.
413
FORMAL SEMIOTICS
The o b j e c t s c o n s i d e r e d now w i l l g e n e r a l l y be r e f e r r e d t o a s s i g n s , which c a r r y ( o r convey, r e p r e s e n t ) some meanings. We b e g i n w i t h t h e s i m p l e s t f o r m a l s y s t e m , and t h e n e n r i c h it i n imposing a d d i t i n n a l s t r u c t u r a l constraints
.
Thus, we c o n s i d e r t h e system, c a l l e d a r e p r e s e n t a t i o n system (5.1) where M and Z a r e s e t s of meanings and s i g n s , and R i s a (non-fuzzy) b i n a r y r e l a t i o n which c o n n e c t s s i g n s and meanings. The symbol zRm s i g n i f i e s t h a t " z r e p r e s e n t s m t r , o r "meaning o f s i g n z i s m f ' . The meaning o f a g i v e n s i g n may be n o t u n i q u e , and
also a g i v e n meaning may be r e p r e s e n t e d by v a r i o u s s i g n s . It i s t h e r e f o r e n a t u r a l t o i n t r o d u c e t h e s e t s R ( z ) = ( m E: M : zRmf
(5.2)
and R-l(m) = ( z E Z : zRm')
.
(5.3)
If R ( z ) # PI, t h e s i g n z i s c a l l e d m e a n i n g f u l , o r i n t e r p r e t a b l e : t h e r e e x i s t s a t l e a s t one meaning m which t h i s s i g n r e p r e s e n t s . N e x t , two s i g n s , z and z ' a r e ( p a r t i a l l y ) synonymous, i f R ( z ) n R ( z f ) # 0, i . e . if t h e y have some meanings i n common. We have f u l l synonymy, i f R ( z ) = R ( z ' ) , and hyponymy, i f R ( z ) C R ( z t ) . F i n a l l y , an i m p o r t a n t c a s e o f p a r t i a l
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synonymy i s e q u i p o l l e n c e , R(z')
-
R ( z ) and R ( z )
0
when a l l s e t s R ( z )
-
R(z'),
R(z') a r e nonempty.
We have h e r e t h e f o l l o w i n g t h e o r e m , which c h a r a c t e r i z e s t h e p r o p e r t i e s o f t h e r e l a t i o n s i n t r o d u c e d above. THEOREM. The r e l a t i o n o f f u l l synonymy i s an e q u i v a l -
e n c e , w h i l e p a r t i a l synonymy i s r e f l e x i v e and symmetri c , b u t n o t n e c e s s a r i l y t r a n s i t i v e . Hyponymy i s r e f l e x i v e and t r a n s i t i v e , b u t n o t symmetric, w h i l e e q u i p o l l e n c e i s symmetric, b u t n e i t h e r t r a n s i t i v e n o r r e f l e x - ive
.
L e t u s now c o n s i d e r t h e s e t s R - l ( m )
defined by ( 5 . 3 ) ,
i . e . s e t s o f a l l r e p r e s e n t a t i o n s o f meaning m
.
I n g e n e r a l , t h e meaning m i s c a l l e d r e p r e s e n t a b l e , i f # PI. The r e p r e s e n t a t i o n i s u n i q u e , i f R- 1(m)
R '(m)
i s a singleton.
On t h e o t h e r hand, i f R-'(m) n R - l ( m f ) # B , t h e meanings m and m ' have some common r e p r e s e n t a t i o n s . I n s u c h a c a s e , any s i g n z E R-'(m) r\ R - ' ( m ' ) i s s a i d to be ambiguous, a s i t r e p r e s e n t s b o t h m and m ' . 1
( m ) p R-'(m') = 0, t h e n m and m ' a r e o r t h o g o n a l : any s i g n which r e p r e s e n t s one o f t h e s e meanings, does n o t r e p r e s e n t t h e o t h e r . F i n a l l y , i f R-
5.2. Multiple representations The f i r s t e x t e n s i o n o f t h e s y s t e m ( 5 . 1 ) w i l l i n v o l v e simultaneous a n a l y s i s of several representation sys-
415
FORMAL SEMIOTICS
tems, or e q u i v a l e n t l y , s e v e r a l t y p e s o f s i g n s . Without loss o f g e n e r a l i t y , we r e s t r i c t t h e a t t e n t i o n t o t h e case of n = 2 representations; t h e generalization t o t h e c a s e n > 2 i s immediate. Thus, t h e s y s t e m now have t h e form (5.4) where Z1 and Z 2 a r e two c l a s s e s o f s i g n s , assumed t o be d i s j o i n t , i . e . Z1o Z 2 = 0 . A s b e f o r e , R i < M % Z i , i = 1,2, a r e r e l a t i o n s which c o n n e c t meanings w i t h t h e corresponding s i g n s of a given t y p e . The s e t s R 1 ( z ) , R 2 ( z ) , RT1(m) and R;'(m) are d e f i n e d a s i n ( 5 . 2 ) and ( 5 . 3 ) , and t h e c o n c e p t s o f synonymy, p a r t i a l and f u l l , e t c . c a r r y o v e r t o e a c h o f t h e s e t s
Z1 and 2 2 s e p a r a t e l y . However, t h e synonymy may a l s o c o n c e r n s i g n s o f v a r i o u s t y p e s , i . e . p a i r s of s i g n s z E Z1 and z ' C Z2. O f p a r t i c u l a r importance h e r e i s e q u i p o l l e n c e : t h e u s e of s i g n s z Z1 and z' Z2 which a r e e q u i p o l l e n t , leads t o s h a r p e n i n g o f meaning, n a r r o w i n g i t down t o t h e i n t e r s e c t i o n R 1 ( z )
n R2(z1).
C o n s i d e r now t h e s e t s R;l(m) and R;'(m), contained r e s p e c t i v e l y i n Z and Z2. L e t u s n o t e t h e f o l l o w i n g 1 theorem.
and z 2 C R ; l ( m ) , z1 C R;'(m) and z a r e ( p a r t i a l l y ) synonymous. 12
THEOREM. z
then the signs
An i m p o r t a n t s p e c i a l c a s e i s c o v e r e d h e r e by t h e folowi n g d e f i n i t i o n , which c o n c e r n s t h e e x p r e s s i n v e power of s i g n s .
CHAPTER 4
416
DEFINITION. Suppose t h a t t h e s e t s R;l(m)
and R ; l ( m )
a r e nonempty, and s u c h t h a t t h e c o n d i t i o n
h o l d s . Then t h e r e p r e s e n t a t i o n system CM, Z1, Rl> i s s a i d t o have more e x p r e s s i v e power f o r m t h a n t h e system (M,
Z2,
R2).
The term i s j u s t i f i e d by t h e f a c t t h a t i f ( 5 . 5 ) h o l d s , t h e n e a c h s i g n z1 which r e p r e s e n t s m i n t h e f i r s t system does s o w i t h l e s s a m b i g u i t y (more p r e c i s i o n ) t h a n a s i g n z 2 i n t h e second s y s t e m .
...,
I n t h e g e n e r a l c a s e , we have n d i s j o i n t s e t s , Z1, Z o f t a x o n o m i c a l t y p e s o f s i g n s , and t h e c o r r e s p o n d n R n . We may t h e n r e g a r d Z1, ..., i n g r e l a t i o n s R1, Z as a p a r i t i t i o n o f t h e s e t n
...,
z
=
z 1 i) z 2
u
,..
zn,
and c o n s i d e r one r e l a t i o n R , b e i n g t h e union o f t h e r e l a t i o n s Ri. Given two p a r t i t i o n s , s a y F and F ' , o f t h e same s e t Z we may d e f i n e t h e i r prods a y , i n t o s e t s Z i and Z;, u c t F n F' as F 0 F' = f Z i
A
T o d e f i n e t h e union
Z'
j'
i = 1
,..., n ,
j = 1
,..., m].
F y F' o f two p a r t i t i o n s , i t i s
most c o n v e n i e n t t o u s e t h e e q u i v a l e n c e r e l a t i o n s c o r r e s p o n d i n g t o F and F ' . Any p a r t i t i o n d e f i n e s as e q u i v a l e n c e r e l a t i o n , two e l e m e n t s b e i n g e q u i v a l e n t ,
41 7
FORMAL SEMIOTICS
i f t h e y a r e i n t h e same s e t o f p a r t i t i o n . L e t t h e s e
- ' . Then t h e r e l a t i o n - F u F t , r e l a t i o n s be -and to be d e n o t e d s i m p l y by .-", i s d e f i n e d as t h e t r a n s i t i v e e x t e n s i o n o f t h e sum o f r e l a t i o n s and - ' , i . e . z rv" z' i f
-
3 zl,...,z t : z z1
-'
-
z
1
z2 and..,
or z and z
-'
-
z
1
and z1
z or z t
N
z2 or z'.
We have t h e n t h e f o l l o w i n g theorem. THEOREM. For a l l p a r t i t i o n s F ,
N e e d l e s s to s a y , b o t h
\J
and n
F' we have
may be d e f i n e d for
more t h a n two t e r m s . While i n t e r s e c t i o n of two p a r t i t i o n s i s a familiar operation, leading t o a f i n e r p a r t i t i o n , obtained by subdividing t h e c a t e g o r i e s , t h e o p e r a t i o n o f union o f p a r t i t i o n s a p p e a r s , a t f i r s t , somewhat a r t i f i c i a l and o f l i t t l e p r a c t i c a l u s e . Howe v e r , t h i s o p e r a t i o n i s of primary i m p o r t a n c e for des c r i b i n g t h e c o g n i t i v e p r o c e s s e s which i n v o l v e a s s o c i a t i o n s . One can namely a r g u e t h a t i f two o b j e c t s ( s i g n s , e t c . ) a r e a s s o c i a t e d i n o n e ' s mind, t h e n t h e y must b e l o n g t o t h e same c l a s s i n a p a r t i t i o n F u F 1 u F". for s u i t a b l e p a r t i t i o n s F , F', F",.. .
.
5.3. Fuzzy meanings L e t u s now i n t r o d u c e a n e x t e n s i o n o f t h e s y s t e m , con-
s i s t i n g o f a f u z z i f i c a t i o n o f t h e r e l a t i o n R , and o f
..
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418
a d d i t i o n o f temporal a s p e c t s . For s i m p l i c i t y , we r e t u r n h e r e t o t h e c a s e o f one r e presentation set Z . Thus, we c o n s i d e r t h e system
where now T i s t h e s e t of c o n s i d e r e d t i m e moments, and e a c h Rt i s a f u z z y r e l a t i o n i n M x Z , i . e . a function Rt:
M
A
Z
-+
[O,ll
.
(5.7)
Here t h e v a l u e R ( m , z ) i s t h e d e g r e e t o which z i s a t r e p r e s e n t a t i o n o f m a t t i m e t. It i s now p o s s i b l e t o c o n s i d e r t h r e e f a m i l i e s o f f u z z y s e t s , o b t a i n e d by t r e a t i n g t h e f u n c t i o n ( 5 . 7 ) as de-
p e n d i n g on one v a r i a b l e o n l y . Thus, for any f i x e d t E T and m t M y t h e f u n c t i o n R t ( m , - ) may be i n t e r p r e t e d a s a membership f u n c t i o n o f a s u b s e t o f Z , namely o f o f a l l r e p r e s e n t a t i o n s o f m a t t. We s h a l l s a y t h a t m i s f u l l y r e p r e s e n t a b l e a t t , i f
sup R t ( m , z ) ztz
(5.8)
= 1
i . e . if t h e s e t o f r e p r e s e n t a t i o n s o f m a t t i s norma l . I n g e n e r a l , t h e number d (m) = 1
t
-
sup
Rt(m,z)
(5.9)
Z E Z
may b e c a l l e d t h e r e p r e s e n t a b i l i t y d e f i c i t o f m a t t .
FORMAL SEMIOTICS
419
N e x t , one may c o n s i d e r R (m,z) as a f u n c t i o n o f m , for t f i x e d t and z . I t t h e n r e p r e s e n t s a fuzzy s e t o f meani n g s a s s o c i a t e d w i t h s i g n z a t time t , i . e . t h e s e t of a l l meanings f o r which z s e r v e s as r e p r e s e n t a t i o n . Again, i f sup
Rt(m,z)
= 1,
m t M
t h e n z may b e c a l l e d normal: t h e r e i s a t l e a s t one meaning which z r e p r e s e n t s i n f u l l d e g r e e . The most i n t e r e s t i n g c o n s t r u c t i o n i s o b t a i n e d when m and z a r e f i x e d , and one c o n s i d e r s R t ( m , z ) as a funct i o n o f t . T h i s g i v e s a f u z z y s e t o f moments, or fuzzy p e r i o d , when z r e p r e s e n t s m, o r a t e m p o r a l t r a c e o f r e p r e s e n t a t i o n o f m by z . We s h a l l s a y t h a t meaning m o f s i g n z e v o l v e s r e g u l a r l y , i f f o r a l l t l c t 2 < t we have
3
T h i s means t h a t t h e f u n c t i o n Rt(m,z) has a t most one
peak. A t y p i c a l behaviour i s t h e r e f o r e such t h a t t h e
d e g r e e t o which z r e p r e s e n t s m f i r s t i n c r e a s e s , and s t a r t s t o d e c r e a s e a f t e r r e a c h i n g t h e maximum. T h i s may be e x e m p l i f i e d by v a r i o u s s i g n s which g r a d u a l l y a c q u i r e r e c o g n i t i o n and e n t e r t h e u s a g e , and subseque n t l y " f a d e " ( e . g . become l e s s f a s h i o n a b l e ) . A c a s e i n p o i n t h e r e may be v a r i o u s s l a n g e x p r e s s i o n s . We have h e r e t h e f o l l o w i n g theorem THEOREM. The c l a s s o f a l l meanings which e v o l v e regul a r l y i s c l o s e d under c o n j u n c t i o n o f meanings.
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For t h e p r o o f , s e e Nowakowska ( 1 9 8 3 a ) .
5.4. R e l a t i o n s h i p s between s t r u c t u r e s I n t h i s and p r e c e d i n g s e c t i o n , t h e b a s i c c o n c e p t u a l scheme c o n s i s t e d of two s e t s o f o b j e c t s o f some k i n d ( o b j e c t s b e i n g r e p r e s e n t e d , and o b j e c t s r e p r e s e n t i n g t h e m ) , and t h e r e l a t i o n o f r e p r e s e n t a t i o n . T h i s sche-
m e i s i n t e r p r e t a b l e i n many w a y s , and t h e c h o i c e of assumptions about t h e o b j e c t s of b o t h s e t s was c o n s t r a i n e d by t h e intended i n t e r p r e t a t i o n . Thus, i n S e c t i o n
4, w e c o n s i d e r e d o b j e c t s which are
r e p r e s e n t e d , as b e l o n g i n g t o some common c l a s s ( e . g . b o o k s ) , and o b j e c t s r e p r e s e n t i n g them were d e s c r i p t i o n s , v e r b a l , n u m e r i c a l , e t c . I n t h i s c a s e , i t was n a t u r a l t o c o n s i d e r t h e s t r u c t u r e o f t h e s e t o f o b j e c t s as induced b y t h e g o a l , and by t h e v a l u e s of a t t r i b u t e s . Another c a s e o c c u r s i n s e m i o t i c s , where o b j e c t s r e p r e s e n t e d a r e meanings, and o b j e c t s which r e p r e s e n t them are signs. S t i l l a n o t h e r p o s s i b i l i t y i s when o b j e c t s b e i n g r e p r e s e n t e d a r e phenomena, and o b j e c t s r e p r e s e n t i n g them a r e t h e i r models or t h e o r i e s . I n p a r t i c u l a r , r e l a t i v e l y s i m p l e s i t u a t i o n s of t h i s k i n d a r e measurement prob-
lems, t o be d i s c u s s e d i n d e t a i l i n n e x t c h a p t e r .
It ought t o b e c l e a r t h a t t h e s t r u c t u r e o f o b j e c t s r e p r e s e n t e d and s t r u c t u r e o f t h e i r r e p r e s e n t a t i o n s , may d i f f e r w i d e l y , depending on t h e i n t e n d e d i n t e r -
pretation.
FORMAL SEMIOTICS
42 1
We s h a l l now d e s c r i b e t h e s i t u a t i o n i n g e n e r a l c a s e ,
r e t a i n i n g t h e symbols M and 2 f o r t h e c l a s s o f o b j e c t s r e p r e s e n t e d and f o r t h e c l a s s o f t h e i r r e p r e s e n t a t i o n s , s i n c e o u r main i n t e n t i o n i s t o e x p l o r e a p p l i c a t i o n s to s e m i o t i c s , where e l e m e n t s o f M a r e meanings, and e l e ments o f Z a r e s i g n s . It ought t o be k e p t i n mind, however, t h a t t h i s i s n o t t h e o n l y p o s s i b l e i n t e r p r e t a tion.
5.4.1. S t r u c t u r e o f r e p r e s e n t e d o b j e c t s . G e n e r a l l y , i t i s n e c e s s a r y t o t a k e i n t o account t h e p o s s i b i l i t y t h a t :
-- t h e s e t M h a s some s t r u c t u r e , as d e s c r i b e d
by var-
i o u s r e l a t i o n s between o b j e c t s ;
--
t h e e l e m e n t s of t h e s e t m may p o s s e s s some i n t e r n -
al structure. A s reagards .the first p o s s i b i l i t y , a c e r t a i n s t r u c t u r e
of t h e s e t o f o b j e c t s was i n t r o d u c e d i n S e c t i o n 4 , namely a n e q u i v a l e n c e ., d e s c r i b i n g " i d e n t i t y from t h e p o i n t o f view o f t h e g o a l " , and r e l a t i o n s i n d u c e d by a t t r i b u t e v a l u e s . However, i n t h i s c a s e , no i n t e r n a l s t r u c t u r e o f o b j e c t s was t a k e n i n t o c o n s i d e r a t i o n . A g e n e r a l d e s c r i p t i o n would i n v o l v e a r e l a t i o n a l s t r u c -
t u r e o f t h e form
where i n t u r n e a c h U(m) i s a system
.,
Qmj>
(5.12)
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A s b e f o r e , M i s a s e t o f o b j e c t s under c o n s i d e r a t i o n , S . a r e some r e l a t i o n s on M ( b i n a r y , t e r n a r y , e t c . ) , 1
and U ( m ) i s a system which d e s c r i b e s t h e i n t e r n a l s t r u c t u r e of o b j e c t m. Here Km i s a s e t o f " c o n s t i t u e n t p a r t s " o f m, w h i l e Q m l , , Q m j a r e some r e l a t i o n s between t h e p a r t s o f m, t h e i r a t t r i b u t e s , e t c .
.. .
The c h o i c e of s p e c i f i c r e l a t i o n s , t h e i r c h a r a c t e r ,
--
e t c . , depends
as a l r e a d y s t r e s s e d
--
on t h e i n t e n d -
ed i n t e r p r e t a t i o n . We g i v e below one example o f such r e l a t i o n s , c o n c e r n i n g t h e c a s e when e l e m e n t s o f m are meanings o r , more g e n e r a l l y , c o n c e p t s , composed o u t of e l e m e n t a r y c o n c e p t s (sememes). I n t u i t i v e l y , we have h e r e t h r e e k i n d s o f r e l a t i o n s : one of them a l l o w s t o o r d e r ( p a r t i a l l y ) t h e meanings
as t o t h e i r
l e v e l s of g e n e r a l i t y ; a n o t h e r c o n c e r n s
t h e meanings on t h e same g e n e r a l i t y l e v e l , and d e s c r i b e s t h e s i t u a t i o n t h a t t h e s e meanings may b e " c l o s e r " o r " f a r t h e r a p a r t " . F i n a l l y , t h e t h i r d concerns t h e meanings ( o r : c o n c e p t s ) on d i f f e r e n t l e v e l s of gener a l i t y , and r e f l e c t s t h e f a c t t h a t t h e l e s s g e n e r a l one may be a c o n s t i t u e n t p a r t o f t h e more g e n e r a l o n e . A c c o r d i n g l y , we c o n s i d e r a t r i p l e t of r e l a t i o n s
&Y
t , P)'
(5.13)
Here g i s a b i n a r y r e l a t i o n on M , assumed t o b e r e f -
l e x i v e and t r a n s i t i v e ; t h e symbol mgm' means t h a t m i s l e s s g e n e r a l ( i n t h e s e n s e o f weak i n e q u a l i t y ) t h a n m'. N a t u r a l l y , some meanings may remain incomparable according t o g.
423
FORMAL SEMIOTICS
I f mgm' and m ' g m ,
we s h a l l w r i t e m = m ' , and s a y g t h a t t h e s e meanings a r e on t h e same l e v e l o f g e n e r a l ity.
N e x t , t i s a q u a r t e r n a r y r e l a t i o n on M , h o l d i n g b e t ween q u a d r u p l e t s which a r e a l l on t h e same l e v e l o f (ml,m2)t(m3,rn4) i s t o be i n t e r g e n e r a l i t y . Formally, p r e t e d t h a t ml and m 2 a r e c l o s e r one t o a n o t h e r t h a n rn3 and m 4 . T h i s r e l a t i o n i s assumed t o be t r a n s i t i v e among t h e p a i r s o f e l e m e n t s o f m , and a l s o s a t i s f y t h e following conditions :
and f o r a l l m,ml,m2
F i n a l l y , t h e r e l a t i o n t i s t o concern only t h e pairs on t h e same g e n e r a l i t y l e v e l , which may be e x p r e s s e d as
The f i r s t two c o n d i t i o n s a s s e r t t h e
symmetry o f
r e l a t i o n t w i t h r e s p e c t t o each p a i r , w h i l e t h e t h i r d a s s e r t s t h a t t h a t t h e " c l o s e n e s s " o f m and m i t s e l f i s t h e highest closeness possible. F i n a l l y , p i s t h e r e l a t i o n " b e i n g a p a r t of'", which h o l d s between some e l e m e n t s o f M which a r e on d i f f e r e n t g e n e r a l i t y l e v e l s . A c c o r d i n g l s , i t i s assumed t h a t
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p i s r e f l e x i v e and t r a n s i t i v e on M , and t h a t whenever
mpm' ( m i s a p a r t o f m f ) , t h e n a l s o rngm': mpm'
+
mgm'.
T h i s means simply t h a t t h e r e , l a t i o n s p and g s a t i s f y
the condition p
c
g.
Thus, under t h e above s t r u c t u r e , e l e m e n t s o f M have h i e r a r c h i c a l c h a r a c t e r , r e p r e s e n t a b l e as a t r e e . Given t h e n o t i o n of c l o s e n e s s , as r e p r e s e n t e d by r e l a t i o n t , one can d e s c r i b e s u c h t r e e s of c o n c e p t s (or meanings) i n t e r m s o f v a r i o u s p a r a m e t e r s , s u c h as w i d t h or depth. Semantic w i d t h may be d e f i n e d as t h e d i a m e t e r o f a floor o f t h e t r e e o f h i e r a r c h y , w h i l e s e m a n t i c d e p t h i s t h e number of floors. It may be p o s t u l a t e d t h a t t h e c o n c e p t s on h i g h e r g e n e r a l i t y l e v e l c o r r e s p o n d t o c o n s t r u c t s , while o t h e r s correspond t o o b j e c t s t h a t are d i r e c t l y observable. If t h e s e c o n s t r u c t s and n o t i o n s a r e of modal t y p e , and
t h e i r i n t e n s i t i e s may be r e p r e s e n t e d i n n u m e r i c a l domain, t h e n t h e f i r s t c a t e g o r y i s a n o b j e c t o f i n t e r e s t o f t e s t t h e o r y , w h i l e t h e second -- o f measurement t h e o r y . A s rega rds numerical r e p r e s e n t a t i o n ,
i t i s merely one
o f many p o s s i b l e i n s e m i o t i c s .
Finally, s t i l l another possible s t r u c t u r e occurs i f we have some d e p e n d e n c i e s between e l e m e n t s of M . Such d e p e n d e n c i e s may be d e s c r i b e d i n v a r i o u s w a y s . Thus,
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FORMAL SEMIOTICS
one may have a mapping, s a y T : M 4 M y w i t h m 1 = T ( m ) b e i n g i n t e r p r e t e d t h a t m 1 depends on m . The f u n c t i o n T may depend on s e v e r a l a r g u m e n t s , s o t h a t m 1 = T(ml,...,mk),
m 1 on m 1 >
'* *
which means a m u l t i p l e dependence o f >mk'
The dependence may a l s o be o f a s t o c h a s t i c c h a r a c t e r ,
as b e s t e x e m p l i f i e d by some n a t u r a l phenomena. E l e ments o f M a r e t h e n o b s e r v a b l e v a r i a b l e s ( o r , more p r e c i s e l y , t h e i r l a b e l s ) , and a r e l a t i o n s h i p between a p a i r ml = X and m2 = Y i s r e p r e s e n t a b l e as a j o i n t d i s t r i b u t i o n o f t h e s e v a r i a b l e s . Thus, t h i s r e l a t i o n s h i p h a s t h e form of t h e j o i n t d i s t r i b u t i o n f u n c t i o n F ( x , y ) = P(X 4 x , Y 4 y ) . O c c a s i o n a l l y , i t makes s e n s e t o r e p r e s e n t such a dependence
numerically, e.g
as a c o r r e l a t i o n c o e f f i c i e n t between X and Y , e t c .
Homomorphisms. G e n e r a l l y , t h e e l e m e n t s o f t h e s e t Z (which a r e used t o r e p r e s e n t o b j e c t s i n M ) , a r e a l s o r e l a t i o n a l s t r u c t u r e s o f some k i n d . Moreover, t h e r e a r e a l s o r e l a t i o n s on t h e s e t Z . The n a t u r e o f
5.4.2.
t h e s e r e l a t i o n s i s , t o a l a r g e e x t e n t , similar t o t h a t of t h e c o r r e s p o n d i n g r e l a t i o n s on M . The above s e n t e n c e r e q u i r e s " s i m i l a r i t y " of M and 2 on " c o r r e s ponding" r e l a t i o n s . When p u t i n a p r e c i s e form, t h i s means e x i s t e n c e o f a homomorphism between t h e s t r u c t u r es. Suppose t h a t t h e s t r u c t u r e o f 2 i s r e p r e s e n t a b l e a s
where
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w i t h s i m i l a r i n t e r p r e t a t i o n as i n t h e c a s e ( 5 . 1 1 ) and
( 5 . 1 2 ) for t h e s t r u c t u r e o f M.
For s i m p l i c i t y o f p r e s e n t a t i o n , suppose t h a t a l l r e l a t i o n s S i , S!1 a r e b i n a r y .
DEFINITION. If t h e r e e x i s t s a f u n c t i o n Z -?M
h:
s u c h t h a t for p a i r s lations , z1S;.z2
implies
J
(Si ,Si ) , 1
1
...,
( S i k y S i k ) of re-
h(zl)Si h(z2) j
for j = l , . . . , k , t h e n w e s a y t h a t h e s t a b l i s h e s a homomorphism o f Z and M w i t h r e s p e c t t o r e l a t i o n s Si . j
T h i s homomorphism c o n c e r n s m u l t i o b j e c t s and m u l t i s i g n s ,
i n t h e s e n s e o f t h e r e l a t i o n s between s i g n s and t h e c o r r e s p o n d i n g r e l a t i o n s between meanings o f t h e s e signs , without e n t e r i n g i n t o t h e i n t e r n a l s t r u c t u r e s o f s i g n s and meanings. A s a n example, one may t h i n k h e r e o f meanings of s i g n s
composed o u t of two or more s i g n s .
,..
1
.,Si r e p r e s e n t s t h e t y p e , or Here t h e s e t {Si t r a c e o f hornomordhism, akd i s c a l l e d t h e p r o j e c t i o n s e t . I n o t h e r words, t h i s s e t c o n t a i n s t h e i n f o r m a t i o n about t h e r e l a t i o n s which a r e p r e s e r v e d under homomorphism.
FORMAL SEMIOTICS
427
DEFINITION. Let h b e a f u n c t i o n which e s t a b l i s h e s a homomorphism between Z and M . I f t h e r e e x i s t s a f u n c t i o n
XlQA(Z)
,i.x2 J
implies
u(x1)QZyi ~ ( x , ) , j
t h e n we s a y t h a t homomorphism h i s s t r u c t u r a l l y i c o n i c a t element z . Thus, a s t r u c t u r a l l y i c o n i c homomorphism i s such t h a t i t n o t o n l y r e p r e s e n t s t h e r e l a t i o n s between o b j e c t s , but also represents t h e i r i n t e r n a l structure, o r at l e a s t some o f i t s f e a t u r e s . The s e t Q i l , . . . , Q Z i c o n t a i n s t h e i n f o r m a t i o n about t h e r e l a t i o n s o f t h e o b j e c t which a r e r e p r e s e n t e d , and i s c a l l e d t h e s e l e c t ion s e t . I n p a r t i c u l a r , suppose t h a t t h e r e a r e f u n c t i o n a l dep e n d e n c i e s between e l e m e n t s o f M , and t h a t t h e y a r e p r e s e r v e d under homomorphism h. T h i s means t h a t t h e r e such t h a t i f T 1 ( z ) = e x i s t s a f u n c t i o n T 1 : Z -+Z, z ’ , then also T[h(z)l = h ( z l ) . I f f u n c t i o n T corresponds t o a c a u s a l r e l a t i o n connect-
i n g p a i r s ( m , T ( m ) ) , t h e n homomorphism h i s s a i d t o be i n d e x i n g , and t h e s i g n s z a r e i n d e x e s o f t h e m’s c o r r e s p o n d i n g t o them under f u n c t i o n h . On t h e o t h e r hand, i f t h e f u n c t i o n h p r e s e r v e s no s t r u c t u r a l a s p e c t s o f e l e m e n t s o f M , and no m u t u a l r e l a t i o n s between them, we have a p u r e l y symbolic re-
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presentation. To g i v e some examples, a p h o t o g r a p h p r e s e r v e s a t l e a s t
p a r t o f t h e s t r u c t u r e o f t h e o b j e c t , and a l s o m u t u a l r e l a t i o n s between them. On t h e o t h e r hand, a map shows mutual r e l a t i o n s , e . g . between towns, b u t g i v e s no i n f o r m a t i o n about t h e s t r u c t u r a l p r o p e r t i e s o f t h e towns. These examples show t h a t s i g n s may o r may n o t be i c o n i c
, and
y e t c o n s t i t u t e a good r e p r e s e n t a t i o n .
S i m i l a r l y , r e p r e s e n t a t i o n of meanings t h r o u g h words, i s p u r e l y s y m b o l i c : e . g . t h e words " s h o r t " and " l o n g " do n o t r e f l e c t t h e r e l a t i o n s between meanings by any r e c o g n i z a b l e f e a t u r e s , s u c h as number o f l e t t e r s , e t c .
5 . 5 . S i g n s and i n f o r m a t i o n L e t u s o b s e r v e f i r s t t h a t t h e above c o n s i d e r a t i o n s can immediately e x t e n d e d t o t h e c a s e o f s e v e r a l t y p e s o f s i g n s , hence t o m u l t i - t y p e r e p r e s e n t a t i o n s , as it was done i n S e c t i o n 4 , by c o n s i d e r i n g p a r t i t i o n s Z = Z1 U U Zn i n t o sign-types. A r e p r e s e n t a t i o n
...
o f a meaning i n form o f a s i g n may be u n i - t y p e , o r homogeneous, i f i t h a s t h e form o f a s i n g l e element o f one o f t h e s e t i n t h e p a r t i t i o n . I n g e n e r a l , i t may be a v e c t o r o f t h e form ( z ~ ,z 2 , . . , z n ) , where e a c h z i i s e i t h e r a n element o f Z i , o r z = 6 , t o i symbolize t h e f a c t t h a t s i g n s o f i - t h t y p e a r e n o t
.
used. G e n e r a l l y , s u c h composite s i g n s are some s t r u c t u r e s , i n which one s i g n a p p e a r s i n a c o n t e x t of other signs, of d i f f e r e n t types. I n a sense, t h e t h e o r y here i s p a r a l l e l t o t h e t h e o r y of multimedia1 communication, and t h e n o t i o n s of s u p p o r t i n g , i n h i b i t -
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FORMAL SEMIOTICS
ion, cancelling, e t c . carry over t o the present case. It i s n o t w o r t h w h i l e t o r e p e a t t h e d e f i n i t i o n s o f t h e s e n o t i o n s ; it i s s u f f i c i e n t t o r e c a l l t h a t t h e s e d e f i n i t i o n s were o b t a i n e d by c o n s i d e r i n g t h e meanings of 2 s i g n , s a y z Zi, i n context of s i g n s z i 1'"" Z Z , z n , by comparing i t w i t h t h e meaning i-1' of zi = 6 i n t h e same c o n t e x t . I n s h o r t , one c o n s i d e r s t h e e f f e c t o f r e p l a c i n g t h e s i g n z by "no-sign" 6 . i
..
Now, s i g n s p r o v i d e i n f o r m a t i o n about meanings, a l t h o u g h t h e i n f o r m a t i o n e l i c i t e d by t h e o b s e r v e r ( u n d e r t h e o b s e r v a b i l i t y c o n s t r a i n t s ) need n o t c o i n c i d e w i t h t h e i n f o r m a t i o n i n t e n d e d by t h e p e r s o n who produced t h e sign. Consider f o r s i m p l i c i t y a single- type s y s t e m , w i t h nonfuzzy r e l a t i o n R , i . e . a s y s t e m ( M , Z , R ) . F o r any s i g n z w e c o n s i d e r t h e n t h e set R ( z ) o f a l l i t s meani n g s , i . e . t h e s e t o f a l l meanings which z r e p r e s e n t s . The s e t R ( z ) may be c a l l e d t h e s e t o f a l l ttSharpll meanings o f z . Suppose t h a t t h e i n t e n d e d meaning was m 0 (2 M , V a r i o u s c o n c e p t s may now be d e f i n e d t h r o u g h and t h e t h e p r o p e r t i e s o f s e t s R(z), l o c a t i o n o f m 0' e q u i v a l e n c e r e l a t i o n IV i n M , which p a r t i t i o n s M i n t o e q u i v a l e n c e c l a s s e s , t o b e d e n o t e d by [ m ] . F i r s t l y , i t i s c l e a r t h a t s i g n z c o n t a i n s more informR(zl). a t i o n , o r i s more i n f o r m a t i v e t h a n z ' , i f R(z) T h i s means s i m p l y t h a t z has s m a l l e r s e t o f p o s s i b l e meanings which may be a t t a c h e d t o i t .
c
If
R(z) c [m] for some meaning m , t h e n we may s a y
t h a t sign z i s s u f f i c i e n t for meaning m s T h i s condit i o n means simply t h a t z c a r r i e s i n f o r m a t i o n which
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may n o t a l l o w t o i d e n t i f y t h e meaning u n i q u e l y , but
a l l meanings m which are i n R ( z ) s a t i s f y t h e c o n d i t i o n
t h a t t h e y a r e e q u i v a l e n t from t h e p o i n t of view o f the goal. A s i g n may be s u f f i c i e n t f o r meaning m , b u t t h i s does
n o t imply t h a t i t s u s e for conveying i n f o r m a t i o n m 0 i s s e n s i b l e . G e n e r a l l y , w e may s a y t h a t z i s f e a s i b l e f o r m o , i f mo G R ( z ) . If R ( z ) C [m,], then z i s adequate f o r m 0' I n most c a s e s , t h e s e t R ( z ) i n t e r s e c t s w i t h a number
[ m ] . Given a s i g n z , i t i s therefore n a t u r a l t o distinguish three classes of meanings: t h o s e which a r e c o n t a i n e d i n R ( z ) t o g e t h e r w i t h t h e i r whole e q u i v a l e n c e c l a s s , t h o s e which a r e e x c l u d e d from R ( z ) , and t h o s e which a r e n e i t h e r cont a i n e d , n o r e x c l u d e d . Thus, w e may p u t of equivalence c l a s s e s
We may to the ion t o t o any
t h e n s a y ' t h a t A ( z ) i s t h e lower a p p r o x i m a t i o n meaning z , w h i l e B ( z ) i s t h e upper approximati.t. The i n t e n d e d meaning, s a y m , may b e l o n g 0 of t h e s e t h r e e c l a s s e s .
I n f u z z y c a s e , when t h e r e l a t i o n
R i s described i n
terms of i t s c h a r a c t e r i s t i c f u n c t i o n Rt ( p o s s i b l y depending on time t , as i n s e c t i o n 5 . 3 ) , we o b t a i n t h e membership f u n c t i o n R t ( z , m ) , e x p r e s s i n g t h e d e g r e e
43 1
FORMAL SEMIOTICS
t o which t h e s i g n z r e p r e s e n t s meaning m . I t i s n o t p o s s i b l e t o d e f i n e s e t s A ( z ) , B ( z ) and C ( z ) , b u t t h e upper and lower a p p r o x i m a t i o n of meaning o f z may be d e f i n e d ( f o r time t ) , as t h e k e r n e l and c a r r i e r o f t h e membership f u n c t i o n R t , s o t h a t we may now p u t A(z) =
im:
Rt(m,z)
= 17
Thus, i n t h i s c a s e , t h e s e t C ( z ) of e x c l u d e d meanings w i l l be t h e s e t C ( z ) = {m: R t ( m , z ) = O f
.
G e n e r a l l y , a meaning m may be c a l l e d e x p r e s s i b l e , or d e s c r i b a b l e , i f t h e r e e x i s t s a s i g n z ( p e r h a p s compos i t e ) , such t h a t A(z)
= B ( z ) = [m]
( h e r e t h e d e f i n i t i o n i s t h e same i n t h e fuzzy and nonf u z z y c a s e , s o t h a t t h e s e t s A ( z ) and B ( z ) may be i n t e r p r e t e d a c c o r d i n g t o e i t h e r o f t h e two f o r m u l a s above). Somewhat more g e n e r a l l y , we s a y t h a t meaning m i s k-describable, (or: k-expressible, k-definable), i f t h e r e e x i s t s a s i g n z such t h a t [ m l R ( z ) and moreov e r , R ( z ) i s a union of e x a c t l y k e q u i v a l e n c e c l a s s e s o f some meanings. We have h e r e t h e f o l l o w i n g theorem. THEOREM. If meaning m i s k - d e f i n a b l e ,
a s i g n z and meanings ml,m2,
... 'mk-l
then t h e r e e x i s t s such t h a t
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m i d mj,
i , j = 1,.
..,k-1,
i # j,
and k- 1
u
~ ( z = ) C ~ I Y
[mil.
i=l
We may s a y t h a t ( u n d e r z ) t h e meaning m i s confounded w i t h k-1 o t h e r non-equivalent
meanings. T h u s , k - d e f i n a b i l i t y means simply confounding w i t h k - 1 o t h e r meanings. I f meaning rn i s k - d e f i n a b l e ,
and i s n o t ( k = l ) - d e f i n a b le, t h e n we s a y t h a t m i s s t r i c t l y k - d e f i n a b l e .
L e t D d e n o t e t h e c l a s s of a l l meanings rn which a r e k - d e f i n a b l e f o r some k . D E F I N I T I O N . The s i g n s z 1 , z 2
a r e s a i d t o obey t h e mean-
i n g t r u n c a t i o n law, if
Thus, under meaning t r u n c a t i o n l a w , a composite s i g n c a r r i e s o n l y t h o s e meanings which a r e common t o b o t h s i g n s ( n a t u r a l l y , t h i s i s a r a r e and r a t h e r s p e c i a l c a s e , s i n c e i n most i n s t a n c e s when two s i g n s a r e composed, a new meaning a r i s e s ) . However, we have
THEOREM. g n d e r meaning t r u n c a t i o n l a w , t h e c l a s s D
is
c l o s e d under a l l s e t - t h e o r e t i c a l o p e r a t i o n s , i . e . u n i o n s ; i n t e r s e c t i o n s and c o m p l e t e n t a t i o n . The assumption o f t h i s theorem, namely t h e t r u n c a t i c n law, may be e x p e c t e d t o h o l d i n c a s e when meanings
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a r e r e p r e s e n t e d as d e s c r i p t i o n s i n t e r m s o f a t t r i b u t e s , i . e . f o r s p e c i a l category of s i g n s . I n such a c a s e , e a c h new i t e m o f i n f o r m a t i o n ( d e s c r i p t o r ) narrows down t h e s e t o f o b j e c t s which a r e r e p r e s e n t e d b y t h e composite sign. On t h e o t h e r hand, i n a r t , t h e assumption o f t r u n c a t i o n o f meaning does n o t o c c u r -- on t h e c o n t r a r y , we have a p p e a r e n c e s of new meanings, due t o c o m b i n a t i o n s of s i g n s . We s a y t h a t s e m i o t i c system (or: i n f o r m a t i o n s y s t e m ) i s k - s e l e c t i v e , i f any meaning i s k - d e f i n a b l e . A s i g n z which meets t h e r e q u i r e m e n t s o f d e f i n a b i l i t y f o r meani n g rn w i l l b e c a l l e d a c h a r a c t e r i s t i c s i g n f o r m. I n c a s e when meaning m i s n o t k - d e f i n a b l e f o r any k , we have t o c o n s i d e r t h e upper and lower a p p r o x i m a t i o n s , as d e f i n e d above. The i n f o r m a t i o n c o n t a i n e d i n any s i g n w i l l t h e n b e ambiguous, i n t h e s e n s e t h a t i t a l w a y s g i v e s a t l e a s t two e q u i v a l e n t meanings, o f which
one i s e x c l u d e d and t h e o t h e r i s n o t . Imagine now t h a t t h e d e s c r i p t i o n , i n form o f a t t r i b u t e v a l u e s , is g i v e n s e q u e n t i a l l y , and l e t R1 '> R2 3
...
be t h e s u c c e s s i v e i n f o r m a t i o n s e t s , a f t e r s p e c i f y i n g t h e f i r s t , s e c o n d , e t c . a t t r i b u t e v a l u e . Thus, if dk
i s t h e d e s c r i p t i o n i n v o l v i n g t h e f i r s t k s e t s B1,
...,
o f a t t r i b u t e v a l u e s ( f u z z y or n o t ) , t h e n we have k Rk = R(d ) . We may t h e n s a y t h a t a t t r i b u t e k i s rek dundant i n t h e c o n t e x t B1,.. ,Bk-l i f Fik =
B
.
Rk-l'
An a t t r i b u t e which i s r e d u n d a n t i n any c o n t e x t , i t i s t o t a l l y r e d u n d a n t . The most common t y p e o f redundancy o c c u r s , i f t h e a t t r i b u t e i s dependent on t h e p r e c e d i n g
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...,
o n e s . Thus, B
i s dependent on B1, B k k-1 i f t h e i n f o r m a t i o n s e t for any o b j e c t x a f t e r dk - i s t h e same as a f t e r d k'
5.6. S t a t i s t i c a l semiotics L e t us now c o n n e c t t h e t h e o r y o f s i g n s and t h e i r meani n g s w i t h t h e t h e o r y o f random o b j e c t s and t h e i r obs e r v a b i l i t y , introduced a t t h e beginning of t h i s c h a p t e r . Thus, s i g n s and meanings a r e now b o t h random o b j e c t s , a p p e a r i n g w i t h some f r e q u e n c i e s . These freq u e n c i e s w i l l , i n g e n e r a l , depend on t h e u s e r o f t h e s y s t e m , i . e . on t h e p e r s o n who p r o d u c e s t h e s i g n ( w e s h a l l d e n o t e t h i s p e r s o n by x). On t h e o t h e r hand, t h e p e r s o n who p e r c e i v e s t h e s i g n , t o b e d e n o t e d b y y , u s e s i n g e n e r a l some mask, and t h e r e f o r e o b t a i n s o n l y some i n f o r m a t i o n about t h e s i g n , p e r h a p s i n s u f f i c i e n t t o i d e n t i f y t h e s i g n completely. The system w i l l now c o n s i s t s o f t h e f o l l o w i n g e l e m e n t s
where M , Z and R are as b e f o r e , IT i s a p r o b a b i l i t y d i s t r i b u t i o n on M , s o t h a t X(m) i s t h e f r e q u e n c y w i t h which t h e s e n d e r x u s e s meaning m , p ( z l m ) i s a f a m i l y o f p r o b a b i l i t y d i s t r i b u t i o n s on Z , where p ( z m ) r e p r e s e n t s t h e p r o b a b i l i t y t h a t sender w i l l choose s i g n z t o r e p r e s e n t meaning m . Next, I i s t h e c l a s s of a l l p o s s i b l e d a t a ( i n f o r m a t i o n ) about z which i s a t t a i n a b l e under a g i v e n mask (assumed t o be f i x e d
FORMAL SEMIOTICS
435
by t h e p e r c e i v e r y ) , and q i s t h e f a m i l y o f p r o b a b i i -
l i t y d i s t r i b u t i o n s q ( z ) I ) , t h a t t h e s i g n was z , g i v e n observation I. Clearly ,
i s t h e u n c o n d i t i o n a l p r o b a b i l i t y t h a t s i g n z w i l l be
produced, s o t h a t p ( z ) i s t h e o v e r a l l f r e q u e n c y o f s i g n z ( i . e . f r e q u e n c y o f random o b j e c t s z). If we d i s r e g a r d t h e c o n t e n t o f t h e s i g n , t h e n one can
u s e t h e Shannon f o r m u l a
as t h e a v e r a g e amount o f i n f o r m a t i o n c a r r i e d by a s i g n . Here t h e amount o f i n f o r m a t i o n c a r r i e d by an i n d i v i d u a l sign i s log l / p ( z ) . N a t u r a l l y , t h i s approach i s of l i m i t e d u s e , s i n c e t h e f r e q u e n c y o f s i g n s has l i t t l e t o do w i t h t h e s e m a n t i c i n f o r m a t i o n which t h e y c a r r y . S i m i l a r l y , one can u s e t h e j o i n t d i s t r i b u t i o n
o f s i g n and meaning, t o d e f i n e t h e j o i n t e n t r o p y , and hence d e f i n e ( a s u s u a l i n i n f o r m a t i o n t h e o r y ) t h e amount o f i n f o r m a t i o n c a r r i e d by one v a r i a b l e about t h e o t h e r ( i n t h i s c a s e one v a r i a b l e i s t h e s i g n , and t h e o t h e r i s t h e meaning).
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I n t h e c a s e under c o n s i d e r a t i o n , t h e assignment o f meaning, or i n t e r p r e t a t i o n s , may b e o b t a i n e d from Bayes' f o r m u l a . Here t h e problem i s t o d e t e r m i n e t h e c o n d i t i o n a l p r o b a b i l i t y of meaning, s a y m , g i v e n t h e 0 o b s e r v e d d a t a I . L e t u s s t a r t from t h e s i m p l e s t c a s e when I = z , s o t h a t z i s o b s e r v a b l e , We have h e r e
Given z , one can a s s i g n t o i t t h a t meaning m
which 0 maximizes t h e above e x p r e s s i o n . T h i s a p p r o a c h has l o n g t r a d i t i o n i n a n a l y s i s o f i n f o r m a t i o n , and we s h a l l omit the details. T o summarize, and t o i n c l u d e t h e i n f o r m a t i o n I , our
random s e m i o t i c s y s t e m c o n s i s t s o f t h e e l e m e n t s o f t h e nonrandom s y s t e m , and t h e a p p r o p r i a t e p r o b a b i l i t y d i s t r i b u t i o n s . The f o l l o w i n g c h a i n of e v e n t s d e s c r i b e s the situation:
-- s e n d e r x wants t o convey meaning m o . T h i s o c c u r s w i t h p r o b a b i l i t y $ ( m o ) , which e x p r e s s e s t h e f a c t t h a t some meanings a r e i n t e n d e d more o f t e n t h a n o t h e r s . --
s e n d e r x t h e n c h o o s e s s i g n z 0' w i t h p r o b a b i l i t y p ( z I m ) . S i g n z o i s a r e p r e s e n t a t i o n of m o , i . e . we 0 0 u s u a l l y w i l l have
mo 6. R ( z o ) .
-- t h e s i g n z o i s p e r c e i v e d by t h e o b s e r v e r y t h r o u g h a mask, which may g i v e ambiguous i n f o r m a t i o n about zo. T h i s a m b i g u i t y i s d e s c r i b e d by t h e p r o b a b i l i t y d i s t r i b u t i o n q(zolI), where I i s t h e i n f o r m a t i o n o b t a i n e d .
FORMAL SEMIOTICS
43 I
The problem i s t o d e t e r m i n e t h e p r o b a b i l i t y t h a t meani n g m was i n t e n d e d , g i v e n t h e i n f o r m a t i o n I . We may 0 write
Thus, w e need t h e p r o b a b i l i t y t h a t s i g n z i s r e a l l y z, g i v e n t h e o b s e r v a t i o n I , and t h e p r o b a b i l i t i e s concerni n g t h e c h o i c e s o f meanings and s i g n s b y t h e s e n d e r . The f i r s t f a c t o r i n t h e l a s t f o r m u l a depends on t h e c h a r a c t e r i s t i c s o f t h e s e n d e r , and t h e second f a c t o r on t h e c h a r a c t e r i s t i c s o f t h e p e r c e i v e r .
--
5 . 7 . Comments I n t h e formal s e m i o t i c s of t h i s s e c t i o n , o b s e r v a t i o n o f an o b j e c t i s r e l a t i v e t o t h e s i t u a t i o n , t o measurement p r o c e d u r e s , and t o mappings i n t h e n u m e r i c a l o r l i n g u i s t i c domains, which a r e t h e p r e r e q u i s i t e s o f f o r m u l a t i n g t h e o b s e r v a t i o n a l judgments. The l a t t e r mapping a l l o w s u s t o r e p l a c e t h e o p e r a t i o n s on o b j e c t s by o p e r a t i o n s on symbols -- e i t h e r i n m a t h e m a t i c a l domain or i n l i n g u i s t i c s p a c e -- o f c l a s s i f i c a t i o n s
o r d e s c r i p t i o n s . T h i s mapping, and i t s c h a r a c t e r , i s d e t e r m i n e d by t h e c l a s s o f t r a n s f o r m a t i o n s o f t h e o b s e r v a t i o n a l s y s t e m , which impose c o n s t r a i n t s on
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on i t s adequacy and i n v a r i a n c e . I n o t h e r words, one may d e a l w i t h t h e whole c l a s s of d e s c r i p t i o n s o r num e r i c a l a s s i g n m e n t s , from which one chooses o n l y t h o s e which a r e c h a r a c t e r i z e d b y s u f f i c i e n t l e v e l o f i n v a r i a n c e . An i m p o r t a n t problem h e r e i s t h e c h o i c e of adeq u a t e r e p r e s e n t a t i o n s o f o b j e c t s and r e l a t i o n s between them. The system o f c a u s a l r e l a t i o n s between o b j e c t s , i n t r o duced p r i m a r i l y f o r o b j e c t s , must be e x t e n d e d t o s e t s o f o b j e c t s which c o n s t i t u t e s i t u a t i o n s , t h e c a u s a l r e l a t i o n s between networks o f o b j e c t s , s e t s o f s e n t e n c e s d e s c r i b i n g f a c t s o r laws, c l a s s e s o f p r o c e d u r e s , n u m e r i c a l or l i n g u i s t i c mappings, and c o n d i t i o n s f o r invariance. T h i s i s a dynamical s y s t e m , n o t o n l y on t h e s e n s e o f
changes of o b j e c t s , and t h e c o r r e s p o n d i n g changes o f descriptions, p o s s i b i l i t i e s of continuous c o l l e c t i o n o f i n f o r m a t i o n , e t c . , b u t a l s o because some m o d i f i c a t i o n s o f o b s e r v a t i o n a l systems i n d u c e t h e improved and e n r i c h e d d e s c r i p t i o n s o f r e l a t i o n s between o b j e c t s . I n consequence, i t also l e a d s t o e l i m i n a t i o n o f e r r o r s , e x t e n s i o n of o l d , or i n t r o d u c t i o n o f new t h e o r y f o r t h e e x t e n d e d s e t of o b s e r v a t i o n a l s e n t e n c e s . T o sum up, t h e dynamics o f r e l a t i o n s between o b s e r v a -
t i o n s and c o l l e c t e d knowledge a b o u t o b j e c t s , on t h e one hand, and p r o d u c t i o n o f t h e i r c o p i e s on a g i v e n l e v e l o f r e p r e s e n t a t i o n , shows t h e m e t h o d o l o g i c a l and i n t e g r a t i v e r o l e of s e m i o t i c c o n c e p t i o n s o f t h e f o u n d a t i o n s of c o g n i t i o n . THera i s a n e s p e c i a l l y i m p o r t a n t problem c o n n e c t e d w i t h t h e above s k e t c h e d s e m i o t i c a l a p p r o a c h , namely t h a t
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of o p t i m i z a t i o n o f o p e r a t i o n s on r e p r e s e n t a t i o n s . The main q u e s t i o n h e r e i s which c o g n i t i v e p r o c e s s e s i n t e r vene and f a c i l i t a t e t h e o p t i m i z a t i o n p r o c e s s e s . Depending on t h e medium and v a r i o u s g o a l s o f o p t i m i z a t i o n , i t i s n e c e s s a r y t o l o o k f o r r e l a t i o n s between c o g n i t i v e p r o c e s s e s and r e p r e s e n t a t i o n s . For t h e soci a l s c i e n c e s , t h e s e m i o t i c a l a p p r o a c h o f f e r s c l e a r and e s t a b l i s h e d l o g i c a l l y r e s e a r c h on s t r u c t u r e o f knowledge and i t s r e l a t i o n t o r e a l i t y . For p h i l o s o p h y , i t o f f e r s u n i f i c a t i o n o f o n t o l o g y and e p i s t e m o l o g y . For t h e l a t t e r , one g e t s a n e x t e n s i o n o f i t s c l a s s i c a l s c o p e . For l o g i c -- one g e t s p r o b a b l y a s i m p l i f i c a t i o n and improvement o f l o g i c a l c a l c u l i . For l i n g u i s t i c s -- i t a l l o w s t o c o n s i d e r u n i f i e d s y s tems l y i n g a t t h e b a s e o f a l l n a t u r a l and f o r m a l l a n guage s
.
Also, f o r t h e f o u n d a t i o n s of i n f o r m a t i o n s c i e n c e , t h e i n t r o d u c t i o n o f s t r u c t u r a l e l e m e n t s i n t o what i s des c r i b e d b o t h i n memory and l a n g u a g e , may l e a d t o new more economical p r i n c i p l e s o f computer o p e r a t i o n s , and new t h e o r e t i c a l s o l u t i o n s .
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4
BIBLIOGRAPHY FOR CHAPTER
and B I R C H , D .
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The Dynamics o f A c t i o n . N e w Y o r k . W i l e y .
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1981
Jongsma, H . J .
and V i n c e p t , J .
"Two D i m e n s i o n a l F i l t e r i n g , O r i e n t e d L i n e D e t e c t o r s a n d F i g u r a l A s p e c t s as D e t e r m i n a n t s o f V i s u a l I l l u s i o n s " . P e r c e p t i o n a n d Psychop h y s i c s . 29; 4 .
352-358.
FELLER. W .
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An I n t r o d u c t i o n t o P r o b a b i l i t y T h e o r y a n d I t s Qplications.
N e w York. Wiley
.
K l i r , G.J.
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Trends i n G e n e r a l Systems Theory. N e w York. Wiley
LEE, S.H.
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(ed.)
1981 O p t i c a l Information Processing. Fundamentals. Berlin. Springer. NOWAKOWSKA , M .
1967
" Q u a n t i t a t i v e A p p r o a c h t o t h e Dynamics o f P e r cept ion". E n e r a 1 Systems, X I I .
81-95.
1973 Language of M o t i v a t i o n a n d Language o f A c t i o n s . The Hague. Mouton.
1 9 7 8 " V e r b a l a n d N o n v e r b a l Communication as a Mult i d i m e n s i o n a l Language". ( i n P o l i s h ) . Vol. I X ,
S t u d i a Semiotyczne.
181-196.
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1979
"FUZZY Concepts: Their Structure and Problems of Measurement". In M.M. Gupta, R.K. Ragade and R.R. Yager (eds.) Advances in Fuzzy Set Theory and Application. Amsterdam. North Holland. 361-387. 1979a "Foundations of Formal Semiotics: Objects and Their Verbal Copies". Ars Semeiotica. Vol. 11, nr. 2. 133-1118. 1980 "Semiotic Systems and Knowledge Representation". Intern. Journal of Man-Machine Studies. 13. 223-257. 1980a "Empirical Semiotics". Ars Semeiotica. Vol. 111, nr. 2. 1981 "Formal Semiotics and Multidimensional Semiotic Systems". Cybernetics and Systems: An International Journal. Vol. 12. 83-100. 1983a Cognitive Sciences. New Perspectives. New York. Academic Press. 1983b "Dynamics of Perception: Some New Models". Intern. Journal of Man-Machine Studies. 18.
175-197. SHAPLEY, L . S . and SHUBIK, M. "A Method of Evaluating the Distribution of 1954 Power in a Committee System". The American Political Science Review. 48. 787-792. ULLMANN, S. 1979 The Interpretation of Visual Motion. Cambridge, Mass. MIT Press. ZIMMERMANN, H.J. and ZYSNO, P. 1980 "Latent Connectives in Human Decision Making". Fuzzy Sets and Systems. 4, 37-51.
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5
MEASUREMENT THEORY
The measurement o f a t r a i t o f o b j e c t s o f a c e r t a i n s e t
c o n s i s t s of a s s i g n i n g numbers t o t h e s e o b j e c t s , so as t o r e f l e c t a d e q u a t e l y t h e measured t r a i t . Measurement t h e o r y i s concerned w i t h f o r m a l c o n d i t i o n s which must be met i n o r d e r f o r such a n assignment o f numbers t o e x i s t . I n t h i s r e s p e c t , measurement t h e o r y ought t o be d i s t i n g u i s h e d from s c a l i n g t h e o r y , which deals w i t h t h e t e c h n i q u e s o f a s s i g n i n g numbers t o o b j e c t s s o as t o r e f l e c t t h e measured t r a i t . The e m p i r i c a l s i t u a t i o n s whose a n a l y s i s became simpli f i e d , made more p r e c i s e ( o r even i n some c a s e s made p o s s i b l e ) by q u a n t i f i c a t i o n , a p p e a r i n numerous d i s c i p l i n e s . T h i s r e s u l t e d i n t h e s t u d i e s of e x i s t e n c e and p r o p e r t i e s o f measurement, conducted t o a l a r g e e x t e n t i n d e p e n d e n t l y i n d i f f e r e n t domains. An a t t e m p t o f s y n t h e s i s o f t h e r e s u l t s c o n c e r n i n g measurement i n v a r i o u s d i s c i p l i n e s , such as p h y s i c s , economics, psychology, e t c . , l e a d s t o t h e c o n c l u s i o n t h a t a l l known problems o f t h e s o - c a l l e d fundamental measurement ( i . e . measurement which i s n o t based on r e s u l t s o f p r e v i o u s measurements) can be reduced t o
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one o f t h r e e problems: o r d i n a l measurement, measurement b y u s i n g s t a n d a r d s e q u e n c e s , and s o l v i n g i n e q u a l i t i e s ( s e e Krantz e t a l . 19-71). T h i s c h a p t e r w i l l c o n t a i n a p r e s e n t a t i o n o f a number
o f measurement axiom s y s t e m s , s o as t o c o v e r t h e most t y p i c a l c a s e s o c c u r r i n g i n p r a c t i c e . I n each c a s e , t h e system w i l l concern some a b s t r a c t r e p r e s e n t a t i o n o f emp i r i c a l r e l a t i o n s . The most i m p o r t a n t o f t h e s e r e l a t i o n s are connected w i t h comparison of o b j e c t s w i t h r e s p e c t t o some p r o p e r t y ( i . e . a s s e r t i n g t h a t one o b j e c t i s "smaller" than t h e o t h e r ) . O t h e r r e l a t i o n s c o n c e r n "composite" o b j e c t s . Here t h e p r o t o t y p e s a r e c o n c a t e n a t i o n s , i n which t h e f e a t u r e s o f t h e o b j e c t s behave i n an a d d i t i v e way ( a s t y p i f i e d b y p u t t i n g r o d s end t o end s o t h a t t h e i r l e n g t h s comb i n e a d d i t i v e l y ) Another k i n d on "composite" o b j e c t s a r e t h o s e c h a r a c t e r i z e d by two or more i n d e p e n d e n t f e a t u r e s , each c o n t r i b u t i n g t o t h e o v e r a l l c h a r a c t e r i s t i c s ( a t e x e m p l i f i e d by u t i l i t i e s o f " b a s k e t s " cont a i n i n g v a r i o u s goods.)
.
T h e r e a r e i m p o r t a n t s p e c i a l c a s e s , a p a r t from p h y s i c s ,
when t h e f i r s t t y p e o f c o n c a t e n a t i o n ( w i t h a d d i t i v i t y ) h o l d s , namely c o n c e r n i n g s u b j e c t i v e p r o b a b i l i t y and u t i l i t y , and c o n c e r n i n g r i s k p e r c e p t i o n . The c o r r e s p o n d i n g axiom systems w i l l be p r e s e n t e d ; as r e g a r d s r i s k , a new model w i l l a l s o be shown, which a c c o u n t s f o r some seemingly i r r a t i o n a l b e h a v i o u r (from t h e p o i n t o f view o f t h e model o f s u b j e c t i v e e x p e c t e d u t i l i t y ) . A s a n i l l u s t r a t i o n of measurement s y s t e m s , a t h e o r y of
s u b j e c t i v e t i m e w i l l be shown, t o g e t h e r w i t h h y p o t h e s e s
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which e x p l a i n t h e d i s t o r t i o n s o f p e r c e p t i o n o f t i m e . F i n a l l y , t h e c h a p t e r w i l l c o n t a i n some r e s u l t s concerni n g a n e m p i r i c a l c o u n t e r p a r t of measurement t h e o r y , namely s c a l i n g theory. 1. BASIC PROCEDURES OF MEASUREMENT
1.1. O r d i n a l measurement
L e t A d e n o t e t h e s e t o f o b j e c t s under c o n s i d e r a t i o n s . It w i l l be assumed t h a t t h e s e o b j e c t s may be compared from t h e p o i n t o f view o f a g i v e n a t t r i b u t e . F o r any a,b E A , l e t a ). b s t a n d f o r t h e f a c t t h a t t h e o b j e c t a i s e v a l u a t e d h i g h e r (from t h e p o i n t o f view o f t h e a t t r i b u t e under c o n s i d e r a t i o n ) t h a n o b j e c t b. S i m i l a r l y , a - w b w i l l mean t h a t a and b are i n d i s t i n g u i s h a b l e from t h e p o i n t of view o f t h e a t t r i b u t e i n q u e s t i o n .
The e l e m e n t s o f t h e s e t A may be r o d s , o r p i e c e s o f a s t r i n g , compared from t h e p o i n t o f view o f t h e i r l e n g t h ; d i n n e r s , compared a c c o r d i n g t o p r e f e r e n c e ; p a i n t i n g s , compared a c c o r d i n g t o t h e i r a e s t h e t i c v a l u e , and so f o r t h . The problem o f measurement o f t h e a t t r i b u t e which un-
d e r l i e s t h e comparisons, c o n s i s t s o f c o n s t r u c t i o n o f a f u n c t i o n f, d e f i n e d on A , s u c h t h a t f ( a ) > f ( b ) i f and o n l y i f a ’ > b (hence a l s o f ( a ) = f ( b ) i f and only if a b).
It ought t o be c l e a r t h a t f o r s u c h a measurement t o be p o s s i b l e , t h e r e s u l t s o f comparisons o f e l e m e n t s o f A , e x p r e s s e d by t h e r e l a t i o n s h a n d & must s a t i s f y some c o n s i s t e n c y c o n d i t i o n s . For i n s t a n c e , t h e r e s u l t s o f
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i n d i f f e r e n c e a s s e r t i o n s a - b must be t r a n s i t i v e : i f a b and b rv c , t h e n a l s o a c ( s i n c e i n t h e opposit e c a s e , we would have f ( a ) = f ( b ) and f ( b ) = f ( c ) , bu t f ( a ) # f ( c ) , which i s i m p o s s i b l e ) . N
L e t us o b s e r v e , however, t h a t i n c a s e s when t h e asserti o n t h a t a . u b i s o b t a i n e d by a n a p p l i c a t i o n o f some e m p i r i c a l p r o c e d u re , t h e t r a n s i t i v i t y of t h e r e l a t i o n i s by no means a u t o m a t i c : t h e r e may e x i s t t r i p l e t s a , b , c such t h a t a and b , as w e l l as b and c, are i n d i s t i n g u i s h a b l e , w h i l e a and c are a l r e a d y d i s t i n g u i s h a b l e . As a n example, one may take t h r e e shades o f some colour.
-
The theorem below p r o v i d e s c o n d i t i o n s imposed on t h e
r e s u l t s o f comparisons which imply t h e e x i s t e n c e o f the function f. Observe f i r s t t h a t I n s t e a d o f c o n s i d e r i n g two r e l a t i o n s 4 and C J ) one can t a k e as a s t a r t i n g p o i n t one r e l a t ion , such t h a t a d b d e n o t e s t h e f a c t t h a t a i s no
4
l a t e r t h a n b ( w i t h r e s p e c t t o t h e a t t r i b u t e under consideration).
4
i s c a l l e d a weak o r d e r i n t h e s e t A , i f i t s a t i s f i e s t h e f o l l o w i n g two c o n d i t i o n s : t r a n s i t i v ity, i . e . The r e l a t i o n
and co n n e cte d n e ss, i.e. either a
4
b,
or
b<
a.
(1.2)
Thus, c o n d i t i o n ( 1 . 2 ) a s s e r t s t h a t e v e r y two el emen t s
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a r e comparable, while (1.1) a s s e r t s t h e c o n s i s t e n c y of t h e s e comparisons. The theorem on e x i s t e n c e of measurement may be formulated a s follows. THEOREM 1.1.
let4
Let
A be a f i n i t e o r countable s e t , and
be a weak o r d e r i n A .Then t h e r e e x i s t s a numerleal f u n c t i o n f , d e f i n e d on A, such t h a t
f ( a > <' f ( b )
iff
a 4 b.
(1.3)
Moreover, i f g i s any f u n c t i o n d e f i n e d on A s a t i s f y i n g (1.31, t h e n t h e r e e x i s t s a s t r i c t l y i n c r e a s i n g f u n c t i o n F, such t h a t f o r every a E A w e have
The b a s i c i d e a o f t h e proof i s as follows. F i r s t l y , we d e f i n e t h e r e l a t i o n s w a n d 4 by p u t t i n g a - b i f and only i f a 4 b and b a , and p u t t i n g a < b i f and only i f a d b and not b d a.
a n d 4 a r e c a l l e d r e s p e c t i v e l y t h e symm e t r i c and asymmetric p a r t of t h e r e l a t i o n . 4 . I f t h e r e l a t i o n 4 i s a weak o r d e r , then-, i s an equivalence (i.e. i s r e f l e x i v e , symmetric and t r a n s i t i v e ) , and t h e i s a n t i r e f l e x i v e , antisymmetric and t r a n s i relation t i v e . Consequently, t h e s e t A p a r t i t i o n s i n t o c l a s s e s of e q u i v a l e n t elements, and t h e s e c l a s s e s may be ordered by r e l a t i o n 4. The r e l a t i o n s
Without loss of g e n e r a l i t y w e may assume t h a t no two elements of A are e q u i v a l e n t ( o r , which amounts t o t h e same, t h a t we c o n s i d e r t h e s e t of a l l equivalence c l a s s e s w i t h r e s p e c t t o -1.
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If t h e s e t A i s f i n i t e , t h e f u n c t i o n f may b e d e f i n e d s i m p l y by o r d e r i n g e l e m e n t s o f A, and a s s i g n i n g t o .If A i s i n s u c c e s s i v e e l e m e n t s t h e numbers 0,1,2,... f i n i t e , such a c o n s t r u c t i o n may be i m p o s s i b l e : it may namely happen t h a t f o r any two e l e m e n t s a,b E A which a r e n o t e q u i v a l e n t t h e r e e x i s t s a n element c i n between ( i - e . such t h a t a 4 c 4 b o r b c .( a , depending whe-
t h e r a 4 b o r b-(
a).
I n s u c h c a s e s , t h e " n e x t " element d o e s n o t e x i s t , and t h e f u n c t i o n f h a s t o be d e f i n e d d i f f e r e n t l y . Without g o i n g i n t o d e t a i l s , t h e i d e a o f t h e proof i s a s f o l l o w s . F i r s t l y , we o r d e r t h e s e t A i n t o a sequence ao,al,. and p u t f ( a o ) = 0. W e t h e n p r o c e e d by i n d u c t i o n : i f f(ao),...,f(an-l) are a l r e a d y d e f i n e d , t h e n t h e v a l u e f ( a n ) depends on t h e l o c a t i o n o f an w i t h r e s p e c t t o t h e p r e c e d i n g e l e m e n t s . If an i s " t o t h e r i g h t " o f a l l ai, i = O,l,...,n-1, we t a k e as f ( a n ) any number I n c a s e when an e x c e e d i n g a l l f ( a i ) , i = O,l,...,n-l. i s t o t h e l e f t of a l l preceding elements, t h e procedure i s analogous. F i n a l l y , i n t h e r e m a i n i n g c a s e , an must
..
l i e between some ai and a j , i . e . aiA a n 4 a , and j an- 1 o t h e r t h a n ai e a c h term o f t h e sequence a O , and a must l i e e i t h e r t o t h e l e f t o f ai o r t o t h e j One may t h e n t a k e f ( a n ) = $ [ f ( a i ) t f ( a j ) ] . right of a j' The f u n c t i o n f so d e f i n e d s a t i s f i e s t h e c o n d i t i o n s o f t h e theorem.
...,
It i s n o t hard t o show by a n example, t h a t t h e asserti o n o f Theorem 1.1 i s n o t v a l i d w i t h o u t t h e a s s u m p t i o n t h a t t h e s e t A i s c o u n t a b l e ( s e e R o b e r t s , 3.979). Even though t h e c a s e o f a f i n i t e s e t A i s o f g r e a t e s t pract i c a l i m p o r t a n c e , i t i s n e v e r t h e l e s s of a c o n s i d e r a b l e t h e o r e t i c a l i n t e r e s t t o f i n d n e c e s s a r y and s u f f i c i e n t
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c o n d i t i o n s under which t h e a s s e r t i o n o f Theorem 1.1 i s t r u e (i.e. under which we have a measurement o f t h e a t t r i b u t e o f e l e m e n t s o f A on a n o r d i n a l s c a l e ) . We s h a l l q u o t e h e r e t h e theorem o f Bi rk h o ff and Milgram ( s e e Ro b e r t s , 1979). We s h a l l s a y t h a t t h e set B C A i s order-dense ( w i t h r e s p e c t t o t h e weak o r d e r 4 ), i f t h e c o n d i t i o n s a , b e A B , a-( b imply t h a t t h e r e e x i s t s an element c c B such t h a t a d c 4 b. We have
-
=<
be a t r a n s i t i v e , asymmetric and conn e c t e d b i n a r y r e l a t i o n i n A. Then t h e r e e x i s t s a funct i o n f on A s a t i s f y i n g (1.3) and ( 1 . 4 ) i f and o n l y i f t h e r e e x i s t s a s e t B 6 A which i s order-dense.
THEOREM 1 . 2 .
The p r o o f may be found i n R o b e rt s (1979).
1.2.
S ta n d ar d sequences
I n t h e preceding s e c t i o n , the r e l a t i o n 4 implied only an o r d e r i n g o f e l e m e n t s o f t h e s e t A , and more p r e c i s e ly an o r d e r i n g o f e q u i v a l e n c e c l a s s e s w i t h r e s p e c t t o t h e induced r e l a t i o n .e.A s a r e s u l t , there was a c o n s i d e r a b l e freedom o f c h o i c e o f t h e f u n c t i o n f . We s h a l l now c o n s i d e r c o n d i t i o n s which r e s t r i c t t h i s freedom.
-
A s b e f o r e , assume t h a t we have a s e t A , whose e l e m e n t s may be compared w i t h r e s p e c t t o t h e a t t r i b u t e under cons i d e r a t i o n (these comparisons a r e d e s c r i b e d by t h e rel a t i o n & ) . I n a d d i t i o n , we assume t h a t el emen t s o f A may be c o n c a t e n a t e d , and t h a t t h e comparisons concern
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t h e c o n c a t e n a t e d e l e m e n t s.
Formally, t h e o p e r a t i o n of c o n c a t e n a t i o n w i l l b e denoted by 0 : t h u s , a o b i s t h e r e s u l t o f c o n c a t e n a t i o n o f e l e m e n t s a and b. A s a n example of c o n c a t e n a t i o n , imagine t h a t t h e elements o f t h e set A a r e r o d s o f v a r i o u s l e n g t h s ; t h e operation of concatenation c o n s i s t s of p u t t i n g t h e r o d s end t o end, w h i l e comparisons are accomplished by p u t t i n g t h e r o d s p a r a l l e l , so t h a t t h e i r ends on one s i d e a g r e e . For i n s t a n c e , o f Fig. 1 we have a o b >- C .
C
c
J
Fig. 1
A s a n o t n e r example, we may t a k e comparison o f w e i g h t s
on a s c a l e ; c o n c a t e n a t i o n c o n s i s t s i n t h i s c a s e o f p u t t i n g two o r more w e i g h t s on t h e same s i d e o f t h e scale. To f o r m u l a t e t h e theorem, i t w i l l be co n v en i en t t o i n c l u d e i n t h e set A a l l c o n c a t e n a t i o n s o f f i n i t e numbers of ele m en t s o f A ( i n o t h e r words, w e assume t h a t t h e s e t A i s c l o s e d under t h e o p e r a t i o n o f c o n c a t e n a t i o n ) . The problem c o n s i s t s o f c o n s t r u c t i n g a f u n c t i o n f , de-
f i n e d on A, such t h a t f i s a measurement i n t h e s e n s e o f t h e p r e c e d i n g theorem, i . e . a 3 b i f and o n l y i f
451
SELECTED TOPICS IN MEASUREMENT THEORY
<
f(a) f ( b ) , and moreover, s a t i s f y i n g t h e c o n d i t i o n o f a d d i t i v i t y : f ( a o b ) = f ( a ) t f ( b ) f o r a l l a,b i n A. B ef o r e f o r m u l a t i n g a theorem which g i v e s t h e c o n d i t i o n s which must be met by t h e r e l a t i o n 4 and o p e r a t i o n o f concatenation o i n order f o r a function f t o e x i s t , it i s worth w h i l e t o d e s c r i b e t h e b a s i c ideas o f measurement i n t h e p r e s e n t c o n t e x t . L e t a be a f i x e d element o f t h e set A. sequence
a,
a
o
a,
a
0
a
0
Consider t h e
a,...;
f o r s i m p l i c i t y , denote a
I,
a = 2a, a o a
o
a = 3a, e t c .
The sequence a, 2a, 3a,...
i s c a l l e d a s t a n d a r d sequenc e based on s t a n d a r d a. A t y p i c a l c a s e here i s a r u l e r , w i t h a s c a l e , e.g. i n milimetres, which a l l o w s t h e comp a r i s o n o f i n t e r v a l s w i t h i n t e r v a l s on t h e r u l e r , which a r e o f t h e form na, where a i s t h e d i s t a n c e between two s u c c e s s i v e marks on t h e r u l e r , e q u a l 1 milimetre. If b i s an element o f t h e set A , and f o r some n t h e
element b f a l l s between n a and ( n t 1 ) a ( i . e . b )= n a, and ( n t 1 ) a 2: b ) , t h e n one can assume t h a t t h e measure f ( b ) o f element b i s c o n t a i n e d between n f ( a ) and ( n t l ) f ( a ) , where f ( a ) i s t h e measure o f element a. The p r o o f o f t h e theorem which a s s e r t s t h e e x i s t e n c e
o f measurement f (which w i l l n o t be p r e s e n t e d h e r e ) i s based on a n i n t u i t i v e l y obvious f a c t t h a t when t h e s t a n d a r d sequences become more "dense", t h e s u c c e s s i v e app r o x i m ati o n s of t h e measure of b w i l l approach t h e v a l u e f ( b ) , and t h a t t h e f u n c t i o n f so d e f i n e d meets
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t h e r e q u i r e m e n t s f o r measurement. N a t u r a l l y , as i n t h e c a s e o f o r d i n a l measurement, i t i s n e c e s s a r y t o impose some c o n d i t i o n s of r e l a t i o n 4 , which describes t h e r e s u l t s o f comparisons, and on t h e operation a of concatenation. We have namely THEOREM 1.3 (see Suppes and Zinnes, 1 9 6 3 ) . L e t u s a s s u -
m e t h a t i n t h e s e t A we have a b i n a r y r e l a t i o n 4 , a b i n a r y o p e r a t i o n 0 : A x A -+ A s a t i s f y i n g t h e followi n g c o n d i t i o n s : f o r any a , b , c G A 1)
ad
2) (a o b )
3)
and
b o
if a 4
b,
c
a 4 c;
b & c, then
4
a o (b o c ) ;
then
a
o
c< b
o
c;
4 ) i f it i s n o t t r u e t h a t a & b , t h e n t h e r e e x i s t s an b and b<, a o c ; element c such t h a t a o c
5) i t i s n o t t r u e 6)
that a
&
a
o b;
a & b , t h e n t h e r e e x i s t s a number n s u c h t h a t nb $ a , where nb d e n o t e s t h e n - f o l d c o n c a t e n a t i o n
...
b o b o
o
b.
Then t h e r e e x i s t s a f u n c t i o n f d e f i n e d on A, s u c h t h a t
f(a)
f(a
5
o
f(b)
i f , and o n l y i f ,
b) = f(a) t f(b).
ad
b
(1.5)
(1.6)
Moreover, i f any f u n c t i o n g d e f i n e d on A s a t i s f i e s t h e
SEUCTED TOPICS IN MEASUREMENT THEORY
453
c o n d i t i o n s (1.5) and (1.61, t h e n t h e r e e x i s t s a conssu c h t h a t
t a n t k > 0,
f o r every a
& A.
I n terms o f t h e most o b v i o u s i n t e r p r e t a t i o n , namely i n t h e c a s e o f comparisons o f l e n g t h s , w i t h c o n c a t e n a t i o n c o n s i s t i n g on j o i n i n g o b j e c t s end t o end, t h e cond i t i o n s 1 - 6 o f t h e theorem above are i n t u i t i v e l y q u i t e c l e a r , and d e s c r i b e t h e most e l e men t ary p r o p e r t i e s o f t h e two o p e r a t i o n s ( i n f a c t , t h e s e c o n d i t i o n s g i v e o n l y one o f t h e p o s s i b l e axiom systems f o r t h e s o - c a l l ed e x t e n s i v e measurement). It i s worth t o p o i n t o u t t h a t t h e main r o l e o f such c o n d i t i o n s r e l i e s on t h e i r i n t u i t i v e s i m p l i c i t y : t h e main o b j e c t of f i n d i n g a n axiom s y s t e m i s t o d e t e rm i n e c o n d i t i o n s which are b o t h i n t u i t i v e l y a c c e p t a b l e , and a t t h e same time imply c e r t a i n less obvious p r o p e r t i e s . Thus, axiom 1 a s s e r t s t h a t comparisons o f l e n g t h s cons t i t u t e a t r a n s i t i v e r e l a t i o n ; t h i s property should n o t c a u s e any q u e s t i o n s , p ro v i d e d , o f c o u r s e , t h a t one remembers t h a t i n c a s e of o b j e c t s a , b , c o f "almost t h e same l e n g t h " t h e comparisons may be s u b j e c t t o some error. Next, axiom 2 a s s e r t s t h a t c o n c a t e n a t i o n i s a s s o c i a t i v e , w h i l e axiom 3 a s s e r t s t h a t i t p r e s e r v e s i n e q u a l i t i e s . Axiom 4 asserts, i n e f f e c t , t h a t t h e r e e x i s t s a " d i f f e r ence" o f two unequal e l e m e n t s. Indeed, i f i t i s n o t t r u e t h a t a & b , t h e n a ? b , hence a i s r e a l l y s h o r t e r t h a n b. According t o axiom 4 t h e r e e x i s t s a n element c
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such t h a t a o c b and b a o c , which i s t h e same as b - a 0 c. Thus, element c i s t h e d i f f e r e n c e b e t ween b and a. Axiom 5 a s s e r t s t h a t c o n c a t e n a t i o n w i t h al?y element e f f e c t i v e l y i n c r e a s e s t h e l e n g t h : a o b i s a l w a y s longe r t h a n a. F i n a l l y , t h e l a s t axiom ( c a l l e d u s u a l l y t h e Archimedean axiom) a s s e r t s i n e f f e c t t h a t t h e r e a r e no i n f i n i t e l y s h o r t and no i n f i n i t e l y l o n g elements. Indeed, o u t o f any o b j e c t b one may b u i l d a c o n c a t e n a t i o n nb = b o b o b ( n times) which w i l l b e a r b i t r a r i l y long, i.e. w i l l exceed any element a.
...
It i s mainly because t h e s e axioms are s o obvious i n c a s e o f b a s i c p h y s i c a l q u a n t i t i e s such as l e n g t h o r w e i g h t , t h a t t h e problems o f measurement i n p h y s i c s d i d not i n s p i r e t h e development o f measurement t h e o r y and a r e s t i l l d e a l i n g w i t h im(physicists dealt provement o f measurement t o o l s , i . e . w i t h c o n s t r u c t i o n o f new methods o f i n c r e a s i n g l y p r e c i s e measurement, r a t h e r t h a n w i t h p h i l o s o p h i c a l f o u n d a t i o n s o f measurement ).
-
-
It i s worth mentioning, however, t h a t axiom 6 (Archimedean) need n o t be a u t o m a t i c a l l y s a t i s f i e d i n t h e case o f p h y s i c a l measurement: a n o n t r i v i a l example i s o b t a i n e d when we c o n s i d e r c o n c a t e n a t i o n s and comparis o n s o f such p h y s i c a l q u a n t i t i e s as v e l o c i t i e s . Physica l l y , t h e c o n c a t e n a t i o n of v e l o c i t i e s i s o b t a i n e d i f one c o n s i d e r s motions o f a system i n which a n o t h e r o b j e c t i s moving. The Archimedean axiom i s n o t s a t i s f i e d , s i n c e one may
SELECTED TOPICS INMEASUREMENT THEORY
455
c o n c a t e n a t e a g i v e n v e l o c i t y w i t h i t s e l f any number o f t i m e s , and y e t , t h e r e s u l t w i l l n o t exceed t h e v e l o c i t y of l i g h t .
1.3.
Solving i n e q u a l i t i e s
F i n a l l y , t h e t h i r d method o f d e t e r m i n i n g t h e measurement f u n c t i o n f is based on t h e e m p i r i c a l data concerni n g c o n c a t e n a t i o n s o f a c e r t a i n number o f o b j e c t s . The s i t u a t i o n i s b e s t e x p l a i n e d on a n example. L e t us
...,
imagine t h a t we have f i v e r o d s rl, r5, which may be, as b e f o r e , c o n c a t e n a t e d and compared. L e t us assume t h a t t h e r e s u l t s of these comparisons are
Such d a t a may appear if f o r some r e a s o n t h e p r o c e s s o f comparison is d i f f i c u l t and has t o be a p p l i e d "econom i c a l l y " , w i t h o n l y a l i m i t e d number of comparisons. = f ( r ) t h e unLet u s d e n o t e by xI = f ( r l ) ,...¶ x5 5 known l e n g t h s o f r o d s rl, r The r e s u l t s o f compar5' i s o n s may t h e n be p r e s e n t e d i n form o f a system o f i n -
...,
equalities XI
+
x5,
x3 +
X4¶
456
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x3
>
5
x2
>
XI'
9
Each s o l u t i o n o f t h i s s y s t e m ( p r o v i d e d i t e x i s t s ) g i v e s p o s s i b l e l e n g t h s of t h e r o d s rl,.. . , r 5' I n s e t t i n g up t h e s e i n e q u a l i t i e s , it was assumed t h a t t h e measurement f u n c t i o n f s a t i s f i e s t h e a d d i t i v i t y
p r o p e r t y ( s o t h a t , f o r example, t h e c o n c a t e n a t i o n o f r and r c o r r e s p o n d s t o t h e .length x1 + x5, e t c . ) , 1 5 corresponds t o i n e q u a l i t y and t h a t t h e r e l a t i o n i n t h e numerical domain.
>
2. AN ABSTRACT PRESENTATION OF MEASUREMENT PROBLEM
Using t h e examples g i v e n above one could t r y t o char a c t e r i z e t h e measurement problems i n g e n e r a l . I n t h e c a s e s under c o n s i d e r a t i o n we d e a l t w i t h a r e l a t , which d e s c r i b e d t h e r e s u l t s o f comparisons o f ion t h e e l e m e n t s o f a c e r t a i n s e t , and w i t h t h e o p e r a t i o n o f c o n c a t e n a t i o n of e l e m e n t s of t h i s s e t . The problem was t o d e t e r m i n e t h o s e p r o p e r t i e s o f t h e r e l a t i o n and o p e r a t i o n o f c o n c a t e n a t i o n , which i m p l y t h e e x i s t ence o f a f u n c t i o n f w i t h c e r t a i n d e s i r e d p r o p e r t i e s . One of t h e s e p r o p e r t i e s was t h a t f u n c t i o n f a s s i g n s higher values t o "larger" ( i n the sense of r e l a t i o n 4 ) elements. The o t h e r p r o p e r t y was a d d i t i v i t y , i . e . t h e requirement t h a t c o n c a t e n a t i o n be r e p r e s e n t e d by t h e summation. Formally, t h i s i s e x p r e s s e d by requirement t h a t f i s a homomorphism o f t h e s e t A w i t h r e l a t i o n 4 and o p e r a t i o n 0 , and t h e numerical domain w i t h i n and a d d i t i o n equalfty
3
<
<
+.
457
SELECTED TOPICS IN MEASUREMENT THEORY
To s i m p l i f y t h e subsequent f o r m u l a t i o n s , l e t us o b s e r v e f i r s t t h a t w e may r e p l a c e t h e n o t i o n o f o p e r a t i o n o by an a p p o p r i a t e r e l a t i o n . Indeed, i n s t e a d o f t h e b i n a r y o p e r a t i o n 0 which maps A X A i n t o A , w e may c o n s i d e r a t h r e e - p l a c e r e l a t i o n , t o be denoted by t h e same symb o l 0 , which h o l d s between t h e e l e m e n t s a,b,c i f and o n l y i f a o b = C. I n a similar way, w e may s p e a k o f t h e r e l a t i o n o f a d d i t i o n i n t h e n u meri cal domain, meani n g a t e r n a r y r e l a t i o n , denoted a l s o by t. Thus, i n t h e examples c o n s i d e r e d , we d e a l t w i t h a s e t A , and e i t h e r a b i n a r y r e l a t i o n o r a p a i r of r e l a t ion s = $ and 0 . G e n e r a l l y , a s y s t e m c o n s i s t i n g o f a set and some r e l a t i o n s d e f i n e d on i t i s c a l l e d a r e l a t i o n a l s y s t e m . I n t h e f i r s t c a s e , t h e r e f o r e , we have a r e l a t i o n a l s y s t e m o f t h e form < A , 4 > , and i n t h e second c a s e , a r e l a t i o n a l s y s t e m o f t h e form
4,
S i n c e t h e r e l a t i o n s 4 and o were d e f i n e d t h r o u g h some e m p i r i c a l p r o c e d u r e s ( o f comparison or c o n c a t e n a t i o n ) , one g e n e r a l l y r e f e r s t o such systems as t o e m p i r i c a l r e l a t i o n a l systems. The theorems g i v e n i n t h e p r e c e d i n g s e c t i o n asserted
t h a t under some c o n d i t i o n s imposed on t h e r e l a t i o n s under c o n s i d e r a t i o n , t h e r e e x i s t s a homomorphism f which maps t h e r e l a t i o n a l system i n t o t h e correspondi n g n u m er i cal r e l a t i o n a l s y s t e m . To f o r m u l a t e t h e probl e m o f measurement i n g e n e r a l , it w i l l be co n v en i en t t o introduce t h e following definitions.
,...,
a=
...,
Let J$ = < A , R1 Rk) and (B, S1, Sk) den o t e two r e l a t i o n a l systems. Thus, A and B are s e t s , R1, Rk are r e l a t i o n s i n t h e s e t A, and S1, sk
...,
...,
458
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are r e l a t i o n s i n t h e s e t B ( t h e s e r e l a t i o n s may have d i f f e r e n t numbers o f a rg u m e n t s). We s a y t h a t t h e f u n c t i o n f: A +B i s a homomorphism between r e l a t i o n a l systems & and @ , i f f o r e v e r y i = l,...,k, t h e e l e m e n t s
o f t h e s e t A a r e i n r e l a t i o n Ri i f &i:G o n l y i f t h e i r images under f a r e i n t h e r e l a t i o n Si. F o r i n s t a n c e , i n c a s e o f b i n a r y r e l a t i o n s Ri and Si, t h i s means t h a t we have xRiy i f and o n l y i f f ( x ) S i f ( y ) . By a measurement f o r a g i v e n e m p i r i c a l r e l a t i o n a l sys-
tem, we mean a homomorphism f between t h i s s y s t e m and t h e c o r r e s p o n d i n g r e l a t i o n a l s y s t e m i n t h e n u meri cal domain.
For i n s t a n c e , Theorem 1.1 asserted t h a t i f t h e r e l a t i o n a l system s a t i s f i e s a p p r o p r i a t e axioms, t h e n t h e r e e x i s t s a measurement f , i.e. a homomorphism f between . Moreover, eve r y two homomorphisms must be connected i n such a way, i.e. g = Ff f o r some s t r i c t l y i n c r e a s i n g f u n c t i o n F.
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I n t h e second c a s e t h e freedom of c h o i c e was c o n s i d e r a b l y lower: i f f i s a homomorphism between (A, o , & ) and 0 i s a l s o such a homomorphism, and t h e c l a s s o f f u n c t i o n s g g e n e r a t e d i n s u ch a way i s t h e c l a s s o f a l l homomorphisms between these systems. The f u n c t i o n f : A --i, R e which e s t a b l i s h e s a homomorphi s m o f a n e m p i r i c a l r e l a t i o n a l s y s t e m and a n u meri cal r e l a t i o n a l s y s t e m i s sometimes c a l l e d a measurement s c a l e . The c l a s s o f a l l a d m i s s i b l e t r a n s f o r m a t i o n , i.e. t r a n s f o r m a t i o n which t r a n s f o r m t h e s c a l e f i n such a way t h a t t h e new f u n c t i o n i s a l s o a s c a l e , d e t e r m i n e s t h e s o c a l l e d t y p e of measurement s c a l e . I n t h e above examples t h e s c a l e s were o f t h e o r d i n a l and r a t i o t y p e r e s p e c t i v e l y . The names o f s c a l e s are i m p l i e d by t h e f a c t t h a t i n t h e
f i r s t c a s e t h e a d m i s s i b l e t r a n s f o r m a t i o n were l e a v i n g i n v a r i a n t only t h e o r d e r o f o b j e c t s o f t h e set A ,
I n t h e second c a s e , when t h e a d m i s s i b l e t r a n s f o r m a t i o n s were s i m i l a r i t i e s g = a f , o n l y t h e r a t i o s o f s c a l e val u e s remain i n v a r i a n t . I n g e n e r a l , t h e most o f t e n a p p e a r i n g t y p e s o f s c a l e s b e s i d e s mentioned already o r d i n a l and r a t i o s c a l e s a r e : ( a ) i n t e r v a l s c a l e , where t h e a d m i s s i b l e a r e a l l p o s i t i v e l i n e a r t r a n s f o r m a t i o n s ( i . e . g = a f + b , when a > 0 ) . The name o f t h i s s c a l e i s due t o t h e f a c t t h a t under t h e s e t r a n s f o r m a t i o n s t h e r a t i o s o f l e n g t h s of i n t e r v a l s remain i n v a r i a n t . ( b ) nominal s c a l e , where a d m i s s i b l e are a l l one to one t r a n s f o r m a t i o n s . Under t h e s e t r a n s f o r m a t i o n s
-
-
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only i d e n t i t y
o f e l e m e n t s remains i n v a r i a n t .
The above t y p e s o f s c a l e s do n o t e x h a u s t , of c o u r s e ,
a l l p o s s i b i l i t i e s ; t h e y were mentioned h e r e because t h e y appear most o f t e n i n e m p i r i c a l s i t u a t i o n s .
To d e f i n e t h e t y p e o f s c a l e , i . e . t o c h a r a c t e r i z e t h e a d m i s s i b l e t r a n s f o r m a t i o n s i s o f b a s i c importance i n u s i n g a s c a l e . It a l l o w s , f i r s t o f a l l , t o choose stat i s t i c s which a r e s e n s i b l e f o r a given s c a l e ( f o r example i t i s s e n s i b l e t o u s e e x p e c t e d v a l u e and v a r i a n c e f o r a t least i n t e r v a l s c a l e s , while i t i s not s e n s i b l e t o use it f o r an o r d i n a l s c a l e ) . Secondly t h e t y p e of s c a l e d e t e r m i n e s t h e c l a s s o f prop o s i t i o n s which one may s e n s i b l y f o r m u l a t e i n terms o f t h e s c a l e values.These are namely t h e p r o p o s i t i o n s whose t r u t h does n o t change under t r a n s f o r m a t i o n s admissib l e f o r a g i v e n measurement s c a l e . F o r example, t h e sent e n c e " o b j e c t A i s t w i c e as heavy as o b j e c t B" i s alwa y s t r u e o r always f a l s e , r e g a r d l e s s o f t h e c h o i c e o f the unit (admissible t r a n s f o r m a t i o n f o r r a t i o s c a l e s ) . On t h e o t h e r hand,the s e n t e n c e " t h e t e m p e r a t u r e t o d a y i s t w i c e a s h i g h as t e m p e r a t u r e y e s t e r d a y " does n o t make s e n s e : t h e r a t i o o f v a l u e s on a t e m p e r a t u r e s c a l e depends on t h e chosen u n i t and z e r o o f t h e s c a l e and t h e r e f o r e i s n o t i n v a r i a n t under t h e admissible t r a n s formations f o r i n t e r v a l scales. Thus, t h e d e t e r m i n a t i o n o f t h e c l a s s o f a l l transforma t i o n s which p r e s e r v e t h e homomorphism i s o f p r i m a r y importance f o r d e t e r m i n i n g t h e c l a s s o f s e n s i b l e propos i t i o n s which one i s e n t i t l e d t o make u s i n g t h e s c a l e v a l u e s . T h i s problem i s known as t h e problem of meaning;fulness
.
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3. THE ROLE AND CHARACTER OF A X I O M S I n t h e theorems from t h e p r e c e d i n g s e c t i o n s , t h e axioms for relation and c o n c a t e n a t i o n 0 were n o t e s p e c i a l l y i n t e r e s t i n g . They e x p r e s s e d t h e most obvious p r o p e r t i e s , u n q u e s t i o n a b l y s a t i s f i e d e.g. i n t h e c a s e o f concaten a t i o n o f rods or comparisons o f t h e i r l e n g t h s ( w i t h t h e e x c e p t i o n o f t h e r e l a t i o n - which may b e i n t r a n s i t i v e under i n s u f f i c i e n t p r e c i s i o n o f comparisons). Due t o g r e a t v a r i e t y o f measurement s i t u a t i o n s , t h e prob l e m s o f e x i s t e n c e o f a measurement s c a l e have t o be c o n s i d e r e d f o r v a r i o u s e m p i r i c a l r e l a t i o n a l systems. I n t h e s e s i t u a t i o n s , t h e axioms a r e not a l w a y s s a t i s f i e d i n a n obvious way; o f t e n (or p e r h a p s even i n most c a s e s ) t h e axioms e x p r e s s some q u a l i t a t i v e e m p i r i c a l laws, and t h e i r v e r i f i c a t i o n may be q u i t e d i f f i c u l t . The t h e o r e t i c a l problem, i n e a c h c a s e , c o n s i s t s of f i n d -
i n g a n axiom s y s t e m (on t h e e m p i r i c a l r e l a t i o n s ) which would imply t h e e x i s t e n c e o f a measurement o f a g i v e n t y p e . C l e a r l y , such a n axiom s y s t e m i s n e v e r determined u n i q u e l y , and t h e r e f o r e it may be worth w h i l e t o a n a l yze which a d d i t i o n a l c o n s t r a i n t s should be imposed on t h e s e t o f axioms. F i r s t o f a l l , t h e number o f axioms i s a r a t h e r i l l u s o r y
parameter, s i n c e a l l axioms may a l w a y s be made i n t o one by means o f c o n j u n c t i o n . N a t u r a l l y , one does n o t do it i n p r a c t i c e , and c o n c e p t u a l l y d i s t i n c t simple p r o p e r t i e s o f r e l a t i o n s o r o p e r a t i o n s are u s u a l l y f o r m u l a t e d a s s e p a r a t e axioms.
From t h e l o g i c a l p o i n t o f view, t h e most i m p o r t a n t pro-
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p e r t y i s c o n s i s t e n c y ( a n i n c o n s i s t e n t axiom s y s t e m i m p l i e s any a s s e r t i o n , hence i t i s u s e l e s s a s a s o u r c e o f theorems 1. Another l o g i c a l r e q u i r e m e n t i s t h e independence o f axioms, i . e . r e q u i r e m e n t t h a t none o f t h e axioms i s a l o g i c a l consequence o f t h e r e m a i n i n g ones. P r o o f s o f i n dependence may sometimes be d i f f i c u l t ; i n some c a s e s i t i s n o t known whether t h e axioms are independent o r n o t . T h i s i s , however, o f a secondary importance o n l y : any a s s e r t i o n which i s a consequence of a non-independ e n t s e t o f axioms i s a l s o a consequence o f a r e d u c e d s e t o f axioms, w i t h t h e dependent axioms removed. I n c a s e o f measurement t h e o r y , t h e s i t u a t i o n i s somewhat d i f f e r e n t t h a n i n c a s e s of o t h e r f o r m a l i z e d t h e o r i e s . The d i f f e r e n c e l i e s i n t h e f a c t t h a t t h e g o a l i n measurement t h e o r y i s n o t t o l o o k f o r more and more consequences of t h e a c c e p t e d system o f axioms, b u t r a t h e r t o choose a s e t o f axioms which would imply a s p e c i f i c a s s e r t i o n ( o f t h e e x i s t e n c e o f measurement o f a given t y p e ) . Consequently, t h e n a t u r a l q u e s t i o n h e r e i s t o d e t e r m i n e which axioms a r e n e c e s s a r y f o r t h e a s s e r t i o n , and which are n o t . The n e c e s s a r y axioms ( i . e . axioms which are i m p l i e d by t h e a s s e r t i o n ) e x p r e s s t h o s e p r o p e r t i e s which, i f n o t m e t , p r e v e n t t h e e x i s t e n c e o f measurement. For i n s t a n c e , i n Theorem 1 . 1 t h e t r a n s i t i v i t y o f relation i s a necessary condition f o r existence of measurement ( i n d e e d , i f f i s a measurement, t h e n f ( a ) > f ( b ) and f ( b ) >, f ( c ) i m p l y f ( a ) >, f ( c ) . Thus, if w e have a & b and b & c , w e must a l s o have a 2. c , . which means t h a t t h e r e l a t i o n % must be t r a n s i t i v e ) .
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On t h e o t h e r hand, t h e assumption t h a t t h e s e t A i s f i n i t e o r c o u n t a b l e i s n o t a n e c e s s a r y axiom: as shown i n Theorem 1.3, t h e r e e x i s t homomorphisms o f (A, &) i n t o (Re,),) w i t h t h e set A not b e i n g c o u n t a b l e . Some axioms which a r e n o t n e c e s s a r y p l a y t h e r o l e which i s d i f f e r e n t t h a n i n t h e p r e c e d i n g example: t h e y s e r v e namely as means o f e l i m i n a t i n g t r i v i a l , u n i n t e r e s t i n g o r p a t h o l o g i c a l s i t u a t i o n s . For i n s t a n c e , one may assume nonemptiness o f some s e t s , o r e x i s t e n c e of a t l e a s t two n o n e q u i v a l e n t e l e m e n t s , e t c . Another i m p o r t a n t group o f axioms may be d i s t i n g u i s h e d by a n a l y s i s o f t h e i r l o g i c a l n a t u r e ; t h e s e are t h e soc a l l e d s o l v a b i l i t y axioms, which a s s e r t t h e e x i s t e n c e o f e l e m e n t s o f t h e s e t A which s a t i s f y c e r t a i n p r o p e r t i e s . A s an example one may t a k e here axiom 4 i n Theorem 1 . 2 a s s e r t i n g e x i s t e n c e o f " d i f f e r e n c e " between b and a (for a b ) , i.e. element c such t h a t a 0 c = b.
>
A s a l r e a d y mentioned,
there are no c r i t e r i a f o r t h e
c h o i c e o f l'good" s e t of axioms. The c h o i c e i s b a s e d , t o a l a r g e e x t e n t , on i n t u i t i o n , e x p e r i e n c e , and moreo v e r , t h e p o s s i b i l i t y of v e r i f i c a t i o n o f axioms i n emp i r i c a l situations. It i s obvious, t h a t t h e d i f f e r e n c e s i n t h e d e g r e e o f d i f f i c u l t y i n v e r i f i c a t i o n o f axioms may be considerabl e . I n some c a s e s , t h e axioms are o b v i o u s l y t r u e , o r o b v i o u s l y f a l s e , s o t h a t one can h a r d l y s p e a k o f any "empirical" procedure of v e r i f i c a t i o n . This i s t h e case o f axioms which a s s e r t f i n i t e n e s s o f t h e s e t A , o r which a s s e r t t h a t A c o n t a i n s at l e a s t two elements. The res e a r c h e r g e n e r a l l y knows whether t h e s e t w i t h which he deals i s f i n i t e o r n o t , o r whether i t c o n t a i n s two non-
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e q u i v a l e n t elements. I n c a s e o f some o t h e r axioms, t h e s i t u a t i o n i s n o t s o simple. G e n e r a l l y , t h e d e g r e e o f d i f f i c u l t y i n v e r i f i c a t i o n depends b o t h on t h e l o g i c a l s t r u c t u r e o f t h e axiom and on t h e i n t e r p r e t a t i o n o f t h e c o n c e p t s which a p p e a r i n t h e axiom. Because of t h e g r e a t v a r i e t y of t y p e s o f axioms, and t h e v a r i e t y o f e m p i r i c a l s i t u a t i o n s , i t does n o t seem possible t o formulate general r u l e s f o r v e r i f i c a t i o n o f axioms. I n c a s e o f axioms which a s s e r t t h a t a c e r t a i n p r o p e r t y h o l d s f o r a l l p a i r s ( s a y ) o f e l e m e n t s of A , one could p e r h a p s i n d i c a t e which r e s u l t s o f exper i m e n t s would i n v a l i d a t e t h e axiom, i . e . l i s t p o s s i b l e h y p o t h e t i c a l p a i r s which would v i o l a t e t h e axiom. Howe v e r , i t may b e v e r y d i f f i c u l t t o d e s c r i b e t h e r e s u l t s o f e x p e r i m e n t s which one c o u l d t a k e as a s u f f i c i e n t premise t h a t t h e axiom h o l d s . A s a simple example, one may t a k e t h e t r a n s i t i v i t y ax-
>.
iom, s a y f o r r e l a t i o n I f one f i n d s even one t r i p l e t a , b , c such t h a t a b c a (where t h i s r e s u l t i s not due t o random p e r t u r b a t i o n s ) , t h e r e l a t i o n i s n o t t r a n s i t i v e , and t h e axiom should be r e j e c t e d . To a c c e p t t h e axiom as e m p i r i c a l l y v e r i f i e d , one needs s t u d i e s i n which one a n a l y s e s t r i p l e t s a,b,c. The d i f f i c u l t y l i e s here i n t h e f a c t t h a t checking t r a n s i t i v i t y on t r i p l e t s a , b , c which d i f f e r v e r y much i s n o t t o o convincing: one should s e l e c t such t r i p l e t s a , b , c f o r which one h a s a " f a i r chance" o f f i n d i n g i n t r a n s i t i v i t y , if t h e axiom i n r e a l i t y does n o t hold ( i . e . i f t h e r e e x i s t s a t l e a s t one i n t r a n s i t i v e t r i p l e t ) . T h i s leads n a t u r a l l y t o t h e s t u d y o f t r i p l e t s which d i f f e r
> > s
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very l i t t l e . I n such c a s e s , however, f i n d i n g i n t r a n s i t i v i t y may be s i m p l y due t o o b s e r v a t i o n e r r o r s . Moreo v e r , t h e l a r g e r number o f such t r i p l e t s a n a l y s e d , t h e h i g h e r chance o f f i n d i n g an i n t r a n s i t i v e one due o n l y t o observation error. For o t h e r t y p e s o f d i f f i c u l t y , one may c o n s i d e r axioms a s s e r t i n g t h e e x i s t e n c e o f s o l u t i o n s o f some e q u a t i o n s o r i n e q u a l i t i e s ( f o r example, axioms 4 and 6 i n Theor e m 1.2). I n such c a s e s , it i s e q u a l l y d i f f i c u l t t o c h a r a c t e r i z e t h e r e s u l t s which would i n d i c a t e t h a t t h e axiom h o l d s , as w e l l as t h e r e s u l t s which i n d i c a t e t h a t i t d o e s not hold. For i n s t a n c e , suppose t h a t t h e axiom a s s e r t s t h a t f o r any two e l e m e n t s t h e r e e x i s t s a t h i r d element, which s a t i s f i e s a c e r t a i n c o n d i t i o n W. If t h e set A i s f i n i t e , and f o r some p a i r a , b o f e l e m e n t s a l l elements are checked, w i t h none s a t i s f y i n g c o n d i t i o n W, t h e n t h e axiom c l e a r l y does n o t hold. I f , however, t h e s e t A i s i n f i n i t e , o r so numerous t h a t s y s t e m a t i c checking o f a l l i t s e l e m e n t s is n o t f e a s i b l e , t h e n t h e s i t u a t i o n i s n o t so simple. The f a c t t h a t f o r some p a i r a , b we f a i l t o f i n d a n element c w i t h p r o p e r t y W does n o t have t o mean t h a t t h e axiom i s n o t v a l i d ; i n such c a s e s , i t i s d i f f i c u l t t o p o i n t o u t t h e c o n d i t i o n s , i n terms o f e m p i r i c a l r e s u l t s , which would s i g n i f y t h a t t h e axiom i s n o t v a l i d . On t h e o t h e r hand, suppose a s e r i e s o f p a i r s checked, and f o r e a c h p a i r an element c w i t h W i s found. S t i l l , one cannot c l a i m t h a t t h e v a l i d : t h e d i f f i c u l t i e s here are t h e same as i n g t h e t r a n s i t i v i t y axiom.
a , b are
property axiom is w i t h test-
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4. EXTENSIVE MEASUREMENT I N PSYCHOLOGY The main d i f f i c u l t y i n a p p l i c a t i o n of e x t e n s i v e measu-
rement i n psychology (and i n s o c i a l s c i e n c e s i n genera l ) i s t h e f a c t t h a t i n most c a s e s one cannot f i n d an a d e q u a t e i n t e r p r e t a t i o n f o r t h e o p e r a t i o n o f concatena t i o n ( a d d i t i o n o f e l e m e n t s ) . I n c a s e s when such a n o p e r a t i o n i s found, i t o f t e n leads t o t r i v i a l r e s u l t s , o r a l t e r n a t i v e l y , t h e axioms which a r e n e c e s s a r y f o r t h e e x i s t e n c e o f measurement a r e o b v i o u s l y f a l s e . Thus, f o r c o n s t r u c t s such as i n t e l l i g e n c e , a t t i t u d e s , a b i l i t i e s , e t c . t h e r e i s no a d d i t i o n o p e r a t i o n . On t h e o t h e r hand, when a n a l y s i n g such phenomena as t h e p e r c e p t i o n o f w e i g h t , o r of t h e d u r a t i o n of s t i m u l i , t h e c o n c a t e n a t i o n may be d e f i n e d i n a n a t u r a l way. However, when t h e experiment i s performed w i t h s u f f i c i e n t prec i s i o n , one r e c o v e r s from t h e s u b j e c t t h e same o r d e r (up t o d i s c r i m i n a t i o n e r r o r s ) which one would o b t a i n by w e i g h t i n g , o r measuring time w i t h s t o p watch, e t c . F i n a l l y , i n some s i t u a t i o n s t h e c o n c a t e n a t i o n i s w e l l d e f i n e d , b u t t h e axioms o f e x t e n s i v e measurement a r e not satisfied. This i s t r u e , f o r instance, i n t h e study o f u t i l i t i e s o f composite o b j e c t s . L e t us c o n s i d e r t h e s i m p l e s t c a s e , when t h e s e t A cons i s t s of p a i r s ( x , y ) , where x and y r e p r e s e n t t h e amou n t s o f some two goods. The c o n c a t e n a t i o n o f ( x l , y l ) and ( x 2 , y 2 ) i s d e f i n e d , i n t h e most n a t u r a l way, as
If now t h e symbol ). s t a n d s f o r p r e f e r e n c e (i.e. t h e
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+
r e l a t i o n (xl,yl) ( x 2 , y 2 ) means t h a t t h e f i r s t p a i r i s p r e f e r r e d t o t h e s e c o n d ) , t h e n one cannot e x p e c t axiom 3 o f Theorem 1 . 2 ( o f m o n o t o n i c i t y ) t o be always t r u e ; t h i s axiom r e q u i r e s t h a t t h e c o n d i t i o n a > b implies that a 0 c b o c f o r any c. F o r example ( s e e Krantz e t a l . , 1971), suppose t h a t t h e f i r s t c o o r d i n a t e s t a n d s f o r t h e number o f s h i r t s , and t h e second f o r t h e number o f p a i r s o f p a n t s . One can
-
>
( 0 , 3 ) , i.e. imagine t h a t f o r someone w e have ( 3 , O ) he p r e f e r s 3 s h i r t s t o 3 p a i r s o f p a n t s . However, i t may w e l l b e t h a t
I n o t h e r words, i t i s n o t t r u e t h a t he p r e f e r s 6 s h i r t s t o t h e s e t c o n s i s t i n g o f 3 s h i r t s and 3 p a i r s o f p a n t s . The above mentioned d i f f i c u l t i e s w i t h f i n d i n g a n ade-
quate i n t e r p r e t a t i o n f o r the operation of concatenation i n psychology l e d t o a l a s t i n g f o r some time o p i n i o n t h a t e x t e n s i v e measurement i n psychology (and more gen e r a l l y - i n the s o c i a l sciences) i s not possible. T h i s c o n v i c t i o n appeared i n c o r r e c t f o r two r e a s o n s .
a s w i l l be shown i n t h e n e x t s e c t i o n s - e x t e n s i v e measurement may e x i s t even i n s i t u a t i o n s i n which t h e r e i s no a d e q u a t e i n t e r p r e t a t i o n f o r c o n c a t e n a t i o n . Firstly
-
Secondly, i n two s i t u a t i o n s i t was p o s s i b l e t o f i n d a n i n t e r p r e t a t i o n for c o n c a t e n a t i o n , l e a d i n g t o e x t e n s i v e measurement i n psychology. One o f such examples i s t h e measurement o f r i s k . Concatenation i s i n t e r p r e t e d here a s c o n v o l u t i o n o f p r o b a b i l i t y d i s t r i b u t i o n s , i . e . a d d i t i o n o f independent random v a r i a b l e s , Another ex-
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ample i s s u b j e c t i v e p r o b a b i l i t y , where c o n c a t e n a t i o n may be r e p r e s e n t e d a s a d d i t i v i t y f o r d i s j o i n t e v e n t s .
4.1.
Measurement o f r i s k
The p o s s i b i l i t y of e x t e n s i v e measurement o f r i s k was
d i s c o v e r e d r e l a t i v e l y r e c e n t l y (see P o l l a t s e k and Tversky 1970). R i s k i s a p r o p e r t y o f s i t u a t i o n s i n which t h e r e s u l t s are d e f i n e d i n terms o f p r o b a b i l i t y d i s t r i b u t i o n s o f t h e i r o c c u r r e n c e s . The i n t u i t i v e f o u n d a t i o n f o r r i s k measurement l i e s i n t h e p o s s i b i l i t y o f e v a l u a t i n g some s i t u a t i o n s - t o b e c a l l e d l o t t e r i e s - as more r i s k y t h a n o t h e r s i t u a t i o n s . F o r example, i n v e s t i n g o n e ' s s a v i n g i n one e n t e r p r i s e may b e more r i s k y t h a n i n v e s t i n g i t i n a n o t h e r e n t e r p r i s e , o r l e a v i n g t h e money i n t h e s a v i n g account. I n such s i t u a t i o n s , t h e dec i s i o n leads t o a l o t t e r y , understood a s a s e t o f pos s i b l e r e s u l t s ( g a i n s o r l o s s e s ) which may o c c u r w i t h given p r o b a b i l i t i e s . For measurement o f r i s k i t i s n e c e s s a r y t o have f o r m a l r e p r e s e n t a t i o n o f each l o t t e r y as a p r o b a b i l i t y d i s t r i b u t i o n on t h e r e a l l i n e , Numbers r e p r e s e n t h e r e t h e val u e s o f p a r t i c u l a r outcomes ( t h e i r u t i l i t i e s ) . F o r example, t h e investment o f $ 1 . 0 0 0 i n some e n t e r p r i s e may lead t o bankrupcy ( l o s s o f $ 1 . 0 0 0 ) w i t h p r o b a b i l i t y 0.1, and g a i n o f 50 I ( a f t e r a y e a r , s a y ) w i t h probabil i t y 0.9. Such an Investment may be d e s c r i b e d as a lot t e r y , i n form o f a p r o b a b i l i t y d i s t r i b u t i o n c o n c e n t r a t ed on two p o i n t s , namely on -1.000 and +1.500, w i t h p r o b a b i l i t i e s r e s p e c t i v e l y 0 . 1 and 0.9. Leaving t h i s money I n t h e bank a t 4.5 % i n t e r e s t leads t o a "degen e r a t e " l o t t e r y , r e p r e s e n t e d as a p r o b a b i l i t y d i s t r i b u t i o n Concentrated a t one p o i n t $ 1.045. Thus t h e s t a r t -
SELECTED TOPICS IN MEASUREMENT THEORY
469
9
i n g p o i n t o f t h e formal c o n s t r u c t i o n i s t h e set of p r o b a b i l i t y d i s t r i b u t i o n s , denoted b y f , g , , . , ; e a c h o f t h e s e d i s t r i b u t i o n s r e p r e s e n t s a l o t t e r y , o r r i s k y sit u a t ion. A s t h e o p e r a t i o n o f c o n c a t e n a t i o n , w e t a k e t h e convolution of probability distributions; thus, f 0 g i s the d i s t r i b u t i o n o f t h e sum o f two independent random vari a b l e s , w i t h d i s t r i b u t i o n s r e s p e c t i v e l y f and g. F o r random v a r i a b l e s o f c o n t i n u o u s t y p e , i.e. when f and g are d e n s i t i e s , f 0 g i s d e f i n e d by
For d i s c r e t e random v a r i a b l e s , i f f i s c o n c e n t r a t e d
...
a t p o i n t s xl, x2, xn assumed w i t h p r o b a b i l i t i e s pl, p2,...,pn, and g i s c o n c e n t r a t e d a t p o i n t s y l , y 2 , *.*, Y, assumed w i t h p r o b a b i l i t i e s ql, q2, qm, then the d i s t r i b u t i o n f o g i s concentrated a t points o f t h e form xi t y , w i t h c o r r e s p o n d i n g p r o b a b i l i t i e s
...,
j
being
Pi9j.
Moreover, we assume t h a t we have a r e l a t i o n & i n t h e set r e f l e c t i n g t h e o r d e r i n g o f d i s t r i b u t i o n s from t h e p o i n t o f view of t h e d e g r e e o f r i s k which t h e y i n v o l v e : f & g means t h a t t h e d i s t r i b u t i o n ( l o t t e r y ) f i s more ( n o t l e s s ) r i s k y t h a n t h e l o t t e r y g.
9,
N a t u r a l l y , f o r measurement o f r i s k , t h e r e l a t i o n a l s y s tem must meet some c o n d i t i o n s , We namely have THEOREM 4.1. Let 9 be a nonempty set o f p r o b a b i l i t y d i s t r i b u t i o n s , .closed w i t h r e s p e c t t o t h e o p e r a t i o n of
CHAPTER 5
410
c o n c a t e n a t i o n , and assume t h a t is a binary r e l a t i o n i n 9 , s a t i s f y i n g t h e following conditions:
-
1.
(e. & is
i s a weak o r d e r
t r a n s i t i v e and con-
nected);
if f 2: g, then f 2 9 , i f it i s not
2. For anx
f,g,hcF,
3. F o r a l l
f,g,h,h'
o
h
&g
o
h;
t r u e that g & f , t h e n t h e r e e x i s t s a n n such t h a t f n o h & gn o h ' (where f n i s t h e n - f o l d c o n c a t e n a t i o n o f f with itself ; Then t h e r e e x i s t s a f u n c t i o n R d e f i n e d on that
9 such
-
(ii) R ( f
og)
= R ( f ) t R(g).
Moreover, if any o t h e r f u n c t i o n R ' ( i i ) ,t h e n R t = a R for some p o s i t i v e a. Here R i s a measure o f r i s k a s s o c i a t e d w i t h a g i v e n d i s t r i b u t i o n ( l o t t e r y ) . The c o n d i t i o n (i) a s s e r t s t h a t t h e numerical value of r i s k i s higher f o r d i s t r i b u t i o n s which a r e more r i s k y ( a s d e s c r i b e d by t h e r e l a t i o n Moreover, t h e r i s k i s a d d i t i v e : t h e r i s k a s s o c i a t e d w i t h a sum of two i n d e p e n d e n t l o t t e r i e s e q u a l s t h e sum o f r i s k s a s s o c i a t e d w i t h t h e components.
2.).
Axioms 1 , 2 and 3 c o n s t i t u t e , i n f a c t , a c e r t a i n v e r s i o n o f axioms f o r e x t e n s i v e measurement; i n p a r t i c u l a r , axiom 3 i s a m o d i f i e d v e r s i o n o f Archimedean axiom g, i.e. f i s s t r i c t ( s e e axiom 6 o f Theorem 1 . 2 ) : i f f l y more r i s k y t h a n g, t h e n under n-fold i n d e p e n d e n t rep e t i t i o n o f f and g, t h e i n e q u a l i t y f")- gn i s n o t
>
47 I
SELECTED TOPICS INMEASUREMENT THEORY
o n l y p r e s e r v e d , b u t i n a se n se , t h e d i f f e r e n c e o f i t s two sides i n c r e a s e s i n d e f i n i t e l y . Indeed, i t i s asserted t h a t f o r some n we s h a l l have f n o h gn o h ' , reg a r d l e s s o f h and h ' . It i s i n t e r e s t i n g t h a t i f one r e s t r i c t s t h e c o n s i d e r -
a t i o n s t o d i s t r i b u t i o n s w i t h f i n i t e v a r i a n c e s , and i m p o s e s some a d d i t i o n a l c o n s t r a i n t s on t h e r e l a t i o n a l s t r u c t u r e
*
To f o r m u l a t e t h ese c o n d i t i o n s , l e t c s t a n d f o r t h e probability d i s t r i b u t i o n concentrated a t the point c ( i . e . t h e p r o b a b i l i t y d i s t r i b u t i o n o f a random v a r i a b l e which assumes t h e v a l u e c w i t h p r o b a b o l i t y 1). Next, l e t af s t a n d f o r t h e p r o b a b i l i t y d i s t r i b u t i o n o b t a i n e d from f b y change of s c a l e i n r a t i o a ( i . e . a f ( t ) = f ( t / a ) / [ a j ) . F i n a l l y , l e t E ( f ) and C ; * ( f ) d en o t e t h e e x p e c t a t i o n and v a r i a n c e o f t h e d i s t r i b u t i o n f . We have t h e f o l l o w i n g
Assume t h a t t h e r e l a t i o n a l system (9, b, s a t i s f i e s t h e c o n d i t i o n s 1 , 2 and 3 o f Theorem 4.1. and moreover, f o r a l l f ,g 9 and r e a l a we have THEOREM 4.2.
4. a*E5';
5. af
€9;
6.if
a ) O , g
*
-
7. I f E ( f )
= E ( g ) = 0, t h e n
(a)
af $
(b)
f
8. If f n
f k f o a
f
a > 1;
g i f , and o n l y i f 4 f, E(fn)
a f > ag ;
2 = E ( f ) , G (fn) =
d
2
(f)
0)
472
CHAPTER 5
then R(fn)
jR(f).
Then t h e r e e x i s t s a c o n s t a n t q ( 0
( 1) such t h a t
The s e n s e o f t h i s theorem i s a s fo l l o ws . AssumptiOnS
4
and 5 a s s e r t t h a t t h e c l a s s 9 c o n t a i n s a l l d e g e n e r a t e p r o b a b i l i t y d l s t r i b u t i o n s , and a l s o t h a t i t c o n t a i n s a l l d i s t r i b u t i o n s o b t a i n e d by a s c a l e change from i t s members. Assumption 6 a s s e r t s t h a t a d d i t i o n o f a const a n t t o a l o t t e r y does n o t i n c r e a s e t h e r i s k i f t h e c o n s t a n t i s p o s i t i v e . The c o n d i t i o n 7 co n cern s " f a i r " l o t t e r i e s , i . e . l o t t e r i e s w i t h mean 0. It a s s e r t s t h a t t h e r i s k o f such a l o t t e r y i n c r e a s e s when t h e s c a l e u n i t s a r e i n c r e a s e d , and a l s o t h a t an i n c r e a s e o f s c a l e u n i t s does n o t change t h e r i s k o r d e r . F i n a l l y , c o n d i t i o n 8 a s s e r t s c o n t i n u i t y o f r i s k under p a s s a g e t o t h e l i m i t f o r sequence of l o t t e r i e s w i t h t h e same means and v a r i ance s
.
The a s s e r t i o n o f t h e theorem i s t h a t t h e r i s k i s a l i n -
e a r combination o f v a r i a n c e and mean. The meaning o f t h i s theorem and t h e s e n s e o f axioms w i l l
become c l e a r e r , i f i n s t e a d o f d i s t r i b u t i o n s , one u s e s t h e random v a r i a b l e s w i t h t h e s e d i s t r i b u t i o n s , s a y X, Y,... , Each such random v a r i a b l e r e p r e s e n t s a c e r t a i n lottery, o r r i s k y situation. I n the sequel, let X b Y stand f o r the f a c t that t h e l o t t e r y X i s at l e a s t a s r i s k y as l o t t e r y Y. To s i m p l i f y f o r m u l a t i o n s , l e t u s assume t h a t a l l random v a r i a b l e s under c o n s i d e r a t i o n s are independent. Thus, f o r i n s t a n c e , X t Y d e n o t e s t h e combined g a i n (loss) r e s u l t i n g from p a r t i c i p a t i o n i n
413
SELECTED TOPICS Ri MEASUREMENT THEORY
two independent l o t t e r i e s X and Y. The f i r s t axiom a s s e r t s t h a t whenever X
@
Y and Y
&
Z,
then X Z , which means t h a t t h e d e g r e e s o f r i s k a r e t r a n s i t i v e , and a l l random v a r i a b l e s a r e comparable (connectivity). The second axiom a s s e r t s t h a t i f X , Y and Z a r e independ-
8
e n t and X Y , t h e n X + Z ) Y + Z. To t a k e p a r t i n b o t h X and Z i s a t l e a s t as r i s k y as t a k i n g p a r t i n b o t h Y and Z. Axiom 3 (Archimedean) s t a t e s t h e f o l l o w i n g p r o p e r t y . L e t X1, X 2 , be a sequence o f independent random vari a b l e s w i t h t h e same d i s t r i b u t i o n , and l e t a l s o Y1, Y 2 , b e a sequence o f independent random v a r i a b l e s w i t h t h e same d i s t r i b u t i o n ( p e r h a p s d i f f e r e n t t h a n t h a t o f the Xi's). Moreover, l e t Z and W d en o t e two l o t t e r i e s , m u t u a l l y independent and a l s o independent o f a l l X i ' s and Y i ' s .
...
...
Then, i f X1> Y1 (i.e. i f t h e l o t t e r y X1 i s s t r i c t l y r i s k i e r t h a n Y1), t h e r e e s i s t s an n such t h a t t h e l o t t e r y X1 t t Xn t Z i s more r i s k y t h a n t h e l o t t e r y Y1 t + Yn t W. I n o t h e r words, as n i n c r e a s e s , t h e d i f f e r e n c e between r i s k i n e s s o f X1 + t Xn and Y1 + t Yn ( i . e . n -fo l d p a r t i c i p a t i o n i n l o t t e r y X1 and n- fo l d p a r t i c i p a t i o n i n l o t t e r y Y1) becomes so l a r g e t h a t i t cannot be compensated by p a r t i c i p a t i o n i n any l o t t e r i e s Z and W.
...
...
...
...
Axiom 4 s t a t e s t h a t one a l s o c o n s i d e r s d e g e n e r a t e l o t t e r i e s , i . e . p r i z e s o r l o s s e s which are c e r t a i n t o o ccu r ( w i t h o u t t h e randomness a s s o c i a t e d w i t h a l o t t e r y ) .
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414
Axiom 5 asserts t h a t one may change t h e s c a l e i n l o t t e r i e s : i f a random v a r i a b l e X b e l o n g s t o t h e c l a s s o f l o t t e r i e s under c o n s i d e r a t i o n , t h e same i s a l s o t r u e f o r t h e random v a r i a b l e ax. Axioms 6 and 7 a r e c r u c i a l f o r t h e a s s e r t i o n o f t h e theorem. Axiom 6 a s s e r t s t h a t a d d i t i o n o f a p o s i t i v e c o n s t a n t does n o t i n c r e a s e t h e r i s k : X i s a t l e a s t as r i s k y as X t c , i f o n l y c > 0. I n o t h e r words: t h e p a r t i c i p a t i o n i n a l o t t e r y X i s more r i s k y t h a n p a r t i c i p a t i o n i n t h e l o t t e r y X, f o r which one i s b e i n g p a i d e. Axiom 7 a s s e r t s t h a t if t h e l o t t e r y X i s f a i r ( i . e . E ( X ) = O), and a 7 1, t h e n t h e l o t t e r y aX i s more r i s k y t h a n l o t t e r y X. Thus, a game w i t h a l l stakes i n c r e a s e d i n t h e same p r o p o r t i o n i s more r i s k y . The second p a r t o f axiom 7 s t a t e s t h a t whenever E ( X ) = E ( Y ) = 0 , and X i s more r i s k y t h a n Y , t h e n t h e same i s t r u e under any p r o p o r t i o n a l change o f stakes. F i n a l l y , axiom 8 i s t h e o n l y one w i t h no d i r e c t and n a t u r a l i n t e r p r e t a t i o n i n terms o f random v a r i a b l e s , and n o t t h e i r d i s t r i b u t i o n s . It a s s e r t s t h a t i f t h e i s such t h a t t h e d i s t r i b u t i o n s sequence X1, X,, o f t h e terms t e n d t o a c e r t a i n d i s t r i b u t i o n o f a random v a r i a b l e X, and a l s o E(X1) = E ( X 2 ) = and Gc(X1) = 2 d (X,) = t h e n t h e sequence of r i s k s a s s o c i a t e d w i t h l o t t e r i e s X1, X 2 , tends t o the r i s k of t h e l o t -
...
...,
...
n
...
t e r y X. The a s s e r t i o n o f t h e theorem is t h a t t h e r i s k o f t h e
l o t t e r y X depends s o l e l y on t h e v a r i a n c e and mean of X, and t h a t t h i s dependence is l i n e a r o f t h e form R ( X ) = qCi'(X)
-
(1
-
q)E(X),
SELECTED TOPICS IN MEASUREMENT THEORY
415
where q i s some c o n s t a n t between 0 and 1. Thus, f o r " f a i r " l o t t e r i e s , 1.e. when E(X) = 0 , t h e r i s k i s prop o r t i o n a l t o t h e v a r i a n c e . On t h e o t h e r hand, f o r a f i x 2 ed v a r i a n c e G (XI, t h e r i s k becomes smaller when t h e l o t t e r y becomes more f a v o u r a b l e , i . e . when t h e mean
E(X) i n c r e a s e s . 4.2.
Q u a l i t a t i v e scale of probability
The second n o n t r i v i a l example o f c o n c a t e n a t i o n o u t s i d e
t h e scope o f p h y s i c a l s c i e n c e s i s t h e c o n s t r u c t i o n o f the s c a l e of probability. The problem may be f o r m u l a t e d as f o l l o w s . Suppose we
c o n s i d e r a f i x e d c l a s s o f e v e n t s A, B, C,..., f o r which there are d e f i n e d p r o b a b i l i t i e s P ( A ) , P ( B ) , P(C),... Then w e may d e f i n e t h e r e l a t i o n i n t h e c l a s s o f eve n t s by p u t t i n g A B i f P ( A ) 2 P ( B ) . Thus, & i s t h e r e l a t i o n "at l e a s t as p r o b a b l e as".
8
Moreover, t h e p r o b a b i l i t y f u n c t i o n s a t i s f i e s t h e a d d i t i v i t y p r o p e r t y : f o r m u t u a l l y e x c l u s i v e e v e n t s A and B w e have P ( A u B) = P ( A )
+
P(B).
Formally, t h e o p e r a t i o n of a d d i t i o n o f m u t u a l l y e x c l u s i ve ( d i s j o i n t ) e v e n t s may be regarded a s a c o n c a t e n a t i o n ; t h e above c o n d i t i o n s t a t e s simply t h a t t h e p r o b a b i l i t y f u n c t i o n must be a d d i t i v e w i t h r e s p e c t t o c o n c a t e n a t i o n . Consequently, t h e f u n c t i o n P i s a measurement f o r t h e rel a t i o n a l s y s t e m c o n s i s t i n g o f a c e r t a i n number o f e v e n t s , relation and t h e o p e r a t i o n o f c o n c a t e n a t i o n . Here
476
CHAPTER 5
>
the relation was d e f i n e d i n terms of P. One can t h e r e f o r e pose a q u e s t i o n , which i s , i n a s e n s e , oppos i t e : suppose t h a t t h e r e l a t i o n ( " i s a t l e a s t a s prob a b l e a s " ) i s a p r i m i t i v e n o t i o n , and ask f o r t h e cond i t i o n s which i t must meet i n o r d e r for t h e e x i s t e n c e o f t h e p r o b a b i l i t y f u n c t i o n P c o n s t i t u t i n g a measurement for this relation. To f o r m a l l y p r e s e n t t h e r e s u l t s , w e s t a r t w i t h i n t r o -
duc in g some d e f i n i t i o n s . L e t E s t a n d f o r a nonempty set. I t s el emen t s w i l l be c a l l e d ele m en t a ry e v e n t s , and t h e s e t E w i l l b e c a l l e d t h e sample s p a c e , o r space o f e l e m en t ary e v e n t s . A nonempty c l a s s y o f s u b s e t s o f E w i l l be c a l l e d an algebra, i f i t s a t i s f i e s t h e f o l l o w i n g c o n d i t i o n s :
1.
If A F Y , t h e n
2.
If
A,Bfy,
E
-A
= -A€Y;
then A u B E Y ;
If, i n a d d i t i o n , t h e c l a s s
f
satisfies the condition 00
then
i s c a l l e d a $ -a l g e b ra .
One can p r o v e t h a t e v e ry a l g e b r a i s c l o s e d a l s o w i t h r e s p e c t t o t h e o p e r a t i o n o f i n t e r s e c t i o n and d i f f e r e n c e o f e v e n t s ( t h i s f o l l o w s a t once from de Morgan laws); on t h e o t h e r hand, e a c h 6 - a l g e b r a i s c l o s e d w i t h resp e c t t o t h e o p e r a t i o n o f c o u n t a b l e i n t e r s e c t i o n s of events. L e t u s o b s e r v e t h a t if
7 i s an
algebra of events, then
417
SELECTED TOPICS IN MEASUREMENT THEORY
E
ef
7,
and B E
( s i n c e if A E then -AEyby condition = E G Y by c o n d i t i o n 2. Moreover, 0 = by c o n d i t i o n 1).
1, and A u (-A)
-E
€7
The s y s t e m (E,
y,
P> w i l l be c a l l e d a f i n i t e l y a d d i t i v e
p r o b a b i l i t y space, i f
(a)
3is
a n algebra o f s u b s e t s o f E;
( b ) P i s a n u m e ri c a l f u n c t i o n , d e f i n e d on
9,
such
that -
( b . 2 ) P ( E ) = 1;
P(A u B ) = P ( A )
+
P(B).
On t h e o t h e r hand, i f (c)
i s a 6-alaebra;
( d ) whenever
A1,
A*,
...E'f
then M
and
Ai n A
j
= 0
for i # j ,
00
2
P ( u An> = P(An), n =1 n= 1 t h e n P i s c a l l e d a countably a d d i t i v e p r o b a b i l i t y . The ele m en t s o f t h e a l g e b r a are c a l l e d e v e n t s f 'un c ti o n P i s c a l l e d p r o b a b i l i t y .
, and
L e t u s c o n s i d e r now a s y s t e m o f t h e form (E, ,&) ; f i s a n algebra o f s u b s e t s
here E i s some nonempty s e t ,
478
CHAPTER 5
y.
i s a b i n a r y r e l a t i o n on A s already meno f E , and t i o n e d , t h e problem l i e s i n f i n d i n g c o n d i t i o n s on t h e s y s t e m (E, , under which t h e r e e x i s t s a f u n c t i o n , P, b e i n g a ( f i n i t e l y a d d i t i v e ) p r o b a b i l i t y on and such t h a t
2.)
P(A)
2
P(B)
7,
if and o n l y i f , A
B.
L e t u s assume f i r s t t h a t such a f u n c t i o n e x i s t s , and t r y t o f i n d the necessary conditions.
>
y,
must o r d e r weakly t h e s e t i.e. it must be t r a n s i t i v e and connected ( t h i s i s a n obvious necessary requirement). F i r s t of all,
Next, s i n c e P ( E ) = 1 and P ( 0 ) = 0 , we must have E & 0. Moreover, s i n c e 0 P ( A ) 5 1 for e v e r y A € , we must have 0 $ A 3 E. Thus, E i s t h e f i r s t , and 0 i s t h e l a s t element i n t h e o r d e r i n g
2
<
>.
L e t u s assume now t h a t A i s d i s j o i n t w i t h B and a l s o C. Then P ( A U B ) = P ( A ) d i s j o i n t w i t h C , and t h a t B t P ( B ) > P ( A ) + P ( C ) = P ( A u C ) . Conversely, i f A u B 2 A u C, t h e n P(A) t P ( B ) = P(A u B ) P(A U C ) = P ( A ) t P ( C ) , hence P ( B ) P ( C ) , or B & C . Consequently, w e must have t h e m o n o t o n i c i t y c o n d i t i o n : i f A i s d i s j o i n t w i t h B and c, t h e n B )r C i f , and o n l y i f , A u B
>
>
3
> A u C .
L e t u s now c o n s i d e r t h e Archimedean axiom. L e t w d e n o t e t h e e q u i v a l e n c e r e l a t i o n induced b y , i.e. A N B i f A ) B and B > A . Thus, Am B means t h a t A and B are
&
equiprobable.
419
SELECTED TOPICS INMEASUREMENT THEORY
7.
L e t now A d e n o t e a n a r b i t r a r y e v e n t i n We s h a l l s a y i s stant h a t t h e sequence A1, A 2 , o f elements o f d a r d w i t h r e s p e c t t o A, i f f o r e v e r y k t h e e v e n t Ak may b e p a r t i t i o n e d i n t o t h e union Ak = B1 u B2 u c, Bk /V Bk o f d i s j o i n t e v e n t s such t h a t B1m B2 .cv A.
...
-
)i
...
H f A1,
A2,
... i s a s t a n d a r d sequence
...
with respect t o A,
t h e n w e have P ( A k ) = P(B1 u
= P(B1)
+
... u Bk)
...
t P(Bk) = kP(A).
>
It f o l l o w s t h a t i f P ( A ) 0, i.e. A 0, t h e n t h e standard sequence must b e f i n i t e ( s i n c e kP(A) = P ( A k ) 1, hence k l / P ( A ) : any s t a n d a r d sequence based on A may have a t most l / P ( A ) e l e m e n t s ) .
<
To sum up t h e c o n s i d e r a t i o n s up t o now, w e may i n t r o d u c e the following definition:
,>) DEFINITION. A s y s t e m (E, a t i v e probability scale, i f 1. 2.
> E
i s a weak o r d e r i n
>0
and A
>0
w i l l be c a l l e d a q u a l i t -
7;
f o r e v e r y A,
3. If A p\ B = 0 and A n C = 0, t h e n B i f A u B b A U C ;
&C
i f and o n l y
>
4. If A 0, t h e n any sequence which i s s t a n d a r d w i t h r e s p e c t t o A must have o n l y a f i n i t e number o f element s. The f i r s t q u e s t i o n t o a s k i s whether t h e c o n d i t i o n s
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480
1
-
4 a r e s u f f i c i e n t t o imply t h e e x i s t e n c e o f probabi-
l i t y f u n c t i o n P on t h e a l g e b r a
3,
which would c o n s t i t u t e a measurement f o r t h e r e l a t i o n 2 . It t u r n s o u t ( s e e Krantz e t a l . , 1971) t h a t t h e answer i s n e g a t i v e : t h e r e e x i s t s y s t e m s o f q u a l i t a t i v e p r o b a b i l i t y (E, , i.e. s y s t e m s s a t i s f y i n g c o n d i t i o n s 1-4, f o r which t h e p r o b a b i l i t y P d o e s n o t e x i s t . We have t h e r e f o r e a probl e m of f i n d i n g a d d i t i o n a l a s s u m p t i o n s , which would s u f f i c e f o r e x i s t e n c e o f P. We p r e s e n t below one s u c h c o n d i t i o n , due t o Savage (1954).
9 ,h )
5. Assume t h a t (E,
9 ,$>
i s a system o f q u a l i t a t i v e
p r o b a b i l i t y . If A , B E 3 and A B, t h e n t h e r e e x i s t s a o f t h e s e t E , such t h a t f o r e v e r y p a r t i t i o n {C1,. ,Cn) i = 1,2,...,n w e have A B u ci.
..
>
B ( b y m o n o t o n i c i t y ) , t h e c o n d i t i o n above S i n c e B U Ci means t h a t whenever A )- B, one can p a r t i t i o n t h e whole s p a c e i n t o e v e n t s C1, Cn, e a c h h a v i n g so small probab i l i t y t h a t e v e r y e v e n t B u Ci i s "between" e v e n t s A and B.
...,
To e x p l a i n b e t t e r t h e meaning o f axiom 5, i t i s w o r t h w h i l e t o p r e s e n t some of i t s consequences.
The q u a l i t a t i v e p r o b a b i l i t y s t r u c t u r e (E, ,&> w i l l be c a l l e d f i n e , if f o r e v e r y A ZS t h e r e e x i s t s a p a r t i t i o n o f t h e set E i n t o C1, , C n such t h a t A Ci n. f o r i = 1,2
,...,
...
+
I n t u i t i v e l y , t h i s p r o p e r t y means t h a t t h e sample s p a c e may be p a r t i t i o n e d i n t o e v e n t s w i t h a r b i t r a r i l y small probabilities. Next, f o r a g i v e n q u a l i t a t i v e p r o b a b i l i t y s t r u c t u r e ,
SELECTED TOPICS IN MEASUREMENT THEORY
let A N
*
B, i f f o r e v e r y C , D G
B n D = 0, w e have A U C
r e (E,
7 ,>>
8B
9 , such t h a t
and B u D
&
A.
481
A
A
C = 0,
The s t r u c t u -
i s c a l l e d t i g h t , i f t h e c o n d i t i o n A*-
*
B
i m p l i e s A N B.
To u n d e r s t a n d t h i s c o n d i t i o n , i t i s b e s t t o t r y t o imagi n e a s t r u c t u r e which i s n o t t i g h t . Then t h e r e e x i s t two * e v e n t s , A and B, such t h a t A N B and A 4 B. S i n c e * B, f o r e v e r y C € such t h a t A n C = 0, t h e union A A rl C i s a t l e a s t as p r o b a b l e as B, i . e . A U C $ B. T h i s means t h a t A i s l e s s p r o b a b l e t h a n B, b u t a f t e r a d d i n g t o A any e v e n t C which i s d i s j o i n t w i t h i t , one o b t a i n s t h e e v e n t A u C which i s a t l e a s t as p r o b a b l e as B. I n s t i l l o t h e r words, it means t h a t t h e s t r u c t u r e does n o t c o n t a i n any e v e n t s which would be more p r o b a b l e t h a n A, b u t l e s s p r o b a b l e t h a n B.
-
3
It t u r n s o u t t h a t t h e two l a s t n o t i o n s completely char a c t e r i z e t h e s t r u c t u r e s a t i s f y i n g axiom 5, namely w e
have THEOREM 4.3. The s t r u c t u r e o f q u a l i t a t i v e p r o b a b i l i t y (E, $ s a t i s f i e s axiom 5 i f . and o n l y i f , i t i s
,b)
f i n e and t i g h t . Moreover, axioms 1-5 are s u f f i c i e n t f o r t h e e x i s t e n c e o f p r o b a b i l i t y s c a l e . We have namely TFEOREM 4.4.
If a q u a l i t a t i v e p r o b a b i l i t y s t r u c t u r e
(E, $ , 2, s a t i s f i e s axiom 5, t h e n t h e r e e x i s t s a ( f i n i t e l y a d d i t i v e ) p r o b a b i l i t y f u n c t i o n P on such t h a t P(A)
2
P(B)
i f - and o n l y i f . A
8 B.
3,
For some o t h e r c o n d i t i o n s implying t h e same a s s e r t i o n , as w e l l a s f o r an e x t e n s i v e l i t e r a t u r e on t h e s u b j e c t ,
482
CHAPTER 5
s e e R o b e r t s (1979).
5. DIFFERENCE MEASUREMENT One o f t h e most commonly e n c o u n t e r e d e m p i r i c a l s i t u a t i o n i n which t h e r e i s no o p e r a t i o n o f c o n c a t e n a t i o n o f o b j e c t s , i s t h e s i t u a t i o n of comparison o f d i f f e r e n c e s (intervals). F o r m a l l y , a n i n t e r v a l i s d e f i n e d a s a p a i r of o b j e c t s , and t h e comparisons c o n c e r n t h e r a t i o o f t h e l e n g t h s of i n t e r v a l s (or: the differences of values of a t t r i b u t es i n p a i r s of o b j e c t s ) . The s i m p l e s t e m p i r i c a l s i t u a t i o n may b e d e s c r i b e d as
f o l l o w s . The s u b j e c t i s p r e s e n t e d w i t h a p a i r o f s t i m u l i (e.g. two l i g h t s w i t h d i f f e r e n t i n t e n s i t i e s , two sounds w i t h d i f f e r e n t p i t c h o r i n t e n s i t y , e t c . ) , and i s asked t o i n d i c a t e ( b y means o f a s p e c i a l equipment) a s t i m u l u s which l i e s " i n t h e middle" between t h e two s t i m u l i p r e s e n t e d , i . e . a s t i m u l u s which l i e s "at t h e same d i s t a n ce" from e a c h o f . t h e m . T h i s s c a l i n g t e c h n i q u e i s c a l l e d b i s e c t ion. Let u s d e n o t e t h e i n i t i a l s t i m u l i b y a and b. Then t h e o b j e c t i s t o f i n d a s t i m u l u s c , such t h a t t h e d i f f e r e n -
c e s between c and a , and between b and c , a r e t h e same. T h i s t y p e o f e x p e r i m e n t assumes i m p l i c i t l y t h a t t h e subj e c t i s c a p a b l e o f comparing t h e d i f f e r e n c e s between p a i r s o f s t i m u l i ( a t l e a s t between p a i r s w i t h one element i n common, i . e . he c a n compare i n t e r v a l s w i t h a common e n d p o i n t ) . The comparison o f i n t e r v a l s need n o t be d i r e c t . Thus,
SELECTED TOPICS IN MEASUREMENT THEORY
483
i n c a s e of c h o i c e from a p a i r ( a , b ) , t h e p r o b a b i l i t y p ( a , b ) of choosing a i s a c e r t a i n measure of " d i f f e r e n ce" between a and b. I f p ( a , b ) = 4, t h e elements a and 4, we may t h i n k t h a t a i s b are equivalent; i f p(a,b) " t o t h e r i g h t " of b, w i t h larger d e v i a t i o n s p ( a , b ) 4 i n d i c a t i n g l a r g e r d i s t a n c e s . Consequently, t h e knowledge o f p r o b a b i l i t i e s such as p ( a , b ) , p ( c , d ) , . . . may s e r v e as a basis f o r i n f e r e n c e about t h e l e n g t h s o f i n t e r v a l s a b , cd,
>
-
... .
Such a t e c h n i q u e based on p r o b a b i l i t i e s p ( a , b ) , l e a d i n g t o a n assignment o f s c a l e v a l u e s t o s t i m u l i , is c a l l e d t h e t e c h n i q u e of j u s t n o t i c e a b l e d i f f e r e n c e s ( j n d ) . A s a r u l e , a u n i t o f l e n g t h ( d i s t a n c e ) i s t a k e n t o be t h e d i s t a n c e between p a i r a,b such t h a t p ( a , b ) = 3/4. Thus, i n t h e t e c h n i q u e of u n f o l d i n g s c a l e s o f Coombs, t h e subject o r d e r s t h e s t i m u l i according t o h i s preference; i t i s assumed t h a t t h e s u b j e c t has a l s o an " i d e a l p o i n t " i n t h e space under c o n s i d e r a t i o n , and t h a t t h e s t i m u l i are l o c a t e d i n t h e same space. The c l o s e r is a s t i m u l u s t o t h e i d e a l p o i n t , t h e more p r e f e r r e d i t is. Thus, t h e model assumes comparisons o f i n t e r v a l s , counted from the ideal point t o the stimuli. The above examples i n d i c a t e t h a t t h e s i t u a t i o n s i n which t h e e m p i r i c a l data a l l o w us t o i n f e r about t h e d i f f e r -
e n c e s i n p a i r s o f s t i m u l i , i s r e l a t i v e l y common. A t t h e same t i m e , t h e y i n d i c a t e t h e formal v a r i e t y o f such sit u a t i o n s (e.g. comparisons may be p o s s i b l e f o r a l l int e r v a l s , o r o n l y f o r i n t e r v a l s w i t h a common e n d p o i n t ; t h e y may concern a l g e b r a i c d i f f e r e n c e s , o r d i s t a n c e s , e t c . ) . Accordingly, t h e r e e x i s t s a v a r i e t y o f axiom s y s tems, implying measurement f o r such s i t u a t i o n s , each of
484
CHAD772R 5
them c o n s t r u c t e d f o r t h e d e s c r i p t i o n o f some e m p i r i c a l setup. I n t h i s s e c t i o n , some o f such axiom systems w i l l b e p r e s e n t e d . The f i r s t w i l l c o n c e rn s i t u a t i o n s i n which a l l comparisons a r e p o s s i b l e ( n o t o n l y i n t e r v a l s w i t h common e n d p o i n t s ) , and t h e d i f f e r e n c e s are t r e a t e d a l g e b r a i c a l l y ( i . e . a b and ba are of o p p o s i t e s i g n s ) .
5.1.
The s t r u c t u r e of a l g e b r a i c d i f f e r e n c e s
The b a s i c r o l e here i s p l a y e d by a b i n a r y r e l a t i o n on t h e s e t o f p a i r s o f d i f f e r e n c e s ( o r p a i r s o f e l e m e n t s of t h e s e t A ) , hence a q u a r t e r n a r y r e l a t i o n on A. The symbol a b >, cd w i l l be i n t e r p r e t e d a s " t h e d i f f e r e n ce between a and b i s a t l e a s t as l a r g e as t h e d i f f e r e nce between c and d". The problem l i e s i n f i n d i n g c o n d i t i o n s imposed on t h e r e l a t i o n 2 , which would i m p l y t h e e x i s t e n c e o f a func-
t i o n f on A, such t h a t ab
cd
i f , and o n l y i f f ( a )
-
f(b)
>, f(c)
-
f(d).
These c o n d i t i o n s are e x p r e s s e d by t h e f o l l o w i n g theorem (see Krantz e t a l . , 1971).
THEOREM 5.1. Let A be a nonempty s e t , and l e t b i n a r y r e l a t i o n i n A X A, hence a q u a r t e r n a r y r e l a t i o n in A Assume t h a t t h e r e l a t i o n a l s y s t e m < A X A , &> s a t i s f i e s t h e f o l l o w i n g c o n d i t i o n s . For any a,b ,c,d , C A: a l , b l , c l C A and a r b i t r a r y sequences al,a2,.
.
..
1.
the relation
)r i s a weak o r d e r i n
A x A;
485
SELECTED TOPICS IN MEASUREMENT THEORY
2.
If
ab
3. If ab
4.
cd, t h e n
8
be
a'b'
g ab ) cd & aa,
that ad' -
cd
N
dc
)r
ba;
b'c',
t h e n ac
theR there exist d',
ale'; d" t A such
d"b;
.
5. g al, a2,.. i s a s t r i c t l y bounded s t a n d a r d sequence, i.e. a i + l a i d a 2 a 1' and i t i s n o t t r u e t h a t a 2 a 1 ,U a 1a 1' and t h e r e e x i s t d ' , d " such t h a t d ' d " a i a l b d"d' f o r e v e r y i, t h e n t h e sequence al,a2, finite.
>
...
Then t h e r e e x i s t s a f u n c t i o n f d e f i n e d on A , such t h a t f o r a l l a , b , c , d E A we have (1)
ab
& cd
i f . and o n l y i f
f(a)-f(b)
2
f(c)-f(d).
Moreover, i f a f u n c t i o n g s a t i s f i e s t h e c o n d i t i o n
-
( i ) ,t h e n g = d f + f f o r some oC> 0
r.
The s e n s e o f t h e s e axioms i s as fo l l o ws . The f i r s t axiom
a s s e r t s t h a t t h e r e l a t i o n & i s connected and t r a n s i t i v e , i . e . t h a t e v e ry two d i f f e r e n c e s may be compared, and t h e r e s u l t s o f t h e s e comparisons are c o n s i s t e n t . The second axiom s t a t e s t h a t t h e d i r e c t i o n o f i n e q u a l i t y changes under t h e change o f s i g n . The t h i r d axiom s t a t e s t h a t i f two i n t e r v a l s are c o n s t r u c t e d by j o i n i n g t h e i n t e r v a l s w i t h common end p o i n t s , and i f t h e components o f one i n t e r v a l are l o n g e r t h a n t h e components o f t h e second i n t e r v a l , t h e n t h e r e s u l b i n g first i n t e r v a l i s l o n g e r t h a n t h e second. The f o u r t h axiom asserts t h a t g i v e n two i n t e r v b l s , one can
c u t a p a r t o f t h e l o n g e r one, e q u a l i n l e n g t h t o t h e
CHAPTER 5
486
s h o r t e r , and t h a t t h i s may be done c o u n t i n g from each o f t h e two ends o f t h e l o n g e r i n t e r v a l . F i n a l l y , t h e l a s t axiom i s one more v e r s i o n o f t h e A r chimedean axiom. The a s s e r t i o n o f t h e theorem i s t h a t t h e r e e x i s t s a sca-
l e f f o r t h e a t t r i b u t e , whose d i f f e r e n c e s are compared
>r.
thro u g h r e l a t i o n Moreover, f i s a n i n t e r v a l s c a l e , i.e. a s c a l e d e t e rm i n e d u n i q u e l y up t o t h e c h o i c e o f zer o and u n i t . 5.2.
Bisections
The axiom s y s t e m i n t h e p r e c e d i n g s e c t i o n concerned t h e
relation which o r d e r e d t h e d i f f e r e n c e s ( i n t e r v a l s ) . I n c a s e o f e x p e ri m e n t s c o n c e rn i n g b i s e c t i o n s , o u t l i n e d i n t h e i n i t i a l p a r t o f t h e p r e s e n t s e c t i o n , i t i s more con v e n i en t t o have axioms i n which t h e p r i m i t i v e n o t i o n i s t h e midpoint. Such axioms are e a s i e r t o v e r i f y empirically. The p r i m i t i v e c o n c e p t s here are a weak o r d e r
& in
the
s e t A , and a f u n c t i o n B, which maps A x A i n t o A, i . e . a s s i g n i n g t o e a c h p a i r a , b G A t h e p o i n t B ( a , b ) t' A , i n t e r p r b t e d as a midpoint between a and b. W e s h a l l n o t p r e s e n t here t h e whole s y s t e m of axioms con cer n in g b i s e c t i o n s (see P f a n z a g l 1968, Krantz e t a l , 1971 o r R o b e r t s 1979). Roughly, t h e s e axioms a s s e r t t h a t B(a,a) = a ( t h e midpoint o f a d e g e n e r a t e i n t e r v a l coincides with t h e endpoints of t h i s i n t e r v a l ) ; i f a b b , t h e n B( 9 , c) >/ B ( b , c ) , which means t h a t when one en d p o i n t i s changed from a t o b , t h e midpoint changes i n t h e same
SELZCTED TOPICS IN MEASUREMENT THEORY
487
d i r e c t i o n ( s e e Fig. 5 . 1 ) .
F i g . 5.1.
Monotonicity o f t h e midpoint.
F i n a l l y , t h e s o - c a l l e d bi-symmetry axiom, a s s e r t s t h a t
The sense of t h i s axiom may be i l l u s t r a t e d g e o m e t r i c a l l y ( s e e Fig. 5.2).
F i g . 5.2.
I l l u s t r a t i o n o f bisymmetry axiom.
I n a d d i t i o n , one assumes t h e c o n t i n u i t y o f t h e m i d p o i n t B(a,b). w i t h r e s p e c t t o a and b. The a s s e r t i o n o f t h e theorem i s t h a t t h e r e e x i s t s a f u n c t i o n f , d e f i n e d on A , and such t h a t
a
2b
i f , and o n l y i f , f ( a )
2
f(b);
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Moreover, i f some f u n c t i o n g s a t i s f i e s t h e above two c o n d i t i o n s , t h e n g = df' f o r some a)O and
tp
P.
Thus, f i s a measurement on a n i n t e r v a l t y p e s c a l e . I n case o f p e r f e c t symmetry i n evaluation o f the midpoint, we have f [ B ( a , b ) ] = $ f ( a ) t $ f ( b ) , i . e . p = $. The t h e orem t a k e s , however, i n t o a c c o u n t t h e p o s s i b i l i t y o f s y s t e m a t i c d e v i a t i o n s ; f o r i n s t a n c e , t h e tendency t o p l a c e t h e m i d p o i n t somewhat c l o s e r t o t h e l e f t e n d p o i n t a, o f t h e i n t e r v a l w i l l b e e x p r e s s e d as t h e p r o p e r t y p etc.
>
5.3. The s c a l e of j u s t n o t i c a b l e d i f f e r e n c e s ( j n d ) A t t h e end, w e s h a l l p r e s e n t two s e t s o f axioms concer-
n i n g d i s c r i m i n a t i o n . I n t h e f i r s t c a s e , t h e problem l i e s i n f i n d a s e t o f axioms f o r c h o i c e p r o b a b i l i t i e s p ( a , b ) , which would i m p l y t h e e x i s t e n c e o f a f u n c t i o n f such t h a t p(a,b)
>
Hepe p ( a , b ) of the p a i r a as h a v i n g (e.g. a may as b r i g h t e r
p(c,d)
i f , and d n l y i f ,
i s t h e p r o b a b i l i t y t h a t upon p r e s e n t a t i o n ( a , b ) , t h e s u b j e c t w i l l i n d i c a t e t h e element more o f t h e a t t r i b u t e under c o n s i d e r a t i o n be p r e f e r r e d t o b , lamp a may b e e v a l u a t e d t h a n lamp b , e t c . ) .
A s b e f o r e , we s h a l l p r e s e n t o n l y t h e b a s i c ideas o f t h e c o n s t r u c t i o n u n d e r l y i n g v a r i o u s axiom s y s t e m s .
SELEC7ED TOPICS IN MEASUREMENT THEORY
489
F i r s t o f a l l , t h e p r o b a b i l i t i e s p ( a , b ) a l l o w u s t o define the relation as f o l l o w s :
3
a
>
b if
and o n l y i f
p(a,b)
3
$.
Thus, we have a m b i f , and o n l y i f p ( a , b ) = $, Generally, if 0 p(a,b) 1 w e speak o f i m p e r f e c t d i s c r i m i n a t i o n between a and b , w h i l e i n c a s e p ( a , b ) = 0 o r 1 we speak o f p e r f e c t d i s c r i m i n a t i o n .
<
<
I f t h e s u b j e c t i s t o choose, w i t h no p o s s 5 b i l i t y of
d e c i s i o n t h a t a and b a r e "equal", and i f t h e c h o i c e p r o b a b i l i t i e s a r e independent of t h e o r d e r o f p r e s e n t a t i o n , t h e n one may assume t h a t
One o f t h e main axioms i s t h e assumption i m p l y i n g t r a n s i t i v i t y o f t h e r e l a t i o n & . There e x i s t s a number o f v e r s i o n s o f such assumption. I f p ( a , b ) $ and p ( b , c ) > $, c , t h e n we must have a k c . T h i s i.e. i f a b and b may b e e n s u r e d bp requirement t h a t
>
(the so-called strong stochastic t r a n s i t i v i k y ) , or
(moderate s t o c h a s t i c t r a n s i t i v i t y ) , o r f i n a l l y p(a,c> (weak
>4
stochastic transitivity).
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One of t h e axioms which a r e necessary f o r t h e e x i s t e n ce of s c a l e f i s t h e so-called guadruple c o n d i t i o n . If t h e s c a l e f e x i s t s , i t must s a t i s f y f(a)
-
f(b)
2
f(c)
-
f(d) i f
and only i f
It f o l l o w s t h a t t h e choice p r o b a b i l i t i e s must s a t i s f y t h e condition
T h i s i s t h e quadruple c o n d i t i o n mentioned above. A s a l r e a d y s t a t e d , t h e r e e x i s t many theorems about t h e
e x i s t e n c e of measurement induced by choice p r o b a b i l i t i e s p ( a , b ) . I n g e n e r a l , t h e a s s e r t i o n s of t h e s e theorems s t a t e t h a t t h e r e e x i s t s f u n c t i o n f such t h a t
Moreover, t h e measurement i s on i n t e r v a l s c a l e , i.e. every o t h e r fun c t i o n g s a t i s f y i n g t h e above c o n d i t i o n i s r e l a t e d t o f u n c t i o n f by a p o s i t i v e l i n e a r transforma t i o n g +f + (3 f o r some A 7 0 and f
.
5.4.
Semiorder s
F i n a l l y , i t i s worth while t o mention here one more axiom system, i n t r o d u c e d by Luce (19561, concerning t h e so-called semiorders.
SELECTED TOPICS IN MEASUREMENT THEORY
491
The a i m o f t h e o r i g i n a l c o n s i d e r a t i o n s o f Luce was t o
c r e a t e a model f o r s i t u a t i o n s when t h e i n d i f f e r e n c e r e l a t i o n need n o t be t r a n s i t i v e . More p r e c i s e l y , one c o n s i d e r s a c e r t a i n set A , and a b i n a r y r e l a t i o n , say D, where t h e symbol aDb i s i n t e r p r e t e d a s ''a i s d i s t i n c t l y dominating o v e r b"; domination means s i m p l y a h i g h e r v a l u e of t h e a t t r i b u t e under study. For example, t h e e l e m e n t s o f A may b e sounds, c o n s i d e r e d from t h e p o i n t of view o f p i t c h . Then aDb means t h a t a i s d i s t i n c e l y h i g h e r i n p i t c h t h a n b. One assumes t h a t t h e r e l a t i o n D s a t i s f i e s t h e followi n g conditions: 1. It i s n o t t r u e t h a t aDa; 2.
If aDb and cDd, t h e n e i t h e r aDd o r cDb;
3. If aDb and bDc, t h e n f o r e v e r y d e i t h e r aDd o r dDc.
A w e have
The i n t e r p r e t a t i o n of t h e s e axioms i s t h e following.
The f i r s t axiom s t a t e s t h a t no o b j e c t dominates i t s e l f . To i l l u s t r a t e t h e meaning o f t h e remaining axioms, l e t u s imagine t h a t t h e e l e m e n t s o f A are p o i n t s on t h e a x i s , and t h a t aDb if t h e p o i n t a i s t o t h e l e f t o f
p o i n t b , a t a d i s t a n c e exgeeding a c e r t a i n t h r e s h o l d ( s a y , 1 cm). Then axiom 2 may be i l l u s t r a t e d as on F i g . 5.3. T h i s f i g u r e g i v e s two p o s s i b l e l o c a t i o n s o f pairs a,b and c,d, w i t h d i s t a n c e i n each p a i r exceeding 1 c e n t i m e t e r . On t h e upper f i g u r e we have aDd, w h i l e on t h e lower f i g u r e we have cDb. The meaning o f axiom 3 may be i l l u s t r a t e d i n a similar way (see Fig. 5.4). Here w e mark t h o s e p a r t s o f t h e axis
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where we have r e s p e c t i v e l y aDd
v
V
"
a
c
b
and
dDc.
"
d
aDd
v
v
v
v
c
a
b
d
cDb Fig. 5.3.
I l l u s t r a t i o n for axiom 2
"
"
\r
a
b
C
Fig.
5.4.
-
I l l u s t r a t i o n for axiom 3.
Let u s now d e f i n e a I b by requirement t h a t i t i s n o t t r u e t h a t aDb, and i t i s n o t t r u e t h a t bDa. The r e l a t i o n I
i s a n i n d i f f e r e n c e : i f a I b , t h e n t h e r e i s no p e r c e p t u a l d i f f e r e n c e between a and b. The axioms 1-3 do n o t i m p l y t r a n s i t i v i t y of t h e r e l a t i o n I: t h e s t i m u l i a , b , c may b e such t h a t a I b , b I c , b u t aDc.
SELECTED TOPICS IN MEASUREMENT THEORY
493
One may, however, d e f i n e t h e e q u i v a l e n c e r e l a t i o n as f o l l o w s : we d e f i n e aEb, i f f o r e v e r y c G A t h e c o n d i t i o n a I c h o l d s i f and o n l y i f b I c . The r e l a t i o n E i s e a s i l y s e e n t o be r e f l e x i v e (aEa),
symmetric ( a E b i f and o n l y i f b E a ) , and t r a n s i t i v e ( a E b and bEc imply aEc). The theorem on e x i s t e n c e o f measurement may b e formul-
ated a s follows: If t h e set A i s f i n i t e , and r e l a t i o n D st i s f i e s axioms 1 , 2 and 3, t h e n t h e r e e x i s t s a f u n c t i o n f d e f i n e d on A such t h a t f o r some p o s i t i v e number 6 , t h e c o n d i t i o n aDb h o l d s i f - and o n l y i f f ( a ) 2 f ( b ) + 6
THEOREM 5.2.
I n t h i s c a s e t h e a s s e r t i o n o f t h e theorem does n o t spec i f y t h e t y p e o f measurement. However, f o r some f u r t h e r r e s u l t s c o n ce rn i n g se m i o rd e rs, see R o b e r t s (1979).
6. C O N J O I N T MEASUREMENT
One o f t h e more i n t e r e s t i n g problems o f measurement t h e o r y i s t h e problem o f c o n j o i n t measurement. We d e a l w i t h such measurement i n s i t u a t i o n s , when t h e o b j e c t s under c o n s i d e r a t i o n may b e decomposed i n t o two o r more components, e a c h i n f l u e n c i n g i n a s p e c i f i c way t h e a t t r i b u t e of the object. A t y p i c a l example may be p ro v i d e d by baskets o f goods, e a c h a p p e a r i n g i n some q u a n t i t y . These b a s k e t s are com-
p a r e d from t h e p o i n t o f view o f p r e f e r e n c e . The l a t t e r i s assumed t o depend i n some way on t h e amounts o f goods. A s a n o t h e r example, one may t a k e t h e t e m p e r a t u r e
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and h u m i d i t y , compared from t h e p o i n t of view o f t h e i r pleasantness
.
A s a n example o f p h y s i c s , one can t h i n k o f comparing momentums, depending on masses and v e l o c i t i e s .
To f o r m a l l y d e s c r i b e s i t u a t i o n s t o which t h e problems o f c o n j o i n t measurement a p p l i e s , l e t A1 and A2 d en o t e t h e s e t s o f p o s s i b l e " l e v e l s " o f t h e two f a c t o r s . I n t h e s e q u e l , t h e l e t t e r s a,b,c,... w i l l s t a n d f o r elements o f A1, and l e t t e r s p,q,r,... w i l l s t a n d f o r elements o f A2. Moreover, t h e e l e m e n t s o f t h e s e t A1 X A*, which c o n s t i t u t e " o b j e c t s " o f measurement, w i l l be den o t e d by a p , bq,...; such n o t a t i o n i s more co n v en i en t than ( g , ~ ) , (b,q)
,... .
>
The p r i m i t i v e n o t i o n here i s a b i n a r y r e l a t i o n on A1 % A 2 , i n t e r p r e t e d as a j o i n t e f f e c t o f two f a c t o r s : ap bq means t h a t an o b j e c t w i t h l e v e l s a and p o f t h e two f a c t o r s i s "not l a t e r t h a n " from t h e p o i n t o f view o f t h e j o i n t a t t r i b u t e , t h a n t h e o b j e c t w i t h l e v e l s b and q o f t h e two f a c t o r s . The r e l a t i o n s
>
-
and o f s t P i c t i n e q u a l i t y and i n d i f f e r e n c e are d e f i n e d i n t h e u s u a l way, as shown i n s e c t i o n 4.
I n t h e t h r e e examples g i v e n above, t h e symbol a p & bq * means t h a t : 1. A b a s k e t w i t h amount a of goods 1, and amount p o f goods 2, i s p r e f e r r e d t o a b a s k e t c o n t a i n i n g amount b ' o f goods 1, and amount q o f goods 2. 2. The atm o sp h e ri c s i t u a t i o n w i t h t e m p e r a t u r e a and
SELECTED TOPICS IN MEASUREMENT THEORY
49 5
h u m i d i t y p is t a k e n as more p l e a s a n t t h a n t h e atmosph-
e r i c s i t u a t i o n w i t h t e m p e r a t u r e b and h u m i d i t y q;
3 . The o b j e c t w i t h mass a and v e l o c i t y p has l a r g e r momentum t h a n o b j e c t w i t h mass b and v e l o c i t y q.
2
Formally, we assume t h a t t h e r e l a t i o n i s d e f i n e d on t h e whole s e t A1 8 A2. The problem w i l l l i e i n f i n d i n g c o n d i t i o n s implying t h e e x i s t e n c e o f measurement s c a l e s b o t h f o r A1 and A*, a s w e l l a s f o r A 1 y A2. T h i s explai n s t h e term " c o n j o i n t measurement". The assumption t h a t t h e r e l a t i o n
&
i s d e f i n e d on t h e A 2 imposes some r e s t r i c t i o n s on t h e c l a s s
whole s e t A1)< o f e m p i r i c a l s i t u a t i o n s t o which t h e t h e o r y o f c o n j o i n t measurement a p p l i e s . It i s t h e r e f o r e worth w h i l e t o c h a r a c t e r i z e such s i t u a t i o n s .
t h e o b s e r v a t i o n o f r e l a t i o n & may concern Firstly, n o t t h e whole s e t A 1 % A 2 , b u t o n l y i t s s u b s e t , because
of t e c h n i c a l d i f f i c u l t i e s . For i n s t a n c e , u s i n g t h e example i n p h y s i c s , t h e o b s e r v a t i o n s o f momentum may be d i f f i c u l t i n t h e domain o f large masses and large v e l o -
cities. Secondly, t h e o b s e r v a t i o n s o f t h e r e l a t i o n may conc e r n o n l y a p a r t o f t h e set A1% A2 because o f some f u n c t i o n a l r e l a t i o n s between l e v e l s i n A1 and A*, expressed, s a y , by a c e r t a i n p h y s i c a l l a w . As an example, l e t u s imagine t h a t we o b s e r v e p i e c e s o f i c e , where A1 r e p r e s e n t s mass, and A2 r e p r e s e n t s t h e volume. I n t h i s c a s e t h e r e i s no p o s s i b i l i t y o f o b s e r v i n g v a r i o u s masses o f i c e w i t h t h e same volume, or v a r i o u s volumes w i t h t h e same mass, s i m p l y because t h e s e two are r e l a t e d by
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a functional relation. The above two s i t u a t i o n s , e s p e c i a l l y t h e second, are
e l i m i n a t e d from t h e c o n s i d e r a t i o n s , s i n c e f o r m a l l y , a s a l r e a d y s t a t e d , i t i s assumed t h a t t h e r e l a t i o n is d e f i n e d on t h e whole s e t A1 % A2.
&
G e n e r a l l y , t h e problem may be f o r m u l a t e d as f o l l o w s , We a r e t o f i n d c o n d i t i o n s imposed on t h e r e l a t i o n a l s y s t e m , under which t h e r e e x i s t f u n c t i o n s f on A1 and g on A 2 , and a f u n c t i o n o f two v a r i a b l e s F ( x , y ) , one-to-one w i t h r e s p e c t t o each o f i t s arguments, such t h a t
I n o t h e r words, w e want t o have numerical s c a l e s on e a c h o f t h e s e t s A1 and A 2 , and a r u l e F o f c o n n e c t i n g t h e s c a l e v a l u e s , which would p r e s e r v e t h e o r d e r
F.
T h i s g e n e r a l f o r m u l a t i o n may b e f u r t h e r s p e c i f i e d by
imposing some c o n d i t i o n s on t h e form o f F. I n p a r t i c u l a r , t h e most developed and a l s o most important from t h e p r a c t i c a l p o i n t o f view, i s t h e t h e o r y o f a d d i t i v e conj o i n t measurement, when F ( x , y ) = x t y. I n t h i s c a s e , t h e problem l i e s i n f i n d i n g , f o r a given r e l a t i o n a l s y s t e m t h e f u n c t i o n s f and g, d e f i n e d on A1 and A 2 r e s p e c t i v e l y , such t h a t (6.1)
ap
bq
i f , and o n l y i f
SELECTED TOPICS LN MEASUREMENT THEORY
6.1.
497
Necessary c o n d i t i o n s
A s i n t h e p r e c e d i n g c a s e s , t o o b t a i n t h e n e c e s s a r y ax-
ioms, l e t u s assume t h a t t h e f u n c t i o n s f and g which s a t i s f y c o n d i t i o n (6.1) e x i s t , and l e t u s i n v e s t i g a t e t h e consequences o f t h i s assumption. Each o f t h e s e cons eq u e n c es i s , o f c o u r s e , a s i n e qua non c o n d i t i o n f o r t h e e x i s t e n c e o f c o n j o i n t measurement. Thus, c o n d i t i o n (6.1) implies, f i r s t of a l l ASSUMPTION 1.
i s a weak o r d e r on A1 x A2.
T h i s assumption i s , t h e r e f o r e , t h e f i r s t o f t h e neces-
s a r y conditions.
Next, under t h e assumption of e x i s t e n c e o f an a d d i t i v e r e p r e s e n t a t i o n (6.11, w e may write t h e f o l l o w i n g c h a i n of equivalences:
Consequently, we must have ASSUMPTION 2.
( A d d i t i v e independence). bq. f o r e v e r y q w e have a l s o aq
&
If a p &
bp, t h e n
An an a lo g o u s c o n d i t i o n must a l s o h ol d f o r t h e second c o o r d i n a t e , so t h a t we have
49 8
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ASSUMPTION 2'. bP)
If
ap @ a q , t h e n f o r e v e r y b w e have
m.
Thus, a n o r d e r i n g o f two o b j e c t s w i t h a common v a l u e o f one a t t r i b u t e d o e s n o t change under t h e change o f this attribute. I n g e o m e t r i c i n t e r p r e t a t i o n ( s e e Fig. 6.1) t h e f i r s t c o n d i t i o n means simply t h a t i f p o i n t s P = ap and Q = bp a r e such t h a t P ) Q , t h e n t h e same r e l a t i o n must be s a t i s f i e d f o r a l l c o r r e s p o n d i n g p a i r s o f p o i n t s P ' and Q' on t h e l i n e s crc, and
13 .
The second assumption h a s analogous i n t e r p r e t a t i o n . o(
'A2
q
1 I I I
I I I
- - - - r - - - 1""' Q '1
P '1
I
P
PI
I
I
I
I I
I I
- - - - r - - -1""' PI
Q '
I I
I
a
b
I
Fig. 6.1. Geometric i n t e r p r e t a t i o n o f t h e axiom of a d d i t i v e independence (see explanation i n t e x t ) .
One o f t h e main consequences o f t h e assumption o f add-
499
SELECTED TOPICS IN MEASUREMENT THEORY
i t i v e independence 2 and 2 ' i s t h a t t h e y a l l o w us t o define ordering relations and %2 on A1 and A2 as follows.
k1
DEFINITION. We p u t a ap
&
bp, and p
b2 q
b1 b ,
if f o r e v e r y P E A2 w e have
i f f o r e v e r y a -C A1 we have ap
& aq.
According t o a s s u m p t i o n s 2 and 2', t h e words " f o r e v e r y " i n t h e above d e f i n i n i t i o n may b e r e p l a c e d by " f o r some". It f o l l o w s from t h e above d e f i n i t i o n t h a t : ( i ) +i i s
a weak o r d e r o f t h e s e t Ai ( i = 1,2);
(iii) i f a t l e a s t one o f t h e i n e q u a l i t i e s i n ( i i ) i s
s t r i c t (i.e. (iv)
if
a
k1 b or
a P w bq,
p >2 q ) , t h e n
then a k1 b
ap
> bq;
i f and o n l y i f q
fi
----
9 -
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I........................
........................ '...................*.*** ........................ I........(................
........I................ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . P . . . . . . . . . . . . . . . . . . . . . . . . .
P '
----
I i
-
J.r...L.d.L.L.L.L.L.L.L,L
I I
I I
I
I
a
Fig. 6.2.
b I l l u s t r a t i o n of c o n d i t i o n
-
' (ii)
A1
g2 p.
500
CHAPTER 5
F o r g e o m e t r i c a l i l l u s t r a t i o n , assume t h a t t h e o r d e r s
i n t h e s e t s A1 and A2 are r e p r e s e n t e d by t h e o r d e r i n g s o f p o i n t s on t h e r e a l l i n e s . C o n d i t i o n ( i i ) means t h a t i f t h e i n e q u a l i t i e s on b o t h a x e s go i n t h e same d i r e c t ions, then the j o i n t inequality (according t o t h e relati o n + ) a l s o goes i n t h e same d i r e c t i o n . I n o t h e r words, a l l p o i n t s (see F i g . 6.2) i n t h e shaded area a r e i n t h e same r e l a t i o n w i t h t h e c o r n e r P. C o n d i t i o n ( i v ) , i m p l i e d d i r e c t l y by ( i i ) , s t a t e s t h a t t h e p o i n t s of i n d i f f e r e n c e must l i e on a c u r v e w i t h n e g a t i v e s l o p e ( s e e F i g . 6.3).
I I
I I
a
b
’
A1
Fig. 6 . 3 . I n d i f f e r e n c e curve
a p t u bq, t h e n t h e r e l a t i o n s between a and b, and between p and q , must be g o i n g i n t h e o p p o s i t e d i r e c t ions. If
The n e x t n e c e s s a r y c o n d i t i o n s f o r t h e c o n j o i n t measure-
SELECTED TOPICS INMEASUREMENT THEORY
501
ment c o n cer n c a n c e l l a t i o n . These c o n d i t i o n s may be o b t a i n e d as fo l l o ws . Assuming t h e e x i s t e n c e o f c o n j o i n t measurement, i.e. assuming c o n d i t i o n (6.11, we may w r i t e
Adding t h e s e i n e q u a l i t i e s we g e t
or
Consequently, a n e c e s s a r y c o n d i t i o n f o r t h e e x i s t e n c e o f c o i n j o i n t measurement i s : (Double c a n c e l l a t i o n ) . For a l l a , b , c C A1 and p , q , r E A*, if a r cq and cp 8 b r , t h e n ap @ bq. ASSUMPTION 3 .
2
The meaning o f t h i s c o n d i t i o n are b e s t i l l u s t r a t e d i f one c o n s i d e r s some o f i t s consequences. If a r & cq and
&-
T h i s means (see Fig. 6.4) t h a t i f t h e p o i n t s P = a r and Q = cq l i e on one i n d i f f e r e n ce cu r v e , and p o i n t s P t = cp and Q t = b r a l s o l i e on
cp
b r , t h e n a p & bq.
one i n d i f f e r e n c e c u rv e , t h e n t h e p o i n t s PI1 = ap and Q" = bq must a l s o l i e on one i n d i f f e r e n c e curve. Thus, assumption 3, c a l l e d a l s o t h e axiom of double
502
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c a n c e l l a t i o n , imposes some c o n s t r a i n t s on t h e f a m i l y o f i n d i f f e r e n c e c u rv e s. The t e r m "double c a n c e l l a t i o n " i s connected w i t h t h e f a c t t h a t i n t h e premises o f assumpt i o n 3 t h e r e a p p e a r two i n d e q u a l i t i e s i n v o l v i n g t h e p o i n t s c and r , which " c a n c e l " t o one i n e q u a l i t y , f r e e of b o t h c and r.
7
b
C
a
A1
Fig. 6.4. I l l u s t r a t i o n of t h e axiom o f double c a n c e l l a t i o n . F i n a l l y , t h e last n e c e s s a r y axiom is t h e Archimedean axiom. D e s p i t e t h e l a c k o f t h e o p e r a t i o n of c o n c a t e n a t i o n , one can b u i l d s t a n d a r d sequences a l o n g each of t h e a x e s , i.e. s t a n d a r d sequences o f e l emen t s o f A1 and A*. Because of t h e symmetry, we s h a l l p r e s e n t here o n l y t h e c o n s t r u c t i o n of a s t a n d a r d sequence a l , a2,.. o f elements o f A1. I n t u i t i v e l y sp e a k i n g , t h e " d i s t a n c e s " b e t -
.
SEL.EClED TOPICS IN MEASUREMENT THEORY
SO3
ween a l and a2, between a 2 and a3,... e t c . must be ident i c a l . T h i s may be e x p r e s s e d by r e q u i r i n g t h a t t h e r e exA2 such t h a t p k2 q and alp ~ a ~ f o+r e~ v e r yq i s t p,q i = 1,2,... T h i s i s i l l u s t r a t e d on Fig. 6.5.
.
I
I
al
Fig. 6.5.
I
I
a2
a3
I
I
a5
’
A1
C o n s t r u c t i o n o f a s t a n d a r d sequence
The above d e f i n i t i o n concerns i n c r e a s i n g sequences.
If
q k2 p a one o b t a i n s i n a similar way a d e c r e a s i n g s t a n d a r d sequence. The s t a n d a r d sequences o f e l e m e n t s o f t h e s e t A2 are d e f i n e d i n a s i m i l a r way.
Now, t h e Archimedean axiom may be formulated as f o l l o w s . ASSUMPTION 4.
Each bounded s t a n d a r d sequence has a t most a f i n i t e number o f elements.
{anla
where a n € Ai Boundedness o f a s t a n d a r d sequence (1 = 1,2) means t h a t t h e r e e x i s t elements x and y i n Ai such t h a t x an Zi y f o r a l l n; t h e i n d e x i e q u a l s 1 or 2 , depending whether w e c o n s i d e r sequences i n A1 or A2.
504
6.2.
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Sufficient conditions
The above f o r m u l a t e d n e c e s s a r y c o n d i t i o n s (assumption
8
that i s a weak o r d e r , assumption o f a d d i t i v e independence, double c a n c e l l a t i o n axiom, and Archimedean axiom) are n o t s u f f i c i e n t t o e n s u r e t h e e x i s t e n c e o f c o n j o i n t measurement. It t u r n s o u t t h a t i t i s n e c e s s a r y t o impose some r e q u i r e m e n t s c o n c e rn i n g s o l v a b i l i t y o f t h e eq u at i o n a p d bq f o r g i v e n any t h r e e among a,b ,p ,q , namely: ASSUMPTION 5.
( s o l v a b i l i t y ) . For any t h r e e among a , b € A1 and p,q Ci A2 t h e r e e x i s t s f o u r t h element such t h a t ap hl bq. For a n i l l u s t r a t i o n , assume t h a t we have a , b and p (see Fig. 6.6).
I F i g . 6.6.
I
I
I
I
b
a
I l l u s t r a t i o n o f s o l v a b i l i t y axiom.
The s o l v a b i l i t y assumption asserts t h a t t h e i n d i f f e r e n c e c u r v e which p a s s e s t h r o u g h t h e p o i n t ap w i l l i n t e r s e c t t h e l i n e d ; t h e coordinate of t h i s p o i n t o f i n t e r -
SELECTED TOPICS IN MEASUREMENT THEORY
505
s e c t i o n i s t h e desired p o i n t q. F i n a l l y , t h e l a s t assumption e l i m i n a t e s t r i v i a l S i t u a t i o n s . It may be fo rm u l a t e d as f o l l o w s : ASSUMPTION 6. Each o f t h e c o o r d i n a t e s i s e s s e n t i a l , i . e .
t h e r e e x i s t a,b
e A1 and
p,q G A2 such t h a t a
h1 b and
p >2 q *
Under as s u m p t i o n s 1-6 t h e c o n j o i n t measurement e x i s t s . More p r e c i s e l y , we have t h e f o l l o w i n g theorem ( s e e Krantz e t a l . , 1971):
&}
THEOREM 6.1. If t h e s y s t e m 0 and
e
>
- p1,
p2
Thus, t h e axioms f o r c o n j o i n t measurement imply a l s o t h e e x i s t e n c e o f measurements on t h e s e t s A1 and A 2 sep a r a t e l y , and t h e s e measurements are on a n i n t e r v a l s c a l e . What is c h a r a c t e r i s t i c here i s t h a t on b o t h a x e s t h e u n i t s must be i d e n t i c a l ( w h i l e t h e z e r o s may be arbitrary),
6.3.
G e n e r a l i z a t i o n : m u l t i n o m i a l c o n j o i n t measurement
There e x i s t s i t u a t i o n s i n which t h e p a r t i c u l a r f a c t o r s i n f l u e n c e t h e f i n a l r e s u l t , b u t n o t i n an a d d i t i v e way.
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Such s i t u a t i o n s o c c u r most o f t e n i n p h y s i c s , where one has laws which are n o t l i n e a r . F o r i n s t a n c e , s u c h f a c t ors as mass and v e l o c i t y determine t h e k i n e t i c energy E e q u a l Jmv2, e t c . P a s s i n g t o examples from psychology, one may mention here s i m i l a r i t y i n d i c e s b u i l t on t h e basis of v a r i o u s a t t r i b u t e s , d e f i n e d as n
are t h e vecwhere 5 = (xl,...,xn) and y = (yl,...,yn) t o r s o f v a l u e s o f t h e a t t r i b u t e s o f 5 and y. A s s t i l l a n o t h e r example from psychology, w e may t a k e t h e m o t i v a t i o n model o f Atkinson (1963), where it i s assumed t h a t t h e a t t r a c t i v e n e s s o f a s i t u a t i o n o f r i s k i s e x p r e s s e d as A = IsPsMs
-
IfPfMf,
where I and If are t h e s o - c a l l e d i n c e n t i v e v a l u e s of S s u c c e s s and f a i l u r e , P s and Pf are s u b j e c t i v e probab i l i t i e s o f s u c c e s s and f a i l u r e , and Ms and Mf a r e t h e g e n e r a l m o t i v e s towards achievement and towards avoidi n g f a i l u r e . Atkinson p o s t u l a t e s t h a t Pf = 1 Ps and Is = 1 Ps, If = -P Consequently, t h e a t t r a c t i v e n e s s S' o f a g i v e n s i t u a t i o n becomes
-
-
A s a n o t h e r example of l a c k o f a d d i t i v i t y of f a c t o r s one may t a k e t h e well-known s t u d i e s of C l i f f (19591,
507
SELECTED TOPICS IN MEASUREMENT THEORY
who i n v e s t i g a t e d t h e m u l t i p l i c a t i v e v a l u e s o f a d v e r b s o f q u a n t i t y , such as v e r y , r a t h e r , somewhat, a l i t t l e , e t c . C l i f f analysed the perceived values of various combinations o f t h e above adverbs i n t h e i r c o n n e c t i o n w i t h a d j e c t i v e s such as p l e a s a n t , bad, e t c . ( s o t h a t t h e combinations were "very p l e a s a n t " , " r a t h e r bad", e t c . ) . Because o f t h e obvious e x i s t e n c e o f t h e z e r o of t h e s c a l e , the a c t i o n o f adverbs differs f o r p o s i t i ve and n e g a t i v e domains, f o r i n s t a n c e v e r y good ) . r a t h e r good very pleasant > r a t h e r p l e a s a n t ,
(6.2)
but (6.3)
r a t h e r unpleasant
very unpleasant.
I n t h i s c a s e one may p o s t u l a t e t h a t t h e r e e x i s t s a scal e f l f o r adverbs of q u a n t i t y , w i t h values f l being always p o s i t i v e , and s c a l e f2 f o r e v a l u a t i o n s , w i t h v a l u e s b o t h p o s i t i v e and n e g a t i v e . The combined e f f e c t i s m u l t i p l i c a t i v e , i. e. f ( r a t h e r u n p l e a s a n t ) = fl(rather)f2(unpleasant),
f ( v e r y good) = f l ( v e r y ) f 2 ( g o o d ) , e t c . The change o f d i r e c t i o n o f i n e q u a l i t y i n (6.2) and (6.3) i s connected w i t h t h e f a c t t h a t f l > 0 f o r a l l v a l u e s o f t h e argument, w h i l e f 2 ( u n p l e a s a n t ) 0, and f2(good) 0, etc.
>
<
The above examples i n d i c a t e t h e need o f c o n s t r u c t i o n
o f t h e t h e o r y o f c o n j o i n t measurement f o r cases o t h e r
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508
than additivity. The t h e o r y here i s r a t h e r c o m p l i c a ted , because t h e form
o f n e c e s s a r y c o n d i t i o n s depends s t r o n g l y on t h e form o f t h e f u n c t i o n which d e t e r m i n e s t h e j o i n t e f f e c t of f a c t o r s . A t t h e same t i m e , however, t h i s v a r i e t y o f cond i t i o n s a l l o w s u s t o b u i l d t h e d i a g n o s t i c schemes (see Krantz and Tversky, 1971) f o r d e t e r m i n i n g t h e form o f t h e f u n c t i o n a l dependence.
More p r e c i s e l y , t h e s e d i a g n o s t i c schemes co n cern t h e and c a s e o f t h r e e f a c t o r s , x E A1, y 6 A2 and z € A 3' a l l o w i n some s i t u a t i o n s t o d e t e rm in e whether t h e r e e x i s t s , and if s o of which form, t h e mu l t i n o mi al r e l a t i o n between t h e f a c t o r s x, y and z . G e n e r a l l y , we may d i s t i n g u i s h here f o u r b a s i c t y p e s of polynomials:
(a) additivity: x t y t z ;
( c ) double d i s t r i b u t i v i t y : x y t z , x z t y and y z t x ; ( d ) m u l t i p l i c a t i v i t x : xyz.
The schemes f o r d i a g n o s i s are a p p l i c a b l e t o s i t u a t i o n s
when some o f t h e s e t s A1, A 2 , A may b e p a r t i t i o n e d 3 i n t o " p o s i t i v e " , " n e u t r a l " and " n e g at i v e" p a r t s . The b a s i c idea here i s a s f o l l o w s . L e t us s t a r t , f o r
s i m p l i c i t y w i t h t h e c a s e o f two v a r i a b l e s x c A1 and y t A2, and a weak o r d e r i n t h e C a r t e s i a n p ro d u ct A 1 & A 2 . For a f i x e d x, w e may c o n s i d e r a n o r d e r i n g bx o f t h e s e t A2, d e f i n e d by y1 )zX y 2 if xyl & xy2. I n a similar way one d e f i n e s t h e o r d e r i n g s 8 of t h e set A1. Y
SELECTBD TOPICS INMEASUREMENT THEORY
509
2,
DEFINITION. If a l l t h e o r d e r i n g s ( o b t a i n e d f o r vari o u s x ) c o i n c i d e (which c o rre sp o n d s , i n t u i t i v e l y , t o t h e s i t u a t i o n when a l l e l e m e n t s of A 1 are of t h e same s i g n ) , we s h a l l s a y t h a t A 2 i s sign-independent of A1. Next, we s a y t h a t A 2 i s sign-dependent on A1, i f t h e t s e t A 1 may be p a r t i t i o n e d i n t o s e t s A1, A o1 and AT, o f which a t l e a s t two a r e nonempty, i n such a way t h a t t the r e l a t i o n i n A 2 depends o n l y on whether x E A1, t x e A! o r x g A;, and moreover, f o r ev ery x E A 1 t h e is an i n v e r s e o f t h e r e l a t i o n Zx f o r x € relation A;, w h i l e 8, f o r x F A! i s d e g e n e r a t e ( i . e . a l l elements a r e i n t h i s r e l a t i o n ) .
2,
kx
t
I n t u i t i v e l y , t h i s c o rre sp o n d s t o t h e s i t u a t i o n when A 1 i s t h e s e t of " p o s i t i v e " x, A; i s t h e s e t o f " n e g a t i v e " x, and A 10 is t h e s e t of " n e u t r a l " , o r "zero" x, i.e. such t h a t a f t e r m u l t i p l i c a t i o n t h e y t r a n s f o r m t h e i n e quality i n t o a n e q u a l i t y . The sign-independence and sign-dependence of A 1 from
A 2 are d e f i n e d i n a similar way. I n t h e g e n e r a l c a s e , t h e s e t s A 1 and A 2 w i l l become p a r t i t i o n e d i n t o A1, t A1, o A;, A2, t A 2o and A;, and one may d e f i n e t h e " p o s i t i v e " , " n e u t r a l " and " n e g a t i v e " p a r t o f t h e C a r t e s i a n p ro d u c t A 1 X A 2 by t a k i n g
(A1
X
t t = (A1 x A 2 >
')2"
0
L,
(A; x A;),
A2) t a k e n as t h e re m a i ni n g p a r t o f t h e C a r t e s i a n p r od u c t A 1 x A2.
with (A1%
510
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t
For p o i n t s a € ( A I Y A 2 > , b € (A1 c. r: A 2 ) - w e have a b
>
X
A2)
0
and c E ( A 1
I n a n obvious g e o m e t r i c a l i n t e r p r e t a t i o n , when A1 ZC A2 is t h e p l a n e w i t h c o o r d i n a t e system x,y (see F i g . 6.71, + c o n s i s t s o f q u a r t e r s I and 111, t h e t h e set ( A 1 % A 2 ) s e t (AIX A 2 ) - c o n s i s t s o f q u a r t e r s I1 and I V , w h i l e (A1 x A 2 ) 0 i s t h e union o f t h e two c o o r d i n a t e axes.
I1
I11
Fig.
TI y 1 1
IV
6.7
The sign-dependence
x
r e l a t i o n i s n o t symm e t r i c . For i n s t a n ce, i n the s i t u a t i o n p r e s e n t e d on Fig. 6.8, x i s sign-depend e n t on y , b u t y i s sign-independent o f X . Here x may assume v a l u e s o f one s i g n only ( i n t h i s case p o s i t i v e ) , hence t h e i n e q u a l i t i e s between
The g e n e r a l i z a t i o n t o t h e case of three sets
presents 3 no c o n c e p t u a l d i f f i c u l t A1,
A2 and A
;x
SELECTED TOPICS IN MEASUREMENT THEORY
511
o r A1 and A 2 % A l e s on t h e t h i r d , e.g. A1 X A2 and A 3' 3 e t c . is d e f i n e d f o r m a l l y by t r e a t i n g t h e p a i r as one v a r i a b l e , and a p p l y i n g t h e d e f i n i t i o n f o r t h e c a s e o f two v a r i a b l e s . On t h e o t h e r hand, t h e sign-independence and sign-dependence o f two v a r i a b l e s , s ay A 1 and A2, i s d e f i n e d by a p p l y i n g t h e d e f i n i t i o n f o r two v a r i a b l e s A and r e q u i r i n g t h a t t h e p a r t f o r fixed value of z 3' i t i o n s i n t o p o s i t i v e , n e u t r a l and n e g a t i v e domains b e independent o f t h e c h o i c e o f 2. The g e n e r a l i d e a o f d i a g n o s t i c scheme f o r t h e t y p e o f
polynomial ( a d d i t i v e , d i s t r i b u t i v e , doubly d i s t r i b u t i ve o r m u l t i p l i c a t i v e ) is based on t h e s t u d y o f s i g n dependences between x, y and z. These d ep en d en ci es may be r e p r e s e n t e d i n form o f m a t r i c e s o f t h e form
where 1 d e n o t e s sign-dependence, and 0 d e n o t e s s i g n independence. Thus, i n t h e above m a t r i x , y is sign-dependent on x, w h i l e z is sign-independent of x, e t c . The d i a g o n a l e l e m e n t s a r e n o t f i l l e d . The t o t a l number o f p o s s i b l e m a t r i c e s o f t h i s form i s
6 4 ( a t e ach o f t h e s i x p l a c e s one may have 1 or 0 , henc e t h e number o f m a t r i c e s is 2 6 = 64 ). Some o f t h e m a t r i c e s cannot o c c u r under any polynomial r e l a t i o n s h i p between x, y and z, w h i l e o t h e r s may o c c u r
512
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o n l y f o r some t y p e s o f polynomials. F o r i n s t a n c e , t h e m a t r i x o f t h e form
may o c c u r o n l y i n t h e c a s e o f doubly d i s t r i b u t i v e poly-
nomial, i . e .
when t h e r e l a t i o n i s of t h e form
(here we have sign-dependence between x and y , and b e t ween y and x, and sign-independence between x and y from Z>.
Similarly, the matrix
Z
i
l
l
-
c o r r e s p o n d s t o a m u l t i p l i c a t i v e polynomial f l ( x ) f 2 ( y ) * f ( z ) , where f l ( x ) and f2(Y) may assume b o t h p o s i t i v e 3 and n e g a t i v e v a l u e s , w h i l e f ? ( z ) assumes o n l y p o s i t i v e values
.
Some m a t r i c e s may correspond t o more t h a n one polynom-
513
SELECTED TOPICS IN MEASUREMENT THEORY
i a l . For i n s t a n c e , t h e n u l l matrix
may r e p r e s e n t e v e r y form o f dependence, as l o n g as t h e v a l u e s x, y and z may be o n l y p o s i t i v e ( i n such a c a s e t h e d i a g n o s t i c scheme does n o t lead t o d e c i s i o n ) . I n t h e c a s e of t h e m a t r i x
t h e polynomials may be e i t h e r o f t h e form f l ( x ) f 2 ( y ) f ( z ) , o r o f t h e form Cf,(x> t f 2 ( y ) l f ( z ) . I n c a s e o f
3 3 m u l t i p l i c a t i v e polynomial t h e r e i s no dependence between x and y , z and x and z and y. T h i s i s connected w i t h t h e f a c t t h a t we have a l w a y s f l ( x ) 0 and f 2 ( y ) 0; it i s o n l y f ( 2 ) which may be b o t h p o s i t i v e and nega3 tive
>
>
.
I n t h e l a t t e r c a s e , t h e d e c i s i o n whether t h e polynomial i s m u l t i p l i c a t i v e o r d i s t r i b u t i v e must be based on add-
514
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i t i o n a l c r i t e r i a . These c r i t e r i a a r e o b t a i n e d from t h e a n a l y s i s of c a n c e l l a t i o n conditions, c h a r a c t e r i s t i c f o r a g i v e n form o f t h e polynomial. These c o n d i t i o n s are r a t h e r complicated. To g i v e a n example, f o r a d i s t r i b u t i v e polynomial t h e c a n c e l l a t i o n c o n d i t i o n s assert t h a t i f
then
The n e c e s s i t y o f t h i s c o n d i t i o n may be proved by assum-
i n g the existence of an d i s t r i b u t i v e representation, i.e. o f t h e form [ f l ( x ) + f 2 ( y ) ] f 3 ( z ) . Then t h e f i r s t two c o n d i t i o n s ( a ) and ( b ) assert t h a t
Adding t h e two s i d e s , one o b t a i n s
SELECTED TOPICS IN MEASUREMENT THEORY
515
On t h e o t h e r hand, by ( c ) w e have
S u b t r a c t i n g t h e l a s t two e q u a l i t i e s , we g e t
which means t h a t x 1y 3z 1 N
X y Z
2 4 2 '
For t h e polynomial o f t h e form f l ( x ) f 2 ( y ) t f 3 ( z ) t h e c a n c e l l a t i o n c o n d i t i o n s are s t i l l more complicated. The c a n c e l l a t i o n c o n d i t i o n s sk e t c h e d above, as w e l l a s t h e sign-dependence c o n d i t i o n s , a r e o f c o u r s e , o n l y nec e s s a r y f o r t h e e x i s t e n c e o f a m u l ti n o mi al r e p r e s e n t a t i o n o f t h e a p p r o p r i a t e form. If t h e s e c o n d i t i o n s are met, t h e n f o r t h e e x i s t e n c e o f a r e p r e s e n t a t i o n one ne e d s some a d d i t i o n a l c o n d i t i o n s ( t h e y assert t h e Archimedean p r o p e r t y , s o l v a b i l i t y o f some e q u a t i o n e , e t c . ) . We s h a l l omit here t h e p r e c i s e f o r m u l a t i o n o f t h e s e c o n d i t i o n s . The reader is r e f e r r e d t o t h e o r i g i n a l p a p e r o f Krantz and Tversky, 1971, as w e l l a s t o t h e monograph o f Krantz e t a l . , 1971.
5 16
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7 . U T I L I T Y AND SUBJECTIVE PROBABILITY
7.1. P r i m i t i v e n o t i o n s and t h e i r i n t e r p r e t a t i o n One of t h e b e t t e r known examples o f measurement i n p s y chology i s t h e c o n s t r u c t i o n o f s c a l e s o f u t i l i t y and s u b j e c t i v e p r o b a b i l i t y . There e x i s t s h e r e a s e r i e s of axiom s y s t e m s . T h e i r p r i n c i p a l g o a l i s t o f i n d c o n d i t i o n s which would imply t h e e x i s t e n c e o f t h e s c a l e of s u b j e c t i v e p r o b a b i l i t y on a c e r t a i n c l a s s of e v e n t s , and t h e s c a l e o f u t i l i t y on t h e c l a s s of consequences o f t h e s e e v e n t s . The r e q u i r e m e n t s h e r e a r e : (1) t o ena b l e t h e e m p i r i c a l d e t e r m i n a t i o n of t h e v a l u e s o f prob a b i l i t i e s and u t l l i t i e s , and ( 2 ) t o have t h e v a l u e s s o d e t e r m i n e d p r e d i c t i v e for t h e a c t u a l c h o i c e s i n r i s k y s i t u a t i o n s , under t h e assumption t h a t i n a c h o i c e b e t ween two r i s k y s i t u a t i o n s , t h e s u b j e c t w i l l choose t h a t s i t u a t i o n whose s u b j e c t i v e e x p e c t e d u t i l i t y (SEU) i s higher. I n o t h e r words, one assumes t h a t i n making a c h o i c e , t h e s u b j e c t behaves as i f he wanted t o maximize t h e s u b j e c t i v e e x p e c t e d u t i l i t y . T h u s , t h e problem l i e s i n f i n d i n g a s e t o f e m p i r i c a l l y t e s t a b l e c o n d i t i o n s , which would imply t h e e x i s t e n c e o f s c a l e s o f s u b j e c t i v e prob a b i l i t y and u t i l i t y a l l o w i n g us t o d e t e r m i n e t h e v a l u e s of SEU. The s y s t e m which w i l l be s k e t c h e d below was i n t r o d u c e d by Luce and Krantz ( 1 9 7 1 ) . T h i s system i s based on p r i m i t i v e n o t i o n s which w i l l be p r e s e n t e d below, t o g e t h e r with their interpretation. The f i r s t p r i m i t i v e n o t i o n s a r e : set
X, an a l g e b r a
517
SELECTED TOPICS IN MEASUREMENT THEORY
of s u b s e t s of X , and a s u b c l a s s J \ r of t h e a l g e b r a
4.
As
i n t h e c o n s t r u c t i o n of s c a l e s of s u b j e c t i v e p r o b a b i l i t y , t h e e l e m e n t s of a r e e v e n t s , t o which p r o b a b i l i t i e s w i l l be a s s i g n e d . The s e t X c o n s t i t u t e s t h e c l a s s of a l l e l e m e n t a r y e v e n t s . F i n a l l y , Jf i s t h e c l a s s of e v e n t s
3
w i t h probability zero ( n u l l events).
The n e x t p r i m i t i v e n o t i o n i s t h e s e t C o f consequences, which may accompany t h e o c c u r r e n c e of p a r t i c u l a r e v e n t s , o r be caused by them (sometimes t h e e l e m e n t s o f C a r e c a l l e d " p a y o f f s " , "outcomes", e t c . )
.
3-x
and C , i . e . t h e nonThe e l e m e n t s of t h e s e t s n u l l e v e n t s and outcomes, w i l l be used t o b u i l d o b j e c t s c a l l e d d e c i s i o n s ( o f t e n r e f e r r e d t o a l s o as a c t i o n s ,
or l o t t e r i e s ) .
3
Formally, f o r a given A G - df, by a d e c i s i o n we mean a f u n c t i o n fA, d e f i n e d on t h e s e t A , w i t h v a l u e s from t h e c l a s s C . To e a c h p o i n t x E A , d e c i s i o n fA ass i g n s t h e v a l u e f A ( x ) from t h e s e t C . The c l a s s o f a l l d e c i s i o n s w i l l be d e n o t e d by D . The above c o n c e p t u a l scheme d i f f e r s from t h e one tr3ad i t i o n a l l y a c c e p t e d by t h e f a c t t h a t one c o n s i d e r s h e r e conditional decisions, r e s t r i c t e d t o the s e t A. This g i v e s a more a d e q u a t e d e s c r i p t i o n o f t h e r e a l c h o i c e s i t u a t i o n s , where a d e c i s i o n d e t e r m i n e s t h e s e t of app r o p r i a t e p o s s i b l e e v e n t s . For i n s t a n c e , t h e d e c i s i o n of t a k i n g p a r t i n a g i v e n r a f f l e l e a d s t o t h e r e s t r i c t i o n t o e v e n t s s u c h as w i n n i n g or n o t w i n n i n g a s p e c i f i c
prize, etc.
If A
r\
B =
0, and fA, fg a r e two d e c i s i o n s , t h e n one
CHAPTER 5
518
may c o n c a t e n a t e t h e m , o b t a i n i n g a new d e c i s i o n , d e n o t e d by f A LI fB, d e f i n e d by
Finally, the last primitive notion i s t h e preference
&,
d e f i n e d on t h e s e t D o f d e c i s i o n s : t h e symmeans t h a t t h e d e c i s i o n f A i s p r e f e r r e d f A 2, f B (weakly) t o d e c i s i o n f
relation bol
B'
7.2.
The form of r e p r e s e n t a t i o n
The problem l i e s now i n f i n d i n g a n a c t i o n s y s t e m f o r
9
, , C , D,%) which t h e r e l a t i o n a l system <X, would imply t h e e x i s t e n c e of a f i n i t e l y a d d i t i v e ( s e e S e c t i o n 4 . 2 ) p r o b a b i l i t y measure P on and a f u n c t i o n
y,
u on D , s u c h t h a t (i)
A L d f i f and only i f P ( A ) = 0 ;
(ii)
f A ;S f B i f and only i f u ( f A )
>
u(fB);
It i s of i n t e r e s t t h a t i n t h i s c a s e t h e domain o f t h e
u t i l i t y f u n c t i o n u a r e e l e m e n t s of D , hence d e c i s i o n s , and n o t t h e e l e m e n t s o f t h e c l a s s C ( c o n s e q u e n c e s ) . I n
a s e n s e , t h e f u n c t i o n u i s a l r e a d y t h e s u b j e c t i v e exp e c t e d u t i l i t y (SEU). I n o t h e r axiom s y s t e m s ( e . g . Sa-
SELECTED TOPICS IN MEASUREMENT THEORY
519
vage 1 9 5 4 , o r Davidson, Suppes and S i e g e 1 1957), t h e u t i l i t y f u n c t i o n u i s d e f i n e d on t h e c l a s s of conseque n c e s , and i s used t o d e t e r m i n e t h e v a l u e s o f S E U .
7 . 3 . The axiom s y s t e m of Luce and Krantz Below we s h a l l g i v e and d i s c u s s t h e axioms f o r t h e rel a t i o n a l system (X, .? , A r , C , D,&.), which imply t h e a s s e r t i o n mentioned i n t h e p r e c e d i n g s e c t i o n . 1. C l o s u r e of t h e c l a s s D . fA,
fB
2-
and l e t
Dm
( a ) If A n B = 0 , (b)
A,B(
& e J
then
fA u f B
E
D;
B C A, then the function f A r e s t r i c t e d t o B
belongs t o D. The r o l e of t h i s axiom i s t h e f o l l o w i n g : i t a l l o w s u s t o r e s t r i c t t h e c o n s i d e r a t i o n t o a "managable" c l a s s D of d e c i s i o n s . The p o i n t i s t h a t e l e m e n t s of D a r e func- .K w i t h v a l u e s i n C . I f a l l s u c h t i o n s d e f i n e d on A E f u n c t i o n s were a l l o w e d , we would e n c o u n t e r some t e c h n i c a l d i f f i c u l t i e s ( c o n n e c t e d , e . g . w i t h t h e s i z e of t h i s c l a s s ) , and a l s o c o n c e p t u a l d i f f i c u l t i e s ( t h e n e c e s s i t y of p r e f e r e n t i a l comparison o f f u n c t i o n s which r e p r e s e n t " a b s u r d " , or i m p o s s i b l e d e c i s i o n s ) . Axiom 1 a l l o w s some s o r t o f i n d u c t i v e d e f i n i t i o n : one i n c l u d e s i n t o D c e r t a i n f u n c t i o n s which one wants t o have i n c l u d e d , and t h e n one u s e s c o n d i t i o n s ( i ) and ( i i ) ,i n c l u d a n g i n t o D t h e c o n c a t e n a t i o n s o f d e c i s i o n s i n c a s e s of d i s j o i n t s e t s , and t r u n c a t i o n s t o s m a l l e r s e t s . Such an a p p r o a c h l e a d s t o r e s t r i c t i o n o f a n a l y s i s t o some d e c i s i o n s of i n t e r e s t
8
520
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and a l s o p r o v i d e s some s t r u c t u r e on t h e s e t D , necessary f o r f u r t h e r c o n s i d e r a t i o n s . 2 . Weak o r d e r i n g . The r e l a t i o n $ o r d e r s weakly t h e F
s e t D.
T h i s axiom i s e v i d e n t l y n e c e s s a r y f o r t h e a s s e r t i o n :
i t i m p l i e s t h a t e v e r y two d e c i s i o n s a r e comparable, and t h a t t h e r e s u l t s o f t h e s e comparisons a r e c o n s i s t e n t .
=
3. I n d i f f e r e n c e f o r t h e sum. I f A , B t: fA f g , t h e n f A u fB & f A .
m,
2-J,
A
B
N
T h i s axiom i s n e c e s s a r y ; as may be checked e a s i l y , i t f o l l o w s from c o n d i t i o n s ( i i ) and ( i i i ) i n S e c t i o n 7 . 4 . f
I n d e e d , i f fA
hence
f A v fB
N
f
B'
t h e n u ( f,)
=
u(fg)
, and
A'
I n t u i t i v e l y , t h i s axiom a s s e r t s t h a t combining two i n d i f f e r e n t d e c i s i o n s i s i n d i f f e r e n t w i t h e a c h of them separately.
4 . Monotonicity.If only i f fA
u
gg
>
A n B =
0, then
fA
> fi
if and
v gg'
T h i s axiom i s a l s o n e c e s s a r y . T o p r o v e i t , c o n s i d e r t h e
SELECTED TOPICS IN MEASUREMENT THEORY
521
difference
Since P ( A ( A u B )
>
0 , t h e l e f t hand s i d e i s o f t h e same
s i g n as t h e d i f f e r e n c e u ( f A )
-
u ( f i ) , which p r o v e s t h e
n e c e s s i t y o f Axiom 4 .
T h i s axiom i s somewhat c o m p l i c a t e d ,
and i t s meaning i s
n o t e a s y t o comprehend. The n e c e s s i t y o f t h i s axiom may be proved as f o l l o w s .
The f i r s t f o u r e q u i v a l e n c e s , namely f A ( i )-,, gg (1) f o r i = 1 , 2 , 3 , 4 imply t h a t
holds i f and o n l y i f t h e a n a l o g o u s i n e q u a l i t y h o l d s f o r d e c i s i o n s gB( i ) ,
i.e. if
522
CHAPTER 5
or
(1)
I n a s i m i l a r w a y , t h e e q u a l i t y hA
u
( l ) d h L 2 ) u gB (2) gB
imp l i e s t h a t
= [u(hL2))
If now f:3)v
kA”2
fL4)u
s i m i l a r w a y , one o b t a i n s
-
u(hL’))]P(AI A u
B).
kA2’, t h e n p r o c e e d i n g i n a
SELECTED TOPICS JTV MEASUREMENT THEORY
523
and u s i n g ( a ) we g e t
S i n c e P(Ai Ad B )
>
0 , we have
As a l r e a d y mentioned, from t h e f i r s t f o u r e q u a l i t i e s i t f o l l o w s t h a t t h e i n e q u a l i t y ( c ) must hold a l s o w i t h fLi)
r e p l a c e d by g i i ) .
Since P ( B \ A u B )
>
Consequently,
0 , one can m u l t i p l y b o t h s i d e s o f t h e
l a s t i n e q u a l i t y by P ( B I A y B ) , o b t a i n i n g
524
CHAPTER 5
A f t e r t r a n s f o r m a t i o n s a n a l o g o u s t o t h o s e which l e d t o
( a ) and ( b ) , b u t performed i n t h e r e v e r s e o r d e r , we obtain
which p r o v e s t h a t
Thus, t h e c o n d i t i o n i n Axiom 5 i s n e c e s s a r y .
6 . Archimedean p r o p e r t y . If A
r\
B =
0, and
N is a
i t i s not t r u e fA ( i t 1)L1 i )" gg( 1) t h a t gB ( 0 ) Iv g i l ) , and moreover, rn g;"' f o r a l l i,itl N , t h e n e i t h e r t h e sequence N is N i s unbounded. f i n i t e , o r t h e sequence f i i ) , i
sequence o f s u c c e s s i v e n a t u r a l numbers,
fi
The Archimedean axiom was a l r e a d y d i s c u s s e d s e v e r a l t i mes i n t h e p r e c e d i n g s e c t i o n s . I n t h i s c a s e , i t i s of some i n t e r e s t t o n o t e t h e c o n s t r u c t i o n of t h e s t a n d a r d sequence f A( i ) : i n some s e n s e , t h e e l e m e n t s gAo) and p l a y t h e r o l e o f a s i n g l e and d o u b l e " s p a c e " i n t h e s t a n d a r d sequence. "Addition" ( i n t h i s c a s e , i n t h e sen s e of o p e r a t i o n U.) of t h e element gA1) ( d o u b l e s p a c e ) (i) g i v e s t h e same as a d d i t i o n o f t h e e l e m e n t gg( 0 ) t o fA ( a s i n g l e space) t o the next elements,
S C
7 . Conditions f o r n u l l s e t s . ( i ) then S E X .
a RC&,
and
R,
(ii) RE A nR =
0
A
i f and o n l y i f t h e c o n d i t i o n s f A \I E D A u R N f A , where f A d e n o t e s t h e r e s t r -
imply f
SELECTED TOPICS IN MEASUREMENT THEORY
525
i c t i o n of f A y R t o t h e s e t A .
T h i s a s s u m p t i o n c h a r a c t e r i z e s t h e c l a s s of n u l l s e t s .
Its f i r s t p a r t a s s e r t s c o m p l e t e n e s s , i . e . t h e f a c t t h a t a s u b s e t o f a n u l l s e t i s a g a i n a n u l l s e t . The second p a r t of t h e axiom c h a r a c t e r i z e s t h e n u l l s e t s as f o l l o w s .
The e v e n t R i s i n , ( (hence i s an e v e n t w i t h s u b j e c t i v e p r o b a b i l i t y z e r o , i . e . a c c e p t e d as i m p o s s i b l e ) , i f an e x t e n s i o n of any d e c i s i o n by i n c l u d i n g R t o i t does n o t change t h e a t t i t u d e towards t h i s d e c i s i o n . F o r i n s t a n c e , suppose t h a t someone t h i n k s t h a t t h e e v e n t R i s i m p o s s i b l e . L e t us c o n s i d e r some d e c i s i o n f A , w i t h A n R = @, e . g . A may s t a n d for t h e s e t o f r e s u l t s of a throw of a d i e , and f A may a s s i g n v a r i o u s outcomes t o t h e r e s u l t s . L e t us now c o n s i d e r t h e a l t e r n a t i v e d e f i n e d as f o l l o w s : i f A u R , and t h e d e c i s i o n f A u R’ an e v e n t from A o c c u r s , t h e consequences w i l l be d e t e r mined a c c o r d i n g t o f A , w h i l e i f R o c c u r s , t h e r e w i l l be some o t h e r consequences ( e . g . a b i g winning, d e a t h , e t c . ) If t h e s u b j e c t s i s convinced t h a t R i s i m p o s s i b l e , he w i l l be i n d i f f e r e n t between f A and f A U R ,r e g a r d l e s s o f t h e outcomes i n c a s e of R .
8 . E n t r i v i a l i t y . ( i ) Y-Jf c o n t a i n s a t l e a s t t h r e e pairwise d i s j o i n t events; ( i i )D c o n t a i n s a t l e a s t two n o n e q u i v a l e n t d e c i s i o n s .
T h i s axiom r e q u i r e s no comments; i t s p u r p o s e i s t o e l i -
minate t r i v i a l s i t u a t i o n s .
9 . S o l v a b i l i t y . ( i )F o r g i v e n A hA t D s u c h t h a t hA d gB ;
and
gg t h e r e e x i s t s
526
CHAPTER 5
then there e x i s t s h
D s u c h t h a t hA
A
u
gB-
f
A u B'
T h i s axiom g u a r a n t e e s t h a t t h e c l a s s D c o n t a i n s s u f f i -
c i e n t l y many e l e m e n t s ( h e n c e t h e name " S o l v a b i l i t y " , which s i g n i f i e s t h a t t h e r e e x i s t d e c i s i o n s which s a t i s f y certain equations). It t u r n s o u t t h a t axioms 1-9 a r e s u f f i c i e n t f o r t h e
e x i s t e n c e o f s c a l e s of u t i l i t y and s u b j e c t i v e p r o b a b i l i t y . More p r e c i s e l y , we have t h e f o l l o w i n g theorem. THEOREM. ~f<x,
Y,d,
is
a system s a t i s f y i n g axioms 1 - 9 , t h e n t h e r e e x i s t s a f u n c t i o n P d e f i n e d on the algebra s u c h t h a t ( X , 7, P ) i s a f i n i t e l y addi t i v e p r o b a b i l i t y measure, and f u n c t i o n u d e f i n e d on D such t h a t C,
7,
( i )A 6
Lk;
( i i )f A
>
if and o n l y i f P ( A ) = 0 ;
gB
( i i i )u ( f A
3
i f and o n l y i f u ( f A ) 3 u ( g g ) ;
g,)
= u(f,)P(A\AUB)
+ u(gB)P(B[A u B)
f o r s e t s A , B s u c h t h a t A 0 B = 0. Moreover, t h e f u n c t i o n P i s d e t e r m i n e d u n i q u e l y , and f u n c t i o n u i s d e t e r m i n e d u n i q u e l y up t o a p o s i t i v e l i n e a r transformation. I n c o n n e c t i o n w i t h t h e above s y s t e m of axioms, i n t r o duced by Luce and Krantz (1971), i t i s w o r t h w h i l e to s k e t c h t h e p r e v i o u s l y s u g g e s t e d s y s t e m s , and p o i n t o u t t h e improvement.
H i s t o r i c a l l y , t h e f i r s t axiomatic approach t o t h e u t i -
SELECTED TOPICS IN MEASUREMENT THEORY
l i t y was g i v e n b y von Neumann and Morgenstern
521
(1944).
A d i s a d v a n t a g e of t h e i r system was t h a t i t concerned
o n l y u t i l i t y , w h i l e p r o b a b i l i t i e s appeared e x p l i c i t e l y i n t h e axioms. Roughly s p e a k i n g , von Neumann and Morgenstern c o n s i d e r e d a c e r t a i n class of objects, a,b,c, and " l o t t e r i e s " of t h e form apb, which l e a d s t o w i n n i n g a w i t h p r o b a b i l i t y p , and winning b w i t h p r o b a b i l i t y 1 - p . More p r e c i s e l y , a winning i n a l o t t e r y may be p a r t i c i p a t i o n i n another l o t t e r y . For instance(apb)p'c denotes a l o t t e r y
...
which l e a d s t o t h e l o t t e r y apb w i t h p r o b a b i l i t y p ' , e t c . The p r e f e r e n c e i s d e f i n e d on t,he s e t o f o b j e c t s and l o t t e r i e s formed o u t of t h e s e o b j e c t s . The b a s i c axioms a s s e r t t h a t : (1)
If
(2) apb
a
> b,
.IJ
then a
> apb
). b;
b(l-p)a;
(4) If a k
b
$ c , then there e x i s t s p
b -ape.
The a s s e r t i o n of t h e theorem i s t h a t t h e r e e x i s t s a funct i o n u, d e f i n e d on t h e s e t of o b j e c t s and l o t t e r i e s , such t h a t u ( a ) > u ( b ) i f and o n l y if a b , and u ( a p b ) = p u ( a ) + ( l - p ) u ( b ) , w i t h u b e i n g d e t e r m i n e d u n i q u e l y up t o a positive linear transformation.
>
The f i r s t axiom system which i m p l i e s a t t h e same t i m e t h e s c a l e s of s u b j e c t i v e p r o b a b i l i t y and u t i l i t y was g i v e n by Savage (1954). The s t a r t i n g p o i n t h e r e i s t h e p r e f e r e n c e on l o t t e r i e s , as i n t h e c a s e of t h e s y s t e m
528
CHAPTER 5
o f v o n Neumann a n d M o r g e n s t e r n , e x c e p t t h a t t h e l o t t e r i e s
are b a s e d on e v e n t s , w i t h o u t p r i o r s p e c i f i c a t i o n o f t h e i r p r o b a b i l i t i e s . T h u s , aAb d e n o t e s a l o t t e r y , w h i c h g i v e s a i f t h e e v e n t A o c c u r s , and g i v e s b i f t h e oppos i t e e v e n t -A o c c u r s . I n s t e a d of p r e s e n t i n g t h e axioms, w e s h a l l o u t l i n e t h e procedure f o r e m p i r i c a l d e t e r m i n a t i o n of s u b j e c t i v e p r o b a b i l i t i e s and u t i l i t i e s , as u s e d by D a v i d s o n , S u p p e s and S i e g e 1 (1957). T h i s p r o c e d u r e i s b a s e d on t h e a x i o m s o f S a v a g e ( r o u g h l y , axioms a r e c h o s e n i n s u c h a way t h a t t h i s p r o c e d u r e y i e l d s c o n s i s t e n t r e s u l t s ; some of t h e axioms, concerning t h e s c a l e of s u b j e c t i v e p r o b a b i l i t y , have b e e n p r e s e n t e d i n S e c t i o n 4 . 2 ) . The i d e a o f measurement o f u t i l i t y i s b a s e d o n t h e fol-
l o w i n g r e a s o n i n g . Suppose t h a t w e s u c c e e d e d i n f i n d i n g a n e v e n t A s a t i s f y i n g t h e c o n d i t i o n P ( A ) = P ( - A ) = 1/2, h e n c e whose s u b j e c t i v e p r o b a b i l i t y i s t h e same as for
. . . denote ... .
i t s negation. Let u ( a ) , u ( b ) , u t i l i t i e s of o b j e c t s a , b ,
the required
S u p p o s e t h a t t h e s u b j e c t has a c h o i c e o f l o t t e r y aAb or cAd, and he t h i n k s t h a t t h e f i r s t i s more p r o f i t a b l e ( i . e . he p r e f e r s t h e f i r s t l o t t e r y t o t h e s e c o n d ) , i . e . aAb
cAd. A c c o r d i n g t o t h e p o s t u l a t e o f SEU, t h e u t i l i t y of t h e f i r s t l o t t e r y i s h i g h e r t h a n t h a t of t h e second, s o that u(a)P(A)
+
u(b)P(-A)
and s i n c e P(A) = P ( - A )
7 u(c)P(A) +
= 1,2,
we get
u(d)P(-A),
SELECTED TOPICS IN MEASUREMENT THEORY
529
or
I n t h i s way one o b t a i n s a n i n e q u a l i t y f o r t h e d i f f e r e n c e o f u t i l i t i e s o f a , b , c and d . W i t h a p p r o p r i a t e o r g a n i z a -
t i o n o f e x p e r i m e n t s , a s e r i e s o f s u c h i n e q u a l i t i e s may p r o v i d e r e l a t i v e l y narrow l i m i t s f o r u t i l i t i e s o f some o b j e c t s ( a c c e p t i n g u t i l i t i e s o f two a r b i t r a r y o b j e c t s as t h e z e r o and u n i t o f t h e s c a l e ) . I n t u r n , g i v e n t h e u t i l i t i e s , one may d e t e r m i n e t h e s u b j e c t i v e p r o b a b i l i t i e s . For i n s t a n c e , i f t h e l o t t e r y a E b , based on t h e e v e n t E i s i n d i f f e r e n t w i t h l o t t e r y cEd, i . e . aEb -4 cEd, t h e n we may w r i t e
which a l l o w s t o d e t e r m i n e P ( E ) as
P(E) =
u(d)-u(b) u(a)-u(b)-u(c)+u(d)
The method s k e t c h e d above i s based on t h e p o s s i b i l i t y
o f a n e f f e c t i v e c o n s t r u c t i o n o f an e v e n t w i t h s u b j e c t i P e p r o b a b i l i t y e q u a l 1 / 2 . Below, w e s h a l l p r e s e n t t h i s construction. An o p e r a t i o n a l d e f i n i t i o n of t h e e v e n t A f o r which P ( A ) = P(-A) = 1 / 2 i s based on t h e c o n d i t i o n t h a t f o r any a , b we s h o u l d have a A b . v bAa. I n t u i t i v e l y , A and -A a r e r e g a r d e d as e q u i p r o b a b l e , i f t h e s u b j e c t i s i n d i f f e r e n t between l o t t e r i e s i n which a i s o b t a i n e d i f A occ-
530
UPS
CHAPTER 5
and b i s o b t a i n e d o t h e r w i s e , o r v i c e v e r s a , b i s
o b t a i n e d when S o c c u r s and a i s o b t a i n e d o t h e r w i s e . It i s of some i n t e r e s t t h a t i t was n o t v e r y e a s y t o
f i n d s u c h a n e v e n t A . The most t y p i c a l e v e n t s , s u c h as t o s s of a c o i n , e t c . d i d not lead t o i n d i f f e r e n c e between t h e l o t t e r i e s aAb and bAa. The e v e n t w i t h subj e c t i v e p r o b a b i l i t y 1 / 2 f o r a l l ( o r a t l e a s t f o r most) s u b j e c t s , was o b t a i n e d by c o n s t r u c t i n g a d i e , w i t h t h r e e s i d e s marked w i t h nonsense s y l l a b e Z O J , and on t h e r e m a i n i n g t h r e e s i d e s - t h e nonsense s y l l a b e Z E J . The c h o i c e o f s y l l a b e s was n o t a r b i t r a r y - t h e s e were t h e s y l l a b l e s which evoked t h e l e a s t number o f a s s o c iations. Thus, i n t h e method of Savage, one d e t e r m i n e s f i r s t t h e u t i l i t i e s , and t h e n t h e s u b j e c t i v e p r o b a b i l i t i e s . On t h e o t h e r hand, t h e axiom system o f Luce and Krantz a l l o w s s i m u l t a n e o u s c o n s t r u c t i o n o f u t i l i t i e s and subj e c t i v e p r o b a b i l i t i e s ( s e e Krantz e t a l . 1971, p . 4 1 0 ) . The most i m p o r t a n t d i f f e r e n c e between t h e a p p r o a c h of
Savage and t h e a p p r o a c h of Luce and Krantz l i e s , howev e r , i n t h e f a c t t h a t i n t h e former c a s e , t h e s e t X ( o f e l e m e n t a r y e v e n t s ) must be i n f i n i t e ( s e e axiom 5 i n Section 4 . 4 , concerning p o s s i b i l i t y of f i n e p a r t i t i o n s o f s e t s ) . On t h e o t h e r hand, due t o t h e r e s t r i c t i o n of a n a l y s i s t o c o n d i t i o n a l d e c i s i o n s , t h e axiom s y s t e m o f Luce and Krantz c o n c e r n s b o t h f i n i t e and i n f i n i t e sets. Moreover, Savage c o n s i d e r s a l l p o s s i b l e d e c i s i o n f u n c t i o n f u n c t i o n s , i . e . f u n c t i o n s d e f i n e d on X w i t h v a l u e s i n C , i n p a r t i c u l a r c o n s t a n t f u n c t i o n s . The l a t t e r are
SELECTED TOPICS IN MEASUREMENT THEORY
531
t h o s e u n d e r w h i c h t h e c o n s e q u e n c e d o e s n o t d e p e n d on t h e e v e n t w h i c h o c c u r s ( i n t e r m i n o l o g y of d e c i s i o n t h e o r y , t h e v a l u e o f t h e f u n c t i o n d o e s n o t d e p e n d on t h e "stat e o f N a t u r e " ) . I n a d d i t i o n , o n e has t o c o n s i d e r a l l d e c i s i o n s , even t h e a b s u r d o n e s , o r s u c h which a r e impossible t o implement, e t c . A s o p p o s e d t o t h a t , i n t h e s y s t e m o f Luce and K r a n t z ,
t h e c o n s i d e r a t i o n s are r e s t r i c t e d t o a c e r t a i n c l a s s D o f c o n d i t i o n a l d e c i s i o n s ; t h i s c l a s s need n o t c o n t a i n c o n s t a n t d e c i s i o n s , and i s u s u a l l y much smaller t h a n t h e c l a s s of a l l l o g i c a l l y p o s s i b l e d e c i s i o n s . This a p p r o a c h i s , t h e r e f o r e , more r e a l i s t i c , a l l o w i n g t o r e s t r i c t t h e considerations t o a c l a s s of "sensible" decisions. F i n a l l y , t h e c o n d i t i o n a l d e c i s i o n s allow us t o t a k e i n t o a c c o u n t t h e f a c t t h a t making a g i v e n d e c i s i o n c a n c a u s e a c h a n g e o f p r o b a b i l i t i e s o f some e v e n t s ( e . g . t h e y may become i m p o s s i b l e ) . T h u s , t h e t h e o r y o f Luce and K r a n t z l e a d s t o a more n a t u r a l f o r m u l a t i o n o f many d e c i s i o n
problems
. 8 . SOME PROBLEMS I N UTILITY THEORY
T h i s s e c t i o n w i l l d e a l w i t h some p r o b l e m s c o n n e c t e d w i t h SEU m o d e l . On t h e one h a n d , c e r t a i n p a r a d o x e s i n d i c a t e t h a t human p r e f e r e n c e s may v i o l a t e c e r t a i n a x i o m s . S e c o n d l y , i t may happen t h a t t h e u t i l i t y o f a n outcome may d e p e n d on i t s p r o b a b i l i t y o f o c c u r r e n c e . I f t h i s i s a l l o w e d , c e r t a i n e n t i r e l y new phenomena may a p p e a r ; t h e s e w i l l be d i s c u s s e d i n S e c t i o n 8 . 2
532
CHAPTER 5
( p r e s e n t i n g new r e s u l t s on b e h a v i o u r under r i s k ; s e e Nowakowska 1 9 8 0 ) . F i n a l l y , i n more complex s o c i a l s i t u a t i o n s , i t may happen t h a t o n e ' s u t i l i t y depends n o t o n l y on t h e d i r e c t outcome o f an a c t i o n , b u t a l s o on " i t e r a t e d p e r c e p t i o n " , i . e . on how t h e d e c i s i o n maker p e r c e i v e s t h a t t h e o t h e r s w i l l p e r c e i v e h i m , i f he make s s u c h and s u c h d e c i s i o n . A c e r t a i n model ( s e e Nowakows k a 1 9 7 6 ) , a l l o w i n g t h e p r e d i c t i o n o f p r o s o c i a l and a n t i s o c i a l b e h a v i o u r w i l l be p r e s e n t e d .
8 . 1 . C e r t a i n paradoxes of u t i l i t y t h e o r y The axioms o f u t i l i t y t h e o r y , s u c h as t h o s e of Luce and K r a n t z , may be i n t e r p r e t e d as p r i n c i p l e s o f r a t i o n a l b e h a v i o u r , i n t h e f o l l o w i n g s e n s e : everyone who a g r e e s w i t h t h e axioms, and t h i n k s t h a t t h e y d e s c r i b e h i s own p r i n c i p l e s o f d e c i s i o n making, s h o u l d a g r e e t h a t t h e r e e x i s t s c a l e s of u t i l i t y and s u b j e c t i v e p r o b a b i l i t y c h a r a c t e r i s t i c f o r him, and t h a t he behaves as i f he wanted t o maximize t h e s u b j e c t i v e e x p e c t e d u t i l i t y (SEU). I n t h i s s e n s e , t h e t h e o r y may s e r v e as a normati v e mode 1 of be h a v i o u r
.
The problem whether or n o t t h i s t h e o r y c o n s t i t u t e s a l s o
a d e s c r i p t i v e model, i s much more d e l i c a t e . It t u r n s o u t , namely, t h a t i t was p o s s i b l e t o f i n d a number o f examples o f b e h a v i o u r i n c o n s i s t e n t w i t h t h e SEU model ( A l l a i s 1953, E l l s b e r g 1 9 6 1 ) . For i n s t a n c e , l e t us cons i d e r f o u r d e c i s i o n s i t u a t i o n s , d e n o t e d by I , 11, I11 and I V , p r e s e n t e d s c h e m a t i c a l l y i n form of t h e following table.
-
SELJ2ClED TOPICS INMEASUREMENT THEORY
Events
A1 (red)
Probabilities
1/3 -.
lottery I l o t t e r y I1 l o t t e r y 111 lottery IV
533
A3
A2
(black)
(white)
2/3
1 00
0
0
0
100
0
1 00
0
1 00
0
1 00
100
F o r i n s t a n c e , i m a g i n e an u r n c o n t a i n i n g 10 r e d b a l l s a n d 20 b a l l s o f w h i c h some a r e b l a c k a n d some a r e whi-
t e , i n unknown p r o p o r t i o n . The p r o b a b i l i t y o f d r a w i n g a r e d b a l l i s 1/3, w h i l e t h e p r o b a b i l i t y o f d r a w i n g a b a l l w h i c h i s e i t h e r b l a c k or+ w h i t e i s 2 / 3 . The p a y o f f s f o r v a r i o u s outcomes a r e p r e s e n t e d on t h e table. The l o t t e r y I g i v e s p r o b a b i l i t y o f w i n n i n g e q u a l 1/3, w h i l e l o t t e r y I1 g i v e s a n unknown p r o b a b i l i t y of w i n n i n g , b e t w e e n 0 a n d 2 / 3 , d e p e n d i n g on t h e number o f b l a c k b a l l s , w h i c h m a y be b e t w e e n 0 and 2 0 . One c a n i m a g i n e a p e r s o n , s a y w i t h a n a v e r s i o n t o r i s k , who would p r e f e r l o t t e r y I t o l o t t e r y 11.
L e t u s c o n s i d e r now l o t t e r i e s I11 and I V . As b e f o r e
l o t t e r y I , now t h e l o t t e r y I V g i v e s a known p r o b a b i l i t y o f w i n n i n g ( 2 / 3 ) , w h i l e l o t t e r y I11 ( a s b e f o r e l o t t e r y 11) g i v e s h i g h e r r i s k : p r o b a b i l i t y o f w i n n i n g i s unknown h e r e , and we know o n l y t h a t i t i s c o n t a i n e d b e t w e e n t h e l i m i t s 1/3 (when t h e r e a r e n o w h i t e b a l l s ) and 1 (when t h e r e a r e n o b l a c k b a l l s ) .
CHAPTER 5
534
Here a g a i n p e r s o n s w i t h an a v e r s i o n t o r i s k m i g h t p r e f e r l o t t e r y IV t o l o t t e r y 111. However t h e p r e f e r e n c e s I ?- I1 and IV 2111 are i n c o n s i s t e n t w i t h t h e " s u r e t h i n g p r i n c i p l e " , one of t h e forms of which i s e x p r e s s -
-
ed by axiom 4: i f f A--. gB , t h e n f A gB f A w gg. In t h i s c a s e , l o t t e r i e s I and 11, as w e l l as l o t t e r i e s I11 and IV do n o t d i f f e r f o r e v e n t A so t h a t t h e 3, d i r e c t i o n of p r e f e r e n c e s s h o u l d depend o n l y on t h e outcomes f o r A1 and A 2 . These outcomes a r e , however, i n i d e n t i c a l c o n f i g u r a t i o n f o r t h e p a i r s 1,II and II1,IV. C o n s e q u e n t l y , i f someone p r e f e r s I t o 11, he s h o u l d also p r e f e r I11 t o IV and v i c e v e r s a . W r i t i n g I and s i m i l a r l y s y m b o l i c a l l y l o t t e r y I as f A A 2 u
fi3,
10
f o r o t h e r l o t t e r i e s , w e have h e r e
I11 I11 I rvf fAlu A 2 A1"A2
IV I1
' fAlv
IV A2-
fA1o A 2
'
Consequently , T
r
r-r
I11
I11
hence
I11
fA1u
IV
A2
). fA1vA2
and
' A ~ ~2 u A f ~ 3
IV f A l ~ A 2
or
IV fA3
'
SELECTED TOPICS IN MEASUREMENT THEORY
535
It i s i n t e r e s t i n g t h a t some ( t h o u g h n o t a l l ) p e r s o n s
who p r e f e r I t o I1 and p r e f e r IV t o 111, c o n f r o n t e d w i t h t h e above r e a s o n i n g showing i n c o n s i s t e n c y o f t h e i r preferences with the sure thing principle, s t i l l r e t a i n t h e i r preferences. Such d a t a i n d i c a t e t h a t some o f t h e axioms - i n p a r t i may b e q u e s t i o n a b l e , and c a s t some c u l a r axiom 4 d o u b t on t h e d e s c r i p t i v e adequacy o f SEU model.
-
It i s n o t known what consequences may be o b t a i n e d i f
one o m i t s t h e s u r e t h i n g p r i n c i p l e . A t any r a t e , t h e f a c t t h a t some p e r s o n s , even when c o n f r o n t e d w i t h t h e i n c o n s i s t e n c y of t h e i r p r e f e r e n c e s w i t h t h i s axiom s t i l l r e t a i n t h e i r p r e f e r e n c e s , i s by i t s e l f a n i n t e r e s t i n g p sy c ho l o g i c a1 p he nome non
.
8 . 2 . A model of c h o i c e b e h a v i o u r The p r e s e n t s e c t i o n w i l l c o n c e r n a c e r t a i n new model ( s e e Nowakowska 1 9 8 0 ) , d e s i g n e d t o c o v e r t h o s e s i t u a t i o n s i n which t h e SEU model does n o t a p p l y , and a l s o t o i n c o r p o r a t e some p s y c h o l o g i c a l f e a t u r e s o f m o t i v a t i o n model of Atkinson ( s e e A t k i n s o n 1963, A t k i n s o n and B i r c h 1970; Heckhausen 1973; Weiner 1 9 7 2 ) . T h i s model w i l l d e a l w i t h b i n a r y c h o i c e s o n l y . The
d e c i s i o n maker i s t o choose between a r i s k y o p t i o n , t o be d e n o t e d by R , and a r i s k l e s s o p t i o n N . The l a t t e r i n v o l v e s no u n c e r t a i n t y , and b r i n g s a reward b . Thus, t h e o p t i o n N s e r v e s mainly as a r e f e r e n c e frame: t h e d e c i s i o n maker w i l l choose t h e r i s k y o p t i o n R , i f i t s e x p e c t e d u t i l i t y ( t o be d e f i n e d l a t e r ) e x c e e d s t h e
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u t i l i t y o f t h e r i s k l e s s o p t i o n N , t h a t i s , i f i t exceeds u ( b ) . Now, t h e o t h e r o p t i o n (R) may l e a d t o o n e of t h e two o u t c o m e s , w h i c h we s h a l l c a l l s u c c e s s and f a i l u r e . The s u b j e c t i v e p r o b a b i l i t i e s o f t h e s e outcomes w i l l be p and 1-p. The s u c c e s s b r i n g s t h e r e w a r d r , and f a i l u r e l e a d s t o a l o s s d (or: "reward" - d ) . The scheme o f t h e c o n s i d e r e d d e c i s i o n s i t u a t i o n may
t h u s be r e p r e s e n t e d as t h e f o l l o w i n g d e c i s i o n t r e e . DECISION
RISKY OPTION R
I R I SKLESS OPTION N
P r o b . 1-p
It i s c l e a r t h a t t h e v a l u e o f t h e r i s k y o p t i o n R m u s t be a f u n c t i o n o f a t l e a s t t h r e e p a r a m e t e r s :
t h e subj e c t i v e p r o b a b i l i t y p o f s u c c e s s , t h e amount r o f r e ward f o r s u c c e s s , and t h e amount d o f l o s s from f a i l ure. The main o b j e c t o f i n t e r e s t w i l l be t o s t u d y t h e de-
SELECTED TOPICS IN MEASUREMENT THEORY
531
pendence on t h e f i r s t p a r a m e t e r , t h a t i s , s u b j e c t i v e p r o b a b i l i t y p o f s u c c e s s . Consequently (remembering t h a t r and d a r e a l s o i n t e r v e n i n g ) we s h a l l d e n o t e t h e r i s k y o p t i o n R w i t h p r o b a b i l i t y p of s u c c e s s by R The symbol than").
P
.
w i l l stand f o r preference ("is b e t t e r
We s t a r t from i n t r o d u c i n g t h e f o l l o w i n g d e f i n i t i o n s . D E F I N I T I O N . The s e t o f a l l p
<
[O,l]
w i l l be c a l l e d t h e r i s k a r e a .
DEFINITION. The r i s k a r e a A i f t h e conditions p
such t h a t R
P
N
6
[O,l] i s c a l l e d normal, A and p ' 7 p imply p ' c A .
N a t u r a l l y , i n t h e above d e f i n i t i o n , t h e r a n g e of p 1 i s a l s o r e s t r i c t e d t o t h e i n t e r v a l [0,1]. I n g e n e r a l , i t w i l l be t a c i t l y assumed t h a t a l l numbers a p p e a r i n g as p r o b a b i l i t i e s w i l l be s o r e s t r i c t e d . From t h e above d e f i n i t i o n i t f o l l o w s t h a t a r i s k a r e a A i s n o t normal, i f f o r some p ' , p" we have p 1 p",
<
p l c A and p"
g
A.
Two t y p e s o f non-normal r i s k a r e a s w i l l be d i s t i n g u i s h e d , c a l l e d c e n t r a l and boundary. DEFINITION. A r i s k a r e a A w i l l be c a l l e d c e n t r a l , i f there e x i s t points p p 1 < p" s u c h t h a t p ' E A , w h i l e The r i s k a r e a A w i l l be c a l l e d boundary, p , p" 4 A . if whenever p E A and p 1 < p , t h e n p 1 6 A . Thus, boundary r i s k a r e a s c o n s i s t o f an i n t e r v a l which s t a r t s a t 0 , and a r e , i n a s e n s e , d u a l t o normal r i s k
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areas. N a t u r a l l y , as i t i s n o t r e q u i r e d t h a t a r i s k a r e a be c o n n e c t e d , t h e above c o n c e p t s do n o t e x h a u s t a l l t y p e s o f r i s k areas. I n t h e s e q u e l , we s h a l l d e n o t e by f ( p ) t h e e x p e c t e d val u e of t h e r i s k y option R It i s t h e n c l e a r , t h a t a l l P r i s k areas a r e normal i f , and o n l y i f t h e f u n c t i o n f ( p ) i s monotone i n c r e a s i n g . T h i s f o l l o w s from t h e f a c t t h a t t h e r i s k a r e a i s t h e s e t o f a l l p o i n t s p f o r which f ( p ) ). u ( b ) , where u ( b ) i s t h e u t i l i t y of t h e r i s k l e s s option N.
.
It would seem t h a t normal r i s k a r e a s s h o u l d be most n a t u r a l , and a p p e a r most o f t e n , i f n o t always: t h e mor e p r o b a b l e i s t h e s u c c e s s , t h e more v a l u a b l e i s t h e risky option (other conditions being equal). To p u t i t d i f f e r e n t l y , non-normal r i s k a r e a s a p p e a r
somewhat p a t h o l o g i c a l : w i t h s u c h a r i s k a r e a , t h e deas b e t t e r P t h a n t h e r i s k l e s s o p t i o n N for some p , b u t when p i s c i s i o n maker s h o u l d choose t h e r i s k y o p t i o n R
i n c r e a s e d t o p ' , he would choose t h e r i s k l e s s o p t i o n N . One might be tempted t o l a b e l s u c h a b e h a v i o u r "irrational". One of t h e main r e s u l t s o f t h i s s e c t i o n , however, w i l l be t o show t h a t -- e x c e p t f o r t h e extreme c a s e o f t h e " c l a s s i c a l " SEU model -- non-normal r i s k a r e a s a r e rat h e r a r u l e t h a n an e x c e p t i o n . More p r e c i s e l y , e x c e p t f o r t h e SEU model, r i s k areas w i l l a p p e a r i n any s i t u a t i o n , f o r s u i t a b l e v a l u e s o f t h e p a r a m e t e r s which r e p r e s e n t r e w a r d s , l o s s e s , and t h e comparison l e v e l .
SELECTED TOPICS Lh' MEASUREMENT THEORY
539
It i s e s s e n t i a l t o p o i n t o u t t h e f o l l o w i n g f a c t . To
d e t e r m i n e whether or n o t t h e non-normal r i s k a r e a s may a p p e a r , one needs t o s t u d y t h e i p r o p e r t i e s o f t h e v a l u e f u n c t i o n f ( p ) . On t h e o t h e r hand, i f t h e non-norm a 1 r i s k areas may a p p e a r , t h e n t h e q u e s t i o n whether o r n o t t h e y w i l l a c t u a l l y a p p e a r depends a l s o on t h e comparison l e v e l . T h i s i s i l l u s t r a t e d on t h e f i g u r e below, a t which f ( p ) i s n o t monotone. Thus, t h e nonnormal r i s k a r e a s may a p p e a r , and t h e y a c t u a l l y do app e a r for comparison l e v e l k , b u t n o t f o r comparison level k'
.
mi
I
I I
I j
z
I I
l
J,xxxlxxxxx3xxx~ xxxxx "
,
Here t h e p a r t of t h e i n t e r v a l [O,l] marked by s i n g l e x i s a normal r i s k a r e a , w h i l e t h e p a r t marked w i t h doubl e x i s a non-normal ( c e n t r a l ) r i s k a r e a . To summarize t h e c o n s i d e r a t i o n s t h u s f a r , we may s a y t h a t t h e r e e x i s t s a c l a s s o f s i t u a t i o n s , and a psychol o g i c a l l y d e f e n s i b l e d e f i n i t i o n of t h e v a l u e f u n c t i o n f ( p ) of r i s k y o p t i o n R s u c h t h a t f o r some p a r a m e t e r s P' r and d t h e f u n c t i o n f ( p ) i s n o t m o n o t o n i c a l l y i n c r e a s i n g . Thus, f o r a p p r o p r i a t e v a l u e s o f t h e comparison
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l e v e l , t h e r i s k a r e a s w i l l b e non-normal. 8 . 2 . 1 . General p o s t u l a t e s We s h a l l a c c e p t t h e f o l l o w i n g p o s t u l a t e :
POSTULATE.
evaluating the risky option R
t h e deP ' ___ c i s i o n maker forms a swn o f v a l u e s o f outcomes weighted by t h e i r p r o b a b i l i t i e s , s o t h a t f ( p ) = v a l u e o f RP = p v ( S ) t ( l - p ) v ( s f )
where v ( S ) failure,
and
(8.1)
v ( S ' ) a r e t h e v a l u e s o f s u c c e s s and
Let u s s e e f i r s t t h e consequences o f t h i s p o s t u l a t e i n t h e " c l a s s i c a l " SEU model. T h i s model i s c h a r a c t e r i z e d by t h e c r u c i a l p o s t u l a t e t h a t t h e v a l u e s v ( S ) and v ( S t ) o f s u c c e s s and f a i l u r e do n o t depend on t h e i r pro-
b a b i l i t i e s of o c c u r r e n c e . Thus, v ( S ) depends o n l y on t h e reward r , and i s u s u a l l y t a k e n t o be t h e u t i l i t y o f r e w a r d , u(1-1. S i m i l a r l y , v ( S t ) depends o n l y on t h e amount d of l o s s , and i s e q u a l t o t h e u t i l i t y u ( - d ) . Consequently, r e l a t i o n ( 8 . 1 ) reduces t o
= p[U(r)
- u(-d)]
t U(-d).
I m p l i c i t i n t h e t e r m s " s u c c e s s " and ' ! f a i l u r e " i s t h e assumption t h a t u ( r )
> u(-d),
so that f ( p ) i s increas-
i n g ( l i n e a r ) f u n c t i o n o f p . We have t h e r e f o r e t h e f o l l o w i n g theorem c h a r a c t e r i z i n g SEU model:
SELECTED TOPICS IN MEASUREMENT THEORY
THEOREM 1.
& I t h e c l a s s i c a l SEU model, a l l
541
r i s k areas
a r e normal. The s i t u a t i o n s when t h e p o s t u l a t e o f independence o f u t i l i t i e s o f s u c c e s s and f a i l u r e on t h e p r o b a b i l i t i e s may be e x p e c t e d t o be s a t i s f i e d a r e a l l c a s e s , when t h e d e c i s i o n maker has no c o n t r o l o v e r t h e s e p r o b a b i l i t i e s . However, t h e r e e x i s t s i t u a t i o n s when t h e o c c u r r e n c e o f s u c c e s s o r f a i l u r e d e p e n d s , even p a r t i a l l y , on t h e dec i s i o n m a k e r , f o r i n s t a n c e on h i s s k i l l , p e r s i s t e n c e , knowledge, e t c . I n s u c h c a s e s , t h e assumption o f t h e independence of u t i l i t i e s of s u c c e s s and f a i l u r e on t h e i r p r o b a b i l i t i e s i s no l o n g e r a d e q u a t e . One may t h e n expect t h a t t h e l e s s l i k e l y i s t h e success, t h e higher i s t h e s a t i s f a c t i o n from i t s o c c u r r e n c e , and t h e more l i k e l y i s t h e f a i l u r e , t h e more s a t i s f a c t i o n from avoiding it. Thus f a r , t h e o n l y model which t o o k s u c h a dependence i n t o a c c o u n t was t h e m o t i v a t i o n model of Atkinson (1963). However, t h i s model n e g l e c t e d a l l Reconomical" v a r i a b l e s , s u c h as amounts o f rewards and losses, and conc e n t r a t e d s o l e l y on t h e c h o i c e o f t a s k d i f f i c u l t y , i . e . p r o b a b i l i t y of success. I n c i d e n t a l l y , one could a r g u e t h a t even i n t h e c a s e o f t o t a l l a c k o f c o n t r o l o f t h e d e c i s i o n maker o v e r t h e p r o b a b i l i t i e s o f s u c c e s s , t h e v a l u e s v ( S ) and v ( s l ) may depend on p . Imagine a p e r s o n who wins a p r i z e , say a c o l o u r TV s e t , on a l o t t e r y . He comes home and
t e l l s h i s w i f e : "Can you b e l i e v e t h a t I won t h i s p r i z e , even though t h e r e was only one chance i n 1 0 0 0 0 " ( s a y ) . Perhaps he would be l e s s e l a t e d i f t h e chances of win-
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n i n g t h e same c o l o u r TV s e t w e r e , s a y , one i n a hundred. The r e a s o n i s t h a t i n t h i s c a s e t h e r e would be l e s s o f t h e f a c t o r due t o something " u n u s u a l " . 8 . 2 . 2 . The g e n e r a l model. L e t u s now c o n s i d e r t h e gener a l model of c h o i c e o u t l i n e d i n t h e p r e c e d i n g s e c t i o n s .
According t o t h e main p o s t u l a t e , t h e v a l u e o f t h e r i s k y o p t i o n R i s t h e weighted a v e r a g e of t h e v a l u e s of t h e P two outcomes. It w i l l be assumed g e n e r a l l y t h a t t h e v a l u e v ( S ) o f
s u c c e s s depends on t h r e e f a c t o r s : (1) t h e amount r of reward; (2) p r o b a b i l i t y p o f s u c c e s s , and ( 3 ) s t r e n g t h M of t h e motive t o s u c c e s s . On t h e o t h e r hand, t h e v a l u e v ( S ' ) o f f a i l u r e w i l l a l s o depend on t h r e e f a c t o r s : (1) t h e amount d o f l o s s ; ( 2 ) p r o b a b i l i t y p o f s u c c e s s (or e q u i v a l e n t l y : probab i l i t y 1-p of f a i l u r e ) , and ( 3 ) s t r e n g t h F o f f e a r of f a i l u r e (motive t o avoid f a i l u r e ) . I n e a c h c a s e , t h e f i r s t two f a c t o r s p r e s e n t no d i f f i c u l t y . Some e x p l a n a t i o n , however, i s needed f o r t h e motive s M and F . These n o t i o n s are due t o Atkinson ( 1 9 6 3 ) ; t h e i r i n t u i t i v e c o n t e n t i s q u i t e s t r a i g h t f o r w a r d , and an i n f o r m a l d e f i n i t i o n i s r a t h e r c l e a r . Thus, M i s t h e g e n e r a l t e n d e n c y o f t h e d e c i s i o n maker t o a c h i e v e s u c c e s s , g u i d i n g h i s b e h a v i o u r towards s i t u a t i o n s i n which achievement i s p o s s i b l e . On t h e o t h e r hand, f e a r o f f a i l u r e F p a r a l y s e s h i s a c t i o n s , c a u s i n g him t o w i t h draw from r i s k y s i t u a t i o n s , which may lead t o f a i l u r e . The main problem c o n s i s t s of measurement of t h e s e motiv-
es.
S i n c e t h e e x a c t v a l u e s a r e , t h u s f a r , beyond r e a c h ,
543
SELECTED TOPICS mMEASUREMENT THEORY
and one has only t e s t s which p r o v i d e us w i t h o r d i n a l d a t a , t h e model below w i l l be c o n s t r u c t e d s o t h a t t h e l a c k of knowledge of e x a c t v a l u e s of M and F w i l l n o t p r e v e n t one from making q u a l i t a t i v e p r e d i c t i o n s . I t w i l l be assumed t h a t M and F a r e some p o s i t i v e v a l u -
e s which e n t e r m u l t i p l i c a t i v e l y t o t h e v a l u e s v ( S ) and v ( S ' ) . Thus,
The f u n c t i o n I s ( p , r ) w i l l be c a l l e d t h e i n c e n t i v e
of s u c c e s s , w h i l e I a f ( p , d ) w i l l be c a l l e d t h e i n c e n t i v e t o a v o i d f a i l u r e ( s e e Atkinson 1 9 6 3 ) . We s h a l l now f o r m u l a t e some h y p o t h e s e s about t h e s e func-
tions. HYPOTHESIS 1. The i n c e n t i v e v a l u e I s ( p , r ) of s u c c e s s s a t i s f i e s the following conditions:
( a ) I ( p , r ) i s a d i f f e r e n t i a b l e f u n c t i o n of p for S any r , and t h e d e r i v a t i v e 31s/l)p i s bounded i n t h e c l o s e d i n t e r v a l [0,1];
for a l l s u f f i c i e n t l y small r > 0 , and p s u f f i c i e n t l y c l o s e t o 1, t h e f u n c t i o n I s ( p , r ) i s r e p r e s e n t a b l e a s a sum (b)
(8.4)
(8.5)
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H(r)
--+0 as r -+ 0 .
(8.7)
C o n d i t i o n ( a ) of H y p o t h e s i s 1 i s o f t e c h n i c a l c h a r a c t e r and s c a r c e l y r e q u i r e s any comments: i t a s s e r t s t h a t t h e r a t e a t which t h e i n c e n t i v e o f s u c c e s s changes w i t h t h e change of p r o b a b i l i t y o f s u c c e s s remains bounded. Observe t h a t t h i s c o n d i t i o n does n o t a s s e r t a n y t h i n g about t h e s i g n of t h e d e r i v a t i v e w i t h r e s p e c t t o p . C o n d i t i o n ( b ) of H y p o t h e s i s 1 s t a t e s g e n e r a l l y t h a t t h e v a r i a b l e s p and r i n t h e i n c e n t i v e I ( p , r ) o f s u c c e s s S s e p a r a t e a d d i t i v e l y i n t h e neighbourhood o f t h e p o i n t p = 1, r = 0 . It s a y s t h e r e f o r e t h a t t h e p s y c h o l o g i c a l
and economical e f f e c t s a r e a d d i t i v e , a t l e a s t i n t h e domain o f small rewards and h i g h l y p r o b a b l e s u c c e s s e s , i . e . i n t h e domain- where t h e s e e f f e c t s a r e weak. I n s t i l l o t h e r words, t h i s a s s u m p t i o n s t a t e s t h a t p and r do n o t i n t e r a c t i n t h e i n c e n t i v e v a l u e of s u c c e s s i n t h e neighbourhood o f p = 1, r = 0. The j u s t i f i c a t i o n o f t h e t h r e e s p e c i f i c a s s u m p t i o n s (8.5), ( 8 . 6 ) and ( 8 . 7 ) i s as f o l l o w s . The i n c e n t i v e v a l u e o f s u c c e s s i s (or may b e ) c o n n e c t e d w i t h overcomi n g some d i f f i c u l t i e s , use of s k i l l , knowledge, e t c . I n any s u c h c a s e , t h e l e s s l i k e l y i s t h e s u c c e s s , t h e more s a t i s f a c t i o n i t b r i n g s . T h i s means, i n t u r n , t h a t I ( p , r ) s h o u l d d e c r e a s e w i t h t h e i n c r e a s e of p , hence S i t s d e r i v a t i v e w i t h r e s p e c t t o p s h o u l d be n e g a t i v e . I n f a c t , Hypothesis 1 r e q u i r e s t h a t t h e i n c e n t i v e of s u c c e s s s h o u l d d e c r e a s e w i t h p o n l y i n t h e neighbour-
SELECTED TOPICS IN MEASUREMENT THEORY
54s
hood o f p = 1 ( c o n d i t i o n ( 8 . 6 ) ) ; no m o n o t o n i c i t y r e q u i rement i s imposed f o r o t h e r v a l u e s o f p . I n t e r p r e t i n g t h e f u n c t i o n s G and H i n c o n d i t i o n ( 8 . 4 ) as t h e e f f e c t s on i n c e n t i v e o f s u c c e s s due r e s p e c t i v e l y t o t h e p r o b a b i l i t y of s u c c e s s ( h e n c e from o c c u r r e n c e of " s u c c e s s as s u c h " , r e g a r d l e s s o f t h e economical reward r ) , and t o t h e economical reward r , one may s a y t h e f o l l o w i n g . C o n d i t i o n ( 8 . 5 ) s t a t e s t h a t t h e satTisf a c t i o n from " s u c c e s s as s u c h " , s e p a r a t e l y from t h e e c o n o m i c a l r e w a r d , becomes n e g l i g i b l e when t h e s u c c e s s becomes c e r t a i n . However, t h e r a t e of change of t h i s e f f e c t as p t e n d s t o 1 i s n o t n e g l i g i b l e ( c o n d i t i o n ( 8 . 6 ) ) : a s p becomes l a r g e r , t h e p s y c h o l o g i c a l s a t i s f a c t i o n from s u c c e s s becomes s m a l l e r a t a r a t e which a p p r o a c h e s some n e g a t i v e v a l u e . F i n a l l y , c o n d i t i o n ( 8 . 7 ) a s s e r t s t h a t t h e e f f e c t on t h e i n c e n t i v e o f s u c c e s s from t h e reward r becomes n e g l i g i b l e when t h e reward a p p r o a c h e s z e r o . The f o l l o w i n g a s s u m p t i o n a b o u t I S ( p , r ) w i l l be needed i n c a s e o f one theorem o n l y ; t h i s e x p l a i n s why i t i s l i s t e d separately. HYPOTHESIS 2 .
If r
> 0 , then
I n t u i t i v e l y , t h i s hypothesis a s s e r t s t h a t t h e incentive v a l u e of s u c c e s s which i s v e r y i m p r o b a b l e , and which b r i n g s a p o s i t i v e r e w a r d , i s p o s i t i v e . A moment o f reflection i s sufficient t o r e a l i z e that condition (8.8) i s v e r y n a t u r a l and p l a u s i b l e .
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L e t u s now t u r n t o t h e h y p o t h e s e s about t h e i n c e n t i v e
v a l u e I a f ( p , d ) o f a v o i d i n g f a i l u r e . It a p p e a r s t h a t f o r t h e e x i s t e n c e of non-normal r i s k a r e a s , t h e a s s u m p t i o n s about t h e f u n c t i o n Iaf a r e s t i l l weaker t h a n t h o s e a b o u t t h e f u n c t i o n I s . Again, we s e p a r a t e t h e c o n d i t i o n which i s needed i n one o f t h e theorems o n l y .
HYPOTHESIS 3 . The f u n c t i o n I a f ( p , d ) i s d i f f e r e n t i a b l e w i t h r e s p e c t t o p f o r e v e r y d , and t h e d e r i v a t i v e 9 1 /Jp i s bounded i n t h e c l o s e d i n t e r v a l [O,l]. af HYPOTHESIS 4 . For a l l d
>0
we have
(8.9) Both of t h e s e h y p o t h e s e s h a r d l y need any i n t u i t i v e j u s t i f i c a t i o n . H y p o t h e s i s 3 i s a g a i n of a t e c h n i c a l c h a r a c t e r , a s s e r t i n g simply a c e r t a i n d e g r e e o f smoothn e s s of f u n c t i o n I ( p , d ) . H y p o t h e s i s 4 s t a t e s t h a t af f o r p s u f f i c i e n t l y c l o s e t o 1, hence f o r f a i l u r e s which a r e h i g h l y improbable, t h e v a l u e I a f ( p , d ) i s n e g a t i v e (at l e a s t f o r f a i l u r e s which r e s u l t i n a loss d > 0 ) . To p u t i t d i f f e r e n t l y : i f d > 0 , t h e f a i l u r e a l w a y s " h u r t s " , and i t h u r t s more when i t i s i m p r o b a b l e . T h i s , i n c i d e n t a l l y , s u g g e s t s t h a t a n a t u r a l requirement on Iaf would be t h a t i t i s a d e c r e a s i n g f u n c t i o n o f p . We do n o t s t a t e i t as a n a s s u m p t i o n , however, s i n c e i t w i l l n o t be needed i n e s t a b l i s h i n g t h e p r o o f o f t h e t heorem. We s h a l l now prove two theorems about t h e e x i s t e n c e of non-normal r i s k a r e a s ,
SELECTED TOPICS IN MEASUREMENT THEORY
541
THEOREM 2 . Assume t h a t t h e i n c e n t i v e f u n c t i o n s I s ( p , r )
and I a f ( p , d ) s a t i s f y r e s p e c t i v e l y Hypotheses 1 and 3. Then for e v e r y F gr& d t h e r e e x i s t s Mo s u c h t h a t whene v e r M > M and r i s s u f f i c i e n t l y s m a l l , t h e r i s k a r e a 0i s non-normal f o r s u i t a b l e reward b f o r r i s k l e s s o p t i o n . I n symbolic n o t a t i o n :
(M > M o ) 3 ( 3 r o ) ( V r ) C ( cr r o ) j
( 3 b ) : r i s k area i s
non-normall. What t h i s theorem a s s e r t s , i n e s s e n c e , i s t h a t i f t h e
motive M t o a c h i e v e s u c c e s s i s s u f f i c i e n t l y h i g h , and t h e reward r o f s u c c e s s i s s u f f i c i e n t l y small, t h e n one c a n f i n d s u c h a reward f o r t h e r i s k l e s s o p t i o n N t h a t t h e r i s k area w i l l be non-normal: t h e d e c i s i o n maker w i l l p r e f e r t h e r i s k y o p t i o n R f o r some p , b u t when P t h e chances p of s u c c e s s w i l l be s u f f i c i e n t l y c l o s e t o 1, he w i l l p r e f e r t h e r i s k l e s s o p t i o n N
.
P r o o f . It s u f f i c e s t o show t h a t t h e v a l u e f u n c t i o n f ( p ) of t h e r i s k y o p t i o n R i s not monotonically i n c r e a s i n g . P F o r t h a t , i n t u r n , i t i s enough t o show t h a t t h e r e e x i s t s a p o i n t a t which t h e d e r i v a t i v e f ' ( p ) i s n e g a t i v e . We s h a l l p r o v e t h a t s u c h a p o i n t e x i s t s i n t h e neighbourhood of p = 1. Using ( 8 . 1 ) and ( 8 . 3 ) w e o b t a i n
L e t u s choose p c l o s e t o 1 and r c l o s e t o 0 , s o t h a t
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t h e r e l a t i o n ( 8 . 4 ) of Hypothesis 1 holds. S u b s t i t u t i o n
of ( 8 . 4 ) i n t o (8.10) y i e l d s
N e x t , l e t us f i x F and d , and l e t (8.12)
Mo = Fqd/c
where c i s t h e c o n s t a n t a p p e a r i n g i n ( 8 . 6 ) , and q d i s the l i m i t i n c o n d i t i o n ( 8 . 9 ) . A t p r e s e n t , i t i s n o t req u i r e d t h a t q d be n e g a t i v e .
S i n c e c c: 0 by a s s u m p t i o n (8.6), w e h a v e , f o r a l l M exceeding Mo t h e r e l a t i o n MC
- Fqd 4 0 .
(8.13)
D i f f e r e n t i a t i o n of (8.11) y i e l d s
+ (1-P)F ? I a f / 3 P .
>
L e t us now f i x M with p
--+1 and
r
(8.14)
Mo and p a s s i n (8,14)
+0.
t o the l i m i t
We have t h e n M G ( p )
-+
0 by
(8.5), M H ( r ) --+. 0 by ( 8 . 7 ) , and ( 1 - p ) F 2 1 a f / a p + 0 by t h e a s s m D t i o n o f boundedness of t h e d e r i v a t i v e o f Iaf ( H y p o t h e s i s 3 ) . N e x t , p M G 7 ( p ) 4 Me by ( 8 . 6 ) , and f i n a l l y , F I a f ( p , d ) 4 qd by ( 8 . 9 ) . C o n s e q u e n t l y , t h e l i m i t o f t h e d e r i v a t i v e ( 8 . 1 4 ) a s p -3 1 and r --3 0 i s e q u a l Mc - F q d y a q u a n t i t y which i s n e g a t i v e i n view o f ( 8 . 1 3 ) . T h i s c o m p l e t e s t h e p r o o f .
SELECTED TOPICS IN MEASUREMENT THEORY
549
We s h a l l now prove THEOREM 3 . I f , i n a d d i t i o n t o t h e assumptions o f Theo-
rem 2 , t h e i n c e n t i v e s Is and Iaf s a t i s f y r e s p e c t i v e l y Hypotheses 2 and 4 , t h e n one can f i n d v a l u e s F and d
>0
z c h t h a t f o r a s u i t a b l e b y t h e r i s k a r e a i s non-
normal and c e n t r a l . P r o o f . By t h e p r e c e d i n g theorem, t h e v a l u e f u n c t i o n f ( p ) i s n o t monotonically i n c r e a s i n g : it d e c r e a s e s i n t h e neighbourhood o f t h e p o i n t p = 1 f o r s u i t a b l e v a l u e s o f t h e p a r a m e t e r s . To prove t h e a s s e r t i o n o f t h e theorem, i t s u f f i c e s t o show t h a t f ( p ) i s n o t monoton i c a l l y d e c r e a s i n g i n t h e whole i n t e r v a l [0,1]. We s h a l l show t h a t f o r s u i t a b l e c h o i c e o f F and d , t h e d e r i v a t i v e f ' ( p ) i s p o s i t i v e i n t h e neighbourhood of p = 0 .
>
Mo d e f i n e d by ( 8 . 1 2 ) f o r ( F o , d o ) , and t h e n choose M 7 Mo and r s u c h
L e t us f i r s t f i x Fo and d o
0 , choose
t h a t f ' ( p ) i s n e g a t i v e i n t h e neighbourhood of p = 1,
a c c o r d i n g t o Theorem 2 . L e t us d i f f e r e n t i a t e ( 8 . 1 0 ) ; we g e t
Let c
O<&
r
>
<
0 be t h e v a l u e o f t h e l i m i t i n ( 8 . 9 ) , and l e t Mc / 3 . From r e l a t i o n ( 8 . 9 ) w e have t h e n
r
f o r a l l p sufficiently s m a l l , say p
< po.
Next, s i n c e
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550
t h e d e r i v a t i v e 3 1 s / a p i s bounded ( H y p o t h e s i s l), one can f i n d p1 5 p o s u c h t h a t f o r a l l p d p1 we have
Consequently, f o r p
=
Mcr-
<'
2L +
p1 we may w r i t e
FQ9
where, f o r s i m p l i c i t y , Q = ( l - p ) 2 1 a f / 2 p view of t h e c o n d i t i o n on
L;,
-
Iaf(p,d). In
we g e t
and t h e l a s t v a l u e i s p o s i t i v e f o r s u f f i c i e n t l y small F , i n view o f t h e boundedness o f t h e d e r i v a t i v e
31 / a p , g u a r a n t e e d by H y p o t h e s i s 3 . af To complete t h e proof i t s u f f i c e s t o o b s e r v e t h a t t h e d e c r e a s e of F does n o t cause any v i o l a t i o n o f t h e cond i t i o n s i m p l y i n g t h a t f l ( p ) < 0 for some p c l o s e t o 1; i n d e e d , under t h e assumption d > 0 o f t h e t h e o r e m , and under H y p o t h e s i s 4 , i f c o n d i t i o n ( 8 . 1 3 ) h o l d s f o r some F o , t h e n i t a l s o h o l d s f o r a l l F 4 F o . T h i s comp-
l e t e s the proof. The c o n d i t i o n s i n t h e above theorems may s t i l l b e somewhat weakened. It i s namely n o t n e c e s s a r y t o r e q u i r e t h e a d d i t i v e r e p r e s e n t a t i o n ( 8 . 4 ) . I n f a c t , one may
SEUCTED TOPICS IN MEASUREMENT THEORY
551
p r o v e t h e f o l l o w i n g theorem. THEOREM 4 . Assume t h a t t h e f u n c t i o n s
Is
and
Iaf a r e
d i f f e r e n t i a b l e w i t h respect t o p f o r every r ( r e s p . d ) . Suppose t h a t
(8.19) and g ( 0 ) g ( r ) i s a continuous f u n c t i o n of r i s negative;
exists.
(8.20) (8.21)
Then t h e r i s k a r e a s a r e non-normal f o r s u i t a b l e v a l u e s of t h e parameters.
Proof. We s h a l l show, as i n t h e p r o o f o f Theorem 2 , t h a t the v a l u e f u n c t i o n f ( p ) i s not monotonically i n c r e a s i n g , because f * ( p ) i s n e g a t i v e f o r some p . L e t u s write
From ( 8 . 1 8 ) - ( 8 . 2 1 ) i t f o l l o w s t h a t t h e r e e x i s t s an r and po 1 s u c h t h a t f o r a l l p w i t h po g p < 1 we have
<
(8.23)
5.52
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M o r e o v e r , as p 3 1 , t h e sm o f t h e l a s t two terms i n ( 8 . 2 2 ) r e m a i n s bounded, s a y by F t . Then f ' ( p ) 4 -Mz
+
Ft
(8.24)
f o r a l l p o < p C 1, and by i n c r e a s i n g M w e c a n a t t a i n the condition f ' ( p ) < 0. The t h e o r e m s o f t h i s s e c t i o n show t h a t non-normal
risk areas a r e r a t h e r a r u l e t h a n a n e x c e p t i o n : u n d e r some v e r y g e n e r a l and q u i t e p l a u s i b l e a s s u m p t i o n s , one w i l l have non-normal r i s k areas for s u i t a b l e c o m p a r i s o n val u e s , a t l e a s t f o r persons w i t h high achievement motive M , and when t h e r e w a r d r i s s m a l l . The SEU model i s a n e x p e c t i o n h e r e ; y e t , many p e r s o n s
a r e t r a i n e d t o t h i n k i n terms o f SEU m o d e l , and a c c e p t i t as a s t a n d a r d o f r a t i o n a l i t y . I n t h i s s e n s e , t h e t h e o r e m s o f t h i s s e c t i o n show t h a t a p p a r e n t l y i r r a t i o n a l b e h a v i o u r may i n f a c t b e s o m e t i m e s e a s i l y e x p l a i n e d by t h e c o n c e p t o f non-normal r i s k a r e a s . 8 . 2 . 3 . Some s p e c i a l c a s e s . I n t h i s s e c t i o n w e s h a l l d i s c u s s b r i e f l y some p l a u s i b l e c a s e s o f v a l u e f u n c t i o n s and t h e r e s u l t i n g r i s k a r e a s . F i r s t l y , l e t u s o b s e r v e t h a t t h e o r i g i n a l model o f A t k i n s o n ( 1 9 6 3 ) f o r a c h i e v e m e n t m o t i v a t i o n was o b t a i n e d by p u t t i n g Is = 1 - p and Iaf = -p ( s o t h a t t h e rewards r and l o s s e s d were n o t t a k e n i n t o a c c o u n t ) . We have then f(P) =
(M - F ) p ( l
-
P)
(8.25)
w h i c h i s a p a r a b o l a , w i t h b r a n c h e s d i r e c t e d upwards o r
SELECTED TOPICS IN MEASUREMENT THEORY
downwards, depending on t h e s i g n o f M - F , and w i t h t h e extremum a t p = c a s e s M 7 F and F > M y t h e s u i t a b l e p a r i s o n l e v e l b y i e l d s a non-normal
553
the difference $. I n each of t h e c h o i c e o f t h e comr i s k area.
The model o f Atkinson concerned t h e extreme c a s e , when t h e i n c e n t i v e s depend o n l y on p r o b a b i l i t i e s ( i . e . on t a s k d i f f i c u l t i e s ) . I n a somewhat more g e n e r a l c a s e , one c o u l d p o s t u l a t e
+
v ( S ) = M I S = M(1-p)
r
(8.26)
d,
(8.27)
and
-
= -Fp
v ( S ' ) = FIaf
where f o r s i m p l i c i t y , we assumed t h a t t h e r e w a r d s and l o s s e s are e x p r e s s e d i n u t i l i t y u n i t s , s o t h a t one can w r i t e s i m p l y r and d i n s t e a d of u ( r ) and u ( - d ) . We have t h e n
-
f ( p ) = -(M
F ) p2 t
(P t d t
M
-
F)p
-
d.
(8.28)
L e t u s f i r s t f i n d c o n d i t i o n s under which a l l r i s k a r e a s
a r e normal. I n c a s e when M
> F,
i . e . when t h e motive t o
a c h i e v e s u c c e s s dominates o v e r t h e f e a r o f f a i l u r e , t h e b r a n c h e s of t h e p a r a b o l s ( 8 . 2 7 ) a r e d i r e c t e d downwards, and f ( p ) w i l l be m o n o t o n i c a l l y i n c r e a s i n g i n t h e i n t e r v a l [O,l] i f and o n l y i f t h e maximum o c c u r s t o t h e r i g h t o f t h e p o i n t p = 1. T h i s means t h a t we must have r + d t M - F 2(M
-
F)
21
(8.29)
554
CHAPT.ER 5
which red-uces t o P
If M
<
+
d ) , M - F.
(8.30)
F , i . e . when t h e f e a r of f a i l u r e i s s t r o n g e r
t h a n t h e motive t o a c h i e v e s u c c e s s , t h e b r a n c h e s o f p a r a b o l s ( 8 . 2 8 ) a r e d i r e c t e d upwards, and f ( p ) i s i n c r e a s i n g i n t h e i n t e r v a l [0,1] i f and o n l y i f t h e mini-
mwn o c c u r s t o t h e l e f t o f t h e p o i n t p = 0 . T h i s means t h a t we must have r + d + M - F 2(M
-
F)
5 0
(8.31:
which r e d u c e s t o r + d < F - M .
(8.32)
We may t h e r e f o r e f o r m u l a t e THEOREM 5 . If t h e i n c e n t i v e s o f s u c c e s s and f a i l u r e a r e g i v e n by ( 8 . 2 6 )
and
(8.27), then a l l r i s k areas are
normal i f and o n l y i f
r t d 2 \ M - FI.
(8.33)
T h i s r e s u l t may be i n t e r p r e t e d as f o l l o w s . I f t h e rewards r a n d / o r l o s s e s d a r e l a r g e , t h e n a l l r i s k areas a r e normal. A l t e r n a t i v e l y , non-normal r i s k a r e a s may a p p e a r ( f o r s u i t a b l e comparison l e v e l s ) o n l y i f b o t h t h e r e w a r d s r and l o s s e s d a r e s u f f i c i e n t l y small. The v a l u e o f
SELECTED TOPICS IN MEASUREMENT THEORY
555
t h i s r e s u l t l i e s , among o t h e r s , i n i t s q u a l i t a t i v e cha-
r a c t e r , l a r g e l y i n d e p e n d e n t o f t h e v a l u e s M and F which a r e , as a l r e a d y mentioned, u n a v a i l a b l e e m p i r i c a l l y . C o n d i t i o n s ( 8 . 2 6 ) and ( 8 . 2 7 ) p o s t u l a t e d t h a t t h e i n c e n t i v e f u n c t i o n s I and Iaf depend on t h e m o t i v e s M and S F : we have h e r e Is = 1 - p t r / M and Iaf = -p - d / F . It i s o f c o n s i d e r a b l e i n t e r e s t t o abandon t h i s a s s u m p t i o n and p o s t u l a t e
( s e e Nowakowska 1979a, b ) . S u b s t i t u t i o n i n t o f ( p ) y i e l d s a g a i n a p a r a b o l a , w i t h - ( M - F ) as a c o e f f i c i e n t a t p 2 . However, t h e r e m a i n i n g c o e f f i c i e n t s now a l s o depend on M and F. The d e t a i l e d d i s c u s s i o n of r i s k areas and t h e i r depende n c e on p a r t i c u l a r p a r a m e t e r s may be found i n t h e c i t e d p a p e r s Nowakowska 1979a,b, as w e l l as f o r some s p e c i a l c a s e s of i n t e r e s t i n Nowakowska 1982. P a p e r o f Nowakows k a ( 1 9 8 1 ) p r e s e n t s i m p l i c a t i o n s of t h e t h e o r y f o r t h e c a se of d e c i s i o n s concerning innovations. Model ( 8 . 3 4 ) opens up s e v e r a l p o s s i b i l i t i e s , of b o t h t h e o r e t i c a l and e m p i r i c a l i n t e r e s t . F i r s t l y , a g l a n c e a t t h e v a l u e f u n c t i o n f ( p ) , e q u a l now f ( p ) = -(M-F)p
2
t [M(r+l)
t F(d-1)lp
- Fd
(8.35)
r e v e a l s t h a t i t i s no l o n g e r symmetric i n M and F , i . e t h e problem c e a s e s t o be i n v a r i a n t under changes of t h e m o t i v e s which l e a v e i n v a r i a n t t h e d i f f e r e n c e M - F ( a s was t h e c a s e i n a l l t h e o r i e s c o n s i d e r e d t h u s f a r ; s e e
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f o r i n s t a n c e A t k i n s o n 1963; Atkinson and B i r c h 1 9 7 0 ; Krantz e t a l . 1971; Weiner 1972; Weiner and S i e r a d z 1973). The a s s y m e t r y of m o t i v e s M and F i s a f e a t u r e o f g r e a t i m p o r t a n c e , and d e s e r v e s e x t e n s i v e e m p i r i c a l r e s e a r c h , s i m p l y because i n many c a s e s one might w i s h t o c o n t r o l t h e r i s k a r e a s of p e r s o n s , e . g . by modifying t h e i r mot i v a t i o n . The t e c h n i q u e s o f c o n t r o l l i n g t h e need o f achievement M a r e q u i t e d i f f e r e n t from t h e t e c h n i q u e s of c o n t r o l l i n g f e a r of f a i l u r e F , hence t h e g r e a t s i g n i f i c a n c e o f t h e knowledge how r i s k areas depend on them separately. Another i m p o r t a n t problem h e r e i s as f o l l o w s . S e p a r a t i n g i n ( 8 . 3 5 ) t h e " A t k i n s o n i a n " t e r m (M-F)p(l-p) , one i s l e f t w i t h t h e l i n e a r t e r m pMr - ( 1 - p ) F d , where rewards and l o s s e s i n t e r a c t w i t h m o t i v e s ( p o s s i b l y s u c h a t e r m i s v a l i d only f o r m o d e r a t e l y small r and d , and f o r p r o b a b i l i t i e s p s u f f i c i e n t l y f a r away from t h e extremes p = 0 and p = 1 ) . I f , a t l e a s t i n t h i s r a n g e , t h e formu l a ( 8 . 3 5 ) were a v a l i d r e p r e s e n t a t i o n of v a l u e , i t would mean t h a t ( r e l a t i v e l y t o p e r s o n ' s w e a l t h ) rewards i n r i s k y o p t i o n s a r e valued d i f f e r e n t l y t h a n analogous r e w a r d s i n r i s k l e s s o p t i o n s (and t h e same w i t h l o s s e s ) . Thus
, mc t i v a t i o n a l
f a c t o r s would produce a s o r t o f
"King Midas E f f e c t t r ( t u r n i n g e v e r y t h i n g i n t o g o l d ) . F o r h i g h M y t h e r e f o r e , even w o r t h l e s s ( r c l o s e t o 0) rewards might a p p e a r d e s i r a b l e , w h i l e t h e same l o s s d may a p p e a r "more t h r e a t e n i n g " f o r p e r s o n s w i t h h i g h e r f e a r of f a i l u r e F . Many examples of m o t i v a t i o n a l e x t r e m e s s i l l u s t r a t i n g t h i s e f f e c t may be found i n t h e l i t e r a t u r e ( e . g . B a l z a c ' s
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F a t h e r G o r i o t ) , as w e l l as h i s t o r y ( e . g . P r i n c e Krishna o r S t . A u g u s t i n e , l e a v i n g t h e i r r i c h e s t o become r e l i g i o u s l e a d e r s , hence r e j e c t i n g s a f e and c o m f o r t a b l e o p t i o n s f o r more e l u s i v e e t h i c a l v a l u e s ) . What i s , however, even more i m p o r t a n t , i s t h a t model
( 8 . 3 4 ) a l l o w s u s t o abandon t h e "businessman" o r "manae r i a l model s u g g e s t e d by SEU, and propose a model o f s u c c e s s and p a s s i o n , p r o v i d i n g , i n f a c t , new i n t u i t i o n s behind t h e n o t i o n o f s u c c e s s . The l a t t e r c o n c e p t was l e f t u n d e f i n e d i n t h e p r e s e n t model, s e r v i n g as a s o r t o f p r i m i t i v e , and t h e main problem was t o a n a l y s e t h e e f f e c t s of c o n t r o l l i n g i t s p r o b a b i l i t y , e . g . b y s k i l l , knowledge, e t c . However, i f one abandons t h e narrow way o f l o o k i n g a t s u c c e s s only i n economical t e r m s , one c a n t h i n k o f reward r i n terms of m o r a l , e t h i c a l , i d e o l o g i c a l , e t c . v a l u e s . V a r i o u s c o n c e p t i o n s and s t y l e s of success cause then assigning t o such successes values, which may be a m p l i f i e d t o a l a r g e e x t e n t by t h e motiva t i o n a l f a c t o r s , l e a d i n g t o t h e i r o v e r e s t i m a t i o n ( o r underestimation).
8 . 2 . 4 . D i s c u s s i o n . Looking a t t h e model of t h i s s e c t i o n from a somewhat more g e n e r a l p o i n t o f view, one c a n f o r m u l a t e t h e b a s i c problem as f o l l o w s . The d e c i s i o n maker f a c e s a d e c i s i o n s i t u a t i o n , i n which he i s t o make t h e c h o i c e o f one o f t h e two o p t i o n s ( t h e g e n e r a l i z a t i o n t o t h e c a s e of more o p t i o n s i s o b v i o u s ) . The dec i s i o n s i t u a t i o n i s d e s c r i b e d by a number of p a r a m e t e r s ; i n t h e p r e s e n t c a s e , t h e p a r a m e t e r s a r e p , r , d and b . I n a d d i t i o n , t h e d e c i s i o n maker h i m s e l f i s a l s o charact e r i z e d by some p a r a m e t e r s ( i n t h e p r e s e n t c a s e , t h e
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m o t i v e s M and F ) . Combining a l l t h e p a r a m e t e r s t o g e t h e r , one o b t a i n s a r e p r e s e n t a t i o n of a p a r t i c u l a r d e c i s i o n s i t u a t i o n ( t o g e t h e r w i t h t h e d e c i s i o n maker) as a p o i n t i n a m u l t i dimensional space. I n t h e p r e s e n t c a s e , w e have t h e r e f o r e s i x p a r a m e t e r s . Generally, a c l a s s of decision s i t u a t i o n s w i t h k opti o n s , R1, R2,..., R i s c h a r a c t e r i z e d by a multidimenk s i o n a l parameter = (sl,. . . , s n ) , t h e c o o r d i n a t e s b e i n g l o s s e s , r e w a r d s , p r o b a b i l i t i e s , e t c . Thus, s C E n , and l e t S C En d e n o t e t h e c l a s s o f a l l a d m i s s i b l e s i t u a t i o n s , as d e t e r m i n e d by v a r i o u s c o n s t r a i n t s on c o o r d i ( e . g . a l l p r o b a b i l i t i e s must be between 0 n a t e s of and 1, e t c . )
s
s
A s o l u t i o n o f t h e c l a s s S of d e c i s i o n s i t u a t i o n s i s a
f a m i l y o f k s e t s , S1
,...,
...
Sk, s u c h t h a t S 1 u LI Sk then = S, with t h e i n t e r p r e t a t i o n being t h a t i f 5 C S j’ t h e b e s t c h o i c e i n t h e s i t u a t i o n d e s c r i b e d by s i s t h e option R j’
Observe t h a t i t i s n o t r e q u i r e d t h a t t h e s e t s S be j disjoint: i f C Si n Sj, t h e n t h e c h o i c e o f o p t i o n Ri i s as good as t h e c h o i c e of o p t i o n R
s
3’
L e t us n o t e t h a t t h e above i s t h e weakest concept o f a s o l u t i o n , s i n c e i t i s r e q u i r e d merely t h a t we know o n l y the best option, but not i t s numerical valuation, nor t h e r a n k i n g s o f o t h e r o p t i o n s . One c o u l d , of c o u r s e , d e f i n e s o l u t i o n s i n a s t r o n g e r s e n s e , e.g. by r e q u i r i n g t h a t t o e a c h ?it h e r e i s a s s i g n e d a r a n k i n g o f t h e s e t o f o p t i o n s , o r ( s t i l l s t r o n g e r ) , a v e c t o r of n u m e r i c a l valuations.
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L e t us o b s e r v e t h a t t h e concept o f t h e d e c i s i o n s p a c e used h e r e d i f f e r s from t h e n o t i o n o f m o t i v a t i o n a l s p a c e i n t r o d u c e d i n Nowakowska ( 1 9 7 3 ) . T h e r e , s u c h a spac e was c h a r a c t e r i z e d s e p a r a t e l y f o r e a c h i n d i v i d u a l , and was spanned by a number of dimensions ( p s y c h o l o g i c a l c o n t i n u a , o r s c a l e s , as r e p r e s e n t e d l i n g u i s t i c a l l y by c l a s s e s of s p e c i a l f u n c t o r s , s u c h as "I b e l i e v e t h a t " , "I a m c e r t a i n t h a t " , "I am happy t h a t " , "I want" , "I p r e f e r " , and s o f o r t h ) . T h e s e f u n c t o r s were o b t a i n e d t h r o u g h a b s t r a c t i o n from t h e c o n t e n t o f s e n t e n c e s which one u s e s t o e x p l a i n , j u s t i f y o r e v a l u a t e t h e d e c i s i o n s which one made, o r i s p l a n n i n g t o make. I n s u c h a s p a c e , e a c h o p t i o n i s r e p r e s e n t e d by a p o i n t , and t h e problem l i e s i n a s s i g n i n g a r a n k i n g o f o p t i o n s , m e e t i n g some " n a t u r a l " c r i t e r i a . It was shown i n Nowakowska (1973), t h a t t h e f o r m a l s t r u c t u r e o f s u c h problem i s t h e same a s of group d e c i s i o n making, and i n cons e q u e n c e , one may a p p l y t h e Arrow I m p o s s i b i l i t y Theorem (Arrow 1 9 6 3 ) . Such a n i n t e r p r e t a t i o n l e a d s t o v a r i o u s p s y c h o l o g i c a l consequences c o n c e r n i n g t h e p r o c e s s o f d e c i s i o n making, simply because p e o p l e must p a r t i t i o n i n t o c a t e g o r i e s , d e p e n d i n g on t h e p a r t i c u l a r axiom o f Arrow which t h e y v i o l a t e . Each of t h e s e c a t e g o r i e s m a y , i n t u r n , be i n t e r p r e t e d i n p s y c h o l o g i c a l t e r m s . R e t u r n i n g t o t h e problem of t h e p r e s e n t model, t h e d e c i s i o n s p a c e i s spanned by s i x d i m e n s i o n s , of which four c o r r e s p o n d t o s c a l e s of i n t e r v a l t y p e o r h i g h e r ( t h e s t a t u s o f ' t h e s c a l e s f o r m e a s u r i n g M and F need n o t be r e s o l v e d h e r e ) . The s e t S o f a d m i s s i b l e s i t u a t i o n s i s d e f i n e d by t h e s e t o f c o n s t r a i n t s
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( b may b e a r b i t r a r y , and t h e l a s t i n e q u a l i t y i s i m p l i c i t i n t h e u s e o f terms " s u c c e s s " and " f a i l u r e " ) . A s o l u t i o n t o t h i s d e c i s i o n problem i s a s e t S
#
C
S of
s
those vectors f o r which t h e r i s k y o p t i o n i s b e t t e r than the riskless option, A c t u a l l y , i n t h i s s e c t i o n t h e o b j e c t o f s t u d y were t h e #
p r o p e r t i e s of t h e i n t e r s e c t i o n s of S w i t h t h e subspace o f S o b t a i n e d by f i x i n g a l l v a l u e s i n except f o r p; t h i s s u b s p a c e i s t h e r e f o r e a n i n t e r v a l o f l e n g t h 1, and
a non-normal r i s k area o c c u r s s i m p l y i f t h i s i n t e r v a l n * i n t e r s e c t s w i t h S , b u t i t s r i g h t hand end i s n o t i n S . The most i m p o r t a n t f e a t u r e o f t h e r e s u l t s o f t h i s s e c t -
i o n l i e s i n t h e i r g e n e r a l i t y . T h e y show t h a t non-normal r i s k areas a r e bound t o o c c u r u n d e r p r a c t i c a l l y a l l c o n d i t i o n s w h i c h a r e n o t c o v e r e d by SEU m o d e l , b e c a u s e o f t h e d e p e n d e n c e o f v a l u e s o f s u c c e s s and f a i l u r e on their probabilities. These f i n d i n g s have some d e e p p s y c h o l o g i c a l concequen-
c e s . They may e x p l a i n some o f t h e o b s e r v e d d e v i a t i o n s o f e m p i r i c a l d a t a from t h e p r e d i c i s o o n s o f t h e SEU m o d e l . S i n c e t h e l a t t e r i s g e n e r a l l y t h o u g h t o f as a s t a n d a r d of r a t i o n a l i t y , t h e t h e o r e m s p r o v e d i n t h i s s e c t e m may e x p l a i n c e r t a i n b e h a v i o u r s w h i c h one would be i n c l i n e d t o term " i r r a t i o n a l " . I n t h i s s e n s e , t h e p r e s e n t model o f f e r s some r a t i o n a l i z a t i o n o f i r r a t i o n ality.
C e r t a i n a p p l i c a t i o n s o f t h i s model w i l l be shown i n t h e c h a p t e r c o n c e r n i n g t h e models of s c i e n t i f i c c a r -
eers.
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8.3. A d e c i s i o n model f o r s o c i a l b e h a v i o u r The c o n s i d e r a t i o n s o f t h i s s e c t i o n w i l l c o n c e r n a dec i s i o n a p p r o a c h t o t h e phenomenon o f p r o s o c i a l i t y , ant i s o c i a l i t y and a s o c i a l i t y . I n a l l e x i s t i n g q u a l i t a t i v e d e s c r i p t i o n s or t h e o r i e s
o f p r o s o c i a l i t y and a n t i s o c i a l i t y , t h e t e n d e n c y t o s u c h t y p e s o f b e h a v i o u r i s r e g a r d e d as a p e r s o n a l i t y t r a i t . A s opposed t o t h a t , t h e p r e s e n t a p p r o a c h makes no assumption about t h e e x i s t e n c e o f such t r a i t s , and a t t r i b u t e s pro- o r a n t i s o c i a l b e h a v i o u r t o b o t h s i t u a t i o n a l and p s y c h o l o g i c a l f a c t o r s . The t h e o r y w i l l be based on t h e assumption t h a t p e o p l e
may be c a t e g o r i z e d a c c o r d i n g t o t h e i r need o f a c c e p t -
ance and f e a r o f r e j e c t i o n , and w i l l p r o v i d e t e s t a b l e h y p o t h e s e s about t h e b e h a v i o u r o f p e r s o n s from v a r i o u s c a t e g o r i e s o f t h e i n t r o d u c e d taxonomy and i n v a r i o u s t y p e s of s i t u a t i o n s .
8 . 3 . 1 . The model.
The s t a r t i n g p o i n t o f t h e t h e o r y ( s e e Nowakowska 1977) i s t h e assumption t h a t i n making a c h o i c e o f a n a c t i o n , p e o p l e t e n d t o maximize b o t h t h e e x p e c t e d d i r e c t reward and a l s o t h e " i n d i r e c t " r e ward, i n form o f e v a l u a t i o n s by o t h e r members o f t h e r e f e r e n c e g r o u p , and c o n s e q u e n t l y , a c c e p t a n c e o r rej e c t i o n b y t h e group. These two t y p e s o f reward a r e t y p i c a l l y n e g a t i v e l y r e l a t e d , i . e . i n c r e a s e o f one b r i n g s about a d e c r e a s e o f t h e o t h e r . To p r o c e e d f o r m a l l y , assume t h a t t h e group under consm ' s i d e r a t i o n c o n s i s t s o f m members l a b e l e d sl,
...,
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It w i l l be assumed t h a t t h e i - t h member o f t h e group w i l l choose h i s a c t i o n s o as t o maximize t h e q u a n t i t y
A i s t h e change ( p o s i t i v e or n o t ) o f i n d i v i d u c a1 s a t i s f a c t i o n , due t o g a i n or loss o f some goods. A v ( Qk .i ) i s t h e change ( a s p e r c e i v e d b y s i , or Next, a n t i c i p a t e d by him) o f e v a l u a t i o n of s i by s k y d u e t o c a u s i n g some e v e n t s which a r e f a v o u r a b l e or h a r m f u l t o s k . F i n a l l y , aii i s t h e weight which si a t t a c h e s t o b r i n g i n g s a t i s f a c t i o n t o h i m s e l f , and aik i s t h e weight which he a t t a c h e s t o t h e o p i n i o n o f s k . where
The main a s s u m p t i o n i s t h a t Qki,
t h e e v a l u a t i o n of si by s k ) i s b a s e d on t h e consequences ( t o s ) o f a l l e v e n t s k caused b y s i . Denoting by xki (1), x ( * ) ,. t h e degrees ki o f " p l e a s u r e " or " d i s c o m f o r t " caused by si t o sk by s u c c e s s i v e e v e n t s produced .by h i m , t h e v a l u e Qki a f t e r n-th e vent i s e q u a l t o
..
where -D i s a r e j e c t i o n t h r e s h o l d (which may depend on i and k ) . T h i s a s s u m p t i o n means t h a t i n e v a l u a t i n g t h e
p e r s o n s i yt h e " j u d g e " s k i s w i l l i n g t o " c a n c e l good a g a i n s e v i l " , i . e . u s e t h e upper p a r t of f o r m u l a ( 8 . 3 7 ) as l o n g as no e v e n t caused by s i was worse t h a n t h e t h r e s h o l d -D. Otherwise, h i s judgment i s o f t h e "nonf o r g i v i n g " t y p e , s p e c i f i e d by t h e second l i n e o f t h e
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formula ( 8 . 3 7 ) . F i n a l l y , f u n c t i o n v i s assumed t o be i n c r e a s i n g , w i t h a jump a t t h e t h r e s h o l d - D , s o t h a t g e n e r a l l y , e v e r y one p r e f e r s t o be v a l u e d h i g h by a l l , or most,members o f t h e g r o u p , and f e e l s p u n i s h e d i f he i s r e j e c t e d . Now, a n a c t i o n may be termed p r o s o c i a l , i f i t y i e l d s a n e g a t i v e v a l u e of Ac, and p o s i t i v e v a l u e s o f b v ( Q k i ) f o r a l l , or most, i n d i c e s k . I t may be termed a n t i s o c i a l , i f the r e l a t i o n i s opposite, i.e. i f i s posit i v e , and a l l (or most) of A v ( Q . ) a r e n e g a t i v e . An ki a s o c i a l a c t i o n i s s u c h which y i e l d s all a v ( Q k i ) e q u a l t o zero.
8 . 3 . 2 . A n a l y s i s of t h e model. A s w i l l be shown, t h e p e r s o n ' s b e h a v i o u r depends p r i m a r i l y on t h e c h a r a c t e r o f t h e f u n c t i o n v . If i t becomes f l a t as t h e argument i n c r e a s e s on t h e p o s i t i v e s i d e ( i . e . w i t h t h e i n c r e a s e o f Q k i ) , t h e p e r s o n a t t a c h e s l e s s and l e s s weight t o b e i n g a c c e p t e d a s h i s l e v e l of a c c e p t a n c e i n c r e a s e s . On t h e o t h e r hand, i f i t becomes more s t e e p as Qki i n c r e a s e s , t h e p e r s o n i s " a c c e p t a n c e hungry". F o r n e g a t i v e v a l u e s t h e s i t u a t i o n i s s i m i l a r . If v becomes f l a t t e r a s Qki d e c r e a s e s , t h e p e r s o n has bounded f e a r of r e j e c t i o n , w h i l e i f i t becomes more s t e e p as Qki d e c r e a s e s , he has unbounded f e a r o f r e j e c t i o n . T h i s g i v e s a c l a s s i f i c a t i o n i n t o four b a s i c psycholo-
g i c a l t y p e s o f p e r s o n s , o b t a i n e d b y combining t h e s e two d i c h o t o m i e s of b e h a v i o u r of t h e f u n c t i o n v for pos i t i v e and n e g a t i v e v a l u e s o f t h e argument. The c l a s s -
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i f i c a t i o n i s p r e s e n t e d on F i g . 8 . 3 . 1 .
-D
/
'ki
1
------+
-D
/
i Fig. 8.3.1.
( a ) L i m i t e d need o f a c c e p t a n c e and bounded f e a r o f r e j e c t i o n ; ( b ) "Acceptance hungry", bouned f e a r o f r e j e c t i o n , ( c ) L i m i t e d need o f a c c e p t a n c e , unbounded f e a r o f r e j e c t i o n , ( d ) "Acceptance hungry", unbounded f e a r o f r e j e c t i o n .
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T o u s e a s i m p l e example, i n c a s e ( c ) , i f a p e r s o n i s
h i g h l y a c c e p t e d by k ( s e e F i g . 8 . 3 . 2 ) , i . e . Qki i s f a r t o t h e r i g h t , he i s l i k e l y t o behave i n a n a n t i s o c i a l way: l o w e r i n g o f Qki b r i n g s o n l y a small d e c r e a s e of v(Qki), so t h a t av(Qki) i s small, s i n c e t h e f u n c t i o n v i s f l a t i n t h e a c c e p t a n c e r e g i o n . Thus a d e c r e a s e i n v ( Q k i ) i s e a s i l y c o u n t e r b a l a n c e d by a p o s i t i v e reward A,. On t h e o t h e r hand, such a t y p e o f p e r s o n , if r e j e c t e d , or t h r e a t e n e d b y r e j e c t i o n , a r e l i k e l y t o behave i n
a p r o s o c i a l way: l o w e r i n g o f Qki,
g i v e n i t i s low a l r e A v ( Q k i ) , which a r e n o t e a s i l y c o u n t e r b a l a n c e d by r e w a r d s h c . ( s e e
a d y , y i e l d s h i g h n e g a t i v e i n c r e m e n t s of
Fig. 8.3.3).
Fig. 8.3.2
Fig. 8.3.3
T h i s p r e d i c t i o n t a k e s i n t o a c c o u n t o n l y one v a l u e o f I n r e a l i t y , as i n d i c a t e d i n formula ( 8 . 3 5 ) , t h e r e Qki w i l l b e as many t e r m s i n t h e sum t o be maximized as
.
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t h e r e a r e members of t h e c o n s i d e r e d group.
The p r e d i c t -
i o n s may be made f o r a l l f o u r t y p e s of p e r s o n s , and t h e y t a k e on t h e f o l l o w i n g form. HYPOTHESIS 1. L e t a p e r s o n ' s s e n s i t i v i t y t o a c c e p t a n c e and r e j e c t i o n be l i m i t e d ( c a s e a ) . I f he i s p e r c e i v e d as a m b i v a l e n t , i . e . h i g h l y a c c e p t e d by some and h i g h l y r e j e c t e d by o t h e r s , t h e n h i s a c t i o n s a r e l i k e l y t o b e a s o c i a l o r a n t i s o c i a l . O t h e r w i s e , h i s a c t i o n s a r e lik e l y t o be p r o s o c i a l . HYPOTHESIS 2 . L e t a p e r s o n be dominated by t h e need o f a c c e p t a n c e , w h i l e h i s f e a r o f r e j e c t i o n i s bounded ( c a s e b ) . I f he i s a c c e p t e d by few ( o r merely t o l e r a t e d ) , and r e j e c t e d by many, h i s b e h a v i o u r i s l i k e l y t o b e asoc i a l o r a n t i s o c i a l . O t h e r w i s e , i t i s l i k e l y t o be prosocial. HYPOTHESIS 3 . L e t a p e r s o n b e dominated by f e a r o f rej e c t i o n , w h i l e h i s need of a c c e p t a n c e i s l i m i t e d ( c a s e c ) . I f he i s h i g h l y a c c e p t e d by most, and r e j e c t e d o n l y by few, h i s b e h a v i o u r i s l i k e l y t o b e a s o c i a l o r a n t i s o c i a l . I n t h e o p p o s i t e c a s e , i t i s l i k e l y t o be prosocial. HYPOTHESIS 4. I f a p e r s o n has i n c r e a s i n g s e n s i t i v i t y t o b o t h r e j e c t i o n and a c c e p t a n c e ( c a s e d ) , he i s l i k e l y t o behave i n a p r o s o c i a l way, e x c e p t i n c a s e s when he i s n e i t h e r r e j e c t e d n o t a c c e p t e d by many ( i . e . he i s p e r c e i v e d i n d i f f e r e n t l y by m o s t ) . The above h y p o t h e s e s , t a k e n j o i n t l y , a s s e r t t h a t a p e r s o n ' s b e h a v i o u r ( c l a s s i f i e d i n terms o f pro- or a n t i s o -
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c i a l i t y ) depends on:
--
h i s p s y c h o l o g i c a l t y p e ( e x p r e s s e d t h r o u g h h i s need
o r a c c e p t a n c e and f e a r o f r e j e c t i o n ) ;
--
h i s p e r c e p t i o n o f t h e group (number o f t h o s e who
a c c e p t h i m , and of t h o s e who r e j e c t h i m ) ;
--
t h e d i r e c t reward which he e x p e c t s to g e t from h i s
action. I n e a c h o f t h e f o u r b a s i c p s y c h o l o g i c a l t y p e s a - d , one may s i n g l e o u t s i t u a t i o n s i n which t h e p e r s o n o f a g i v e n t y p e i s l i k e l y t o behave i n a n a s o c i a l o r a n t i s o c i a l way. C o n s e q u e n t l y , t h e t e n d e n c i e s t o p r o s o c i a l i t y o r a n t i s o c i a l i t y a r e - as s t a t e d a t t h e b e g i n n i n g -- n o t t h e i n h e r e n t c h a r a c t e r i s t i c s of a person.
8.3.3. D i s c u s s i o n . E x t e n s i o n s o f t h e model. An i m p o r t a n t a s p e c t of t h e c o n s i d e r e d problem c o n c e r n s t h e q u e s t i o n o f e v a l u a t i o n s o f Qki. The model s p e c i f i e s t h a t t h e b e h a v i o u r o f p e r s o n s i i s guided by h i s e v a l u a t i o n s v(Q k .i ) o f e v a l u a t t i o n s Qki o f him by o t h e r members o f t h e group. Such a n a s s u m p t i o n p r e s u p p o s e s t h a t si knows a l l v a l u e s Qki, i f not e x a c t l y , t h e n a t l e a s t i n s u f f i c i e n t d e g r e e to s e r v e him as a g u i d e f o r b e h a v i o u r . One t h e r e f o r e needs to complement t h e s u g g e s t e d t h e o r y by a n a p p r o p r i a t e t h e o r y o f communication, which a c c o u n t s f o r s u c h knowledge. Such a t h e o r y o f communication, where v a r i o u s meanings o f h i g h e r o r d e r , s u c h as " f r i e n d l i n e s s " , " r e j e c t i o n " , " a c c e p t a n c e " , e t c . a r e e x p r e s s e d on d i f f e r e n t media o f
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communication, has been s u g g e s t e d b y Nowakowska (1979). The main i d e a s ( s e e a l s o c h a p t e r on s e m i o t i c s ) a r e a s follows. C o n s i d e r a s e t of communication media, e . g . v e r b a l medium, f a c i a l e x p r e s s i o n s , g e s t u r e s , body movements, e t c . , and a " m u l t i d i m e n s i o n a l language" o f communicati o n on s u c h media. To s i m p l i f y t h e p r e s e n t a t i o n , cons i d e r o n l y two p e r s o n s , A and B . Suppose t h a t A conveys t h e message, and B r e c e i v e s i t . The p e r s o n A h a s h i s " a l p h a b e t o f a c t i o n s " , which c o m p r i s e s a c t i o n s on d i f f e r e n t media. The v e c t o r o f a c t i o n s , one on e a c h medium ( e . g . s a y i n g " H e l l o " , s m i l e on medium o f f a c i a l e x p r e s s i o n s , t i p p i n g t h e h a t on t h e medium o f body movements, and s o o n ) forms a u n i t o f communication a c t i o n s . S t r i n g s o f s u c h u n i t s form e x p r e s s i o n s i n t h e " m u l t i d i m e n s i o n a l language o f communication b e h a v i o u r " . Thus, e a c h s t r i n g i s i n f a c t a bundle o f p a r a l l e l s t r i n g s , one on e a c h medium. O f c o u r s e , n o t e v e r y s t r i n g i s admissible; a primitive notion of t h e system i s t h a t of the s e t of a l l admissible s t r i n g s . The s p e c i f i c a t i o n o f t h i s c l a s s d e t e r m i n e s t h e s y n t a x o f communication b e h a v i o u r : i n p a r t i c u l a r , i t a l l o w s to d e f i n e a d m i s s i b i l i t y o f a u n i t , and e x p r e s s v a r i o u s c o n s t r a i n t s which must b e met i n f o r m i n g t h e u n i t s . A s r e g a r d s meanings, one c o n s i d e r s here t h e s e t o f mean-
i n g s of h i g h e r o r d e r , as e x e m p l i f i e d above. From t h e r e c e i v e r ' s p o i n t o f v i e w , t h e meanings may b e f u z z y , and one has t o c o n s i d e r t h e d e g r e e t o which a g i v e n u n i t o r s t r i n g of u n i t s e x p r e s s e s a g i v e n meaning. Within t h i s c o n c e p t u a l framework, i t i s p o s s i b l e t o
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569
e x p r e s s v a r i o u s c o n c e p t s r e l e v a n t f o r v e r b a l and nonv e r b a l communication, e s p e c i a l l y t h o s e which r e f e r t o meanings e x p r e s s e d t h r o u g h n o n v e r b a l media, c o v e r i n g ( f i r s t o f a l l ) t h e most i n t e r e s t i n g c a s e when t h o s e meanings c o n t r a d i c t t h e u t t e r e d s e n t e n c e s . The c o n c e p t s h e r e are t h o s e o f g e n e r a t i o n , s u p p o r t i n g , i n h i b i t i o n and c a n c e l l a t i o n of meanings, as w e l l as c o n c e p t s r e l a t e d t o i n t e r c h a n g e a b i l i t y of media, i . e . t h o s e conc e r n i n g t h e problem o f t h e p o s s i b i l i t y o f e x p r e s s i n g some meanings on c e r t a i n media b u t n o t t h e o t h e r s . Even from t h e s e c u r s o r y remarks i t s s h o u l d be o b v i o u s t h a t the outlined theory provides i n f a c t a conceptual framework which g i v e s a c c e s s t o a s s e s s m e n t o f Qki, which ( a s a r u l e ) a r e n o t communicated v e r b a l l y . I n c o n n e c t i o n w i t h t h e p r e s e n t s y s t e m , one may s u g g e s t t h e f o l l o w i n g promising l i n e s of r e s e a r c h , both theo-
r e t i c a l and e m p i r i c a l . F i r s t l y , causing
v a r i o u s e v e n t s may r e q u i r e d i f f e r e n t c o s t s , i . e . e x p e n d i t u r e o f e f f o r t , e t c . I n o t h e r words, some e v e n t s may be more d i f f i c u l t t o a t t a i n t h a n o t h e r s . Inasmuch as one c o n s i d e r s e v e n t s caused s o l e l y by one p e r s o n , t h i s c o s t may b e i n c o r p o r a t e d i n t o t h e u t i l i t y i n d e x o f t h e p e r s o n c a u s i n g t h e e v e n t ; hence one may s a y t h a t t h i s a s p e c t has a l r e a d y been i n c l u d e d i m p l i c i t l y i n t o t h e s y s t e m . However, as r e g a r d s t h e e v e n t s which r e q u i r e some c o o p e r a t i o n w i t h o t h e r s , t h e s i t u a t i o n i s n o t s o s t r a i g h t f o r w a r d , and one s h o u l d i n t r o duce o n t o t h e s y s t e m t h e c o n c e p t o f " b a r g a i n i n g " , or s i d e payments. These s i d e payments may have t h e form o f c o o p e r a t i o n , i n t h e s e n s e o f r e s i g n i n g from some h i g h l y r e w a r d i n g e v e n t s for t h e p r i c e o f g e t t i n g a
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reward from some o t h e r e v e n t s , caused by someone e l s e , who a l s o r e s i g n s from e v e n t s most r e w a r d i n g f o r him. The s i d e payments i n q u e s t i o n are r e q u i r e d t o i n d u c e t h e o t h e r s t o j o i n c o a l i t i o n s , o r b r e a k away from some c o a l i t i o n s , i n o r d e r t h a t c e r t a i n e v e n t s could be caused or a v o i d e d . The f o r m a l a s p e c t s of t h e s i t u a t i o n a r e q u i t e complex, and a s y e t , t h e r e i s no s a t i s f a c t o r y t h e o r y of c o a l i t i o n which would e x p l a i n t h e mechanisms of c o a l i t i o n f o r m a t i o n i n a n a d e q u a t e way. C l e a r l y , however, any a d e q u a t e t h e o r y of p r o s o c i a l b e h a v i o u r must e v e n t u a l l y i n c o r p o r a t e t h e s e a s p e c t s . Another a s p e c t w o r t h i n v e s t i g a t i n g i s t h e reward f o r m y s t i f i c a t i o n , or m i s r e p r e s e n t a t i o n of o n e ' s own u t i l i ty, s o as t o i n d u c e o t h e r s t o perform some a c t i o n s , o r a v o i d p e r f o r m i n g some a c t i o n s . Such m i s r e p r e s e n t a t i o n s , one may a r g u e , a r e r a t h e r a common p r o c e d u r e . They o c c u r a l w a y s when one t r i e s t o p r e s e n t h i s behavi o u r as p r o s o c i a l , as a s t r a t e g i c p r e p a r a t i o n f o r some f u t u r e reward; more o f t e n t h e y a r e simply s p u r i o u s , i . e . a c t i o n s which a r e s o c i a l l y f u t i l e , w h i l e h a v i n g an a p p e a r e n c e of b e i n g p r o s o c i a l . Most o f t e n , however , s u c h m i s r e p r e s e n t a t i o n s have t h e form of v e r b a l behavi o u r which i s t o camouflage, o r draw a t t e n t i o n away from, o n e ' s t r u e a c t i o n s and m o t i v e s . The s t r a t e g i c a s p e c t s o f m i s r e p r e s e n t a t i o n of o n e ' s
u t i l i t y a r e a g a i n v e r y complex from t h e g a m e - t h e o r e t i c p o i n t of view (for some r e s u l t s , s e e F r e i m e r and Yu,
19741. One s h o u l d mention here t h a t m i s r e p r e s e n t a t i o n need n o t be n e c e s s a r i l y n e g a t i v e : a f t e r a l l , even s i m p l e p o l i t e n e s s i s a m i s r e p r e s e n t a t i o n of some s o r t , made
571
SELECTED TOPICS IN MEASUREMENT THEORY
i n o r d e r t o i n d u c e t h e o t h e r p e r s o n t o do something or r e f r a i n from d o i n g s o m e t h i n g . I n c o n n e c t i o n w i t h t h e above d e c i s i o n model f o r prosoc i a 1 b e h a v i o u r , one can a l s o t r y t o c a r r y o v e r t h e main ideas t o t h e c o n s t r u c t i o n o f a g e n e r a l d e c i s i o n model, n o t r e s t r i c t e d t o t h e c a s e of s o c i a l d e c i s i o n s and u t i l i t y i n d e x c o n n e c t e d w i t h t h e judgments. The main f e a t u r e o f t h e model l i e s i n t h e f a c t t h a t t h e u t i l i t y i n d e x c o n c e r n s t h e " u t i l i t y changes" r a t h e r t h a n u t i l i t i e s o f f i n a l s t a t e s . I n o t h e r words, when contemplating a l t e r n a t i v e courses of a c t i o n s , the pers o n compares v a r i o u s changes i n h i s l e v e l o f s a t i s f a c t i o n ( u t i l i t y i n d e x , e t c . ) It a p p e a r s t h a t t h i s a p p r o a c h may e x p l a i n t h e w e l l known f a c t o f i n t r a n s i t i v i t y o f preferences. The g e n e r a l o u t l i n e o f s u c h a d e c i s i o n model might be
..
as f o l l o w s . L e t u s d e n o t e g e n e r a l l y by x , y , z , . the p o s s i b l e " p r e s e n t " u t i l i t y l e v e l s , i .e . t h e w e l f a r e , d e g r e e o f s a t i s f a c t i o n , e t c . ( d e p e n d i n g on t h e c o n t e x t ) of t h e p r e s e n t s t a t e , i . e . o f s t a t u s quo. Suppose t h a t a,b,c, are t h e p o s s i b l e c o u r s e s of a c t i o n s , one of which i s t o be s e l e c t e d . The i d e a i s t o assume t h a t t h e u t i l i t y i n d e x of t h e s e a c t i o n s depends on t h e p r e are t h e u t i l i s e n t s t a t e , S O t h a t u ( x , a ) , u ( x , b ) ,. t i e s (or e x p e c t e d u t i l i t i e s ) o f a c t i o n s a , b , . i n sit u a t i o n x.
...
..
..
The s p e c i f i c form o f u ( x , a ) depends on t h e c h a r a c t e r
of t h e s i t u a t i o n and a l t e r n a t i v e s . The p o s t u l a t e of c h o i c e ,
i s simply that
or g e n e r a l l y , o f p r e f e r e n c e ,
572
CHAPmR 5
a
>ix
b
iff u(x,a)
> u(x,b),
t h a t i s , a i s p r e f e r r e d t o be i n s i t u a t i o n x i f t h e
u t i l i t y o f a e x c e e d s t h a t of b . F a i l u r e t o take t h e s i t u a t i o n x i n t o account, i . e . assignment o f u t i l i t y i n d i c e s t o a l t e r n a t i v e s a , b , c o n l y , r e s u l t s i n n o n - t r a n s i t i v e p r e f e r e n c e s , s i n c e we may have a b c and c a f o r some s i t u a t i o n s x,y 2nd z .
...
kxb, kY
tZ
T h i s g e n e r a l o u t l i n e l e a v e s open t h e q u e s t i o n o f t h e
appropriate definition of u t i l i t y function u(x,a). This d e f i n i t i o n depends on t h e c h a r a c t e r o f t h e o p t i o n a . The most i m p o r t a n t c a s e s , which a l l o w e m p i r i c a l v a l i d a t i o n o f v a r i o u s h y p o t h e s e s , a r e t h o s e i n which x i s a n u m e r i c a l w e l f a r e i n d e x , and a , b , c , are either r i s k l e s s r e w a r d s , o r r i s k y l o t t e r i e s , i n v o l v i n g monet a r y r e w a r d s d e p e n d i n g on t h e o c c u r r e n c e o f some e v e n t s
...
When a i s a r i s k l e s s monetary r e w a r d , t h e n u ( x , a ) i s simply t h e u t i l i t y i n d e x o f r e c e i v i n g t h e sum a ( p o s s i b l y n e g a t i v e ) f o r a p e r s o n w i t h wealth x. The o b v i o u s p r o p e r t i e s a r e h e r e u(x,O) = 0 , w i t h u ( x , a ) b e i n g a n i n c r e a s i n g f u n c t i o n o f a . The f i r s t o f t h e s e p r o p e r t i e s r e q u i r e s measurement o f u t i l i t y on a r a t i o s c a l e , a l l o w i n g b o t h p o s i t i v e and n e g a t i v e v a l u e s , w i t h t h e z e r o r e p r e s e n t i n g " n e u t r a l " reward. Another p r o p e r t y , p o s t u l a t e d a l r e a d y by B e r n o u l l i i n e i g h t e e n t h c e n t u r y i s t h a t u ( x , a ) i s a d e c r e a s i n g funct i o n o f x f o r every a. It a p p e a r s t h a t one may j u s t i f y t h e a s s u m p t i o n t h a t u ( x , a ) , f o r any f i x e d x, i s " s t e e p e r " on t h e n e g a t i v e
SELECTED TOPICS lN MEASUREMENT THEORY
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s i d e t h a n on t h e p o s i t i v e s i d e , i . e . t h a t l o s s o f t h e amount a a p p e a r s " l a r g e r " t h a n t h e g a i n o f t h e same amount. C o n s e q u e n t l y , f o r r i s k l e s s o p t i o n s , t h e u t i l i t y may be assumed t o be a f a m i l y o f f u n c t i o n s o f t h e form p r e s e n t e d on F i g . 8 . 3 . 4 , where more f l a t f u n c t i o n s correspond t o l a r g e r x.
Fig. 8.3.4. Concerning t h e r i s k y o p t i o n s , c o n s i d e r t h e s i m p l e s t c a s e when t h e r e a r e two complementary e v e n t s , A and A ' , and a s s o c i a t e d w i t h them r e w a r d s a and a t . A s b e f o r e , l e t x s t a n d f o r t h e " p r e s e n t " s t a t e o f w e a l t h . Then t h e t o t a l u t i l i t y of s u c h a n o p t i o n ( i n s t a n d a r d n o t a t i o n s , t h e o p t i o n o f p a r t i c i p a t i n g i n t h e l o t t e r y a A a t ) must be a f u n c t i o n o f t h e arguments
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where m ( A ) and m ( A ' ) are some ( a s y e t u n s p e c i f i e d ) f u n c t i o n s o f t h e e v e n t s A and A ' ( d e p e n d i n g , among o t h e r t h i n g s , on t h e i r s u b j e c t i v e p r o b a b i l i t i e s ) . It a p p e a r s n a t u r a l t o p o s t u l a t e t h a t t h e dependence of t h e above f u n c t i o n U on arguments x , a and a ' i s t h r o u g h t h e r i s k l e s s u t i l i t i e s u ( x , a ) and u ( x , a ' ) . I t i s a l s o p l a u s i b l e , i n view of t h e p a r t i a l s u c c e s s o f SEU model, t h a t U i s ].inear, i n t h e s e n s e t h a t
Concerning t h e f u n c t i o n s m ( A ) and m ( A ' ) , one can make t h e s i m p l i f y i n g a s s u m p t i o n t h a t t h e y depend o n l y on the subjective p r o b a b i l i t i e s of the events i n question, so t h a t m ( A ) = m(p), m ( A ' )
= m(p')
where p and p ' a r e s u b j e c t i v e p r o b a b i l i t i e s o f A and A ' . It c o u l d b e a r g u e d , however, t h a t m ( A ) may depend on some o t h e r f e a t u r e s of t h e e v e n t A , i . e . i t may happen t h a t m ( A ) # m ( B ) , even though s u b j e c t i v e probab i l i t i e s o f A and B a r e e q u a l (some e v e n t s may a p p e a r more " p l e a s i n g " t o t h e s u b j e c t , i r r e s p e c t i v e of t h e rewards a s s o c i a t e d w i t h t h e m ) . To complete t h e s e c o n s i d e r a t i o n s ,
one can now o u t l i n e two p o s s i b l e a p p r o a c h e s t o t h e problem. One would cons i s t o f f i n d i n g a d e f e n s i b l e s e t of p o s t u l a t e s which imply a p a r t i c u l a r form of f u n c t i o n U . Another i s t o u s e e x p e r i m e n t a l t e c h n i q u e s , combining them w i t h i d e n t i f i c a t i o n p r o c e d u r e s developed i n systems t h e o r y , t o d e t e r m i n e whether U i s , i n f a c t , l i n e a r i n m ( A ) and m ( A ' ) , and t o e s t i m a t e its form.
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SELECTED TOPICS IN MEASUREMENT THEORY
9 . A NEW THEORY OF TIME
T h i s s e c t i o n w i l l be d e v o t e d t o a n a p p l i c a t i o n o f t h e
i d e a s o f measurement t h e o r y t o t h e c o n s t r u c t i o n of a t h e o r y o f t i m e , b o t h o b j e c t i v e and s u b j e c t i v e . The t h e o r y w i l l be based on ( f u z z y ) r e l a t i o n d e s c r i b i n g t h e subjective perception of t i m e , i n t h e sense of t h e r e l a t i o n "occurs e a r l i e r t h a n " .
We s h a l l f o r m u l a t e p o s t u l a t e s which w i l l imply t h e e x i s t e n c e o f two t i m e s c a l e s , o b j e c t i v e and s u b j e c t i v e , b o t h o f a n i n t e r v a l t y p e . L a t e r , some h y p o t h e s e s o f p s y c h o l o g i c a l c h a r a c t e r w i l l b e f o r m u l a t e d which w i l l e x p l a i n t h e d i s t o r t i o n s o f time p e r c e p t i o n .
9 . 1 . The b a s i c f o r m a l scheme The t h e o r y below i s based on t h e f o l l o w i n g s e t o f p r i mitive notions
(9.1) w i t h t h e i n t e n d e d i n t e r p r e t a t i o n as f o l l o w s . F i r s t l y , Z i s a s e t , whose e l e m e n t s w i l l be i n t e r p r e t e d as e v e n t s , u n d e r s t o o d a s i n s t a n t e n o u s changes o f s t a t e of some f r a g m e n t s o f r e a l i t y . Next, T w i l l b e t h e s e t , whose e l e m e n t s w i l l be e v e n t u a l l y i n t e r p r e t a b l e as moments o f t i m e ; a t t h e s t a r t , however, no s t r u c t u r e i s assumed i n the s e t T. The t h i r d p r i m i t i v e c o n c e p t o f t h e system (g.l), W,
namely
i s a c l a s s of fuzzy b i n a r y r e l a t i o n s i n Z , s o t h a t
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e a c h w i n W i s a f u n c t i o n which maps Z % Z i n t o C0,ll. The r e l a t i o n s i n W a r e assumed t o be indexed by t d T , so t h a t formally
W
=
{wt,
t E T’f
(9.2)
-+ [O,l].
(9.3)
where f o r e a c h t
wt:
z
Y
z
Moreover, t h e f u n c t i o n s ( 9 . 3 ) may be d e f i n e d o n l y for some p a i r s ( a , b ) G Z ; i n o t h e r words, w may be a p a r t t i a l f u n c t i o n on 2 x 2. The i n t e n d e d i n t e r p r e t a t i o n h e r e i s s u c h t h a t i f t h e p a i r ( a , b ) i s i n t h e domain o f t h e d e f i n i t i o n o f w t , t h e n t h e v a l u e w ( a , b ) i s t h e d e g r e e t o which ( a s judged t a t t i m e t ) , t h e e v e n t a p r e c e d e d t h e e v e n t b ( i. e . occurr e d e a r l i e r t h a n b ) . C l e a r l y , i f w t ( a , b ) = 1, t h e n a t
t , t h e e v e n t a i s remembered as undoubtedly e a r l i e r t h a n b; i f w t ( a , b ) = 0 , t h e n b i s remembered as e a r l i e r than a , while t h e intermediate cases 0 ( wt(a,b) < 1 r e p r e s e n t d e g r e e s of d o u b t . It s h o u l d b e remarked t h a t t h e v a l u e s w t ( a , b ) may chan-
ge i n t i m e , s o t h a t one may have w t ( a , b ) = 1 f o r some t , w h i l e w (al,.b) = 0 f o r some t ’ . T h i s means t h a t t’ t h e p e r c e p t i o n ( o r memory) o f e v e n t s a and b changed d r a s t i c a l l y between t i m e t and t ’ . F i n a l l y , i t ought t o be remarked t h a t t h e o r d e r o f occurrence i s understood here i n t h e s t r i c t s e n s e ; t h e concept r e l a t i n g t o s i m u l t a n e o u s o c c u r r e n c e o f a and b w i l l be i n t r o d u c e d l a t e r t h r o u g h t h e f u n c t i o n s w t .
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SELECED TOPICS IN MEASUREMENT THEORY
The l a s t p r i m i t i v e concept o f t h e s y s t e m ( 9 . 1 ) , namely H, W i l l be assumed t o be a q u a r t e r n a r y r e l a t i o n i n T ,
t ) i n t e r p r e t e d as t h e 3' 4 f a c t t h a t t h e i n t e r v a l o f t i m e between t l and t 2 i s no l o n g e r t h a n t h e i n t e r v a l of t i m e between t and t 4 . 3
w i t h t h e symbol ( t l , t 2 , ) H ( t
9.2.
The o b j e c t i v e c a r d i n a l and o r d i n a l t i m e
A s mentioned b e f o r e , t h e f u n c t i o n s w
t may be p a r t i a l .
It i s t h e r e f o r e n a t u r a l ' t o i n t r o d u c e t h e i r domain o f
d e f i n i t i o n , namely Mt
= ((a,b) E Z:
wt(a,b)
i s defined').
(9.4)
We s h a l l impose t h e f o l l o w i n g p o s t u l a t e which w i l l allow us t o introduce t h e ordinal objective time. POSTULATE 1. For a l l t , t o r Mt
I
T we have e i t h e r Mt C Mt
,
C Mt.
We may now d e f i n e t h e f o l l o w i n g r e l a t i o n on t h e s e t T :
we p u t t
< t',
5 t'
iff
Mt C M t l .
(9.5)
we s h a l l s a y t h a t t h e moment t o c c u r r e d e a r l i e r (or: n o t l a t e r ) t h a n t h e moment t ' . The i n t u i t i v e j u s t i f i c a t i o n here i s as f o l l o w s : i f t h e p a i r ( a , b ) o f e v e n t s d o e s no b e l o n g t o t h e omain of d e f i n i t i o n o f w t , t h e n i t i s i m p o s s i b l e t o compare a and b as t o t h e o r d e r o f t h e i r o c c u r r e n c e . T h i s may happen i f one o r b o t h o f t h e s e e v e n t s d i d n o t y e t o c c u r a t time t ( o n e or b o t h of them a r e i n t h e f u t u r e a t t i m e t ) . If t
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A s t i m e g o e s o n , more and more e v e n t s c e a s e t o b e l o n g
t o t h e f u t u r e ( t h e y o c c u r ) . Thus, i f t h e s e t o f e v e n t s which a r e i n t h e domain o f w i s l e s s t h a n t h e s e t of t t h o s e which a r e i n t h e domain o f w i t means t h a t t t r y must b e e a r l i e r t h a n t ' .
<
We s h a l l r e f e r t o t h e r e l a t i o n on T ( o b v i o u s l y , conn e c t e d and t r a n s i t i v e ) as t h e o r d i n a l time. The r e l a t i o n of s t r i c t o r d e r and t h e r e l a t i o n = of s i m u l t a n e i t y , a r e d e f i n e d i n t h e u s u a l way. Put now Ft = { a E Z :
( 3 b ) w i t h ( a , b ) r f Mt
1.
(9.6)
The s e t Ft i s t h e r e f o r e a c l a s s o f e v e n t s i n Z , w i t h t h e p r o p e r t y t h a t a t l e a s t f o r one o t h e r e v e n t i t i s
i m p o s s i b l e t o t e l l ( e v e n i n a f u z z y w a y ) which o f them o c c u r r e d e a r l i e r . According t o t h e i n t e r p r e t a t i o n above, i t i s n a t u r a l t o t r e a t e v e n t s i n F as t h o s e which bet l o n g t o t h e f u t u r e a t time t ( h a v e n o t y e t o c c u r r e d ) . Thus, t h e s e t ( 9 . 6 ) w i l l be c a l l e d f u t u r e a t t . The complement Z - F
w i l l be t h e r e f o r e t h e u n i o n o f t p a s t a t t , and p r e s e n t a t t ( t h e s e t s of t h o s e e v e n t s which o c c u r r e d p r i o r t o t i m e t , and t h o s e which o c c u r e x a c t l y a t t ) . L e t u s d e n o t e t h e p a s t a t t by Pt , and t h e p r e s e n t a t t by N t ( f r o m "now"), s o t h a t , t a k i n g n e g a t i o n i n ( 9 . 6 ) w e have
ac
Pt
cl
Nt
iff
( f f b ) : ( a , b ) E Mt.
(9.7)
I n o t h e r words, t h e e v e n t a b e l o n g t o e i t h e r p a s t o r p r e s e n t a t t , i f for any o t h e r e v e n t , one can d e t e r m i n e
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SELECTED TOPICS IN MEASUREMENT THEORY
( p o s s i b l y o n l y i n a fuzzy way) whether a o r b o c c u r r e d earlier. To s e p a r a t e p a s t from p r e s e n t , i t w i l l b e n e c e s s a r y t o a s s i g n t o an e v e n t t h e t i m e o f i t s o c c u r r e n c e , and def i n e t h e p r e s e n t a t t as t h e c l a s s o f a l l e v e n t s whose t i m e o f o c c u r r e n c e i s t . For s u c h a c o n s t r u c t i o n , i t i s n e c e s s a r y t o impose more s t r u c t u r e on t h e s e t T , u t i l i z i n g t h e p r i m i t i v e concept H . We may namely d e f i n e an o r d e r i n g r e l a t i o n
c_' i n
T , by
putting
I n t u i t i v e l y , t h i s means t h a t t ,I p r e c e d e s t,L' ( i n t h e o r d e r s t ) , i f t h e d i s t a n c e between t l and t 2 i s l a r g e r t h a n t h e z e r o d i s t a n c e between t and t i t s e l f . T h i s means t h a t t h e d i s t a n c e between t l and t 2 i s n o n n e g a t i v e , i . e . t h a t t 1 p r e c e d e s ( i s no l a t e r t h a n ) t 2 . We now impose t h e f o l l o w i n g p o s t u l a t e c o n n e c t i n g t h e r e l a t i o n H and t h e r e l a t i o n s w t : POSTULATE 2 .
The r e l a t i o n s 4 and
<'
coincide.
POSTLULATE 3. The r e l a t i o n H i s s u c h t h a t t h e r e e x i s t s
a mapping f : T
+R
( s e t of r e a l s ) s a t i s f y i n g t h e
condition :
( t1 ,t 2 ) H ( t 3 ,t 4
*
iff f(t2)-f(tl) 5 f(t4)-f(t3).(9.9)
Moreover, i f f i s any o t h e r f u n c t i o n s a t i s f y i n g t h e * c o n d i t i o n ( 9 . 9 ) , t h e n f = o(f t p f o r some A, 0 .
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P o s t u l a t e 3 r e p l a c e s , i n f a c t , a s e t o f p o s t u l a t e s which imply t h e e x i s t e n c e o f a n i n t e r v a l s c a l e r e p r e s e n t a t i o n f o r t h e r e l a t i o n H , such as r e q u i r e m e n t of t r a n s i t i v i t y , c o n n e c t e d n e s s , Archimedean axiom, and s o f o r t h . P o s t u -
l a t e 3 a s s e r t s t h a t t h e f l o w o f t i m e as d e f i n e d by t h e o r d i n a l r e l a t i o n I i s t h e same as t h e flow o f t i m e as d e f i n e d by t h e r e l a t i o n C ' .
-
Without loss o f g e n e r a l i t y we may assume t h a t t h e funct i o n f appearing i n Postulate 3 i s such t h a t f ( t ) = t ; t h i s means simply t h a t t h e t i m e s c a l e f i s s e l e c t e d i n s u c h a way t h a t i t s z e r o c o i n c i d e s w i t h t h e z e r o on t h e r e a l l i n e , and t h e u n i t o f t i m e e q u a l s 1. We may now d e f i n e t h e t i m e of o c c u r r e n c e o f t h e e v e n t a (
Zas T(a) = inf
it: a G! "1.
(9.10)
T h i s means t h a t T ( a ) i s t h e e a r l i e s t moment t a t which
a does n o t b e l o n g t o t h e f u t u r e a t t . T h i s d e f i n i t i o n a g r e e s w i t h t h e i n t u i t i o n , as shown by t h e f o l l o w i n g theorems. THEOREM. The f u t u r e s Ft a r e monotone d e c r e a s i n g , t h a t
is, Ft >
F t l whenever t 5 t ' .
THEOREM.
Let
#
. * >t .
T(a) = t
a h Ft f o r a l l t
Then a C Ft f o r a l l t
<
#
t
and
F i n a l l y , we impose a p o s t u l a t e which a l l o w s t h e s e p a r a t i o n o f p a s t and p r e s e n t . POSTULATE 4 . I f a
4 Ft and
b
c F t , t h e n ( a , b ) c Mt and
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SELECTED TOPICS IN MEASUREMENT THEORY
w (a,b) t
= 1.
(9.11)
T h i s p o s t u l a t e a s s e r t s t h a t e v e n t s which a r e i n t h e
p a s t o r p r e s e n t a r e always d i s t i n g u i s h a b l e from t h e
e v e n t s which a r e i n t h e f u t u r e , and t h e r e s u l t s o f comp a r i s o n s a r e not f u z z y . I n o t h e r words, one can have some d o u b t s as t o t h e o r d e r of e v e n t s which have a l r e a d y o c c u r r e d , o r as r e g a r d s t h e e v e n t s which have n o t y e t o c c u r r e d ( e . g . e v e n t s s u c h as a = b i r t h of t h e f i r s t d a u g h t e r o f M r X , and b = b i r t h of t h e second son o f M r . Y , i n c a s e when none o f t h e s e c h i l d r e n i s b o r n as y e t ) . However, i f one e v e n t i s i n t h e f u t u r e , and t h e o t h e r i n t h e p a s t , t h e y may a l w a y s be d i s t i n g ~ i s h e d w i t h o u t any f u z z i n e s s . We a l s o impose t h e f o l l o w i n g p o s t u l a t e . POSTULATE 5 . For e v e r y t t h e r e e x i s t s a t l e a s t one e v e n t a t: Z w i t h T ( a ) = t. T h i s p o s t u l a t e means t h a t a t any moment some e v e n t t a k e s p l a c e : t h e e v e n t s a r e "dense" i n t h e o r d e r i n g ,C
.
One may now s e p a r a t e p a s t and p r e s e n t , d e f i n i n g them
as Pt
=
(a: T ( a )
<
t)
, Nt
=
la:
T(a) =
tj,
(9.12)
s o t h a t w e have t h e r e l a t i o n
Z = P t u Ntw
Ft
(9.13)
w i t h all t h r e e s e t s b e i n g d i s j o i n t , and moreover ( i n
view o f p o s t u l a t e 5 ) , N t
# 0.
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9.3. S u b j e c t i v e t i m e and d i s t o r t i o n s o f t i m e p e r c e p t i o n The c o n s t r u c t i o n from t h e p r e c e d i n g s e c t i o n i s somewhat t o o r e s t r i c t i v e i n view o f P o s t u l a t e 5: t h e c l a s s Z o f a l l e v e n t s i s simply t o o r i c h f o r one o b s e r v e r . I n t h e s e q u e l , t h e r e f o r e , i t w i l l b e n e c e s s a r y to r e s t r i c t t h e a n a l y s i s t o a s u b c l a s s Z 1 o f e v e n t s , t o be i n t e r p r e t e d as t h e c l a s s of a l l e v e n t s which t h e o b s e r v e r n o t i c e s ( t h e c l a s s e s Z' may be d i f f e r e n t f o r v a r i o u s o b s e r v e r s ) . We shall c o n s i d e r now a s i n g l e o b s e r v e r , and assume t h a t t h e e v e n t s f o r Z' a r e " r e c o r d e d " by h i m and s t o r e d i n h i s memory f o r a l o n g e r o r s h o r t e r p e r i o d . We s h a l l now make more e x t e n s i v e use of t h e r e l a t i o n s wt (assumed t o r e p r e s e n t t h e judgments of t h e o b s e r v e r a t time t ) . L e t t be f i x e d , and l e t us d e f i n e
a Ct b
iff
wt(a,b)
>
i.
(9.14)
Thus, t h e r e l a t i o n 4, h o l d s o n l y f o r p a i r s ( a , b ) which a r e i n t h e s e t Mt o f d e f i n i t i o n of w t ' POSTULATE 6 . I f ( a , b ) f M t ,
then (b,a)
c
Mt.
This p o s t u l a t e i s of t e c h n i c a l c h a r a c t e r : it asserts
simply t h a t i f t h e p a i r ( a , b ) i s d i s t i n g u i s h a b l e i n t h e s e n s e o f r e l a t i o n w t , t h e n t h e same i s t r u e f o r t h e p a i r (b,a).
Moreover, we impose
POSTULATE 7. For all t and all ( a , b )
G Mt
w e have
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SELECTED TOPICS IN MEASUREMENT THEORY
The case of s t r i c t i n e q u a l i t y h e r e may o c c u r i f t h e e v e n t s a and b are judged ( a t time t ) as s i m u l t a n e o u s . Thus, we may p u t St(a,b) = 1
-
Cwt(a,b) + w t ( b , a ) l
(9.16)
+ 3St(a,b).
(9.17)
and
*
= wt(a,b)
wt(a,b)
From t h e i n t e r p r e t a t i o n o f w t , i t f o l l o w s t h a t S t ( a , b ) i s t h e d e g r e e t o which ( a s judged a t t ) , t h e e v e n t s a * and b a r e s i m u l t a n e o u s . S i m i l a r l y , w t ( a , b ) i s t h e d e g r e e t o which t h e e v e n t a p r e c e d e s ( i n t h e s e n s e o f weak order) t h e event b.
<
*
b i f f wt(a,b) 2 I n t h e u s u a l way we d e f i n e a t a w t b i f f a C t b and b 6, a .
4
and
We have t h e f o l l o w i n g theorem.
THEOREM. a
dt
b
iff
wt(a,b) = wt(b,a)
*
I n d e e d , w e have a w t b i f f w t ( a , b ) = 3 , hence i f * * w ( a , b ) = w t ( b , a ) . Consequently, b y ( 9 . 1 7 ) we o b t a i n t
*
wt ( a , b ) = w t ( a , b ) + 4 S t ( a , b )
S i n c e t h e f u n c t i o n St i s symmetric w i t h r e s p e c t t o i t s arguments, t h e a s s e r t i o n f o l l o w s .
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584
The f o l l o w i n g p o s t u l a t e c o r r e s p o n d s t o t h e p r o p e r t y o f
s t r o n g s t o c h a s t i c t r a n s i t i v i t y , known from p s y c h o l o g i c a l literature: POSTULATE 8 .
*
If w t ( a , b )
Let ( a , b ) , (a,c)and ( b , c ) , b e l o n g t o Mt. * $, t h e n
2 $ and w t ( b , c )
*
wt(a,c)
>
max C w t ( a , b ) , w t ( b , c ) l .
(9.18)
We have t h e n t h e f o l l o w i n g theorem THEOREM. The r e l a t i o n C~
i s t r a n s i t i v e and c o n n e c t e d .
C o n s e q u e n t l y , f o r e v e r y t we o b t a i n a n o r d e r i n g o f a l l e v e n t s i n t h e p a s t o r p r e s e n t a t t a c c o r d i n g t o t h e relation T h i s o r d e r i n g w i l l be c a l l e d t h e s u b j e c t i v e ordinal t i m e
<,.
.
It i s w o r t h o b s e r v i n g t h a t w h i l e t h e o b j e c t i v e t i m e
i s i n v a r i a n t under t h e c h o i c e o f r e f e r e n c e p o i n t t ( i . e . whenever T ( a ) < T ( b ) f o r some judgment t i m e t , t h e same i s t r u e f o r a l l o t h e r judgment t i m e s ) , t h i s i s i n g e n e r a l f a l s e f o r s u b j e c t i v e t i m e s : one may have a b f o r some t , and b < , , a f o r some t and t ' .
<,
Given t h e n o t i o n o f s u b j e c t i v e t i m e , one can d e f i n e one d i s t o r t i o n o f i t s p e r c e p t i o n , namely t h e r e v e r s a l :
i t o c c u r s , f o r e v e n t s a and b y i f T ( a ) < T ( b ) , b u t b < t a . I n o t h e r words, o b j e c t i v e l y a p r e c e d e s b , b u t as judged a t t i m e t , t h e s i t u a t i o n i s r e v e r s e , i . e . one j u d g e s b t o p r e c e d e a . Such m i s p e r c e p t i o n may be due t o v a r i o u s f a c t o r s , s u c h as memory l a p s e s , e m o t i o n a l d i s t u r b a n c e s , and s o on.
SELECTED TOPICS IN MEASUREMENT THEORY
585
G e n e r a l l y , one may e x p e c t t h a t r e v e r s a l s o c c u r a c c o r d i n g t o t h e following hypotheses. HYPOTHESIS 1. The c l o s e r one t o a n o t h e r i n t i m e a r e t h e e v e n t s a and b , i . e . t h e s m a l l e r i s t h e d i f f e r e n c e I T ( a ) - T ( b ) l , t h e more p r o b a b l e i s t h e r e v e r s a l o f e v e n t s a and b . HYPOTHESIS 2 . The l o n g e r i s t h e p a s s a g e of t i m e from e v e n t s a and b t o t h e moment when t h e judgment i s made, i . e . t h e l a r g e r i s t h e d i f f e r e n c e t - max C T ( a ) , T ( b ) l , t h e more chances f o r a r e v e r s a l . HYPOTHESIS 3. The more e v e n t s c C Z ' f a l l i n g between a and b , t h e l e s s chances f o r a r e v e r s a l o f t h e o r d e r o f a and b . The t h r e e e f f e c t s d e s c r i b e d i n t h e s e h y p o t h e s e s c o n c e r n
" p r o x i m i t y e f f e c t " ( H y p o t h e s i s 1), " p a s s a g e o f t i m e e f f e c t " ( H y p o t h e s i s 2 ) , and " s e p a r a t i o n e f f e c t " (Hypot h e s i s 3 ) . I n o t h e r words, t h e e v e n t s which a r e o b j e c t i v e l y c l o s e one t o a n o t h e r i n t i m e , o r which b o t h o c c u r r e d l o n g t i m e b e f o r e t h e judgment, a r e l i k e l y t o b e c o n f u s e d as r e g a r d s t h e o r d e r o f t h e i r o c c u r r e n c e . The l a s t h y p o t h e s i s s t a t e s t h a t i f t h e r e were many remembered e v e n t s i n between a and b , t h e chances f o r a r e v e r s a l are l e s s . To f o r m u l a t e f u r t h e r h y p o t h e s e s , i t i s n e c e s s a r y t o i n t r o d u c e t h e s u b j e c t i v e t i m e measured on a n i n t e r v a l 'ip t y p e s c a l e . T h i s may be a t t a i n e d b y imposing on wt t h e c o n d i t i o n s a n a l o g o u s t o t h o s e imposed on t h e s o - c a l l e d c h o i c e p r o b a b i l i t i e s ( s e e e . g . Luce 1959, Debreu 1 9 5 8 ) .
5 86
CHAPTER 5
A s i n t h e c a s e of r e l a t i o n H , i n s t e a d o f f o r m u l a t i n g e x p l i c i t l y s u c h s e t s of c o n d i t i o n s , we s i m p l y f o r m u l a t e
the following postulate. POSTULATE 9. F o r e v e r y t , t h e f u n c t i o n
t h e r e e x i s t s a mapping
s * Pt --3 t' satisfying the conditions:
*
wt i s s u c h t h a t
R (set of reals)
and 0
*
*
= wt(c,d) 4 1
4 wt(a,b)
implies
(9.20)
*
Moreover, i f st i s any o t h e r f u n c t i o n s a t i s f y i n g * f o r some d > ( 9 . 1 9 ) and ( 9 . 2 O ) , t h e n s t = d s t t
e
0.
The i n t e r p r e t a t i o n o f s t ( a ) i s t h a t i t i s t h e t i m e of
o c c u r r e n c e o f t h e e v e n t a , as remembered a t t i m e t . I t i s t h e r e f o r e n a t u r a l t o measure t h e d i s t o r t i o n o f p e r c e p t i o n o f t i m e by t h e r a t i o (9.21)
Obviously, i f f t ( a , b ) i s n e g a t i v e , t h e n a r e v e r s a l o f a and b o c c u r s a t t i m e t . I f f t ( a , b ) i s p o s i t i v e , t h e r e i s no r e v e r s a l ; however, t h e v a l u e o f t h i s f r a c t i o n may s e r v e as a measure of d i s t o r t i o n o f p e r c e p t i o n of t i m e : i f t h e p e r c e p t i o n i s n o t d i s t o r t e d , t h i s r a t i o ought t o be c l o s e t o 1 (assuming t h a t b o t h o b j e c t i v e and sub-
SELECTED TOPICS IN MEASUREMENT THEORY
587
j e c t i v e t i m e are measured i n t h e same u n i t s ) . Thus, . K c h o o s i n g some c o n s t a n t s k and K s u c h t h a t 0 c k C l L w i t h k and K c l o s e t o 1, we may s a y t h a t p e r c e p t i o n of e v e n t s a and b a t time t i s w o r t i o n f r e e , i f k
C, f t ( a , b )
<
K.
We s a y t h a t t h e p e r c e p t i o n o f a and b a t t i m e t i s p o s i t i v e l y d i s t o r t e d , if f t ( a , b ) > K , and n e g a t i v e l y d i s t o r t e d , i f 0 c f t ( a , b ) 4 k. Thus, a p o s i t i v e d i s t o r t i o n means t h a t t h e s u b j e c t i v e t i m e i s " s t r e t c h e d o u t " : t h e i n t e r v a l between a and b seems l o n g e r t h a n i t was i n r e a l i t y ( i n d e e d , t h e t r u e d i s t a n c e between a and b i s T ( a ) - T ( b ) , w h i l e t h e subj e c t i v e l y perceived distance, at time t , i s s t ( a ) st(b). On t h e o t h e r hand, a n e g a t i v e d i s t o r t i o n means t h a t t h e t e m p o r a l d i s t a n c e between a and b seems s h o r t e r t h a n i t was i n r e a l i t y , i . e . t h e t i m e seemed t o f l o w faster. To f o r m u l a t e t h e h y p o t h e s e s , l e t us a g r e e t o s p e a k o f s u b j e c t i v e p e r c e p t i o n of l o c a l t i m e , i f b o t h moments T ( a ) and T ( b ) a r e c l o s e t o t , w h i l e i f max [ T ( a ) , T ( b ) l i s much l e s s t h a n t , we s h a l l s p e a k on p e r c e p t 5 o n o f distant past. Assume f o r s i m p l i c i t y t h a t T ( a ) < T ( b ) , s o t h a t i n r e a l i t y a p r e c e d e s b , and c o n s i d e r a g a i n t h e c l a s s Z ' o f e v e n t s "of i n t e r e s t " f o r t h e o b s e r v e r . HYPOTHESIS 4 . Suppose t h a t t h e number o f e v e n t s c d Z 1 with T ( a ) 4 T(c)
< T(b)
is small. Then t h e l o o a l t i m e
w i l l t e n d t o be p o s i t i v e l y d i s t o r t e d , w h i l e t h e d i s t a n t p a s t w i l l t e n d t o be n e g a t i v e l y d i s t o r t e d .
588
CHAPER 5
A s an example, one can imagine a p e r s o n who a r r i v e s a t
some p l a c e ( e v e n t a ) and w a i t s f o r t h e e v e n t b ( e . g . a r r i v a l o f a f r i e n d ) . Suppose t h e r e a r e no f u r t h e r e v e n t s o c c u r r i n g between a and b . Then t h e r e a l w a i t i n g time T(b)
-
T(a)
w i l l seem l o n g d u r i n g w a i t i n g and
s h o r t l y a f t e r w a r d s , but i n r e t r o s p e c t i o n , s u c h w a i t i n g
t i m e w i l l seem s h o r t . The n e x t h y p o t h e s i s a s s e r t s t h e e f f e c t o p p o s i t e t o t h e
above. HYPOTHESIS 5 . Suppose t h a t t h e number o f e v e n t s c t Z' w i t h T ( c ) s a t i s f y i n g T ( a ) [ T ( c ) < T ( b ) i s l a r g e . Then t h e l o c a l t i m e w i l l t e n d t o be n e g a t i v e l y d i s t o r t e d , w h i l e t h e d i s t a n t p a s t w i l l t e n d t o be p o s i t i v e l y d i s -
torted. A s a n example, imagine a n i n t e r e s t i n g and e v e n t f u l pe-
r i o d i n l i f e . The t i m e " l o c a l l y " f l o w s f a s t , b u t i n r e t r o s p e c t i o n , s u c h a p e r i o d seems l o n g e r t h a n i t was i n reality. One may a m p l i f y t h e l a s t h y p o t h e s i s by HYPOTHESIS 6 . The h i g h e r i s t h e d e n s i t y o f t h e even% from Z' between a and b , t h e s t r o n g e r a r e t h e d i s t o r t i o n s a s s e r t e d i n Hypothesis 5 . The c l a s s of e v e n t s which d e t e r m i n e t h e o b j e c t i v e t i m e , t h a t i s , c l a s s Z , i s assumed l a r g e r t h a n t h e c l a s s Z' o f e v e n t s which d e t e r m i n e t h e s u b j e c t i v e t i m e . Here one may i n t r o d u c e s t i l l s m a l l e r c l a s s , s a y Z " C Z ' , o f pointer events. I n t u i t i v e l y , pointer events are the e v e n t s which are remembered, and which s e r v e as "sepa-
SELECTED TOPICS IN MEASUREMENT THEORY
589
r a t i o n p o i n t s " between o t h e r e v e n t s . One may t h u s det e r m i n e t h e numbers of e v e n t s from Z ' f a l l i n g between j - t h and ( j t 1 ) - s t p o i n t e r e v e n t . We may assume t h a t t h e o c c u r r e n c e o f e v e n t s from any g i v e n c l a s s forms a P o i s s o n p r o c e s s . I f k ( t ) i s t h e i n t e n s i t y of t h i s p r o c e s s ( p o s s i b l y time-dependent), t h e n exp ( u ) d u l i s t h e p r o b a b i l i t y t h a t no e v e n t ( f r o m t h e c l a s s under c o n s i d e r a t i o n ) o c c u r s between S and t . T h e r e f o r e
[-3
1 - exp C - J
t (u)dul
S
i s t h e p r o b a b i l i t y d i s t r i b u t i o n of t h e w a i t i n g t i m e at s f o r t h e next e v e n t .
One may c o n j e c t u r e t h a t t h e i n t e n s i t y x(u) i s i t s e l f random, i n which c a s e we hav.e a doubly s t o c h a s t i c proc e s s . Whether t h e o c c u r r e n c e s o f v a r i o u s t y p e s o f e v e n t s f o r a p e r s o n ( e . g . e v e n t s s u c h as " h a v i n g a good d i n n e r " , " t h i n k i n g about t r a v e l " , e t e . ) form a n o r d i n a r y o r doubly s t o c h a s t i c p r o c e s s i s a n i n t e r e s t i n g q u e s t i o n f o r an empirical research. The i n t u i t i v e argument for d o u b l e s t o c h a s t i c i t y i s t h a t
p e o p l e may t e n d t o change t h e i r environment i n a random manner. The main problem which i s w o r t h i n v e s t i g a t i n g concerns t h e deformation of s u b j e c t i v e time i n c a s e s when t h e e v a l u a t i o n s c o n c e r n t i m e i n two or more nonhomogeneous i n t e r v a l s , i . e . i n t e r v a l s i n which one has p a r t s w i t h h i g h , and p a r t s w i t h low d e n s i t y o f e v e n t s from Z 1 . One may c o n j e c t u r e t h a t we have here HYPOTHESIS 7 . I n e v a l u a t i n P t h e s u b j e c t i v e l e n g t h of
590
CNAPTER 5
a n i n t e r v a l , one t a k e s i n t o a c c o u n t o n l y t h e t o t a l number o f e v e n t s from Z', and n o t t h e i r l o c a t i o n s . N a t u r a l l y , t h i s h y p o t h e s i s may b e e x p e c t e d t o h o l d o n l y f o r r e l a t i v e l y short i n t e r v a l s . I n essence, t h i s i s t h e h y p o t h e s i s advanced b e f o r e , and i t t e l l s t h a t some s o r t o f "averaging" t a k e s p l a c e . O f c o u r s e , i n e v a l u a t i n g a homogeneous p a r t o f a non-homogeneous i n t e r v a l , one h a s p o s i t i v e o r n e g a t i v e d e f o r m a t i o n s , depending on t h e numbers of e v e n t s i n t h e s e s u b i n t e r v a l s . A s regards p o i n t e r events, especially occurring period-
i c a l l y and s t a n d a r d i z e d , t h e y p l a y c r u c i a l r o l e i n s t r u c t u r i n g t h e f u t u r e . Each p r e s e n t d e t e r m i n e s , f o r a given observer, a d i s t r i b u t i o n of possible f u t u r e s ( o r : p o s s i b l e w o r l d s ) i n which t h e s u b j e c t c o n s t r u c t s e x p e c t ed o r d e r s o f e v e n t s o f i n t e r e s t f o r h i m , u s i n g t h e p o i n t e r s y s t e m s . For e a c h f i x e d f u t u r e t h e s u b j e c t may choose a d i s t r i b u t i o n of t h e f u t u r e s s t a r t i n g from i t , and b y i t e r a t i n g t h i s p r o c e s s , he c a n b u i l d up t h e f u t u r e up t o some h o r i z o n . O b v i o u s l y , t h e f u r t h e r one g o e s , t h e more d i f f i c u l t i t becomes t o c o n s t r u c t t h e i t e r a t i o n s around t h e p e r i o d i c p o i n t e r e v e n t s .
9.4, Time and f r e q u e n c y of e v e n t s There i s a n i n t e r e s t i n g a l t e r n a t i v e p o s s i b i l i t y o f c o n n e c t i n g t h e r e l a t i o n " o c c u r s e a r l i e r t h a n " used t o d e f i n e t i m e , and "more f r e q u e n t t h a n " , used to d e f i n e t h e s c a l e of s u b j e c t i v e p r o b a b i l i t y . The i n t u i t i v e i d e a i s t h a t when one waits f o r an o c c u r r e n c e o f two e v e n t s , t h e n ( o n t h e a v e r a g e ) t h e more f r e q u e n t e v e n t w i l l occur e a r l i e r .
SELECTED TOPICS IN MEASUREMENT THEORY
591
To f o r m u l a t e t h e s e i n t u i t i v e n o t i o n s i n t o a p r o p e r
s e t u p , l e t u s p r o c e e d as f o l l o w s . C o n s i d e r a f a m i l y o f P o i s s o n p r o c e s s e s , e a c h w i t h i t s own “ t y p e ” o f e v e n t s , i n d e x e d by some p a r a m e t e r , s a y q , s o t h a t f o r e a c h q we have a P o i s s o n p r o c e s s of e v e n t s a9 ; l e t u s d e n o t e t h i s p r o c e s s by H q . Thus, H q i s a sequence o f e v e n t s , or e q u i v a l e n t l y , a sequence o f random t i m e s of t h e i r o c c u r r e n c e
...
Tql
T:
< T;< . . .
(9.22)
such t h a t t h e successive differences
Xy
= Tq
-
Tq i- 1
(9.23)
a r e i n d e p e n d e n t and i d e n t i c a l l y d i s t r i b u t e d w i t h t h e exponential distribution (9.24) where i s some p o s i t i v e c o n s t a n t c h a r a c t e r i z i n g t h e p r o c e s s q Hq
.
It i s w e l l known t h a t t h e e x p e c t e d t i m e between e v e n t s
i n t h e p r o c e s s Hq e q u a l s l/r
9’
i.e.
Moreover, t h e P o i s s o n p r o c e s s i n unique among p o i n t p r o c e s s e s i n t h e s e n s e of h a v i n g t h e ” f o r g e t t i n g p r o ? p e r t y ” ( s e e F e l l e r 1 9 5 7 ) : g i v e n any t , t h e w a i t i n g t i m e f o r t h e next event a f t e r t i s a l s o e x p o n e n t i a l l y d i s t r i b u t e d a c c o r d i n g t o ( 9 . 2 4 ) . Thus, t h e random v a r i a b l e
592
CHAPTER 5
Y ( t ) d e f i n e d as
Q
Y ( t ) = min ) T q q
= min
-
IT?:
t : Tq > - t ] T:
(9.26)
2 tj - t
has t h e p r o p e r t y P[Y
Q
( t )<
Consequently
, E[Y
XI 4
= 1
-
e
- rq
(9.27)
( t) I = l/Jq.
C o n s i d e r now two P o i s s o n p r o c e s s e s , w i t h e v e n t s a'' and a q f 1 ,and f o r f i x e d t c o n s i d e r t h e p r o b a b i l i t y t h a t t h e n e a r e s t e v e n t from t h e p r o c e s s H q l w i l l o c c u r e a r l i e r t h a n t h e n e a r e s t e v e n t from t h e p r o c e s s H '". T h i s means t h a t t h e w a i t i n g t i m e , c o u n t i n g from t , f o r an e v e n t from H q ' w i l l b e s h o r t e r t h a n t h e w a t i n i n g t i m e f o r an e v e n t from Hq". We assume t h a t t h e p r o c e s s e s a r e independent. 'Thus,
Using t h i s formula we may t h e r e f o r e w r i t e
(9.29)
593
SELEC'IED TOPICS INMEASUREMENT THEORY
Thus, t h e r e q u i r e d p r o b a b i l i t y i s e x p r e s s e d t h r o u g h t h e r a t i o o f t h e mean i n t e r a r r i v a l t i m e s o f t h e two p r o c e sses. If E ( X 7 ' )
= E(XYI'), i . e .
t h e n p ( q ' , q " ) = $,
t h e s e a v e r a g e s a r e t h e same,
as e x p e c t e d . When E ( X q ' ) / E ( X q l ' )
is
c l o s e t o z e r o , i . e . t h e e v e n t s from t h e p r o c e s s Hq' a r e much more f r e q u e n t t h a n t h o s e from t h e p r o c e s s H q I'
,
w e g e t p ( q ' , q " ) c l o s e t o 1, i . e . i t i s much more l i k e l y t h a t t h e f i r s t e v e n t w i l l be from t h e p r o c e s s H". On t h e o t h e r hand, i f E ( X q ' ) / E ( X q " ) i s v e r y l a r g e , s o t h a t t h e e v e n t s from Hq" o c c u r more o f t e n t h a n t h o s e
f r o m H q ' , w e g e t p ( q ' , q " ) c l o s e t o 0: i t is u n l i k e l y t h a t a n e v e n t from Hq' w i l l p r e c e d e a n e v e n t from H q " . The problem may now be f o r m u l a t e d s o t h a t from o b s e r v a t i o n of many i n s t a n c e s i n which e i t h e r a q ' o r a q I' o c c u r s e a r l i e r , one g e t s a n i d e a as t o t h e p r o b a b i l i t y I n t h e u s u a l way, one c a n d e f i n e t h e r e l a t i o n o f " b e i n g on t h e a v e r a g e e a r l i e r t h a n " for p a r a m e t e r s
p(ql,q").
q : t h e r e l a t i o n q l < q " h o l d s i f and o n l y i f p ( q ' , q " )
exceeds
3.
L e t u s now t a k e q l , q 2 and q = a , p(q2,q3) = b, with a,b
p(s1,q3).
3
and assume t h a t p ( q l , q 2 )
2 4.
L e t us e s t i m a t e
We have
(9.30) and s i n c e
we o b t a i n
and
Consequently ,
and we proved THEOREM. The p r o b a b i l i t i e s p ( q ' , q ' ' ) s a t i s f y s t r o n g
s t o c h a s t i c t r a n s i t i v i t y , and c o n s e q u e n t l y ion 4 i s t r a n s i t i v e .
,
the relat-
L e t u s observe t h a t t h i s r e l a t i o n orders parameters
i n a way c o n s i s t e n t w i t h t h e f u n c t i o n
kq,
q i n t h e sen-
s e t h a t whenever q 1 3 q 2 , t h e n 1/h < l / A q , i . e . lq 3192 I n o t h e r words, t h e i n t e n9s 1i t y q g f P o i s s o n p r J c e s s e s under c o n s i d e r a t i o n i s a monotone d e c r e a s i n g f u n c t i o n o f q : h i g h e r i n t e n s i t i e s imply s h o r t e r wait-
.
i n g times. The above c o n s t r u c t i o n may be used t o o b t a i n an i n t e r v a l t y p e s c a l e o f t i m e , by b u i l d i n g i t from i n c r e m e n t s o f t h e a v e r a g e w a i t i n g t i m e . I n d e e d , we have
SELECTED TOPICS IN MEASUREMENT THEORY
Thus, i f we t a k e a r b i t r a r i l y as a u n i t mean i n t e r a r r i v a l t i m e i n t h e p r o c e s s
595
f time t h e
t h e n we o b t -
ain
E(X q 2 ) - p ( q 1 , q 2 ) / C 1
-
P(q1'q2)1
(9.32)
.
as t h e a v e r a g e i n t e r a r r i v a l t i m e f o r e v e n t s i n H92 Proceeding i n t h i s way, a l l average i n t e r a r r i v a l t i m e s a r e e x p r e s s e d i n t h e same u n i t s . L e t us now a n a l y s e t h e same s i t u a t i o n i n a more r e a l i s t i c s e t u p , c o n s i d e r i n g some r e s t r i c t i o n s on o b s e r v a b i l i t y . F i r s t l y , t h e o b s e r v e r may o n l y have f i n i t e t i m e T a v a i l a b l e for e x p e r i m e n t a t i o n . Thus, i f a n e v e n t from e i t h e r o f t h e p r o c e s s e s H q ' and Hq" o c c u r s b e f o r e t h e t i m e t + T (where t i s t h e t i m e when t h e o b s e r v a t i o n s t a r t s ) , t h e outcome i s r e c o r d e d and used as p a r t of
for e v a l u a t i o n o f p ( q ' , q " ) . However, i t may happen t h a t no e v e n t of t h e s e p r o c e s s e s o c c u r s b e f o r e t t T . One may c a l l h e r e t h e i n t e r v a l ( t , t t T ) a " t r a p " s e t a t t i m e t , and i t may be p o s s i b l e t h a t n o t h i n g i s "caught
dath
i n the trap". The second p o s s i b l e c o n s t r a i n t on o b s e r v a b i l i t y i s t h a t
t h e o b s e r v e r may omit some e v e n t s for whatever r e a s o n s ( o v e r l o o k i n g , f a i l u r e of r e c o r d i n g d e v i c e s , e t c . ) Now, w i t h f i n i t e n e s s o f t h e t r a p t h e r e i s n o t much of
a problem. The p r o b a b i l i t y t h a t no e v e n t from e i t h e r of t h e p r o c e s s e s o c c u r s b e f o r e t + T e q u a l s
(9.33) Thus,
1 - p';(q',q")
i s t h e p r o b a b i l i t y that t h e experi-
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ment w i t h t h e t r a p o f l e n g t h T w i l l r e c o r d which e v e n t i s f i r s t , i . e . w e have
(9.34) N e x t , we have
(9.35) and d i v i d i n g by t h e p r o b a b i l i t y o f t h e c o n d i t i o n w e o b t a i n t h e p r e v i o u s f o r m u l a for p ( q ' , q ' ' ) . The o n l y d i f f e r e n c e i s t h a t t h e f i n i t e n e s s of t h e t r a p slows down t h e p r o c e s s of e s t i m a t i o n : t h e s h o r t e r i s t h e t r a p ( t i me T a l l o w e d f o r o b s e r v a t i o n s ) , t h e l a r g e r f r a c t i o n o f experiments w i t h inconclusive r e s u l t s . Suppose now t h a t e a c h o f t h e e v e n t s i n t h e p r o c e s s Bq ' i s u n o b s e r v a b l e w i t h p r o b a b i l i t y d , and e a c h o f t h e e v e n t s i n t h e p r o c e s s Hq" i s u n o b s e r v a b l e w i t h p r o b a b i lity , independently of o b s e r v a b i l i t y o f o t h e r e v e n t s Our a i m w i l l be t o compute t h e p r o b a b i l i t y t h a t t h e f i r s t o b s e r v e d e v e n t w i l l be a q ' , p r e c e d i n g t h e f i r s t o b s e r v e d e v e n t aq 'I
p
.
n L e t Y ( t ) w i l l be t h e t i m e o f t h e f i r s t o b s e r v e d e v e n t 9' i n t h e p r o c e s s Hq' a f t e r t i m e t . We may t h e n w r i t e
SELEClED TOPICS IN MEASUREMENT THEORY
-1 = e
4
,(l-d)x
597
(9.36)
*
Thus, t h e w a i t i n g t i m e Y ( t ) i s s t i l l e x p o n e n t i a l , q’ with parameter , replaced by (1-00. By t h e same kg * q argument, t h e random v a r i a b l e Y ( t ) i s e x p o n e n t i a l q It w i t h parameter k 1- ) C o n s e q u e n t l y , we may p r o c e e d q P as b e f o r e , o b t a i n i n g for t h e p r o b a b i l i t y p ( q , q ” ) t h e expression I\
.
-
1 (9.37)
I n p a r t i c u l a r , i f 6 = (3 , i . e . i f t h e d i s t u r b a n c e o f o b s e r v a b i l i t y c o n c e r n s b o t h p r o c e s s e s i n t h e same way, t h e n t h e r e s u l t i n g p ( q ’ , q ” ) i s t h e same as for t h e undisturbed processes.
1 0 . PROBLEMS OF SCALING 1 0 . 1 . I n t r o d u c t o r y remarks
The t e c h n i q u e s o f s c a l i n g , t h a t i s , a s s i g n i n g n u m e r i c a l v a l u e s t o o b j e c t s ( s t i m u l i ) were s t u d i e d a l r e a d y by Fechner ( 1 8 6 0 ) o v e r a c e n t u r y ago ( a n i n f a c t even e a r l i e r t h a n t h a t , s i n c e some problems o f p s y c h o l o g i c a l measurement were a n a l y s e d i n X V I I c e n t u r y ; s e e Ramul 1960).
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The development o f s c a l i n g t e c h n i q u e s proceeded i n two main d i r e c t i o n s . One o f t h e m was s t a r t e d i n t h e 1 9 2 0 ' s b y T h u r s t o n e ( s e e T h u r s t o n e 1927a,b; s e e a l s o T h u r s t o -
ne 1 9 5 9 ) . These t e c h n i q u e s a r e based m o s t l y on o r d e r i n g s , and b e c a u s e o f l a c k o f m e t r i c i n f o r m a t i o n , t h e y use some assumptions n e c e s s a r y f o r s c a l e c o n s t r u c t i o n s , mostly assumptions concerning t h e o r i g i n of v a r i a b i l i t y o f r e s u l t s f o r t h e same s t i m u l i . The second d i r e c t i o n i s connected p r i m a r i l y w i t h t h e work o f S t e v e n s ( s e e S t e v e n s 1936, and Harper and S t e vens 1 9 4 8 ) , who i n t r o d u c e d t h e methods o f d i r e c t eval u a t i o n of t h e stimulus by t h e subject ( e . g . "twice
as l a r g e as t h e s t a n d a r d " ) , as w e l l as t h e methods i n which t h e s u b j e c t p r o d u c e s a g i v e n s t i m u l u s ( e . g . a s t i m u l u s "midway" between two s t i m u l i , o r s t i m u l u s l'twice a s l a r g e " as a g i v e n s t i m u l u s , e t c . ) . T h i s d i r e c t i o n of r e s e a r c h i s c a r r i e d o u t most e x t e n s i -
v e l y i n Sweden, by t h e s t u d e n t s o f Ekman ( s e e Ekman and SjBberg 1 9 6 5 ) . An e x h a u s t i v e r e v i e w of methods o f s c a l i n g may be found i n T o r g e r s o n ( 1 9 5 8 ) ; s e e a l s o Luce, Bush and G a l a n t e r 1 9 6 3 , as w e l l as Coombs, Dawes and Tversky 1970.
10.2. ;elations
t o t h e o r i e s o f measurement
I n many c a s e s of s c a l i n g t e c h n i q u e s , t h e c o r r e s p o n d i n g a x i o m a t i c systems f o r measurement have not a s y e t been c o n s t r u c t e d . These t e c h n i q u e s a r e t h e n d e s c r i b e d i n terms o f t h e r e q u i r e d numerical r e p r e s e n t a t i o n d i r e c t l y ( i n s t e a d of i n t e r m s of axioms about t h e r e l a t i o n a l s y s t e m , i m p l y i n g t h e e x i s t e n c e of t h e d e s i r e d s c a l e ) .
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599
I n some c a s e s , t h e s i t u a t i o n i s o p p o s i t e : t h e axiom s y s t e m i m p l y i n g t h e e x i s t e n c e of a s c a l e i s known, and may be checked t o h o l d i n a g i v e n e m p i r i c a l s i t u a t i o n , b u t it i s n o t known how t o d e t e r m i n e t h e s c a l e v a l u e s ( e . g . i n many c a s e s of c o n j o i n t measurement). F i n a l l y , even i n t h e c a s e when t h e axiom systems a r e known, and t h e theorems g i v e i n d i c a t i o n how t o d e t e r m i ne t h e s c a l e v a l u e s , i t seldom happens t h a t t h e e m p i r i c a l d a t a s a t i s f y e x a c t l y t h e axioms; i n g e n e r a l , t h e d a t a a r e s u b j e c t t o e r r o r s , and t h e a i m o f s c a l i n g t e c h n i q u e s i s t o d e t e r m i n e t h e s c a l e v a l u e s which would approximate t h e model as e x a c t l y as p o s s i b l e . I n e s s e n c e , e a c h s c a l i n g t e c h n i q u e i s based on t h e a s s u m p t i o n o f e x i s t e n c e of t h e s c a l e of a g i v e n t y p e , and on t h e a s s u m p t i o n s which e x p r e s s r e l a t i o n s between t h e s c a l e v a l u e s and t h e o b s e r v e d e m p i r i c a l d a t a . I n o t h e r words, one p o s t u l a t e s t h e e x i s t e n c e of t h e s c a l e and imposes a s s u m p t i o n s which d e s c r i b e t h e way i n which t h e d a t a are formed. The problem l i e s i n d e t e r m i n i n g t h e s c a l e v a l u e s o f s t i m u l i , i . e . on d e t e r m i n i n g t h e n u m e r i c a l r e p r e s e n t a t i o n for t h e s e s t i m u l i . I n t h e n e x t s e c t i o n , some o f t h e most i m p o r t a n t s e t s o f a s s u m p t i o n s o f t h i s t y p e w i l l be p r e s e n t e d , t o g e t h e r w i t h t h e corresponding s c a l i n g techniques. It i s i m p o r t a n t t o mention t h a t a s u c c e s s i n a p p l y i n g
a g i v e n s c a l i n g t e c h n i q u e ( i . e . a s s i g n i n g numbers t o s t i m u l i ) does n o t , by i t s e l f , c o n s t i t u t e a l o g i c a l argument t h a t t h e axioms for t h e e x i s t e n c e of t h e s c a l e aTe s a t i s f i e d . It i s p r e c i s e l y t h e a t t e m p t s t o f i n d means of v e r i f y i n g t h i s a s s u m p t i o n t h a t l e d t o t h e
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development of measurement t h e o r y . N a t u r a l l y , one may a p p l y s c a l i n g t e c h n i q u e s as a means of v e r i f y i n g a g i v e n model, i n t h e s e n s e t h a t a n e g a t i v e r e s u l t i n d i c a t e s t h a t some of t h e a s s u m p t i o n s a r e n o t m e t . T h i s works, however, i n one way o n l y : t h e s u c c e s s i n a p p l y i n g a s e a l i n g t e c h n i q u e does n o t mean t h a t t h e model i s v a l i d .
10.3. C l a s s i f i c a t i o n of s c a l i n g t e c h n i q u e s A t p r e s e n t , a s u f f i c i e n t l y g e n e r a l s c a l i n g t h e o r y does
n o t e x i s t . Roughly, t h e s c a l i n g t e c h n i q u e s a r e c a t e g o r i z ed w i t h r e s p e c t t o t h e most o b v i o u s c r i t e r i o n , namely w i t h r e s p e c t t o t h e t y p e of e m p i r i c a l d a t a . One o f such
c l a s s i f i c a t i o n s has been i n t r o d u c e d b y Coombs (1964), namely i n t o t h e d a t a of dominance t y p e , and d a t a o f prox i m i t y t y p e . For t h e d i s c u s s i o n o f s c a l i n g t e c h n i q u e s , i t w i l l be somewhat more c o n v e n i e n t t o i n t r o d u c e a more r i c h c l a s s i f i c a t i o n , c o n c e r n i n g t h e d a t a used a s a b a s i s f o r s c a l i n g (and n o t , a s i n t h e c a s e o f Coombs, o f a l l psychological data). One may namely d i s t i n g u i s h t h r e e b a s i c c l a s s e s o f d a t a : ( a ) p a i r comparison; (b) magnitude s c a l i n g ; ( c ) r a t i n g and c a t e g o r y s c a l e s . While e a c h d a t a may be c a t e g o r i z e d t o one o f t h e above t y p e s , t h e b a s i c t r o u b l e l i e s i n t h e f a c t t h a t such i n f o r m a t i o n i s n o t v e r y f r u i t f u l : e a c h of t h e c a t e g o r i e : and c o m p r i s e s t e c h n i q u e s which have v e r y l i t t l e i n common. i s h i g h l y non-homogeneous,
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60 1
10.3.1. P a i r comparison. A s mentioned, t h i s c l a s s o f t e c h n i q u e s i s h i g h l y h e t e r o g e n e o u s . It i s c h a r a c t e r i s t i c here t h a t i n f o r m a t i o n c o n c e r n s p a i r s o f s t i m u l i and i s e x p r e s s e d i n terms of t h r e e q u a l i t a t i v e c a t e g o r i e s , " l a r g e r " , " s m a l l e r " , and " e q u a l " . N a t u r a l l y , t h e s e t e r m s ought t o be r e p l a c e d by t h e a p p r o p r i a t e t e r m s i n c o n c r e t e c a s e s ; t h u s , i n c a s e s o f comparisons o f l i g h t i n t e n s i t y , w e s h a l l have c a t e g o r i e s " b r i g h t e r " , and " d a r k e r " , i n c a s e s of comparison o f w e i g h t s , we s h a l l have " h e a v i e r " and " l i g h t e r " , and s o f o r t h . The e m p i r i c a l d a t a a r e n o t n e c e s s a r i l y i n e x p l i c i t form o f p a i r comparisons: t h e y may have t h e form o f a n o r d e r i n g o f a s e t o f stimuli ( e . g . p r e f e r e n t i a l ) ; o b v i o u s l y , however, a n o r d e r i n g i s i n e s s e n c e a s e t of p a i r comparisons. Some o f t h e s c a l i n g t e c h n i q u e s c o n n e c t e d w i t h p a i r comp a r i s o n s have a l r e a d y been d i s c u s s e d i n c o n n e c t i o n w i t h t h e a p p r o p r i a t e measurement systems ( e . g . t h e problems o f s u b j e c t i v e p r o b a b i l i t y and u t i l i t y , and t e c h n i q u e of u n f o l d i n g o f s c a l e s of Coombs). I n t h e f i r s t c a s e , t h e comparisons concerned l o t t e r i e s , and i n t h e s e c o n d , t h e d a t a had t h e form o f a s e t o f p r e f e r e n t i a l o r d e r i n g s of a c l a s s of o b j e c t s . I n t h i s s e c t i o n we s h a l l p r e s e n t one o f t h e b a s i c t e c h n i q u e s o f s c a l i n g based on p a i r comparisons, namely t h e s o - c a l l e d t e c h n i q u e j n d (from j u s t n o t i c a b l e d i f f e r e n c e s ) , and t h e r e l a t e d laws of Weber, F e c h n e r and Thurstone. The s t a r t i n g p o i n t h e r e i s t h e p s y c h o m e t r i c f u n c t i o n , which i s a n i d e a l i z e d outcome of p a i r c o m p a r i s o n s .
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Let u s c o n s i d e r a f i x e d s e t of s t i m u l i which d i f f e r o n l y b y one a t t r i b u t e , w i t h v a l u e s on a c e r t a i n p h y s i c a l continuum ( e . g . s t i m u l i b e i n g l i g h t s w i t h v a r i o u s i n t e n s i t i e s , sounds w i t h d i f f e r e n t p i t c h or i n t e n s i t y , e t c . ) . The s u b j e c t i s p r e s e n t e d w i t h t h e s t i m u l i i n p a i r s , and i s a s k e d t o p o i n t o u t i n e a c h p a i r t h a t s t i m u l u s which has "more" o f a g i v e n a t t r i b u t e ( i n t h e s e q u e l , we s h a l l use t h e t e r m s " l a r g e r " and " s m a l l e r " , remembering t h a t t h e s e t e r m s may have t o be r e p l a c e d b y some o t h e r s i n concrete s i t u a t i o n s ) . The comparisons a r e c a r r i e d o u t many t i m e s , e s p e c i a l l y f o r p a i r s of s t i m u l i f o r which t h e e v a l u a t i o n s a r e n o t c o n s i s t e n t ( i . e . f o r p a i r s where t h e a t t r i b u t e v a l u e s a r e c l o s e , hence f o r which t h e comparison i s d i f f i c u l t ) . For a s u f f i c i e n t number of o b s e r v a t i o n s o f a g i v e n p a i r ( a , b ) of s t i m u l i , one may t h e n e s t i m a t e t h e p r o b a b i l i t y p ( a , b ) t h a t s t i m u l u s a w i l l be i n d i c a t e d a s l a r g e r t h a n s t i m u l u s b . A s a r u l e , i n s u c h s t u d i e s one d o e s n o t a l l o w t h e a s s e r t i o n " e q u a l " , i . e . one f o r c e s t h e c h o i c e one way o r a n o t h e r , even if t h e s t i m u l i a r e t h e same. N a t u r a l l y , t h e p r o b a b i l i t i e s p ( a , b ) are e s t i m a t e d on t h e b a s i s of t h e f r e q u e n c i e s o f a s s e r t i o n s t h a t a i s larger than b. It t u r n s o u t t h a t i f one o f t h e s t i m u l i , s a y b , i s k e p t f i x e d , and one c o n s i d e r s t h e p r o b a b i l i t y p ( x , b ) f o r v a r i o u s x , t h e n t h i s p r o b a b i l i t y forms t h e s o - c a l l e d p s y c h o m e t r i c f u n c t i o n , w i t h s h a p e as p r e s e n t e d on F i g . 10.1.
I n o t h e r words, for s t L m u l i x much l e s s t h a n b , w e s h a l l have p ( x , b ) z 0 : i n a l l , o r almost a l l c a s e s , b w i l l be d e c l a r e d l a r g e r t h a n x. If x i s much l a r g e r t h a n b , t h e n
SELECTED TOPICS IN MEASUREMENT THEORY
603
x w i l l be d e c l a r e d l a r g e r t h a n b i n most c a s e s . If x i s c l o s e t o b y t h e v a l u e s p ( x , b ) w i l l p ( x , b ) N, 1, i . e .
l i e s t r i c t l y between 0 and 1, and w i l l be a n i n c r e a s i n g f u n c t i o n of b .
b F i g . 10.1. Psychometric f u n c t i o n I n p r a c t i c e , t h e v a l u e s of t h e p s y c h o m e t r i c f u n c t i o n a r e known o n l y f o r some s e l e c t e d p o i n t s x and b . One assumes u s u a l l y t h a t f o r e a c h f i x e d b y t h e p s y c h o m e t r i c f u n c t i o n p ( x , b ) i s d e f i n e d f o r a l l v a l u e s of t h e a r g u ment x ( w i t h i n t h e r a n g e of o b s e r v a b i l i t y of t h e stimuli of a g i v e n k i n d ) , and i s s t r i c t l y i n c r e a s i n g i n t h e
i n t e r v a l for which 0
<
p(x,b)(
1.
The p s y c h o m e t r i c f u n c t i o n a l l o w s us t o d e f i n e t h r e e
q u a n t i t i e s , which p l a y b a s i c r o l e i n p s y c h o p h y s i c s . The f i r s t i s t h e s o - c a l l e d PSE ( p o i n t of s u b j e c t i v e e q u a l i t y ) , d e f i n e d as a p o i n t x , ( b ) s u c h t h a t we have 2 t h e r e l a t i z n p ( x , ( b ) , b ) = 4. A c l o s e l y r e l a t e d q u a n t i t y z i s CE ( c o n s t a n t e r r o r ) , d e f i n e d a s x 4 ( b ) - b. Despite e m p i r i c a l evidence o f t h e e x i s t e n c e o f e r r o r CE i n most c a s e s , one u s u a l l y assumes t h a t p ( b , b ) = i , i . e . t h a t CE = 0 . The most i m p o r t a n t q u a n t i t y d e f i n e d t h r o u g h p s y c h o m e t r i c
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function p(x,b) i s t h e so-called jnd ( j u s t noticable d i f f e r e n c e ) . The v a l u e o f j n d a t t h e p o i n t b , i . e . j n d ( b ) i s d e f i n e d as (10.1)
where t h e v a l u e s a p p e a r i n g i n t h i s formula a r e d e f i n e d
as
The c o n s t r u c t i o n o f j n d and CE i s p r e s e n t e d on F i g u r e
10.2.
2 j-n-d-- -( b-) - .) &l -- --
F i g . 1 0 . 2 . Geometric i n t e r p r e t a t i o n o f j n d and CE
The v a l u e s o f j n d f u n c t i o n have been d e t e r m i n e d exper i m e n t a l l y for a s e r i e s of p s y c h o p h y s i c a l c o n t i n u a . It a p p e a r e d t h a t w i t h r e a s o n a b l e a p p r o x i m a t i o n ( u n d e r
SELECTED TOPICS lN MEASUREMENT THEORY
605
s u i t a b l e s c a l i n g o f s t i m u l i b ) one may assume t h a t t h i s function is linear, i . e . j n d ( b ) = Kb.
(10.3)
T h i s i s t h e s o - c a l l e d Weber l a w , and t h e c o n s t a n t K i s
c a l l e d t h e Weber c o n s t a n t . Connected c l o s e l y w i t h t h e Weber law i s t h e problem o f s c a l i n g due t o F e c h n e r . F i r s t of a l l , l e t u s d e f i n e a n o t i o n more g e n e r a l t h a n j n d , r e p l a c i n g na.mely i n t h e d e f i n i t i o n o f j n d t h e v a l u e s 1/4 and 3 / 4 by t and 1-t. More p r e c i s e l y , f o r 0 < t 4 1, d e f i n e D ( t , b ) = x t ( b ) - b , where x ( b ) i s t given by
(10.4) The f u n c t i o n s o d e f i n e d i s c a l l e d t h e Weber f u n c t i o n . The g e n e r a l i z e d Weber l a w a s s e r t s t h a t D ( t , b )
i s li-
near i n b, i . e . that D(t,b)
= K(t)b t C ( t ) ,
where K ( t ) and C ( t ) a r e c o n s t a n t s i n d e p e n d e n t of b ( t h e y may depend on t ) .
The problem of F e c h n e r c o n s i s t s o f f i n d i n g a s t r i c t l y monotone f u n c t i o n u , i n d e p e n d e n t o f t , s u c h t h a t UCb t D ( t , b ) I
-
U(b)
= g(t)
(10.5)
where g ( t ) i s a s t r i c t l y i n c r e a s i n g f u n c t i o n of t ,
606
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and g does n o t depend on b . I n some s i m p l i f i c a t i o n , t h e problem l i e s i n f i n d i n g a t r a n s f o r m a t i o n u o f s c a l e v a l u e s , under which t h e v a l u e s o f j n d f u n c t i o n a r e i n d e p e n d e n t o f t h e magnit u d e of t h e s t i m u l u s . One can prove ( s e e Luce and G a l a n t e r , 1963, p . 2 1 1 ) t h a t i f t h e problem of Fechner has a s o l u t i o n , i . e . i f there e x i s t s a function u satisfying (10.5), then u i s u n i q u e l y d e t e r m i n e d up t o t r a n s l a t i o n , i . e . e v e r y o t h e r f u n c t i o n u ' s a t i s f y i n g ( 1 0 . 5 ) i s s u c h t h a t u' = u t k , where k i s some c o n s t a n t . A r e l a t i o n between t h e g e n e r a l i z e d Weber l a w and t h e
s o l u t i o n t o problem o f F e c h n e r i s d e s c r i b e d by t h e f o l lowing theorem ( s e e Luce and G a l a n t e r , 1963, p . 2 1 2 ) : THEOREM.
I f t h e g e n e r a l i z e d Weber law h o l d s , w i t h
t ) , then the solution t o t h e Fechner problem i s g i v e n b y t h e f u n c t i o n C(t)/K(t)
= m (independent o f
u(b) = A log (b where A
0
and
+
m) t B,
B are constants.
Thus, t h i s theorem a s s e r t s t h a t i n some s i t u a t i o n s t h e Fechner s c a l e i s l o g a r i t h m i c . A somewhat d i f f e r e n t a p p r o a c h t o t h e problem o f s c a l -
i n g connected w i t h p s y c h o m e t r i c f u n c t i o n was s u g g e s t ed by T h u r s t o n e ( 1 9 2 7 b ) . He assumed namely t h a t w i t h t h e s t i m u l i a and b t h e r e a r e a s s o c i a t e d two random v a r i a b l e s X and Y , c a l l e d d i s c r i m i n a n t p r o c e s s e s , and t h e a s s e r t i o n t h a t a i s l a r g e r t h a n b i s given i n t h e
SELECTED TOPICS IN MEASUREMENT THEORY
607
>
c a s e when X Y , i . e . when X - Y > 0 . A t t h e n e x t t r i a l t h e v a l u e s o f X and Y may be d i f f e r e n t , and t h e oppos$t e i n e q u a l i t y may h o l d . The s c a l e v a l u e s a r e h e r e t h e e x p e c t a t i o n s u ( a ) and u ( b ) o f t h e random v a r i a b l e s X and Y . Thus, w e have
and t h e problem r e d u c e s t o making a s s u m p t i o n s under which one may d e t e r m i n e t h e p r o b a b i l i t y d i s t r i b u t i o n o f t h e d i f f e r e n c e X - Y. T h u r s t o n e assumed t h a t t h e d i s t r i b u t i o n s o f t h e d i s c r i minant p r o c e s s e s X and Y a r e normal, w i t h means u ( a ) 2 and u ( b ) , v a r i a n c e s CT: and G b , and c o r r e l a t i o n c o e f f i c i e n t rab. I n t h i s c a s e t h e d i f f e r e n c e X - Y has a l s o normal d i s t r i b u t i o n , w i t h mean u ( a ) - u ( b ) and v a r i a n c e h
h
h
The s t a n d a r d i z e d random v a r i a b l e
u = ‘ab
has t h e normal d i s t r i b u t i o n N(0,l).
L e t u s d e n o t e by
z ( a , b ) t h e v a l u e d e t e r m i n e d by t h e e q u a t i o n
Then we have X
-
Y
>
0 i f and o n l y i f
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N e x t , w e ha ve ( b y symmetry o f t h e n o r ma l d i s t r i b u t i o n )
and i t f o l l o w s t h a t
or
The l a s t e q u a t i o n i s t h e s o - c a l l e d
cornparat i v e j u d g m e n t s
T h u r s t o n e ' s law o f
.
To e a c h p a i r o f s t i m u l i t h e r e c o r r e s p o n d s o n e e q u a t i o n of t h e f orm ( 1 0 . 6 ) . N a t u r a l l y , a l l e q u a t i o n s t a k e n t o g e t h e r c o n t a i n more unknowns t h a n e q u a t i o n s , and t o s o l v e them one has t o impose some a d d i t i o n a l a s s u mp ti o n s . G e n e r a l l y , i f n s t i m u l i a r e c o n s i d e r e d , t h e numb e r o f e q u a t i o n s i s n ( n - l ) / 2 , w h i l e t h e number o f unknowns e q u a l s : a ) n means, o f wh ich one may b e t a k e n a r b i t r a r i l y , t h u s f i x i n g t h e z e r o o f t h e s c a l e ; t h i s g i v e s n -1 unknowns ; b ) n v a r i a n c e s , o f wh ich one may b e t a k e n a r b i t r a r i l y , t h u s f i x i n g t h e u n i t of t h e s c a l e ; t h i s gives n- 1 unlknowns ; c ) n ( n t 1 ) / 2 c o r r e l a t i o n s ( i n c l u d i n g p a i r s of s t i m u l i o f t h e form ( a , a ) ) .
SELECTED TOPICS IN MEASUREMENT THEORY
609
To r e d u c e t h e number o f unknowns, T h u r s t o n e made sever a l s i m p l i f y i n g a s s u m p t i o n s . The most common s e t o f a s s u m p t i o n s i s t h e s o - c a l l e d c a s e V o f T h u r s t o n e , wher e i t i s assumed t h a t a l l v a r i a n c e s a r e e q u a l ( i . e . 2 -Da2 = d b = G 2 ) , and a l s o t h a t a l l i n t e r c o r r e l a t i o n s are equal ( i . e . r = r = = r ) . Also, i n geneab ac r a l one t a k e s r = 0 , which c o r r e s p o n d s t o independence of discriminant processes.
...
...
S i n c e t h e v a l u e G which d e t e r m i n e s t h e u n i t s of t h e s c a l e may b e t a k e n a r b i t r a r i l y , one may p u t d = l/f-, hence 5 = l/fi when r = 0 . Then rab= 1 and t h e e q u a t i o n ( 1 0 . 6 ) t a k e n on t h e form u(a)
-
u(b) = z(a,b).
I n t h i s c a s e , f o r n s t i m u l i we have n ( n - 1 ) / 2 e q u a t i o n s w i t h n-1 unknowns ( t h e m e a n s ) . For a s o l u t i o n t o e x i s t , t h e unknowns must s a t i s f y c e r t a i n c o n d i t i o n s . F o r i n s tance z(a,b) = u(a) - u(b) = u(a)
- u(c)
t u(c)
-
u(b)
and i t f o l l o w s t h a t p r o b a b i l i t i e s p ( a , b ) are d e t e r m i n e d u n i q u e l y by p r o b a b i l i t i e s p ( a , c ) and p ( c , b ) . I n practical situations, the probabilities p(a,b) are known o n l y up t o e r r o r s o f t h e i r e s t i m a t i o n s . I n s u c h cases, i n order t o find the scale values u(a) - u(b) one h a s t o a p p l y a p p r o p r i a t e methods o f s t a t i s t i c a l es-
610
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t i m a t i o n . F o r a r e v i e w o f such methods, s e e T o r g e r s e n 1958, or F l e i s s 1981. To sum up, t h e main i d e a of T h u r s t o n e i s t o r e p r e s e n t s t i m u l i as random v a r i a b l e s , which r e f l e c t i n s t a n t e n u o u s f l u c t u a t i o n s of p e r c e i v e d m a g n i t u d e s . The c h o i c e i s conn e c t e d w i t h t h e v a l u e s of t h e s e random p r o c e s s e s a t t h e moment of making t h i s c h o i c e . T h u r s t o n e made v a r i o u s a s s u m p t i o n s about t h e d i s t r i b u t i o n s o f t h e d i s c r i m i n a n t p r o c e s s e s , among which h i s c a s e V p r e s e n t e d above i s t h e b e s t known. H i s s c a l i n g method c o n c e r n s t h e c h o i c e s o f a s i n g l e p e r s o n ; i n p r l a c t i c e , however, e s t i m a t e s o f t h e c h o i c e p r o b a b i l i t i e s p ( a , b ) a r e made on t h e b a s i s o f c h o i c e s i n a group of' s u b j e c t s . T h i s f a c t c o m p l i c a t e s t h e proc e d u r e s f o r t e s t i n g t h e adequacy of T h u r s t o n e models. Nevertheless, h i s i d e a s played an i n s p i r i n g r o l e f o r much o f f u r t h e r r e s e a r c h , b o t h t h e o r e t i c a l and empir i c a l . T h i s r e s e a r c h proceeded i n two main d i r e c t i o n s . One o f them i s e x e m p l i f i e d by t h e method of Coombs of unfolding scales. Coombs (1958; s e e a l s o Coombs, Dawes and Tversky 1970) assumed t h a t t o s t i m u l i ( a n a l y s e d from t h e p o i n t o f view o f p r e f e r e n c e s ) t h e r e c o r r e s p o n d d i s c r i m i n a n t proc e s s e s X and Y , as i n t h e model o f T h u r s t o n e , and moreo v e r , t h e s u b j e c t h a s h i s " i d e a l p o i n t " , which i s a l s o r e p r e s e n t e d as a random v a r i a b l e I . P r o b a b i l i t y p ( a , b ) o f c h o o s i n g t h e s t i m u l u s a from t h e p a i r ( a , b ) depends on t h e mutual d i s t a n c e s o f t h e i d e a l p o i n t from X and Y . More p r e c i s e l y , i t i s assumed t h a t
SELECTED TOPICS IN MEASUREMENT THEORY
61 1
s o t h a t a w i l l be chosen i f i t i s c l o s e r t o t h e i d e a l point than b. T h i s model a l l o w s t o p r e d i c t t h e f r e q u e n c y of i n t r a n s i -
t i v e t r i p l e t s of a l t e r n a t i v e s , i . e . such t r i p l e t s a , b , c f o r which a A, b 4 c 4 a . The s i g n i f i c a n c e of Coombs’ r e s u l t s l i e s i n p o i n t i n g o u t t h a t i n some c a s e s t h e c h o i c e p r o b a b i l i t i e s p ( a , b ) cannot be d i r e c t l y t r a n s f o r m e d i n t o s c a l e v a l u e s , and one has t o i n t r o d u c e t h e n o t i o n o f i d e a l p o i n t . The second main development of i d e a s o f T h u r s t o n e i s due t o Luce (1959). I n s t e a d o f making a s s u m p t i o n s a b o u t t h e mechanisms u n d e r l y i n g t h e c h o i c e s , he t o o k t h e c h o i c e p r o b a b i l i t i e s p ( a , b ) d i r e c t l y as t h e s t a r t i n g p o i n t . More p r e c i s e l y , he t o o k a s h i s p r i m i t i v e n o t i o n t h e p r o b a b i l i t y P(R,T) t h a t under a p r e s e n t a t i o n of t h e s e t T of a l t e r n a t i v e s , an a l t e r n a t i v e from t h e s e t R T w i l l be s e l e c t e d ( s o t h a t p ( a , b ) = P ( j a 7 , { a , b l ) ) . Luce a c c e p t s t h e f o l l o w i n g axiom, c a l l e d sometimes t h e c h o i c e axiom: for x Q R T we have
For a n i l l u s t r a t i o n , l e t T be t h e s e t of i t e m s o n a menu, and l e t R d e n o t e t h e s e t of s e a f o o d i t e m s . F i n a l l y , l e t x s t a n d f o r some s p e c i f i c s e a f o o d i t e m , s a y c r a b l e g s . The L u c e ‘ s axiom a s s e r t s t h a t t h e p r o b a b i l i t y of c h o o s i n g c r a b l e g s from t h e whole menu e q u a l s t h e p r o d u c t of t h e p r o b a b i l i t y o f c h o o s i n g c r a b l e g s from t h e s e t o f a l l s e a f o o d d i s h e s , times t h e p r o b a b i l i t y o f
c h o o s i n g a s e a f o o d i t e m from t h e whole menu.
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The axiom o f c h o i c e i m p l i e s s e v e r a l consequences, o f which t h e most i m p o r t a n t a r e t h e f o l l o w i n g ( s e e Luce 1959; Luce and G a l a n t e r 1963, or Coombs, Dawes and Tversky 1970):
I. I f the considerations are r e s t r i c t e d t o probabili-
t i e s s t r i c t l y between 0 and 1, w e have t h e f o l l o w i n g identi t 2:
11. There e x i s t s a f u n c t i o n v , d e f i n e d on t h e s e t o f s t i m u l i , such t h a t
(10.9) and more g e n e r a l l y
w i t h v b e i n g a measurement on a r a t i o s c a l e .
The i d e n t i t y ( 1 0 . 8 ) a l l o w s us t o compare t h e models of Luce and T h u r s t o n e ( a t l e a s t i n t h e o r y ) . U n f o r t u n a t e l y , t h e p r o b a b i l i t i e s p ( a , b ) d e t e r m i n e d from p ( a , c ) and p ( c , b ) on t h e b a s i s of ( 1 0 . 8 ) d i f f e r s o s l i g h t l y from t h e p r o b a b i l i t i e s d e t e r m i n e d from T h u r s t o n e ’ s model, t h a t ( s e e Luce, 1959, P . 5 6 ) , t h e e x p e r i m e n t a l d i f f e r -
SELECTED TOPICS IN MEASUREMENT THEORY
613
e n t i a t i o n of t h e s e two models does n o t seem p o s s i b l e . Moreover, t h e c h o i c e p r o b a b i l i t i e s p ( a , b ) f o r v a r i o u s p e r s o n s may s a t i s f y t h e c h o i c e axiom, b u t t h e a v e r a g e p r o b a b i l i t i e s i n a group may v i o l a t e t h i s axiom ( s e e Luce 1 9 5 9 , p . 8 ) Thus, a n e m p i r i c a l v e r i f i c a t i o n of L u c e ' s model i s d i f f i c u l t , s i n c e i n p r a c t i c a l s i t u a t i o n s one i s bound t o use t h e e s t i m a t i o n s o f p r o b a b i l i t i e s from groups o f p e r s o n s . It i s w o r t h w h i l e t o s t r e s s t h a t u n t i l now t h e r e i s s t i l l a l a c k o f a n a d e q u a t e g e n e r a l t h e o r y of c h o i c e . It seems t h a t t h e r e a s o n i s t h a t v a r i o u s d e c i s i o n r u l e s
a r e used i n v a r i o u s c o n t e x t s . T h e r e f o r e a g e n e r a l t h e o r y would have t o d e t e r m i n e t h e c o n d i t i o n s of a p p l i c a b i l i t y of e a c h model, and t h e n t a k e i n t o a c c o u n t t h e p e r ception o f t h e choice s i t u a t i o n s .
Magnitude s c a l i n g . One o f t h e main f e a t u r e s of t h e s c a l i n g t e c h n i q u e s d i s c u s s e d i n t h e p r e c e d i n g s e c t i o n i s t h a t t h e s o u r c e o f i n f o r m a t i o n i s t h e incons i s t e n c y o f c h o i c e s from p a i r s which are c l o s e on t h e continuum. Thus, t h e s c a l e i s b u i l t l o c a l l y , on t h e f r a g m e n t s of t h e continuum i n which t h e d i s c r i m i n a t i o n i s d i f f i c u l t , and t h e p i e c e s a r e j o i n e d t o g e t h e r b y use of t h e assumption t h a t t h e p e r c e p t i o n o f 1 j n d i s t h e same on t h e whole continuum. 10.3.2.
These t e c h n i q u e s l e a d t o s c a l e s o f a n i n t e r v a l t y p e , where t h e z e r o and t h e u n i t a r e s e l e c t e d a r b i t r a r i l y . I n many c a s e s , however, t h e a r b i t r a r i n e s s o f t h e c h o i c e of zero i s debatable: o f t e n the zero i s distinguished i n a n a t u r a l way, as l a c k o f t h e t r a i t , and n e g a t i v e
614
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v a l u e s make no s e n s e ( e . g . i n c a s e o f b r i g h t n e s s , w e i g h t , l e n g t h , e t c . ) . One can e x p e c t t h e r e f o r e t h a t f o r many c o n t i n u a t h e r e s h o u l d e x i s t s c a l e s o f a r a t i o t y p e , and not only of i n t e r v a l t y p e . S t a r t i n g from a c r i t i q u e , a c c o r d i n g t o which t h e incons i s t e n c i e s s h o u l d n o t s e r v e as a b a s i s f o r c o n s t r u c t i o n o f t h e s c a l e , S t e v e n s o r i g i n a t e d a r e s e a r c h program, aimed a t c r e a t i n g b e t t e r s c a l i n g methods t h a n t h o s e o f Fechner and T h u r s t o n e . S t e v e n s t o o k t h e d a t a c o n c e r n i n g t h e d i r e c t comparison o f t h e s t i m u l u s w i t h t h e s t a r i d a r d s t i m u l u s , or more g e n e r a l l y , comparison o f magnitudes o f two s t i m u l i . I n some v e r s i o n s o f t h e e x p e r i m e n t a l s e t u p , t h e s u b j e c t s were asked t o produce a s t i m u l u s ( b y u s i n g a s p e c i a l equipment, e . g . c o n t r o l o f l i g h t i n t e n s i t y ) m e e t i n g c e r t a i n c r i t e r i a ( s u c h as s t i m u l u s which would be “ t w i c e a s 1 a r g e ” a s t h e one which was d i s p l a y e d , e t c . ) I n a sense, t h e s c a l e values here a r e supplied d i r e c t l y by t h e s u b j e c t s . The o n l y t r o u b l e l i e s i n t h e p o s s i b i l i t y of i n t e r n a l incosistency: i f t h e subject claims that
s t i m u l u s b i s t w i c e as l a r g e as a , and c i s t h r e e t i m e s a s l a r g e as b , t h e n a s h o u l d be claimed t o be 6 t i m e s
as l a r g e a s c . G e n e r a l l y , i f t h e r a t i o of magnitudes u ( a ) and u ( b ) i s claimed t o be r , and t h e r a t i o o f magnitudes u ( b ) and u ( c ) i s claimed t o be s , t h e n t h e r a t i o of magnitudes u ( a ) and u ( c ) s h o u l d be r s . O f c o u r s e , t h e e m p i r i c a l d a t a do n o t have t o meet t h i s cond i t i o n , and t h e problem l i e s i n d e t e r m i n i n g t h e s c a l e v a l u e s which approximate i n t h e b e s t p o s s i b l e way t h e e m p i r i c a l d a t a . The t e c h n i q u e s a r e d e s c r i b e d i n d e t a i l i n T o r g e r s o n (1958), and w i l l be o m i t t e d h e r e .
615
SELECTED TOPICS INMEASUREMENT THEORY
The most i m p o r t a n t r e s u l t of S t e v e n s and h i s s c h o o l i s e s t a b l i s h i n g the f a c t that f o r t h e so-called p r o t e t i c c o n t i n u s ( r o u g h l y , t h o s e c o n t i n u a which a r e measurable on a r a t i o s c a l e , such as w e i g h t , b r i g h t n e s s , e t c . ) t h e r e l a t i o n between t h e s c a l e v a l u e s u ( x ) and t h e i n t e n s i t i s x of t h e s t i m u l i has a power c h a r a c t e r :
u(x) = ax
b
(10.11)
where a and b a r e some p o s i t i v e c o n s t a n t s . I n t h i s form u l a , t h e c o n s t a n t a may be chosen a r b i t r a r i l y , w h i l e t h e exponent b i s c h a r a c t e r i s t i c f o r a g i v e n continuum. Thus, t h e law ( 1 0 . 1 1 ) i s i n a d i s t i n c t i n c o n s i s t e n c y w i t h t h e F e c h n e r law, which c l a i m s t h a t t h e v a l u e s u on a p s y c h o p h y s i c a l s c a l e a r e r e l a t e d l o g a r i t h m i c a l l y t o t h e i n t e n s i t y x of t h e stimulus. The main c r i t i q u e o f S t e v e n s ' a p p r o a c h i s t h a t t h e
n u m e r i c a l custom o f p e r s o n s may d i s t u r b t h e e v a l u a t i o n s of t h e p e r c e i v e d r a t i o s . To c o u n t e r t h a t c r i t i q u e , S t e v e n s i n t r o d u c e d t h e t e c h n i q u e o f c r o s s - m o d a l i t y matchi n g , i n which t h e s u b j e c t e v a l u a t e s t h e p e r c e i v e d r a t i o s o f magnitudes o f s t i m u l i and e x p r e s s e s i n on a n o t h e r continuum. For i n s t a n c e , t h e s u b j e c t may b e a s k e d t o e v a l u a t e t h e r a t i o o f w e i g h t s o f two o b j e c t s , and e x p r e s s t h i s p e r c e i v e d r a t i o by s e t t i n g t h e i n t e n s i t y of a l i g h t s o that i t s perceived r a t i o t o the s t a n d a r d i s t h e same ("matches") t h e p e r c e i v e d r a t i o of weights. Such e x p e r i m e n t s s h o u l d l e a d t o c o n s i s t e n t outcomes b i n t h e following sense. I f u(x) = ax i s the perceived i n t e n s i t y of a s t i m u l u s on one p s y c h o p h y s i c a l c o n t i -
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nuum ( e . g . w e i g h t ) , measured w i t h r e s p e c t t o a s t a n d a r d , and v ( y ) = cyd i s t h e i n t e n s i t y o f a n o t h e r s t i m u l u s , ( a g a i n w i t h r e s p e c t to a s t a n d a r d ) on a n o t h e r continuum ( e . g . b r i g h t n e s s ) , and i f s u b j e c t i v e l y t h e s e i n t e n s i t i e s a r e t h e same, t h e n ax
b
= cy
d
,
hence Y = (a/c)
l / d .b/d
It f o l l o w s t h a t when m o d a l i t i e s a r e matched a c r o s s c o n t i n u a , one s h o u l d o b s e r v e a g a i n a power law, w i t h t h e exponent b / d .
Such t y p e s o f r e l a t i o n s have been e s t a b l i s h e d f o r about 2 0 p s y c h o p h y s i c a l c o n t i n u a , and i n v a r i a b l y t h e v a l u e s of t h e exponents coincided w i t h t h e p r e d i c t i o n s ( s e e S t e v e n s 1959, 1 9 6 1 ) . It i s w o r t h to mention t h a t t h e t e c h n i q u e o f cross-mod a l i t y matching may be a p p l i e d a l s o i n c a s e s when o n l y
one o f t h e c o n t i n u a i s o f p s y c h o p h y s i c a l c h a r a c t e r , w h i l e t h e o t h e r i s o f p s y c h o l o g i c a l c h a r a c t e r , and has no p h y s i c a l c o u n t e r p a r t ( e . g . s e r i o u s n e s s o f c r i m e s ) . T h i s l e a d s t o d e t e r m i n i n g s c a l e v a l u e s for c e r t a i n psychological properties. One s h o u l d s t r e s s t h a t w h i l e t h e t e c h n i q u e of S t e v e n s was f o r a l o n g t i m e a n o b j e c t o f i n t e n s i v e s t u d i e s , i t s t h e o r e t i c a l f o u n d a t i o n s ( f r o m measurement t h e o r e t i c a l p o i n t o f view) were d e v e l o p e d r e l a t i v e l y r e c e n t l y ( s e e K r a n t z e t a l . 1 9 7 1 ) . The a x i o m a t i c system which
SELECTED TOPICS IN MEASUREMENT THEORY
617
i m p l i e s t h e e x i s t e n c e o f t h e s c a l e and s p e c i f i e s i t s c h a r a c t e r i s almost t h e same as t h e system f o r a l g e b r a i c differences, presented i n t h e e a r l i e r parts of t h i s chapter. The s t a r t i n g p o i n t t h e r e was a r e l a t i o n 2 on a s e t o f p a i r s of o b j e c t s from a s e t A , i . e . a q u a r t e r n a r y r e l a t i o n on t h e s e t A , i n t e r p r e t e d as f o l l o w s : ab cd i f
>
t h e p e r c e i v e d d i f f e r e n c e between a and b exceeds t h e p e r c e i v e d d i f f e r e n c e between c and d . To d e s c r i b e t h e c r o s s - m o d a l i t y m a t c h i n g , one c o n s i d e r s a c e r t a i n number o f s e t s A1, ..., Am , r e p r e s e n t i n g t h e s e t s of s t i m u l i on v a r i o u s c o n t i n u a . We have a l s o an o r d e r i n g r e l a t i o n 3 d e f i n e d on t h e s e t
The symbol a . b . 2 a ! b ! , where ai,bi Ai, a t , b j € A j 1 1 J J j i s i n t e r p r e t e d as " t h e r a t i o o f i n t e n s i t i e s o f s t i m u l i ai and b i i s a t l e a s t as l a r g e as t h e r a t i o of i n t e n s -
-
i t i e s o f s t i m u l i a! and b ' " . J j
Here w e may have A
i
= A
j
.
The axioms a r e almost i d e n t i c a l as i n t h e c a s e o f t h e s t r u c t u r e of algebraic differences: they a s s e r t t h a t the relation
2
i s a weak o r d e r , and t h a t aibi
i f and o n l y i f b ' a ! j J
2
5. a j b j
(change of d i r e c t i o n of i n e q u a l i t y under t a k i n g r e c i p r o c a l s ) . The t h i r d axiom a s s e r t s t h a t a i b i ,> a J! b !J and b i c i >- ,b j' c j' imply a c 2 % ! c ! ( m o n o t o n i c i t y ) . There i s a n a d d i t i o n a l i i J J axiom o f s o l v a b i l i t y : f o r any a i , b .i E A 1 . there exist c l , d l E A1 s u c h , t h a t a . b . 2 c l d l . T h i s means t h a t e v e r y biai
1 1
o f s t i m u l i on continuum A may be matched b y a i r a t i o o f some s t i m u l i on continuum A 1' The r e m a i n i n g
ratio
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two axioms a s s e r t t h a t t h e o r d e r i n g 2 r e s t r i c t e d t o a f i x e d continuum Ai s a t i s f i e s t h e s o l v a b i l i t y and Archimedean axiom. The a s s e r t i o n o f t h e theorem i s what was d i s c o v e r e d e m p i r i c a l l y b y S t e v e n s , namely t h a t t h e r e e x i s t s p o s i t i v e f u n c t i o n s u l,..., u m' d e f i n e d on r e s p e c t i v e s e t s A i , such t h a t
These f u n c t i o n s a r e d e t e r m i n e d u n i q u e l y up t o a power u' a r e some o t h e r funct r a n s f o r m a t i o n , i . e . i f ui m t i s n s s a t i s f y i n g t h e above c o n d i t i o n , t h e n t h e r e e x i s t p o s i t i v e c o n s t a n t s d l ,. . . , d m and 8 s u c h t h a t
,...,
u'i =
diui
lr f o r
i = 1,2,
... ,m.
A t t h e end, l e t us mention t h a t d e s p i t e t h e s o l u t i o n
o f m e a s u r e m e n t - t h e o r e t i c a l problems c o n n e c t e d w i t h S t e v e n s ' a p p r o a c h , h i s laws s t i l l c a u s e some c o n t r o v e r s i e s . They a r e r e l a t e d , f i r s t o f a l l , w i t h t h e f a c t t h a t h i s laws were o b t a i n e d by means o f s c a l i n g p r o c e d u r e s , and n o t by c h e c k i n g t h a t t h e axioms o f t h e c o r r e s p o n d i n g s y s t e m s h o l d . The p r o c e d u r e s a r e based on e s t i m a t e s o f r a t i o s f r o b samples o f s u b j e c t s . S i n c e a psychophys i c a l s c a l e i s t o r e f l e c t t h e p e r c e p t i o n of a s i n g l e p e r s o n , it i s n o t c l e a r how t o i n t e r p r e t t h e r e s u l t s o b t a i n e d from a v e r a g i n g t h e answers o f groups o f p e r s o n s . F i n a l l y , more p r e c i s e a n a l y s e s i n d i c a t e t h e e x i s t ence o f some s y s t e m a t i c d e v i a t i o n s i n S t e v e n s ' laws: t h e s c a l e v a l u e s a r e approximated w e l l by t h e laws of t h e form u ( x ) = a ( x - c ) b or u ( x ) = axb t c , b u t n o t b y
SELECTED TOPICS IN MEASUREMENT THEORY
619
t h e law u ( x ) = a x b . Thus f a r , t h e r e i s no s a t i s f a c t o r y e x p l a n a t i o n f o r t h e s e d e v i a t i o n s ( f o r some h y p o t h e s e s see K r a n t z e t a l . , 1971, p . 5 1 9 ) .
1 0 . 3 . 3 . R a t i n g s c a l e s , and c l a s s i f i c a t i o n i n t o e q u a l l y s p a c e d c a t e g o r i e s . T h i s t y p e o f s c a l i n g t e c h n i q u e s was o r i g i n a t e d b y T i t c h e n e r ( 1 9 0 5 ) , under t h e name of method of s i n g l e s t i m u l i . T h e y a r e a p p l i c a b l e t o s t i m u l i which a r e o r d e r e d on a continuum, and one may e x p e c t t h a t t h e answers w i l l r e f l e c t i n some way t h i s o r d e r i n g . The e s s e n t i a l f e a t u r e i s t h a t t h e s t i m u l i a r e p r e s e n t e d t o t h e s u b j e c t one by one. I n c a s e of r a t i n g s c a l e s , t h e s u b j e c t i s asked t o c l a s s i f y t h e s t i m u l u s i n t o one o f t h e c a t e g o r i e s , d e s c r i b e d i n t e r m s of some a d j e c t i v e s , s u c h as v e r y q u i e t , q u i e t , l o u d , v e r y l o u d , o r p l e a s a n t , n e u t r a l , unpleasant, e t c . In case of the c l a s s i f i c a t i o n i n t o e q u a l l y s p a c e d c a t e g o r i e s , t h e p e r s o n i s asked t o c l a s s i f y t h e s t i m u l u s i n t o c a t e g o r i e s which are d e s c - r i b ed by numbers, s u c h as from -3 t o i-3, from 1 t o 1 0 0 , e t c .
The e m p i r i c a l d a t a have t h e form of f r e q u e n c i e s of c l a s s e s f o r t h e same s t i m u l u s under r e p e t i t i o n . T h e y a l l o w t o estimate t h e p r o b a b i l i t y p ( k \ s ) t h a t s t i m u l u s s w i l l be c l a s s i f i e d t o c a t e g o r y numbered k ( r e g a r d l e s s o f t h e a c t u a l names of c a t e g o r i e s , one may always assume t h a t t h e y a r e numbered from 1 t o m). The s i m p l e s t s c a l i n g t e c h n i q u e , t h e s o - c a l l e d mean c a t e g o r y s c a l e , c o n s i s t s of d e f i n i n g t h e s c a l e v a l u e o f t h e s t i m u l u s s as
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A more c o m p l i c a t e d t e c h n i q u e i s based on a g e n e r a l i z a t -
i o n o f t h e law o f c o m p a r a t i v e judgment o f T h u r s t o n e . Let us c o n s i d e r , f o r e a c h s t i m u l u s s , t h e p r o b a b i l i t y d i s t r i b u t i o n f u n c t i o n o f i t s c l a s s i f i c a t i o n to v a r i o u s categories, that is P(r,s)
=
f
p(k\s).
k= 1
Here P ( r , s ) i s t h e p r o b a b i l i t y t h a t s t i m u l u s s w i l l be
c l a s s i f i e d t o one of t h e c a t e g o r i e s 1 , 2 , . . . , r . N e x t , one a s s u m e s t h a t w i t h e a c h s t i m u l u s s and w i t h t h e upper l i m i t o f c a t e g o r y r , t h e r e a r e c o n n e c t e d two random v a r i a b l e s , S and T r . The s t i m u l u s s w i l l be
c l a s s i f i e d t o one o f t h e c a t e g o r i e s 1 , 2 , . . . , r Thus , P(P,s)
= P[Tr
- S
>
if S
< Tr.
01.
A s f o r p a i r comparisons,
one assumes t h a t t h e random v a r i a b l e s S and Tr have normal d i s t r i b u t i o n , w i t h means i d e n t i f i e d respectively with the scale value of the s t i m u l u s , and t h e b o r d e r o f t h e c a t e g o r y , and some v a r i a n c e s and c o r r e l a t i o n s . To perform e f f e c t i v e s c a l i n g , i . e . to d e t e r m i n e t h e s c a l e v a l u e s o f s t i m u l i , one needs to make Some a s s u m p t i o n s a b o u t v a r i a n c e s and c o r r e l a t i o n s , a n a l o g o u s as t h e a s s u m p t i o n s i n c a s e V of Thurstone. An i m p o r t a n t s p e c i a l c a s e o f T h u r s t o n e ' s t e c h n i q u e i s t h e s o - c a l l e d method o f s u c c e s s i v e i n t e r v a l s , s u g g e s t e d by Adams and Messick ( 1 9 5 8 ) . One assumes h e r e t h a t t 2, between s u c c e s s i v e c l a s s e s t h e boundaries t
.. .
SELECED TOPICS IN MEASUREMENT THEORY
62 1
are n o t random; t h e s t i m u l u s i s r e p r e s e n t e d , as b e f o r e , by a random v a r i a b l e S ( d i s c r i m i n a n t p r o c e s s ) , w i t h normal d i s t r i b u t i o n . The mean o f t h i s d i s t r i b u t i o n i s t h e s c a l e v a l u e of t h e s t i m u l u s , and t h e v a r i a n c e i s unknown. Thus, t h e p r o b a b i l i t i e s p ( r l s ) a r e e q u a l t o p r o b a b i l i t i e s of o b s e r v i n g a v a l u e S i n t h e i n t e r v a l (tr+J Y that is +
where u(s) and 6S are t h e mean and s t a n d a r d d e v i a t i o n of t h e d i s c r i m i n a n t p r o c e s s S . Adams and Messick (1958) gave n e c e s s a r y and s u f f i c i e n t
c o n d i t i o n s under which t h e r e e x i s t common v a l u e s ( f o r a l l s t i m u l i ) t l , t 2 , . . . o f b o u n d a r i e s of c a t e g o r i e s . These c o n d i t i o n s a r e e x p r e s s e d t h r o u g h p r o b a b i l i t i e s p ( r [ s ) of categories. I n c o n n e c t i o n w i t h s c a l i n g t e c h n i q u e s based on e q u a l l y p l a c e d c a t e g o r i e s and t e c h n i q u e s based on magnitude s c a l i n g one s h o u l d mention a v e r y i n t e r e s i n g phenomenon d i s c o v e r e d when t h e s e t e c h n i q u e s are a p p l i e d t o v a r i o u s continua. S t e v e n s ( 1957) i n t r o d u c e d a p a r t i t i o n of p s y c h o p h y s i c a l c o n t i n u a i n t o p r o t e t i c and m e t a t e t i c . The f i r s t c a t e g o ry comprises c o n t i n u a where t h e c o r r e s p o n d i n g p h y s i c a l continuum a l l o w s measurement on a r a t i o s c a l e ( e , g . l e n g t h , w e i g h t , e t c . ) . The second c a t e g o r y , o f m e t a t e t i c c o n t i n u a , comprises t h o s e where t h e p h y s i c a l c o n t i n u u m ’ a l l o w s measurement o n l y on a n i n t e r v a l s c a l e ( e . g . t e m -
622
CHAPTER 5
perature, pitch, etc. ) It a p p e a r s ( s e e S t e v e n s and G a l a n t e r 1957) t h a t when one compares t h e r e s u l t s o f s c a l i n g b y two t e c h n i q u e s
(magnitude s c a l i n g , and c a t e g o r y r a t i n g s ) , t h e y a r e d i f f e r e n t f o r p r o t e t i c and m e t a t e t i c c o n t i n u a . F o r t h e l a t t e r t h e dependence i s l i n e a r , w h i l e f o r p r o t e t i c cont i n u a we have t h e f o l l o w i n g p i c t u r e : i f t h e s c a l e v a l u e s o b t a i n e d by magnitude s c a l i n g a r e r e p r e s e n t e d b y t h e h o r i z o n t a l a x i s , and t h o s e by c a t e g o r y s c a l i n g - b y t h e v e r t i c a l a x i s , t h e r e a p p e a r s a curve w i t h "negative a c c e l e r a t i o n " , as on F i g . 1 0 . 3 .
magnitude r a t i o s c a l e Fig. 10.3. T h i s phenomenon a p p e a r s i n a l l c a s e s of p r o t e t i c con-
t L n u a , and i t s n a t u r e has n o t y e t been e x p l a i n e d adequately. One o f t h e most s e r i o u s problems c o n n e c t e d w i t h c a t e g o r y s c a l i n g i s t h a t o f i n v a r i a n c e o f t h e s c a l e under change of e x p e r i m e n t a l c o n d i t i o n s . It t u r n s o u t t h a t t h e s c a l e depends i n some r e l a t i v e l y low d e g r e e on t h e number of c a t e g o r i e s , b u t i s h i g h l y s e n s i t i v e t o t h e changes i n t h e s e t o f s t i m u l i and o r d e r o f p r e s e n t a t i o n . We
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have h e r e t h e e f f e c t t h a t t h e s c a l e v a l u e s i n c r e a s e more s t e e p l y i n t h e neighbourhood of t h o s e i n t e n s i t i e s which a r e p r e s e n t e d more f r e q u e n t l y . A q u a l i t a t i v e e x p l a n a t i o n , which t a k e s i n t o a c c o u n t t h e
m o t i v a t i o n a l f a c t o r s , i s r e l a t i v e l y s i m p l e : i f t h e maj o r i t y of p r e s e n t e d s t i m u l i come from a fragment of t h e s c a l e c o m p r i s i n g r e l a t i v e l y few c a t e g o r i e s ( s o t h a t n e i g h b o u r i n g c a t e g o r i e s a r e empty, or have o n l y few c l a s s i f i c a t i o n s ) , t h e n t h e n a t u r a l tendency t o p r e c i s e d i s c r i m i n a t i o n w i l l c a u s e t h e usage of t o o many c a t e g o r i e s . T h i s w i l l l e a d t o a s t e e p e r i n c r e a s e of s c a l e val u e s ( h e n c e more e x a c t d i s c r i m i n a t i o n ) i n t h e neighbourhood o f more f r e q u e n t s t i m u l i . T h u s f a r , t h e r e i s no s a t i s f a c t o r y t h e o r y which would t a k e i n t o a c c o u n t b o t h t h e m o t i v a t i o n a l f a c t o r s and t h e e f f e c t s of f r e q u e n c y o f s t i m u l i on s c a l e v a l u e s . The problem l i e s i n a c o n s t r u c t i o n o f a n a d e q u a t e c h o i c e model f o r c l a s s i f i c a t i o n of s t i m u l i , which would l e a d t o s c a l e v a l u e s t h a t a r e i n v a r i a n t under v a r i a t i o n s and m o d i f i c a t i o n s of e x p e r i m e n t a l p r o c e d u r e s . A s d i s t i n c t from measurement t h e o r y , s c a l i n g t h e o r y
l a c k s i n most c a s e s a s a t i s f a c t o r y t h e o r e t i c a l foundat i o n s , which would a c c o u n t a d e q u a t e l y for v a r i o u s phenomena o b s e r v e d i n u s i n g d i f f e r e n t s c a l i n g t e c h n i q u e s . I n a b s t r a c t p r e s e n t a t i o n , t h e main i s s u e s h e r e a r e a s f o l l o w s . It i s assumed t h a t t o e a c h s t i m u l u s i n t e n s i t y t h e r e c o r r e s p o n d s a s c a l e v a l u e , which r e p r e s e n t s t h e p r o p e r t i e s of p e r c e p t i o n of t h i s s t i m u l u s . Each s c a l i n g t e c h n i q u e u s e s t h e d a t a e l i c i t e d from s u b j e c t s ' respons e s , s p e c i f i c for a g i v e n t e c h n i q u e . The problem l i e s ,
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most g e n e r a l l y ,
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i n c o n s t r u c t i o n of a model which would
c o n n e c t t h e i n t e n s i t i e s of s t i m u l i , t h e i r v a l u e s on p s y c h o p h y s i c a l s c a l e , and p r o b a b i l i t i e s of d i f f e r e n t responses. I n o t h e r words, one wants a model which would l e a d t o p r o b a b i l i t i e s of r e s p o n s e s for s t i m u l i o f a g i v e n i n t e n s i t y ( i . e . p r o b a b i l i t i e s p ( r j s ) i n cases of category r a t i n g , p r o b a b i l i t i e s p ( a , b ) i n c a s e of p a i r comp a r i s o n s , e t c . ) . These p r o b a b i l i t i e s s h o u l d depend on t h e v a l u e s on p h y c h o p h y s i c a l s c a l e , i n a way p e r m i t t i n g t h e e s t i m a t i o n of t h e s e v a l u e s on t h e b a s i s of e s t i m a t es of p r o b a b i l i t i e s . Moreover, a s a t i s f a c t o r y model should be u n i v e r s a l , i n t h e sense t h a t i t should lead t o t h e same p s y c h o p h y s i c a l s c a l e , r e g a r d l e s s o f t h e s c a l i n g t e c h n i q u e used. Some a t t e m p t s i n t h i s d i r e c t i o n a r e due t o Luce (1959), who i n t r o d u c e d v a r i o u s c h o i c e models. I n p a r t i c u l a r , f o r p s y c h o p h y s i c a l phenomena ( n o t o n l y s c a l i n g , b u t a l s o d e t e c t i o n and d i s c r i m i n a t i o n ) he i n t r o d u c e d t h e concept o f g e n e r a l i z a t i o n f u n c t i o n , which e x p r e s s e s t h e p r o b a b i l i t y t h a t s t i m u l u s s w i l l appear " l i k e s t i mulus t " . The second n o t i o n o f Luce i s t h a t of b i a s f u n c t i o n , d e f i n e d on t h e s e t of r e s p o n s e s , d e s c r i b i n g t h e s y s t e m a t i c d i s t o r t i o n s o f r e s p o n s e s . Thus, Luce assumes two p s y c h o l o g i c a l mechanisms which d e t e r m i n e t h e response: w i t h a c e r t a i n p r o b a b i l i t y , stimulus s i s p e r c e i v e d as s t i m u l u s t ( " g e n e r a l i z e d t o s t i m u l u s t " ) , w i t h v a l u e u ( t ) on t h e a p p r o p r i a t e p s y c h o p h y s i c a l s c a l e . T h i s p e r c e p t i o n , i n t u r n , l e a d s t o r e s p o n s e r, d i s t o r t e d by bias f u n c t i o n b . Under c e r t a i n a s s u m p t i o n s one can d e r i v e t h e o r e t i c a l
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f o r m u l a e for p r o b a b i l i t y of r e s p o n s e r , g i v e n t h e s t i mulus s . However, w i t h few e x c e p t i o n s , t h e s e f o r m u l a e do n o t a l l o w e s t i m a t i o n o f p s y c h o p h y s i c a l f u n c t i o n . Moreover, L u c e ' s model d o e s n o t t a k e i n t o a c c o u n t s u c h f a c t o r s as i n f l u e n c e of i n s t r u c t i o n s , o r d e r and f r e q u ency of p r e s e n t a t i o n s , e t c . , s o t h a t t h e q u e s t i o n o f a n a d e q u a t e model i s s t i l l open. It i s n o t even c l e a r what i s t h e n e c e s s a r y number o f p s y c h o l o g i c a l mechanisms i n v o l v e d ( i . e . if t h e two mechanisms of Luce, o f gener a l i z a t i o n and b i a s , are s u f f i c i e n t ) .
1 0 . 4 . P e r c e p t i o n o f s c a l e o f s o c i a l power
Most t y p i c a l examples o f s c a l i n g concern t h e c o n t i n u a where t h e s t i m u l u s i n t e n s i t y i s d e t e r m i n e d b y i t s phys i c a l p r o p e r t i e s , and t h e s c a l e v a l u e s r e f l e c t p e r c e p t i o n o f t h e s e p r o p e r t i e s . The s i t u a t i o n becomes more c o m p l i c a t e d , b o t h t h e o r e t i c a l l y and e m p i r i c a l l y , when t h e u n d e r l y i n g continuum has a more " e l u s i v e " c h a r a c t e r I n t h i s s e c t i o n , we s h a l l p r e s e n t a n example o f s u c h s i t u a t i o n , when t h e p e r c e p t i o n c o n c e r n s t h e s c a l e o f s o c i a l power, and i n p a r t i c u l a r , t h e c o n d i t i o n s which tend t o d i s t o r t t h i s perception. O b v i o u s l y , t h e s t u d y o f d i s t o r t i o n s r e q u i r e s t h e knowl e d g e o f " t r u e " powers, t o be compared w i t h t h e a c t u a l c h o i c e s by t h e p e r c e i v i n g s u b j e c t s . T h i s r e q u i r e m e n t f o r c e d t h e r e s t r i c t i o n of c o n s i d e r a t i o n s t o some i d e a l i z e d s i t u a t i o n s , namely t o v o t i n g committees d e s c r i b e d i n terms of t h e i r c o n s t i t u t i o n s , where t h e power i n d e x of e v e r y member c o u l d be computed a p r i o r i . These i n d i -
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ces p r o v i d e t h e r e f e r e n c e base i n t h e experiments. The r e s u l t s i n d i c a t e t h a t p e r c e p t i o n of c e r t a i n power yielding privileges, e.g. the r i g h t t o veto, tend t o be s e r i o u s l y d i s t o r t e d .
1 0 . 4 . 1 . The c o n c e p t o f power. The concept o f power i s one of t h e c e n t r a l n o t i o n s i n t h e s o c i a l s c i e n c e s . A t t h e same t i m e , it a p p e a r s t o b e one o f t h e more e l u s i v e c o n c e p t s , and i n most c a s e s , d e f i e s t h e a t t e m p t s o f g i v ing it a precise d e f i n i t i o n . T h i s s e c t i o n w i l l p r o v i d e a b r i e f r e v i e w of some o f t h e
d e f i n i t i o n s of s o c i a l power, and i n d i c a t e t h e r e a s o n s why t h e r e seems t o be o n l y one s u g g e s t e d t h u s f a r , which i s a p p l i c a b l e f o r t h e p u r p o s e of t h e s t u d y o f d i s t o r t i o n s o f p e r c e p t i o n of s o c i a l power, namely t h e v o t i n g power i n d e x o f Shapley and Shubik (1954). Before p r e s e n t i n g t h i s i n d e x , it i s worth while t o o u t l i n e some o t h e r a t t e m p t s t o d e f i n e power. The non-mathematical a p p r o a c h e s t o power y i e l d n o t h i n g which c o u l d be u s e f u l for t h e p r e s e n t p u r p o s e , i n t h e s e n s e t h a t t h e n u m e r i c a l i n d e x may be e l i c i t e d from them, n o r can one p r e c i s e l y d e t e r m i n e who i s more p o w e r f u l i n a given s i t u a t i o n . To i l l u s t r a t e t h e d i f f i c u l t i e s e n c o u n t e r e d i n mathematic a l d e f i n i t i o n s , l e t us c o n s i d e r two p e r s o n s , A and B , who may be " f o r " o r " a g a i n s t " a c e r t a i n i s s u e . The f i n a l d e c i s i o n depends a l s o on some o t h e r f a c t e o r s , i . e . o t h e r members of t h e group, o r some random e f f e c t s may i n f l u e n c e t h e outcome, e t c .
SELECTED TOPICS INMEASUREMENT THEORY
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The s i t u a t i o n may be r e p r e s e n t e d i n t h e form o f t h e following matrix. P r o b a b i l i t y of a p o s i t i v e outcome:
for
against
p2 1
p22
I n t h i s c a s e , a l l d e f i n i t i o n s ( s e e Wittman 1 9 7 7 f o r a r e v i e w of t h e l i t e r a t u r e on t h e s u b j e c t ) a g r e e : A i s more p o w e r f u l t h a n B y i f P12
>
p21.
T h i s , o f c o u r s e , i s a n a t u r a l and e a s i l y d e f e n s i b l e de-
f i n i t i o n : A i s more p o w e r f u l t h a n B y i f t h e p r o b a b i l i t y of p o s i t i v e outcome i s g r e a t e r if A i s for and B i s a g a i n s t i t , t h a n if A i s a g a i n s t and B i s f o r . I n more g e n e r a l s i t u a t i o n s , t h e m a t r i x a l l o w s for more a c t i o n s of b o t h A and B y and t h e d e f i n i t i o n s a r e based on s u c h p r o p e r t i e s as t h e a b i l i t y of one p e r s o n t o res t r i c t t h e r a n g e of p o s s i b i l i t i e s o f t h e o t h e r , r e s t r i c t h i s outcomes, o r on a b i l i t y t o harm. F o r t h e r e v i e w o f l i t e r a t u r e s e e Wittman 1977; f o r o t h e r s o u r c e m a t e r i a l , see f o r i n s t a n c e Dahl 1957, H a r s a n y i 1956, March 1957, or Nagel 1975. A l l t h e s e d e f i n i t i o n s a r e d e f e n s i b l e , b u t for p r a c t i c a l
a p p l i c a t i o n s t h e c r u c i a l p o i n t i s t o de’;ermine t h e res-
628
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p e c t i v e p r o b a b i l i t i e s P i j . I n d e e d , even i n t h e s i m p l e s t c a s e above, i f A and B a r e members o f a l a r g e r d e c i s i o n making body, i t i s n o t c l e a r how t o d e t e r m i n e P ij
'
Thus, for t h e p u r p o s e o f t h e p r e s e n t s t u d y , where t h e c r u c i a l p o i n t was t o a s s e s s t h e a p r i o r i powers, which c o u l d be compared w i t h t h e c h o i c e s o f t h e s u b j e c t , t h e c o n s i d e r a t i o n s had t o be r e s t r i c t e d t o v o t i n g committees.
V o t i n g power. Assume t h a t we have a v o t i n g bod y , e . g . a committee, board o f d i r e c t o r s , e t c . We a s s u 10.4.2.
me t h a t e a c h member may v o t e " y e s " o r "no" , t h a t i s , he may be " f o r ' ' o r " a g a i n s t " some i s s u e , and t h a t t h e r e e x i s t s a w e l l d e f i n e d r u l e which s p e c i f i e s how t h e v o t es o f d i f f e r e n t members a r e a g g r e g a t e d i n t o t h e f i n a l decision. The s i m p l e s t example o f s u c h a r u l e i s t h e m a j o r i t y
r u l e , or 2 / 3 m a j o r i t y r u l e , when e a c h member has j u s t one v o t e , and t h e i s s u e p a s s e s when t h e r e i s a t l e a s t
a c e r t a i n number of v o t e s " y e s " . I n s u c h a c a s e , o f c o u r s e , a l l members have t h e same a p r i o r i power because o f symmetry. N a t u r a l l y , i n a c t u a l s i t u a t i o n s , v a r i o u s members o f s u c h committees may have d i f f e r e n t r e a l powers, depending on who t h e y a r e , whom t h e y know, how p e r s u a s i v e t h e y may b e , e t c .
These a s p e c t s , however, are n o t e a s i l y r e p r e s e n t a b l e n u m e r i c a l l y , and t h e r e f o r e t h e c o n s i d e r a t i o n s had t o b e r e s t r i c t e d t o t h e a p r i o r i power connected w i t h t h e p o s i t i o n w i t h i n a committee, r a t h e r t h a n w i t h t h e a c t -
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u a l p e r s o n who o c c u p i e s t h i s p o s i t i o n . The c a s e when t h e r e i s a s y m m e t r y i s of l i t t l e i n t e r e s t . However, t h e r e a r e c a s e s of i n t e r e s t when t h e r e i s no symmetry, and t h e a p r i o r i v o t i n g powers o f v a r i o u s members a r e d i f f e r e n t . I n p a r t i c u l a r , t h e f o l l o w i n g
f e a t u r e s a r e common i n p r a c t i c e , and t h e r e f o r e w o r t h studying:
-- v a r i o u s members may have d i f f e r e n t numbers o f v o t e s ;
---
some members may have v e t o power; v o t i n g may b e i n d i r e c t : for i n s t a n c e , t h e r e may b e s e v e r a l subcommittees, and t h e f i n a l d e c i s i o n depends on t h e r e s u l t s of v o t i n g s i n t h e subcommittees; -- a p e r s o n may b e l o n g t o s e v e r a l subcommittees, and t h e r e f o r e may i n e f f e c t have s e v e r a l i n d i r e c t v o t e s . The q u e s t i o n i s t o d e v i s e a measure which would adequat e l y r e p r e s e n t t h e v o t i n g power of d i f f e r e n t members of t h e committee, as g u a r a n t e e d t o them by t h e c o n s t i t u t i o n of it. T h i s q u e s t i o n was s o l v e d by Shapley and Shubik
(1954).
The i n d e x of s o c i a l power d e v i s e d by them i s based on t h e concept of minimal winning c o a l i t i o n , i . e . minimal group of p e r s o n s who may e n s u r e t h e f i n a l d e c i s i o n " y e s " r e g a r d l e s s of t h e v o t e s of t h e r e m a i n i n g members. We p r e s e n t t h e d e f i n i t i o n o f Shapley and Shubik i n t h e
form which a t t h e same t i m e p r o v i d e s a method o f numeri c a l d e t e r m i n a t i o n o f v o t i n g power ( s e e , a l s o Kemeny, S n e l l and Thompson 1957, o r Luce and R a i f f a 1957). C o n s i d e r a f i x e d p e r m u t a t i o n o f v o t e r s , and imagine t h a t t h e members v o t e ' ' y e s " one a f t e r a n o t h e r , accord-
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i n g t o t h i s p e r m u t a t i o n . Then t h e r e w i l l b e a moment when t h e i s s u e w i l l p a s s , even b e f o r e a l l t h e v o t e s a r e c a s t . The l a s t member, whose v o t e was d e c i s i v e i s c a l l ed t h e p i v o t i n t h i s p e r m u t a t i o n . O b v i o u s l y , t h e p i v o t p l u s a l l members who p r e c e d e him i n t h e g i v e n permutat i o n form a minimal winning c o a l i t i o n . Thus, e a c h p e r m u t a t i o n has e x a c t l y one p i v o t , and t h e power of a v o t i n g member i s d e f i n e d a s t h e number o f p e r m u t a t i o n s i n which he a p p e a r s as p i v o t . For normal-i z a t i o n , one d i v i d e s t h e t o t a l number o f p e r m u t a t i o n s i n which x i s a p i v o t by n! ( t h e t o t a l number o f permut a t i o n s of n members o f a c o m m i t t e e ) . Thus, i f d ( x ) i s t h e v o t i n g power i n d e x o f x , t h e n
# of p e r m u t a t i o n s i n which x i s a p i v o t
d(x) =
-
n! To u s e a s i m p l e example, c o n s i d e r t h e c a s e o f a committ e e w i t h f o u r members, A , B , C and D . Assume t h a t A has 2 v o t e s , and t h a t o t h e r members have one v o t e e a c h , s o t h a t t h e r e a r e a l t o g e t h e r 5 v o t e s . The i s s u e i s dec i d e d by a s i m p l e m a j o r i t y o f v o t e s - a t l e a s t t h r e e v o t e s "yes" , b u t w i t h t h e a d d i t i o n a l p r o v i s i o n t h a t B has v e t o power: h i s v o t e must be " y e s " , or o t h e r w i s e t h e i s s u e does n o t p a s s . The l i s t of all 2 4 p e r m u t a t i o n s , w i t h p i v o t a l members u n d e r l i n e d , i s g i v e n i n t h e t a b l e below.
For example, i f t h e p e r m u t a t i o n i s A B C D , t h e p i v o t i s B : a f t e r A g i v e s two v o t e s , and B g i v e s h i s one v o t e , t h e i s s u e pasSes, s i n c e t h e r e a r e a l r e a d y t h r e e v o t e s
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" y e s " and v o t e o f B among them. S i m i l a r l y , i n t h e p e r m u t a t i o n B C D A , t h e p i v o t i s D -- he c a s t s t h e t h i r d v o t e , and B h a s a l r e a d y v o t e d l t y e s l l . T a b l e of p e r m u t a t i o n s w i t h p i v o t a l members A B C D
B A C D
C A B D
D A B C
A B D C
B A D C
C A D B
D A C E
A C E D
B C A D
C B A D
D B A C
A C D B
B C D A
C B D A
D B C A
A D B C
B D A C
C D A B
D C A B
A D C B
B D C A
C D B A
D C B A
A l t o g e t h e r , A a p p e a r s 6 t i m e s as a p i v o t , B - 1 4 t i m e s , and C and D e a c h 2 t i m e s . The powers a r e t h e r e f o r e d ( A ) = 6 / 2 4 = 1/4,
d ( B ) = 1 4 / 2 4 = 7/12
d ( C ) = d ( D ) = 2 / 2 4 = 1/12.
A s may be s e e n , d e s p i t e h a v i n g o n l y t w i c e as many v o t e s
as C o r D , member A has t h r e e times as much power. A l s o , t h e r i g h t o f v e t o g i v e s v e r y h i g h power: B h a s o n l y one v o t e , s i m i l a r l y a s C and D , b u t s e v e n t i m e s as much power.
1 0 . 4 . 3 . The h y p o t h e s e s . The e x p e r i m e n t s i n v o l v e d subj e c t i v e e v a l u a t i o n s o f s o c i a l power o f d i f f e r e n t memb e r s o f h y p o t h e t i c a l committees p r e s e n t e d t o t h e subj e c t . Each committee was d e s c r i b e d by s p e c i f y i n g i t s c o n s t i t u t i o n , i . e . by c h a r a c t e r i z i n g how t h e v o t e s o f
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d i f f e r e n t members i n t e r v e n e i n t h e f i n a l d e c i s i o n , To f a c i l i t a t e g r a s p i n g t h e power s t r u c t u r e o f a committee, i t was a l s o p r e s e n t e d i n form o f a diagramme ( s e e t h e following s e c t i o n ) . The s u b j e c t was asked t o a s s e s s which o f t h e two spec i f i e d members of a committee has more power, and a l s o g i v e h i s numerical e v a l u a t i o n of t h e r a t i o of t h e i r powers. I n t h e second s t a g e o f t h e e x p e r i m e n t , t h e committees were p r e s e n t e d i n p a i r s , and t h e s u b j e c t was a s k e d t o choose t h a t committee, which he f e l t more l i k e l y t o v o t e i n h i s f a v o u r . The c o n t r o l l e d f a c t o r i n t h i s s t a g e o f experiment was t h e i n f o r m a t i o n about t h e v o t e s o f some ( b u t n o t a l l ) members o f t h e commitee. Thus, i n e f f e c t , t h e c h o i c e was made n o t s o much between commit t e e s , as between p a i r s c o n s i s t i n g o f a commitee and s p e c i f i c i n f o r m a t i o n ( p o s s i b l y none) a b o u t t h e v o t e s o f some o f i t s members. To f o r m u l a t e t h e b a s i c h y p o t h e s e s , suppose t h a t B are two members o f t h e committee which are t o a s s e s s e d , and l e t d(A) and d ( B ) be t h e i r s o c i a l F u r t h e r , l e t p(A,B) be t h e p r o b a b i l i t y t h a t t h e
A and be
powers. subject
w i l l p o i n t o u t A as more p o w e r f u l i n t h e p a i r ( A , B ) , and l e t r ( A , B ) be t h e s u b j e c t i v e l y e v a l u a t e d r a t i o d ( A ) / d ( B ) o f powers o f A and B .
HYPOTHESIS 1. P e r c e p t i o n o f s o c i a l power i s o r d e r - c o r r e c t , i n t h e s e n s e t h a t p(A,B) 7 3 i f and o n l y i f we have d(A) '7 d ( B ) .
-
HYPOTHESIS 2 . T h e r e e x i s t s a t h r e s h o l d i n p e r c e p t i o n
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of s o c i a l power, i n t h e s e n s e t h a t p(A,B) i s s i g n i f i c a n t l y greater t h a n $ only i f t h e d i f f e r e n c e d(A) - d ( B ) exceeds t h e t h r e s h o l d . HYPOTHESIS 3. I n g e n e r a l , p ( A , B ) t e n d s to be l a r g e r for l a r g e r v a l u e s o f t h e d i f f e r e n c e d(A) - d ( B ) , but the r e l a t i o n i s n o t f u n c t i o n a l : p(A,B) depends n o t o n l y on d(A) - d ( B ) , b u t a l s o on t h e complexity o f t h e committee. HYPOTHESIS 4. P e r c e p t i o n of s o c i a l power i s b i a s e d , i n t h e s e n s e t h a t t h e e x p e c t a t i o n of r ( A , B ) t o the r a t i o d(A)/d(B).
i s not equal
HYPOTHESIS 5 . The b i a s t e n d s t o be l a r g e r f o r some t y p e s o f s o c i a l r o l e s ; i n p a r t i c u l a r , power due t o t h e r i g h t o f v e t o , and t h e power due t o t h e r i g h t o f c a s t i n g more t h a n one v o t e , t e n d t o be u n d e r e s t i m a t e d . T o f o r m u l a t e f u r t h e r h y p o t h e s e s , l e t Q ( C , x , J ) be t h e
p r o b a b i l i t y t h a t t h e committee C w i l l d e c i d e i n f a v o u r o f some i s s u e , g i v e n t h a t t h e members s p e c i f i e d i n J w i l l v o t e as s p e c i f i e d , and t h a t t h e r e m a i n i n g members w i l l v o t e randomly and i n d e p e n d e n t l y , w i t h p r o b a b i l i t y x o f v o t e Ityes". To u s e a s i m p l e example, l e t C s t a n d for t h e committee d e s c r i b e d i n t h e preceding s e c t i o n ( c o n s i s t i n g of f o u r members A , B , C and D , o f which A has two v o t e s , B has v e t o power, and t h e i s s u e i s d e c i d e d by s i m p l e m a j o r i t y ) . I f J s t a n d s f o r t h e i n f o r m a t i o n "A w i l l v o t e a g a i n s t t h e i s s u e " , t h e n Q ( C , x , J ) = x3, s i n c e t h e i s s u e may p a s s o n l y i f B , C and D w i l l a l l v o t e " y e s " ( a s t h e r e a r e a l r e a d y two v o t e s a g a i n s t ) . S i m i l a r l y , i f J
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stands f o r t h e information t h a t B w i l l vote "yes", then Q(C,x,J) = x + ( 1 - x ) x 2 I n d e e d , i f B v o t e s " y e s " , t h e i s s u e w i l l pass i f e i t h e r A ( w i t h 2 v o t e s ) w i l l v o t e l r f o r l r ( p r o b a b i l i t y x ) , i n which c a s e C and D may v o t e i n an a r b i t r a r y way, or e l s e , i f A v o t e s no ( p r o b a b i l i t y 1-x), and b o t h C and D v o t e " y e s " ( p r o b a b i l i t y x 2 ) .
.
L e t now ( C , J ) and ( C 1 , J 1 ) be two committees w i t h g i v e n i n f o r m a t i o n about t h e v o t e s . Denote by p ( C , J ; C ' , J ! ) t h e p r o b a b i l i t y t h a t t h e s u b j e c t w i l l p o i n t o u t ( C , J ) as more f a v o u r a b l e committee ( t h e one he f e e l s would be more l i k e l y t o v o t e i n h i s f a v o u r ) . HYPOTHESIS 6 . I n a s s e s s i n g committees a c c o r d i n g t o t h e i r " f a v o u r a b l e n e s s " , t h e s u b j e c t s t e n d t o e v a l u a t e them a s i f assuming t a c i t l y t h a t t h e members o f t h e committee a r e u n p r e j u d i c e d and do n o t form c l i q u e s : p ( C , J ; C 1 , J 1 ) > 4 i f and o n l y i f Q ( C , ; , J ) Q(Ct,$,J1), and i n general, p ( C , J ; C 1 , J 1 ) i s an i n c r e a s i n g f u n c t i o n of t h e d i fference Q(C,$,J) - Q(C1,$,J1).
>
The c o n s i s t e n c y o f t h e c h o i c e s w i t h t h e o r d e r i n d u c e d by Q ( C , x , J ) c o r r e s p o n d s t o t h e e x p e c t a t i o n o f t h e l a c k of c l i q u e s , s i n c e Q ( C , x , J ) i s c a l c u l a t e d under t h e a s s u m p t i o n o f independence o f v o t e s o f members o f t h e committee. The c o n s i s t e n c y w i t h Q ( C , x , J ) f o r t h e p a r t i c u l a r value x = $ corresponds t o t h e expectation t h a t members o f t h e committee w i l l be u n p r e j u d i c e d ( w i l l b e e q u a l l y l i k e l y v o t i n g " f o r " as " a g a i n s t " ) . HYPOTHESIS 7 . The c h o i c e p r o b a b i l i t i e s p ( C , J ; C ' , J 1 ) a r e c o n s i s t e n t i n t h e s e n s e o f weak s t o c h a s t i c t r a n s i t i v i t y ( s e e for i n s t a n c e Coombs, Dawes and Tversky
1970).
SELECTED TOPICS IN MEASUREMENT THEORY
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HYPOTHESIS 8 . The c h o i c e p r o b a b i l i t i e s s a t i s f y t h e cond i t i o n of s t r o n g s t o c h a s t i c t r a n s i t i v i t y , i f r e s t r i c t e d t o c h o i c e s from t r i p l e t s ( C , J ) , ( C l , J ' ) , ( C ' ' , J ' ' ) with c = C ' = C " , i . e . i f e v a l u a t i o n s concern t h e same commit e e under d i f f e r e n t i n f o r m a t i o n . HYPOTHESIS 9. The p e r c e p t i o n o f s o c i a l power i s b i a s e d i n the sense t h a t there exist t r i p l e t s ( C , J ) , ( C 1 , J 1 ) , -( C ' ' , J ' ' ) of s u f f i c i e n t l y complex s t r u c t u r e , s u c h t h a t t h e c h o i c e p r o b a b i l i t i e s do n o t s a t i s f y s t r o n g s t o c h a s t i c transitivity. The e x p e r i m e n t s r e p o r t e d below c o n c e r n o n l y t h e hypot h e s e s 1-5; t h e r e m a i n i n g h y p o t h e s s e s 6-9 , as w e l l as t h e H y p o t h e s i s 1 0 below, r e q u i r e d more e l a b o r a t e d experiments. The n a t u r a l c o n t i n u a t i o n o f t h e r e s e a r c h on h y p o t h e s e s
1-9 i s t o s t u d y t h e p e r c e p t i o n o f "composite" s o c i a l power, due t o membership i n more t h a n one decision-maki n g body. The e x p e r i m e n t s here c o n c e r n c h o i c e s made b y g r o u p s o f s u b j e c t s f a c i n g t h e t a s k o f c h o o s i n g one o f s e v e r a l members of h y p o t h e t i c a l v o t i n g committees. Each commit t e e makes t h e " v o t i n g " , where t h e member s e l e c t e d b y t h e group ( " b r i b e d " ) v o t e s ' ' y e s " , and t h e r e m a i n i n g v o t e s a r e d e t e r m i n e d b y some chance mechanism. I f t h e v o t e comes o u t f a v o u r a b l y f o r t h e s u b j e c t s , t h e y r e c e i v e some p r i z e . The chance mechanism h e r e i s a c o n t r o l v a r i a b l e : i n som e e x p e r i m e n t s i t i s simply t o s s i n g a c o i n f o r e a c h member s e p a r a t e l y , c o r r e s p o n d i n g t o e x p e c t a t i o n of l a c k o f p r e j u d i c e and l a c k of c l i q u e s . I n o t h e r v e r s i o n s o f
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e x p e r i m e n t , t h e i n f o r m a t i o n a b o u t t h e c l i q u e s and t h e p r o b a b i l i t i e s a r e fuzzy. The c r u c i a l p o i n t i s t h a t t h e committees a r e n o t d i s -
j o i n t , w h i l e t h e c h o i c e o f t h e member who v o t e s ''YeS" for t h e group i s t o be made for a l l committees a t o n c e . The s u b j e c t s f a c e t h e r e f o r e a dilemma, s i n c e a c a n d i d a t e may p l a y more i m p o r t a n t r o l e i n some committees, and a l e s s i m p o r t a n t r o l e i n o t h e r committees. The h y p o t h e s i s t e s t e d a s s e r t s t h a t s o c i a l power, a s
measured by t h e i n d e x of Shapley and Shubik, i s e a s i e r t o p e r c e i v e t h a n p r o b a b i l i t i e s o f complex e v e n t s , and t h a t t h e s o c i a l power r e s u l t i n g from membership i n s e v e r a l v o t i n g committees i s p e r c e i v e d a d d i t i v e l y ( i f t h e i s s u e s t o be d e c i d e d by a l l committees a r e of t h e same i m p o r t a n c e ) . To f o r m u l a t e t h e h y p o t h e s i s p r e c i s e l y , l e t C 1' C2>". s t a n d for t h e committees. Suppose t h a t t h e c h o i c e i s made between p e r s o n s A and B , e a c h of them belongand i n g t o a l l t h e committees.. L e t d l ( A ) , d 2 ( A ) , dl(B), d2(B), be t h e s o c i a l powers of A and B i n t h e committees C1, C2, , and l e t p l ( A ) , p 2 ( A ) , and be t h e p r o b a b i l i t i e s of v o t e " y e s " b y pl(B), p2(B), committees C 1, C2, under t h e scheme o f d e t e r m i n i n g t h e v o t e s used i n p a r t i c u l a r v e r s i o n o f t h e e x p e r i m e n t . Thus, i f J = "member A w i l l v o t e y e s " , and t h e chance mechanism b e i n g i n d e p e n d e n t c o i n t o s s i n g , we have pi(A) = Q ( C i , 2, J).
... ...
...
...
... ...
...
Finally, l e t D(A) = dl(A) t d2(A) t and D(B) = d l ( B ) t d2(B) t be t h e sums o f s o c i a l powers o f A and B i n a l l cominittees. L e t P ( A ) = pl(A) t p 2 ( A ) +
...
...
SELECTED TOPICS INMEASUREMENT THEORY
and P ( B ) = p l ( B ) t p 2 ( B ) t
...
631
b e t h e sums o f p r o b a b i l i -
t i e s o f p o s i t i v e d e c i s i o n s . Thus, P ( A ) and P ( B ) a r e prop o r t i o n a l t o t h e expected t o t a l p r i z e s f o r a l l v o t i n g s , i f e a c h committee v o t e s f o r t h e same p r i z e , and i f A ( r e s p . B ) i s chosen b y t h e group. HYPOTHESIS 1 0 . I n c h o o s i n g between members A and B y t h e d e c i s i v e q u a n t i t i e s are t h e i r s o c i a l powers D ( A ) and D ( B ) , and n o t t h e e x p e c t e d p r i z e s P ( A ) and P ( B ) . As s i b l e e x c e p t i o n o c c u r s i n t h e c a s e when (1) t h e probab i l i t i e s o f v o t e s " y e s ' ' are g i v e n e x p l i c i t l y , ( 2 ) memb e r s do n o t form complex c l i q u e s , and ( 3 ) a l l committees are o f a v e r y s i m p l e s t r u c t u r e . T h i s h y p o t h e s i s may be t e s t e d by o b s e r v i n g c h o i c e s b e t -
ween A and B i n s i t u a t i o n s when D ( A ) 7 D ( B ) and P ( A ) < P ( B ) or v i c e v e r s a . The s o c i a l power i s s t r o n g l y c o r r e l a t e d w i t h t h e probab i l i t y o f p o s i t i v e f i n a l d e c i s i o n ( g i v e n t h a t t h e member i n question votes "yes"). Therefore t h e sets of committees for which t h e sums of powers D ( A ) and D ( B ) and e x p e c t e d p r i z e s P ( A ) and P ( B ) s a t i s f y o p p o s i t e i n e q u a l i t i e s , a r e n o t e a s y t o c o n s t r u c t . However, some s u c h p a i r s have been f o u n d , and may s e r v e as a b a s i s f o r t e s t i n g Hypothesis 1 0 .
The e x p e r i m e n t a l m a t e r i a l . I n t h i s s e c t i o n w e s h a l l p r e s e n t t h e s i x committees which were used i n t h e experiments 10.4.4.
.
Committee 1. The d e s c r i p t i o n of t h e v o t i n g r u l e i s here as f o l l o w s :
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1. The committee c o n s i s t s of 6 members, among them A and B; 2 . A i s t h e chairman;
3. B has t h e r i g h t of c a s t i n g 3 v o t e s , w h i l e t h e r e m a i n i n g members, i n c l u d i n g A , c a s t s one v o t e e a c h ;
4. The d e c i s i o n
i s t a k e n by s i m p l e m a j o r i t y ( t h e r e
are 8 votes altogether), with the provision that i n the c a s e of s p l i t y 4-4 t h e c h a i r m a n ' s v o t e i s d e c i s i v e . The scheme of t h e committee i s as f o l l o w s .
t h e v o t e of A i s decisive
'Ti ;i
,I ,! :
A
B
j
x
x
x
x
Scheme o f committee 1
Thus, i n committee 1, t h e power o f A l i e s i n h i s r i g h t t o b r e a k t i e s , w h i l e t h e power of B l i e s i n h i s r i g h t t o cast three votes. The s u b j e c t s were asked whether A o r B has more power, and g i v e t h e i r a s s e s s m e n t o f t h e powers o f A and B , i f one assumes t h a t t h e power o f e a c h o f t h e r e m a i n i n g members (marked x ) i s t a k e n as 1.
SELECTED TOPICS IN MEASUREMENT THEORY
639
I t i s r e l a t i v e l y easy t o e v a l u a t e t h e powers o f members
of t h i s committee. Member A a p p e a r s as a p i v o t i n 1 4 4 p e r m u t a t i o n s , B - i n 288 p e r m u t a t i o n s , and e a c h o f t h e r e m a i n i n g members - i n 7 2 p e r m u t a t i o n s . S i n c e t h e r e a r e
6!
=
720 p e r m u t a t i o n s , t h e powers a r e
T h u s , B has t w i c e as much power as A , w h i l e t h e powers o f A and B e x p r e s s e d i n u n i t s e q u a l d ( x ) a r e r e s p e c t i -
v e l y 2 and 4 . L e t now P A ( z ) be t h e p r o b a b i l i t y of p o s i t i v e d e c i s i o n
o f Committee 1, i f member A v o t e s " y e s " , and t h e remaini n g members v o t e ''yes" i n d e p e n d e n t l y w i t h p r o b a b i l i t y z . We have t h e n P,(r)
=
z
t
(l-2)[z4
4-
423 ( 1 - z ) l .
I n d e e d , i f A v o t e s " y e s " , t h e r e are two p o s s i b i l i t i e s : e i t h e r B v o t e s a l s o " y e s " ( p r o b a b i l i t y z ) , i n which c a s e t h e i s s u e p a s s e s , as t h e r e are a l r e a d y f o u r v o t e s " y e s " , and t h e chairman A h a s t h e r i g h t t o b r e a k a t i e , or B v o t e s "no" ( p r o b a b i l i t y 1 - z ) . I n t h e l a t t e r c a s e , one needs e i t h e r t h r e e o r f o u r v o t e s o f t h e r e m a i n i n g members. P r o b a b i l i t y o f a l l f o u r v o t e s b e i n g " y e s " i s z 4 , w h i l e p r o b a b i l i t y o f t h r e e v o t e s "yes" and one v o t e "no" i s 42 3 ( 1 - z ) . T h i s p r o v e s t h e formula f o r
S i m i l a r l y , we have
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The argument i s s i m i l a r as b e f o r e . I f B v o t e s r l y e s r l , and s o d o e s A ( p r o b a b i l i t y z) t h e i s s u e p a s s e s , which g i v e s t h e f i r s t t e r m . I f A v o t e s "no" ( p r o b a b i l i t y
1-z), t h e n a t l e a s t two o f t h e r e m a i n i n g members must v o t e ''yes'' f o r t h e i s s u e t o p a s s . The t h r e e t e r m s i n t h e square bracket give respectively t h e probability o f 4 , 3 and 2 v o t e s " y e s " . We s e e t h a t i n t h i s c a s e we have a l w a y s P,(z)
< P,(z),
which a g r e e s w i t h t h e f a c t t h a t B h a s more power t h a n A : t h e p r o b a b i l i t y o f a p o s i t i v e d e c i s i o n i s always h i g h e r when B v o t e s ' r f o r t r t h a n when A v o t e s " f o r " . Committee 2 The d e s c r i p t i o n o f t h e v o t i n g r u l e i s as f o l l o w s . 1. The committee c o n s i s t s of 6 members, among them C and D ; 2 . C i s t h e chairman;
3 . The r e m a i n i n g f i v e members form two s u b c o m i t t e e s o f t h r e e p e r s o n s e a c h ; D b e l o n g s t o b o t h subcommittees;
4 . I n e a c h subcommittee, t h e d e c i s i o n r e q u i r e s simpl e majority ( a t least 2 out of 3 v o t e s ) ;
5 . The f i n a l d e c i s i o n r e q u i r e s t h e m a j o r i t y o f t h r e e v o t e s : t h a t of t h e chairman, and t h o s e o f t h e subcommittees. The scheme o f t h i s committee i s p r e s e n t e d below. Here t h e power o f C i s due t o t h e f a c t t h a t h i s v o t e e n t e r s d i r e c t l y i n t o t h e f i n a l d e c i s i o n , w h i l e t h e power of D l i e s i n t h e f a c t t h a t he b e l o n g s t o two sub-
SELECTED TOPICS INMEASUREMENT THEORY
641
c o m m i t t e e s , hence he h a s i n e f f e c t two i n d i r e c t v o t e s .
I
I
I
decides
X
1
majority
C
I
D
X
dec:
X
X
Scheme o f Committee 2
I n t h i s c a s e , t h e d e c i s i o n whether C o r D has more power i s much more d i f f i c u l t t h a t i n t h e c a s e o f Committee 1, and most o f t h e s u b j e c t made a wrong c h o i c e ( c l a i m i n g t h a t t h e chairman i s more p o w e r f u l t h a t D ) . I n f a c t , t h e s i t u a t i o n i s j u s t t h e o p p o s i t e : enumeration o f p e r m u t a t i o n s i n which C , D o r x i s t h e p i v o t y i e l d s t h e powers : d(C) = 0.2,
d(D)
=
4/15 = 0.2666, d ( x ) = 0.1333.
Thus, i f power o f x i s t a k e n as 1, t h e n t h e power o f chairman C i s 1 . 5 , w h i l e power of member D i s 2 .
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CHAPTER 5
I n t h i s c a s e , when C v o t e s ''yes", we have
P,(z)
=
Z[l-(l-Z) 41
t
(1-z)(2z 2 - z 4) .
Here t h e r e a s o n i n g i s as f o l l o w s . If D v o t e s ''yestr ( p r o b a b i l i t y z i n t h e f i r s t f a c t o r ) , one needs j u s t one more v o t e of t h e r e m a i n i n g members, s i n c e t h e n a t l e a s t one subcommittee w i l l have a m a j o r i t y . The probab i l i t y of a l l f o u r r e m a i n i n g members v o t i n g "no" i s
( l - ~ ) ~s,o t h a t w i t h p r o b a b i l i t y 1 -
at l e a s t one v o t e w i l l be p o s i t i v e . T h i s e x p l a i n s t h e f i r s t t e r m 0-2)~
i n t h e formula. I f D v o t e s "no" ( p r o b a b i l i t y 1-z as a f a c t o r i n t h e second t e r m ) , one needs two v o t e s "yes" i n a t l e a s t one subcommittee. P r o b a b i l i t y o f two v o t e s " y e s r r i n a subcommittee i s z2, and d e n o t i n g by U and V t h e e v e n t s t h a t t h e s e subcommittees v o t e we have P(U u V) 2 = P(U) t P(V) - P(U n V). Here P(U) = P(V) = z , as a l r e a d y s t a t e d , w h i l e P(U n V) i s z4, s i n c e for t h e v o t e t o be "yes" i n b o t h subcommittees, a l l f o u r memb e r s must v o t e " y e s " . T h i s g i v e s t h e whole f o r m u l a . Next, we have
The f i r s t t e r m i s e x p l a i n e d a s b e f o r e : i f b o t h C and D v o t e ''yes" ( p r o b a b i l i t y z ) , t h e n one needs a t l e a s t one m r e v o t e " y e s " . On t h e o t h e r hand, i f C v o t e s "no"
we need a t l e a s t one v o t e "yes" i n e a c h o f t h e two subcommittees. The p r o b a b i l i t y o f a n e g a t i v e v o t e i n a subcommittee i s ( l - ~ ) ~s , o that l-(l-~)~ i s t h e probabi
SELECTED TOPICS IN MEASUREMENT THEORY
643
l i t y of a p o s i t i v e v o t e , and s q u a r i n g w e g e t t h e prob a b i l i t y of p o s i t i v e v o t e s i n b o t h subcommittees.
Committee 3 The v o t i n g r u l e h e r e i s as f o l l o w s . 1. The committee c o n s i s t s of 5 members, among
them E and F
.
2 . E has t h e r i g h t o f c a s t i n g t h r e e v o t e s , w h i l e
e a c h of t h e r e m a i n i n g members c a s t s one v o t e e a c h ;
3. F h a s t h e v e t o power; 4 . The d e c i s i o n r e q u i r e s s i m p l e m a j o r i t y : t h e i s s u e p a s s e s i f a t l e a s t f o u r v o t e s o u t of s e v e n , among them t h e vote of F, support
the issue.
The scheme of t h e committee i s as f o l l o w s .
majority decides;
1
positive vote of F i s necessary
E
F
X
X
I
X
Scheme of committee 3
The power of E i s due t o t h e r i g h t o f c a s t i n g t h r e e v o t e s , w h i l e t h e power of F i s due t o h i s r i g h t of v e t o .
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Here t h e powers a r e e q u a l d(E) = 0.3, d ( F ) = 0.55, d ( x ) = 0.05. T h i s means t h a t if t h e power of x i s t a k e n as a u n i t , t h e n t h e power of E e q u a l s 6 , w h i l e t h e power of F i s e q u a l 11. I n o t h e r words, a member w i t h t h r e e t i m e s as many v o t e s t h a n " o r d i n a r y " member h a s 6 t i m e s as much power, w h i l e a member w i t h v e t o power i s 11 t i m e s as p o w e r f u l as o r d i n a r y member. I n c a s e o f t h i s committee we have
s i n c e i f E v o t e s " y e s " , a l l one needs i s a p o s i t i v e v o t e o f F (who h a s t h e v e t o p o w e r ) . I f F v o t e s "no", t h e i s s u e cannot p a s s . On t h e o t h e r hand, P,(z)
=
z
t (1-z)z 3
.
I n d e e d , i f F v o t e s " y e s " , we need e i t h e r t h e t h r e e vot e s o f E ( p r o b a b i l i t y z ) , o r i f E v o t e s "no" ( p r o b a b i l i t y 1-z), we need a l l t h r e e v o t e s of t h e r e m a i n i n g
members. Thus, we have here P,(z) v a l u e of z .
<
P F ( z ) , r e g a r d l e s s of t h e
Committee 4 The d e s c r i p t i o n of t h e v o t i n g r u l e
i s as follows.
SELECTED TOPICS IN MEASUREMENT THEORY
645
1. The committee consists o f 8 members, among them G and H. They form two subcommittees.
2. Subcommittee 1 comprises 5 members, among them G.
3. Subcommittee 2 comprises 3 members, among them H.
4. If subcommittee 1 votes unanimously "yes'', the issue passes, regardless o f the votes o f subcommittee 2.
5.
I f subcommittee 1 is not unanimous, the positive
decision requires simple majority in each of the subin subcommittee committees (i.e. at least 3 votes llyesIt 1, and at least 2 votes "yes" in subcommittee 2). The scheme of this committee is as f o l l o w s .
a '
"
if:
f o r"
least one against I' H
G
x
x
x
x
Scheme of committee 4
Y
Y
646
CHAPTER 5
Here t h e power o f G i s due t o t h e membership i n a s u b committee which may d e c i d e i n d e p e n d e n t l y o f t h e v o t e s of t h e o t h e r subcommittee, w h i l e power o f H i s due t o h i s membership i n a s m a l l e r subcommittee, hence p r e s u mably e a s i e r t o a t t a i n t h e m a j o r i t y . The s u b j e c t s were asked t o d e c i d e whether G or H has more power, and a l s o assess t h e powers o f G , H and y , i f power of x i s t a k e n as 1.
It i s i n t e r e s t i n g t h a t n o t a l l t h e s u b j e c t s n o t i c e d t h a t G h a s t h e same power as any o f t h e members o f h i s subcommittee, and t h e same h o l d s f o r H . The powers i n t h i s case are equal d ( G ) = d ( x ) = 0.12142, d ( H ) = d ( y ) = 0 . 1 3 0 9
s o t h a t d ( G ) / g ( x ) = 1, d ( H ) / d ( x ) = d ( y ) / d ( x ) = 1 . 0 7 8 . Thus, i t i s somewhat b e t t e r t o b e l o n g t o a s m a l l e r subcommittee. F o r t h i s committee w e have
pG(z) = z4 t [4z3(1-z) t 6z2(1-z)2][z3
+ 322 (1-z)l.
Here t h e r e a s o n i n g i s as f o l l o w s . I f G v o t e s t h e n t h e i s s u e may p a s s b e c a u s e o f t h e unanimous v o t e i n t h e committee c o n t a i n i n g G . T h i s o c c u r s i f a l l f o u r 4 On members v o t e hence w i t h p r o b a b i l i t y z
.
t h e o t h e r hand, i n t h e o p p o s i t e c a s e w e need a majori t y i n GIs subcommittee, w i t h a t l e a s t one v o t e "no", which means t h a t e i t h e r one o r two members m u s t v o t e Ilno" T h i s g i v e s t h e f i r s t s q u a r e b r a c k e t i n t h e formula f o r PG(z). I f t h e G I s subcommittee h a s a m a j o r i t y ,
.
SELECTED TOPICS INMEASUREMENT THEORY
647
one needs a m a j o r i t y i n H's subcommittee, which means t h a t e i t h e r 3 o r 2 v o t e s must b e " y e s " ; t h i s g i v e s t h e second s q u a r e b r a c k e t . Next,
PH(Z) = z5
+
[5z 4 (1-z)
+ 1oz 3 (1-z)2 111 - (1-z)2 1 .
I n d e e d , t h e i s s u e p a s s e s , i f t h e G I s subcommittee v o t e s unanimously " y e s " , which happens w i t h p r o b a b i l i t y z5
.
O t h e r w i s e , G's subcommittee must have a m a j o r i t y , b u t a t l e a s t one "no", i . e . 4 o r 3 v o t e s " y e s " . T h i s g i v e s t h e f i r s t s q u a r e b r a c k e t . The second s q u a r e b r a c k e t c o r r e s p o n d s t o t h e p r o b a b i l i t y of a m a j o r i t y i n t h e HIS subcommittee, i f i t i s known t h a t H v o t e s " y e s " . Committee 5 The d e s c r i p t i o n o f t h e v o t i n g r u l e
i s as f o l l o w s .
1. The committee c o n s i s t s o f 7 members, among them
I and J ; 2 . Two members, among them I , form t h e p r e s i d i u m .
3. The r e m a i n i n g f i v e members form two subcommittees o f t h r e e persons each, with J belonging t o both subcommittees;
4 . I n e a c h subcommittee, t h e d e c i s i o n r e q u i r e s s i m p l e m a j o r i t y (two v o t e s o u t o f t h r e e ) ;
5. F o r f i n a l d e c i s i o n , one t a k e s t h e v o t e s o f subcommittees, and t h e v o t e s o f members o f t h e p r e s i d i u m ( a l t o g e t h e r 4 v o t e s ) . P o s i t i v e d e c i s i o n r e q u i r e s maj o r i t y i n these 4 votes, with t h e provision t h a t i n
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CHAPTER 5
c a s e s o f t h e s p l i t 2-2 t h e v o t e o f chairman I i s decisive. The scheme o f t h i s committee i s as f o l l o w s : I i
1
majority decides; i n case of s p l i t
'
i I
majority
I
Y
X
X
J
X
X
Scheme o f committee 5 Here t h e power o f I i s due t o t h e f a c t t h a t h i s v o t e e n t e r s d i r e c t l y t o t h e f i n a l d e c i s i o n , and a l s o t o t h e f a c t t h a t he b r e a k s t h e t i e . The power o f J l i e s i n t h e f a c t t h a t he b e l o n g s t o two subcommittees, henc e has two v o t e s . The s u b j e c t s were asked t o d e c i d e whether I o r J i s more p o w e r f u l , and a l s o a s s e s s t h e powers o f I , J and y , i f t h e power o f a member d e n o t e d by x i s t a k e n a s 1.
Here t h e v a l u e s o f powers a r e d(1) = 0.4, d ( y ) = 0.2,
d ( J ) = 0.1333,
d(x) = 0.066,
SELECZED TOPICS IN MEASUREMENT THEORY
649
s o t h a t t h e r a t i o s t o be a s s e s s e d a r e
We have h e r e p (z) = z
I
+ (1-z)[z(l-(1-z) 4) + (1-z)(2z 2 - z 4>I.
Here t h e r e a s o n i n g i s as f o l l o w s . C o n s i d e r two c a s e s : y ( t h e d e p u t y chairman) e i t h e r v o t e s " y e s " or "no".
The f i r s t c a s e o c c u r s w i t h p r o b a b i l i t y z, and t h e n n o t h i n g more i s needed, s i n c e t h e r e a r e a l r e a d y two v o t e s " y e s " ( o f I and y ) , and I has t h e r i g h t t o b r e a k t h e t i e . I f y v o t e s ''no" (which a c c o u n t s f o r t h e f a c t o r 1-z b e f o r e t h e s q u a r e b r a c k e t ) , one needs a t l e a s t one m a j o r i t y i n one o f t h e subcommittees. Here a g a i n we c o n s i d e r two c a s e s : e i t h e r J v o t e s " y e s " , or he v o t e s "no" I n t h e f i r s t c a s e , i t s u f f i c e s t h a t o n l y
.
one o f t h e r e m a i n i n g members v o t e s " y e s " , which o c c u r s
-
O t h e r w i s e , we need two v o t e s " y e s " i n a t l e a s t one subcommittee, which o c c u r s with probability 1
(1-z)'.
w i t h p r o b a b i l i t y 2z2 - z4, as a l r e a d y e s t a b l i s h e d i n a n a l y s i s o f committee 2. I f t h e "deputy chairman" y v o t e s ''yes", we have p (z) = z Y
+
(l-z)[z(l-(l-z)
2 2 )
-t
(1-z)z
41.
Here t h e r e a s o n i n g i s as f o l l o w s . The chairman I e i t h e r v o t e s ' ' y e s t t or "no" I n t h e f i r s t c a s e ( p r o b a b i l i t y z)
.
n o t h i n g e l s e i s needed. I n t h e second c a s e ( f a c t o r 1-z b e f o r e t h e s q u a r e b r a c k e t ) , one ,needs m a j o r i t y i n b o t h subcommittees, t o overcome t h e t i e - b r e a k i n g power o f I . Again, J may v o t e " y e s " , i n which c a s e one must
CHAPTER 5
650
have a t l e a s t one v o t e "yes'' i n e a c h subcommittee. 2 A t l e a B t one v o t e " y e s " o c c u r s w i t h p r o b a b i l i t y 1-( 1 - z ) , and s q u a r i n g w e o b t a i n t h e f i r s t t e r m i n t h e b r a c k e t s .
I f J v o t e s "no" , t h e i s s u e w i l l p a s s only i f a l l f o u r r e m a i n i n g members w i l l v o t e " y e s " , which has p r o b a b i l i ty z
4.
F i n a l l y , w e have PJ(Z) = z2 t z(l-z)[l-(l-z)
4 1 t z ( l - z ) t - l - ( 1 - z ) 2 12 -
Here t h e argument i s as f o l l o w s . F i r s t l y , we may have b o t h I and y v o t e " y e s " , and t h e n t h e i s s u e p a s s e s . T h i s 2 occurs with probability z Clearly, for the issue t o p a s s , a t l e a s t one o f t h e p r e s i d i u m members ( I o r y )
.
must v o t e " y e s " . The r e m a i n i n g two terms i n t h e f o r m u l a c o r r e s p o n d now t o t h e c a s e when I v o t e s ''yes1' and y vot e s "no" and v i c e v e r s a . Both t h e s e c a s e s have probab i l i t y z ( 1-z),
appearing i n f r o n t of each of t h e square
brackets. I n t h e f i r s t c a s e , when I v o t e s " y e s " , a l l one needs i s j u s t one v o t e " y e s " among members o f t h e subcommittees o t h e r t h a n J (who v o t e s "yes" by d e f i n i t i o n ) . T h i s occurs with p r o b a b i l i t y 1 - (1-z)
4.
I f I v o t e s "no" and y v o t e s " y e s r 1 ,
one needs m a j o r i t y
i n e a c h of t h e subcommittees, t o overcome t h e t i e - b r e a k i n g power of I . S i n c e J v o t e s " y e s " , a l l t h a t i s needed i s a t l e a s t one v o t e i n e a c h o f t h e subcommittees. T h i s happens w i t h p r o b a b i l i t y 1 - ( l - ~ )f o~r any o f t h e subcommittees, which a c c o u n t s f o r t h e s q u a r i n g i n t h e l a s t term o f formula f o r P,(z).
SELECTED TOPICS IN MEASUREMENT THEORY
65 1
Committee 6 The description of the voting rule is as follows: 1. The committee consists of 1 3 members, among them K and L; 2. The members are divided into two subcommittees; subcommittee 1 consists of 5 members, among them K, while subcommittee 2 consists of 8 members, among them L.
3. K has the right to cast 3 votes, L has the right to cast four votes, while each of the remaining persons has the right to cast one vote each.
4. In each subcommittee, the issue is decided by majority (in committee 1, at least four votes out of seven, while in subcommittee 2, at least six votes out of eleven);
5. Positive final decision requires positive decision in at least one subcommittee. Here the power of K is connected with the fact that to enforce the issue he needs to convince one person out of four, while the power of L lies in the fact that he needs to convince two persons out of seven. The powers in this committee are: d(K)
=
0.3, d(x) = 0.05, d(L) = 0.25, d(y)
=
0.0357
so that the member K of the smaller committee has somewhat more power. The ratios to be estimated by the sub-
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-
I
required positive decision i n at l e a s t one subcommittee
i I
majority decides
K
x
x
x
x
Ill I ! L
Y
Y
IIII
Y
Y
Y
Y
! I Y
Scheme o f committee 6 .
j e c t s ( r e l a t i v e t o power o f x) a r e
I n t h i s c a s e we have
I n d e e d , i f K v o t e s " y e s " , t h e n t h e i s s u e passes i f a t l e a s t one of t h e four members o f h i s subcommittee v o t e s " y e s " . I n t h i s c a s e , one may d i s r e g a r d t h e v o t e s i n
SELECTED TOPICS IN MEASUREMENT THEORY
653
t h e second subcommittee. On t h e o t h e r hand, i f a l l four members o f K 1 s subcommittee v o t e "no" (which a c c o u n t s for t h e f a c t o r (1-z)4 b e f o r e t h e b r a c e s ) , a m a j o r i t y i s needed i n t h e second subcommittee. The c a s e s t o c o n s i d e r are when L v o t e s " y e s " ( i n which c a s e one needs a t l e a s t two v o t e s "yes'' of o t h e r members, which g i v e s t h e f i r s t s q u a r e b r a c k e t ) , o r L v o t e s ''no'' , i n which c a s e one needs 6 o r 7 v o t e s "yes" of o t h e r m e m b e r s, which g i v e s t h e second s q u a r e b r a c k e t . Finally ,
Here t h e argument i s s i m i l a r as b e f o r e . If L v o t e s " y e s " , t h e i s s u e may p a s s b e c a u s e of a m a j o r i t y i n t h e second subcommittee, r e g a r d l e s s o f t h e v o t e of t h e f i r s t subcommittee. T h i s o c c u r s w i t h p r o b a b i l i t y 1-(1-z) 7 -7~(l-z)~.I n t h e o p p o s i t e c a s e (which o c c u r s w i t h t h e p r o b a b i l i t y g i v e n i n t h e f i r s t s q u a r e b r a c k e t ) , w e need a m a j o r i t y i n t h e f i r s t subcommittee. The two t e r m s i n t h e b r a c e s g i v e t h i s p r o b a b i l i t y , depending on w h e t h e r K v o t e s " y e s " o r "no".
1 0 . 4 . 5 . The r e s u l t s . T h i s s e c t i o n c o n t a i n s p r e s e n t a t i o n of some r e s u l t s o f t h e experiment aimed a t t e s t i n g t h e h y p o t h e s e s d e s c r i b e d i n s e c t i o n 10.4.3. A l t o g e t h e r , 30 s u b j e c t s were t e s t e d , e a c h o f them t w i c e i n t h e i n t e r v a l o f two weeks.
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654
The f i r s t h y p o t h e s i s a s s e r t e d t h a t p e o p l e a r e g e n e r a l l y
a b l e t o t e l l c o r r e c t l y who h a s h i g h e r power. The t a b l e below shows t h a t t h i s i s not a l w a y s t r u e ; ranked i n t h e o r d e r o f t o t a l number o f c o r r e c t c h o i c e s made i n two t e s t s , t h e r e s u l t s i n d i c a t e t h a t i t i s sometimes d i f f i c u l t t o a s s e s s c o r r e c t l y t h e o r d e r o f powers. C ommi t t e e
both choices correct
28 25 18 16 13 3
one c h o i c e c o r r e c t and one wrong
both choices wrong
2
0
4 7 7
1
11
4
4 7 6 23
A s may b e s e e n , i n Committee 2 , t h e c h o i c e s were i n
most c a s e s wrong: p e o p l e t e n d t o t h i n k t h a t b e i n g a b l e t o v o t e d i r e c t l y g i v e s more power t h a n membership i n two subcommittees. H y p o t h e s i s 2 c o n j e c t u r e d t h a t t h e r e i s some t h r e s h o l d i n p e r c e p t i o n o f s o c i a l power, namely t h a t p e o p l e a r e a b l e t o make t h e c o r r e c t c h o i c e i f t h e power d i f f e r e n c e i s t o o small. T h i s , a g a i n , does n o t seem t o h o l d , as i l l s u t r a t e d i n F i g . 1 0 . 4 . 1 . T h i s f i g u r e g i v e s t h e p e r c e n t a g e of c o r r e c t c h o i c e s a g a i n s t t h e power d i f f e r e n c e o f members among whom t h e s u b j e c t s were t o make a c h o i c e . T h i s f i g u r e p r o v i d e s a l s o some e v i d e n c e f o r H y p o t h e s i s
SELECTED TOPICS IN MEASUREMENT THEORY
65 5
% of correct choice 100
90
@
#5
€4
#l
80
#6 7O
#4 €4
011
60
#3
50 40
30 20 011
#2
10 G
I -
0.1
-
" 0.2
d i f ferens -77-c e o f
0.3
powers
F i g . 1 0 . 4 . 1 . D i s c r i m i n a t i o n of s o c i a l power
3 , which a s s e r t s t h a t t h e r e i s some p o s i t i v e a s s o c i a t i o n between t h e d i f f e r e n c e o f powers and t h e probab i l i t y o f c o r r e c t c h o i c e . T h i s e v i d e n c e , however,, i s n o t c o n c l u s i v e : t h e c o r r e l a t i o n between power d i f f e r n c e and f r e q u e n c y o f c o r r e c t c h o i c e i s p o s i t i v e , b u t s t a t i s t i c a l l y not significance ( r = 0.56, t h e corresponding v a l u e o f t i s 1.37 f o r 4 d e g r e e s o f f r e e d o m ) .
Hypotheses 1-3 concerned o n l y t h e c o r r e c t n e s s o f c h o i c -
e s . T u r n i n g now t o t h e a s s e s s m e n t o f s o c i a l power, i t a p p e a r s t h a t Hypotheses 4 and 5 a r e t r u e . These
656
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h y p o t h e s e s a s s e r t t h a t t h e r a t i o o f powers i s a s s e s s e d i n a b i a s e d way, and s p e c i f y t o some e x t e n t t h a t t h i s b i a s may be e x p e c t e d t o be l a r g e s t i n a s s e s s m e n t o f
v e t o power and o f t h e r i g h t o f c a s t i n g more t h a n one vote. The t a b l e below g i v e s t h e t r u e r a t i o s and t h e mean e s t i m a t e d r a t i o s , ordeded a c c o r d i n g t o t h e amount o f bias. Committee
members
true ratio
mean e s t i m a t e d ratio
A:x y:x
2
D:x y:x J:x y:x
2 1.078 2
1.98 0.86 2.23 1.32 2.26
3
2,77
G:x
1
1.55
B:x H:x c :x L:x I:x
4
3.27 1.80
K:x E:x F:x
0.71
1.078 1.5
5 6 6 6 11
3.23 3.25 4.10
3.65 3.25 4.11
The r e g r e s s i o n s l o p e h e r e i s 0.296, which i s s i g n i -
f i c a n t l y l e s s t h a t 1 (which c o r r e s p o n d s t o l a c k o f b i a s ) . It may be s e e n t h a t r a t i o s l e s s t h a n 2 a r e o v e r e s t i m a t e d , w h i l e a l l r a t i o s above 2 a r e u n d e r e s t i mated.
SELECTED TOPICS INMEASUREMENT THEORY
657
I t seems t h e r e f o r e t h a t p e o p l e t e n d t o a t t a c h t o o much
weight t o t h e advantage which one p e r s o n has o v e r ano t h e r , as l o n g as t h i s a d v a n t e g e i s n o t t o o l a r g e ( a p p r o x i m a t e l y , up t o t h e p o i n t when one p e r s o n i s t w i c e as p o w e r f u l as a n o t h e r ) . On t h e o t h e r hand, when one p e r s o n i s much more p o w e r f u l t h a n t h e o t h e r , t h i s f a c t i s not p r o p e r l y a p p r e c i a t e d . A s r e g a r d s h y p o t h e s i s 5 , t h e c o n j e c t u r e was t h a t p e o p l e t e n d t o u n d e r e s t i m a t e t h e power due t o t h e r i g h t o f v e t o and due t o t h e r i g h t o f c a s t i n g more t h a n one v o t e . A s r e g a r d s t h e v e t o power, t h i s i s e v i d e n t : memb e r F i n committee 3 has t h e r i g h t t o v e t o , which g i v e s h i m 11 t i m e s as much power as an o r d i n a r y member, w h i l e t h e mean e s t i m a t e s o f t h i s r a t i o was o n l y 4 . 1 1 . A s regards
t h e r i g h t o f c a s t i n g more t h a n one v o t e , t h e s i t u a t i o n i s somewhat more complex. Those members who have t h e r i g h t of c a s t i n g more t h a n one v o t e i n t h e same v o t i n g a r e : B i n committee 1, E i n committee 3, and K and L i n committee 6 . I n a l l t h e s e c a s e s , t h e power i s u n d e r e s t i m a t e d . However, t h e r e i s a l s o a n o t h e r way i n which one may c a s t two or more v o t e s , namely by b e l o n g i n g t o two or more subcommittees. Such p e r s o n s a r e : D i n committee 2 and J i n committee 5 . I n t h e s e two c a s e s , t h e power i s somewhat o v e r e s t i m a t ed.
It i s i n t e r e s t i n g t o mention t h a t i n c h o o s i n g a more " p r o f i t a b l e " o r " p r e f e r r e d " committee i n e a c h possi b l e p a i r o f committees, p e o p l e t e n d t o make c o n s i s t -
e n t c h o i c e s , i . e . c h o i c e s which y i e l d a t r a n s i t i v e r e l a t i o n . Moreover, t h i s o r d e r i n g , i n most c a s e s , coi n c i d e s w i t h t h e o r d e r of p r o b a b i l i t i e s o f winning,
658
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c a l c u l a t e d under t h e a s s u m p t i o n of i n d e p e n d e n c e and u n b i a s e d n e s s of t h e v o t e r s . This order i s presented i n t h e following t a b l e . Committee
6
P(K) = 0.969
1
P(B) = 0.844
P(L) = 0.969 P(A) = 0.656
5
P(1) = 0.844
P ( J ) = 0.625
2
P ( D ) = 0.750
P ( C ) = 0.678
3 4
P ( F ) = 0.562
P(E) = 0 . 5 0 0 P(G) = 0.375
P ( H ) = 0.382
of a p o s i t i v e d e c i s i o n , g i v e n t h a t t h e i n d i c a t e d member of t h e co-
The l e f t column g i v e s t h e p r o b a b i l i t y mmittee v o t e s " y e s " ,
and t h e r e m a i n i n g members v o t e
i n d e p e n d e n t l y , w i t h p r o b a b i l i t y $ for v o t e " y e s r r . The r i g h t hand s i d e column g i v e s t h e same p r o b a b i l i t i e s i f t h e marked member ( n o t t h e most p o w e r f u l i n t h e c o m m i t t e e ) v o t e s " y e s " , and t h e r e m a i n i n g members v o t e a g a i n "yes" i n d e p e n d e n t l y w i t h p r o b a b i l i t y $
.
For i n s t a n c e , i n committee 1, i t i s most p r o f i t a b l e t o have t h e most p o w e r f u l member, B , v o t e " y e s " . T h i s g i v e s t h e chance o f w i n n i n g a b o u t 84 % .On t h e o t h e r h a n d , i.f A v o t e s " y e s " , t h e c h a n c e s of w i n n i n g a r o n l y a b o u t 66
%.
It i s a l s o w o r t h t o m e n t i o n t h a t t h e s u b j e c t s were a s k e d t o s t a t e , i f p o s s i b l e , t h e r e a s o n s for t h e i r c h o i c e s . The comments c o l l e c t e d i n d i c a t e t h a t t h e exp e r i m e n t a l t a s k , r e q u r i n g a n a n a l y s i s of t h e s t r u c t u r e
o f f o r c e s o p e r a t i n g w i t h i n a c o m m i t t e e , was r a t h e r d i f f i c u l t . Sometimes t h e comments on t h e second t e s t -
SELECTED TOPICS LN MEASUREMENT THEORY
659
i n g was o f t h e k i n d "Oh, I remember t h i s s i t u a t i o n ! I remember how d i f f i c u l t i t was t o a n a l y s e i t ! "
Also, i t appeared t h a t j u d g i n g t h e committee 3 , where one member has t h e v e t o power, t h e s u b j e c t s d i s p l a y e d c o n s i d e r a b l e r e s e n t m e n t towards t h i s member, and t o t h e whole s i t u a t i o n i n g e n e r a l . The comments were s o metimes of t h e s o r t : "Veto power s h o u l d be a b o l i s h e d ! " o r t h e l i k e . It would b e o f some i n t e r e s t t o s e e if t h i s t y p e o f r e s e n t m e n t i s somehow s p e c i f i c f o r P o l i s h s u b j e c t s , who have a l w a y s been t a u g h t a t s c h o o l t h a t t h e v e t o power was t h e most prominent s i n g l e f a c t o r responsible f o r the troubles i n the Polish history. I f t h a t i s s o , t h e n t h e a s s e s s m e n t of committee 3 by s u b j e c t s o f o t h e r n a t i o n a l i t i e s s h o u l d n o t evoke s u c h s t r o n g resentment. G e n e r a l l y , t h e r e s u l t s of e m p i r i c a l s t u d i e s here i n d i c a t e t h a t s o c i a l power, even r e s t r i c t e d t o t h e s i m p l e s t c a s e s , i s p e r c e i v e d w i t h s e r i o u s d i s t o r t i o n s . The s o u r c e of t h e s e d i s t o r t i o n s seems t o l i e i n t h e d i f f i c u l t y i n p r o p e r v i s u a l i z i n g and a p p r e c i a t i o n o f t h e f o r c e s and r e l a t i o n s which t e n d t o enhance or d i m i n i s h t h e s o c i a l power. T h i s s u g g e s t s t h a t i n r e a l s i t u a t i o n s , t h e p e r c e p t i o n o f s o c i a l power o f v a r i o u s p e r s o n s i s l i k e l y t o be even more d i s t o r t e d . I n i d e a l i z e d s i t u a t i o n s , as i n d i c a t e d by t h e r e s u l t s o b t a i n e d t h u s f a r , t h e d i s t o r t i o n s concern p r i m a r i l y t h e r o l e o f b l o c k i n g ( v e t o ) and m i s a p p r e c i a t i o n o f m u l t i p l e means o f i n f l u e n c i n g t h e d e c i s i o n . One could t h e r e f o r e c o n j e c t u r e t h a t t h e same f a c t o r s w i l l t e n d t o e x e r t more i n f l u e n c e on a s s e s s m e n t i n r e a l s i t u a t ions.
660
10.4.6.
CHAPTER 5
Discussion.
I n connection with t h e findings
p r e s e n t e d i n t h e p r e c e d i n g s e c t i o n , one could t r y t o g e t more i n f o r m a t i o n about t h e n a t u r e of p e r c e p t i o n of s o c i a l power by a n a l y s i n g v e r b a l r e p r e s e n t a t i o n s of s i t u a t i o n s by t h e s u b j e c t s , g o i n g beyond t h e a n a l y s i s o f t h e comments which t h e y make d u r i n g t h e e x p e r i m e n t -
a l task. The e s s e n t i a l p o i n t i s t h a t t h e p r o c e s s of d e c i s i o n of s u b j e c t s ( e . g . i n c h o o s i n g a "more f a v o u r a b l e " commit t e e ) i s of t h e t y p e of p r o c e s s e s s t u d i e d i n d e c i s i o n t h e o r y , where t h e c h o i c e a l t e r n a t i v e s a r e e v a l u a t e d on m u l t i p l e c r i t e r i a . Here t h e f u t u r e a n a l y s i s might proceed a l o n g t h e l i n e s s u g g e s t e d i n Nowakowska ( 1 9 7 3 ) . The i d e a t h e r e was t o c o n s i d e r t h e s p a c e spanned by t h e d i mensions ( s c a l e s , p s y c h o l o g i c a l c o n t i n u a ) , s u c h as s u b j e c t i v e p r o b a b i l i t y o r u t i l i t y ) c o r r e s p o n d i n g to v a r i ous c r i t e r i a , s o t h a t t h e c h o i c e a l t e r n a t i v e s may be o r d e r e d a l o n g e a c h o f t h e d i m e n s i o n s . Such s p a c e was c a l l e d m o t i v a t i o n a l s p a c e i n Nowakowska ( 1 9 7 3 ) . It was a l s o shown t h e r e t h a t t h e f o r m a l s t r u c t u r e o f i n d i v i d u a l d e c i s i o n s i s t h e same as t h a t o f a group d e c i s i o n , and c o n s e q u e n t l y , Arrow I m p o s s i b i l i t y Theorem ( s e e A r -
row 1963) must a p p l y t o i t . It t u r n s o u t t h a t s u c h an i n t e r p r e t a t i o n y i e l d s c e r t a i n important psychological consequences ( t o be d i s c u s s e d i n n e x t c h a p t e r ) , s i m p l y by o b s e r v i n g t h a t p e o p l e must d i v i d e i n t o v a r i o u s c a t e g o r i e s , depending on which o f t h e f i v e p o s t u l a t e s of Arrow t h e y b r e a k . Each o f t h e s e c a t e g o r i e s , i n t u r n , a c q u i r e s a s u i t a b l e p s y c h o l o g i c a l i n t e r p r e t a t i o n . The t h e o r e t i c a l and e m p i r i c a l i m p o r t a n c e o f s u c h an i d e n t i f i c a t i o n of i n d i v i d u a l and g r o u p s d e c i s i o n making w i l l be d i s c u s s e d i n n e x t c h a p t e r .
66 1
SELECTED TOPICS IN MEASUREMENT THEORY
11. LINGUISTIC MEASUREMENT T h i s s e c t i o n w i l l be d e v o t e d t o some d i f f e r e n t a s p e c t s
of r e p r e s e n t a t i o n o f o b j e c t s o r s i t u a t i o n s , namely t o t h e problems o f l i n g u i s t i c mesurement. The s t a r t i n g p o i n t o f t h e a n a l y s i s w i l l be c l a s s i f i c a t i o n schemes.
11.1. C l a s s i f i c a t i o n schemes
The c e n t r a l r o l e h e r e w i l l be p l a y e d b y t h e s y s t e m
(11.1) t h e r e m a i n i n g c o n c e p t s w i l l v a r y depending on t h e s i t u a t i o n under c o n s i d e r a t i o n . The n o t i o n s i n (11.1) a r e t h e s e t X o f o b j e c t s , o r : t h e n u n i v e r s e o f c l a s s i f i e d o b j e c t s , and t h e s e t C o f c a t e g o r i e s , where
c
*
= jC1,
1.
C*,".
A c l a s s i f i c a t i o n of a n o b j e c t x
(11.2) t X i s , by d e f i n i t i o n ,
*
.
a s s i g n i n g t o i t a n element o f C Thus, a c l a s s i f i c a t i o n establishes a function (possibly p a r t i a l )
t:
x+c
*
(11.3)
w i t h t ( x ) b e i n g t h e c l a s s t o which x i s a s s i g n e d . I f t i s d e f i n e d f o r a l l e l e m e n t s o f X , we speak o f f u l l c l a s s i f i c a t i o n ; g e n e r a l l y , t w i l l be d e f i n e d o n l y on some s e t X ' C X .
662
CHAPTER 5
The f u n c t i o n t i s d e f i n e d e m p i r i c a l l y ; t ( x ) = C . means J
t h a t t h e person under c o n s i d e r a t i o n , i n a s p e c i f i c i n s -
tance, c l a s s i f i e d x t o category C
j
.
T h i s means t h a t when t h e o c c a s i o n o r p e r s o n c h a n g e s ,
w e may have a d i f f e r e n t f u n c t i o n t ( i . e . t h e same o b j e c t may be c l a s s i f i e d t o v a r i o u s c a t e g o r i e s on d i f f e r e n t
o c c a s i o n s , and b y d i f f e r e n t p e r s o n s ) . We s h a l l now i n t r o d u c e two t y p e s o f c l a s s i f i c a t i o n s .
Type 1 c o m p r i s e s t h o s e c l a s s i f i c a t i o n s , i n w h i c h i t i s p o s s i b l e t o d e f i n e t h e notion of " t r u e category" of a n o b j e c t . Type 2 c o m p r i s e s c a s e s i n w h i c h t h e n o t i o n o f t r u e c a t e g o r y makes no s e n s e . T h e e x a m p l e s o f c l a s s i f i c a t i o n s o f Type 1 a r e q u i t e ob-
v i o u s . A s r e g a r d s Type 2 , t h e e x a m p l e s may be a n s w e r s t o q u e s t i o n n a i r e i t e m s , where t h e s u b j e c t i s asked t o c l a s s i f i y h i s " i n t e r n a l s t a t e " , as s p e c i f i e d by t h e i t e m , t o one o f t h e c a t e g o r i e s , u s u a l l y d e s c r i b e d i n l i n g u i s t i c t e r m s , s u c h as " s e l d o m " ,
" v e r y much", e t c .
F o r c l a s s i f i c a t i o n s o f Type 1 i t i s n a t u r a l t o i n t r o d u ce t h e f u n c t i o n
w:
x -c
*
(11.4)
where w ( x ) i s t h e " t r u e " c a t e g o r y o f o b j e c t x. T h i s ,
i n t u r n a l l o w s one t o d e f i n e t h e n o t i o n o f e r r o r i n c l a s s i f i c a t i o n : o b v i o u s l y , a n e r r o r was made, if t ( x ) d i f f e r s from w ( x ) , i . e . i f t h e o b j e c t was c l a s s i f i e d t o a category d i f f e r e n t t h a t i t s t r u e category. One o f t h e main ways o f b u i l d i n g c l a s s i f i c a t i o n schemes
SELECTED TOPICS IN MEASUREMENT THEORY
663
o f Type 1 i s t o c o n s t r u c t a s c a l e o f measurement of some a t t r i b u t e o f o b j e c t s from t h e s e t X , and t h e n d i v i d e t h e s c a l e i n t o s e t s ( i n t e r v a l s , s a y ) , corresponding t o various classes. The e r r o r c o n s i s t i n g o f c l a s s i f y i n g a n o b j e c t from cat e g o r y C ( i . e . s u c h t h a t w ( ~ )= C.) t o some o t h e r caj J t e g o r y C ( i . e . t h e d e c i s i o n t ( x ) = C . ) may have d i f f e r i 1 e n t c o n s e q u e n c e s , depending on b o t h j and i . To d e s c r i be this f a c t , one s h o u l d i n t r o d u c e t h e m a t r i x B =
Cbijl
where b i s the numerical expression of seriousness ij o f e r r o r s . O f c o u r s e , bii = 0 , and t h e m a t r i x need n o t b e symmetric, i . e . i n g e n e r a l b # b j i . The c h o i c e o f ij m a t r i x B depends on t h e p u r p o s e o f c l a s s i f i c a t i o n . One o f t h e c e n t r a l problems of c l a s s i f i c a t i o n s o f Type 1 i s t o f i n d methods o f m i n i m i z a t i o n o f p r o b a b i l i t y o f e r r o r s . T h i s may be a c h i e v e d b y means o f c o n s t r u c t i o n o f a new c l a s s i f i c a t i o n f u n c t i o n t , based on f u n c t i o n s tl, t2, o b t a i n e d from d i f f e r e n t p e r s o n s who c l a s s i f y i n d e p e n d e n t l y t h e same s e t o f o b j e c t s .
...
To u s e an example, one of s u c h f u n c t i o n s may be d e f i n -
ed as f o l l o w s . L e t u s i n t r o d u c e a n a d d i t i o n a l c a t e g o r y C t , c o r r e s p o n d i n g t o " o b j e c t l e f t u n c l a s s i f i e d " . Given t h r e e f u n c t i o n s , t l , t 2 and t one can d e f i n e f u n c t i o n 3' t using t h e "majority" p r i n c i p l e , i . e . t(x) = C
J
i f a t l e a s t two o u t o f t h e t h r e e
values t , ( x ) ,
t,(x)
and t , ( x )
is
C
j
664
CHAPTER 5
and
are a l l distinct. Thus, a c c o r d i n g t o t h e r u l e t , t h e o b j e c t i s p u t i n t o i f a t l e a s t two o u t of t h r e e p e r s o n s a g r e e category C J' t h a t i t ought t o be p u t t h e r e . If t h e t h r e e c l a s s i f i c a t i o n s a l l disagree, t h e o b j e c t remains u n c l a s s i f i e d
.'
It i s i n t u i t i v e l y o b v i o u s t h a t t h e f u n c t i o n t d e f i n e d
above s h o u l d l e a d t o f e w e r e r r o r s t h a n e a c h o f t h e f u n c t i o n s t l , t 2 , t s e p a r a t e l y , a t l e a s t i n some su3 f f i c i e n t l y r e g u l a r c a s e s . One c a n t h e n pose a problem o f f i n d i n g an optimal c l a s s i f i c a t i o n r u l e , b u i l t out of t h r e e ( o r some o t h e r number) o f f u n c t i o n s t i . The o p t i m a l i t y i s d e f i n e d h e r e t h r o u g h t h e e r r o r m a t r i x €3. The s o l u t i o n depends h e r e n o t o n l y on m a t r i x B , b u t a l s o on t h e c l a s s i f i c a t i o n p r o b a b i l i t i e s , i . e . on t h e values p j ( x ) = P ( o b j e c t x w i l l be c l a s s i f i e d t o C.). J These p r o b a b i l i t i e s may depend on t h e s u b j e c t s on a l s o
on o c c a s i o n s . I n c a s e of c l a s s i f i c a t i o n s o f Type 2 , t h e r e i s no t r u e c a t e g o r y , hence i t i s m e a n i n g l e s s t o s p e a k of e r r o r s of c l a s s i f i c a t i o n . Thus, i t i s n o t p o s s i b l e t o d e f i n e here t h e f u n c t i o n w , n o r m a t r i x B .
...
one may u s e o n l y f u n c t i o n s t l , t 2 , A measure o f q u a l i t y o f c l a s s i f i c a t i o n i n t h i s c a s e A s a consequence,
SELECTED TOPICS IN MEASUREh4ENT THEORY
66 5
rcay be t h e p r o b a b i l i t y t h a t t h e o b j e c t w i l l be c l a s s i f i e d t o t h e same c a t e g o r y under r e p e t i t i o n o f c l a s s i f i -
cation, that i s
where t i and t If P [ t i ( x )
=
j
Ckl
a r e two i n d e p e n d e n t c l a s s i f i c a t i o n s . = p k ( x ) d o e s n o t depend on t h e i n d e x
i (of t h e classifying subject and/or occasion), then
(11.6) I n t h e g e n e r a l c a s e , when i and j s i g n i f y two s u b j e c t s o r two o c c a s i o n s f o r t h e same s u b j e c t , and t h e probab i l i t i e s P [ t i ( x ) = C k ] = p k ( x , i ) , w e have s i m i l a r l y a ( x ; i , j ) = PCti(x) = t j ( x ) l
(11.7) k Now, t h e q u a n t i t y 1 - a ( x ) , which i s e m p i r i c a l l y a c c e s s i b l e , may be r e g a r d e d as a v a l u e o f membership f u n c t i o n i n a c e r t a i n f u z z y s e t , namely t h e f u z z y s e t o f imprecise classifications. I n d e e d , a ( x ) = 1 i f and o n l y i f t h e d i s t r i b u t i o n p,(x) i s c o n c e n t r a t e d a t one c a t e g o r y , s o t h a t t h e r e i s no c o n f u s i o n i n c l a s s i f i c a t i o n s . On t h e o t h e r hand, t h e c l o s e r i s t h i s d i s t r i b u t i o n t o t h e uniform d i s t r i b u t i o n , t h e smaller becomes t h e v a l u e a ( x ) , and i t t e n d s t o 0 when t h e number o f c a t e g o r i e s i n c r e a s e s .
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When one c o n s i d e r s a f a m i l y o f a l l c l a s s i f i c a t i o n schemes, some of them a r e more i m p r e c i s e t h a n o t h e r s . To g i v e
examples, c o n s i d e r on t h e one hand, c l a s s i f i c a t i o n s o f Type 1, s u c h a s c l a s s i f y i n g p e r s o n s i n t o males and f e -
males, e t c . , and on t h e o t h e r hand, c l a s s i f i c a t i o n s which a r e common i n everyday p r a c t i c e , e x p r e s s e d i n n a t u r a l language b y a d j e c t i v e s . Thus, one book may b e " i n t e r e s t i n g " , and t h e o t h e r " d u l l " ( a c l a s s i f i c a t i o n w i t h c a t e g o r i e s s u c h as " d u l l " , " i n t e r e s t i n g " and pos s i b l y o t h e r s ) ; s i m i l a r l y , c i t i e s may be " v e r y l a r g e " , " l a r g e " , " s m a l l " e t c . , Each o f t h e s e c l a s s i f i c a t i o n schemes o f t h e k i n d d e s c r i b e d above b e l o n g s t o t h e f a m i l y o f c l a s s i f i c a t i o n s o f Type 2 , and one may d i s t i n g u i s h among them a f u z z y s u b s e t , c o n s i s t i n g of t h o s e c l a s s i f i c a t i o n s which a r e " i m p r e c i s e " . The v a l u e l - a ( x ) , o r r a t h e r t h e a v e r a g e v a l u e of t h i s q u a n t i t y n (11.8)
.
t a k e n o v e r a s u i t a b l y s e l e c t e d s e t x l,.. ,xn of o b j e c t s i s a v a l u e o f membership f u n c t i o n o f t h e c l a s s i f i c a t i o n
scheme i n q u e s t i o n i n t h e c l a s s of i m p r e c i s e schemes. I n c o n n e c t i o n w i t h t h i s a p p r o a c h , i t i s of c o n s i d e r a b l e i n t e r e s t t o a n a l y s e t h e r e l a t i o n s between i m p r e c i s i o n o f t h e c l a s s i f i c a t i o n schemes, a s measured b y t h e v a l u e of membership f u n c t i o n 7, and r i c h n e s s o f l i n g u i s t i c representation of categories. C o n s i d e r namely a l i n g u i s t i c s c a l e , as d e f i n e d by a p a i r of antonyms , s u c h as " l a r g e - s m a l l "
f u l " , e t c . Denote t h e s e antonyms by a
,
"ugly-beautiand A , and l e t
m s t a n d f o r a moderator, such as ''verytr. One can t h e n form a p o t e n t i a l l y i n f i n i t e number o f c a t e g o r i e s , hence
SELECTED TOPICS IN MEASUREMENT THEORY
667
i n f i n i t e l y many c l a s s i f i c a t i o n schemes, w i t h s e t s o f c a t e g o r i e s s u c h as ( a , A ) , ( m a , a , A , m A ) , ( m m a , ma, a , A , m A , m m A ) , e t c . , where t h e i t e r a t i o n mm o f t h e moder a t o r m may o r may n o t have i t s own l i n g u i s t i c r e p r e s e n t a t i o n ( e . g . "very v e r y l a r g e " as "huge", e t c . ) A s t h e number o f c a t e g o r i e s i n c r e a s e s , t h e c l a s s i f i c a t -
i o n scheme becomes, i n t h e o r y a t l e a s t , more r i c h by p r o v i d i n g b e t t e r means f o r d i s t i n g u i s h i n g t h e o b j e c t s . On t h e o t h e r hand, i t seems p l a u s i b l e t h a t i n most cases t h e q u a n t i t i e s 1 - a ( x ) , hence a l s o ?, w i l l i n c r e a s e , s o t h a t t h e c l a s s i f i c a t i o n s w i l l become more and more i m p r e c i s e . T h i s n e g a t i v e r e l a t i o n between r i c h n e s s and i m p r e c i s -
i o n a c c o u n t s f o r t h e tendency t o w a r d s n u m e r i c a l r e p r e s e n t a t i o n s , accompanied w i t h r e l i a b l e measurement proc e d u r e s . I n d e e d , i t i s hoped t h a t such r e p r e s e n t a t i o n s would p r o v i d e l e s s i m p r e c i s e c l a s s i f i c a t i o n schemes. I n c a s e o f t h e s o c i a l s c i e n c e s , t h i s tendency r e s u l t s i n abundance o f i n d i c e s f o r measurement of such o r o t h e r phenomena. However, w i t h o u t t h e p r o p e r measurement-theor e t i c a l f o u n d a t i o n , such i n d i c e s u s u a l l y provide only s p u r i o u s p r e c i s i o n : i n f a c t , t h e c l a s s i f i c a t i o n remains o n l y as p r e c i s e a s i s a l l o w e d i n t h e n a t u r a l l a n g u a g e , and d i s c r i m i n a t i n g a b i l i t y o f s u b j e c t s .
1 1 . 2 . Dynamic c l a s s i f i c a t i o n s
I n p r a c t i c a l s i t u a t i o n s one d e a l s n o t w i t h o n e , b u t w i t h a f a m i l y o f c l a s s i f i c a t i o n schemes, o f t y p e s 1 and 2 , e a c h f i t t i n g some s e t o f o b j e c t s . Thus, we may now c o n s i d e r a f a m i l y
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(x, c
* ( i ) ,i
= 1,2
,... >
(11.9)
A s b e f o r e , X i s t h e u n i v e r s e of c l a s s i f i e d o b j e c t s ; i t
w i l l a l w a y s be assumed l a r g e enought t o comprise t h e
domains of a1.l c l a s s i f i c a t i o n schemes i n ( 1 1 . 9 ) . The f o l l o w i n g " l i n g u i s t i c " a p p r o a c h t o c l a s s i f i c a t i o n schemes may be o f some i n t e r e s t . One can namely v i s u a l i z e t h e p r o c e s s o f s u c c e s s i v e c l a s s i f i c a t i o n s as a d o u b l e p r o c e s s : t h e f i r s t s t e p c o n s i s t s of c h o o s i n g t h e c l a s s i f i c a t i o n scheme, and t h e second - a c a t e g o r y i n t h i s scheme. F o r m a l l y , we d e a l here w i t h t h e proc e s s o f t h e form (11.10) where i l , i a r e i n d i c e s o f s e l e c t e d schemes o f 2'"' c l a s s i f i c a t i o n ( w i t h p o s s i b l e r e p e t i t i o n s ) , and t i ; ( x ) i s t h e c a t e g o r y o f x on j - t h o c c a s i o n , i n t h e c l a s s i f i c a t i o n scheme w i t h i n d e x i j'
*
We may now i n t r o d u c e t h e monoid I of a l l f i n i t e s t r i n g s * i , and monoid C o f a l l f i n i t e s t r i n g s o f il, i 2 , n categories.
...,
Then a s t r i n g ( l l . 1 0 ) may be r e g a r d e d a s a n element o f * * I x C , i . e . a p a i r of s t r i n g s . Across s t r i n g s o f o c c a s i o n s , t h i s e s t a b l i s h e s a mapping
* *
f: x
4
2 I xc
(11.11)
which t o e a c h x i n X a s s i g n s a s e t f(x) o f d o u b l e s t r i n g s
SELECTED TOPICS IN MEASUREMENT THEORY
669
* *.
i n I XC L e t f ( x ) = CA(x), B ( x ) l , where A ( x ) i s t h e c l a s s o f a l l c a t e g o r i e s from I which may a p p e a r i f x i s c l a s s i f i e d , and s i m i l a r l y for B ( x ) . Finally, let (11.12)
and
Then L1, L and L may be c a l l e d c l a s s i f i c a t i o n l a n g u a g e s . 2 A double s t r i n g u i s admissible ( i s i n L ) i f t h e r e e x i s t s an o b j e c t x which on some o c c a s i o n w i l l be d e s c r i b e d by some p e r s o n t h r o u g h t h e d o u b l e s t r i n g u o f c a t e g o r i e s . N a t u r a l l y , s u c h a c o n c e p t o f a d m i s s i b i l i t y i s f u z z y , and c o n s e q u e n t l y , t h e language L i s a l s o f u z z y . The same a p p l i e s t o l a n g u a g e s L1 and L 2 ; i n t h e s e c a s e s , however, t h e membership f u n c t i o n for L1 and L 2 i s d e t e r m i n e d b y t h e membership f u n c t i o n f o r L . D e n o t i n g for s i m p l i c i t y by ( u , v ) d o u b l e s t r i n g s , w i t h
*
*
u E I and v G C , t h e membership f u n c t i o n , say g L ( u , v ) f o r L d e t e r m i n e s t h e memberships i n L1 and L 2 t h r o u g h
a possibility distribution. Thus, for u C I (u) = gL 1 and
*
SUP*
VE
c
w e have g,(u,v)
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L e t now A be a f u z z y s u b s e t o f t h e u n i v e r s e X o f d i s c o u r s e , d e f i n e d t h r o u g h a membership f u n c t i o n g A ( x ) . We may t h e n a s k q u e s t i o n s s u c h a s : g i v e n t h e c l a s s i f i c a t i o n o f t h e (unknown) x t o be ( u , v ) , what i s t h e deg r e e t o which x i s i n A? Denote t h e l a s t q u a n t i t y by q ( A ; u , v ) . Then, from t h e construction of the p o s s i b i l i t y d i s t r i b u t i o n s , it f o l lows t h a t q ( A ; u , v ) = sup
min C g A ( x ) , q ( x ; u , v ) l
(11.14)
XGX
The knowledge o f
q ( A ; u , v ) f o r a l l A would, o f c o u r s e ,
d e t e r m i n e q ( x ; u , v ) , t h e q u a n t i t y which i s o f i n t e r e s t . However, as a r u l e q ( A ; u , v ) i s known o n l y f o r a l i m i t ed number of s e t s A , and t h e q u e s t i o n may be f o r m u l a t e d
as f o l l o w s : l e t
A ' be t h e c l a s s o f s e t s A f o r which
q ( A ; u , v ) i s d e f i n e d . What i s t h e minimal and m a x i m a l solution q(x;u,v) of equation (11.14)? t
D e n o t i n g t h e s o l u t i o n s by q ( x ; u , v ) and q - ( x ; u , v ) ,
we
may i n t e r p r e t them a s f o l l o w s . The i n f o r m a t i o n i s t h a t some unknown o b j e c t was c l a s s i f i e d , and t h a t i t s "comp o s i t e c a t e g o r y " i s ( u , v ) . We a r e t h e n g i v e n t h e informa t i o n how much p o s s i b l e i t i s t h a t t h e d e s c r i b e d o b j e c t
i s i n some s e t A from t h e c l a s s A ' .
We want t o e v a l u a t e
q ( x ; u , v ) , i . e . t h e p o s s i b i l i t y t h a t t h e d e s c r i b e d obj e c t was x . Unless t h e c l a s s A ' i s s u f f i c i e n t l y r i c h , t h e exact knowledge o f q ( x ; u , v ) i s n o t a t t a i n a b l e ; one c a n , howe v e r , f i n d t h e maximal and minimal s o l u t i o n , and t h e n
w e have
SELECTED TOPICS IN MEASUREMENT THEORY
The i d e a l c a s e , o f c o u r s e , i s when q
t
= q
-
671
, and
= 1 f o r some x o J w i t h q ( x ; u , v ) = 0 f o r a l l x
c o r r e s p o n d s t o unambiguous and non-fuzzy ion of x
q(xo;u,v)
# x 0' T h i s
identificat-
on t h e b a s i s o f ( u , v ) . I n g e n e r a l , however, ot
# q-, s o t h a t a t l e a s t f o r some x t h e d e g r e e o f p o s s i b i l i t y t h a t t h e o b j e c t c l a s s i f i e d as ( u , v ) was x i s d e t e r m i n e d o n l y up t o c e r t a i n bounds.
w e have q
The i m p o r t a n c e of t h e above c o n s i d e r a t i o n s l i e s i n t h e
f a c t t h a t i t s t r e s s e s a t l e a s t t h r e e s o u r c e s of uncert a i n t y i n determining x given t h e information ( u , v ) . One o f them i s i n h e r e n t i n any c l a s s i f i c a t i o n scheme, and c o n s i s t s simply of t h e f a c t t h a t t h e i n v e r s e image o f ( u , v ) may c o n t a i n more t h a n one e l e m e n t . The second s o u r c e o f u n c e r t a i n t y l i e s i n i m p r e c i s i o n , and can i n p r i n c i p l e be removed: i t i s c o n n e c t e d w i t h t h e f a c t t h a t t h e p o s s i b i l i t y d i s t r i b u t i o n i s d e f i n e d o n l y f o r some t
s e t s A . T h i s , i n g e n e r a l , g i v e s q # q-, a t l e a s t f o r some a r g u m e n t s , and makes i t i m p o s s i b l e t o d e t e r m i n e t h e membership f u n c t i o n q ( x ; u , v ) . F i n a l l y , t h e t h i r d source i s i n h e r e n t i n t h e vagueness + o f t h e c o n c e p t s u s e d . Even i f q = q , s o t h a t q ( x ; u , v ) i s u n i q u e l y d e t e r m i n e d , i t may s t i l l happen t h a t we have 0 ;q ( x ; u , v ) 1 f o r some x . Then t h e s e t o f o b j e c t s
x d e s i g n a t e d by q ( x ; u , v ) i s i n h e r e n t l y f u z z y . It i s w o r t h t o mention t h a t t h e r e a r e s t r i n g s o f c a t e -
g o r i e s o f c l a s s i f i c a t i o n which a r e t h e m s e l v e s c a t e g o r i e s i n a h i g h e r o r d e r c l a s s i f i c a t i o n s . Thus, t o a s t r i n g (u,v) of categories there corresponds a c l a s s Ci j i n i - t h c l a s s i f i c a t i o n ; t h i s c l a s s , i n general, i s fuzzy, which c o r r e s p o n d s t o fuzzy c o n c e p t s , as d i s c u s s e d i n Chapter 3.
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1 1 . 3 . L i n g u i s t i c measurement L e t u s now c o n s i d e r some l i n g u i s t i c problems c o n n e c t e d
w i t h c l a s s i f i c a t i o n and measurement. F i r s t o f a l l , i t i s c l e a r t h a t w e d e a l here n o t w i t h o n e , b u t w i t h many
c l a s s i f i c a t i o n schemes a t t h e same t i m e , as i n t h e p r e c e d i n g s e c t i o n . Moreover, many o f t h e s e schemes form " s c a l e s " , i . e . c a t e g o r i e s a r e l i n e a r l y o r d e r e d i n some n a t u r a l w a y . I n a number o f c a s e s , t h e s e c a t e g o r i e s may be m o d i f i e d by " s t r e t c h i n g " o r " e n r i c h i n g " t h e s c a l e s (by a p p l y i n g m o d i f i e r s , as mentioned i n t h e p r e c e d i n g s e c t i o n s ) . F i n a l l y , t h e r e i s a p o s s i b i l i t y of forming new c a t e g o r i e s "ad hoc", and a l s o i n t r o d u c e o p e r a t l o n s on c a t e g o r i e s , such as a l t e r n a t i v e or c o n j u n c t i o n , leadi n g t h e r e f o r e t o composite c a t e g o r i e s . The s e n t e n c e s e x p r e s s i n g c l a s s i f i c a t i o n s may be j o i n e d by o p e r a t i o n s o f c o n j u n c t i o n , i m p l i c a t i o n , e t c . ,
and a l s o p r e c e d e d b y q u a n t i f i e r s ( e . g . " f o r almost a l l x").
An i m p o r t a n t s p e c i a l c a s e c o n c e r n s c l a s s i f i c a t i o n of events or sentences according t o categories expressing possibilities or truth. F o r m a l l y , we have h e r e a system o f t h e form
where X i s a g a i n t h e u n i v e r s e o f c l a s s i f i e d o b j e c t s . The r e m a i n i n g t h r e e e l e m e n t s c o n s t i t u t e a l i n g u i s t i c s p a c e , w i t h t h e s t r u c t u r e as d e s c r i b e d below. Depending on t h e c o n t e x t , t h e e l e m e n t s o f X may be obj e c t s , r e a l or imagined, e v e n t s , p s y c h o l o g i c a l s t a t e s ,
SELECTED TOPICS IN MEASUREMENT THEORY
e t . Also; t h e o b j e c t s
673
x i n X may t h e m s e l v e s form com-
plex r e l a t i n n a l structures. The l i n g u i s t i c s p a c e lows.
(W,
G , GI) i s i n t e r p r e t e d as fol-
F i r s t of a l l , W i s a s e t , whose e l e m e n t s are
a d m i s s i b l e names of c a t e g o r i e s . The s e t W c o n t a i n s exp r e s s i o n s s u c h as " b l a c k " , " l o n g " , e t c . , as w e l l a s composite e x p r e s s i o n s , e . g . "white as snow", and s o o n . I t i s i m p o r t a n t t o o b s e r v e t h a t W may a l s o c o n t a i n v e r b s ,
as c l a s s i f i e r s of a c t i o n s . N e x t , G i s a f a m i l y of c l a s s i f i c a t i o n s y s t e m s , formed o u t of e l e m e n t s of W . Thus,
where e a c h
C(i)
i s of t h e form
w(i) i n W . The c a t e g o r i e s need n o t be o r d e r e d i n j any n a t u r a l way.
with
i s t h e name of The i n t e r p r e t a t i o n h e r e i s t h a t w (i' j j - t h c a t e g o r y i n t h e c l a s s i f i c a t i o n C ( i ) . The examples h e r e may be t h e names of c o l o u r s , names o f f a m i l y r e l a t i o n s h i p s , c l a s s e s o f v e r b s of m o t i o n , e t c ,
F i n a l l y , GI i s a s u b f a m i l y of G , c o n s i s t i n g o f t h o s e c l a s s i f i c a t i o n schemes i n which t h e c a t e g o r i e s a r e o r d e r e d i n a n a t u r a l w a y . The examples h e r e may be s y s tems of c a t e g o r i e s s u c h as ( s m a l l , medium l a r g e ) , o r ( v e r y small, s m a l l , medium, l a r g e , v e r y l a r g e ) , e t c . The e l e m e n t s o f G ' may be r e l a t e d one to a n o t h e r ; t h e
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614
most i m p o r t a n t r e l a t i o n s h e r e a r e s t r e t c h i n g and r e f i nement o f s c a l e s . F o r m a l l y , t h e s e r e l a t i o n s may be d e s c r i b e d a s f o l l o w s . Let C =
? W ~ ~ . . . , W and ~ ~ C f
b e two e l e m e n t s
={W~,...~W;]
o f G', w r i t t e n i n t h e i r n a t u r a l o r d e r .
,...,
<
m , and t h e r e of C' s u c h t h a t w1
We s a y t h a t C i s embedded i n C f , i f n
e x i s t s a subsequence w 1 wi' n = w ' w' y . . . , il wn in' Moreover, C ' i s a n e x t e n s i o n o f C, i f t h e f o l l o w i n g
=
c o n d i t i o n s h o l d : ( a ) C i s embedded i n C f , ( b ) i l y . . . y in i s a sequence of c o n s e c u t i v e numbers; C ' i s a s i m p l e r e f i n e m e n t of C, if ( a ) C i s embedded i n C f , and ( b ) i l
-- 1, in = m. The i n t u i t i v e c o n t e n t o f t h e s e d e f i n i t i o n s i s as f o l lows. Embedding o f one s y s t e m i n a n o t h e r means simply t h a t t h e s y s t e m embedded i s l i n g u i s t i c a l l y p o o r e r , i n t h e s e n s e t h a t i t c o n t a i n s o n l y some of t h e c a t e g o r i e s o f t h e r i c h e r system. The l a t t e r i s a n e x t e n s i o n , i f t h e a d d i t i o n a l categories (not appearing i n C ) occur o n l y a t t h e end p o i n t s o f t h e s c a l e . I f t h e e n d s remain t h e same, w h i l e t h e new c a t e g o r i e s a p p e a r i n between t h e o l d o n e s , we have s i m p l e r e f i n e m e n t . The most f r e q u e n t l y o c c u r r i n g r e f i n e m e n t of a l i n g u i s t i c s c a l e i s connected w i t h t h e use o f m o d i f i e r s , e . g . t t v e r y " , mentioned b r i e f l y i n t h e p r e c e d i n g s e c t i o n . C l e a r l y , we have THEOREM. The r e l a t i o n of embedding i s t r a n s i t i v e , i . e . i f C i s embedded i n C f
C i s embedded i n C".
, and
C' i s embedded i n C", t h e n
SELECTED TOPICS IN MEASUREMENT THEORY
675
Another o p e r a t i o n i s g r o u p i n g t h e c a t e g o r i e s , which c o r r e s p o n d s t o d i s j u n c t i o n o f c l a s s e s , and p o s s i b l y ass i g n i n g a new name t o t h e group. The most common o p e r a t i o n , however, i s c r o s s i n g two ,wn\and C ' = { w i , . classifications C = )w 1,' Then C x C ' i s o b t a i n e d as t h e C a r t e s i a n p r o d u c t o f ,w &w; Such an o p e r a t i o n t h e form w l & w i , w & w ' 1 2'"' n l e a d s t o a n element o f G , even i f b o t h C and C' a r e i n G I , i . e . t h e r e e x i s t s no n a t u r a l o r d e r i n g of e l e m e n t s o f c x C'.
. . ,"A>.
..
.
C o n s i d e r now a g a i n t h e p r o c e s s o f c l a s s i f i c a t i o n , i . e . c o n s i d e r t h e c l a s s i f i c a t i o n s d y n a m i c a l l y , as t h e p r o connected w i t h t h e obc e s s ( i k ,t i ( x ) ) , k = 1 , 2 , , k j e c t x.
..
I n o r d e r t o i s o l a t e s i t u a t i o n s which a r e amenable f o r m a t h e m a t i c a l and s t a t i s t i c a l a n a l y s i s , l e t u s t r y t o f i n d what t y p e of r e g u l a r i t i e s may be e x p e c t e d i n ( a t l e a s t some) of t h e p r o c e s s e s . L e t u s f i x some o b j e c t x , and c o n s i d e r t h e p r o c e s s o f i when x i s d i s c u s s e d ( m e n t i o n e d , moments i 1, c l a s s i f i e d t o c a t e g o r y t (x)).
*,...
ik
We make t h e f o l l o w i n g i d e a l i z e d a s s u m p t i o n s . The o b j e c t
x may b e c a t e g o r i z e d on N c l a s s i f i c a t o r y schemes, and t h e r e a r e no r e p e t i t i o n s , i n t h e s e n s e t h a t once a g i v e n c l a s s i f i c a t o r y scheme i s u s e d , i t cannot be used a g a i n . Thus, t h e p r o c e s s . m u s t end a f t e r a t most N t i m e s , A t e n t a t i v e h y p o t h e s i s which might a c c o u n t f o r t h e tem-
...
p o r a l c h a r a c t e r i s t i c s o f t h e p r o c e s s , s a y T1, T 2 , where Tk i s t h e t i m e o f a p p e a r e n c e o f i k , i s as follows.
616
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A t each u n i t o f t i m e t h e person ( s p e a k e r ) s e l e c t s an
element from an u r n c o n t a i n i n g N e l e m e n t s marked l , . . . , N and M b l a o k : ? b a l l s ( r e p r e s e n t i n g c l a s s i f i c a t i o n s o f other objects). Each t i m e one o f t h e marked N eleme'nts i s s e l e c t e d , i t i s checked whether t h i s element has been a l r e a d y s e l e -
c t e d b e f o r e . If i t h a s , i t i s r e t u r n e d , o t h e r w i s e t h e speaker u t t e r s a sentence concerning t h e c l a s s i f i c a t i o n o f x a c c o r d i n g t o t h e sampled c l a s s i f i c a t i o n scheme (and t h e sampled element i s r e t u r n e d ) . If a b l a c k b a l l i s drawn, it i s simply r e t u r n e d t o t h e u r n .
Suppose t h a t t h e count o f t i m e s t a r t s a t 0 . Then P(T1 = k ) = [M/(MtN)]k-l[N/(M+N)]. A t t h e c h o i c e o f t h e f i r s t marked element ( a t t i m e T1),
t h e number o f b a l l s f a v o u r i n g t h e mention o f x d e c r e a s e s t o N - 1 , w h i l e t h e number o f r e m a i n i n g e l e m e n t s i n c reases t o M t l . Therefore
P(T2
= k+j!T1
= k) =
G e n e r a l l y , f o r m = 1,2, P(Tm = k t j
I Tm-l
[(M+l)/(M+N)]'-l[(N-l)/(M+N)I.
...,N-I
= k)
= [(Mtm-l)/(MtN)]j-l
[(N-mtl)/(N+M)l.
Thus, t h e " i t e r - c l a s s i f i c a t i o n t i m e s " Tm g e o m e t r i c a l l y d i s t r i b u t e d , w i t h mean
-
Tm-l
are
SELECTED TOPICS LV MEAS UREMENT THEORY
677
C o n s e q u e n t l y , t h e w a i t i n g time f o r rn-th c l a s s i f i c a t i o n of o b j e c t x equals
...
t
4).
N
I f b o t h M and N a r e l a r g e , t h e sum i n p a r e n t h e s e s may
be approximated by log N - l o g ( M - m ) ,
s o t h a t we have
S i m i l a r f o r m u l a s may a l s o b e d e r i v e d for t h e v a r i a n c e o f Tm and v a r i a n c e of Trn - T m - l * C o n s e q u e n t l y , o b s e r v a t i o n s o f c o n s e c u t i v e mentions of o b j e c t x i n a c o n v e r s a t i o n ( s a y ) might p r o v i d e means for e s t i m a t i n g t h e c o n s t a n t s M and N ; t h e l a t t e r cons t a n t s r e p r e s e n t t h e " r i c h n e s s " o f o b j e c t x , and r i c h n e s s o f t h e whole d i s c o u r s e . The above model c o i n c i d e s w i t h t h e s o - c a l l e d c o l l e c t o r model ( s e e F e l l e r 1957), e x c e p t t h a t we have h e r e t h e inclusion o f constant M. A very important s p e c i a l c a s e of l i n g u i s t i c c l a s s i f i c a -
t i o n schemes i s when t h e c l a s s i f i e d o b j e c t s a r e t h e c l a s s i c a t o r y p r o p o s i t i o n s , and c a t e g o r i e s r e f e r t o v a r i o u s p s y c h o l o g i c a l c o n t i n u a , s u c h as s u b j e c t i v e probab i l i t y , u t i l i t y , d e o n t i c c o n t i n u a , e t c . We have h e r e two c l a s s i f i c a t i o n schemes o f t h e t y p e s d e s c r i b e d abo* v e . The f i r s t i s as b e f o r e , o f t h e form ( X , C ) , w i t h * X b e i n g t h e s e t o f o b j e c t s , and C b e i n g t h e c l a s s o f c a t e g o r i e s . The second scheme i s s u c h t h a t i t s o b j e c t s and c a t e g o r i e s a r e p r o p o s i t i o n s o f t h e form x E C i' The g e n e r a l form a r e v a r i o u s modal frames M I , M2,
... .
678
CHAPTER 5
of a c l a s s i f i c a t i o n i s t h e n "x
Cilf
C- M
more c o n v e n i e n t l y w r i t t e n as M.Cx 6 C i l .
j'
which i s
J
The a n a l y s i s of f o r m a l s t r u c t u r e of such schemes, i n which t h e c a t e g o r i e s M1, M2, a r e s p e c i a l t y p e s of modal f r a m e s , namely t h e m o t i v a t i o n a l f u n c t o r s , i s giv-
...
e n i n Nowakowska ( 1 9 7 3 ) . The f u n c t o r s c o n s i d e r e d t h e r e were o b t a i n e d by a b s t r a c t i o n from t h e c o n t e n t o f sent e n c e s used i n e x p l a i n i n g , j u s t i f y i n g , e v a l u a t i n g , planning, e t c . the actions i n t h e past o r future. In constructing the motivational calculus, the basic n o t i o n was t h a t of s e m a n t i c i m p l i c a t i o n . The c a l c u l u s c o n t a i n s c e r t a i n i m p l i c a t i o n s , as e x e m p l i f i e d by Dp j T ( - p ) & - C r ( - p ) , which i s t o be r e a d t h a t "I doubt p " i m p l i e s t h a t "I t h i n k t h a t n o t p , and I a m n o t c e r t a i n that not p". By c o n s t r u c t i o n o f a s e t - t h e o r e t i c a l model, i t was po-
s s i b l e t o p r o v e t h e c o n s i s t e n c y of t h e c a l c u l u s . The i m p l i c a t i o n s of m o t i v a t i o n a l c a l c u l u s may be r e g a r d ed as r u l e s of t r a n s f o r m a t i o n of s e n t e n c e s , which p r e s e r v e t h e i r meaning. T h i s , i n f a c t , was t h e f i r s t work i n t h e f o u n d a t i o n s of approximate r e a s o n i n g i n t h e nat u r a l language. I n t h i s a n a l y s i s , f o r t h e f i r s t time t h e meaning of a s e n t e n c e was n o t a n a l y s e d t h r o u g h t h e concept of t r u t h , b u t t h r o u g h t h e n o t i o n o f s e m a n t i c i m p l i c a t i o n , b e i n g an e x t e n s i o n of t h e m a t e r i a l i m p l i cation. It i s i m p o r t a n t t o remark t h a t m o t i v a t i o n a l c a l c u l u s
a l l o w s f o r i t e r a t i o n of m o t i v a t i o n a l f r a m e s , and g i v e s r u l e s o f t r a n s f o r m a t i o n of some frames i n t o o t h e r s . Also, v a r i o u s groups o f frames a r e e l e m e n t s o f GI, i . e
SELECTED TOPICS IN MEASUREMENT THEORY
679
t h e y form c e r t a i n s c a l e s , b e i n g r e p r e s e n t a t i o n s o f d i f f e r e n t p s y c h o l o g i c a l c o n t i n u a . Taken j o i n t l y , t h e s e continua c o n s t i t u t e a multidimensional cognitive space, c a l l e d motivational space. G e n e r a l l y , a c l a s s i f i c a t o r y scheme from GI c o r r e s p o n d s t o a p e r c e p t u a l s c a l e . I n most c a s e s , i t c o n s i s t s of f o u r groups of e l e m e n t s : (1) a b a s i s , formed o u t o f a p a i r o f antonyms, s u c h as "small-large",
e t c . , ( 2 ) mod i f i c a t i o n s o f t h e b a s e o b t a i n e d by a p p l y i n g m o d i f i e r s , i n t e n s i f i e r s , o r m o d e r a t o r s , s u c h as " v e r y " or " r a t h e r " , e t c . , ( 3 ) e x p r e s s i o n s b u i l t o u t o f e l e m e n t s o f (2), and (4) a d d i t i o n a l e x p r e s s i o n s , s u c h as a n a l o g i e s or metap h o r s , which a r e formed i n ways o t h e r t h a n t h o s e d e s c ribed i n ( 2 ) . L e t M and m d e n o t e t h e i n t e n s i f i e r and m o d e r a t o r . The k i t e r a t i o n s o f them w i l l be d e n o t e d by m k and M , s o t h a t m 0 x = x , M 0 x = x, and m (k+l)x = m(m k x), M ( k + l ) x = k M ( M x ) , where x i s e i t h e r a or A , i . e . t h e b a s e . I n a d d i t i o n t o p o s t u l a t i n g t h a t a < A , we p o s t u l a t e hek k r e t h a t t h e s e q u e n c e s M A and m a a r e s t r i c t l y i n c r e k k a s i n g , w h i l e t h e s e q u e n c e s M a and M a a r e s t r i c t l y dek
c r e a s i n g . Moreover, f o r a l l k and r we have m a
< m1"A .
N a t u r a l l y , h i g h o r d e r i t e r a t i o n s are d i f f i c u l t , i f n o t i m p o s s i b l e t o d i s c r i m i n a t e one from a n o t h e r . Assuming, f o r i n s t a n c e , t h a t k = 2 i s t h e maximal admis s i b l e number o f i t e r a t i o n s , we g e t a n o r d i n a l s c a l e with 10 points
MMa, Ma, a , m a , mma,
mmA, m A , A , M A , MMA.
The most i n t e r e s t i n g h e r e i s t h e o p e r a t i o n o f t h e func-
680
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t o r o f n e g a t i o n . While for e l e m e n t s of G ( u n o r d e r e d c l a s s i f i c a t o r y s y s t e m s ) , t h e n e g a t i o n l e a d s t o t h e complement o f t h e s e t of c a t e g o r i e s n e g a t e d , t h e n e g a t i o n o f a n e x p r e s s i o n on t h e s a c l e under c o n s i d e r a t i o n g i v e s a p a r t o f t h e s c a l e e x t e n d i n g from t h e n e g a t e d f u n c t o r towards t h e o t h e r antonym. T h i s a l l o w s u s t o b u i l d " i n t e r v a l " by c o n j u n c t i o n of n e g a t i o n s . F o r i n s t a n c e -ma = x '> ma-) , and -mA = f x : x < mA], s o t h a t
1.:
-ma & -mA
=
ix: m a
<
x
< mA.3
.
A s an example, one may compare e x p r e s s i o n s such as
"not b l u e " ( i n which c a s e i t may be o f any c o l o u r ) , and "not l a r g e " ( i n which c a s e t h e v a l u e s t o t h e r i g h t from " l a r g e " , s u c h as " v e r y l a r g e " are e x c l u d e d ) .
1 1 . 4 . C o n s t r u c t i o n of l i n g u i s t i c s c a l e s : s n a l o g i e s and metaphors The e l e m e n t s o f t h e l i n g u i s t i c s c a l e d i s c u s s e d t h u s
f a r , d e s p i t e the t h e o r e t i c a l p o s s i b i l i t y of b u i l d i n g a n i n f i n i t e number o f e x p r e s s i o n s , are n e v e r t h e l e s s r a t h e r p o o r , s i n c e t h e i t e r a t i o n s o f m o d i f i e r s and i n t e n s i f i e r s do n o t a l w a y s g i v e new meaning. F o r f u r t h e r e x t e n s i o n of t h e s c a l e , i n t h e s e n s e of i n -
c r e a s i n g t h e number of c a t e g o r i e s and i n c r e a s e of d i s c r i m i n a t i v e n e s s , one u s e s a n a l o g i e s , and t h e i r s p e c i a l forms, namely metaphors. The c l a s s o b t a i n e d by u s e o f a n a l o g i e s i s o f t h e form "such as N", where N i s a name of some o b j e c t , which i s t o be compared w i t h t h e o b j e c t x w i t h r e s p e c t t o some c h a r a c t e r i s t i c . T h i s
SELECTED TOPICS IN MEASUREMENT THEORY
68 1
amounts t o i n t r o d u c i n g a d d i t i o n a l c l a s s i f i c a t i o n c a t e g o r i e s , l a b e l e d by names o f o b j e c t s N and t h e f e a t u r e s w i t h r e s p e c t t o which t h e comparisons a r e t o be made. Formally, . w e w r i t e x = N , which means t h a t x i s an,q "such a s N" w i t h r e s p e c t t o t h e f e a t u r e q . The e q u a l i t y i n t h e above f o r m u l a i s a f u z z y r e l a t i o n : i t s membership f u n c t i o n may b e d e s c r i b e d as f o l l o w s . If q ( x ) and q ( N ) a r e t h e v a l u e s o f t h e t r a i t i n q u e s t i o n f o r t h e o b j e c t x and o b j e c t N , t h e n t h e d e g r e e t o which x i s a n a l o g o u s t o N w i t h r e s p e c t t o q , i s some f u n c t i o n F o f t h e arguments q ( x ) and q ( N ) , t h a t i s , F [ q ( x ) , q ( N ) ] . C l e a r l y , i f q ( x ) = q ( N ) , t h e n F = 1. It i s w o r t h t o o b s e r v e , however, t h a t F need n o t be symm e t r i c w i t h r e s p e c t t o i t s arguments. T h i s means t h a t i s x i s a n a l o g o u s t o N w i t h r e s p e c t t o q , t h e n i t need n o t be t r u e t h a t N i s analogous t o x w i t h r e s p e c t t o q I n t r o d u c t i o n o f a n a l o g i e s t o a g i v e n s c a l e , i n form of categories N N2, enriches t h e s c a l e , by increasing i t s discriminativeness.
...,
However, as i n t h e c a s e o f i n t e n s i f i e r s , a l s o h e r e t h e r e e x i s t s a l i m i t beyond which t h e s c a l e c e a s e s t o be e n r i c h e d . One can f o r m u l a t e a h y p o t h e s i s t h a t t h e r i c h n e s s o f l i n g u i s t i c r e p r e s e n t a t i o n o f a g i v e n s c a l e depends on t h e p o s s i b i l i t i e s o f p e r c e p t i o n , and more p r e c i s e l y -- on t h e p o s s i b i l i t i e s o f d i s c r i m i n a t i n g s t i m u l i with respect t o a given c h a r a c t e r i s t i c s . For example, f o r v i s u a l s t i m u l i , c l a s s i f t c a t i o n on s c a l e s i s u s u a l l y r i c h . On t h e o t h e r hand, f o r d e o n t i c s c a l e s ( d e g r e e s t o which one ought t o do s o m e t h i n g ) , t h e d i s c r i m i n a t i o n i s r a t h e r p o o r , and t h i s i s r e f l e c t e d i n t h e rather poor l i n g u i s t i c r e p r e s e n t a t i o n of such sca-
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l e . However, d e o n t i c s c a l e may b e e n r i c h e d b y expr e s s i o n s which r e f e r to s i t u a t i o n s i n which a g i v e n a c t i o n i s necessary o r i m p o r t a n t . F o r a thorough d i s c u s s i o n o f t h i s and o t h e r s c a l e s , s e e Nowakowska (1973). Somewhat more g e n e r a l l y , one may c o n s i d e r t h e r e l a t i o n o f analogy and i t s l i n g u i s t i c r e p r e s e n t a t i o n as follows. F i r s t l y , l e t u s o b s e r v e t h a t t h e r e a r e a t l e a s t two
a p p r o a c h e s p o s s i b l e h e r e . One i s t o t r e a t t h e r e l a t i o n o f a n a l o g y a s a p r i m i t i v e n o t i o n , and t r y t o c h a r a c t e r i z e it through postulated p r o p e r t i e s . Thus, d e n o t i n g by X t h e c l a s s o f o b j e c t s under c o n s i deration, w e introduce t h e r e l a t i o n A (analogy) i n X , s o t h a t xAy means t h a t x i s a n a l o g o u s t o y .
Out o f t h r e e b a s i c p r o p e r t i e s , r e f l e x i v i t y , symmetry and t r a n s i t i v i t y , o n l y t h e f i r s t may b e assumed to h o l d : xAx, which means t h a t x i s a n a l o g o u s t o x . I f one p o s t u l a t e s symmetry o f A , t h e n analogy becomes a t o l e r a n c e r e l a t i o n ; i n a d d i t i o n , i f one assumes t r a n s i t i v i t y , one g e t s a n e q u i v a l e n c e . However, symmetry i s d e b a t a b l e , and a t b e s t may be t a k e n o n l y a s a c r u d e a p p r o x i m a t i o n . On t h e o t h e r hand, t r a n s i t i v i t y i s too s t r o n g , and one c a n e x p e c t some k i n d o f " d i s s i p a t i o n " o f a n a l o g y : i f xAy and yAz t h e n n o t n e c e s s a r i l y xAz.
,
The second a p p r o a c h t o t h e c o n c e p t o f analogy c o n s i s t s o f t r y i n g to b u i l d i t o u t o f some o t h e r r e l a t i o n s ; i n t h i s c a s e , t h e p r o p e r t i e s of t h e c o n s t r u c t e d analogy r e l a t i o n w i l l be consequences o f t h e assumed p r o p e r t -
SELECTED TOPICS INMEASUREMENT THEORY
683
i e s of t h e c o n s t i t u e n t r e l a t i o n s . Here t h e most p r o m i s i n g d e f i n i t i o n seems t h e f o l l o w i n g . We p o s t u l a t e t h a t e l e m e n t s o f X a r e systems o f some s o r t . We s a y t h a t x and x ' are a n a l o g o u s , i f t h e r e exi s t s u b s y s t e m s x l c x and x'; c x f w i t h x1 and x i b e i n g i s o m o r p h i c . The l a r g e s t subsystems x l , x i w i t h t h i s p r o p e r t y i n d i c a t e t h e d e g r e e o f analogy between x and X'.
Under s u c h a d e f i n i t i o n o f a n a l o g y , i t i s c l e a r l y a t o l e r a n c e r e l a t i o n , i . e . i t i s r e f l e x i v e and symmetric. Whether or n o t i t i s t r a n s i t i v e depends on f u r t h e r cond i t i o n s imposed i n d e f i n i t i o n , e . g . e x i s t e n c e o f a c o r e o f a n a l o g y , i . e . a common subsystem i n a c h a i n o f pairwise analogous s y s t e m s . F o r m a l l y , x and X I a r e a n a l o g o u s w i t h c o r e z , i f t h e r e e x i s t s u b s y s t e m s x 2 c x 1 c x and X I2 c X I1 c x f , s u c h t h a t x2 x; z , and x w i t h N standing f o r 1' isomorphism. I n t h i s c a s e , d e n o t i n g by A Z t h e r e l a t i o n o f a n a l o g y w i t h c o r e z , we have t r a n s i t i v i t y , i . e . i f x1AZx2 and x2AZx3, t h e n x1AZx3. N
-
xi
11.5. Dimensionality of descriptions C o n s i d e r a g a i n d e s c r i p t i o n s o f o b j e c t s , i n form of S t r i n g s
of c l a s s i f i c a t o r y Such d e s c r i p t i o n s 1 1 . 2 , where u was mes, and v -- t h e
schemes and c o r r e s p o n d i n g c a t e g o r i e s . were d e n o t e d by (u',v) i n S e c t i o n t h e s t r i n g o f c l a s s i f i c a t o r y schecategories.
Suppose now t h a t w e c o n s i d e r n o t o n l y t h e c l a s s X of
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o b j e c t s , but a l s o t h e Class, s a y P , o f s u b j e c t s / o c c a s i o n s . F o r e a c h p i n P and x i n X , we may t h e n have a d e s c r i p t i o n , r ( x ) = (u ( x ) , v ( x ) ) , o f o b j e c t x , as P P P s p e c i f i e d by p e r s o n / o c c a s i o n p . Suppose f o r a moment t h a t we may i n t r o d u c e a d i s t a n c e , s a y d , i n t h e c l a s s o f a l l d e s c r i p t i o n s r ( x ) ; moP r e g e n e r a l l y , w e may assume t h a t d i s some measure o f d i s s i m i l a r i t y , not n e c e s s a r i l y s a t i s f y i n g t h e postul a t e s for distance function. One may t h e n e x p e c t t h a t d [ r ( x ) , r ( y ) ] w i l l r e f l e c t P q b o t h t h e d i f f e r e n c e s between o b j e c t s x and y , and a l s o between p e r s o n s and o c c a s i o n s . Thus, i n g e n e r a l , d [ r p ( x ) , r (x)] need n o t be e q u a l z e r o , and w i l l r e f q l e c t t h e d i f f e r e n c e s i n p e r c e p t i o n o f x, as w e l l as l i n g u i s t i c h a b i t s of p e r s o n s , e t c . Suppose now t h a t w e have t h e d a t a on v a l u e s o f d from a s e t o f c l a s s i f y i n g s u b j e c t s , and f o r a s e t o f o b j e c t s . Such d a t a may be used f o r m u l t i d i m e n s i o n a l s c a l i n g , as d e s c r i b e d i n C h a p t e r 2 . Here one c o u l d c o n j e c t u r e t h a t (a) t h e r e s u l t i n g dimensionality of t h e space o f c l a s s i f i c a t i o n s w i l l b e a t l e a s t as l a r g e as t h e d i m e n s i o n a l i t y of t h e s p a c e o f o b j e c t s ; ( b ) t h e e f f e c t due t o t h e d i m e n s i o n a l i t y o f p e r s o n s p a c e w i l l be cons t a n t a c r o s s s e t s o f o b j e c t s , The l a s t c o n j e c t u r e i s due t o t h e i n t u i t i o n , a c c o r d i n g t o which t h e s e t o f p e r s o n s has some " i n h e r e n t " d i f f e r e n t i a t i o n , which i n f l u e n c e t h e r e s u l t , and d i f f e r e n t i a t i o n imposed on i t by t h e e x p e r i m e n t e r s h o u l d have no e f f e c t on t h e result. I f s u c h a c o n j e c t u r e were t r u e , i t c o u l d be c a l l e d l a w o f r e l a t i v e s e m a n t i c i n v a r i a n c e o f judgments.
685
SELECTED TOPICS IN MEASUREMENT THEORY
1 1 . 6 . Measurement by analogy I n t h i s s e c t i o n w e s h a l l t r e a t i n some g e n e r a l i t y t h e i n f o r m a t i o n about t h e o b j e c t s u p p l i e d by measurement b y a n a l o g y . Somewhat more g e n e r a l l y , we s h a l l c o n s i d e r An o f a t t r i b u t e s e t X o f o b j e c t s x , and s e t s A 1 , . . . , v a l u e s . Each o b j e c t x ( X w i l l be assumed t o p o s s e s s a v a l u e o f e a c h of t h e a t t r i b u t e s ; t h i s means t h a t for e a c h i = 1,. , n w e have a f u n c t i o n
..
fi:
X -3 A i ,
w i t h f i ( x ) b e i n g i n t e r p r e t e d as t h e v a l u e o f i - t h a t -
t r i b u t e of o b j e c t x .
...,
...
Let F = ( f l , ,fn), so that F(x) = ( f l ( x ) , fn(x)) i s t h e v e c t o r o f a t t r i b u t e v a l u e s o f x. Of c o u r s e , A1 x . . . x A n . F(x) We may now d e f i n e x
y, i f F(x) = F(y), so that the r e l a t i o n r v i d e n t i f i e s a l l o b j e c t s w i t h t h e same v a l u e s of a t t r i b u t e s .
...
Moreover, for z = ( Z l , . . . y ~ n ) G A1 x x An, l e t [z] be t h e c l a s s of a l l o b j e c t s w i t h F ( x ) = z , i . e . objects identical w i t h respect t o a l l a t t r i b u t e s . L e t G ( x ; z ) be t h e p o s s i b i l i t y d i s t r i b u t i o n on [ z l , s o
t h a t G ( x ; z ) i s t h e d e g r e e of p o s s i b i l i t y t h a t t h e o b j e c t d e s c r i b e d b y z I s x.
More g e n e r a l l y , i f C i s a ( c r i s p ) s e t o f v e c t o r s z, then G(x;C) =
sup G ( x ; z ) z
cc
686
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i s t h e p o s s i b i l i t y t h a t t h e o b j e c t i s x, g i v e n i t s
at.tributes being i n C. If t h e s e t C i s fuzzy, w i t h membership f u n c t i o n f C ( z ) - j t h e n G ( x ; C ) i s d e f i n e d as G(x;C) =
min [ f C ( z ) , G ( x ; z ) l .
sup Z
Suppose now t h a t we have a l s o some o t h e r s e t o f o b j e c t s , whose e l e m e n t s w i l l be d e n o t e d g e n e r a l l y b y N , w i t h o r w i t h o u t i n d i c e s . These o b j e c t s w i l l s e r v e t o b u i l d sentences containing analogies. We s h a l l a n a l y s e a somewhat more g e n e r a l s i t u a t i o n t h a n
i n v o l v i n g a n a l o g i e s , namely t h e s e n t e n c e s o f t h e form x gi N , where a r e t o be i n t e r p r e t e d as "x i s l e s s o r e q u a l t h a n N w i t h r e s p e c t t o a t t r i b u t e i". Now, w i t h e v e r y s u c h s e n t e n c e s one may a s s o c i a t e a fuzzy s u b s e t Bi o f t h e s e t Ai o f v a l u e s o f i - t h a t t r i b u t e . T h i s s u b s e t may, b u t need n o t , c o r r e s p o n d t o t h e v a l u e s of i - t h a t t r i b u t e o f t h e o b j e c t N ; i t i s t h e s e t o f v a l u e s which a r e "evoked" b y N . T o mention some examples, "powerful a s a l i o n " o r " c l e v e r as a fox" d e t e r m i n e some s e t s of v a l u e s on t h e s c a l e s o f power and c l e v e r n e s s r e s p e c t i v e l y , n o t n e c e s s a r i l y a p p l i c a b l e t o t h e animals i n q u e s t i o n . A s a consequence, a d e s c r i p t i o n ( l i n g u i s t i c measurement)
i n form o f a c o n j u n c t i o n o f k s t a t e m e n t s
Cil
H ;x
N1
&
x
&
... &
x
S.
ikN k
i s r e d u c i b l e t o a v e c t o r ( B l,...,B ) of f u z z y s e t s , k In the sequel, the where B i s a s u b s e t o f t h e s e t A i j
membership f u n c t i o n o f t h e s e t Bi
.
Jill
be bi(zi)
,
Z i f
Ai
SELECTED TOPICS IN MEASUREMENT THEORY
For r = l , . . . , n ,
687
define
T h i s d e f i n e s a fuzzy s u b s e t of t h e s e t Ar
of a t t r i b u t e v a l u e s . We may now d e f i n e t h e C a r t e s i a n p r o d u c t DH b y DH(z) = DH(zl,.
..,Zy)
The s e t DH r e p r e s e n t s t h e i n f o r m a t i o n about t h e o b j e c t ( o r , r a t h e r , about i t s a t t r i b u t e s ) , g i v e n t h e d e s c r i p t i o n H . One may now u s e t h e s e t DH t o make i n f e r e n c e about t h e o b j e c t x g i v e n i t s d e s c r i p t i o n . We have here t h e f o l l o w i n g c h a i n : i n f o r m a t i o n H g i v e s the possibility distribution DH(z) t h a t the attribute v a l u e s a r e z . Given z , t h e p o s s i b i l i t y t h a t t h e o b j e c t i s i n f a c t x i f G ( x ; z ) , and t h e p o s s i b i l i t y t h a t t h @ object described i s i n a s e t M C X i s P o s s ( M I H ) = sup
xtM
sup z
rnin CG(x;z), DH(z)l.
We may now s a y t h a t i n f o r m a t i o n H q - i m p l i e s t h a t x C M is POSS
(-M\H)
i
1-q.
T h i s means t h a t i n f o r m a t i o n H q - i m p l i e s t h a t x E M ,
if
f o r any x n o t i n M , and any z , e i t h e r G ( x ; z ) or D H ( z ) i s less t h a n 1-q. If w e i n t e r p r e t t h e v a l u e s l e s s t h a n 1-q as " p r a c t i c a l i m p o s s i b i l i t y " , t h e n we may say t h a t
d e s c r i p t i o n H q-implies t h a t t h e d e s c r i b e d o b j e c t i s i n
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t h e s e t M , i f whenever i t i s not p r a c t i c a l l y impossibl e t h a t t h e a t t r i b u t e v a l u e s of t h e o b j e c t ( g i v e n i t s d e s c r i p t i o n ) a r e z , t h e n i t s i s p r a c t i c a l l y impossibl e t h a t the object with a t t r i b u t e values z i s i n M. A t t h e end i t i s w o r t h s t r e s s i n g t h a t t h e s u g g e s t e d
s y s t e m o f l i n g u i s t i c measurement i s w i d e r and r i c h e r t h a n t h e t o p i c s s t u d i e d b y Zadeh (1978). I n t h e l a t t e r c a s e , t h e s i t u a t i o n i s as f o l l o w s . Given some c l a s s i f i c a t i o n i n t o f u z z y c a t e g o r i e s , one t r i e s t o d e t e r m i n e t h e m m e r i c a l r e p r e s e n t a t i o n of f u z z i n e s s . I n t h e c i t ed p a p e r , Zadeh c o n s t r u c t s a s e m a n t i c language PRUF ( P o s s i b i l i t i c R e l a t i o n a l U n i v e r s a l F u z z y ) , which a l l o w s c a l c u l a t i o n o f n u m e r i c a l v a l u e s f o r some l i n g u i s t i c v a r i a b l e s , s u c h as s e n t e n c e s , e x p r e s s i o n s o r words, and pass t o composite l i n g u i s t i c v a r i a b l e s a g a i n . A s opposed t o t h a t , i n t h i s s e c t i o n
t h e t o p i c i s rat h e r c o g n i t i v e t h a n a l g o r i t h m i c , and c o n c e r n s a w i d e r c l a s s o f l i n g u i s t i c v a r i a b l e s . Also, it allows f o r an e m p i r i c a l a c c e s s t o c l a s s i f i c a t i o n s , t h r o u g h t h e funct i o n a ( x ) introduced i n t h e preceding section.
SELECTED TOPICS IN MEASUREMENT THEORY
689
BIBLIOGRAPHY TO CHAPTER 5
.-
I
, E.W.,
""-
AUA1"lY
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L e comportement d e l'homme r a t i o n n e l d e v a n t l e r i s q u e , C r i t i q u e d e p o s t u l a t s e t axioms d ' e c o l e a m e r i c a i n e . Econornetrica. 2 1 . 503-546.
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On t h e u s e o f i n c o n s i s t e n c y o f p r e f e r e n c e i n p s y c h o l o g i c a l measurement. J o u r n a l o f E x p e r i m e n t a l P s y c h o l o g y . 55. 1-7.
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YU, P.L.
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P r o s o c i a l b e h a v i o u r : a d e c i s i o n model. P o l i s h P s y c h o l o g i c a l B u l l e t i n . 8 . 177-186.
1 9 7 9 a New i d e a s i n d e c i s i o n t h e o r y . I n t e r n . J o u r n a l o f Man-Machine S t u d i e s . 11. 213-234. 1979b P s y c h o l o g i c a l f a c t o r s i n d e c i s i o n making: N e w d e c i s i o n models. I n G . F a n d e l and T . Gal ( e d s . )
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A g e n e r a l i z e d model of c h o i c e b e h a v i o r u n d e r r i s k . M a t h e m a t i c a l S o c i a l S c i e n c e s . v o l . 1, nr. 2
1 9 8 1 P s y c h o l o g i s c h e Aspekte d e r I n n o v a t i o n i n Wisse n s c h a f t und T e c h n i k
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FORMAL THEORY OF ACTIONS
T h i s c h a p t e r w i l l c o n t a i n a p r e s e n t a t i o n of t h e f o r m a l
t h e o r y of a c t i o n s . The f i r s t v e r s i o n of t h i s t h e o r y was p u b l i s h e d i n Nowakowska ( 1 9 7 3 ) ; f o r t h e r e f e r e n c e t o v a r i o u s s u b s e q u e n t e x t e n s i o n s and m o d i f i c a t i o n s s e e Bibliography. I n t h i s c h a p t e r we p r e s e n t a m o d i f i c a t i o n aimed a t b r i n g i n g t o f o c u s t h e r e l a t i o n s between t h e o r y of a c t i o n s and p s y c h o l o g i c a l d e c i s i o n t h e o r y . The e x t e n s i o n w i l l c o n s i s t mainly o f i n c l u s i o n o f p r o b a b i l i s t i c e l e m e n t s and f u z z i f i c a t i o n . The aim w i l l be t o c o n s t r u c t a p o s s i b l y most g e n e r a l d e s c r i p t i o n of human b e h a v i o r . The main i s s u e s which w i l l b e c o v e r e d by t h e formalism and e x p l i c a t e d by means of a n a n a l y s i s of a network o f i n t e r r e l a t e d concepts concerning a c t i o n s , a r e t h e f o l lowing:
---
t h e s e q u e n t i a l c h a r a c t e r o f human a c t i o n s , and t h e a n a l o g i e s between c l a s s e s of s t r i n g s of a c t i o n s and languages ;
---
t h e s t r u c t u r e of g o a l s , t h e i r a t t a i n a b i l i t y , and
various optimization c r i t e r i a ;
---
t h e c o n s t r a i n t s on i n f o r m a t i o n a v a i l a b l e t o t h e de-
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c i s i o n maker, and t h e i r e f f e c t s on a t t a i n a b i l i t y ;
---
t h e p s y c h o l o g i c a l components i n v o l v e d i n c h o i c e o f a c t i o n s , and t h e r e s u l t i n g a p p a r e n t s u b o p t i m a l i t y ;
---
t h e a c t i o n s of s e v e r a l s u b j e c t s ( g r o u p a c t i o n s ) .
1. THE B A S I C SCHEME 1.1. The d e t e r m i n i s t i c c a s e
The t h e o r y w i l l be based on t h e f o l l o w i n g q u i n t u p l e t of p r i m i t i v e concepts
(1.1) where
S
and
A
a r e s e t s of s t a t e s and a c t i o n s , w i t h
# b e i n g "no a c t i o n " , r i s t h e a d m i s s i b i l i t y f u n c t i o n ,
and p i s t h e t r a n s i t i o n f u n c t i o n . The i n t u i t i o n s u n d e r l y i n g t h i s scheme are t h e f o l l o w i n g . The s u b j e c t a c t s i n an environment which - i f l e f t w i t h o u t i n t e r f e r e n c e - e v o l v e s randomly ( o r perhaps p a r t i a l l y r a n d o m l y ) . I n a n extreme c a s e , t o be c o n s i d e r e d i n t h i s s e c t i o n , t h e environment e v o l v e s i n a d e t e r m i n i s t i c manner. A t some moments (assumed e q u a l l y s p a c e d ) t h e s u b j e c t has a chance t o a c t , t h a t i s , choose a n a c t i o n from t h e s e t of a c t i o n s which a r e a v a i l a b l e t o him a t t h e moment, and perform t h i s action. The a v a i l a b i l i t y of a c t i o n s d e p e n d s , i n g e n e r a l , on t h e p r e v i o u s h i s t o r y , t h a t i s , on t h e p r e v i o u s s t a t e s and a c t i o n s .
699
FORMAL THEORY OF ACTIONS
When a n a c t i o n i s performed, t h e environment p a s s e s t o a new s t a t e , a c c o r d i n g t o t h e t r a n s i t i o n law of t h e s y s t e m , and t h e n t h e s i t u a t i o n r e p e a t s .
I n t h i s way, a sequence of c o n s e c u t i v e a c t i o n s c a u s e s c o n s e c u t i v e t r a n s i t i o n s between s t a t e s . The c o n t r o l c o n s i s t s t h e r e f o r e of c h o o s i n g t h e a c t i o n s s o a s t o a t t a i n a g i v e n g o a l . The n a t u r e of g o a l s w i l l be speci f i e d l a t e r . I n t u i t i v e l y , however, e v e r y g o a l w i l l be e x p r e s s i b l e t h r o u g h " e l e m e n t a r y " g o a l s , namely " t o a t t a i n a s p e c i f i e d s t a t e a t a s p e c i f i e d time". I n t h i s section, w e shall discuss t h e nature of t h e c o n t r o l p r o c e s s , s t a r t i n g from t h e s i m p l e s t c a s e o f det e r m i n i s t i c s i t u a t i o n s , and g r a d u a l l y e n r i c h i n g i t . F i r s t l y , S i s t h e s e t of s t a t e s o f t h e environment, o f t e n a l s o r e f e r r e d t o a s " s y s t e m " . The f o r m a l r e q u i r e ment of t h e i n t e r p r e t a t i o n i s t h a t e l e m e n t s of S a r e e x h a u s t i v e and e x c l u d e one a n o t h e r . T h i s means t h a t a t e v e r y t i m e t h e environment i s i n e x a c t l y one s t a t e s from S . The s t a t e o f t h e system a t time t w i l l be denoted by s ( t ) . An i n f o r m a l r e q u i r e m e n t i s t h a t t h e knowledge a b o u t t h e s t a t e a t a given time p r o v i d e s a l l i n f o r m a t i o n r e l e v a n t f o r t h e p u r p o s e o f t h e d e s c r i p t i o n . To a t t a i n t h i s , one u s u a l l y t a k e s v e c t o r - v a l u e d d e s c r i p t i o n s , t h a t i s , t a k e S t o be a C a r t e s i a n p r o d u c t
s
= SIX
... x
Sk
(1.2)
...,
where S1, S a r e s e t s o f p o s s i b l e v a l u e s of some k k attributes. T h i s w i l l be i l l u s t r a t e d on an example, d e l i b e r a t e l y
700
CHAPTER 6
o v e r s i m p l i f i e d , s o as t o p o i n t out only t o t h e p r i n c i p l e s i n v o l v e d . Suppose t h a t t h e "environment" i s some k i n d of a d e v i c e which has two lamps; e a c h o f them may b e on (t) or o f f ( - ) . We may t h e n t a k e as S1 and S2 t h e s e t s of v a l u e s o f a t t r i b u t e s o f t h e l e f t and r i g h t lamp r e s p e c t i v e l y , namely S1 = S2 = It,-?. The s e t S o f s t a t e s w i l l be e q u a l S1 y S 2 , and w i l l t h e r e f o r e c o n s i s t of f o u r e l e m e n t s , (t,t), (t,-), (-,t) and (-,-) This d e s c r i p t i o n i s s u f f i c i e n t , i f the only information
needed c o n c e r n s t h e s t a t u s of t h e two lamps. One may a l s o r e s i g n from i n f o r m a t i o n about one o f t h e l a m p s ; f o r i n s t a n c e , " l e f t lamp on" w i l l be r e p r e s e n t e d as t h e group of s t a t e s {(t,t), (t,-)j. If some o t h e r i n f o r m a t i o n i s needed t o d e s c r i b e t h e
s t a t e , i t has t o be i n c l u d e d by e n l a r g i n g t h e s e t S . For i n s t a n c e , suppose t h a t i n a d d i t i o n t o lamps, t h e d e v i c e has a d i a l , which may be s e t a t any o f t h e l e t t e r s A , B y C or D . Then we may t a k e S = f A , B , C , D ) , 3 and S = S1 X S2 X S w i l l c o n s i s t of 1 6 e l e m e n t s ,
(+,+,A),
3 (+,+,B),.-., (-,-,I)).
It ought t o be c l e a r t h a t t h i s method o f c h a r a c t e r i z -
i n g s t a t e s may a l w a y s be a p p l i e d t o p r o v i d e t h e d e s c r i p t i o n of as many f e a t u r e s as one l i k e s ( a t t h e c o s t o f making t h e s e t o f s t a t e s v e r y l a r g e , t h o u g h ) . Also, i f i n a g i v e n i n s t a n c e one i s n o t i n t e r e s t e d i n a g i v e n a t t r i b u t e , one may s t i l l u s e t h e l a r g e r s e t of s t a t e s ( i n c l u d i n g t h i s a t t r i b u t e ) , and group t h e s t a t e s i n an a p p r o p r i a t e w a y , as i l l u s t r a t e d above. I n t h e s e q u e l , i t w i l l a l w a y s be t a c i t l y assumed t h a t e l e m e n t s o f S p r o v i d e a l l i n f o r m a t i o n needed f o r t h e purpose of d e s c r i p t i o n . T h i s convention i s important, s i n c e l a t e r we s h a l l e x p r e s s t h e g o a l s t h r o u g h s e t s o f
701
FORMAL THEORY OF ACTIONS
states. S e c o n d l y , A w i l l be t h e s e t o f a l l p o s s i b l e a c t i o n s , w i t h t h e d i s t i n g u i s h e d element # E A i n t e r p r e t e d as " r e m a i n i n g i n a c t i v e " ( i . e . "no a c t i o n " ) . To d e s c r i b e t h e l a s t two p r i m i t i v e c o n c e p t s of t h e s y s t e m ( l . l ) ,namely r and p , we need t o i n t r o d u c e a n a d d i t i o n a l n o t i o n , namely t h a t o f a h i s t o r y . I n t u i t i v e l y , a h i s t o r y i s a d e s c r i p t i o n o f t h e p a s t (up t o a g i v e n t i m e ) s t a t e s and a c t i o n s . We imagine t h a t t h e s t a t e s and a c t i o n s a l t e r n a t e , i n t h e s e n s e t h a t a n a c t i o n c a u s e s t h e t r a n s i t i o n t o a new s t a t e , and g i v e n a new s t a t e , one chooses t h e n e x t a c t i o n . We a s s m e t h a t i n i t i a l l y , a t t = 0 , t h e environment i s
i n some s t a t e s ( O ) , and t h e f i r s t a c t i o n i s u n d e r t a k e n a t t i m e t = 1, c a u s i n g t h e . i n s t a n t e n u o u s t r a n s i t i o n t o s t a t e s(1). I n o t h e r words, we n e g l e c t h e r e t h e tempor a l a s p e c t s s u c h as d u r a t i o n o f a c t i o n s , and t h e time i t t a k e s f o r a system t o pass t o a new s t a t e . D e f i n e now
HA = S,
(1.3)
f o r n = 1,2, ..., and
HI1 '= S
)c
A, Hi
=
S
%
A
(1.4)
... .
The e l e m e n t s o f HA w i l l be c a l l e d f o r n = 2,3, s - h i s t o r i e s , and e l e m e n t s of Hi w i l l be c a l l e d a-hist o r i e s (of length n ) .
Thus, an s - h i s t o r y i s a s t r i n g o f a l t e r n a t i n g s t a t e s and a c t i o n s , which e n d s w i t h a s t a t e , s o t h a t a n e l e ment of HA w i l l be o f t h e form
702
CHAPTER 6
with a ( i ) being the
a c t i o n u n d e r t a k e w a t time t = i .
S i m i l a r l y , an a - h i s t o r y w i l l be a s t r i n g o f s t a t e s and a c t i o n s e n d i n g w i t h an a c t i o n , s o t h a t a n element of
:H
w i l l have t h e form
The d i s t i n c t i o n between s - h i s t o r i e s and a - h i s t o r i e s i s convenient f o r t h e f ollow ing r easons: an s - h i s t o r y i s t h e i n f o r m a t i o n which w i l l t y p i c a l l y be used when choosing t h e next a c t i o n , while a n a - h i s t o r y determine s the next t r a n s i t i o n . Accordingly, t h e a d m i s s i b i l i t y f u n c t i o n r w i l l a s s i g n t o e v e r y s - h i s t o r y t h e s e t of a c t i o n s a v a i l a b l e a f t e r t h i s s-history,
so that
where 2A i s t h e c l a s s of a l l s u b s e t s of A . The v a l u e r(hljl) o f t h e f u n c t i o n r a t t h e s - h i s t o r y h; w i l l a l s o b e d e n o t e d by A h l . Ahl
c
A;
this is the set
09
From ( 1 . 7 ) i t f o l l o w s t h a t a l l a c t i o n s which a r e ad-
r n i g s i b l e a f t e r t h e s - h i s t o r y h:. We s h a l l assume t h a t #
E A
hA
for a l l n and a l l h:11 € H:,I 1
I n o t h e r words, t h e " n o - a c t i o n "
(1.8)
i s always admissible.
The n o t i o n o f a d m i s s i b i l i t y may h e r e be i n t e r p r e t e d i n a v a r i e t y o f w a y s , depending on t h e g o a l o f a n a l y s i s .
FORMAL THEORY OF ACTIONS
703
The most i m p o r t a n t i n t e r p r e t a t i o n s a r e connected w i t h the following: (1) P h y s i c a l a d m i s s i b i l i t y . T h i s l e a d s t o s t r u c t u r a l a n a l y s e s of v a r i o u s s p e c i f i c a c t i o n s i t u a t i o n s , i n p a r t i c u l a r t h o s e s t u d i e d i n psychology o f work.
( 2 ) P s y c h o l o g i c a l a d m i s s i b i l i t y f o r a given person.
Here t h e c l a s s o f a l l s t r i n g s of a c t i o n s which a r e adm i s s i b l e ( t h e a d m i s s i b i l i t y of a s t r i n g of a c t i o n s b e i n g d e f i n e d below, t h r o u g h t h e n o t i o n o f a d m i s s i b i l i t y o f a n a c t i o n ) may c o n s t i t u t e a b a s i s f o r c r e a t i n g a s p e c i a l p e r s o n a l i t y t h e o r y , i n which t h e concept of p e r s o n a l i t y would be e x p l i c a t e d and taxonomized i n terms o f "grammatical r u l e s " f o r v a r i o u s " l a n g u a g e s o f a c t i o n s " , t h e l a t t e r ( d i s c u s s e d i n next s e c t i o n )
being s e t s of admissible s t r i n g s o f actions.
( 3 ) S o c i a l a d m i s s i b i l i t y , u n d e r s t o o d as c o n s i s t e n c y w i t h a g i v e n r o l e . Here t h e a n a l y s i s might p r o v i d e
s t r u c t u r a l c h a r a c t e r i s t i c s and taxonomy o f s o c i a l r o l e s .
(4) O r g a n i z a t i o n a l a d m i s s i b i l i t y , i . e . c o n s i s t e n c y w i t h a g i v e n s e t o f r u l e s and r e g u l a t i o n s i n a n o r g a -
n i z a t i o n under s t u d y . T h i s a n a l y s i s might l e a d t o a f o u n d a t i o n o f o r g a n i z a t i o n t h e o r y , based on l i n g u i s t i c t h e o r y o f a c t i o n s . As opposed t o " l a n g u a g e s o f s o c i a l o r d e r " , based m o s t l y on r e j e c t i o n or a c c e p t a n c e o f a g i v e n a c t i o n or s t r i n g o f a c t i o n s as c o n s i s t e n t or i n c o n s i s t e n t w i t h a g i v e n s o c i a l r o l e , here t h e c r i t e r i a a r e e x p r e s s e d more f o r m a l l y t h r o u g h norms, r e g u l a t i o n s and customs o f a g i v e n o r g a n i z a t i o n .
( 5 ) A d m i s s i b i l i t y for a g i v e n o b j e c t . The i d e a h e r e i s t h a t each o b j e c t g e n e r a t e s a set of a p p r o p r i a t e
704
CHAPTER 6
s t r i n g s o f a c t i o n s , a d m i s s i b l e i n view o f i t s s t r u c t u r e . I n o t h e r words, one may speak of a c t i o n s which a r e " g r a mmatical" f o r a g i v e n o b j e c t and t h o s e which a r e not grammatical ( a s a n example, one may t h i n k h e r e of a n o b j e c t s u c h as a computer, and a c t i o n s o f o p e r a t i n g i t ) . I n c a s e o f c e r t a i n c o n f i g u r a t i o n s o f o b j e c t s which a r e r e l a t e d i n some way, t h e r e a r i s e s a problem o f r e l a t i o n between "lang,uages" o f t h e s e o b j e c t s , p o s s i b l y e x p r e s s i b l e i n t e r m s of s e t - t h e o r e t i c a l o p e r a t i o n s . It might seem t h a t t h e l a r g e r i s t h e c o n f i g u r a t i o n o f o b j e c t s , and more i n t r i c a t e a r e t h e i r i n t e r r e l a t i o n s h i p s , t h e more c o n s t r a i n t s on a d m i s s i b i l i t y o f a c t i o n s , hence more s t r u c t u r e i n t h e language o f a c t i o n s . A p a r t i c u l a r c o n f i g u r a t i o n o f o b j e c t s i n t h e environ-
ment; i t g e n e r a t e s t h e r e f o r e t h e s e t o f a p p r o p r i a t e a c t i o n s . A l l o t h e r a c t i o n s , from o u t s i d e o f t h i s s e t , a r e t h e r e f o r e diagnostic, i n the sense t h a t t h e y repres e n t t h o s e a s p e c t s o f a c t i v i t y which a r e r e l a t e d t o i n t e r n a l f a c t o r s of t h e a c t i n g p e r s o n , h i s m o t i v a t i o n , p l a n of a c t i o n s , e t c .
( 6 ) A d m i s s i b i l i t y f o r a g i v e n methodology or t h e o r y . T h i s i n t e r p r e t a t i o n i s e s p e c i a l l y a p p l i c a b l e when one
t r e a t s s c e i e n t i f i c a c t i v i t y w i t h i n t h e framework o f t h e p r e s e n t system. One may namely i n t e r p r e t s c i e n t i f i c methods as s e t s of w e l l d e f i n e d and s t r u c t u r e d p r o c e d u r e s (i.e. s t r i n g s of a c t i o n s ) . T h i s allows t h e considera t i o n o f e v e r y method as a s e p a r a t e language o f a c t i o n s , o r a l t e r n a t i v e l y , t o t r e a t i t as a sublanguage of a " u n i v e r s a l " 1.anguage of a c t i o n s o f a g i v e n t h e o r y o r s e t o f t h e o r i e s . Such a n a p p r o a c h l e a d s t o c o n s t r u c t i o n of a formal t heor y of r e s e a r c h .
705
FORMAL THEORY OF ACTIONS
The l a s t c o n c e p t , t h e t r a n s i t i o n l a w p ,
assigns
the
n e x t s t a t e t o a n a - h i s t o r y . Formally
Hi
p:
n
+S ,
(1.9)
where p(h:) i s t h e s t a t e t o which t h e environment passe s a t t h e t e r m i n a t i o n o f a - h i s t o r y h.: I n o t h e r words, p ( h i ) = s ( n ) , i . e . t h e s t a t e a t time t = n . We may now d e f i n e t h e c l a s s of p o s s i b l e h i s t o r i e s , t o
I n t u i t i v e l y , a h i s t o r y h ( a n s-hist o r y , o r an a - h i s t o r y ) i s p o s s i b l e , i f each of i t s a c t i o n s b e l o n g s t o t h e s e t o f a d m i s s i b l e a c t i o n s , and e a c h s t a t e i s o b t a i n e d b y a p p l i c a t i o n of t h e t r a n s i t i o n l a w p . F o r m a l l y , l e t h = ( s ( O ) , a ( l ) , s ( l ) , . . . ) be a h i s t o r y . We s h a l l s a y t h a t h i s p o s s i b l e , if f o r every n = 1 , 2 , b e d e n o t e d by H .
...
.., s ( n ) ) ,
a(nt1) 6 Ah,, n
where h:
= (s(O),.
s(n)
where h:
= (s(0) , . . . , a ( n ) ) .
(1.10)
and
1.2.
= p(h:),
(1.11)
Language o f a c t i o n s
It i s c l e a r t h a t t h e s y s t e m (1.1) i n t h e d e t e r m i n i s t i c
c a s e c o n s i d e r e d i s somewhat s u p e r f l u o u s . The p o i n t i s t h a t an a c t i o n d e t e r m i n e s u n i q u e l y t h e n e x t s t a t e , which i n t u r n d e t e r m i n e s t h e s e t of a v a i l a b l e a c t i o n s . Consequently , t h e i n i t i a l s t a t e and t h e sequence o f consecutive actions determine uniquely t h e next availa b l e a c t i o n , and a l s o t h e s t r i n g o f c o n s e c u t i v e s t a t e s .
706
CHAPTER 6
Thus, i n a p o s s i b l e h i s t o r y h = ( s ( O ) , a ( l ) , s ( l ) ,... ) we may omit a l l s t a t e s e x c e p t t h e f i r s t , s i n c e t h e y a r e d e t e r m i n e d u n i q u e l y by t h e a c t i o n s , and c o n s i d e r , a ( n ) , performa d m i s s i b l e s t r i n g s o f a c t i o n s a ( 1), ed i n t h e s i t u a t i o n s(0).
. ..
The c l a s s L = L[s(O)] of a l l s t r i n g s o f a c t i o n s which a r e p o s s i b l e i n t h e s i t u a t i o n s(0) i s t h u s d e f i n e d by the conditions (1.12)
a(n) G Ah,, n
n = 1,2 ,...,
(1.13)
where h; i s t h e s - h i s t o r y ( s ( O ) , a ( l ) , s ( l ), . . . , s ( n ) ) w i t h s ( i ) bei.ng t h e s t a t e r e s u l t i n g from a c t i o n a ( i ) . The c l a s s L [ s ( O ) ]
of a l l s t r i n g s o f a c t i o n s d e f i n e d i n t h i s way c o n s t i t u t e s t h e language o f a c t i o n s f o r t h e s i t u a t i o n s(O), i n t h e s e n s e d e f i n e d i n Nowakowska
(1973). Moreover, t o e a c h s t r i n g of a c t i o n s i n L [ s ( O ) ] t h e r e c o r r e s p o n d s a unique s t r i n g o f s t a t e s o f t h e e n v i r o n ment. These s t a t e s a r e t h e outcomes of t h e s t r i n g of actions i n question, the state a t t i m e t = n being c a l l e d t h e time-event a t t i m e t = n . T h i s d e t e r m i n i s t i c scheme o f a c t i o n s was e x p l o r e d i n
d e t a i l i n Nowakowska (1973) and also i n s e v e r a l sub-
s e q u e n t p a p e r s , by u s i n g t h e methods o f m a t h e m a t i c a l l i n g u i s t i c s , n o t a b l y t h e n o t i o n s from t h e t h e o r y of d i s t r i b u t i v e classes, e.g. p a r a s i t i c s t r i n g s , e t c . The main p r i n c i p l e i n v o l v e d i n t h i s a n a l y s i s was t h e analogy between t h e c l a s s o f a l l s t r i n g s o f a d m i s s i b l e
FORMAL THEORY OF ACTIONS
707
a c t i o n s , and a l a n g u a g e , r e g a r d e d as t h e c l a s s of a l l a d m i s s i b l e s t r i n g s formed o u t o f t h e v o c a b u l a r y ( s e n tences). Moreover, t h i s a n a l o g y may be e x t e n d e d f u r t h e r : t h e r e s u l t s ( t i m e - e v e n t s ) o f s t r i n g s o f a c t i o n s p l a y t h e same r o l e as meanings of s e n t e n c e s , hence c o n s t i t u t e t h e sem a n t i c s of language of a c t i o n s .
1.3. A l g e b r a i c a p p r o a c h t o g o a l s t r u c t u r e Before proceeding w i t h t h e i n t r o d u c t i o n of p r o b a b i l i s t i c e l e m e n t s and f u z z i f i c a t i o n , l e t us u s e t h e p r e s e n t s e t u p for d e f i n i t i o n s p e r t a i n i n g t o t h e s t r u c t u r e o f goals. We s h a l l assume t h a t w e a r e i n t e r e s t e d i n t h e outcomes o c c u r r i n g no l a t e r t h a n some f i n i t e t i m e h o r i z o n T . L e t ST = S X X S (T t i m e s ) . Moreover, l e t us d e f i ne L T [ s ( 0 ) ] as t h e c l a s s o f a l l s t r i n g s o f a c t i o n s p o s s i b l e i n t h e i n i t i a l s i t u a t i o n s ( O ) , which a r e o f l e n g t h e x a c t l y T , w i t h t h e c o n v e n t i o n t h a t any s t r i n g i n L [ s ( O ) ] o f l e n g t h l e s s t h a n T i s complemented by a d d i n g a s t r i n g o f # , so as t o o b t a i n a s t r i n g o f t h e t o t a l l e n g t h T.
...
We may a s s o c i a t e w i t h any s t r i n g of a c t i o n s i n L , [ s ( O ) ] t h e s t r i n g o f s t a t e s i n ST, which r e s u l t s i f t h i s s t r i n g o f a c t i o n s i s performed, and t h e i n i t i a l s t a t e i s s(0). Denote t h i s assignment by R , s o t h a t
(1.14)
708
CHAPTER 6
We s h a l l s a y t h a t R(u) i s t h e outcome o f s t r i n g u ( i n s i t u a t i o n ~(0)). We s h a l l a l s o w r i t e i t as
w i t h s u ( i ) being t h e s t a t e a t time t = i , r e s u l t i n g
from t h e s t r i n g u ( i n i n i t i a l s i t u a t i o n ~(0)). We may s a y t h a t s u ( n ) i s t h e t i m e - r e s u l t o f t h e s t r i n g u a t time t . We now d e f i n e a n e v e n t a s a s u b s e t of ST, i . e . a s e t o f s t r i n g s o f s t a t e s . An e v e n t A C ST i s s a i d t o o c c u r a t t i m e t , i f t h e r e e x i s t s a set B C S such t h a t A = l(s(l),...,s(T)):
s ( t ) t .)B
(1.16)
T h i s d e f i n i t i o n a p p e a r s t o cover w e l l t h e i n t u i t i o n :
even though t h e e v e n t s are d e f i n e d as s e t s of s t r i n g s o f s t a t e s s p a n n i n g t h e p e r i o d from 1 t o T , t o d e t e r m i n e whether an e v e n t o c c u r r e d a t t i m e t , t h e o n l y i n f o r m a t i o n needed i s t h e t - t h c o o r d i n a t e o f t h e s t r i n g ( o t h e r c o o r d i n a t e s a r e not e s s e n t i a l ) . T e c h n i c a l l y , an event o c c u r r i n g a t t i s a s p e c i a l case of a c y l i n d e r s e t i n t h e p r o d u c t s p a c e ST. L e t Z be t h e c l a s s o f a l l e v e n t s , and l e t Z t be t h e c l a s s o f a l l e v e n t s which o c c u r a t time t . Then e a c h o f t h e c l a s s e s Z , Z t forms a n a l g e b r a , i n t h e s e n s e of c l o s u r e under a l l s e t - t h e o r e t i c a l o p e r a t i o n s : t h e union and i n t e r s e c t i o n o f two e v e n t s o c c u r i n g a t t ( s a y ) i s a g a i n a n e v e n t o c c u r r i n g a t t , and a l s o t h e complement o f a n e v e n t o c c u r r i n g a t t i s a n e v e n t occurring at t .
709
FORMAL THEORY OF ACTIONS
We s h a l l now i d e n t i f y a g o a l (or, more p r e c i s e l y , a
E-
m
g o a l ) o f a c t i o n s w i t h an e v e n t . I n t u i t i v e l y , t h i s means t h a t a c t i o n s a r e performed s o as t o cause a n o c c u r r e n c e of a n e v e n t . C l e a r l y , t h i s d e f i n i t i o n i s broad enough t o c o v e r a l l c a s e s of g o a l s which a r e o f " a l l - o r - n o t h i n g " t y p e . We assumed namely t h a t t h e n o t i o n o f s t a t e i s d e f i n e d i n s u c h a way as t o c o v e r all r e l e v a n t a s p e c t s , and t h a t t h e time h o r i z o n T i s chosen s o l a r g e t h a t w e a r e n o t i n t e r e s t e d i n a n y t h i n g beyond t h e t i m e T . Thus, a l l which may happen and which i s o f any i n t e r e s t f o r t h e d e c i s i o n maker i s e x p r e s s i b l e as a c e r t a i n s t a t e o r group o f s t a t e s , a t some t i m e or t i m e s , i . e . a n e v e n t . Moreover, s i n c e " t o avoid a n e v e n t A" i s t h e same as " t o c a u s e t h e e v e n t - A " , one may a l w a y s assume t h a t t h e g o a l i s t o c a u s e t h e o c c u r r e n c e of some e v e n t . A g o a l may o f t e n be decomposed and e x p r e s s e d i n terms
o f some s i m p l e r g o a l s , by means of s e t - t h e o r e t i c a l oper-
a t i o n s o f u n i o n s and i n t e r s e c t i o n s . A l t e r n a t i v e l y , one may t h i n k o f a c l a s s o f some s i m p l e r g o a l s , and a "comp o s i t e " g o a l c o n s t r u c t e d o u t o f t h e s i m p l e r ones. T h i s t y p e of c o n s t r u c t i o n p l a y s an i m p o r t a n t r o l e i n e x p l i c a t i o n o f t h e n a t u r e of g o a l s , and w i l l be t h e r e f o r e p r e s e n t e d i n some d e t a i l . Let A
*
= $A1,
..., An 7
(1.17)
be some f i x e d c l a s s of e v e n t s , which w i l l be r e f e r r e d t o as f t e l e m e n t a r y e v e n t e " o r " e l e m e n t a r y g o a l s " . L e t f(xl, x n ) be a p r o p o s i t i o n a l f u n c t i o n of n v a r i a b l e s , i n v o l v i n g o n l y t h e f u n c t o r s of c o n j u n c t i o n and
...,
710
CHAPTER 6
*
d i s j u n c t i o n . We s h a l l d e n o t e b y f ( A ) t h e s e t ( e v e n t ) o b t a i n e d by s u b s t i t u t i n g i n f t h e e v e n t s Ai arguments x
for the
and r e p l a c i n g t h e f u n c t o r s o f c o n j u n c t i o n
i'
and d i s j u n c t i o n by t h e s e t - t h e o r e t i c a l o p e r a t i o n s of i n t e r s e c t i o n s and u n i o n s r e s p e c t i v e l y .
*
The e v e n t f ( A ) s o o b t a i n e d w i l l b e c a l l e d t h e compos i t e g o a l g e n e r a t e d by f from t h e c l a s s A
*.
I n p a r t i c u l a r , we d e n o t e ) = x
f1(x1,
".,X
f2(x1,
...,xn)
=
*)
Ai
n
1
...
& x2 &
'
(1.18)
'n
and
fl
... v
x l v x2 v
xn
.
(1.19)
i s i n t e r p r e t e d as " t o a t t a i n a l l * elementary g o a i s Ail1, w h i l e f ( A ) = Ai i s i n t e r 2 p r e t e d as " t o a t t a i n a t l e a s t one o f khe g o a l s A i T 1 . Then f l ( A
=
u
F o r o t h e r examples, suppose t h a t n = 3, and l e t f ( x 1 , x 2 , x 3 ) = x1 & ( x 2 v x 3 ) .
(1.20)
*
Then f ( A ) i s t h e g o a l " t o a t t a i n A1 and a t l e a s t one of t h e r e m a i n i n g e l e m e n t a r y g o a l s " . L e t us now i n t r o d u c e t h e f o l l o w i n g d e f i n i t i o n , DEFINITION.
The e v e n t A i s s a i d t o b e a t t a i n a b l e , i f
and u n a t t a i n a b l e , i f R-'(A)
=
0.
711
FORMAL THEORY OF ACTIONS
Thus, a g o a l A i s a t t a i n a b l e , i f t h e r e e x i s t s a t l e a s t one s t r i n g s of a c t i o n s u s u c h t h a t R(u) 6 A . One s h o u l d remember t h a t t h i s d e f i n i t i o n i s r e l a t i v e t o t h e i n i t i a l state s ( O ) , since the function R defined by ( 1 . 1 4 ) depends on t h e i n i t i a l s t a t e . To s t r e s s t h i s f a c t , i f n e c e s s a r y , we s h a l l s p e a k o f s ( 0 ) - a t t a i n a b i l i t y and s ( 0 ) - u n a t t a i n a b i l i t y .
L e t now A
* = f A1,..
.,Anj
be a f i x e d c l a s s o f e l e m e n t -
a r y goals.
*
DEFINITION. We s h a l l s a y t h a t t h e s e t A i s s t r o n g l y * a t t a i n a b l e , i f t h e g o a l f l ( A ) i s a t t a i n a b l e , and * weakly a t t a i n a b l e , i f t h e g o a l f 2 ( A ) i s a t t a i n a b l e , where f l and f 2 a r e g i v e n by ( 1 . 1 8 ) and ( 1 . 1 9 ) . F o r any g o a l A , t h e s e t R - l ( A )
i s the c l a s s of a l l
s t r i n g s o f a c t i o n s which c a u s e t h e o c c u r r e n c e of A . To s t r e s s t h e dependence o f t h e i n i t i a l s t a t e , w e s h a l l d e n o t e t h i s s e t by C S A , and s h a l l r e f e r t o i t as t h e s e t o f means o f a t t a i n i n g A i n s . S i n c e t h e i n v e r s e images p r e s e r v e a l l s e t - t h e o r e t i c a l o p e r a t i o n s , we have t h e f o l l o w i n g theorem.
Let f ( A
*)
be a composite g o a l . Then t h e s e t o f a l l means o f a t t a i n i n g f ( A * ) in s i s o b t a i n e d by s u b s t i t u t i n g the s e t s Cs(Ai) i n p l a c e o f arguments xi i n f , and r e p l a c i n g t h e f u n c t o r s o f c o n j u n c t i o n and THEOREM.
d i s j u n c t i o n by o p e r a t i o n s o f i n t e r S e c t i o n and u n i o n . Symbolically, (1.22)
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where C s ( A )
*
= (CsA1,
...,
CsAn).
T h i s theorem e s t a b l i s h e s an isomorphism between t h e
s t r u c t u r e of a composite g o a l and t h e s t r u c t u r e o f t h e c l a s s o f a l l s t r i n g s o f a c t i o n s which a t t a i n t h i s g o a l ( f o r more d e t a i l s , s e e Nowakowska 1 9 7 6 ; s e e a l s o Robbel
1977). B e f o r e p r o v i n g n e x t theorem, l e t us i n t r o d u c e some new concepts. F i r s t l y , denote
with
R g i v e n by ( 1 . 1 4 ) , s o t h a t R
n
i s t h e c l a s s of a l l
s t r i n g s o f s t a t e s i n ST which may be a t t a i n e d , s t a r t i n g from s ( O ) , by a s t r i n g of a c t i o n s . n , An? o f e v e n t s , and any e v e n t Given any s e t A = ? A 1 , . * B C ST, we s a y t h a t t h e e v e n t s i n A are i n d e p e n d e n t
..
r e l a t i v e t o B , i f any i n t e r s e c t i o n of t h e form (-1)k lA l n w i t h ki
...
n
(-1)kn A n 0 B
(1.24)
e q u a l 0 or 1, i s n o t empty.
We may now i n t r o d u c e t h e f o l l o w i n g d e f i n i t i o n .
*
DEFINITION. The c l a s s A of e v e n t s i s s a i d t o a l l o w * complete p o s s i b i l i t y , i f t h e e v e n t s i n A a r e independent r e l a t i v e t o R
*
.
We s h a l l prove t h a t a c l a s s of e v e n t s a l l o w s t h e comp l e t e p o s s i b i l i t y i f , and o n l y i f , t h e r e i s no i n c l u s i o n between any i n t e r s e c t i o n of e v e n t s of t h e form
713
FORMAL THEORY OF ACTIONS
I n t u i t i v e l y , complete p o s s i b i l i t y means t h a t one can * a t t a i n any combination of e v e n t s from A , and a v o i d * a l l o t h e r e v e n t s from A .
..
To f o r m u l a t h e t h e theorem, l e t I = 1 1 , 2 , , , n ) be t h e n* s e t of a l l i n d i c e s o f e v e n t s i n A . F o r any s e t D C In let AD =
0D
~~n
R
*
(1.25)
i C
and (1.26) Then t h e f o l l o w i n g theorem h o l d s . THEOREM.
* A
Let
A
%
= {A1,..
. ,Anl
with n
>
1. Then t h e s e t
d o e s n o t a l l o w complete p o s s i b i l i t y i f , and o n l y i f t h e r e e x i s t s e t s D1, D 2 C I n , D1" D2 = 0, w i t h a t l e a s t one o f t h e s e s e t s nonempty, and a n i n d e x j al D1 u D 2 , such t h a t
F o r t h e p r o o f , s e e Nowakowska ( 1 9 7 6 ) . The i n t u i t i o n connected w i t h t h i s theorem i s s i m p l y t h i s : i f any o f t h e i n c l u s i o n s ( 1 . 2 7 ) h o l d s , t h e n one * can c o n s t r u c t a combination o f e v e n t s from A and t h e i r complements, which i s n o t a t t a i n a b l e . The n o t i o n o f complete p o s s i b i l i t y becomes i m p o r t a n t when one c o n s i d e r s a c t i o n s o f groups of p e r s o n s w i t h d i v e r g i n g g o a l s . It may t h e n happen t h a t some e v e n t s o f t h e form ( 1 . 2 4 ) s e r v e as g o a l s f o r d i f f e r e n t p e r -
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s o n s , and t h e r e a r i s e s t h e q u e s t i o n o f a c h i e v i b g a compromise s o l u t i o n . NOW, t h e set R - l ( A )
i n d e f i n i t i o n ( 1 . 2 1 ) i s a s e t of s t r i n g s which c a u s e A to o c c u r , and o r i g i n a t e from s(O), i.e.
We may d e n o t e , more g e n e r a l l y , for a f i x e d s t r i n g z = ( s ( 0 ) , . . . , s u ( r ) ) s a t i s f y i n g ( 1 . 1 2 ) and ( 1 . 1 3 )
where zv s a t i s f i e s ( 1 . 1 2 ) and ( 1 . 1 3 ) .
Suppose now t h a t R - l ( A ) # 0 and a l s o - R - l ( A ) # 0, so t h a t A may be a t t a i n e d by some s t r i n g of a c t i o n s , and a l s o A may be a v o i d e d by some o t h e r s t r i n g o f a c t i o n s . Le? Del(A) = max 4 z : R i l ( A )
-1 # 0 , -RZ ( A ) # 0j.
Then Del ( A ) i s t h e l o n g e s t t i m e d u r i n g which one can d e l a y t h e d e c i s L o n whether t o a c h i e v e o r t o a v o i d A , i . e . t h e l o n g e s t t i m e d u r i n g which one may have a f r e e dom w i t h r e s p e c t t o A . I n o t h e r words, a t t i m e Del ( A ) one must p e r f o r m a n a c t i o n which w i l l e i t h e r c a u s e A ( r e g a r d l e s s o f t h e s u b s e q u e n t a c t i o n s ) , or w i l l e l i m i n a t e A , a g a i n r e g a r d less o f the subsequent a c t i o n s . C l e a r l y , we have here t h e f o l l o w i n g i d e n t i t i e s connect-
i n g D e l ( A ) and Del (B):
FORMAL THEORY OF ACTIONS
Del ( A Del ( A
f~
3
715
B.) = min CDel(A), D e l ( B ) l ,
B) = max CDel(A), D e l ( B ) ] .
I n c o n n e c t i o n w i t h t h e n o t i o n o f f u n c t i o n Del which i n forms u s f o r how l o n g maximally one may p o s t p o n e t h e d e c i s i o n r e g a r d i n g a t t a i n i n g or a v o i d i n g A , i t i s n a t u r a l t o c o n s i d e r more s p e c i f i c a l l y t h e a c t i o n s which e i t h e r d e t e r m i n e o r e l i m i n a t e some s i m p l e o r composite event A. Suppose t h a t Ril(A)
# 0 and
for s i m p l i c i t y , l e t
US
-Ril(A)
# 0;
agree t o say t h a t i n t h i s case,
z l e a v e s A-freedom.
which a r e p o s s i b l e i n t h e s t a t e which i s a t t a i n e d when z i s performed s t a r t i n g from t h e g i v e n i n i t i a l s t a t e s ( O ) , a c c o r d i n g t o ( 1 . 1 2 ) and ( 1 . 1 3 ) . Then, w i t h r e s p e c t t o t h e e v e n t A , t h e s e t DZ may be p a r t i t i o n e d i n t o t h r e e d i s j o i n t s u b s e t s as f o l l o w s : one s e t w i l l comprise a l l t h o s e a c t i o n s which d t e r m i n e A t o o c c u r r e g a r d l e s s o f what i s done n e x t ; t h e o t h e r s e t c o m p r i s e s t h o s e a c t i o n s which make A i m p o s s i b l e t o o c c u r r e g a r d l e s s o f a l l subseque n t a c t i o n s , and t h e t h i r d s e t comprises t h e r e m a i n d e r . F o r m a l l y , we d e f i n e Let D Z be t h e c l a s s of a l l a c t i o n s
DetZ(A) =
la6
D Z : zau
L
3 zau
R-'(A)]
and E l i m Z ( A ) = f a 6 DZ:
zau d R - I ( A )
for a l l u')
.
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Clearly, i f a
D e t Z ( A ) i s performed a f t e r s t r i n g z , t h e n A o c c u r s r e g a r d l e s s of t h e s u b s e q u e n t a c t i o n s u , s o t h a t such an a c t i o n a ( i n context z ) determines A . S i m i l a r l y , i f a 4 E l i m ( A ) i s performed a f t e r s t r i n g z , Z t h e n A does n o t o c c u r , a g a i n r e g a r d l e s s o f s u b s e q u e n t actions, so t h a t A i s eliminated. N a t u r a l l y , one o r b o t h o f t h e s e t s D e t Z ( A ) and E l i m z ( A ) may be empty.
We have t h e f o l l o w i n g t h e o r e m . THEOREM. Suppose t h a t z l e a v e s A-freedom, = Del ( A ) . Then Det A u E l i m A = D Z
Z
Z
.
and t h a t
Izi
T h i s means t h a t i f a s t r i n g z l e a v e s A-freedom,
but i s t h e l o n g e s t p o s s i b l e among s u c h s t r i n g s , t h e n t h e s e t s
Det A and E l i m A e x h a u s t a l l a c t i o n s , i . e . e v e r y a c t i o n Z
Z
i n D Z must e i t h e r d e t e r m i n e o r e l i m i n a t e A . Generally, we say that a s t r i n g z leads t o a decisive moment, i f D e t A u E l i m A 0. A d e c i s i v e moment may Z Z be p o s i t i v e , i f DetzA # 0 o r n e g a t i v e , i f E l i m z A # 0. I f b o t h t h e s e c o n d i t i o n s h o l d , t h e d e c i s i v e moment i s mixed, and i f DetzA i / E l i m z A = D Z , t h e d e c i s i v e moment i s c a l l e d u l t i m a t e ( i t must t h e n be m i x e d ) .
The n o t i o n s of d e c i s i v e moments and t h e i r t y p o l o g y a r e o f obvious r e l e v a n c e for any d e c i s i o n a n a l y s i s , i f i t i s n e c e s s a r y t o i n c l u d e t h e s e q u e n t i a l a s p e c t s o f dec i s i o n making. One may a l s o c o n s i d e r o t h e r s t r u c t u r a l c o n s t r a i n t s on a c t i o n s , c o n n e c t e d w i t h t h e s i z e o f s e t s D Z for v a r i o u s z , and s i z e s o f t h e s e t s R-'(A) for Z v a r i o u s z and A .
FORMAL THEORY OF ACTIONS
717
Thus, one may i n t r o d u c e t h e f o l l o w i n g d e f i n i t i o n s . DEFINITION. Suppose t h a t t h e s e t D Z of a c t i o n s which a r e a d m i s s i b l e a f t e r t h e s t r i n g z c o n s i s t s o f one e l e ment o n l y , s o t h a t D Z = $a)..We s h a l l s a y t h a t one i s t h e n i n an e n f o r c e d s i t u a t i o n . I n o t h e r words, a f t e r p e r f o r m i n g z , o n l y one a c t i o n a i s p o s s i b l e t o p e r f o r m , s o t h a t a l l s t r i n g s i n L which b e g i n w i t h z must b e o f t h e form zau f o r some u. S i m i l a r l y , i t may happen t h a t f o r some s t r i n g z and e v e n t A w e have t h e r e l a t i o n :
T h i s means t h a t w h i l e t h e s e t D Z may c o n t a i n more t h a n
one e l e m e n t , a l l s t r i n g s which a r e i n R i l ( A ) have a c t i o n a performed a f t e r t h e s t r i n g z . I n o t h e r words, i n t h e s i t u a t i o n when z i s a l r e a d y performed, i f one wants t o a t t a i n A , i t i s n e c e s s a r y t o p e r f o r m a c t i o n a. We may s a y t h a t i n t h i s c a s e , s t r i n g z l e a d s t o a n only-exit s i t u a t i o n with respect t o A . C l e a r l y , i n t h i s c a s e w e must have
which e x p r e s s e s t h e p r o p e r t y t h a t a f t e r p e r f o r m i n g z , any a c t i o n which i s a d m i s s i b l e and d i f f e r e n t t h a n a , eliminates t h e event A . Observe t h a t i t does n o t f o l l o w t h a t a E D e t Z ( A ) : i t may happen t h a t a f t e r p e r f o r m i n g t h e a c t i o n a , one
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s t i l l h a s , a t some l a t e r s t a g e , a p o s s i b i l i t y of p e r -
f o r m i n g a n a c t i o n which w i l l c a u s e A , and a l s o a p o s s i b i l i t y of e l i m i n a t i n g A .
1 . 4 . A f u z z i f i c a t i o n of a d m i s s i b i l i t y of a c t i o n s L e t us now r e l a x t h e r e q u i r e m e n t t h a t t h e f u n c t i o n r ,
d e f i n e d on t h e c l a s s o f s - h i s t o r i e s , d e t e r m i n e s a c r i s p s u b s e t o f t h e s e t o f a c t i o n s A , as p o s t u l a t e d i n ( 1 . 7 ) . I n s t e a d , we s h a l l assume t h a t f o r any s - h i s t o r y h ' , t h e s e t A h l of a c t i o n s admissible a f t e r h' i s fuzzy. We s h a l l u s e t h e same symbol r f o r a d m i s s i b i l i t y funct i o n , e x c e p t t h a t now we s h a l l p o s t u l a t e t h a t
(1.28) where r ( h l , a ) i s t h e d e g r e e t o which a c t i o n a i s admissible after s-history h'. This i s a generalization of t h e scheme c o n s i d e r e d p r e v i o u s l y , i n t h e s e n s e t h a t i n S e c t i o n 1.1, t h e f u n c t i o n r assumed o n l y v a l u e s 0 and 1. I n a c c o r d a n c e w i t h ( 1 . 8 ) w e s h a l l assume t h a t
s o t h a t t h e n o - a c t i o n # i s always a d m i s s i b l e i n t h e
d e g r e e 1. Suppose now t h a t i n t h e i n i t i a l s t a t e s ( 0 ) a s t r i n g o f a c t i o n s u = ( a ( l ) , a ( 2 ) , . . . , a ( n ) ) was performed.
We s h a l l s t i l l r e t a i n t h e a s s u m p t i o n t h a t t h e t r a n s i t i o n s are d e t e r m i n i s t i c . D e f i n e now t h e f o l l o w i n g sequences
719
FORMAL THEORY OF ACTIONS
(1.30)
and
...
Thus, s ( O ) , s(l), i s t h e sequence of c o n s e c u t i v e s t a t e s i n d u c e d by t h e s t r i n g of a c t i o n s u , w h i l e zl, z2,... i s t h e sequence of numbers, r e p r e s e n t i n g t h e a d m i s s i b i l i t i e s o f c o n s e c u t i v e a c t i o n s i n u. We s h a l l now u s e some i d e a s o f t h e t h e o r y o f d - b i l i t y
( s e e Nowakowska 1 9 8 2 ) t o d e f i n e t h e o v e r a l l a d m i s s i b i l i t y o f t h e s t r i n g u.
L e t namely d be a r e a l - v a l u e d f u n c t i o n o f two real v a r i a b l e s , s a t i s f y i n g the following conditions: 1. SYMMETRY:
(1.32) 2 . MONOTONICITY: d(X,y)
i s increasing i n x f o r every y
(1.33)
( h e n c e a l s o i s i n c r e a s i n g i n y for e v e r y x); a c t u a l l y , w e s h a l l r e q u i r e somewhat weaker c o n d i t i o n , namely t h a % d i s n o n - d e c r e a s i n g i n e a c h of t h e arguments.
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3. BOUNDARY CONDITIONS: o((x,o) = 0 ,
(1.34)
d ( 1 , l ) = 1;
4. TRANSITIVITY:
F o r g i v e n f u n c t i o n d s a t i s f y i n g t h e above c o n d i t i o n s , * w e d e f i n e i t s d u a l o( ( x , y ) by t h e r e l a t i o n
* (x,y)
o(
= 1
-
(1.36)
o((l-x, 1 - y ) .
A s examples of s u c h f u n c t i o n s , one may t a k e
used by Zadeh i n h i s p o s s i b i l i t y t h e o r y , w i t h d u a l be* i n g d ( x , y ) = max ( x , y ) , o r
whose d u a l g i v e s p r o b a b i l i s t i c summation x t y - xy.
* d
(x,y) =
Somewhat more g e n e r a l l y , we may t a k e i n s t e a d of ( 1 . 3 8 ) the functions (1.39) or (1.40)
721
FORMAL THEORY OF ACTZONS
r=
which b o t h reduce t o ( 1 . 3 8 ) when 1. The f u n c t i o n ( 1 . 3 9 ) was s u g g e s t e d by Hamacher ( 1 9 7 5 ) , w h i l e t h e func t i o n ( 1 . 4 0 ) was i n t r o d u c e d by Dubois and Prade ( 1 9 8 0 ) . A s o t h e r f u n c t i o n s one may t a k e , f o r i n s t a n c e
(Sugeno 1 9 7 4 ) , o r d ( x , y ) = 1 - min [l, (1-x)' (Yager
t (1-y)']
l/q ,970
(1.42)
1980).
L e t us now assume t h a t a f u n c t i o n satisfying z above p o s t u l a t e s i s chosen, and l e t z l , n a d m i s s i b i l i t i e s of a c t i o n s i n t h e s t r i n g u, i n i a l s i t u a t i o n s ( 0 ) . Define now t h e sequence v l ,
...,
the be t h e the initv 2 , ...
as
v1 = zl,
(1.43) and p u t V(U,S(O))
=
Vn'
(1.44)
I n t h i s way we a s s i g n e d a v a l u e V(u,s(O))
t o the s t r i n g u and i n i t i a l s i t u a t i o n s ( 0 ) ; t h i s v a l u e w i l l be c a l l e d d - a d m i s s i b i l i t y of t h e s t r i n g u ( i n s i t u a t i o n ~ ( 0 ) ) . I n p a r t i c u l a r , i f < ( x , y ) = min ( x , y ) , we have V(u,s(O)) = min z so t h a t t h e assignment of a d m i s s i b i l i t y f o l i' lows t h e "weakest l i n k " p r i n c i p l e : t h e a d m i s s i b i l i t y
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of a s t r i n g of a c t i o n s e q u a l s t h e l o w e s t o f a d m i s s i b i l i t i e s of i t s actions.
1.5. Stochastic transitions L e t us now add t h e f i n a l g e n e r a l i z a t i o n to o u r s y s t e m , namely assume t h a t t h e t r a n s i t i o n s to t h e n e x t s t a t e a r e ( a t l e a s t p a r t i a l l y ) random. Thus, a n a c t i o n a p p l i e d i n a g i v e n s i t u a t i o n , as s p e c i f i e d by h f , does n o t d e t e r m i n e t h e n e x t s t a t e : i t may o n l y i n f l u e n c e t h e p r o b a b i l i t i e s of n e x t s t a t e s . I n o t h e r words, t h e cont r o l c o n s i s t s of c h o o s i n g a c t i o n s which make c e r t a i n s u b s e q u e n t s t a t e s more p r o b a b l e , and c e r t a i n o t h e r s t a t e s less probable. F o r m a l l y , t h e t r a n s i t i o n law p i n system (1.1) has now t h e form of a p r o b a b i l i t y d i s t r i b u t i o n on S , a s s i g n e d t o every a - h i s t o r y h", o r e q u i v a l e n t l y , t o every p a i r ( h f , a ) c o n s i s t i n g of a n s - h i s t o r y h ' and a c t i o n a ( i n t h e s e t of a d m i s s i b i l e a c t i o n s ) . We s h a l l w r i t e
(1.45) f o r t h e p r o b a b i l i t y t h a t t h e n e x t s t a t e w i l l be s , g i v e n t h a t a c t i o n a was a p p l i e d i n s - h i s t o r y h l . Nat u r a l l y , w e have for e v e r y ( h l , a )
(1.46) S i n c e i n t h e p r e s e n t s e t u p t h e p e r s o n does n o t know t h e f u t u r e s t a t e s u n t i l t h e y o c c u r , he c a n n o t , i n gener a l , d e c i d e i n advance which s p e c i f i c a c t i o n s he w i l l
723
FORMAL THEORY OF ACTIONS
t a k e (among o t h e r r e a s o n s , because t h e a d m i s s i b i l i t y
o f a g i v e n a c t i o n a t some f u t u r e t i m e may depend on t h e e v e n t s which w i l l o c c u r i n b e t w e e n ) . Thus, one must a p p l y a d e c i s i o n r u l e , i . e . s p e c i f y what he w i l l do i n e v e r y c i r c u m s t a n c e which may o c c u r . Such a d e c i s i o n rul e c o r r e s p o n d s t o t h e n o t i o n of s t r a t e g y i n t h e t h e o r y of games: t h e l a t t e r s p e c i f i e s f u t u r e a c t i o n s i n e v e r y p o s s i b l e o c c a s i o n which may a r i s e . We s h a l l now g i v e t h e n e c e s s a r y f o r m a l i s m , and i n t r o -
duce some c o n c e p t s which w i l l a l l o w u s t o d i s t i n g u i s h some i m p o r t a n t c l a s s e s of d e c i s i o n r u l e s . A s mentioned, t h e d e c i s i o n r u l e must s p e c i f y t h e a c t i o n
t o be t a k e n i n e v e r y s i t u a t i o n which may a r i s e . Formall y , t h i s means t h a t t o e v e r y s - h i s t o r y h ' , t h e r u l e , say a s s i g n s t h e c h o i c e of a n a c t i o n , i . e . is a
q,
9
mapping
9:
HA -7A
(1.47)
n
q(hJ,)
where t o r y hh.
i s t h e a c t i o n t o be t a k e n i n c a s e of s - h i s -
L e t 'Q= ( f l , f 2 ,
... ) ,
i s a function where f n : HJ,--+A which maps t h e c l a s s o f a l l s - h i s t o r i e s of l e n g t h n i n t o t h e c l a s s of a c t i o n s . L e t u s assume now t h a t t h e a d m i s s i b i l i t y , as e x p r e s s e d by t h e f u n c t i o n r i s n o t f u z z y . We may now i n t r o d u c e
t h e following d e f i n i t i o n .
q=
(fl, f2, DEFINITION, The r u l e i f f o r e v e r y n and e v e r y hA c HJ,
... )
i s admissible,
(1.48)
724
CHAPTER 6
I n o t h e r words, an a d m i s s i b l e r u l e would a l w a y s choose an a d m i s s i b l e a c t i o n . I n c a s e o f f u z z y a d m i s s i b i l i t y , t h e r e q u i r e m e n t of adm i s s i b i l i t y of a r u l e must be f o r m u l a t e d d i f f e r e n t l y ;
i n f a c t , i t may be done i n s e v e r a l w a y s . A c c o r d i n g l y ,
we introduce t h e following d e f i n i t i o n . DEFINITION. A r u l e sible i f ---.-.-, rCh;,
fn'(h;)l
\9=
(fl, f2,.
..)
i s s t r o n g l y admis-
= 1 for a l l n and a l l h;
(1.49)
and w e a k l y a d m i s s i b l e , i f r[h;,
fn(li;)]
7 0 for a l l n and a l l h;.
(1.50)
Thus, a s t r o n g l y a d m i s s i b l e r u l e would a l w a y s choose a n a c t i o n which i s a d m i s s i b l e i n d e g r e e o n e , w h i l e a weakly a d m i s s i b l e r u l e would n e v e r choose a n a c t i o n whose a d m i s s i b i l i t y i s z e r o . Now, a t r i p l e c o n s i s t i n g o f t h e i n i t i a l s t a t e s(0) = s
0' d e c i s i o n r u l e '-f and t r a n s i t i o n p r o b a b i l i t i e s p determines u n i q u e l y t h e p r o b a b i l i t y d i s t r i b u t i o n on t h e c l a s s o f a l l h i s t o r i e s . I n o t h e r words, t h e h i s t o r y becomes a stochastic process.
Lo
For g i v e n so, t h e r u l e determines t h e choice of the f i r s t a c t i o n b y a l = f1 (s0 ) . Next, f o r g i v e n ( s o , a l ) , the state at t = 1 i s s with probability p(sl(ao,sO). 1 Then t h e a c t i o n a2 i s s e l e c t e d from t h e e q u a t i o n a2 = f 2 ( s 0 , a l , s 1 ) , and s o on. Observe t h a t t h e a c t i o n a2 i s random, s i n c e i t depends on t h e random c h o i c e o f s l . Generally, the probability of the history
FORMAL THEORY OF ACTZONS
725
i s g i v e n by t h e p r o d u c t of p r o b a b i l i t i e s o f s u c c e s s i v e
choices of states, that i s ,
where a
j
= f . (so,al,. J
. . ,sj ) .
The p r o c e s s s o d e f i n e d i s t h e g e n e r a l d i s c r e t e time c o n t r o l p r o c e s s , as a n a l y s e d i n c o n t r o l t h e o r y . Under s u c h g e n e r a l i t y , however, n o t much may be s a i d a b o u t the p r o p e r t i e s of t h i s process , finding optimal control p o l i c i e s 'f , and s o on. To o b t a i n a scheme which would be manageable for a n a l y s i s , one u s u a l l y assumes ( e . g . i n c o n t r o l t h e o r y , dy-
namic programming, e t c . ) t h a t t h e t r a n s i t i o n p r o b a b i l i t i e s are Markovian, i . e . t h e y depend o n l y on t h e l a s t s t a t e and a c t i o n . F o r m a l l y , we i n t r o d u c e t h e following definition.
DEFINITION. The t r a n s i t i o n p r o b a b i l i t i e s p a r e s a i d t o be Markovian, or of Markov t y p e , i f for e v e r y n
where p l , p 2 , .
..
are f u n c t i o n s
pn : S x A + S such that f o r every ( s , a ) C S x A
(1.54)
126
CHAPTER 6
(1.55) Moreover, t h e Markov t r a n s i t i o n p r o b a b i l i t i e s a r e s a i d t o be s t a t i o n a r y , i f p1 = p 2 = p.
...
The a d v a n t a g e of d e a l i n g w i t h Markov p r o b a b i l i t i e s , and e s p e c i a l l y w i t h s t a t i o n a r y Markov p r o b a b i l i t i e s , l i e s i n t h e f o l l o w i n g f a c t . G e n e r a l l y , t h e c o n t r o l (de-
9)
i s chosen s o a s t o o p t i m i z e some g o a l . cision rule Now, t h e d e c i s i o n r u l e i s of t h e form ‘P= ( f l y f*’ ) o f a sequence of f u n c t i o n s , where t h e n - t h f u n c t i o n f depends on t h e whole h i s t o r y up to t h e t i m e t = n . n I f t h e t r a n s i t i o n p r o b a b i l i t i e s a r e Markovian, one can u s u a l l y show ( u n d e r some m i l d a s s u m p t i o n s ) t h a t i n s e a r c h i n g f o r a n optimum one may r e s t r i c t t h e c o n s i d e r a t i o n s t o Markov p o l i c i e s ( r u l e s ) , namely s u c h r u l e s ‘Q f o r which t h e n - t h f u n c t i o n f n depends only on t h e s t a t e sn a t tlime t = n ( i . e . one may d i s r e g a r d p r e v i o u s s t a t e s and a c t i o n s ) .
.. .
Moreover, i n c a s e of s t a t i o n a r y Markov t r a n s i t i o n prob a b i l i t i e s , t h e o p t i m a l d e c i s i o n r u l e i s o f t e n of t h e form
y
= (f, f,
...)
(1.56)
where f: S-+A
(1.57)
s o t h a t t h e c h o i c e o f a c t i o n a t e v e r y moment depends o n l y on t h e s t a t e a t t h e g i v e n t i m e . T h i s f a c t r e d u c e s t h e complexity o f t h e problem, t o t h e problem o f f i n d i n g t h e optimal f u n c t i o n f . T h i s i s o f t e n s o l v a b l e by
FORMAL THEORY OF ACTIONS
727
s t a n d a r d methods, e . g . b y s o l v i n g Bellman e q u a t i o n .
2 . TEMPORAL RELATIONS BETWEEN EVENTS
I n t h i s s e c t i o n we s h a l l modify somewhat t h e b a s i c scheme (1.1). We s h a l l namely omit t h e components conc e r n i n g t h e a c t i o n s of t h e p e r s o n ( i . e . A , # and r ) , and c o n c e n t r a t e on t h e problems o f d i s t i n g u i s h i n g c e r t a i n important categories of events, t h e i r observabilit y , e t c . I n p a r t i c u l a r , we s h a l l i n v e s t i g a t e one i m p o r t a n t a t t r i b u t e of e v e n t s , namely t h e time of t h e i r occurrence. Accordingly, w e s h a l l refer t o t h e person i n q u e s t i o n as o b s e r v e r ( a s w e now d i s r e g a r d h i s a c t i o n s ) . We may c o n s i d e r t h e a n a l y s i s of t h i s s e c t i o n a s a " m i -
c r o s c o p i c " p i c t u r e o f t h e p e r i o d between two a c t i o n s o f t h e o b s e r v e r - t h a t i s , t h e p e r i o d when he c o l l e c t s t h e i n f o r m a t i o n about t h e t r a n s i t i o n t o t h e next s t a t e which s e r v e s h i m as a b a s i s for d e c i d i n g a b o u t t h e next action. I n t h e s u b s e q u e n t c o n s i d e r a t i o n s , t h e symbol t , w i t h or w i t h o u t s u b s c r i p t s , w i l l s t a n d f o r t h e o b j e c t i v e t i m e ( c l o c k , c a l e n d a r , e t c . , depending on t h e c o n t e x t a n a l y s e d . We assume t h a t t i s a one-dimensional numeri c a l i n d e x , r e p r e s e n t i n g t h e r e l a t i o n "A o c c u r s e a r l i e r t h a n B", and " t h e t e m p o r a l d i s t a n c e between A and B i s l a r g e r t h a n t h a t between C and D". Moreover, w e assume t h a t t i s m e a s u r a b l e on an i n t e r val type scale. The o b s e r v e r , t o g e t h e r w i t h t h e p a r t o f environment
728
CHAPTER 6
which he p e r c e i v e s , form a c e r t a i n s y s t e m , which p a s s e s t h r o u g h c e r t a i n s t a t e s . The i n t e r n a l s t a t e s of t h e obs e r v e r may a l s o c o n s t i t u t e t h e components of t h e s t a t e description.
2.1. The t e m p o r a l t r u t h s y s t e m s L e t S be t h e c l a s s o f a l l p o s s i b l e s t a t e s of t h e s y s tem, and l e t I be t h e t i m e i n t e r v a l under c o n s i d e r a t i o n . We d e f i n e f i r s t a h i s t o r y as a f u n c t i o n h: I
j
S
(2.1)
w i t h h ( t ) b e i n g t h e s t a t e of t h e system a t t i m e t .
Observe t h a t t h e p r e s e n t d e f i n i t i o n o f h i s t o r y a g r e e s w i t h t h e one i n t r o d u c e d i n t h e p r e c e d i n g s e c t i o n : s i n c e the actions of the observer are disregarded, the hist o r y needs only t o s p e c i f y the states at v a r i o u s times, which i s g i v e n i n (2.1), w i t h t h e o n l y d i f f e r e n c e t h a t now t i m e may be c o n t i n u o u s . I n r e a l i t y , t h e t r u e h i s t o r y o f t h e system i s u s u a l l y n o t known i n f u l l , due t o f a c t o r s s u c h as p a r t i a l obs e r v a b i l i t y o f t h e s t a t e , d i s c o n t i n u i t i e s o f t h e obs e r v a t i o n p r o c e s s , and s o on. We s h a l l f o r m a l i z e t h e c o n s t r a i n t s on o b s e r v a b i l i t y due t o s u c h f a c t o r s i n n e x t s e c t i o n , when w e c o n s i d e r t h e knowledge o f t h e o b s e r v e r . A t p r e s e n t , we assume o n l y t h a t t h e t r u e h i s t o r y ho i s a n element o f a c l a s s o f a l l a d m i s s i b l e h i s t o r i e s H . The a d m i s s i b i l i t y h e r e r e f e r s t o some l o g i c a l , p h y s i c a l , e t c . c o n s t r a i n t s on h , known t o hold r e g a r d l e s s o f t h e p r o c e s s of o b s e r v a t i o n s .
729
FORMAL THEORY OF ACTIONS
L e t SI d e n o t e t h e c l a s s o f a l l f u n c t i o n s which map I
i n t o S . We s h a l l i n t r o d u c e t h e f o l l o w i n g d e f i n i t i o n . DEFINITION. By a t e m p o r a l t r u t h system we s h a l l mean a quadruplet
(2.2) where I i s t h e p e r i o d o f o b s e r v a t i o n , S i s t h e s e t o f s t a t e s o f t h e s y s t e m , H C SI i s t h e c l a s s of a d m i s s i b l e h i s t o r i e s , and ho H is the "true" history. I n t h e s e q u e l , w e s h a l l t a k e I t o be t h e i n t e r v a l C O , T l , w i t h T s u f f i c i e n t l y l a r g e , s o t h a t w e may r e g a r d a l l e v e n t s o f i n t e r e s t as o c c u r r i n g b e f o r e T .
2.2.
Events
I n t h e p r e c e d i n g s e c t i o n , e v e n t s were d e f i n e d as s e t s o f h i s t o r i e s . T h i s d e f i n i t i o n , which may be termed c o n s t r u c t i v e , i s c o n v e n i e n t f o r some t y p e o f a n a l y s i s . We s h a l l now u s e somewhat d i f f e r e n t a p p r o a c h , l e a d i n g t o a descriptive d e f i n i t i o n of an event.
Ac
Let S be a s e t of s t a t e s . We s h a l l s a y t h a t e v e n t A occurs i n h i s t o r y h, i f h ( t ) S f o r some t E I . T h i s g e n e r a l d e f i n i t i o n o f a n e v e n t has t o be made s p e c i f i c s o as t o e n a b l e one t o d i s t i n g u i s h c e r t a i n c l a s s e s o f e v e n t s u s e f u l i n subsequent c o n s i d e r a t i o n s . Let t f
I and A C S . Assume t h a t h ( t f ) 4 A . We now
d e f i n e t h e t i m e of t h e f i r s t e n t r a n c e t o A ( i n h i s t o r y h) as
after t'
CHAPTER 6
730
t'(A,t') h
= inf
It
E I: t
>
t', h(t)
w i t h t h e convention t h a t t i ( A , t r )
all t
>
ations)
>
e
AT
T i f h(t)
(2.3)
g
A for
t ' (here T i s t h e end o f t h e p e r i o d o f observ-
.
S i m i l a r l y , i f h ( t ' ) E A , w e d e f i n e t h e time of f i r s t e x i t from A a f t e r t ' a s
th(A,t')
= inf
f
t
<
I: t
>
t ' , h ( t ) d A[.
(2.4)
O b v i o u s l y , i f h ( t ' ) E A , t h e f i r s t e x i t from A a f t e r t ' i s t h e same as t h e f i r s t e n t r a n c e a f t e r t ' t o t h e complement of A . Thus, t h e c o n c e p t of t h e f i r s t e x i t i s r e d u c i b l e t o t h e c o n c e p t of f i r s t e n t r a n c e ; accordi n g l y , we s h a l l c o n c e n t r a t e on t h e l a t t e r . Generally, a t r i p l e (A,t',h) with h ( t ' ) A w i l l be c a l l e d a n e v e n t A a f t e r t l i n h i s t o r y h . We s h a l l s a y t h a t t h i s event occurs, i f
i . e . i f the f i r s t entrance t o A a f t e r t ' occurs i n h i s t o r y h.
If t k ( A , t t )
>
t " , t h e n t h e e v e n t s ( A , t ' , h ) and ( A , t " , h )
w i l l be i d e n t i f i e d .
I n t h e s e q u e l , we s h a l l s i m p l i f y t h e c o n s i d e r a t i o n s t f = 0; t h e n a n e v e n t w i l l be i d e n t i f i e d w i t h a s e t A s u c h t h a t h ( 0 ) Q A , and w i l l be i n t e r p r e t e d simply as t h e f i r s t e n t r a n c e t o A i n h i s t o r y h; s u c h a n e v e n t w i l l b e denoted simply by S'. We s h a l l a l s o d e n o t e by A- t h e e v e n t d e s c r i b e d as " f i r s t e x i t by t a k i n g
FORMAL THEORY OF ACTIONS
73 1
from A a f t e r t = 0 " . The c h a r a c t e r i s t i c f e a t u r e of t h e t e v e n t s A and A- i s t h a t i f t h e y o c c u r , t h e n t h e i r t i mes o f o c c u r r e n c e a r e w e l l d e f i n e d . The g e n e r a l i z a t i o n t o t h e c a s e t ' ward and w i l l be o m i t t e d .
>
0 i s straightfor-
A l s o , f o r s i m p l i f i c a t i o n of t h e s u b s e q u e n t d e f i n i t i o n s , t i t w i l l be c o n v e n i e n t t o d e n o t e t h e time t h ( A , O ) of t h e t f i r s t e n t r a n c e t o A i n h by t ( A , h ) , and l e t t ( A - , h ) t be t h e t i m e t h [ A , t h ( A , O ) ] o f t h e f i r s t e x i t from A a f t e r t h e first entrance t o A. DEFINITION. We s h a l l s a y t h a t t h e e v e n t A i s t h i n , or instantenuous, i n h, i f t(At,h) = t(A-,h)
(2.6)
t
whenever t ( A , h ) < T . Otherwise, t h e e v e n t A w i l l be called thick, or lasting. DEFINITION. The e v e n t A w i l l be c a l l e d t h i n , or i n s t a n t e n u o u s , i f i t i s t h i n i n e v e r y h i s t o r y from H i n which i t o c c u r s , i . e . s u c h t h a t h ( 0 ) d A and t ( A t , h ) < T . Thus, a s e t A ( e v e n t A ) i s t h i n i n h , i f t h e e v e n t associated w i t h i t (provided i t occurs) i s instantenuous: t h e h i s t o r y h s a t i s f i e s t h e c o n d i t i o n h ( t ) € A o n l y a t scme i s o l a t e d p o i n t o r p o i n t s , and n o t i n a n i n t e r v a l o f moments of t i m e . On t h e o t h e r hand, i f A i s thick, then h ( t ) c A i n the i n t e r v a l [t(At,h),t(A-,h)l o f some p o s i t i v e d u r a t i o n . Assume now t h a t h(0) d A for a l l h G H . The e v e n t A w i l l be c a l l e d n e c e s s a r y , o r u n a v o i d a b l e , i f
732
CHAPTER 6
s u p F ( At , h ) : h E H I
(2.7)
so t h a t a necessary event occurs i n every h i s t o r y hEH. F u r t h e r , t h e e v e n t A w i l l be c a l l e d n e c e s s a r i l y t r a n s -
m, i f
t
whenever t ( A , h ) 4 T , t h e n t ( A - , h )
<
T.
I n p a r t i c u l a r , every t h i n event i s necessarily transient.
2 . 3 . R e l a t i v e n e c e s s i t y of e v e n t s Let B C A . C l e a r l y , i f t h e f i r s t e n t r a n c e t o B o c c u r s , s o d o e s t h e f i r s t e n t r a n c e t o A , b u t t h e c o n v e r s e need n o t be t r u e : a h i s t o r y may e n t e r t h e l a r g e r s e t A w i t h out p r i o r entrance t o t h e smaller s e t B. S i m i l a r l y , i f t h e f i r s t e x i t from A o c c u r s , t h e n s o does t h e f i r s t e x i t from B y b u t n o t c o n v e r s e l y : a h i s t o r y may e x i t from a s m a l l e r s e t B , w i t h o u t l e a v i n g the larger s e t A. The c a s e when t h e above c o n v e r s e s are t r u e i s o f con-
s i d e r a b l e i n t e r e s t . Moreover, i t i s n o t n e c e s s a r y t o r e q u i r e t h a t B C A . Accordingly, w e i n t r o d u c e t h e f o l lowing d e f i n i t i o n . D E F I N I T I O N . L e t A,B be two s u b s e t s o f S . W e s h a l l say
that B i s necessary r e l a t i v e t o A , i f t h e condition t(At,h) < T implies t(Bt,h) < T. t
Moreover, i f t ( A t , h ) < T i m p l i e s t ( A t , h ) 5 t ( B , h ) , then B necessarily follows A, while i n the opposite c a s e , we say t h a t B n e c e s s a r i l y p r e c e d e s A . F i n a l l y , < T , t h e n A and B a r e s a i d
i f t(At,h) 4 T i f f t(B+,h)
FORMAL THEORY OF ACTIONS
733
t o be i n s e p a r a b l e . To g i v e some examples of t h e n o t i o n s i n t r o d u c e d , suppose t h a t growth of a c h i l d i s d e s c r i b e d by a f u n c t i o n h(t) = [fl(t), f2(t), I, where t h e f i r s t two c o o r d i n a t e s d e s i g n a t e r e s p e c t i v e l y h e i g h t and weight a t t i m e t . S u p p o s e a l s o t h a t t h e s e two c o o r d i n a t e s must i n c r e a s e i n a c o n t i n u o u s and s t r i c t l y monotone way ( t h i s i s g e n e r a l l y t r u e for h e i g h t , up t o a c e r t a i n a g e , and may be r e g a r d e d a l s o as t r u e f o r w e i g h t i n normal c i r cumstances).
...
Then a s t a t e A c h a r a c t e r i z e d by f i x i n g t h e f i r s t coord i n a t e i s t h i n , and i s a l s o u n a v o i d a b l e for c e r t a i n v a l u e s of t h i s c o o r d i n a t e . On t h e o t h e r hand, t h e s e t o b t a i n e d by f i x i n g t h e f i r s t two c o o r d i n a t e s i s a l s o t h i n , b u t n o t u n a v o i d a b l e : t h e growth may be s u c h t h a t h e i g h t and weight w i l l n e v e r t a k e up s i m u l t a n e o u s l y t h e p r e a s s i g n e d v a l u e s , even i f e a c h o f t h e s e v a l u e s must be assumed.
2.4.
Knowledge r e p r e s e n t a t i o n
Thus f a r , we have n o t made any use of t h e l a s t p r i m i t i v e c o n c e p t o f t h e s y s t e m (2.2), namely t h e t r u e h i s t o r y ho; t h e n o t i o n s i n t r o d u c e d concerned h i s t o r i e s i n general. Now, i n most c i r c u m s t a n c e s , t h e o b s e r v e r does n o t know h o . I n f a c t , he may o b s e r v e some e v e n t s , which g i v e him t h e i n f o r m a t i o n t h a t t h e t r u e h i s t o r y ho i s s u c h that at times t l , t2,. of o b s e r v a t i o n i t belongs t o I n f a c t , even s u c h an assumption some s e t s A1, A 2 ,
... .
..
734
CHAPTER 6
i s o f t e n t o o s t r o n g , and one has t o use i t s f u z z i f i e d
.
t 2,.. t h e v a l u e s v e r s i o n : a t times a p p r o x i m a t e l y t o f t h e t r u e h i s t o r y were i n t h e f u z z y s e t s A1, A 2 , . .
.
A t y p i c a l s i t u a t i o n f a l l i n g i n t o t h i s scheme o c c u r s
when t h e o b s e r v e r d o e s n o t remember e x a c t l y t h e t i m e of o b s e r v a t i o n , n o r t h e s t a t e o f h i s t o r y a t t h i s t i m e . The o b s e r v a t i o n s , as r e c a l l e d l a t e r , may c o n c e r n n o t
t h e t i m e of a s i n g l e e v e n t , b u t t h e o r d e r of two e v e n t s ( e . g . " A o c c u r r e d somewhat e a r l i e r t h a n B " , e t c . )
.
I n g e n e r a l , t h e knowledge a b o u t t h e t i m e h i s t o r y ho may be r e p r e s e n t e d i n form of a s t r i n g of p r o p o s i t i o n s , s a y P I , P 2 3 * * Y 'n 9 e a c h i n d u c i n g a f u z z y s u b s e t o f H. Before p r o c e e d i n g f u r t h e r , i t i s w o r t h w h i l e t o i n t r o duce some i m p o r t a n t and most common t y p e s of s u c h prop o s i t i o n s , and show how t h e y may be r e p r e s e n t e d a s f u zzy s e t s of h i s t o r i e s .
-
Suppose f i r s t t h a t we have a non-fuzzy p r o p o s i t i o n p = " A t t i m e t , t h e s t a t e belonged t o A"
(2.8)
where n e i t h e r t i m e t n o r t h e s e t A a r e f u z z y . F o r m a l l y , w e may t h e n i n t r o d u c e t h e s e t H
P
= f h C H: h ( t )
A1
(2..
9)
and r e p r e s e n t t h e s t a t e m e n t ( 2 . 8 ) as e q u i v a l e n t t o t h e a s s e r t i o n t h a t ho E H P A t y p i c a l c a s e o f t h e s e t A h e r e i s t h a t o f t e n t h e obs e r v e r knows t h e v a l u e ( o r p e r h a p s t h e approximate val u e ) of some o f t h e c o o r d i n a t e s o f t h e s t a t e d e s c r i p t i o n , w i t h o u t knowing o t h e r c o o r d i n a t e s ; s u c h knowledge
.
FORMAL THEORY OF ACTIONS
73 5
d e t e r m i n e s t h e n t h e s e t of p o s s i b l e s t a t e s . G e n e r a l l y , t h e s e t A may be f u z z y , and hence g i v e n i n terms of a membership f u n c t i o n , s a y g A ( s ) , s E S , w i t h g A ( s ) b e i n g t h e d e g r e e t o which t h e s t a t e s b e l o n g s to t h e s e t A . C o n s e q u e n t l y , w e may d e f i n e t h e f u z z y s e t H o f h i s t o r i e s , connected w i t h t h e p r o p o s i t i o n p , P w i t h membership f u n c t i o n U ( h ) , h E: H , g i v e n b y P
I n words, t h e d e g r e e t o which t h e h i s t o r y h b e l o n g s t o t h e s e t H e q u a l s t h e d e g r e e t o which t h e s t a t e h ( t ) P of h b e l o n g s t o t h e f u z z y s e t A . Suppose now t h a t n o t o n l y t h e s e t A , b u t a l s o t h e t i m e t o f o c c u r r e n c e o f A i s f u z z y , e x p r e s s e d by some f u z z y p h r a s e , s u c h as " a t a p p r o x i m a t e l y t i m e t " , e t c . T h i s f u z z i n e s s i s e x p r e s s i b l e i n t e r m s o f a membership funct i o n v t ( t 7 ) o f " t i m e s a p p r o x i m a t e l y t " . Here v t ( t T ) i s t h e d e g r e e t o which t t i s " a p p r o x i m a t e l y e q u a l t " . Then as t h e membership f u n c t i o n U ( h ) o f t h e f u z z y s e t P of h i s t o r i e s H c o r r e s p o n d i n g t o t h e p r o p o s i t i o n p , P one may t a k e
(2.11) Thus, t h e d e g r e e U ( h ) t o which t h e h i s t o r y h s a t i s f P i e s t h e c o n s t r a i n t s imposed b y t h e p r o p o s i t i o n " a t t i m e a p p r o x i m a t e l y t , t h e s t a t e was i n t h e f u z z y s e t A " , s a t i s f i e s t h e c o n d i t i o n U ( h ) 2 k i f , and o n l y i f , P t h e r e e x i s t s a t i m e t ' which i s " a p p r o x i m a t e l y e q u a l t " i n t h e d e g r e e a t l e a s t k , and a l s o s u c h t h a t t h e d e g r e e t o which t h e s t a t e a t t ' i s i n A i s a t l e a s t k .
736
CHAPTER 6
A s a n o t h e r t y p e o f p r o p o s i t i o n , suppose t h a t p a s s e r t s t h a t A o c c u r r e d much e a r l i e r t h a n B . Here a g a i n A and
B
a r e f u z z y s e t s o f s t a t e s , c h a r a c t e r i z e d by membership
f u n c t i o n s g A ( s ) and g , ( s ) . Let v ( t l , t 2 )
be t h e membership f u n c t i o n i n t h e f u z z y
s e t o f p a i r s o f moments t l y t 2 s u c h t h a t t l i s much e a r l i e r t h a n t 2 . Then as t h e membership f u n c t i o n U (h)
P
one may t a k e
I
L
(2.12) L e t now tion
o(
)t
0~
be t h e d u a l , as d e f i n e d by ( 1 . 3 6 ) , t o a func-
satisCying conditions (1.32)-(1.35).
For examp-
le, w e may have h e r e Zadeh‘s f u n c t i o n
o r p r o b a b i l i s t i c summation
n
( x , y ) = x -I- y
-
xy.
(2.14)
The d u a l s t o o t h e r f u n c t i o n s g i v e n i n S e c t i o n 1 are as f o l l o w s . The d u a l t o ( 1 . 3 9 ) i s
(2.15) aual t o (1.40)
is
x t y - xy
n (X,Y)
=
dual t o (1.41) i s
-
max ( 1 - x ,
min ( x , y , I - 2 / )
1-y
,S )
J
(2.16)
737
FORMAL THEORY OF ACTIONS
*
4 (x,y)
+ i j - x y ) , ' ~ ' ~-1;
= min (1, x t y
(2.17)
and f i n a l l y , d u a l t o ( 1 . 4 2 ) i s (2.18)
Suppose now t h a t we have a f u n c t i o n f:
x
-4
[O,ll
(2.19)
where X i s some a r b i t r a r y s e t . F o r a f i x e d f u n c t i o n d the
e(
*- e x t e n s i o n
*
of f, t o be d e n o t e d by f N n , w i l l be
a function
fd
*:
2x
--+
[O,ll
(2.20)
d e f i n e d as f o l l o w s :
(b)
f ,({XI) oc
= f ( x ) f o r every x
c X I
(2.22)
...,
If A = fxo,xl, x n j i s a f i n i t e s u b s e t of X , (c) t h e n fd,(A) = e n , where t h e sequence c o , cl, is d e f i n e d i n d u c t i v e l y by co =
f(x&
'k+l
= d
* [c,,
...
(2.23)
f(xktl)],
k = O,l,
...
(2.24)
( d ) f o r a n arbitrary s u b s e t A o f X , we p u t fd,(A)
= sup Ifd,(B):
B
c
A, B f i n i t e ) .
(2.25)
738
CHAPTER 6
C l e a r l y , we have t h e f o l l o w i n g theorem. THEOREM.
The
*
4 -extension
i s determined uniquely.
I n d e e d , t h e only f a c t which one must v e r i f y i s t h a t condition ( c ) determines uniquely t h e value f for d* any f i n i t e s e t A , r e g a r d l e s s o f t h e o r d e r i n g o f t h i s s e t . To prove t h i s a s s e r t i o n , i t i s enough t o show * t h a t 6 i s symmetric and t r a n s i t i v e . The s y m m e t r y of * i s a n obvious consequence of t h e symmetry o f o c , and o( * To show t r a n s t h e d e f i n i t i o n of t h e d u a l f u n c t i o n d * i t i v i t y o f o( , one may w r i t e t h e f o l l o w i n g c h a i n of e q u a l i t i e s , which i s based on t h e d e f i n i t i o n o f t h e d u a l f u n c t i o n , and on t r a n s i t i v i t y o f o c :
.
= 1
- d[l-x,
= 1 - d[l-X, = 1
-
o(
1-d
*(Y,Z)l
d(l-Y,l-Z)I
[&(l-x,l-y),
T h i s shows t h a t
* K
1-zl
i s t r a n s i t i v e , and completes t h e
p r o o f of t h e theorem. We a r e now r e a d y t o f o r m u l a t e t h e laws of knowledge representation. Assume t h a t t h e knowledge of t h e o b s e r v e r ( a t a g i v e n t i m e ) has t h e form of a s t r i n g o f p a i r s
739
FORMAL THEORY OF ACTIONS
(2.26) where pi i s a c e r t a i n p r o p o s i t i o n , and Mi
i s a modal
frame, r e p r e s e n t i n g t h e o b s e r v e r ' s a t t i t u d e towards t h e p r o p o s i t i o n pi ' Examples of modal frames ( s e e Nowakowska 1973 for a c a l c u l u s o f modal frames f o r m i n g a l i n g u i s t i c r e p r e s e n t a t i o n of m o t i v a t i o n ) a r e "I t h i n k t h a t " , "It i s p o s s i b l e t h a t " , "I am a l m o s t c e r t a i n t h a t " , "I b e l i e v e t h a t " , etc. Now, we assume t h a t w i t h e a c h modal frame M,I t h e r e i s * associated a function di, w i t h t h e properties described above. F o r i n s t a n c e , w i t h t h e modal frame "It i s p o s s i b l e t h a t " , t h e r e i s a s s o c i a t e d (Zadeh 1978) t h e f u n c t i o n
*
( x , Y ) = max ( x , ~ ) .
Roughly, i t means t h a t t h e p o s s i b i l i t y o f a s e t of h i s t o r i e s i s e q u a l t o t h e maximal p o s s i b i l i t y o f t h e i n dividual histories. O t h e r modal frames may be r e p r e s e n t e d by o t h e r f u n c t i o n s * 4 , which r e f l e c t t h e b e h a v i o u r of t h e modal frame when a p p l i e d t o a s e t of elements. Now, w i t h e v e r y pi we a s s o c i a t e i t s s k e l e t o n , which has Ai2,. .') o f d i s j o i n t t h e form of a f a m i l y Vi = f A i l , s e t s o f h i s t o r i e s ( s o t h a t A i j C H f o r e v e r y i , j ) , and a f u n c t i o n f i : V. ---+ [0,1]. The i n t e r p r e t a t i o n i s s u c h 1 that each A r e p r e s e n t s a c e r t a i n s e t of h i s t o r i e s ij 8 i s i t s ,L - b i l i t y . ( a n e v e n t ) , and f i ( A . . )
.
1J To u s e a n example, suppose t h a t t h e modal frame Mi
is
"I am a l m o s t c e r t a i n t h a t " , and pi d e s c r i b e s some p r o p e r t y of h i s t o r i e s , s a y "Event A o c c u r r e d s h o r t l y bef o r e e v e n t B". The s e t s A i j r e p r e s e n t c l a s s e s of h i s -
CHAPTER 6
740
t o r i e s , whose d e g r e e of c e r t a i n t y one i s a b l e t o a s s e s s .
For i n s t a n c e , Ail
may be t h e c l a s s o f h i s t o r i e s s u c h
t h a t A o c c u r r e d no more t h a n 5 m i n u t e s b e f o r e B ( s o that f . (A.
) would be c o n s i s t e n t w i t h " a l m o s t c e r t a i n t y "
11
1
of pi,
and t h e r e f o r e assume a h i g h v a l u e ) ; n e x t Ai2
may
be t h e c l a s s o f a l l h i s t o r i e s i n which B p r e c e d e d A ,
s o t h a t fi(Ai2)
would be s m a l l , e x p r e s s i n g low d e g r e e
of c e r t a i n t y t h a t t h e t r u e h i s t o r y i s i n Ai2,
and s o
on. Intuitively,
t h e s k e l e t o n i s composed o u t o f " t e s t i n g
e v e n t s " , which t h e s u b j e c t may a s s e s s and e x p r e s s t h e i r v a l u e a c c o r d i n g t o t h e c o r r e s p o n d i n g modal f r a m e . Given t h e s k e l e t o n ( V i , f i ) , of a l l s u b s e t s o f Vi,
*
we d e n o t e by Vi
the class
i . e . t h e c l a s s of a l l subsets of
H which a r e u n i o n s of s e t s i n V , ,
and J * t o Vi.
*
we d e n o t e by
t h e 6 - e x t e n s i o n of f i ,i F i n a l l y , we d e n o t e by U ( h ) any f u n c t i o n d e f i n e d on Pi H , whose & - e x t e n s i o n t o - t h e c l a s s of a l l s u b s e t s of
*
'd*
*
Y
DEFINITION. The knowledge K ,
g i v e n by ( 2 , 2 6 ) and t h e
a s s o c i a t e d s k e l e t o n s , w i l l be r e p r e s e n t e d by a f u z z y s e t HK o f h i s t o r i e s , w i t h t h e membership f u n c t i o n
where
U
*
*
( h ) a r e t h e 6 - e x t e n s i o n s d e s c r i b e d above.
pi
2.5. C o n s i s t e n c y and adequacy o f knowledge U s i n g t h e p r o p e r t i e s of t h e s e t s HK one may now d e f i n e
741
FORMAL THEORY OF ACTIONS
v a r i o u s p r o p e r t i e s o f knowledge, s u c h as i t s c o n s i s t e n c y . On t h e o t h e r hand, u s i n g also t h e concept o t " t r u e " h i s t o r y h o , one may e x p r e s s p r o p e r t i e s of knowledgr r e l a t e d t o i t s t r u t h , adequacy, p r e c i s i o n , e t c . DEFINITION, The knowledge K w i l l be c a l l e d c o n s i s t e n t , if
sup U K ( h ) = 1
(2.28)
h
i . e . when t h e fuzzy s e t
HK
i s normal.
T h i s means t h a t f o r c o n s i s t e n t s e t s HK t h e r e e x i s t s a t
l e a s t one h i s t o r y h k H which meets a l l t h e c o n s t r a i n t s imposed b y K i n f u l l d e g r e e . On t h e o t h e r hand, i f U K ( h ) = 0 f o r a l l h C H , t h e n t h e knowledge K w i l l be c a l l e d t o t a l l y i n c o n s i s t e n t . Weakening t h e above e x t r e m e s , w e may s a y t h a t K i s r-inconsistent , i f SUP
UK(h) = 1
-
t.
(2.29)
h F o r € = 0 we o b t a i n h e r e c o n s i s t e n c y , w h i l e f o r c = 1 we g e t t o t a l i n c o n s i s t e n c y . Even t h o u g h K may be i n c o n s i s t e n t , some o f i t s s u b s e t s may be c o n s i s t e n t . I n p a r t i c u l a r , l e t (2.30)
*
Denote U K l ( h ) = min U (h). W e t h e n s a y t h a t K' i s pi. maximally c o n s i s t e h t , i f J sup U K l ( h ) = 1, w h i l e sup U K l , ( h ) < 1 f o r any K" 3 Kl w i t h K' # K".
742
CHAPTER 6
The above c o n c e p t s concerned t h e i n t e r n a l p r o p e r t i e s o f knowledge K , e x p r e s s e d o n l y t h r o u g h t h e p r o p e r t i e s of t h e f u z z y membership f u n c t i o n U K ( h ) . Using t h e n o t i o n o f t r u e h i s t o r y h o , we may now i n t r o duce t h e f o l l o w i n g d e f i n i t i o n s . DEFINITION, The knowledge K i s s a i d t o b e a d e q u a t e , i f
u K ( hO ) 2
UK(h) f o r a l l h C H
(2.31)
and s t r i c t l y a d e q u a t e , i f U K ( h o ) > U K ( h ) for a l l h # h o .
(2.32)
I n p a r t i c u l a r , i f UK(ho) = 1, w e s a y t h a t knowledge K i s t r u t h f u l ( o b s e r v e t h a t t r u t h f u l knowledge i s a l -
ways a d e q u a t e , though n o t n e c e s s a r i l y s t r i c t l y a d e q u a t e ) . I f K i s t r u t h f u l and s t r i c t l y a d e q u a t e , i t w i l l be c a l l -
ed p r e c i s e . Generally, t h e sets
(2.33) and {h: UK(h) = O j
(2.34)
represent the certainty i n K: the first s e t r e f l e c t s a p o s i t i v e and c e r t a i n knowledge a b o u t t h e h i s t o r y , w h i l e t h e second s e t - t h e n e g a t i v e and c e r t a i n knowl e d g e about t h e h i s t o r y h . C o n s e q u e n t l y , t h e s e t
FORMAL THEORY OF ACTIONS
743
may be c a l l e d t h e a m b i g u i t y s h e l l of K : i t i s what i s
p o s s i b l e i n v a r y i n g d e g r e e s , b u t n o t c e r t a i n , about t h e h i s t o r y K. I n order t o represent the s i t u a t i o n i n a quantitative way, one would need a measure on t h e c l a s s of a l l h i s -
t o r i e s , t o express t h e r e l a t i v e s i z e s of s e t s (2.33), ( 2 . 3 4 ) and ( 2 . 3 5 ) .
3. DEVELOPMENT The c o n c e p t s i n t r o d u c e d i n t h e p r e c e d i n g s e c t i o n a r e u s e f u l i n c h a r a c t e r i z i n g t h e p r o c e s s of development, or p r o g r e s s . I n t u i t i v e l y , a development i s a h i s t o r y which p r o c e e d s i n a " p o s i t i v e " d i r e c t i o n , s o t h a t as t h e t i m e p r o g r e s s e s , t h e h i s t o r y assumes more and more "advanced" s t a t e s . T h i s n e c e s s i t a t e s an o r d e r i n g i n t h e s e t of a l l s t a t e s , w i t h r e s p e c t t o t h e r e l a t i o n " i s more d e v e l o p ed t h a n " ( " r e p r e s e n t s h i g h e r development t h a n " , "repr e s e n t s more p r o g r e s s t h a n " , e t c . ) . N a t u r a l l y , s u c h a r e l a t i o n depends on t h e p a r t i c u l a r c o n t e x t ' o f s t u d y , and may b e d e f i n e d i n v a r i o u s w a y s . By a d e v e l o p m e n t a l system, w e s h a l l mean a q u i n t u p l e t
where I and S a r e t h e same as i n system ( 2 . 2 ) , i . e . t h e t i m e i n t e r v a l under s t u d y and s e t o f s t a t e s , s o i s a s e l e c t e d element of S , c a l l e d t h e i n i t i a l s t a t e ,
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HoC H i s a c l a s s o f a d m i s s i b l e h i s t o r i e s s a t i s f y i n g t h e c o n d i t i o n h ( 0 ) = s o , and d i s a b i n a r y r e l a t i o n i n S , s a t i s f y i n g t h e c o n d i t i o n s o f r e f l e x i t i v i t y and t r a n s i t i v i t y : f o r every s,sT,s" S we have s d s , and i f s d s ' and s ' d s " , t h e n s d s " .
(3.2)
The symbol s d s ' w i l l be i n t e r p r e t e d as " s t a t e s ' r e p r e s e n t s e i t h e r t h e same o r h i g h e r development t h a n s " . L e t us o b s e r v e t h a t we do n o t r e q u i r e t h a t e v e r y two s t a t e s b e comparable a c c o r d i n g to r e l a t i o n d : t h e r e may e x i s t p a i r s o f s t a t e s s , s '
such t h a t n e i t h e r s d s ' n o r s ' d s . I n o t h e r words, t h e r e l a t i o n d i s a p a r t i a l o r d e r on t h e s e t o f s t a t e s . I n t h e s t a n d a r d way we may d e f i n e s
s' i f s d s ' and s ' d s , and a l s o sd s' i f s d s ' and n o t s ' d s , s o t h a t t r e p r e s e n t s e q u a l development, and d i s " s t r i c t l y l e s s developed t h a n " . Y
#
We may now def'ine f o r m a l l y development as f o l l o w s : DEFINITION. A h i s t o r y h c1 Ho i s c a l l e d d - c o n s i s t e n t , i f f o r a l l t , t f w i t h t < t ' we have h ( t ) d h ( t ' ) . h i s t o r y i s s u c h t h a t a t l a t e r moments i t r e p r e s e n t s no r e g r e s s , i . e . t h e development a t l e a s t as h i g h as a t p r e v i o u s moments. A d-consistent
DEFINITION. A d - c o n s i s t e n t h i s t o r y i s a development , if f o r some t , t ' w i t h t < t ' we have h ( t ) d * h ( t ' ) . Thus, f o r development, we r e q u i r e d - c o n s i s t e n c y , and also a s t r i c t i n c r e a s e i n development between some
745
FORMAL THEORY OF ACTIONS
moments
.
n L e t Hd and Hd be t h e c l a s s e s of h i s t o r i e s which a r e r e s p e c t i v e l y d - c o n s i s t e n t , and r e p r e s e n t i n g development. N a t u r a l l y , we have n Hd
<
(3.3)
Hd C H o .
For f u r t h e r a n a l y s i s i t w i l l be n e c e s s a r y t o s p e c i f y i s some more d e t a i l t h e c h a r a c t e r o f t h e s t a t e s p a c e S and t h e r e l a t i o n d , t o impose more s t r u c t u r e on t h e system ( 3 . 1 ) . We s h a l l namely assume t h a t t h e s t a t e d e s c r i p t i o n i s m u l t i d i m e n s i o n a l , s o t h a t e l e m e n t s of S a r e v e c t o r s s = (x,,
...,x,) c
s1
x
...
.x
Sk
(3.4)
w i t h c o o r d i n a t e s r e p r e s e n t i n g v a l u e s o f some a t t r i b u t -
es. C o n s e q u e n t l y , a h i s t o r y i s a v e c t o r of f u n c t i o n s
where h i ( t ) c S i , t h e l a t t e r b e i n g t h e s e t of a l l possible values of i - t h a t t r i b u t e .
We s h a l l a l s o assume t h a t e a c h s e t Si i s l i n e a r l y ord e r e d b y a r e l a t i o n d i , r e p r e s e n t i n g development on i - t h a t t r i b u t e . I n o t h e r words, we assume t h a t f o r ever y i , we have a b i n a r y r e l a t i o n d on S such t h a t i i’ f o r a l l x , y , z e Si
(a)
e i t h e r xdiy o r ydixy
746
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(b)
x d1 .x,
(c)
i f xd.y and y d . z , t h e n x d . z . 1
A s before, we define
1
1
*
xdiy and ydix, and xdiy i f x d . y and n o t y d i x . Apart from c o n d i t i o n ( a ) o f con1 n e c t e d n e s s o f r e l a t i o n s d i , t h e r e q u i r e m e n t s are t h e * same a s f o r the r e l a t i o n d . I n p a r t i c u l a r , di i s t h e s t r i c t o r d e r i n g on i - t h a t t r i b u t e ( a c c o r d i n g t o t h e development). X U . ~by
1
We may now d e f i n e r e l a t i o n d on S , by p u t t i n g , f o r s
(x,,
...,x k )
and s ' = ( x i ,
=
...,x i ) ...,k .
s d s ' i f x d x' f o r i = 1, i i i
(3.6)
G e n e r a l l y , i f s and s' a r e s u c h t h a t x . d . x ' and x!d.x i i i J J j for some i and j, t h e n s and s ' a r e n o t comparable according t o t h e r e l a t i o n d , i . e . n e i t h e r sds' nor s'ds.
F i n a l l y , we impose one more c o n d i t i o n , which w i l l make the relation d linear. POSTULATE. There e x i s t s a f u n c t i o n F d e f i n e d on t h e s e t of a l l s t a t e s S , which g e n e r a t e s r e l a t i o n d i n t h e sense t h a t s d s ' i f , and o n l y i f , F ( s )
< F(s').
(3.7)
O b v i o u s l y , t h e f u n c t i o n F ( s ) = F ( x l , . . . , ~ k ) must be monotone w i t h r e s p e c t a l l of i t s arguments, i n t h e f o l l o w i n g s e n s e : for a l l i = l , . . . , k , and a l l f i x e d * * * * v a l u e s x l , . . . , x i-l, X i + l , . . . , x k o f a l l arguments e x c e p t
141
FORMAL THEORY OF ACTZONS
*
*
*
*
t h e i - t h y t h e f u n c t i o n F ( xl y . . . y x i - l , ~ i y ~ i + l y . . . , x k ) i s non-decreasing w i t h r e s p e c t t o t h e r e l a t i o n d i'
I n t h i s c a s e , t h e s e t S p a r t i t i o n s i n t o iso-development c o n t o u r s , d e f i n e d by t h e c o n d i t i o n F ( s ) = c o n s t . T o i l l u s t r a t e t h e s u b s e q u e n t c o n c e p t s , w e s h a l l con-
a two-dimensional d e s c r i p t i o n ; t h e c o o r d i n a t e s w i l l be d e n o t e d by x and y , s o t h a t s = ( x , y ) , and a h i s t o r y i s a p a i r o f f u n c t i o n s h ( t ) = [ x ( t ) , y ( t ) ] , where t 6 I . sider t h e case k = 2 , i . e .
Without loss o f g e n e r a l i t y , assume t h a t t h e r e l a t i o n s d l and d c o i n c i d e w i t h n u m e r i c a l o r d e r i n g s on x - a x i s 2 and y - a x i s . We a l s o assume t h a t t h e i n i t i a l s t a t e i s s = (0,O). 0 and t h a t a l l Moreover, we assume t h a t S = S 2 = [ O , l ] , 1 h i s t o r i e s end a t t h e same p o i n t [ x ( T ) , y ( T ) ] = ( 1 , l ) . Now, t h e o r d e r i n g d i s r e p r e s e n t e d by iso-development c u r v e s F ( s ) = F ( x , y ) = c o n s t , and i t f o l l o w s t h a t t h e s e c u r v e s must be n e g a t i v e l y i n c l i n e d . Consequently, a d- consistent h i s t o r y i s a p a t h l e a d i n g from (0,O) t o ( 1 , l ) i n a n i n c r e a s i n g way; s i n c e (0,O) * = h ( 0 ) d h ( T ) = ( l , l ) ,each d - c o n s i s t e n t h i s t o r y i s i n t h i s c a s e also a development. F i g u r e 1 i l l u s t r a t e s t h e s e c o n c e p t s . The c u r v e s ( a ) and ( b ) a r e iso-development c u r v e s F ( x , y ) = c , w i t h c o n s t a n t c h i g h e r f o r curve ( b ) , r e p r e s e n t i n g h i g h e r l e v e l of development. The c u r v e ( A ) r e p r e s e n t s a development, w h i l e c u r v e (B) r e p r e s e n t s a' h i s t o r y which i s n o t a development, s i n c e i t d e c r e a s e s i n x - c o o r d i n a t e . S i m i l a r l y , c u r v e ( C ) does n o t r e p r e s e n t a development, since it decreases i n y-coordinate.
748
CHAPTER 6
Y
/--'
j
I
\
X
1
F i g u r e 1. Iso-development c u r v e s and development h i s t o r i e s We may now use t h e c o n c e p t s c o n c e r n i n g t h e t i m e o f t h e f i r s t e n t r a n c e and f i r s t e x i t from a s e t , t o i n t r o d u c e t h e n o t i o n of a p h a s e , o r stadium of development. t
L e t us r e c a l l t h a t t h e symbol t h ( A , t ' ) d e f i n e d by ( 2 . 3 ) was used t o d e n o t e t h e t i m e o f t h e f i r s t v i s i t i n A a f t e r time t ' i n h i s t o r y h , while t h ( A , t ' ) denoted t h e t i m e of f i r s t e x i t from A a f t e r rime t l i n h i s t o r y h . t It w i l l be c o n v e n i e n t t o w r i t e t h ( A , t l ) = @ and t ; ( A , t ' ) = ti! i f t h e f i r s t e n t r a n c e t o A a f t e r t ' ( r e s p . t h e f i r s t e x i t from A a f t e r t l ) does n o t o c c u r i n h i s t o r y h. We s h a l l now d e f i n e a p h a s e of development. I n t u i t i v e l y ,
a p h a s e i s a s e t of s t a t e s w i t h t h e p r o p e r t y t h a t a development may p a s s t h r o u g h i t a t most once. Accordingl y , we s h a l l i n t r o d u c e h e r e t h e f o l l o w i n g d e f i n i t i o n s , c o v e r n i n g t h e c a s e s of i n i t i a l , t e r m i n a l and t r a n s i e n t
FORMAL THEORY OF ACTIONS
749
phases. DEFINITION. A s e t P
C
S w i l l be c a l l e d a p h a s e o f deve-
lopment, i f any o f t h e f o l l o w i n g t h r e e s e t s o f c o n d i t i o n s holds :
( a ) (0,O) t P , (1,l) &' P . I f h i s a development and t f = t-(P,O), then t i ( P , t l ) = @ ( i . e . the f i r s t entranh c e t o P a f t e r t ' does n o t o c c u r ) . I n t h i s c a s e , P w i l l be c a l l e d a n i n i t i a l p h a s e .
( b ) (0,O) $ P , '(1,l) 4 P . If h i s a development, and t 1 = t+h(P,O) # @ (so t h a t a l s o t" = t h ( P , t t ) # @ ) , then t t ( P , t " ) = @ ( i . e . i f P was e n t e r e d by h i s t o r y h , t h e n h
i t was a l s o e x i t e d , and t h e r e was no second e n t r a n c e ) . I n t h i s c a s e , P w i l l be c a l l e d a t r a n s i e n t p h a s e .
cf
(1,l) C P . I f h i s a development, and t t = t +h( P , O ) , t h e n t , ( P , t t ) = @ (observe t h a t i n t h i s c a s e t ' # @ ) . Thus, a f t e r t h e f i r s t e n t r a n c e t o P , t h e r e i s no e x i t from i t . I n t h i s c a s e , t h e p h a s e P w i l l be c a l l e d t e r m i n a l . ( c ) (0,O)
P,
Moreover, we i n t r o d u c e t h e f o l l o w i n g i m p o r t a n t t y p e of phases. DEFINITION. A t r a n s i e n t p h a s e P w i l l be c a l l e d n e c e s s a PJ, or u n a v o i d a b l e , i f f o r e v e r y development h , we have tL(P,O) #
@.
I n o t h e r words, a t r a n s i e n t p h a s e i s u n a v o i d a b l e , i f any development must p a s s t h r o u g h t h i s p h a s e ( n o t e t h a t any i n i t i a l o r t e r m i n a l p h a s e i s u n a v o i d a b l e i n
CHAPTER 6
750
t h i s sense). We may now f o r m u l a t e t h e f o l l o w i n g theorem. THEOREM. If a l l developments a r e c o n t i n u o u s , t h e n a
t r a n s i e n t phase P i s u n a v o i d a b l e , i f i t c o n t a i n s any complete iso-development
curve.
The above n o t i o n s a r e i l l u s t r a t e d on F i g . 2. Y
F i g u r e 2 . V a r i o u s t y p e s of development p h a s e s . Here t h e development r e p r e s e n t e d b y t h e lower c u r v e
p a s s e s t h r o u g h p h a s e s Ply P2, P 3 , P5’ P6’ P and P8’ 7 w h i l e t h e development r e p r e s e n t e d b y t h e upper c u r v e p a s s e s t h r o u g h p h a s e s P1, P3, P4, P5, P
7 and Pa.
C l e a r l y , PI i s an i n i t i a l p h a s e , w h i l e Pa i s a t e r m i n -
a l phase; a l l o t h e r p h a s e s a r e t r a n s i e n t . Moreover, phases P P and P a r e u n a v o i d a b l e . 3’ 5 7
751
FORMAL THEORY OF ACTIONS
We may a l s o i n t r o d u c e t h e f o l l o w i n g more g e n e r a l d e f i n i t i o n , of c o n d i t i o n a l u n a v o i d a b i l i t y . DEFINITION. The p h a s e P i s s a i d t o be c o n d i t i o n a l l y unP a v o i d a b l e , g i v e n t h e phase P , i f t h e r e l a t i o n h ( t ) for some t i m p l i e s t h a t h ( t ' ) C P I for some t ' > t . T h i s means t h a t if t h e development p a s s e s t h r o u g h t h e
p h a s e P, i t must l a t e r p a s s t h r o u g h t h e p h a s e PI.
Suppose now t h a t w e have s e v e r a l s y s t e m s of p h a s e s , s a y El, , !'I r ' Each s y s t e m ' I i i s a c e r t a i n p a r t i t i o n of t h e s t a t e s p a c e S i n t o p h a s e s , s a y
r2,. ..
-
4
(3.8) such t h a t
for j # k ,
(3.9)
and (3.10)
*
Since for e v e r y i , t h e p h a s e s i n t h e s y s t e m are d i s j o i n t and e x h a u s t i v e , t h e development h must b e l o n g , a t e a c h t i m e t , t o e x a c t l y one p h a s e of i - t h s y s t e m . L e t us d e n o t e t h e i n d e x of t h i s p h a s e b y j ( i , h , t ) , s o & a t , by d e f i n i t i o n
L e t now h b e a n a r b i t r a r y development i n Hd.
Ti
hence a l s o
752
CHAPTER 6
C o n s e q u e n t l y , a development p r o c e s s p a s s e s t h r o u g h a s e q u e n c e of p h a s e s i n e a c h p h a s e s y s t e m
Ti, hence
t h r o u g h a s e q u e n c e of "compound" p h a s e s , e a c h b e i n g a n i n t e r s e c t i o n of p h a s e s from v a r i o u s s y s t e m s . D E F I N I T I O N . An i n t e r s e c t i o n of t h e form
(3.13) w i l l be c a l l e d a compound p h a s e .
L e t Q d e n o t e t h e c l a s s o f a l l compound p h a s e s . C l e a r l y , t h e number of e l e m e n t s i n Q i s nln
*... n r .
D E F I N I T I O N . We s h a l l s a y t h a t t h e compound p h a s e
i s a d m i s s i b l e , i f t h e r e e x i s t s a development jlj2i.. j r s u c h t h a t f o r some t we have h c. Hd
P
(3.14) Let Q
*
be t h e c l a s s of a l l a d m i s s i b l e compund p h a s e s .
The d i f f e r e n c e
Q - Q
*
expresses a c e r t a i n t y p e of
c o n s t r a i n t s on p h a s e s from v a r i o u s s y s t e m s , o r i n o t h e r words, a c e r t a i n t y p e o f r e l a t i o n s h i p between t h e s y s tems of d e v e l o p m e n t a l p h a s e s . T h i s t y p e of c o n s t r a i n t s
i s c o n n e c t e d w i t h t h e p o s s i b i l i t y of s i m u l t a n e o u s coe x i s t e n c e of p h a s e s from v a r i o u s s y s t e m s : a c o m b i n a t i o n
of p h a s e s , one from e a c h s y s t e m , i s i n a d m i s s i b l e , i f t h e r e i s no development h which would b e l o n g a t t h e
same t i m e t o a l l t h e s e p h a s e s .
153
FORMAL THEORY OF ACTIONS
To e x p l o r e a n o t h e r t y p e o f c o n s t r a i n t s , on s e q u e n c e s of p h a s e s , one may proceed as f o l l o w s . L e t D be t h e c l a s s of a l l f i n i t e s t r i n g s of e l e m e n t s
o f Q , s u c h t h a t no two n e i g h b o r i n g e l e m e n t s a r e e q u a l . Thus, e a c h element of D i s a s t r i n g B1B2.. .Bn, where e a c h B i s a compound p h a s e ( p r o d u c t of t h e form (3.13)1, j and s u c h t h a t Bi # Bi+l for i = 1,. , n-1.
..
DEFINITION. We s h a l l s a y t h a t a s t r i n g B1B2.. , B n G D
i s a d m i s s i b l e , i f t h e r e e x i s t s a development h and t i mes t l ;t 2 < , ' tn such t h a t
...
h(tk) Let D
*
<
Bk,
k = 1
,...,n.
(3.15)
d e n o t e t h e c l a s s of a l l a d m i s s i b l e s t r i n g s .
S i m i l a r l y as for Q
*
,
the difference D
-
D
*
expresses
a c e r t a i n t y p e of c o n s t r a i n t s on development: i f a * s t r i n g u i s not admissible ( i s i n D - D ), then t h e r e i s no h i s t o r y h which would r e p r e s e n t development, and a l s o p a s s t h r o u g h t h e p a r t i c u l a r sequence of compound p h a s e s , as s p e c i f i e d i n t h e s t r i n g u. Thus, e l e m e n t s o f D
*
r e f l e c t t h e c o n s t r a i n t s on tempo-
r a l s e q u e n c i n g of p h a s e s .
*
We s h a l l r e f e r t o t h e s e t Q o f a d m i s s i b l e compound * p h a s e s as t h e p h a s e v o c a b u l a r y , w h i l e t h e s e t D w i l l be c a l l e d t h e p h a s e l a n g u a g e . The second t e r m i s conn e c t e d w i t h t h e f a c t t h a t language i s , f o r m a l l y speaki n g , a c l a s s o f a d m i s s i b l e s t r i n g s formed o u t of a c e r t a i n v o c a b u l a r y . I n t h i s c a s e , t h e e l e m e n t s of t h e c a b u l a r y a r e compound p h a s e s . Some of t h e c o n s t r a i n t s e x p r e s s e d by Q
*
and D
*
VO-
a r e of
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a s y n t a c t i c c h a r a c t e r , w h i l e some o t h e r a r e r e l a t e d t o t h e c o n t e n t of t h e phase systems under c o n s i d e r a t i o n ( o n e may s a y t h a t t h e s e c o n s t r a i n t s a r e of s e m a n t i c character). I n p a r t i c u l a r , we have h e r e t h e f o l l o w i n g theorem, which e x p r e s s e s some s y n t a c t i c c o n s t r a i n t s . THEOREM.
If
BIB 2 . . . B
T h i s theorem a s s e r t s
<
n
J
D
t
, -t h e n
B
j
;Q
*
for e v e r y j .
t h a t i f a s t r i n g o f compound
p h a s e s i s a d m i s s i b l e , t h e n e a c h o f t h e compound p h a s e s a p p e a r i n g i n i t m u s t be a d m i s s i b l e . N a t u r a l l y , one cannot e x p e c t t h e c o n v e r s e t o be t r u e : some s t r i n g s formed o u t o f admissiible compound p h a s e s may n o t be a d m i s s i b l e . We may a l s o p r o v e t h e f o l l o w i n g theorem. THEOREM.
Suppose t h a t u = BIB 2 . . . B n
6
D i s such t h a t
= Bk f o r some j and k (j # k ) . Then t h e s t r i n g u j not a dmi ssi bl e.
B
F o r t h e p r o o f , suppose t h a t B
= Bk = B for some j ', k . j By d e f i n i t i o n of t h e s e t D , B . and Bk c a n n o t b e n e i g h J b o r i n g e l e m e n t s of t h e s t r i n g u , s o t h a t k - j -,1.
It f o l l o w s t h a t t h e r e e x i s t s a compound p h a s e Bm # B
such t h a t j
<
m
<
k.
Assune t h a t t h e s t r i n g u i s a d m i s s i b l e . Then for some development h and some t l L t 2 i t 3 w e have h ( t l ) 6 B , h ( t 2 ) < Bm and h ( t ) c B . T h i s , i n t u r n i m p l i e s t h a t 3 f o r some phase s y s t e m I \i ' t h e development h p a s s e s t h r o u g h a c e r t a i n phase i n s y s t e m f l i ( a t t l ) , l e a v e s c
FORMAL THEORY OF ACTIONS
155
i t ( a t t i m e t 2 ) , and r e t u r n s t o i t a g a i n ( a t t i m e t ) . 3 T h i s c o n t r a d i c t s t h e d e f i n i t i o n o f a p h a s e , and t h u s completes t h e p r o o f of t h e theorem. Some o t h e r n e c e s s a r y c r i t e r i a f o r a d m i s s i b i l i t y o f s t r i n g s may be o b t a i n e d as f o l l o w s . Suppose t h a t t h e phase P ( i ) i n system Ti i s unavoidj a b l e . L e t R b e t h e c l a s s of a l l i n t e r s e c t i o n s P
Jl.. . j n
o f t h e form (3.13), whose i - t h component i s j i - j . We have t h e n t h e f o l l o w i n g theorem. THEOREM.
If u = B1...B -
n
* , then B
6 D
c R for some k .
k -
E q u i v a l e n t l y , w e may s a y t h a t i f a s t r i n g u = B 1 . . . B n i s s u c h t h a t B . d' R f o r j = l , . . . , n ,
missible.
t h e n u i s n o t ad-
J
Even though e a c h compound p h a s e i n Q
n
may a p p e a r f o r
some development, some o f them may a p p e a r more o f t e n t h a n o t h e r s . T h i s w i l l happen i f t h e r e i s a c e r t a i n element o f randomness i n t h e development h . S i m i l a r l y , f o r t h e same r e a s o n , some a d m i s s i b l e s t r i n g s 1 i n D may be more p r o b a b l e t h a n o t h e r s . Thus, i n a d d i t i o n t o l o g i c a l c o n s t r a i n t s on e l e m e n t s n 32 o f Q and D , one has a l s o c e r t a i n s t a t i s t i c a l c h a r a c teristics. A f o r m a l d e s c r i p t i o n of t h e s e c h a r a c t e r i s t i c s i n v o l v e s
t r e a t i n g t h e development ( o r , more g e n e r a l l y ,
history)
as a random p r o c e s s . Thus, we may r e g a r d t h e h i s t o r y as a s t o c h a s t i c p r o c e s s , governed by a n a p p r o p r i a t e l a w , s o t h a t for e v e r y t i n I , and s e t A of s t a t e s , we can speak o f t h e p r o b a b i l i t y f ( t , A ) = P ( h ( t ) 6 A ) t h a t a h i s t o r y w i l l assume a v a l u e i n A a t t i m e t .
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Taking as A a p a r t i c u l a r p h a s e , from one o f t h e systems -c
tii,
o r a compound p h a s e , one can o b t a i n t h e p r o b a b i l i t y
of t h i s p h a s e a t t i m e t . T y p i c a l l y , i f A denotes a t r a n s i e n t p h a s e , t h e probabil i t y f ( t , A ) s h o u l d f i r s t i n c r e a s e w i t h t , and t h e n dec r e a s e . T h i s simply means t h a t for s m a l l t , o n l y a
small f r a c t i o n of h i s t o r i e s w i l l be s u c h t h a t t h e y a t t a i n phase A a t t i m e t . On t h e o t h e r hand, for l a r ge t , o n l y a small f r a c t i o n o f h i s t o r i e s w i l l be i n phase A , because most o f them w i l l have e n t e r e d and l e f t t h i s p h a s e by t i m e t . T y p i c a l l y t h e r e f o r e , t h e r e w i l l be a unique t i m e t A a t which t h e f u n c t i o n f ( t , A ) a t t a i n s i t s maximum. T h i s v a l u e may be c a l l e d a t y p i c a l t i m e f o r phase A .
4 . MULTIPLE GOALS L e t us now r e t u r n t o t h e problem o f g o a l s o f a c t i o n and c o n s i d e r t h e c a s e of a c t i o n s performed i n o r d e r t o a t t a i n a number of d i v e r s e g o a l s . To e x h i b i t t h e e s s e n t i a l s t r u c t u r a l a s p e c t s o f s u c h s i -
t u a t i o n s , we s h a l l i n t r o d u c e c o n s i d e r a b l e s i m p l i f i c a t i o n s i n o t h e r a s p e c t s o f t h e a c t i o n s t r u c t u r e s . Thus, we d i s r e g a r d t h e s e q u e n t i a l c h a r a c t e r o f d e c i s i o n s , as w e l l as s t o c h a s t i c a n d / o r f u z z y c h a r a c t e r o f t h e r e l a t i o n s d e s c r i b i n g dependence of outcomes on a c t i o n s . We simply assume t h a t w e have an a c t i o n s p a c e , s a y A , s u c h t h a t e x a c t l y one o f t h e a c t i o n s from A i s t o be chosen ( c o n s e q u e n t l y , t h e elements of A a r e t y p i c a l l y s t r i n g s o f e l e m e n t a r y a c t i o n s ) . Moreover, l e t Z be t h e c l a s s
FORMAL THEORY OF ACTIONS
151
o f a l l t i m e - e v e n t s , and l e t R be t h e r e l a t i o n o f causa t i o n , connecting elements of A w i t h elements of Z . The symbol aRz w i l l d e n o t e t h e f a c t t h a t p e r f o r m i n g t h e a c t i o n a t A c a u s e s t h e o c c u r r e n c e of time-event z . Consequently, R ( a ) = I z c Z : aRz1
and
a r e r e s p e c t i v e l y t h e c l a s s of a l l t i m e - e v e n t s c a u s e d by a c t i o n a , and t h e c l a s s o f a l l a c t i o n s which c a u s e t h e time-event z . Let Q be t h e s u b s e t o f Z d e f i n e d by q
9 if
a , a ' E A w i t h aRq and - a l R q .
The s e t Q c o n s i s t s t h e r e f o r e o f c o n t r o l l a b l e t i m e - e v e n t s , i . e . e v e n t s which may be a t t a i n e d by a n a p p r o p r i a t e c h o i c e o f a n a c t i o n , and can a l s o be a v o i d e d b y a c h o i c e o f a n a c t i o n . Observe t h a t t i m e - e v e n t s from Z - Q are o f l i t t l e i n t e r e s t : they e i t h e r never o c c u r , o r always occur, regardless of actions undertaken. In p r a c t i c a l s i t u a t i o n s , a goal i s usually defined i n some broad t e r m s . To combine i t w i t h t h e c o n s i d e r e d s y s t e m , one might b e tempted t o a s s o c i a t e w i t h e a c h t i m e - e v e n t t h e d e g r e e t o which t h i s time-event s a t i s f i e s t h e g o a l . Such a n a p p r o a c h , however, has some e s s e n t i a l drawbacks. F i r s t l y , i t would i m p l i c i t l y assu-
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m e t h a t t i m e - e v e n t s a r e comparable w i t h r e s p e c t t o t h e d e g r e e t o which t h e y meet a g i v e n g o a l . T h i s i s a f a i r l y s t r i n g e n t c o n d i t i o n , and one might wish t o weaken i t somewhat. S e c o n d l y , v a r i o u s t i m e - e v e n t s i n t e r a c t i n some w a y s , b o t h w i t h r e s p e c t t o a t t a i n a b i l i t y , and w i t h respect t o s a t i s f a c t i o n of t h e goal.
4.1.
General s t r u c t u r e of a goal
Let now A E Q be a c l a s s of t i m e - e v e n t s .
qeA
Define
q&A
s o t h a t R”(A) i s t h e c l a s s o f a l l a c t i o n s which c a u s e t h e o c c u r r e n c e o f a l l t i m e - e v e n t s i n A , and non-occurence of a l l time-events o u t s i d e A . F u r t h e r , l e t
so t h a t
Q
*
i s t h e c l a s s o f a l l s e t s A which are a t t -
ainable. D E F I N I T I O N . By a g o a l , we s h a l l u n d e r s t a n d a r e l a t i o n -
a l s y s t e m o f t h e form
where Q i s as d e f i n e d above, and D1,,.., D
*
r are bi-
n a r y r e l a t i o n s on Q , assumed t o be t r a n s i t i v e , r e f l e x i v e and c o n n e c t e d .
159
FORMAL THEORY OF ACTIONS
*
F o r A , B t Q , t h e c o n d i t i o n ADiB w i l l be i n t e r p r e t e d as t h e f a c t t h a t t h e composite e v e n t A i s b e t t e r ( i n t h e weak s e n s e ) t h a n t h e composite e v e n t B , from t h e p o i n t of view of i - t h a s p e c t of t h e g o a l G ( o r : from t h e p o i n t o f view o f i - t h c r i t e r i o n ) . The a s s u m p t i o n of c o n n e c t e d n e s s of r e l a t i o n s D k c o r responds t o t h e i m p l i c i t assumption t h a t t h e a s p e c t s of t h e g o a l a r e s u f f i c i e n t l y w e l l i s o l a t e d t o e n s u r e a l i n e a r r e p r e s e n t a t i o n . N a t u r a l l y , when t h e s e a s p e c t s a r e combined, t h e r e s u l t i n g r e l a t i o n need no l o n g e r be connected. We d e f i n e t h e s t r i c t p r e f e r e n c e D; and i n d i f f e r e n c e I . corresponding t o t h e r e l a t i o n D i n t h e u s u a l way. 1 i This system i s s u f f i c i e n t l y f l e x i b l e t o cover p r a c t i -
c a l l y a l l c o n t i n g e n c i e s which may a p p e a r i n r e a l s i t u ations
.
Observe f i r s t l y t h a t t h e g o a l which one wants t o a t t a i n may u s u a l l y b e f o r m u l a t e d as " b r i n g t h e environment t o a c e r t a i n desired state". This s t a t e i s defined as a p a r t i c u l a r c o n f i g u r a t i o n of a t t r i b u t e s , r e l a t i o n s , e t c . o f t h e o b j e c t s which c o n s t i t u t e t h e e n v i r o n m e n t . Such c o n f i g u r a t i o n s , o c c u r r i n g a t a g i v e n t i m e o r t i mes, c o n s t i t u t e a composite t i m e - e v e n t Thus, t h e b a s i c material f o r d e f i n i n g formally t h e g o a l s a r e * elements of Q
.
.
I n t h e s i m p l e s t c a s e , t h e g o a l may be t h e o c c u r r e n c e o f a s i n g l e t i m e - e v e n t , and any o t h e r time-event i s not s a t i s f a c t o r y ( i . e . t h e goal i s of "all-or-nothing" t y p e ) . I n t h i s c a s e t h e r e i s j u s t one r e l a t i o n , D1, and A D i B whenever t h e time-event i n q u e s t i o n i s i n
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A b u t n o t i n B , and A I I B
i f t h i s t i m e event e i t h e r
b e l o n g s t o b o t h A and B , o r t o none o f t h e s e s e t s . Another c a s e may o c c u r when a g a i n , t h e r e i s a d e s i r ed time-event;, s a y g , which r e p r e s e n t t h e g o a l , b u t t h e l a t t e r i s no o f " a l l - o r - n o t h i n g ' ' t y p e . Thus, we s t i l l have ADIB if q E A and g 6 B , b u t t h e r e l a t i o n A D i B ' m a y hold a l s o i n o t h e r c a s e s . I n s h o r t , g satis-
f i e s t h e goal. i n t h e h i g h e s t d e g r e e , b u t t h e r e a r e o t h e r t i m e - e v e n t s which s a t i s f y t h i s g o a l i n l e s s e r degree. The above examples covered t h e c a s e when t h e g o a l was d e s c r b e d d i r e c t l y as a n o c c u r r e n c e o f some t i m e - e v e n t T h i s need n o t be t h e c a s e , and t h e g o a l may be d e f i n e d
.
i n t u i t i v e l y i n terms from o u t s i d e o f t h e system ( e . g . be connectdd w i t h t h e p r e v i o u s s i t u a t i o n s , e t c . ) , w h i l e e a c h c o m b i n a t i o n o f t i m e - e v e n t s from Q i n v o l v e s some d e g r e e t o which t h e g o a l i s a t t a i n e d . I n s u c h cases t h e r e may n o t e x i s t "complete" s a t i s f a c t i o n o f t h e g o a l , and v a r i o u s s e t s o f t i m e - e v e n t s may o n l y be comp a r e d w i t h one a n o t h e r a c c o r d i n g t o d i f f e r e n t c r i t e r i a , e x p r e s s i n g how much t h e y a r e " s a t u r a t e d " w i t h t h e g o a l . L e t u s o b s e r v e t h a t t h e concept o f c r i t e r i o n , i . e . t h e r e l a t i o n D k , was l e f t u n s p e c i f i e d . T h i s was d e l i b e r a t e ,
s o as t o a l l o w g r e a t e r f l e x i b i l i t y o f i n t e r p r e t a t i o n s . I n p a r t i c u l a r c a s e s , t h e r e l a t i o n s D k may, f o r i n s t a n c e , r e f l e c t t h e r e s u l t s o f comparisons of t h e e v e n t s and t h e i r c o m b i n a t i o n s on v a r i o u s p s y c h o l o g i c a l con* t i n u a , e . g . f o r A,B Q one may have A D I B i f A i s more wanted t h a n B , AD2B i f ' one i s m o r e - o b l i g e d t o c a u s e A t h a n B, AD B if A i s e x p e c t e d t o make t h e ' .
3
d e c i s i o n maker more happy t h a n B , and s o f o r t h . I n s u c h a c a s e , e a c h o f t h e r e l a t i o n s D k would have i t s
FORMAL THEORY OF ACTIONS
761
own l i n g u i s t i c r e p r e s e n t a t i o n , i n form o f a c l u s t e r o f functors. Another p o s s i b l e i n t e r p r e t a t i o n o f t h e r e l a t i o n Dk might c o n c e r n v a r i o u s economical o r e n v i r o n m e n t a l c r i t e r i a . Thus, D1 may be ( i n a n a p p r o p r i a t e s e n s e ) " t o be more p r o f i t a b l e t h a n " , D may mean " t o be b e t t e r from t h e 2 p o i n t o f view o f e n v i r o n m e n t a l p r o t e c t i o n " , and s o o n . F i n a l l y , one more p o s s i b i l i t y h e r e i s o f extreme importa n c e . I n many r e a l l i f e s i t u a t i o n s , one i s f a c i n g n o t w i t h o n e , b u t w i t h a s e r i e s o f g o a l s , which may b e , a t l e a s t p a r t i a l l y , i n c o m p a t i b l e . Then t h e r e l a t i o n s ill,
D2,... may c o r r e s p o n d t o p a r t i a l s u b g o a l s , s o t h a t
D 1 may be " b e t t e r from t h e p o i n t o f view o f t h e f i r s t s u b g o a l " , and s o o n . To p u t i t d i f f e r e n t l y , one may s a y t h a t i f G1 = (W,D1, D,) and G 2 = (W,Di, D'), then the j o i n t goal m G = G n G i s d e f i n e d as t h e system (W,D1,...,D ,Dt 1 2 r 1'
...,
...,
. . . ,D'm ) .
4.1.1. The c a s e o f one c r i t e r i o n . A d d i t i v i t y . L e t u s c o n s i d e r t h e s i m p l e s t c a s e when r = 1, i . e . t h e g o a l i s o f t h e form
The r e l a t i o n D i s simply a n o r d e r i n g o f e l e m e n t s o f Q from t h e " b e s t " from t h e p o i n t o f view o f t h e c r i t e rion D, t o the worst. L e t u s assume t h a t Q elements onl y.
i
c o n s i s t s o f a f i n i t e number o f
*
,
162
CHAPTER 6
DEFINITION. We s a y t h a t r e l a t i o n D a l l o w s a d d i t i v e re-
presentation, i f t h e r e e x i s t s a real-valued function f d e f i n e d on Q , s u c h t h a t f o r a l l A , B c Q
*
w e have
It i s w o r t h w h i l e t o o b s e r v e t h a t t h e r e l a t i o n D does not determine t h e f u n c t i o n f uniquely.
Moreover, n o t e v e r y r e l a t i o n D a l l o w s a d d i t i v e r e p r e sentation.
The f o l l o w i n g theorem g i v e s a n e c e s s a r y
condition f o r additive representation. THEOREM. I n o r d e r f o r D t o a l l o w a n a d d i t i v e r e p r e s e n t -
a t i o n , i t i s n e c e s s a r y t h a t f o r any q Q , if (A-{q3)DtA * f o p some s e t A C Q , t h e n t h e same h o l d s f o r a l l s e t s n A C Q , and i f B D ' ( B - { q ] ) f o r some s e t B y t h e n t h e sa-
me i s t r u e f o r a l l s e t s B . To p r o v e t h i s t h e o r e m , o b s e r v e t h a t i f we had t h e cond i t i o n ( A - f q l ) D ' A f o r some A , and B D t ( B - { q J ) f o r some o t h e r s e t B , t h e n we would have f ( q )
>0
and f ( q )
<
0,
which i s i m p o s s i b l e . A s a n example o f a r e l a t i o n D which does n o t a l l o w
a d d i t i v e representation, l e t Q consist of 3 elements, s a y a , b and c . Suppose t h a t Q comprises a l l s u b s e t s , and l e t D be s u c h t h a t a l l two-element s u b s e t s a r e on t h e t o p o f t h e o r d e r i n g , f o l l o w e d b y one-element s u b s e t s ,
*
empty s e t , and t h e n f u l l s e t . Consequently , we have {a,c3Dr{aj,
and [ a , b ) D 1
5 a,b,c) ,
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FORMAL THEORY OF ACTIONS
and c o n s e q u e n t l y , f(a)
+
f(c)
> f(a),
which y i e l d s f ( c )
>
0,
and
which g i v e s a c o n t r a d i c t i o n . We may now i n t r o d u c e t h e f o l l o w i n g d e f i n i t i o n . DEFINITION. A time-event
q with f ( q )
d e s i r a b l e , and an e v e n t q w i t h f ( q ) undesirable.
>
<
0 w i l l be c a l l e d 0 w i l l be c a l l e d
Observe t h a t t h i s d e f i n i t i o n c o n c e r n s o n l y t h e g o a l s i n which t h e r e l a t i o n D a l l o w s a n a d d i t i v e r e p r e s e n t a tion. S i n c e t h e f u n c t i o n f i s n o t d e t e r m i n e d u n i q u e l y by t h e g o a l , one needs t o show t h a t t h e above d e f i n i t i o n i s m e a n i n g f u l , i n t h e s e n s e t h a t d e s i r a b i l i t y or u n d e s i r a b i l i t y o f a n e v e n t d o e s n o t depend on t h e c h o i c e o f the representation. I n d e e d , we have
i s d e s i r a b l e f o r some f u n c t i o n Suppose t h a t Then f , which g i v e s an. add t ve r e p r e s e n t a t i o n o f D . -
THEOREM.
q i s also C e s i r a b l e for any o t h e r f u n c t i o n f
t
which
g i v e s an a d d i t i v e r e p r e s e n t a t i o n o f D . Proof.
Suppose t h a t q i s d e s i r a b l e f o r some f u n c t i o n
CHAPTER 6
164
f , i.e.
f(q)
>
0 . Let f
*
be some o t h e r a d d i t i v e r e p r e -
s e n t a t i o n o f D . By a s s u m p t i o n , f o r any s e t A which d o e s n o t c o n t a i n q , we have ( A U
*
we have
*
which y i e l d s f ( 4 )
>
0 , and c o m p l e t e s t h e p r o o f .
DEFINITION. A t i m e - e v e n t
q i s called neutral with res-
p e c t t o g o a l G which a l l o w s a d d i t i v e r e p r e s e n t a t i o n , i f f ( q ) = 0. O b v i o u s l y , as b e f o r e w e have THEOREM. N e u t r a l i t y i s i n v a r i a n t w i t h r e s p e c t t o t h e
choice of r e p r e s e n t a t i o n f .
4.1.2.
Goals and t h e i r i d e a l s . L e t u s c o n s i d e r a g a i n
t h e c a s e o f one c r i t e r i o n D . T h i s t i m e , however, we do n o t assume t h a t t h e s e t Q
*
*
is finite.
*
*
w e have A DA f o r a l l * A c Q , we s h a l l s a y t h a t g o a l G i s c l o s e d , and any * A w i t h t h e above p r o p e r t y w i l l be c a l l e d t h e i d e a l o f t h e g o a l G . O t h e r w i s e , t h e g o a l G w i l l be c a l l e d o p e n . DEFINITION. I f f o r some A
C Q
We have THEOREM.
If
A
*
B
*
* *
I n d e e d , w e have A DB
* A
and
* B
* *
are two i d e a l s o f G , t h e n A I B
* *
and B I A
.
by t h e a s s u m p t i o n t h a t
are i d e a l s , which i m p l i e s t h e a s s e r t i o n .
FORMAL THEORY OF ACTIONS
765
Next, w e have a l s o THEOREM.
If
Q
*
i s f i n i t e , then the goal G i s closed.
I n d e e d , l e t A1 be a r b i t r a r y , and l e t
G
S1 = $ A
*
Q : AIDA].
* -
S1 = 0, t h e n A1 i s t h e i d e a l , and t h e g o a l i s * c l o s e d . O t h e r w i s e , t a k e A 2 e Q - S l y and d e f i n e If Q
S2
=iA
*
G Q : A~DA]
.
By t r a n s i t i v i t y , w e have S1 C S2 and S1 Z S2. Again,
*
-
S2 = 0, t h e g o a l i s c l o s e d , s i n c e A 2 i s an i d e * a l . O t h e r w i s e , we can choose A 3 G Q - S 2 and p r o c e e d a s b e f o r e . I n t h i s way, we c o n s t r u c t a n i n c r e a s i n g se* quence S c S C . E Q w i t h a l l elements d i s t i n c t . 2, * I n c a s e when Q i s f i n i t e , we must have Q = Sk f o r some k , and t h e a s s e r t i o n f o l l o w s . if Q
..
4 . 1 . 3 . The e f f e c t o f m u l t i d i m e n s i o n a l i t y . L e t us now c o n s i d e r t h e c a s e r > 1, i . e . t h e c a s e o f more t h a n one c r i t e r i o n . N a t u r a l l y , t h e s i t u a t i o n i s of i n t e r e s t only i f t h e c r i t e r i a a r e , a t l e a s t ; p a r t i a l l y , inconsistent. W e s t a r t from t h e followi.ng d e f i n i t i o n :
The p o i n t A i f the conditions
DEFINITION.
BD A , k = 1, k
6
. . ,r
Q
*
i s c a l l e d Pareto optimal,
CHAPTER 6
766
with
BDiA
f o r a t l e a s t one k , imply t h a t B d Q
*.
Thus, a c o n g i g u r a t i o n A of t i m e - e v e n t s i s P a r e t o o p t i m a l , i f one cannot f i n d any c o n f i g u r a t i o n which would be a t l e a s t as good w i t h r e s p e c t t o a l l c r i t e r i a , and be s t r i c t l y b e t t e r on a t l e a s t one c r i t e r i o n . I n o t h e r words, any improvement a l o n g a l l c r i t e r i a must lead o u t s i d e t h e r a n g e of a d m i s s i b i l i t y . Let
*
Q,
A i s Pareto optimal
= {A:
j
.
We have t h e n t h e f o l l o w i n g t h e o r e m .
If
THEOREM.
* and
A , B E Qp
for all j .
AD.B for a l l j , then A 1 . B J J
DEFINITION. The g o a l G = d Q , D A ,
*
closed, i f there e x i s t s A G Q
*
A D A k
f o r e v e r y k = 1,.
...,
Dr)
i s s a i d t o be
such t h a t
. . ,r
and e v e r y
A6 Q
*
.
C l e a r l y , i f a g o a l i s c l o s e d , t h e n any s u b g o a l < Q , D k ) c o r r e s p o n d i n g t o one c r i t e r i o n i s also c l o s e d . The converse i s , i n general, f a l s e , t h a t i s , closure of a l l subgoals
does n o t imply t h e g e n e r a l c l o s u r e .
I n d e e d , c l o s u r e o f < Q , D k ) means t h a t t h e r e e x i s t s an i d e a l A k ; b u t A k need n o t be t h e i d e a l f o r o t h e r subgoals. DEFINITION, The g o a l G = ( Q , D l , . . . , D r ) l o c a l l y c o n s i s t e n t a t maximum, Q**C Q* s u c h t h a t
w i l l be c a l l e d
i f there exists a class
167
FORMAL THEORY OF ACTIONS
if A
C Q
**
and B d Q
** , t h e n
ADkB f o r a l l k ,
and for all i,j then AD.B. J
= 1,
... , r ,
if
A,B
6
Q
** , and
ADiB,
O t h e r w i s e , t h e g o a l w i l l be c a l l e d i n c o n s i s t e n t . The i n t u i t i o n behind t h i s d e f i n i t i o n i s as f o l l o w s . All
t i m e - e v e n t s must b e p a r t i t i o n e d i n t o two c a t e g o r i e s : t h o s e " c l o s e " t o t h e g o a l , and t h o s e " d i s t a n t " from i t . Here t h e f i r s t c o n d i t i o n i n t h e d e f i n i t i o n j u s t i f i e s t h e u s e of t h e term " c l o s e " and " d i s t a n t " . F o r " d 2 s t a n t " t i m e - e v e n t s , no assumption i s made about t h e c o n s i s t ency of r e l a t i o n s D k , w h i l e f o r c l o s e t i m e - e v e n t s , t h e se r e l a t i o n s must c o i n c i d e .
4.2. The problems o f a g g r e g a t i o n The c o n c e p t o f l o c a l c o n s i s t e n c y a t maximum i s o f a
v e r y l i m i t e d a p p l i c a b i l i t y , s i m p l y because i n m o s t c a s e s which o c c u r i n p r a c t i c e , t h e g o a l s a r e n o t cons i s t e n t . A s a r u l e , i f a combination A of time-events s a t i s f i e s t h e g o a l w i t h r e s p e c t t o one c r i t e r i o n , i t d o e s n o t s a t i s f y i t from t h e p o i n t o f view o f o t h e r c r i t e r i a . T h i s i s a phenomenon which h a s been e n c o u n t e r e d i n many d o m a i n s , and was e x t e n s i v e l y s t u d i e d i n t h e l i t e r a t u r e , b e g i n n i n g w i t h P a r e t o . The problem r e d u c e s simply t o t h e q u e s t i o n o f deter mining t h e c o n d i t i o n s under which one can combine a s e t o f b i n a r y r e l a t i o n s i n t o one " o v e r a l l " o r d e r i n g r e l a t i o n .
768
CHAPTER 6
There i s no u n i v e r s a l s o l u t i o n t o t h i s problem, and
t h e s i t u a t i o n i s , i n a s e n s e , h o p e l e s s . The s o l u t i o n s which had been s u g g e s t e d t h u s f a r ( e a c h s u i t a b l e i n a p p r o p r i a t e c i r c u m s t a n c e s ) a r e as follows.
( a ) F u n c t i o n a l r e p r e s e n t a t i o n . I n many c a s e s , D . may J be r e p r e s e n t e d n u m e r i c a l l y , on s c a l e s o f s u f f i c i e n t l y h i g h o r d e r ( i n t e r v a l o r r a t i o s c a l e s ) , To be more p r e c i s e , such a representation requires t h e existence of some r e l a t i o n s s t r o n g e r t h a n D namely some ( u s u a l l y j' ternary o r quarternary) r e l a t i o n s , describing e i t h e r t h e p r o p e r t i e s of m i d p o i n t s , o r r e s u l t s o f comparisons o f l e n g t h s o f i n t e r v a l s ( s e e C h a p t e r 5 ) . If t h e s e r e l a t i o n s meet t h e a p p r o p r i a t e axioms o f measurement t h e o r y , then t h e r e e x i s t scale representations of t h e c r i t e r i a , unique up t o ( s a y ) l i n e a r t r a n s f o r m a t i o n s . I n s u c h a c a s e , one u s u a l l y a c c e p t s as a r e p r e s e n t a t i o n of t h e goa l a c e r t a i n f u n c t i o n o f s c a l e v a l u e s of t h e c r i t e r i a , and one b a s e s t h e judgment on t h e o r d e r induced by t h i s judgment. T h i s i s t h e a p p r o a c h u s e d , s a y , i n l i n e a r programming.
...
F o r m a l l y , w e have here r f u n c t i o n s f l , ,f,, e a c h den f i n e d on Q , s u c h t h a t f i ( A ) '2f i ( B ) i f f ADiB. The u n i v e r s a l ordering r e l a t i o n i s then defined by a function F of r v a r i a b l e s , so t h a t
gives t h e s c a l e v a l u e of A . V e r y o f t e n t h e f u n c t i o n F i s assumed l i n e a r , t h a t i s
r
FORMAL THEORY OF ACTIONS
where
a
i
169
a r e some p o s i t i v e c o n s t a n t s .
( b ) Acceptance o f a h i a r a r c h y . O f t e n t h e approach con-
s i s t i n g of f i n d i n g one f u n c t i o n m e a s u r i n g t h e o v e r a l l degree of a t t a i n m e n t of t h e g o a l i s not a p p l i c a b l e . One might t h e n t r y t o impose a c e r t a i n h i e r a r c h y o f a.nd l o o k s f o r a l e x i c o g r a p h i c a l o r d e r i n j’ duced by t h i s h i e r a r c h y . A s a n o t h e r p o s s i b i l i t y , one relations D
may mention h e r e s u c h t h a t one l o o k s
f o r a n optimum
w i t h r e s p e c t t o one c r i t e r i o n , g i v e n t h a t t h i s optimum
meets c e r t a i n c o n s t r a i n t s d e s c r i b e d i n terms o f t h e remaining c r i t e r i a . The
l a t t e r approach i s e x e m p l i f i e d by t h e t h e o r y o f
s t a t i s t i c a l t e s t s , where one wants t o minimize t h e prob a b i l i t i e s o f e r r o r s o f t h e f i r s t and second k i n d ( r e j e c t i n g t h e h y p o t h e s i s when i t i s t r u e , and a c c e p t i n g t h e h y p o t h e s i s when i t i s f a l s e ) . The p r o b a b i l i t i e s of
t h e s e two k i n d s o f e r r o r s a r e u s u a l l y n e g a t i v e l y r e -
l a t e d , i n t h e s e n s e t h a t a d e c r e a s e of one p r o b a b i l i t y b r i n g s an i n c r e a s e o f t h e o t h e r . A t y p i c a l s o l u t i o n would t h e n c o n s i s t o f f i x i n g t h e p r o b a b i l i t y o f e r r o r o f one t y p e ( u s u a l l y , o f t h e f i r s t k i n d ) , and l o o k i n g
for a p r o c e d u r e t h a t minimizes t h e p r o b a b i l i t y o f e r r o r o f t h e second k i n d , among p r o c e d u r e s which have f i x e d p r o b a b i l i t y o f e r r o r o f t h e f i r s t k i n d . ( i . e . one wants t o maximize power o f t h e t e s t , f o r g i v e n s i g n i f i n a n c e level). Actually, t h e s i t u a t i o n i s s o simple only i n t h e case o f t e s t i n g a simple h y p o t h e s i s a g a i n s t a simple a l t e r n a t i v e ; i n more g e n e r a l c a s e , t h e r e i s n o t o n e , b u t a f a m i l y o f p r o b a b i l i t i e s o f error o f second k i n d (power functions).
710
CHAPTER 6
N a t u r a l l y , s o l u t i o n s of t y p e s ( a ) and ( b ) may be comb i n e d , when some c r i t e r i a a r e c o n s t r a i n e d , and o t h e r s a r e a g g r e g a t e d i n t o a f u n c t i o n a l form. ( c ) I m p o s s i b i l i t y o f a g e n e r a l s o l u t i o n . The g e n e r a l s o l u t i o n , as shown by Arrow ( 1 9 6 3 ) i s n o t p o s s i b l e , i . e . t h e r e i s no o v e r a l l r e l a t i o n which would " f a i t h f u l l y 1 ' represent a l l partial relations D where t h e t e r m j' " f a i t h f u l l y " i s e x p l i c a t e d i n form o f s e v e r a l n a t u r a l and i n t u i t i v e l y a c c e p t a b l e c o n d i t i o n s . I n d e e d , f o r a f o r m a l p o i n t of view t h e s i t u a t i o n h e r e i s v e r y much t h e same as i n t h e c a s e o f Arrow's Impos s i b i l i t y Theorem. The r e l a t i o n s Dk c o r r e s p o n d t o i n d i v i d u a l p r e f e r e n t i a l o r d e r i n g s of a l t e r n a t i v e s ( e l e -
*
ments o f Q ) , and t h e o v e r a l l o r d e r i n g a s s i g n e d t o them c o r r e s p o n d s to o r d e r i n d u c e d by t h e s o c i a l w e l f a r e func-
t ion. ( d ) R e s i g n a t i o n from c o m p a r a b i l i t y . 1 n s t e a d on i n s i s t i n g on t h e e x i s t e n c e o f one l i n e a r r e l a t i o n which "sumone may i n t r o d u c e t h e conmarizes" t h e r e l a t i o n s D j' c e p t o f d o m i n a t i o n , w i t h one o b j e c t d o m i n a t i n g a n o t h e r i f i t p r e c e d e s i t w i t h r e s p e c t to a l l r e l a t i o n s . The c l a s s o f a l l o b j e c t s which a r e n o t dominated i s t h e Pareto optimal s e t , defined i n t h e preceding section. One t h e n r e s i g n s from t h e r e q u i r e m e n t t h a t t h e t h e o r y should s i n g l e o u t an optimal s o l u t i o n .
4 . 3 . An a l g e b r a o f c r i t e r i a . C o n f l i c t t h e o r y Suppose now t h a t we have r c r i t e r i a ( o r : s u b g o a l s ) , c h a r a c t e r i z e d by t h e r e l a t i o n s D1,
...,D r ,
a l l defined
771
FORMAL THEORY OF ACTIONS
on t h e same s e t Q
*.
Assume t h a t e a c h of t h e g o a l s s e p a r a t e l y i s c l o s e d , s o * such t h a t w e that fo P there exists A n r any k = 1, k have A D B f o r e v e r y B . k k n * L e t Q be t h e c l a s s of a l l i d e a l s o f k - t h g o a l . If Q j *k b o t h g o a l s j and k a r e j o i n t l y a t t a i n a b l e , P Qk # 0
...,
*
w h i l e i f Q . n Qk = !3, t h e g o a l s a r e j o i n t l y u n a t t a i n J * Qj n Q k s a t i s f i e s b o t h a b l e . I n t h e f i r s t c a s e , any A
*
goals. T h i s leads t o the following d e f i n i t i o n . DEFINITION. A system of g o a l s , d e f i n e d by r e l a t i o n s D is satisfiable (or: jointly attainable), if
D1,...,
r
Q1
A
... n Q k #
0.
Assume now t h a t a l l s u b g o a l s a r e n o t a t t a i n a b l e j o i n t l y ; s u c h a s i t u a t i o n may be c a l l e d c o n f l i c t i n g , s i n c e i n s u c h a c a s e one must choose some of t h e s u b g o a l s t o be m e t , and r e s i g n from some o t h e r s u b g o a l s . are j o i n t l y satis,... D 16 . ,..., D . 3 of g o a l s w i l l
Assume now t h a t g o a l s Di f i a b l e . Then t h e s e t D f l = { D i
be c a l l e d maximal, if f o r any 'goal DiI k d D ' , i s not satisf$&%le. set C D , ~ ,D ,D 1 ik ik+l
. ..
1
the
To somehow s i m p l i f y t h e n o t a t i o n s , d e n o t e by M t h e c l a s s of a l l c r i t e r i a ( g o a l s ) D1,.. ,Dr L e t Hs de-
.
.
n o t e t h e c l a s s of a l l s u b s e t s o f M which a r e j o i n t l y s a t i s f i a b l e , and l e t H-s = M - H S Y s o t h a t H - S comprise s a l l s u b s e t s which are n o t j o i n t l y s a t i s f i a b l e . * F u r t h e r m o r e , l e t Hs d e n o t e t h e c l a s s o f a l l maximal j o i n t l y s a t i s f i a b l e sets.
CHAPTER 6
112
DEFINITION. We s h a l l s a y t h a t t h e s u b g o a l s D i , D . E M J a r e i r r e c o n c i l i a b l e , i f f o r any s e t S C M , we have
I f a c o n f l i c t i s g e n e r a t e d s o l e l y by t h e e x i s t e n c e o f
i r r e c o n c i l i a b l e p a i r s , we s h a l l c a l l i t l o c a l . The examples o f l o c a l c o n f l i c t s a r e easy t o f i n d : suppose
t h a t t h e s e t M c o n s i s t s o f t h r e e g o a l s , D1, D 2 and D 3’ * where t h e c l a s s H o f maximal s a t i s f i a b l e s e t s c o n i s i s t s S and ( D D ) . Thus, D1 and D2 e x c l u d e of p a i r s (D1,D2) 1’ 3 one a n o t h e r , w h i l e D may be combined w i t h e a c h o f 3 them. A s a n o t h e r t y p e o f c o n f l i c t , w e may d i s t i n g u i s h t h e
c a s e when t h e r e e x i s t s a c o n s t r a i n t on t h e s a t i s f i a b l e s e t s , g e n e r a t e d by a c e r t a i n i n e q u a l i t y . T h i s s i t u a t i o n i s d e s c r i b e d by t h e f o l l o w i n g d e f i n i t i o n , DEFINITION. We say t h a t t h e c o n f l i c t i s g e n e r a t e d by
a g l o b a l c o n s t r a i n t , i f t h e r e e x i s t s a nonnegative f u n c t i o n f d e f i n e d on t h e s e t o f g o a l s , and a c o n s t a n t K such t h a t r
<
max f ( D k ) k
K
<
f(Di) i= l
and s u c h t h a t t h e s e t and o n l y i f
(Di
,..., D
1
) is satisfiable i f
ik
k ) 5
f(Di
J =1
K.
3
A t y p i c a l c a s e o f s u c h c o n f l i c t a r i s e s when s a t i s f y i n g
a g i v e n g o a l i n v o l v e s some c o s t , and t h e t o t a l f u n d s
FORMAL THEORY OF ACTIONS
113
a v a i l a b l e a r e n o t s u f f i c i e n t t o s a t i s f y a l l g o a l s . Here t h e c o n d i t i o n msx f ( D ) 5 K means t h a t e a c h s u b g o a l k separately i s a t t a i n a b l e . It i s o f some i n t e r e s t t o o b s e r v e t h a t n o t e v e r y conf l i c t which i s l o c a l i s g e n e r a t e d b y a g l o b a l c o n s t r a i n t .
I n o t h e r words, t h e r e e x i s t c o n f l i c t s f o r which t h e f u n c t i o n f and c o n s t a n t K g i v e n i n t h e above d e f i n i t i o n do n o t e x i s t .
..
I n d e e d , l e t D1,. ,D be j o i n t l y n o t s a t i s f i a b l e , and 5 suppose t h a t t h i s s e t c o n t a i n s two u n r e c o n c i l i a b l e p a i r s , namely ( D l J D 2 ) and (D D ) . Then t h e maximally a t t a i n 3’ 4 a b l e s e t s a r e t h o s e which c o n t a i n D and one element 5’ * from e a c h o f t h e i r r e c o n c i l i a b l e p a i r s . Thus, H S cons i s t s of t r i p l e t s
We s h a l l assume t h a t t h i s c o n f l i c t i s g e n e r a t e d by a
g l o b a l c o n s t r a i n t , and show t h a t t h i s a s s u m p t i o n leads t o a contradiction. I n d e e d , suppose t h a t a f u n c t i o n f e x i s t s , and l e t f(D1) = a , f ( D 2 ) = b , f ( D 3 )
= c, f(D4) = d, f ( D5) = e .
S i n c e ( D D ,D ) i s a t t a i n a b l e , and (D1,D2) 1’ 3 5 must have a t c t e C K ,
a t b > K ,
which y i e l d s b ) c t e .
i s n o t , we
7 74
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S i m i l a r l y , w e must have
b + d + e c K a n d c t d ) K , which y i e l d s c , b + e . However, b > c + e and c > b t e i m p l y e 4 0 , which c o n t r a d i c t s t h e assumption of n o n n e g a t i v i t y o f f . It i s a l s o worth w h i l e t o s t r e s s t h a t l o c a l c o n f l i c t s ,
and c o n f l i c t s g e n e r a t e d b y a g l o b a l c o n s t r a i n t do n o t e x h a u s t t h e t y p e s of c o n f l i c t s . To show t h i s , c o n s i d e r a c o n f l i c t w i t h s e v e n s u b g o a l s ,
D1,. . . ,D7, and assume t h a t i n e a c h o f t h e t r i p l e t s (D1,D2 ,D3 ) and (D4,D5,D6) e v e r y two, b u t n o t a l l t h r e e goals, are j o i n t l y a t t a i n a b l e . T h i s gives t h e class of a l l m a x i m a l l y a t t a i n a b l e s e t s c o n s i s t i n g o f a l l quintuplets containing D and two p a i r s o u t o f e a c h 7’ * o f t h e above t r i p l e t s . C o n s e q u e n t l y , H c o n s i s t s o f S
C l e a r l y , t h e c o n f l i c t i s n o t l o c a l , s i n c e t h e r e are no irreconciliable pairs. To show t h a t t h e c o n f l i c t i s n o t g e n e r a t e d b y a g l o b a l
c o n s t r a i n t , we p r o c e e d as b e f o r e , and assume t h a t t h e r e e x i s t s a p o s i t i v e f u n c t i o n f , and d e r i v e a c o n t r a -
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FORMAL THEORY OF ACTIONS
d i c t i o n . L e t f(D1) = a , f ( D 2 ) = b , f ( D ) = c , f ( D L i ) = d ,
3
f ( D 5 ) = e , f ( D ) = f . f ( n ) = g . We have t h e n
7
6
a t b t d t e
+
g < K and
a t b t c
>
K,
hence d
c )
+
e t g.
By s y m m e t r y , w e may w r i t e t h e f o l l o w i n g 9 i n e q u a l i t i e s :
a > d t e t g ,
a ) d + f t g ,
a > e + f + g
b > d t e t g,
b
> d t f t g,
c > e t f t g,
c > d t e + g ,
c > d t f t g ,
c > e t f + g .
Adding, w e o b t a i n
3(a
t b t c)
>
6 ( d t e t f ) t 9g.
+
>
6 ( a t b + c ) + 9g,
Similarly, 3(d
e t f)
hence
18g+ 3 ( a t b t c + d t e + f ) ( O which r e q u i r e s t h a t some o f t h e v a l u e s must be nega-
t i v e , thus lesding t o a contradiction. One can i n t r d d u c e a u s e f u l taxonomy o f c o n f l i c t s ( s e e
CHAPTER 6
116
Nowakowska
1973), by i d e n t i f y i n g t h e
t y p e of c o n f l c t
w i t h t h e c l a s s o f maximally a t t a i n a b l e s e t s . Such a t a xonomy i s u n e x p e c t e d l y r i c h , as shown by t h e f o l l o w i n g theorem. THEOREM.
I n c a s e of 3 j o i n t l y u n a t t a i n a b l e motives (ao-
a l s , e t c . ) t h e number o f t y p e s of c o n f l i c t i s 8 , w h i l e i n c a s e of 4 j o i n t l y u n a t t a i n a b l e g o a l s , t h e number o f c o n f l i c t s e q u a l s 113. We s h a l l p r o v e t h e t h e o r e m i n t h e more d i f f i -
Proof.
c u l t case of
4
motives ( g o a l s ) , s a y A,B,C
and D . We
s h a l l e n u m e r a t e t h e c a s e s , d e p e n d i n g on t h e a t t a i n a b l e subsets. Altogether there a r e
4
t r i p l e t s , o f which some may be
a t t a i n a b l e , and some n o t .
a ) There i s 1 c a s e w i t h a l l t r i p l e t s a t t a i n a b l e , s o
*
t h a t t h e c l a s s Hs e q u a l s ( A B C , ABD, ACD, BCD). b ) There are
4
c a s e s w i t h t h r e e of t h e t r i p l e t s a t t a i n -
*
a b l e . A t y p i c a l c l a s s Hs would b e ( A B C , ABD, A C D ) . c ) Suppose now t h a t o n l y two t r i p l e t s a r e a t t a i n a b l e ,
s a y ABC and ABD. T h i s d e t e r m i n e s t h e a t t a i n a b i l i t y o f
a l l p a i r s e x c e p t C D . We have t h e r e f o r e two p o s s i b l e
*
c l a s s e s Hs, d e p e n d i n g on a t t a i n a b i l i t y o f CD namely ( A B C j ABD, C D )
or
(ABC, A B D ) .
777
FORMAL THEORY OF ACTIONS
4
S i n c e t h e two t r i p l e t s may be chosen i n ( 2 ) = 6 w a y s , and e a c h may b e combined w i t h a t t a i n a b i l i t y o r u n a t t a i n a b i l i t y of t h e one p a i r i n q u e s t i o n , we have 1 2 p o s s i bilities. d ) Imagine now t h a t only one t r i p l e t , s a y A B C ,
is attain-
a b l e . T h i s d e t e r m i n e s t h e s t a t u s of a t t a i n a b i l i t y o f
p a i r s AB, A C and B D , but l e a v e s t h e t h r e e p a i r s which
i n v o l v e D , namely A D , BD and CD undetermined as t o t h e i r a t t a i n a b i l i t y . Each of t h e t h r e e p a i r s may be a t t a i n a b l e o r n o t , which g i v e s 2 3 = 8 c o m b i n a t i o n s . A s a t i s f i a b l e t r i p l e t may b e chosen i n 4 w a y s , which g i v e s a l t o g e t h e r 4 . 8 = 32 p o s s i b i l i t i e s . The t y p i c a l c l a s s e s o f maximally a t t a i n a b l e s e t s here a r e : ( A B C , A D , BD, C D ) , ( A B C , A D , B D ) ' , ( A B C , A D ) o r ( A B C , D). The l a s t s i t u a t i o n c o r r e s p o n d s t o t h e c a s e when none o f t h e p a i r s i n v o l v i n g D i s a t t a i n a b l e , s o t h a t t h e maximal s e t i n v o l v i n g D i s j u s t D i t s e l f . e ) I n c a s e s when t h e r e no t r i p l e t s which would be a t t a i n 4 C D . Each o f a b l e , we have ( 2 ) = 6 p a i r s , AB, A C , them may be a t t a i n a b l e o r n o t , which g i v e s 2 6 = 64 cases.
...,
C o n s e q u e n t l y , t h e t o t a l number o f d i s t i n c t t y p e s o f c o n f l i c t i s 1 + 4 t 1 2 + 32 t 64 = 113, which completes the proof. T o summarize t h e c o n s i d e r a t i o n s o f t h e l a s t two s e c t -
i o n s i n i n t u i t i v e terms, one may s a y ' t h a t t h e g o a l , e x p l i c a t e d f o r m a l l y a s a r e l a t i o n a l s t r u c t u r e , was def i n e d as a t t a i n i n g a s t a t e which meets c e r t a i n c r i t e r i a . If t h i s i s possible, t h e goal i s a t t a i n a b l e , otherwise it i s not a t t a i n a b l e .
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If w e have s e v e r a l g o a l s , t h e y may be combined i n t o a
composite junctions structure structure
g o a l , by o p e r a t i o n s o f c o n j u n c t i o n s and d i s o f e l e m e n t a r y g o a l s . It a p p e a r s t h a t t h e o f t h e composite g o a l i s isomorphic t o t h e of t h e s e t of a l l ways of a t t a i n i n g i t .
The g o a l s ( c o m p o s i t e or n o t ) may be a t t a i n a b l e p a r t i a l l y , s o t h a t e a c h a c t i o n (or s t r i n g o f a c t i o n s ) may be a s s i g n e d t h e g r a d e t o which i t s a t i s f i e s a g o a l . T h i s approach l e a d s i n a n a t u r a l way t o t h e n o t i o n o f an i d e a l o f t h e g o a l , b e i n g t h e s e t o f a l l s t a t e s which meet t h e g o a l i n h i g h e s t d e g r e e . The most p e r p l e x i n g problem o c c u r s if one has a number
o f g o a l s which i n t e r a c t i n some w a y , s o t h a t a t t a i n i n g one o f them i n h i g h e r d e g r e e i m p l i e s a t t a i n i n g some o t h e r s i n l e s s e r d e g r e e . Two a p p r o a c h e s a r e p o s s i b l e h e r e . The t r a d i t i o n a l one c o n s i s t s o f a g g r e g a t i o n o f t h e g o a l s i n t o one j o i n t g o a l . U n f o r t u n a t e l y , t h e r e e x i s t s no method o f a g g r e g a t i o n which would be s a t i s factory i n a l l cases. Another p o s s i b i l i t y , e x p l o r e d i n some d e t a i l h e r e , l i e s i n abandoning t h e I d e a o f a g g r e g a t i o n , and r e g a r d i n g t h e c a s e of m u l t i p l e g o a l s as a s i t u a t i o n o f a n i n t e r n a l c o n f l i c t of some s o r t , when one must r e s i g n from some g o a l s i n o r d e r t o a t t a i n o t h e r s . One i s l e d h e r e i n a n a t u r a l way t o t h e n o t i o n o f maximal s e t s o f g o a l s which a r e j o i n t l y a t t a i n a b l e , o r e q u i v a l e n t l y , t o maxim a l s e t s o f m o t i v e s which a r e j o i n t l y s a t i s f i a b l e . A r a t h e r unexpected r e s u l t h e r e i s t h e l a s t theorem,
which a s s e r t s t h a t a r e l a t i v e l y s i m p l e c o n f l i c t o f f o u r competing m o t i v e s ( g o a l s ) which cannot be s a t i s f i e d j o i n t l y may t a k e on one of 113 n o n e q u i v a l e n t forms ,
FORMAL THEORY OF ACTIONS
719
depending on which p a i r s o r t r i p l e t s o f m o t i v e s a r e s a t i s f i a b l e and which a r e n o t .
I n p s y c h o l o g i c a l a p p l i c a t i o n s , t h e most t y p i c a l scheme i s a c o n f l i c t o f f o u r m o t i v e s , which may be g e n e r a l l y
described as: -- o c c u r r e n c e of a n e v e n t which one wants t o o c c u r ; -- non-occurrence of an e v e n t which one wants t o a v o i d ; -- p e r f o r m i n g a n a c t i o n which one ought t o perform;
--
n o t p e r f o r m i n g an a c t i o n which one s h o u l d n o t p e r f o r m
T h i s a l l o w s a c h a r a c t e r i z a t i o n of i n t e r n a l c o n f l i c t s
o f t h e t y p e appro a c h- a p p r o a c h , "appro a c h- avo i d a n c e I' and "avoidance-avoidance", a s w e l l as c o n f l i c t s w i t h i n t h e I d domain, o r between t h e I d and Superego domains i n dynamic psychology. It i s i n t e r e s t i n g t o compare t h e 113 t h e o r e t i c a l l y po-
s s i b l e t y p e s of c o n f l i c t s w i t h t h e number o f t h o s e , f o r which i t i s p o s s i b l e t o f i n d r e a l l i f e examples. The l a t t e r c l a s s i s v e r y l i m i t e d : i t c o n t a i n s o n l y few conf l i c t s , and t h e a t t e m p t s t o f i n d examples of s i t u a t i o n s ( i n psychology) o f a c o n f l i c t of a given t y p e l e a d t o s i t u a t i o n s which a r e a r t i f i c i a l and d i f f i c u l t t o i m a g i n e . Thus, d e s p i t e t h e r i c h n e s s o f r e a l s i t u a t i o n s , and t h e t h e o r e t i c a l p o s s i b i l i t y of 113 c a t e g o r i e s , t h e r e seems t o e x i s t o n l y a v e r y l i m i t e d number o f t y p e s of internal conflicts. T h i s l i m i t a t i o n i s probably related t o r e s t r i c t i o n s imposed by "grammars of s i t u a t i o n s " and b y l e a r n i n g , which l e a d t o a tendency o f r e d u c i n g o f a l l s i t u a t i o n s t o "binary" c o n f l i c t s . Of
c o u r s e , t h i . s l i m i t a t i o n c o n c e r n s o n l y t h e psycholog-
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i c a l c o n f l i c t s ; i n o t h e r c a s e s , e . g . p l a n n i n g or s o l v i n g o r g a n i z a t i o n a l problems (where t h e c o n f l i c t i s genera t e d by l a c k of r e s o u r c e s , e t c . ) , one can e a s i l y cons t r u c t examples o f e a c h of t h e 113 t y p e s of c o n f l i c t s . If a g i v e n composite g o a l i s a t t a i n a b l e , t h e n t h e r e may
u s u a l l y be more t h a n one way o f a t t a i n i n g i t . I n such c a s e s i t i s of some i n t e r e s t t o impose a p p r o p r i a t e c r i t e r i a of o p t i m a l i t y , o r v a l u a t i o n s o f s t r i n g s of a c t i o n s , s o as t o be a b l e t o s e l e c t t h o s e which a r e " b e s t " i n some s e n s e or o t h e r . Such a c o n s t r u c t i o n i s of importance b o t h f o r n o r m a t i v e p u r p o s e s , when t h e system i s used t o d e s c r i b e some a c t i o n s i t u a t i o n , and one i s i n t e r e s t e d i n a t t a i n i n g a c e r t a i n g o a l , and a l s o f o r d e s c r i p t i v e p u r p o s e s , when one s t u d i e s t h e b e h a v i o u r i n a g i v e n s i t u a t i o n , comparing i t w i t h t h e o p t i m a l behaviour. One o f s u c h c r i t e r i a i s p r o v i d e d b y p r a x i o l o g y ; r o u g h l y , i t t e l l s t o a t t a i n a g i v e n g o a l by p e r f o r m i n g o n l y t h e a c t i o n s which a r e e s s e n t i a l (economy of a c t i o n s ) , and by s t r i n g s of a c t i o n s w i t h t h e s h o r t e s t d u r a t i o n (economy of t i m e ) . A c c o r d i n g l y , i n t h e s e t of a l l s t r i n g s of a c t i o n s which b r i n g a b o u t t h e g o a l , one may d i s t i n g u i s h t h e s u b s e t s P r a x and Prax min ( s e e Nowakowska
1 9 7 3 , P P . 138-141).
5 . VERBAL AND NONVERBAL ACTIONS: MOTIVATION AND C H O I C E One g e n e r a l l y e x p e c t s t h a t i n t h e i r b e h a v i o u r p e o p l e are c o n s i s t e n t w i t h t h e i r u t t e r e d p l a n s , i n t e n t i o n s , d e s i r e s , m o t i v e s , e t c . , a t l e a s t t o t h e e x t e n t t o which
FORMAL THEORY OF ACTIONS
781
t h e s e u t t e r a n c e s are j o i n t l y s a t i s f i a b l e , i . e . t h e r e e x i s t s t r i n g s of a c t i o n s which a r e c o n s i s t e n t w i t h a l l utterances. I n o r d e r t o d e v e l o p t h e o r e t i c a l f o u n d a t i o n s for s t u d y o f c o n s i s t e n c y between v e r b a l and n o n v e r b a l b e h a v i o u r , one may d i s t i n g u i s h two c l a s s e s o f a c t i o n s , v e r b a l and n o n v e r b a l o n e s . T h i s amounts t o d i s t i n g u i s h i n g i n t h e " v o c a b u l a r y of a c t i o n s " o f a p e r s o n a s p e c i a l s u b c l a s s o f a c t i o n s , namely v e r b a l o n e s . S p e c i f i c a l l y , t h e s e
v e r b a l a c t i o n s c o n s i s t of u t t e r a n c e s i n which a p e r s o n evaluates, plans, j u s t i f i e s , explains, etc. his past or future actions. These i d e a s were i n t r o d u c e d and e x p l o r e d i n Nowakowska
(1973), d i r e c t i n g ( f o r t h e f i r s t t i m e i n t h e l i t e r a t u r e , dominated t h e n by t h e i d e a s o f Chomsky) t h e a n a l y s i s o f v e r b a l b e h a v i o u s towards t h e s u r f a c e o f l a n g u a g e . The example o f l i n g u i s t i c r e p r e s e n t a t i o n o f m o t i v a t i o n shows t h e methodology of s t u d y i n g v a r i o u s c l a s s e s o f s u c h r e p r e s e n t a t i o n s , b o t h i n t h e i r a s p e c t s o f semantic o r g a n i z a t i o n , t h e a s p e c t s o f l o g i c a l laws o f i n f e r e n c e from t h e n a t u r a l l a n g u a g e , as w e l l as v a r i o u s a s p e c t s o f approximate r e a s o n i n g , which may be o b s e r v e d i n t h e n a t u r a l language. A s r e g a r d s semantic o r g a n i z a t i o n , t h e l i n g u i s t i c r e p r e -
s e n t a t i o n o f s u c h c o g n i t i v e s c a l e s ( p s y c h o l o g i c a l cont i n u a ) as s u b j e c t i v e p r o b a b i l i t y , i n form of e p i s t e m i c f u n c t o r s , t h e r i c h n e s s and d i s c r i m i n a b i l i t y o f t h i s r e p r e s e n t a t i o n , e t c . were a n a l y s e d i n d e c i s i o n a l cont e x t f o r t h e f i r s t t i m e i n Nowakowska ( 1 9 7 3 ) .
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5.1. L i n g u i s t i c r e p r e s e n t a t i o n o f m o t i v a t i o n a l s p a c e
If one a b s t r a c t s from t h e s p e c i f i c c o n t e n t o f t h e u t t e r -
a n c e s i n which p a s t o r f u t u r e a c t i o n s a r e j u s t i f i e d , e x p l a i n e d , e t c . , one o b t a i n s s e n t e n c e schemes, c h a r a c t e r i z e d by the use of c e r t a i n f u n c t o r s , c a l l e d i n t h e s e q u e l m o t i v a t i o n a l f u n c t o r s . They may be p a r t i t i o n e d i n t o t h e following categories:
( a ) e p i s t e m i c f u n c t o r s , s u c h a s "I know t h a t " , "I am c e r t a i n t h a t " , "I d o u b t t h a t " , e t c . (b) e m o t i o n a l f u n c t o r s , s u c h as "I am g l a d t h a t " , e t c .
( c ) p r o p e r m o t i v a t i o n a l f u n c t o r s , s u c h as "I w a n t " , I r I prefer", etc. (d) deontic functors,
s u c h as "I o u g h t to", "1 m u s t " ,
"I s h o u l d " , e t c . One p o s t u l a t e s h e r e e x i s t e n c e o f a c e r t a i n number of s c a l e s , f o r w h i c h t h e s e f u n c t o r s form a l i n g u i s t i c r e p r e s e n t a t i o n . T h e s e s c a l e s , by d e f i n i t i o n , f o r m a mot i v a t i o n a l s p a c e . It i s assumed t h a t i n any c h o i c e s i t u a t i o n , t h e o p t i o n s a r e e v a l u a t e d on t h e s c a l e s o f m o t i v a t i o n a l s p a c e , and f i n a l d e c i s i o n i s determined by t h e s e e v a l u a t i o n s through a corresponding d e c i s i o n
function. V a r i o u s d e c i s i o n models proposed i n psychology, s u c h as SEU ( S u b j e c t i v e E x p e c t e d U t i l i t y ) m o d e l , o r model o f m o t i v a t i o n o f A t k i n s o n (1963), f a l l u n d e r t h i s gen e r a l scheme. To d e s c r i b e t h e s t r u c t u r e o f a d e c i s i o n m o d e l , one may
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accept t h e following hypotheses. HYPOTHESIS 1. The s c a l e s o f m o t i v a t i o n a l s p a c e w i t h t h e l o w e s t l e v e l o f g e n e r a l i t y ( i . e . s u c h which c a n n o t b e decomposed i n t o s u b s c a l e s ) c o r r e s p o n d t o b a s i c u n i t s of self-perception. HYPOTHESIS2 . I n v a r i o u s s t i m u l u s of various
situations, the roles
s c a l e s may d i f f e r , w i t h some s c a l e s domi-
n a t i n g ( t h e d o m i n a t i n g s c a l e may depend h e r e on t h e s p e c i f i c features of decision s i t u a t i o n ) . The a r g u m e n t s f o r H y p o t h e s i s 2 a r e as f o l l o w s . F i r s t l y , some e x p e r i m e n t s o f T v e r s k y ( 1 9 6 9 ) show t h a t o f t e n t h e p e r s o n s t a k e o n l y one a s p e c t o f t h e o p t i o n i n t o a c c o u n t , w i t h o u t t r y i n g (or w i t h o u t b e i n g a b l e t o t r y ) t o t a k e i n t o account o t h e r a s p e c t s . S e c o n d l y , o n e may show t h a t H y p o t h e s i s 2 i s , i n a s e n s e , a n u n a v o i d a b l e c o n s e q u e n c e o f Arrow I m p o s s i b i l i t y Theorem (Arrow 1 9 6 3 ) . The argument h e r e i s a s f o l l o w s . I n i t s o r i g i n a l f o r m u l a t i o n , Arrow's theorem c o n c e r n s s o c i a l d e c i s i o n s , i . e . s i t u a t i o n s i n which e a c h member of t h e s o c i e t y ( s o c i a l group, l e g i s l a t i v e assembly, v o t i n g b o d y , e t c . ) v o t e s , or e x p r e s s e s h i s p r e f e r e n c e s i n some o t h e r w a y . The a l t e r n a t i v e s , s a y A , B , C ,
...
may
b e c a n d i d a t e s , v a r i a n t s o f b u d g e t , a n d s o f o r t h . The problem i s i n t h e e x i s t e n c e o f s o c i a l choice f u n c t i o n ( a l s o r e f e r r e d t o as s o c i a l w e l f a r e f u n c t i o n ) , w h i c h would r e f l e c t a d e q u a t e l y ( i n a " d e m o c r a t i c " w a y ) t h e p r e f e r e n c e s ot members o f t h e s o c i e t y . The c o n c e p t o f "democracy" i s e x p l i c a t e d h e r e i n form o f a number o f p o s t u l a t e s , whose names a r e d e r i v e d from t h e i r i n t e r pretation.
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It ought t o be c l e a r , t h a t f o r m a l l y t h e s i t u a t i o n h e r e i s t h e same as i n t h e c a s e of i n d i v i d u a l c h o i c e based on e v a l u a t i o n s on v a r i o u s s c a l e s of m o t i v a t i o n a l s p a c e : t h e s e e v a l u a t i o n s p l a y t h e r o l e of p r e f e r e n c e s o f v a r i o u s members of t h e s o c i e t y , and f i n a l d e c i s i o n p l a y s t h e r o l e of s o c i a l decision. It i s r e q u i r e d t h a t t h e s o c i a l c h o i c e f u n c t i o n ( f i n a l
o r d e r i n g of a l t e r n a t i v e s of t h e c h o i c e s i t u a t i o n ) s a t i s f i e s t h e f o l l o w i n g c o n d i t i o n s , i n which
2X
stands f o r
stands f o r the t h e p r e f e r e n c e of p e r s o n x, and 2 SOC s o c i a l p r e f e r e n c e ( a l l t h e s e r e l a t i o n s a r e , by assumpt i o n , t r a n s i t i v e and c o n n e c t e d , and t h e r e l a t i o n s of
>
-
>sot'
i n d i v i d u a l and s o c i a l s t r i c t p r e f e r e n c e xJ and a r e defined i n the usual way). indifference 4 XJ
SOC
a ) Lack of d i c t a t o r . No member of t h e s o c i e t y i s a d i c t a t o r , i n t h e s e n s e t h a t h i s p r e f e r e n c e s dominate o v e r t h e p r e f e r e n c e s o f o t h e r s . Formally, x i s a d i c t a t o r ,
>sot
i f t h e c o n d i t i o n A >x B i m p l i e s A B regardless of p r e f e r e n c e s of o t h e r members of t h e s o c i e t y . b ) S o v e r e i g n t y . The group may d e c i d e t h a t A
>sot
By and i t may a l s o d e c i d e t h a t B A by a p p r o p r i a t e votes ( t h a t i s : it i s not t r u e t h a t A B y even i f B x) A f o r a l l members x; a n a l t e r n a t i v e f o r m u l a t i o n of t h i s p o s t u l a t e i s t h a t t h e s o c i a l c h o i c e f u n c t i o n
>sot
>sot
i s n o t imposed).
c ) C o n s i s t e n c y between i n d i v i d u a l and s o c i a l c h o i c e . Suppose t h a t f o r some s e t of p r e f e r e n c e s w e have A )sot B y and t h a t f o r some r e a s o n t h e v o t i n g i s r e p e a t e d . Assume t h a t i n t h e new v o t i n g , a l l who v o t e d A > x B s t i c k to t h e i r o p i n i o n , w h i l e some of t h o s e
785
FORMAL THEORY OF ACTIONS
who p r e v i o u s l y v o t e d B > x A a r e now v o t i n g A
>x B.
Then i n new v o t e , t h e s o c i a l c h o i c e s h o u l d a g a i n be
A
>
SOC
B.
A r u l e which does n o t meet t h i s c o n d i t i o n a p p e a r s
h i g h l y u n n a t u r a l : i t s h i f t s from A
>sot
B to B
>sot
A
when t h e number o f v o t e s f o r A i n c r e a s e s . d)
Independence of i r r e l e v a n t a l t e r n a t i v e s . Suppose
t h a t some v o t i n g gave t h e r e s u l t A
>sot
B , and t h a t
f o r some r e a s o n t h e v o t i n g i s r e p e a t e d , w i t h new o p t i o n C added. Suppose t h a t i n t h e new v o t i n g , e v e r y p e r s o n who p r e v i o u s l y v o t e d A
> x B v o t e s t h e same way ( w i t h
o p t i o n C b e i n g e i t h e r b e t t e r t h a n A , worse t h a n B , o r between A and B ) . S i m i l a r l y , assume t h a t e v e r y p e r s o n who p r e v i o u s l y v o t e d B > x A s t i l l v o t e s t h e same w ay, w i t h C p l a c e d i n a n a r b i t r a r y w a y . Then i n t h e s o c i a l
c h o i c e f u n c t i o n w e s h o u l d a g a i n have A
>
SOC
B.
I n o t h e r words, t h e mutual o r d e r i n g o f A and B i n t h e s o c i a l c h o i c e f u n c t i o n s h o u l d depend o n l y on t h e o r d e r i n g s o f A and B i n i n d i v i d u a l o r d e r i n g s , and n o t on t h e o r d e r i n g s o f any o t h e r a l t e r n a t i v e . e ) Freedom. There a r e a t l e a s t t h r e e p e r s o n s i n t h e soc i e t y , and a t l e a s t t h r e e a l t e r n a t i v e s , which may be o r d e r e d i n any p o s s i b l e way. The theorem o f I m p o s s i b i l i t y o f Arrow a s s e r t s t h a t t h e p o s t u l a t e s a ) - e ) are mutually i n c o m p a t i b l e , t h a t i s , e v e r y s o c i a l c h o i c e f u n c t i o n must v i o l a t e a t l e a s t one o f t h e s e condi.tions. The consequences of Arrow theorem for t h e problem of
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i n d i v i d u a l c h o i c e based on e v a l u a t i o n s on s c a l e s o f m o t i v a t i o n a l space a r e a s f o l l o w s . A s a l r e a d y mentioned, we may i n f e r t h a t a t l e a s t one o f t h e p o s t u l a t e s o f Arrow theorem must be v i o l a t e d . It a p p e a r s t h a t t h e l i k e l y c a n d i d a t e s for t h a t a r e p o s t u l a t e s a ) , b) and d ) , i . e . l a c k o f d i c t a t o r s h i p , s o v e r e i g n t y and i n d e pendence o f i r r e l e v a n t a l t e r n a t i v e s . The r e m a i n i n g p o s t u l a t e s are p r o b a b l y s a t i s f i e d , a t l e a s t i n d e c i s i o n s of r a t i o n a l p e r s o n s . One can now t r y t o d e s c r i b e t h e d e c i s i o n p r o c e s s e s o f p e r s o n s f o r whom t h e d e c i s i o n f u n c t i o n s do n o t s a t i s f y t h e p o s t u l a t e s a ) , b ) or d ) . I n c a s e when a ) i s v i o l a t e d , t h e d e c i s i o n f u n c t i o n i s d i c t a t o r i a l , i n t h e s e n s e t h a t one s c a l e dominates o v e r t h e r e m a i n i n g o n e s . I n o t h e r words, t h e f i n a l p r e f e r e n t i a l o r d e r i n g c o i n c i d e s w i t h t h e o r d e r i n g on t h e dominating scale. There a r e two n a t u r a l c a n d i d a t e s f o r such d o m i n a t i n g
continuum, namely t h e s c a l e s r e p r e s e n t i n g Superego, and s c a l e s r e p r e s e n t i n g I d . One can t h e r e f o r e e x p e c t t h a t among p e r s o n s whose c h o i c e f u n c t i o n s do not s a t i s f y a ) one may d i s t i n g u i s h two c a t e g o r i e s : p e r s o n s f o r whom t h e d o m i n a t i n g continuum b e l o n g s t o Superego f a m i l y , r e p r e s e n t e d b y f u n c t o r s s u c h a s "I ought t o " , "I must", e t c . , and p e r s o n s for whom t h e d o m i n a t i n g continuum b e l o n g s t o t h e I d f a m i l y , r e p r e s e n t e d b y f u n c t o r s such as "I want", "I p r e f e r " , e t c . The f i r s t group w i l l comprise p e r s o n s g e n e r a l l y d e s c r i b e d a s r e l i a b l e , r i g h t e o u s , h o n e s t , e t c . , w h i l e t h e second group compr i s e s h e d o n i s t s . From t h e c l i n i c a l p o i n t o f view, t h e f i r s t group w i l l c o n t a i n compulsive p e r s o n s , and t h e second - p s y c h o p a t s .
FORMAL THEORY OF ACTIONS
787
The second p o s s i b i l i t y i s t h a t t h e c h o i c e f u n c t i o n d o e s n o t s a t i s f y p o s t u l a t e b ) o f s o v e r e i g n t y . Thus , t h e d e c i s i o n ( a t l e a s t w i t h r e s p e c t t o some a l t e r n a t i v e s ) i s imposed. T h i s means t h a t t h e c h o i c e a l w a y s c o i n c i d e s w i t h some a c c e p t e d s t a n d a r d , even i f a l l o r d e r i n g s on
p s y c h o l o g i c a l continua are Contrary t o t h i s s t a n d a r d . It seems t h a t i n t h i s c a t e g o r y one may f i n d o n l y e x t r e -
me c a s e s , of p e r s o n s w i t h weak p e r s o n a l i t y , whose s t r e n g t h i s d e r i v e d from b l i n d o b e d i e n c e ( e . g . SS-men). F i n a l l y , t h e r e may e x i s t p e r s o n s whose c h o i c e f u n c t i o n does n o t s a t i s f y t h e p o s t u l a t e d ) o r independence from i r r e l e v a n t a l t e r n a t i v e s . From p s y c h o l o g i c a l p o i n t of view, t h i s grpup w i l l comprise n e u r o t i c s , who a r e prone t o change t h e d e c i s i o n when a new a l t e r n a t i v e becomes a v a i l a b l e , even i f t h i s a l t e r n a t i v e i s n o t p r e f e r r e d on any continuum: j u s t t h e mere p r e s e n c e o f new o p t i o n , however u n d e s i r a b l e , makes a p e r s o n t o change h i s mind. I n extreme c a s e s , w e have here a t y p i c a l " a l l o r n o t h i n g " a t t i t u d e o f some n e u r o t i c s . The above groups r e p r e s e n t p e r s o n s f o r whom t h e c h o i c e f u n c t i o n g i v e s t r a n s i t i v e and c o n n e c t e d f i n a l p r e f e r e n c e s . I n o t h e r words, t h e theorem o f Arrow i n t h e present case a s s e r t s t h a t a l i n e a r ordering of a l t e r n a t i v e s may be a c h i e v e d o n l y a t t h e c o s t o f v i o l a t i n g one o f t h e p o s t u l a t e s (most l i k e l y a ) , b ) o r d ) ) . There remains a p o s s i b i l i t y o f p e r s o n s f o r whom t h e c h o i c e f u n c t i o n does n o t y i e l d a l i n e a r o r d e r i n g , o n l y a p a r t i a l o r d e r , or i n t r a n s i t i v i t y . P s y c h o l o g i c a l l y , not
much may be s a i d about t h i s group, e x c e p t t h a t , as i n d i c a t e d by e x p e r i m e n t s of Tversky ( 1 9 6 9 ) -- t h e group o f p e r s o n s w i t h i n t r a n s i k i v e p r e f e r e n c e s may be i n f a c t more numerous t h a n one might e x p e c t .
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One s h o u l d mention t h a t t h e e a s e w i t h which one o b t a i n s t h e above consequences o f Arrow theorem i s c o n n e c t e d w i t h i t s n e g a t i v e c h a r a c t e r . I n d e e d , t h e only assumption which i s used h e r e i s t h a t t h e j o i n t e v a l u a t i o n i s based on some e v a l u a t i o n s on p a r t i c u l a r s c a l e s , e a c h c o n c e r n i n g some a s p e c t of t h e a l t e r n a t i v e s . The e x a c t n a t u r e of t h e s e a s p e c t s i s i r r e l e v a n t f o r t h e a s s e r t i o n , and one needs no e m p i r i c a l t e s t s t o v e r i f y t h e consequences. I n o t h e r words, i t t u r n s o u t t h a t t h e above i n t e r p r e t a t i o n y i e l d s c e r t a i n i m p o r t a n t p s y c h o l o g i c a l consequenc e s about t h e p r o c e s s o f d e c i s i o n making, s i m p l y b y o b s e r v i n g t h a t p e o p l e must d i v i d e i n t o v a r i o u s c a t e g o r i e s , depending on which o f t h e f i v e p o s t u l a t e s of Arrow t h e y v i o l a t e . Each o f t h e s e c a t e g o r i e s , i n t u r n , a c q u i r e s a n i n t e r p r e t a t i o n i n p s y c h o l o g i c a l terms. The t h e o r e t i c a l and e m p i r i c a l i m p o r t a n c e o f i d e n t i f i c a t i o n of i n d i v i d u a l and group d e c i s i o n , and t h e r e s u l t i n g a p p l i c a t i o n o f Arrow theorem i s as f o l l o w s . F o r t h e o r e t i c a l r e s e a r c h i t s t r e s s e s t h e importance o f o r d i n a l t y p e s o f c o n t i n u a , which may i n t e r v e n e i n t h e p r o c e s s of d e c i s i o n , and which a r e t r a d i t i o n a l l y l e f t o u t i n d e c i s i o n models. S e c o n d l y , i t l e a d s (among o t h e r s ) t o t h e d e f i n i t i o n and taxonomy o f i n t e r n a l c o n f l i c t s mentioned above. For e m p i r i c a l r e s e a r c h , i t suggests ( a ) t o i d e n t i f y t h e c l a s s e s of s i t u a t i o n s when t h e c r i t e r i a a r e n o t a g g r e g a t e d i n t o a t r a n s i t i v e r e l a t i o n , and ( b ) i f t h e y a r e so aggregated, t o search f o r those p a r t i c u l a r axioms o f Arrow which have been v i o l a t e d i n o r d e r to a t t a i n t r a n s i t i v i t y . The u l t i m a t e p u r p o s e o f t h e l a t t e r
FORMAL THEORY OF ACTIONS
789
s t u d y i s t o i d e n t i f y t h e t y p e s o f d e c i s i o n problems which a r e most l i k e l y t o l e a d t o v i o l a t i o n o f a g i v e n axiom; such knowledge would, i n t u r n , a l l o w t o r e s t r i c t t h e c l a s s o f d e c i s i o n c r i t e r i a which c o u l d b e employed by t h e d e c i s i o n maker.
5.2.
Motivational calculus
If one l o o k s a t l i n g u i s t i c r e p r e s e n t a t i o n o f t h e s c a l e s of m o t i v a t i o n a l s p a c e , one o b t a i n s a fragment o f n a t u r a l l a n g u a g e . The a n a l y s i s o f t h i s fragment i s o f cons i d e r a b l e i n t e r e s t f o r i t s own s a k e ; e s p e c i a l l y importa n t here a r e t h e r u l e s o f i n f e r e n c e , which c o n s t i t u t e the motivational calculus. A s r e g a r d s m o t i v a t i o n a l c a l c u l u s , i t p r o v i d e s schemes of' i n f e r e n c e from s e n t e n c e s i n n a t u r a l language c o n t a i n -
i n g m o t i v a t i o n a l f u n c t o r s . The b a s i c p r i m i t i v e n o t i o n o f t h e t h e o r y i s t h e c o n c e p t o f i n a d m i s s i b i l i t y of a
s e n t e n c e . I n t u i t i v e l y , an e x p r e s s i o n i s i n a d m i s s i b l e , i f t h e r e i s no c o n t e x t i n which i t could be meaningfull y a p p l i e d , or' i f it c o u l d be used o n l y i n " u n n a t u r a l "
s i t u a t i o n s . T h i s c o n c e p t a l l o w s us t o d e f i n e t h e n o t i o n o f semantic i m p l i c a t i o n : A 2 B ( A implies semantically B), i f t h e s e n t e n c e "A and n o t B" i s i n a d m i s s i b l e . M o t i v a t i o n a l c a l c u l u s c o m p r i s e s a number o f s e m a n t i c i m p l i c a t i o n s tletween s e n t e n c e s c o n t a i n i n g m o t i v a t i o n a l f u n c t o r s . These i m p l i c a t i o n s a l l o w i n f e r e n c e from t h e s e n t e n c e s i n n a t u r a l l a n g u a g e . To g i v e some examples, "I a m g l a d t h a . t p" i m p l i e s t h a t t h e p e r s o n e i t h e r knows t h a t p , or he t h i n k s t h a t p ( i s t r u e ) . S y m b o l i c a l l y , t h i s may be w r i t t e n as Gp Kp J T p . A s a n o t h e r ex-
s/
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CHAPTER 6
ample o n e may t a k e i m p l i c a t i o n s s u c h a s W(p&q) >S Wp & Wq, where W s t a n d s f o r "I w a n t " , o r D ( p & q ) S> Dp V Dq, where D s t a n d s f o r "I d o u b t t h a t " . S i m i l a r l y , t h e l a t t e r f u n c t o r D may be r e l a t e d t o o t h e r e p i s t e m i c f u n c t o r t h r o u g h t h e i m p l i c a t i o n Dp --+ T ( - p ) & - C r ( - p ) , S w i t h C r s t a n d i n g f o r "I a m c e r t a i n t h a t " . T h e s e i m p l i c a t i o n s a s s e r t t h a t i f someone w a n t s a conj u n c t i o n o f two e v e n t s , t h e n he w a n t s e a c h o f them separately , too; t h e implications involving t h e functor "I d o u b t " s t a t e t h a t i f someone d o u b t s a c o n j u n c t i o n , t h e n he d o u b t s a t l e a s t o n e o f i t s c o m p o n e n t s , a n d a l s o when someone d o u b t s ( t h e t r u t h o f ) p , t h e n he t h i n k s t h a t not-p, but i s n o t c e r t a i n t h a t not-p. By c o n s t r u c t i n g a s e t - t h e o r e t i c a l m o d e l , i t i s shown i n Nowakowska (1973) t h a t t h e o b t a i n e d c a l c u l u s (comp r i s i n g some 40 i m p l i c a t i o n s ) i s l o g i c a l l y c o n s i s t e n t . I n e f f e c t , t h e s e semantic i m p l i c a t i o n s ( i n form o f t r a n s f o r m a t i o n r u l e s from s e n t e n c e s i n v o l v i n g m o t i v a t i o n a l f u n c t o r s ) c o n s t i t u t e r u l e s of approximate reasoni n g , t h e a p p r o x i m a t i o n r e s u l t i n g from t h e f u z z i n e s s o f t h e underlying concept of semantic a d m i s s i b i l i t y . I n the construction of motivational calculus, i n order t o a c h i e v e g r e a t e r g e n e r a l i t y , t h e t r a n d i t i o n a l way o f l i s t i n g t h e p r i m i t i v e s a n d l a y i n g down t h e r u l e s o f c o n s t r u c t i o n o f c o m p o s i t e s e n t e n c e s was d e l i b e r a t e l y o m i t t e d . It s h o u l d b e c l e a r , h o w e v e r , t h a t g i v i n g s u c h r u l e s would r e s t r i c t t h e c o n s i d e r a t i o n s t o a s p e c i f i c l a n g u a g e , and would a l s o r e q u i r e a s p e c i f i c a t i o n o f the sense of motivational functors (the l a t t e r t a s k i s
s t i l l n o t c o m p l e t e l y s o l v e d f o r any s i n g l e f u n c t o r , l e t
FORMAL THEORY OF ACTIONS
791
a l o n e f o r t h e whole s e t o f f u n c t o r s a t o n c e ) . It d i d a p p e a r more p r o m i s i n g t o p r o v i d e d i r e c t l y a s e t o f i n f e r e n c e r u l e s , an d p r o v e t h e c o n s i s t e n c y o f t h i s s e t . One c a n a r g u e t h a t s u c h an a p p r o a c h i s c l o s e r t o t h e n a t u r a l l a n g u a g e , s i n c e l a y i n g down t h e c o n s t r u c t i o n r u l e s f o r c o m p o s i t e s e n t e n c e s wo u la a l l o w l o n g s e n t e n c e s , acceptable l o g i c a l l y but not l i n g u i s t i c a l l y . The laws o f t h e m o t i v a t i o n a l c a l c u l u s a l l o w f o r m u l a t i o n
o f t h e laws o f m o t i v a t i o n a l c o n s i s t e n c y , by c o mb in in g d e c i s i o n s w i t h u t t e r a n c e s ( t h i s t o p i c w i l l be analysed i n n e x t s u b s e c t i o n ) . Such laws l e a d t o c l a s s e s o f conc l u s i o n s ( u t t e r a n c e s o r a c t i o n s ) w h ic h a r e c o n s i s t e n t w i t h g i v e n u t t e r a n c e , h en ce a l s o a l l o w some p r e d i c t i o n s o f b e h a v i o u r on t h e b a s i s o f u t t e r a n c e s . A n o t h e r p o s s i b i l i t y i s t o u s e t h e laws o f m o t i v a t i o n a l c o n s i s t e n c y and m o t i v a t i o n a l c a l c u l u s f o r d i r e c t i n f e r e n c e a b o u t m o t i v a t i o n . Knowing t h e p e r s o n ' s u t t e r a n c e s and h i s d e c i s i o n s ( a c t i o n s ) , an d a s s u m i n g t h a t he i s m o t i v a t i o n a l l y c o n s i s t e n t , t h e problem l i e s i n adding t a c i t premise:; t o h i s u t t e r a n c e s i n s u c h a way t h a t h i s d e c i s i o n s become m o t i v a t i o n a l l y c o n s i s t e n t w i t h h i s u t t e r a n c e s ( t a c i t and n o t t a c i t ) . I n s u c h c a s e , t h e added p r e m i s e s r e f l e c t t h e ( u n - u t t e r e d ) m o t i v e o f a p e r s o n . C l e a r l y , t h e l e s s freed o m i n s u c h a n a d d i t i o n of p r e m i s e s , t h e more c e r t a i n i s t h e i n f e r e n c e a b o u t p e r s o n ' s motives. The r u l e s of i n f e r e n c e o u t l i n e d abo v e may p r o v i d e forma l methods o f i n f e r e n c e i n p s y c h o l o g i c a l d i a g n o s i s .
It i s also w o r t h w h i l e t o m e n t i o n t h a t t h e s t u d y of l a n g u a g e o f m o t i v a t i o n an d m o t i v a t i o n a l c a l c u l u s , b a s e d
792
CHAPTER 6
on common and everyday s e n s e and i n t u i t i o n , i s a neces s a r y p r e r e q u i s i t e f o r t h e c o n s t r u c t i o n of a model o f
language u s e r . To sum up, t h e c e n t r a l i d e a o f m o t i v a t i o n a l c a l c u l u s ,
which seems t o have some i m p o r t a n c e f o r s e m a n t i c and psycholinguistic s t u d i e s , i s the introduction of l i n g u i s t i c r e p r e s e n t a t i o n o f s i t u a t i o n a b o u t which one wants t o make t h e i n f e r e n c e , and s u c c e s s i v e t r a n s f o r m a t i o n s which a r e a p p l i e d t o t h i s s i t u a t i o n , based on l i n g u i s t i c competence and i n t u i t i o n . It i s assumed h e r e t h a t t h e l e x i c a l u n i t s have c e r t a i n a d m i s s i b l e l o g i c a l p r o p e r t i e s , which a l l o w o b s e r v a t i o n and s t u d y o f d i f f e r e n c e s between l o g i c a l and l i n g u i s t i c s c o p e o f inference. Finally, t h e studies of l i n g u i s t i c inference s u p p l y arguments a g a i n s t s y n t a c t i c c o n c e p t i o n s o f t h e l a n g u a g e , b y s t r e s s i n g t h e f a c t t h a t i n t h e dynamics o f l i n g u i s t i c transformations t h e important r o l e i s
p l a y e d by b o t h s y n t a c t i c and s e m a n t i c a s p e c t s , o r -- i n o t h e r words
--
i n t e r a c t i o n s between l o g i c a l and seman-
t i c properties. I n Nowakowska ( 1 9 8 3 ) m o t i v a t i o n a l c a l c u l u s was e n r i c h e d by i n c l u s i o n o f t e m p o r a l a s p e c t s , i . e . l i n g u i s t i c rep r e s e n t a t i o n o f time. This i n t r o d u c e s an a d d i t i o n a l d i m e n t i o n t o a l r e a d y m u l t i d i m e n s i o n a l concept o f l i n g u i s t i c r e p r e s e n t a t i o n o f m o t i v a t i o n , and a l l o w s f o r a more dynamic t r e a t m e n t of d e c i s i o n s p a c e , i d e n t i f i e d ( t h r o u g h Arrow Theorem) w i t h c o g n i t i v e o r m o t i v a t i o n a l s p a c e , i n which one makes t h e judgments about c o u r s e s of a c t i o n s i n d e c i s i o n s i t u a t i o n s . I n t h i s w a y , t h e s t r i n g s o f a c t i o n s are combined w i t h c o g n i t i v e p r o c e s s e s and l i n g u i s t i c d e s c r i p t i o n s i n t o
FORMAL THEORY OF ACTIONS
193
one u n i f i e d system o f t h o u g h t and a c t i o n , i n which one c o n s i d e r s r a t h e r r e a s o n s f o r a c t i o n s t h a n more e l u s i v e c a u s e s of a c t i o n s . T h i s e x t e n d e d a c t i o n system i s based on t h e c o n c e p t i o n
-- d e r i v e d from n a t u r a l language -- of l i n g u i s t i c measurement ( s e e Chapter 5 l , which a l l o w s as e x t e n s i o n of m e a s u r e m e n t - t h e o r e t i c a l c o n c e p t s t o s u c h n o t i o n s as measurement by a n a l o g y , e t c .
5.3. Linguistic representation of time i n motivational calculus The t o p i c o f t i m e i s d i s c u s s e d e x t e n s i v e l y i n C h a p t e r
5 , h e r e w e c o n s i d e r o n l y some s p e c i a l a s p e c t s , namely t h e l i n g u i s t i c r e p r e s e n t a t i o n of t i m e . We s t a r t h e r e from t h e g e n e r a l c o n c e p t u a l framework o f
f u z z y a t t r i b u t e s which change i n t i m e , similar t o t h a t used i n C h a p t e r 5. I n t h e s e q u e l , t h e symbol t w i l l s t a n d f o r a moment of t i m e , w i t h t h e c o n v e n t i o n t h a t t = 0 s i g n i f i e s t h e p r e s e n t , being t h e time of t h e u t t e r a n c e of t h e sentenc e . P o s i t i v e v a l u e s o f t s i g n i f y f u t u r e moments, w h i l e n e g a t i v e v a l u e s of t denote p a s t . N e x t , w e i n t r o d u c e t h e s e t X o f o b j e c t s under c o n s i d e r a t i o n , i t s elements b e i n g x , y , z , . .
. .
We s h a l l be concerned w i t h a t t r i b u t e s of o b j e c t s i n X ;
t h e s e a t t r i b u t e s may be f u z z y , and a l s o may change i n time.
794
CHAPTER 6
To i n t r o d u c e t h e a p p r o p r i a t e c o n v e n i e n t f o r m a l i s m , l e t U , V,
...
be t h e a t t r i b u t e s under c o n s i d e r a t i o n . W i t h e a c h a t t r i b u t e we may a s s o c i a t e t h e s e t of i t s p o s s i b l e v a l u e s , s a y AU, AV, ...
.
The b a s i c c o n c e p t which w i l l be used f o r e x p l i c a t i n g various concepts concerning l i n g u s t i c representation o f t i m e w i l l be t h a t o f p o s s i b i l i t y , as p e r c e i v e d by AU o f t h e o b s e r v e r s , t h a t o b j e c t x has v a l u e xu a t t r i b u t e U a t t i m e t . We s h a l l assume t h a t t h i s p o s s i b i l i t y i s g r a d e d , and a c c o r d i n g l y , d e n o t e i t by t h e symbol f S ( x , t , x u ) . I n t h e s e q u e l , we s h a l l c o n s i d e r t h e c a s e o f a f i x e d o b s e r v e r s , and t o s i m p l i f y t h e n o t a t i o n s , we s h a l l o f t e n omit t h e s u b s c r i p t s , w r i t i n g s i m p l y f ( x , t , x U ) . Thus,
-
0 c f(x,t,xu)
f
1,
with f ( x , t , x ) = 1 representing the subjective certainty
U
t h a t t h e v a l u e of a t t r i b u t e U o f o b j e c t x a t t i m e t
e q u a l s xu. C o n s i d e r now an e x p r e s s i o n " x i s u a t t i m e t " . Here u i s some d e s c r i p t i o n , p e r t a i n i n g t o a t t r i b u t e U . As a n example, we may t a k e as U t h e a t t r i b u t e s u c h a s h e i g h t , and as u t h e e x p r e s s i o n s such a s " t a l l " , e t c . Now, w i t h e v e r y d e s c r i p t o r u w e may a s s o c i a t e a f u z z y s u b s e t o f t h e s e t AU o f a t t r i b u t e v a l u e s o f U. Let t h i s fuzzy s u b s e t be d e s c r i b e d by membership f u n c t i o n g u . We may t h e n d e f i n e t h e p o s s i b i l i t y f ( x , t , u ) t h a t x i s u a t t i m e t , as
795
FORMAL THEORY OF ACTIONS
f(x,t,u) =
sup
min C f ( x , t , x U ) , g u ( x , ) l .
(5.1)
xu (5 AU Given t h e f u n c t i o n f ( x , t , u ) , we may d e f i n e , for e v e r y 0 5 o( G 1
fixed& with
w i t h t h e conventions t h a t
t (x,u) = -
w
if f ( x , t , u ) l o <
f o r every t
(5.3)
i f f ( x , t ,u) < &
for e v e r y t .
(5.4)
d
and
t (x,u) =
oc
*
Thus, t , ( x , u ) i s t h e time o f t h e f i r s t o c c u r r e n c e o f t h e e v e n t "x i s u i n d e g r e e a t l e a s t & < " . We may now s t a r t e x p l i c a t i n g e x p r e s s i o n s which i n d i c a t e t i m e , e i t h e r d i r e c t l y or i n d i r e c t l y , s u c h as ''a w h i l e ago" , "soon" , " r e c e n t l y " , e t c . These e x p l i c a t i o n s w i l l b e based on f u z z y d e s c r i p t i o n s o f c e r t a i n s e t s o f t i m e moments d e f i n e d t h r o u g h f u n c t i o n f ( x , t , u ) , and a l s o times t d ( x y u ) . I n t h e d e f i n i t i o n s below, we s h a l l need c e r t a i n f u z z y s e t s o f t i m e moments. These w i l l g e n e r a l l y be d e n o t e d by c a p i t a l l e t t e r s ; e . g . n G LARGE means t h a t n i s a l a r g e nur*l.ber, t h e a t t r i b u t e " l a r g e " b e i n g f u z z y , and c h a r a c t e r i z e d by a n a p p r o p r i a t e membership f u n c t i o n . To be more p r e c i s e , t h e f u z z y s e t s i n q u e s t i o n w i l l depend a l s o on t h e c o n t e x t , i n t h i s c a s e t h e p a i r w i l l mean t h a t n i s a (x,u), s o t h a t n G L A R G E ( x ,u) l a r g e number i n t h e c o n t e x t s p e c i f i e d by o b j e c t x and
796
CHAPTER 6
f u z z y a t t r i b u t e u. Thus, x was u r e c e n t l y
<+
(5.5)
( 3 t < 0 ) : f ( x , t , u ) = 1 and I t [ c SMALL(x,u). T h i s means t h a t a t some t i n t h e p a s t o b j e c t x had
a t t r i b u t e u i n f u l l d e g r e e , and t h e t i m e t i s small, i n t h e context (x,u). It i s w o r t h t o mention t h a t t h e r o l e o f c o n t e x t h e r e i s e s s e n t i a l . I n d e e d , "I v i s i t e d him r e c e n t l y " i n d i c a t -
e s t h e time of perhaps s e v e r a l days a f t e r t h e event t o o k p l a c e . However, "We r e c e n t l y had I c e Age" i n d i c a t e s t h e t e m p o r a l d i s t a n c e o f some 2 0 000 y e a r s , which i s a small number i n t h e g e o l o g i c a l c o n t e x t , b u t n o t necessarily otherwise. Replacing i n c o n d i t i o n ( 5 . 5 ) t h e requirement t h a t f ( x , t , u ) = 1 by f ( x , t , u ) 2 o( f o r some O C , w e o b t a i n a weaker v e r s i o n o f t h e same d e f i n i t i o n . However, i n t h e ! s e q u e l , t h e d e f i n i t i o n s w i l l b e f o r m u l a t e d f o r o( = 1. A future counterpart of recently is
soon, which
may
b e d e f i n e d as f o l l o w s :
x w i l l soon by u @
( 3t
> 0):
(5.7)
f ( x , t , u ) = 1 and It1 CSMALL(x,U).
A s r e g a r d s t e m p o r a l q u a n t i f i e r s , we may w r i t e
x has a l w a y s been u CJ ( V t ( 0 ) : f ( x , t , u ) = 1. ( 5 . 8 )
797
FORMAL THEORY OF ACTIONS
I n f a c t , we may r e s t r i c t h e r e t h e g e n e r a l q u a n t i f i e r , s o t h a t t h e r i g h t hand s i d e o f ( 5 . 8 ) becomes (
3
T
< O)('vt):T L t C and
0 =) f ( x , t , u ) = 1
T 6 LARGE(x,u).
(5.9)
Here T i s t h e t i m e o f r e f e r e n c e , which depends on t h e
context. Similarly, x has n e v e r been u
(V t
c
0):
f ( x , t , u ) = 0.(5.10)
A s r e g a r d s t h e e x i s t e n t i a l q u a n t i f i e r "Sometimes", i t h a s two u s a g e s : t e m p o r a l ( e . g . "It sometimes g e t s
r a t h e r c o l d i n New Y o r k " ) and f r e q u e n t i a l ( e . g . "Apples
a r e sometimes : r o t t e n " ) . Thus ( s e e a l s o Suppes 1979, who s t u d i e d q u a n t i f i e r s s u c h as " a l l " "some", e t c . ) x i s sometimes u
f ( x , t j , u ) = 1 and f ( x , t t , u ) = 0 j
and n
c
NOT VERY SMALL(x,U).
(5.11)
Observe t h a t t h e r e q u i r e m e n t o n l y t h a t f ( x , t , u ) = 1 j f o r some t i m e s t i s n o t s u f f i c i e n t : i t i s a l s o necej s s a r y t h a t f ( x , t t , u ) = 0 f o r some p o i n t s t t . I n d e e d , j j i f we had f ( x , t , u ) = 1 f o r all t , t h e u s e o f "sometimes"
798
CHAPTER 6
would n o t be a p p r o p r i a t e . I n o t h e r words, t h e s e n t e n ce "x i s sometimes u" i m p l i e s t h a t x i s sometimes n o t U.
A s r e g a r d s f r e q u e n t i a l q u a n t i f i e r s , such a s " o f t e n " ,
"seldom" , e t c
.
we have
x i s often u
f ( x , t j , u ) = 1 and f ( x , t l , u ) = 0 and n j
g~ n / [ t n l C LARGE
(x,u)
gc u n i t s o f time
LARGE
(x,u)
C APPROPRIATE
(x,u)
Only t h e l a s t c o n d i t i o n r e q u i r e s some comments. While n / [ t n l measures t h e f r e q u e n c y o f o c c u r r e n c e o f s i t u a t i o n s when x i s u p e r u n i t o f t i m e , i n o r d e r f o r h a v i n g a j u s t i f i a b l e usage o f " o f t e n " , t h e u n i t must be a p p r o p r i a t e l y chosen. I n d e e d , i f a v o l c a n o e r u p t s once i n e v e r y t e n y e a r s , i t e r u p t s " o f t e n " . I f , howev e r , John i s l a t e f o r work o n l y once i n a b o u t e v e r y 1 0 y e a r s , t h e f r e q u e n c y i s t h e same, b u t " o f t e n " i s not appropriate. T o d e f i n e r e l a t i o n a l t e r m s , such as " e a r l i e r " , " b e f o r e " , " p r i o r t o " , " l a t e r " , e t c . , we w i l l u s e t h e t i m e s t , ( x , u ) d e f i n e d by ( 5 . 2 ) . We s h a l l c o n s i d e r t h e c a s e d = 1 o n l y , and f o r s i m p l i c i t y d e n o t e t ( x , u ) = t , ( x , u ) . Firstly
, we
have
x was u b e f o r e y was v e t ( x , u )
<
t(y,v)
and s i m i l a r l y f o r o t h e r r e l a t i o n a l t e r m s .
(5.12)
-
799
FORMAL THEORY OF ACZ7ONS
For i n s t a n c e , " l a t e r " may be used w i t h u n c o n d i t i o n a l t i m e r e f e r e n c e , s u c h as " l a t e r t h a n on Tuesday", a s w e l l as w i t h c o n d i t i o n a l t i m e r e f e r e n c e , s u c h as " l a t e r t h a n when y becomes v" ( i . e . a f t e r y becomes v ) . Thus, x w i l l be u l a t e r t h a n t
+>t ( x , u ) >
t
(5.13)
while
x w i l l be u a f t e r y becomes v
@
F i n a l l y , t h e r e e x i s t i n t e r v a l r e f e r e n t i a l t i m e s , conn e c t e d w i t h t h e u s e of terms s u c h as " t h r o u g h o u t " , " s i n c e " , " d u r i n g " , e t c . I f T = [ t 1' t 2 1 i s t h e t i m e i n t e r v a l i n q u e s t i o n , we must c o n s i d e r two c a s e s , t 2 < 0 and t 2 = 0 . I n t h e f i r s t c a s e , we u s e Simple P a s t , w h i l e i n t h e second - P r e s e n t P e r f e c t . F o r i n s t a n c e
x was u t h r o u g h o u t T ( t J t ) :t E T + f ( x , t , u )
= 1.
(5.15)
The u s e o f " s i n c e " i s b a s e d on t h e same p r i n c i p l e , w h i l e
" d u r i n g " may have o n l y i n d i r e c t t i m e r e f e r e n c e : x was u d u r i n g t h e t i m e when y was v
&
( 3 t l ) : tl < t ' c t 2
& f(X,t',U) = 13.
The f u n c t i o n f ( x , t , u ) d e s c r i b e s t h e p o s s i b i l i t y t h a t
CHAPTER 6
800
x i s u a t t i m e t . N a t u r a l l y , i f u 1 C u i s a s u b s e t of t h e s e t u of v a l u e s of a t t r i b u t e , t h e n t h e p o s s i b i I i 5 t y t h a t x i s u 1 a t t i s a t most f ( X , t , u ) , i . e .
I n a s i m i l a r w a y , t h e f u n c t i o n f ( x , t , u ) may be used t o e x p r e s s temporal p o s s i b i l i t i e s : t h e p o s s i b i l i t y t h a t x was u i n t h e i n t e r v a l T i s F(x,T,u)
=
sup f ( x , t , u ) . t €T
(5.17)
L e t u s now assume t h a t f ( x , t , u ) d e s c r i b e s t h e " t r u e " h i s t o r y , i . e . t h e e v o l u t i o n of t h e fragment of t h e world under c o n s i d e r a t i o n ( i . e . t h e s e t o f o b j e c t s X , categorized according t o the a t t r i b u t e s i n question, d u r i n g t h e i n t e r v a l of t i m e under s t u d y ) . Suppose now t h a t a t t i m e t = 0 ( p r e s e n t ) a s e n t e n c e i s u t t e r e d , r e f e r r i n g t o t h e p a s t , and a s s e r t i n g s o m e t h i n g about t h e a t t r i b u t e s of c e r t a i n o b j e c t s a t some t i m e s . ( s u c h as " Y e s t e r d a y i t r a i n e d h e a v i l y i n San F r a n c i s c o , and P e t e r g o t w e t " ) . Each s u c h s e n t e n c e may be r e d u c e d t o i t s normal form, i . e . t o a c o n j u n c t i o n of s e n t e n c e s of t h e form, d e n o t e d h e r e by S ( x , t , u , d ) , where t h i s symbol s t a n d s for a n a s s e r t i o n t h a t a t t i m e t , o b j e c t x was u i n t h e degree a t least d . A c o n j u n c t i o n o f such s e n t e n c e s , w i t h x , t , u and oc r a n g i n g o v e r t h e r e s p e c t i v e s e t s X , R , AU and [O,l1 i s t h e n r e p r e s e n t a b l e as
FORMAL THEORY OF ACTIONS
where Q i s a s u b s e t of X
*
R Y A
U
X
80 1
[0,1].
*
We may t h e n say, as i n C h a p t e r 5 , t h a t S i s f a i t h f u l , i f for e a c h ( x , t , u , d ) .E. Q w e have f ( x , t , u ) , d . We may a l s o c o n s i d e r i n t e r v a l or p o i n t f a i t h f u l n e s s , depending whether t h e set; Q i n v o l v e s d i s c r e t e p o i n t s i n t i m e , or an i n t e r v a l o f time. L e t u s now c o n s i d e r some examples o f s e m a n t i c i m p l i c a t i o n s i n v o l v i n g t e m p o r a l l y i n d e x e d s e n t e n c e s and motivational functors. Thus , we have f ' o r i n s t a n c e -Com [a,P)
~3
(3t 7
0 ) : possible mt(a,P)
&
I n words, i t means t h a t t h e s e n t e n c e "I cannot make a t o b e P now" i m p l i e s t h a t i t i s g e n e r a l l y p o s s i b l e t o make a t o be P a t some t i m e , and whenever t h e p e r s o n w i l l b e a b l e t o make a t o be P , and w i l l want i t , he w i l l a c t u a l l y make i t . A s a n o t h e r example, we have O[m(a,P)]
2
when I have t i m e
necess m(a,E')
& mt,(b,Q)l
K3f(
3 t)(t70)
.
T h i s means that. when someone says t h a t he ought t o make
a t o be P when he has t i m e , t h e n he knows t h a t he does n o t have t h e t i m e now, and t h a t he must have done some-
802
CHAPTER 6
t h i n g i n t h e p a s t which n e c e s s i t a t e s making a t o b e P . We have a l s o I w i l l make a t o be P b e f o r e T E
K{(3t)
(0
<
t <'TI
&
(tfb # a ) ( e Q # P ) : -Otm(b,Q)].
Thus, i f someone s a y s t h a t he w i l l do something b e f o r e t i m e T , t h e n he knows t h a t t h e r e w i l l be some t i m e bef o r e T s u c h t h a t he w i l l n o t b e o b l i g e d t o do a n y t h i n g e l s e which might i n t e r f e r e w i t h making a t o be P ( s o
t h a t he w i l l b e a b l e t o f u l f i l l h i s p r o m i s e ) . Next,
(3t
<
0 ) Wt(a,P)
CP (necess. - ( a , P ) )
0
("I wanted a t o
& -K(necess.
& -OW
0
-(a,P))
->S
(a,€')
b e P , b u t d i d n o t know t h a t a can n e v e r
be P" i m p l i e s t h a t t h e s p e a k e r i s now c e r t a i n t h a t a can n e v e r be P , and t h a t he s h o u l d n o t want now a t o be P ) . T h i s i m p l i c a t i o n shows how t h e f o u n d a t i o n s of r a t i o n a l i t y of m o t i v a t i o n a r e s t r u c t u r e d , and how t h e d e f e n s e mechanisms i n t h e s e n s e o f Freud a r e formed. They a l l o w
f u n c t i o n i n g i n t h e c o n d i t i o n s of s t r e s s and d e p r i v a t i o n . Finally, Some t i m e ago I wanted a t o be P
5
FORMAL THEORY OF ACTIONS
803
( t h e s p e a k e r d o e s n o t know now whether he wants a t o be P o r n o t ) .
I n t h i s example one may s e e how we u s e ( i n t h e l a n g u a g e ) t h e weak ( o r v a g u e ) n e g a t i o n , where t h e t r u t h condit i o n s a r e r e s t r i c t e d t o e a r l i e r t i m e s and t h e s e n t e n c e does n o t ( b u t may) be t r u e a t l a t e r t i m e s . T h i s i m p l i e s t h e e x p e c t a t i o n o f change connected w i t h s u c h e v e n t s which o c c u r r e d i n t h e meantime, and c o u l d c a u s e changes i n t h e s t r u c t u r e o f knowledge ("I t h o u g h t t h a t a i s PI', which d o e s n o t imply t h a t I now t h i n k s o ) . This implication presupposes t h a t t h e l i s t e n e r i s not
c e r t a i n w h e t h e r now t h e s p e a k e r i s g o i n g t o s u p p o r t h i s p r e v i o u s s t a t e o f mind, p r e f e r e n c e , e t c . or deny i t , o r announce t h e s t a t e o f c o n f u a i o n , r e q u i r i n g c l a r i f i c a t i o n and j u s t i f i c a t i o n . Such a s s e r t i o n s u n d e r l i e t h e o b s e r v e d and e x p e c t e d c h a n g e a b i l i t y . A s s e r t i n g a g i v e n v a l u e s o f a n a t t r i b u t e o f a n o b j e c t i n t h e past p l a y s i n t h e l a n g u a g e t h e r o l e o f something l i k e vague n e g a t i o n , which l a t e r s h o u l d b e and may be s h a r p e n e d i n t h e discourse. L e t u s now remark on t h e p o s s i b l e f u z z i n e s s o f a p p l i -
c a t i o n o f f u n c t o r s t o t h e i r arguments. Thus, l e t F Z , ,... b e f u n c t o r s F , G ,... w i t h z i n d i c a t i n g t h e d e g r e e , s o t h a t F Z p s t a n d s for p i s F i n d e g r e e z . GZ
Such f u n c t o r s may b e i t e r a t e d , i f p i t s e l f i s a f u z z y e x p r e s s i o n , s o t h a t we may have F Z p , F Z ( G Z , p ) , F Z I G Z l ( H Z l , p ) ]and s o on. The n e g a t i o n o f s u c h f u n c t o r s i s o b t a i n e d from t h e r u l e -FZp = F1-z ( - p ) , s o t h a t
CHAPTER 6
804
I f w e have a time-indexed p r o p o s i t i o n p , and T i s t h e
t i m e a x i s , t h e n f o r e a c h p o s s i b l e world W we may cons t r u c t a f a m i l y o f f u n c t i o n s which map pand W i n t o 2 T , i n d e x e d b y v a r y i n g t i m e , namely showing a t which mome n t s t h e p r o p o s i t i o n p was t r u e i n W . Thus, i f P i s a * s e t o f p r o p o s i t i o n s , and W . i s t h e c l a s s o f p o s s i b l e w o r l d s , t h e n w e have f: P x
w*
->2,
T
,
where f(p,W) i s t h e s e t o f p o i n t s i n t i m e when p i s t r u e i n W . T h i s may be c a l l e d a h i s t o r y o f p i n W . I n c a s e o f t h e s e c o n s i d e r a t i o n s , as w e l l as t h o s e o f C h a p t e r 5 , i n s t e a d o f t h e t e r m " t r u t h " , t h e terms " f a i t h f u l n e s s " and " e x a c t n e s s " were i n t r o d u c e d . These n o t i o n s a r e based on t h e a s s u m p t i o n o f e x i s t e n c e o f t r u e s t a t e o f a n o b j e c t , e x p r e s s e d by i t s a t t r i b u t e s , and r e l a t i o n s between them and t h e p r o p e r t i e s o f descr i p t ion. The d e s c r i p t i o n o f an o b j e c t has t o c o v e r t h e a t t r i b u t e s
which are e s s e n t i a l ( f o r t h e p u r p o s e i n q u e s t i o n ) on a n a p p r o p r i a t e l e v e l o f e x a c t n e s s . The g o a l o f t h e d e s c r i p t i o n i s t h e r e f o r e t h e second c o n s t r a i n i n g r e f e r e n t i a l f a c t o r . C o n s e q u e n t l y , one can have d e s c r i p t i o n s w i t h v a r y i n g d e g r e e s o f f a i t h f u l n e s s and e x a c t n e s s , which r e f l e c t s v a r i o u s d e g r e e s o f t r u t h about t h e o b j e c t . One c o u l d c o n s i d e r t h e c o n d i t i o n s imposed on d e s c r i p t i o n s under which one c o u l d s p e a k o f t h e f u l l t r u t h o f d e s c r i p t i o n . The c o n c e p t s o f f a i t h f u l n e s s and e x a c t n e s s are t h e r e f o r e b a s e d on a c e r t a i n o n t o l o g y of the objects.
FORMAL THEORY OF ACTTONS
805
5 . 4 . M o t i v a t i o n a l c o n s i s t e n c y and s t r u c t u r a l p r o p e r t i e s o f s e t s o f s t r i n g s o f a c t i o n s which f u l f i l l o r b r e a k a promise One o f t h e b a s i c e l e m e n t s o f t h e d e s c r i p t i o n o f p e r s o n ' s b e h a v i o u r i s t h e m o t i v a t i o n a l c o n s i s t e n c y (or i t s l a c k ) , u n d e r s t o o d i n most common s e n s e as t h e c o n s i s t e n c y b e t ween p e r s o n ' s u t t e r a n c e s and a c t i o n s . T h i s element o f d e s c r i p t i o n pr ovides an important c l u e t o t h e personali t y of t h e person evaluated. The m o t i v a t i o n a l c o n s i s t e n c y , a f o u n d a t i o n o f p r o s o c i a l
b e h a v i o u r , c a n m a n i f e s t i t s e l f i n v a r i o u s ways, s u c h as t r u t h f u l n e s s , k e e p i n g p r o m i s e s , e t c . , or as t h e a b i l i t y o f r e l i a b l e and c o n v i n c i n g e x p l a n a t i o n o f o n e ' s behaviour. There i s no need t o s t r e s s t h e i m p o r t a n c e o f t h i s con-
c e p t f o r t h e i n t e r p e r s o n a l r e l a t i o n s , i n s c i e n c e as w e l l as o u t s i d e o f i t . C l e a r l y , t o d e t e r m i n e w h e t h e r a g i v e n b e h a v i o u r i s mot i v a t i o n a l l y consistent w i t h a given utterance r e q u i r e s a s e p a r a t e a n a . l y s i s i n e a c h c a s e . There a r i s e s a spec i a l problem, t o which e x t e n t one may c h a r a c t e r i z e t h e s t r u c t u r a l a s p e c t s o f t h e b e h a v i o u r which i s c o n s i s t e n t ( o r i n c o n s i s t e n t ) w i t h a given utterance. In o t h e r words, t h e problem i s t o answer t h e q u e s t i o n whether t h e c l a s s o f a l l b e h a v i o u r c o n s i s t e n t (or i n c o n s i s t e n t ) w i t h a g i v e n u t t e r a n c e can be c h a r a c t e r i z e d t h r o u g h i t s structural properties. Some r e s u l t s i n t h i s d i r e c t i o n a r e g i v e n i n t h i s s e c t i o n . The p r e s e n t a t i o n r e q u i r e s t h e usage o f c e r t a i n
806
CHAPTER 6
n o t i o n s o f m a t h e m a t i c a l l i n g u i s t i c s ( t h e s e n o t i o n s may be f o u n d , f o r i n s t a n c e , i n Chomsky, 1957). The b a s i c i d e a s may be p r e s e n t e d a s f o l l o w s . Suppose t h a t we c o n s i d e r t h e a c t i o n s o f a c e r t a i n p e r s o n , and t h a t t h i s person, p r i o r t o t h e beginning of t h e s t r i n g o f a c t i o n s , u t t e r e d a s e n t e n c e which c o n t a i n s a promis e . A l l subsequent s t r i n g s of a c t i o n s ( p o t e n t i a l l y p o s s i b l e ) may t h e n b e c l a s s i f i e d i n t o t h o s e which f u l f i l l t h e p r o m i s e , and t h o s e which b r e a k t h e promis e . The o b j e c t w i l l be t o c h a r a c t e r i z e t h e s e s e t s o f s t r i n g s of a c t i o n s . Before t h a t , however, one may p r o c e e d somewhat more g e n e r a l l y , and a n a l y s e t h e n o t i o n o f t h e c o n s i s t e n c y between v e r b a l and n o n v e r b a l b e h a v i o u r . We s h a l l t r y t o d i s t i n g u i s h i n t h e s e n t e n c e and i n t h e b e h a v i o u r , t h o s e e l e m e n t s which are r e s p o n s i b l e f o r
c o n s i s t e n c y or f o r i t s l a c k ; t h e c l a s s e s o f f u n c t o r s of m o t i v a t i o n a l c a l c u l u s w i l l p l a y h e r e an e s s e n t i a l role. G e n e r a l i z i n g t h e f o r m a l i s m i n t r o d u c e d i n Nowakowska (1973), one may proceed a s f o l l o w s . L e t M s t a n d f o r t h e * c l a s s of a l l m o t i v a t i o n a l u t t e r a n c e s , and l e t M b e t h e c l a s s of a l l f i n i t e s t r i n g s o f e l e m e n t s o f M . F u r t h e r , l e t L d e n o t e t h e l a n g u a g e of a c t i o n s under c o n s i d e r a t i o n , i . e . L = LT [s(O)], f o r some t i m e h o r i zon T and some i n i t i a l s t a t e s ( 0 ) . p r i m i t i v e notion here i s that o f consistency between an element of M ( m o t i v a t i o n a l s e n t e n c e ) and a s t r i n g o f a c t i o n s . We s h a l l d e n o t e t h i s r e l a t i o n by 6, s o t h a t t h e s y m b o l uhm w i l l mean t h a t t h e s t r i n g of actions u i s consistent w i t h t h e sentence m. The
FORMAL THEORY OF ACTIONS
807
We p u t
c-l(m)
= {u
c
L : u$mJ
,
s o t h a t C-'(m) i s t h e c l a s s o f a l l s t r i n g s of a c t i o n s which are c o n s i s t e n t w i t h t h e u t t e r a n c e m . The main point; i s t h a t some s e n t e n c e s i n M r e f e r t o t h e p a s t , some r e f e r t o t h e f u t u r e , and some r e f e r t o b o t h
past and f u t u r e . I f t h e dynamic a s p e c t s o f m o t i v a t i o n a l
c o n s i s t e n c y a r e t o be c o n s i d e r e d , one must r e q u i r e t h a t a s e n t e n c e r e l ' e r r i n g t o t h e f u t u r e , s a y , be c o n s i s t e n t w i t h t h e s t r i n g o f a c t i o n which f o l l o w s t h e u t t e r a n c e of t h i s s e n t e n c e . S i m i l a r r e s t r i c t i o n s concern t h e p a s t , and i t i s c l e a r t h a t one must a n a l y s e c o n s i s t e n c y b e t ween a s e n t e n c e and a s u b s t r i n g o f t h e s t r i n g of a c t i o n s i n which t h i s s e n t e n c e i s embedded. To e x p r e s s i t f o r m a l l y , and a t t h e same t i m e l a y t h e f o u n d a t i o n s f o r t h e theorems which d e s c r i b e t h e r e g u l a r i t i e s o f s e t s o f s t r i n g s o f a c t i o n s which a r e c o n s i s t e n t w i t h some u t t e r a n c e s , one may p r o c e e d a s f o l l o w s . Let M = M
+
t J
M- u M- be t h e p a r t i t i o n o f t h e c l a s s o f
a l l m o t i v a t i o n a l s e n t e n c e s i n t o t h o s e which r e f e r t o t
t h e f u t u r e o n l y (M ) , t o t h e p a s t o n l y (M-), and t o t b o t h p a s t and f u t u r e (M-).
*
L e t us c o n s i d e r t h e monoid (M u A ) , where A i s t h e c l a s s o f a c t i o n s under c o n s i d e r a t i o n . Thus, e l e m e n t s o f t h i s monoid a r e s t r i n g s o f a c t i o n s and u t t e r a n c e s of s e n t e n c e s from M . With e v e r y s t r i n g from ( M
*
one may a s s o c i a t e t h e s t r i n g of a c t i o n s o b t a i n e d by removing a l l u t t e r a n c e s , I/
A)
808
CHAPTER 6
as w e l l as s t r i n g of u t t e r a n c e s , o b t a i n e d b y o m i t t i n g
a l l a c t i o n s . These two s t r i n g s , for a g i v e n s t r i n g z * i n ( M u A ) , w i l l b e d e n o t e d by a ( z ) and u ( z ) , s o t h a t n * a ( z ) 6 A and u ( z ) G M We s h a l l assume t h a t t h e f i r s t of t h e s e s t r i n g s , a ( z ) , i s a d m i s s i b l e , t h a t i s , b e l o n g s to t h e language o f a c t i o n s L .
.
C l e a r l y , t h e mappings a and u a r e homomorphisms w i t h respect t o concatenation, i . e . i f z = z'z", then a ( z ) = a ( z ( ) a ( z " ) and u ( z ) = u ( z l ) u ( z " ) . -1
We s h a l l c o n s i d e r t h e s e t a ( L ) , i . e . t h e c l a s s o f a l l mixed s t r i n g s , s u c h t h a t t h e s t r i n g of a c t i o n s i s i n L.
w
and l e t MZ be t h e set o f a l l e l e m e n t s o f M which a p p e a r i n z . F o r m E M Z , l e t z = ummvm ' so t h a t um i s t h e s t r i n g p r e c e d i n g m y and vm i s t h e s t r i n g f o l l o w i n g m. Let z C (M u A)
DEFINITION. We s h a l l s a y t h a t t h e s t r i n g z . i s m o t i v a t i o n a l l y c o n s i s t e n t , i f f o r e v e r y m € MZ w e have a ( u m ) & m + + whenever m E M , a ( vm ) d m whenever m E M , and for + m G M- w e have [ a ( u m ) a ( v m ) l G m . T h i s g e n e r a l scheme
must be somewhat s i m p l i f i e d , i n o r d e r to a l l o w some m e a n i n g f u l and i m p o r t a n t c o n c l u s i o n s . We s h a l l namely p r e s e n t theorems ( s e e Nowakowska 1973) c o n c e r n i n g s t r u c t u r a l p r o p e r t i e s of t h e c l a s s o f a l l m o t i v a t i o n a l l y c o n s i s t e n t s t r i n g s i n c a s e of some t y p e s of utterances. We s h a l l namely c o n s i d e r s t r i n g s of t h e form mu, where t
m i s a c e r t a i n promise ( s o t h a t m t M ) , and u i s a s t r i n g o f a c t i o n s f o l l o w i n g m y so t h a t a ( u ) = u.
809
FORMAL THEORY OF ACTIONS
Let
Lm be t h e c l a s s o f a l l s t r i n g s which s a t i s f y t h e
promise m , i . e . s u c h s t r i n g s u t h a t u6m. We s h a l l c o n s i d e r two t y p e s o f p r o m i s e s , t o be c a l l e d p o s i t i v e and n e g a t i v e . A p o s i t i v e promise i s c h a r a c t e r i z e d by t h e e x i s t e n c e o f a t i m e - h o r i z o n ( a n example h e r e would be "I w i l l c a l l him tomorrow"), w h i l e a neg a t i v e promise has t h e p r o p e r t y t h a t i t may o n l y b e b r o k e n , b u t n e v e r f u l f i l l e d ( e . g . "I s h a l l n e v e r do t h a t again'' )
.
I n c a s e o f p o s i t i v e promises w e assume the, f o l l o w i n g property :
If u c
Lm
uu'
c
L,
then
uu'
g
Lm.
What t h i s c o n d i t i o n a s s e r t s i s t h a t once a promise i s
f u l f i l l e d , no s u b s e q u e n t a c t i o n s may change t h i s f a c t . L e t us s a y t h a t t h e s t r i n g u G Lm d i s c h a r g e s t h e o b l i g -
a t i o n contained i n m , i f the c o n d i t i o n s u = ufu" w i t h 1u"i > 0 imply u ' k Lm. Thus, u f u l f i l l s t h e promise m , b u t no s u b s t r i n g o f i t has t h i s p r o p e r t y . F u r t h e r m o r e , l e t L' be t h e c l a s s of a l l s t r i n g s which m d i s c h a r g e t h e o b l i g a t i o n i n m , and l e t h(m) = max
1lul :
u E LA
1 .
The v a l u e h ( m ) w i l l be c a l l e d t h e m o t i v a t i o n a l h o r i z o n o r t h e promise m . One can p r o v e t h e f o l l o w i n g theorem ( s e e Nowakowska 1973): THEOREM. If t h e c l a s s of a l l n o n - p a r a s i t i c s t r i n g s i n
810
CHAPTER 6
is a Lm is L
c o n t e x t - f r e e l a n g u a g e , and h ( m ) i s f i n i t e , t h e n a c o n t e x t - f r e e language.
A s t r i n g u i s c a l l e d p a r a s i t i c i n L , i f uluu" & L f o r any s t r i n g s u l , u " , s o t h a t a p a r a s i t i c s t r i n g cannot be a p a r t of any s t r i n g i n L . F o r a p r e c i s e d e f i n i t i o n o f c o n t e x t - f r e e l a n g u a g e , s e e f o r i n s t a n c e Ginsburg
1968.
,
t h e s i t u a t i o n i s somewhat d i f f e r * e n t . Denote namely by Lm t h e c l a s s o f a l l s t r i n g s o f a c t i o n s i n L which b r e a k t h e promise m , and assume t h e following property:
For n e g a t i v e promises
If u have
*
G Lm,
then f o r a l l u',u"
u'uu" G
* Lm.
such t h a t u l u u "
G
L,
we
I n o t h e r words, i f t h e s t r i n g u b r e a k s t h e p r o m i s e , t h e n any s t r i n g which c o n t a i n s u as a s u b s t r i n g has a l s o t h e same p r o p e r t y ( o b s e r v e t h a t t h i s c o n d i t i o n i s d i f f e r e n t from t h e one f o r p o s i t i v e p r o m i s e s : i n t h e l a t t e r c a s e , o n l y c o n c a t e n a t i o n s from t h e r i g h t were used). We omit t h e d e t a i l s h e r e ; r o u g h l y , one i n t r o d u c e s f i r s t t h e c l a s s of a l l s h o r t e s t s t r i n g s which b r e a k t h e p r o m i s e . The theorem a s s e r t s t h a t i f t h e maximum l e n g t h * o f s u c h s t r i n g s i s f i n i t e , t h e n Lm i s a c o n t e x t - f r e e language. I n a somewhat l o o s e i n t e r p r e t a t i o n , p e r h a p s c l o s e r t o a metaphor t h a n t o bounds imposed by t e c h n i c a l i t i e s o f d e f i n i t i o n s , one may s a y t h a t t h e p r o p e y t i e s d e s c r i b e d by t h e s e theorems c o n s t i t u t e a f o r m a l r e p r e s e n t a t i o n
FORMAL THEORY OF ACTIONS
81 1
of t h e o b s e r v e d f a c t t h a t s o c i a l norms ( e s p e c i a l l y e t h i c a l o n e s ) a r e , as a r u l e , f o r m u l a t e d w i t h o u t r e f e r e n c e t o s p e c i f i c s i t u a . t i o n s which might s e r v e as c o n t e x t s of t h e i r a p p l i c a b i l i t y . Thus , t a k i n g r e l i g i o n as an example, a s i n i s d e f i n e d as a s p e c i f i c a c t , and i t s p r e s e n c e i n a s t r i n g of a c t i o n s d e t e r m i n e s t h e q u a l i f i c a t i o n of t h i s s t r i n g ( w i t h f e w e x c e p t i o n s ) r e g a r d l e s s of t h e p r e c e d i n g and s u b s e q u e n t a c t i o n s , i . e . r e g a r d l e s s o f t h e c o n t e x t i n which t h e s i n o c c u r s . Law may p e r h a p s b e t h o u g h t o f as somewhat more f l e x i b l e ,
i n t h e s e n s e t h a t i t o c c a s i o n a l l y p r o v i d e s c l a s s e s of c o n t e x t s i n which a g i v e n a c t i o n c o n s t i t u t e s a n o f f e n s e , and c l a s s e s of c o n t e x t s i n which i t i s n o t s o . T h i s amounts t o s p e c i f y i n g s t r i n g s o f a c t i o n s t o g e t h e r w i t h t h e s u r r o u n d i n g c i r c u m s t a n c e s as o f f e n s e s , r a t h e r t h a n i s o l a t e d a c t i o n s . Again, however, t h e p r e s e n c e of a s t r i n g c o n s t i t u t i n g a n o f f e n s e q u a l i f i e s t h e s t r i n g of a c t i o n s i n which t h e o f f e n s e i s embedded. The p r e c e d i n g o r s u b s e q u e n t a c t i o n s may s e r v e as a f a c t o r d e t e r m i n i n g t h e punishment, b u t as a r u l e , n o t f o r t h e q u a l i f i c a t i o n of t h e o f f e n s e as s u c h . The most s e r i o u s o f f e n s e s , s u c h as murder, r a p e , a r s o n , e t c . a r e q u a l i f i e d r e g a r d l e s s of t h e c o n t e x t i n which t h e y a p p e a r . I n t h i s s e n s e , one can s a y t h a t t h e above theorems r e f l e c t c e r t a i n gen e r a l r e g u l a r i t i e s i n t h e t r e a t m e n t o f t h e norms of s o c i a l o r d e r , namely t h e r e l a t i v e u n i v e r s a l i t y of cont e x t s i n which t h e y a p p l y . These r e s u l t s may t h e r e f o r e be of some i n t e r e s t f o r d e o n t i c l o g i c .
6 . STRUCTURE OF SETS OF TIME-EVENTS I n t h i s s e c t i o n we s h a l l a n a l y s e t h e s t r u c t u r e o f c l a s s es o f t i m e - e v e n t s , i n p a r t i c u l a r c l a s s e s of t h e form
CHAPTER 6
812
R ( u ) , i . e . . s e t s of t i m e - e v e n t s of a c t i o n s u.
caused b y a g i v e n s t r i n g
More s p e c i f i c a l l y , l e t u s c o n s i d e r some f i x e d f a m i l y F of s u b s e t s of t h e s t a t e s p a c e , s a y F = {El,. ' * Y E m j and a s s o c i a t e d w i t h them t i m e - e v e n t s . We s h a l l namely s a y t h a t t h e time-event ( E i , t ) o c c u r r e d , i f t h e s t a t e o f t h e system a t t i m e t i s i n t h e s e t Ei. We s h a l l be a n a l y s i n g t h e s e t R F ( u ) , d e f i n e d as t h e c l a s s o f a l l pairs (Ei,t),
E.
1
F , which o c c u r i f u i s p e r f o r m e d .
Let Horn u = ( E :
(E,t)
e
R F ( u ) f o r some t
1
s o t h a t Hom u i s t h e c l a s s o f a l l e v e n t s i n F which o c c u r a t some t i m e i f u i s performed. Using l i n g u i s t i c a n a l o g i e s , a c c o r d i n g t o which a s t r i n g o f a c t i o n s p l a y s t h e r o l e o f an e x p r e s s i o n and t h e outcomes p l a y t h e r o l e o f t h e meaning o f a n e x p r e s s i o n , t h e s i z e o f t h e s e t Hom u r e f l e c t s t h e llhomonymity" o f e x p r e s s i o n u: if t h i s s e t i s l a r g e , t h e n u i s more "ambiguous". I n a s i m i l a r way, f o r a f i x e d E h F we may d e f i n e TrEu = { t :
(E,t)
E
RF(u)l
and
Tr u
=
(t:
We have t h e n
(E,t)
E R F ( u ) f o r some E
i Fj
.
FORMAL THEORY OF ACTIONS
813
The s e t TrEu i s t h e “ t r a c e ” o f e v e n t E when u i s p e r formed, i . e . t h e s e t of a l l moments a t which E o c c u r s ( E may o c c u r more t h a n once as a r e s u l t o f u; t o g i v e a n example, u may be “ p a y i n g a s u b s c r i p t i o n “ , and E may b e t h e e v e n t “ r e c e i v i n g t h e s u b s c r i b e d j o u r n a l ” ) . C l e a r l y , t h e most i n t e r e s t i n g s e t s h e r e a r e Horn u and T r u. The f i r s t o f t h e s e d i s r e g a r d s t h e t e m p o r a l asE
p e c t s , and a1l.o.w~ comparison o f s t r i n g s o f a c t i o n s w i t h r e s p e c t t o t h e “ s c o p e ” o f t h e i r outcomes. The second s e t g i v e s t h e t e m p o r a l c h a r a c t e r i s t i c s o f t h e occurrences of t h e event E . The t i m e - t r a c e s o f e v e n t s a r e o f s p e c i a l i m p o r t a n c e f o r t h o s e everits which o c c u r more t h a n once as a r e s u l t
o f p e r f o r m i n g t h e s t r i n g of a c t i o n s u. Here one may d i s t i n g u i s h some c l a s s e s o f e v e n t s , u s i n g t h e p r o p e r t i e s o f t h e t r a c e and i t s l o c a t i o n w i t h r e s p e c t t o t h e moment o f t e r m i n a t i o n o f t h e s t r i n g u . Thus, i f
t h e n t h e f i r s t o c c u r r e n c e of t h e e v e n t E l i e s a f t e r t h e t e r m i n a t i o n o f t h e s t r i n g u. We may c a l l s u c h a n e v e n t E d e l a y e d . On t h e o t h e r hand, i f t h e l a s t o c c u r ence of t h e event E l i e s bef or e t h e t e r m i n a t i o n of t h e s t r i n g u. t h a t i s max
it: t
t TrEu’f C l u \ ,
t h e e v e n t E ma.y be c a l l e d t r a n s i e n t .
814
CHAPTER 6
I n t h e r e m a i n i n g c a s e , when f o r some t l , t 2 : i n T r Eu w e have t l < IuJ< t 2 , we s p e a k o f p e r s i s t e n t e v e n t s (cons e q u e n c e s of u)
.
These n o t i o n s d e s c r i b e o n l y t h e l o c a t i o n o f t h e s e t T r u w i t h r e s p e c t t o t h e moment of t e r m i n a t i o n o f u.
E
They do n o t , however , p r o v i d e any i n f o r m a t i o n about t h e s t r u c t u r e o f t h e t r a c e o f u. A s r e g a r d s t h e l a t t e r , t h e most i m p o r t a n t and i n t e r e s t i n g c o n c e p t i s t h a t o f
periodicity. S i n c e we d e a l h e r e m o s t l y w i t h f i n i t e s e t s , t h e n o t i o n of p e r i o d i c i t y has t o be d e f i n e d i n a s p e c i a l w a y . I n o t h e r words, t h e u s u a l d e f i n i t i o n o f p e r i o d i c i t y , as i n v a r i a n c e under some t r a n s l a t i o n s , has t o be appropr i a t e l y modified. We s h a l l p r e s e n t h e r e t h e r e l e v a n t concept and r e s u l t s from Nowakowska (1973). Given any s e t o f numbers B , l e t S t ( B ) d e n o t e i t s t r a n s l a t i o n by t, i . e . St(B) = { s :
s = t
-k
x, x
B].
We have t h e n St [S, ( B ) 1 = St +t (B). 2 1 1 2 A s e t B w i l l be c a l l e d p e r i o d i c , i f it i s r e p r e s e n t a b l e
as a u n i o n o f a c e r t a i n number o f t r a n s l a t i o n s o f a g i v e n s e t , c a l l e d c a r r i e r of p e r i o d i c i t y . F o r m a l l y , a s e t B of t i m e moments i s s a i d to be p e r i o d i c , i f t h e r e e x i s t s a s e t N , and two numbers, k > 1 and T 7 0 s u c h t h a t
FORMAL THEORY OF ACTIONS
81 5
provided t h a t max i t : t
c
N\
< min It:
t E s ~ ( N ), ~
I n s u c h a c a s e , t h e s e t N i s c a l l e d t h e c a r r i e r of per i o d i c i t y , and T i s c a l l e d t h e p e r i o d . We a l s o s h a l l s a y t h a t t h e s e t i s of t h e t y p e ( T , k ) . N a t u r a l l y , t h e s e t N i t s e l f may b e p e r i o d i c a c c o r d i n g t o t h e above d e f i n i t i o n . I n p a r t i c u l a r , any s e t c o n s i s t i n g o f j u s t two p o i n t s i s p e r i o d i c . It i s o f i n t e r e s t t o a n a l y s e c o n d i t i o n s under which
a u n i o n o f two p e r i o d i c s e t s i s a l s o p e r i o d i c . The t h r e e theorems below p r o v i d e s u f f i c i e n t c o n d i t i o n s under which a union o f p e r i o d i c s e t s i s p e r i o d i c . F o r t h e p r o o f s , s e e Nowakowska (1973). THEOREM. Assume t h a t t h e s e t s B1
and
B2 a r e p e r i o d i c ,
and -
(1) B1
and
B 2 have t h e same p e r i o d T ;
( 2 ) The c a r r i e r N N
of 1-
B1
Q c
>
Then t h e s e t B1
of B
i s a t r a n s l a t i o n of t h e c a r r i e r
2 2 0 , t h a t i s , N 2 = Sc(N1).
3 B2
i s p e r i o d i c , i f i n a d d i t i o n one
of t h e fol l owing c o n d i t i o n s holds:
( a ) max B1 4 min E and t h e s e t s B1,B2 a r e o f t h e 2’ same t y p e ( T , k ) ; ( b ) max B 7 min B and t h e number c i s a m u l t i p l e of 1/ 2
816
CHAPTER 6
t h e common p e r i o d T .
B2 i s B1, t h e p e r i o d e q u a l s c , and t h e number o f r e p e t i t i o n s i s 2 . I n c a s e ( a ) , t h e c a r r i e r o f t h e sum B1
I n c a s e ( b ) , t h e c a r r i e r i s N1,
t~
t h e p e r i o d r e m a i n s equ-
a l T , and t h e number o f r e p e t i t i o n s depends on t h e numbers kl,
k2 and c .
N e x t , we have a l s o t h e f o l l o w i n g THEOREM. N1
and
B2 w i t h c a r r i e r s
If t h e p e r i o d i c s e t s B1
N 2 a r e o f t h e same t y p e ( T , k )
t h e n t h e s m B1
ir
and
B2 i s p e r i o d i c w i t h c a r r i e r N1 u N 2
and o f t y p e ( T , k ) ( i . e .
b o t h t h e p e r i o d and t h e number
of r e p e t i t i o n s a r e t h e same as f o r e a c h component). F i n a l l y , w e have a l s o t h e f o l l o w i n g t h e o r e m . THEOREM.
:i.f B1 and B2
and ( T 2 , k 2 ) ,
a r e periodic of types (Tl,kl)
w i t h c a r r i e r s N1
and
N2 such t h a t
i s an i n t e g e r m u l t i p l e of t h e p e r i o d T2 i . e . T1 = mT2 for some m;
the period
TI,
( b ) The number k l e q u a l s mk2;
( c ) the set N
*
= N 2 d N1
0
ST1(N1)
satisfies t h e condition
J
S2T (N1)LJ...;I S (m- 1)T (N1) 1
FORMAL THEORY OF ACTIONS
max N
*<
817
n
min ST (N ) , 2
t h e n t h e sum 131 v B2 i s p e r i o d i c . The c a r r i e r o f t h e sum i s N , and b o t h t h e p e r i o d and t h e number o f r e p e t i t i o n s a r e t h e same as for t h e s e t w i t h t h e g r e a t e r p e r i o d , i . e . t h e y a r e e q u a l T2 and k 2 r e s p e c t i v e l y .
w
The a n a l y s i s of p e r i o d i c i t y o f t i m e t r a c e s i s e s p e c i a l -
l y i m p o r t a n t i n c a s e s when some a c t i o n s need t o b e
synchronized i n or der t o achieve a c e r t a i n r e l a t i o n between t i m e - r e s u l t s . To g i v e a s i m p l e example, i f a n a s s e m b l y l i n e i s t o produce a c a r i n e v e r y t m i n u t e s , a n o t h e r assembly l i n e must s u p p l y t h e f i r s t w i t h a w h e e l ' i n e v e r y t / 5 m i n u t e s , a d o o r i n e v e r y t / 2 (or t / 4 ) m i n u t e s , and s o on. To a c h i e v e t h i s w i t h o u t t h e u n n e c e s s a r y d e l a y s o f accum u l a t i o n of p a r t s , i t i s b e s t i f t h e moments o f t e r m i n a t i o n of p r o d u c t i o n o f a g i v e n t y p e o f p a r t form a per i o d i c s e t , w i t h p e r i o d b e i n g a m u l t i p l e o f some common t i m e module. L e t us c o n t i n u e w i t h t h e c e n t r a l a n a l o g y o f t h i s chap-
t e r , namely t h a t between s e t s o f s t r i n g s o f a c t i o n s , and l a n g u a g e s ( t r e a t e d as s e t s of s t r i n g s formed o u t of a c e r t a i n v o c a b u l a r y ) . We may t h e n i d e n t i f y outcomes ( t i m e - e v e n t s ) w i t h meanings o f s t r i n g s o f a c t i o n s . I n t h e p r e s e n t c a s e , t h e s e m a n t i c s o f language o f a c t i o n s L (for a f i x e d i n i t i a l s i t u a t i o n s , and f i x e d adm i s s i b i l i t y l e v e l ) i s d e f i n e d t h r o u g h sequences o f states ( h i s t o r i e s ) . L e t u s c a l l t h e h i s t o r y caused by a s t r i n g o f a c t i o n s t h e z e r o - l e v e ' l meaning o f t h i s s t r i n g . Thus, z e r o - l e -
818
CHAPTER 6
v e l meanings e x h i b i t t h e same s t r u c t u r e as l a n g u a g e s : they a r e c l a s s e s of s t r i n g s of s t a t e s . The meanings s o d e f i n e d a r e n o t u n i q u e l y a s s o c i a t e d w i t h s t r i n g s o f a c t i o n s : t h e r e may e x i s t v a r i o u s a c t -
i o n s t r i n g s c a u s i n g t h e same h i s t o r y . All such s t r i n g s may be c a l l e d synonymous. However, we seldom want t o c a u s e a p a r t i c u l a r s t r i n g o f s t a t e s ; more o f t e n we want t o c a u s e j u s t one o f t h e s t r i n g s which h a s c e r t a i n d e s i r e d p r o p e r t i e s . S i n c e e a c h p r o p e r t y may be i d e n t i f i e d w i t h a s e t , and s e t s o f s t r i n g s of s t a t e s a r e e v e n t s , we may s a y t h a t e v e n t s c o n s t i t u t e a h i g h e r l e v e l semantic of a c t i o n language. I n f a c t , l e t u s choose a p a r t i t i o n o f t h e c l a s s o f a l l
states i n t o d i s j o i n t s e t s
C o n s e q u e n t l y , t o e a c h h i s t o r y ( s t r i n g o f s t a t e s ) we may a s s i g n i n a unique way t h e s t r i n g o f c l a s s e s o f t h e above p a r t i t i o n , s a y
(i,)
i f t h e h i s t o r y i s h = sosl... s T , t h e n s k a l l k.
%
for
S i n c e t o e a c h element o f L t h e r e c o r r e s p o n d s a h i s t o r y h , t h e r e c o r r e s p o n d s a l s o a s t r i n g o f t h e form ( * ) .
T h i s d e f i n e s a "semantic l a n g u a g e " o f a g i v e n s i t u a t i o n (on a given l e v e l of a d m i s s i b i l i t y ) , a s s o c i a t e d with a given p a r t i t i o n .
Changing t h i s p a r t i t i o n one o b t a i n s
FORMAL THEORY OF ACTIONS
819
v a r i o u s semantic languages. I n p a r t i c u l a r , i f t h e p a r t i t i o n becomes f i n e r , one o b t a i n s a more c o n c r e t e and t h e r e f o r e more i n f o r m a t i v e s e m a n t i c l a n g u a g e . O f c o u r s e , o u t o f two p a r t i t i o n s , n e i t h e r may be f i n e r t h a n t h e o t h e r , i n which c a s e t h e l a n g u a g e s a r e n o t comparable. L e t u s now f i x some s e m a n t i c l a n g u a g e . One may now de-
f i n e v a r i o u s r e l a t i o n s between e v e n t s (meanings o f s t r i n g s o f a c t i o n s ) . F o r i n s t a n c e , l e t A,B and C be three events
.
DEFINITION. We s a y t h a t B i s based on A, i f whenever t h e s t r i n g cau.ses A, i t must a l s o c a u s e B a t some l a t e r moment. S i m i l a r l y , A e x c l u d e s B, i f whenever A o c c u r s ,
B cannot o c c u r . Suppose now t h a t A , B , C
a r e such t h a t t h e i r i n t e r s e c t i o n s
a r e a l s o e l e m e n t s of t h e p a r t i t i o n i n q u e s t i o n ( i . e . A,B and C may be sums of e l e m e n t s o f t h e p a r t i t i o n ( * ) which s e r v e s a.s a b a s i s f o r l a n g u a g e ) . DEFINITION. WE. s a y t h a t A t r a n s f o r m s B i n t o C , i f whenever t h e r e a p p e a r s A appear A
A
B , t h e r e must a f t e r w a r d s
C.
As a n o t h e r t y p e of t r a n s f o r m a t i o n , we may r e q u i r e t h a t whenever A P E3 o c c u r s , t h e r e must a l s o o c c u r A ' r , C . The above n o t i o n s were t a k e n as p r i m i t i v e r e l a t i o n t y p e s b y Leniewicz (1975) i n h i s taxonomy o f r e l a t i o n s between e v e n t s . It i s of some i n t e r e s t t h a t i n t h e p r e s e n t s y s tem, h i s p r i m i t i v e n o t i o n s may be f o r m a l l y d e f i n e d , s o t h a t t h i s s y s t e m may be t a k e n as a model f o r t h e n o t i o n s of L e n i e w i c z .
820
CHAPTER 6
The s e m a n t i c l a n g u a g e s may b e a n a l y s e d b y means o f t h e n o t i o n s o f mathematical l i n g u i s t i c s . O f p a r t i c u l a r i n t e r e s t h e r e i s t h e n o t i o n s of p a r a s i t i c s t r i n g s . A s t r i n g v ,of e l e m e n t s o f p a r t i t i o n ( * ) i s c a l l e d p a r a s i t i c , i f f o r any u,w, t h e s t r i n g uvw i s n o t a d m i s s i b l e Thus, a p a r a s i t i c s t r i n g cannot be a p a r t o f a s t r i n g i n a l a n g u a g e . Using t h i s n o t i o n , we may c h a r a c t e r i z e t h e d e f i n i t i o n s a b o v e , by r e q u i r i n g p a r a s i t i c i t y o f the strings
a)
AD1D 2 . . . D k
b)
AD1..
C)
A
d) A
f o r D1 # B y
...,
D k # B;
.DkB;
n
B , D l...Dk, Di
r\
B y D1...Dky
# A Di
4
C;
# A'
A
C.
I n d e e d , if a s t r i n g o f t h e form ( a ) i s p a r a s i t i c , t h e n i n o r d e r f o r a s t r i n g t h a t b e g i n s w i t h A t o be n o t par a s i t i c , i t must l a t e r c o n t a i n B. I n a s i m i l a r w a y , one can s e e t h a t p a r a s i t i c i t y o f s t r i n g s o f t h e form ( b ) , ( c ) and ( d ) i s e q u i v a l e n t t o t h e r e m a i n i n g r e l a t i o n s between e v e n t s . The a n a l o g y between s e m a n t i c s o f n a t u r a l l a n g u a g e s and " s e m a n t i c l a n g u a g e s " b u i l t o u t o f e v e n t s , may be pursued f u r t h e r . One may namely a p p l y h e r e such n o t i o n s
as synonymy, autonymy and hyponymy. F o r i n s t a n c e , two s t r i n g s a r e synonymous w i t h r e s p e c t t o a p a r t i t i o n o f t h e form (*I, i f t h e y a r e a s s i g n e d t h e same s t r i n g s o f c l a s s e s i n t g i s p a r t i t i o n . A s a consequence, t h e r e may e x i s t many l e v e l s o f yynonimity, b e g i n n i n g w i t h t h e synonimity c o r r e s p o n d i n g t o t h e improper p a r t i t i o n i n t o j u s t one c l a s s . I n t h i s c a s e , a l l s t r i n g s of
FORMAL THEORY OF ACTIONS
821
a c t i o n s a r e synonymous. On t h e o t h e r e x t r e m e , we have t h e p a r t i t i o n o f S i n t o s i n g l e t o n s ; t h e n synonymous s t r i n g s o f a c t i o n s are s u c h which y i e l d i d e n t i c a l h i s tories, F o r t h e c o n c e p t of homonymy, one needs t o i n t r o d u c e a
p a r t i t i o n o f t h e c l a s s of a l l a c t i o n s , and a s s i g n t o each s t r i n g of a c t i o n s ala2,..aT t h e corresponding s t r i n g of c a t e g o r i e s o f t h e p a r t i t i o n . Then any two s t r i n g s w i t h t h e same a s s i g n e d s t r i n g s o f c a t e g o r i e s w i l l be homonymous ( d e s p i t e t h e f a c t t h a t t h e y may lead t o d i f f e r e n t outcomes).
The n o t i o n o f homonymity i n t h i s form i s n o t t o o i n t e r e s t i n g , as opposed t o s y n o n i m i t y . I n d e e d , two s t r i n g s a r e synonymous, i f i t does n o t m a t t e r which o f them
w i l l be p e r f o r m e d , as l o n g as we are i n t e r e s t e d o n l y i n d e s c r i p t i o n s of t h e r e s u l t s ( h i s t o r i e s ) only i n terms o f e l e m e n t s of t h e g i v e n p a r t i t i o n . N a t u r a l l y , under f i n e r p a r t i t i o n , s u c h s t r i n g s may c e a s e t o be synonymous. T h e n o t i o n s o f autonymy and hyponymy may be a n a l y s e d i n a s i m i l a r way; w e omit t h e d e t a i l s of the definitions.
7 . GROUP ACTIONS I n t h i s s e c t i o n t h e a n a l y s i s w i l l be concerned w i t h an e x t e n s i o n o f t h e s y s t e m from t h e p r e c e d i n g s e c t i o n s , s o as t o c o v e r t h e c a s e of s i m u l t a n e o u s a c t i o n s of s e v e r a l p e r s o n s . Many n o t i o n s w i l l , of c o u r s e , be t h e same as f o r t h e s y s t e m s o f a c t i o n s o f s i n g l e p e r s o n s , b u t t h e r e w i l l a l s o be some new e f f e c t s , due t o t h e i n t e r a c t i o n s between t h e a c t i n g p e r s o n s . We s h a l l t r y t o c a p t u r e two a s p e c t s h e r e : one i s t h a t some a c t i o n s
CHAPTER 6
822
may e x c l u d e c e r t a i n o t h e r a c t i o n s of t h e r e m a i n i n g p e r -
s o n s , w h i l e some a c t i o n s may be performed o n l y j o i n t l y . The second a s p e c t c o n c e r n s t h e i n t e r a c t i o n between s t a t e s c o r r e s p o n d i n g t o systems o f p a r t i c u l a r p e r s o n s . T o c o n s t r u c t a n a p p r o p r i a t e f o r m a l i s m , we i n t r o d u c e
f i r s t t h e system
(7.1) where K i s i n t e r p r e t e d a s t h e s e t o f p e r s o n s under cons i d e r a t i o n . We s h a l l d e n o t e t h e number o f p e r s o n s i n t h e s e t K by k , and assume t h a t k 7 1. Let
s = slx
...
XSk
(7.2)
.
s o t h a t e l e m e n t s of S , d e n o t e d by = ( s ~ , . . ,sk) are d e s c r i p t i o n s o f s t a t e s o f t h e j o i n t system o f a c t i o n s of a l l per sons. G e n e r a l l y , not every s t a t e i n S i s p o s s i b l e ; i n o t h e r words, some s t a t e s i n s y s t e m s S c a n n o t be combined i
w i t h one a n o t h e r .
The n a t u r e of t h e c o n s t r a i n t s which make some v e c t o r s s i m p o s s i b l e may depend on t h e c o n t e x t . For i n s t a n c e , ' t h e r e may be some p h y s i c a l c o n s t r a i n t s which make c e r t a i n s t a t e s incompatible with o t h e r s , e t c .
...,
I n t u i t i v e l y , a s t a t e 2 = ( s ~ , sk ) i s a d m i s s i b l e , i f i t can be a t t a i n e d by a sequence of j o i n t a c t i o n s , s t a r t i n g from t h e i n i t i a l s t a t e ~ ( 0 ) .
823
FORMAL THEORY OF ACTIONS
To e x p l i c a t e i t f o r m a l l y , we n e e d some a d d i t i o n a l n o t -
i o n s . T h u s , as i n S e c t i o n 1, w e d e n o t e b y A t h e s e t of a c t i o n s , a nd b y # a d i s t i n g u i s h e d a c t i o n c a l l e d " p a u s e " or " n o - a c t i o n "
.
Next, l e t = A x
. .. r
A
(1; t i m e s )
..
k
being vectors g = (al,. ,ak), int e r p r e t e d as s i m u l t a n e o u s p e r f o r m a n c e o f a c t i o n a 1 by p e r s o n 1, a c t i o n a 2 by p e r s o n 2 ¶ . . . , a c t i o n ak by p e r son k. w i t h elements of A
A s i n the c a s e of s t a t e s , not a l l a c t i o n s
a
w i l l b e adm i s s i b l e , because of v a r i o u s c o n s t r a i n t s . For i n s t a n c e , some a c t i o n s may r e q u i r e c o o p e r a t i o n o f members o f t h e gr oup ( a t y p i c a l c a s e m i g h t b e a n a c t i o n o f moving a n o b j e c t t o o heavy t o b e moved by o n e p e r s o n ) . O t h e r a c t i o n s may e x c l u d e o n e a n o t h e r ( t y p i c a l l y , when a n a c t i o n r e q u i r e s u s i n g a t o o l i n s i t u a t i o n s when o n l y o n e t o o l i s a v a i l a b l e , an d has t o be s h a r e d ) . To e x p l i c a t e t h e f o r m a l s t r u c t u r e o f t h e c o n s t r a i n t s on b o t h a c t i o n s and s t a t e s , o n e n e e d s t h e n o t i o n s o f a d m i s s i b i l i t y f u n c t i o n r an d t r a n s i t i o n f u n c t i o n p , as i n S e c t i o n 1. One may namely i n t r o d u c e h e r e t h e conc e p t o f s - h i s t o r y an d a - h i s t o r y , as i n t h e c a s e o f a c t i o n s o f one p e r s o n . Formally, an :;-history
i s a s t r i n g of t h e form
824
CHAPTER 6
a - h i s t o r y i s a s t r i n g o f t h e form
s o t h a t a n a - h i s t o r y ends w i t h a n a c t i o n , w h i l e an sh i s t o r y ends w i t h a s t a t e . F o r m a l l y , t h e a d m i s s i b i l i t y f u n c t i o n r i s a mapping r:uHA->2
Ak
,
n
(7.3)
where i s t h e c l a s s o f a l l s - h i s t o r i e s of l e n g t h n , s o t h a t t h e v a l u e r(h;) i s t o be i n t e r p r e t e d as t h e s e t of a l l j o i n t a c t i o n s 2 6 A k which a r e a d m i s s i b l e a f t e r h i s t o r y h;. S i m i l a r l y , t h e t r a n s i t i o n f u n c t i o n p i s a mapping p : u n
:H
A
S
(7.4)
where Hi i s t h e c l a s s of a l l a - h i s t o r i e s of l e n g t h n . Here t h e v a l u e p ( h i ) i s t o be i n t e r p r e t e d as t h e s t a t e of t h e j o i n t system a f t e r t h e h i s t o r y h i . Again, i n analogy w i t h t h e a n a l y s i s i n S e c t i o n 1, w e d e f i n e a p o s s i b l e h i s t o r y ( i n s t a t e ~ ( 0 ) )as a s t r i n g
of a l t e r n a t i n g s t a t e s and a c t i o n s
such t h a t
FORMAL THEORY OF ACTIONS
825
and s o f o r t h . The v a l u e s o f t h e s t a t e s g ( i ) are u n i q u e l y d e t e r m i n e d by t h e i n i t i a l s t a t e ~ ( 0 ) and a c t i o n s ~ ( l g) ( ,2 ) , ..., * s o t h a t we may d e f i n e t h e language o f a c t i o n s L = * L [s(O)]as t h e c l a s s o f a l l s t r i n g s o f j o i n t a c t i o n s which a p p e a r i n p o s s i b l e h i s t o r i e s ; such s t r i n g s of a c t i o n s w i l l a l s o be c a l l e d admissible.
*
Each s t r i n g o f j o i n t a c t i o n s i n L d e t e r m i n e s t h e c o r r e s p o n d i n g s t r i n g o f s t a t e s , b e i n g an outcome o f p e r forming t h i s s t r i n g of a c t i o n s . The above n o t i o n s may be e a s i l y f u z z i f i e d , and one may i n t r o d u c e s t o c h a s t i c t r a n s i t i o n s , as i t was done a t t h e b e g i n n i n g o f t h i s c h a p t e r . We s h a l l n o t r e p e a t t h e s e d e f i n i t i o n s , s i n c e t h e y do n o t d i f f e r from one-dimensional c ase. To e x p l o r e d e e p e r t h e e f f e c t s o f m u l t i d i m e n s i o n a l i t y , w e s h a l l c o n s i d e r t h e non-fuzzy and d e t e r m i n i s t i c s i t u a t i o n , and a l s o i n t r o d u c e some f u r t h e r s i m p l i f y i n g assumptions. We assume namely t h a t t h e a d m i s s i b i l i t y f u n c t i o n r and t r a n s i t i o n f u n c t i o n p a r e Markovian, i n t h e s e n s e t h a t t h e v a l u e o f r depends o n l y on t h e c u r r e n t s t a t e , and t h e v a l u e o f p depends o n l y on t h e c u r r e n t s t a t e and most r e c e n t a c t i o n . Moreover, we assume t h a t t h e s e f u n c t i o n s a r e t i m e - i n v a r i a n t . F o r m a l l y , t h i s means t h a t t h e r e e x i s t f u n c t i o n s d and f s u c h t h a t
826
CHAPTER 6
and
(7.6) where d and f a r e some f u n c t i o n s s u c h t h a t
(7.7) and f : S % A k +S.
(7.8)
I n s h o r t , d ( g ) i s t h e (nonempty) s e t o f j o i n t a c t i o n s which may be u n d e r t a k e n when t h e system i s i n t h e s t a t e 2, and 2' = f(s,a) i s t h e s t a t e t o which t h e s y s t e m p a s s e s from 2 , i f t h e j o i n t a c t i o n a i s p e r f o r m e d . The language o f a c t i o n s d e f i n e d above as t h e s e t o f a l l p o s s i b l e s t r i n g s o f a c t i o n s depended on t h e i n i t i a l s t a t e ~ ( 0 ) .We s h a l l now e x t e n d t h i s d e f i n i t i o n by a l l o w i n g more freedom r e g a r d i n g t h e i n i t i a l s t a t e s . L e t namely S ' L S be a s u b s e t of S , t h e e l e m e n t s o f S' being i n t e r p r e t e d as p o s s i b l e i n i t i a l s t a t e s of t h e system.
. . . , a( n ) ]
be a s t r i n g of j o i n t a c t i o n s . F o r a g i v e n 2 6 S we d e f i n e i n d u c t i v e l y L e t [a(l),a(2),
827
FORMAL THEORY OF ACTIONS
Thus, e a c h f u n c t i o n g ( j ) maps t h e s e t S i n t o S : t h e vaU
l u e gF)(s-) i s t h e s t a t e a t time t = j , i f s t r i n g u i s performed, and t h e i n i t i a l s t a t e was 2. D E F I N I T I O N . We s a y t h a t t h e s t r i n g of j o i n t a c t i o n s
u = [ g ( l ),..., ~ ( n ) i] s S ' - f e a s i b l e , s 6 S' s u c h t h a t
i f there exists
where gLi) are d e f i n e d by ( 7 . 9 ) .
**
**
DEFINITION. The c l a s s L = L (S') of a l l S ' - f e a s i b l e s t r i n g s of a c t i o n s w i l l be c a l l e d t h e S ' - l a n g u a g e of actions.
,...
Suppose now t h a t u = [ ~ ( l ) , a ( n ) ] an'd v = [~'(l),.. a l ( m ) ] are two s t r i n g s o f j o i n t a c t i o n s . T h e i r concat e n a t i o n uv w i l l be d e f i n e d as
.
¶-
where g ( n + j ) = g'(j) f o r j = 1 THEOREM.
Let u,v G
L
**
b e l o n g s to S' f o r any 2
(Sl).
<
,... . , m .
Then uv
L
**
if
g(n)(s-) U
S'.
T h i s c o n d i t i o n means simply t h a t a f t e r p e r f o r m i n g t h e
s t r i n g u, s t a r t i n g from a n a r b i t r a r y i n i t i a l s t a t e i n S', w e must a g a i n be i n s e t S' when t h e s t r i n g v s t a r t s b e i n g performed. Somewhat more g e n e r a l l y , l e t u s i n t r o d u c e t h e f o l l o w ing definition.
828
CHAPTER 6
DEFINITION. The s t r i n g u o f l e n g t h n i s S ' - p r e s e r v i n g , i f g ( j ) ( S l ) C S' f o r j = l , . . . , n gU( j ) U m a p s s' i n t o
( i . e . i f each f u n c t i o n
sl).
We may now f o r m u l a t e
** (S') i s c l o s e d under c o n c a t e n a t ** s t r i n g u G L (S') is S ' - p r e s e r v i n g .
THEOREM. The c l a s s L
i o n , i f every
**
DEFINITION. The l a r g e s t s e t S ' c S f o r which L ( S t ) i s c l o s e d under c o n c a t e n a t i o n w i l l b e c a l l e d i n i t i a l c o n d i t i o n - f r e e language o f a c t i o n s o f t h e s y s t e m s . I n t h e s e q u e l , we s h a l l assume t h a t some c l a s s S' o f p o s s i b l e i n i t i a l s t a t e s was s e l e c t e d , and we s h a l l
**
*t
write s i m p l y L f o r L ( S f ) . The language L be c l o s e d under c o n c a t e n a t i o n .
**
need n o t
We s h a l l now t r y t o make u s e o f t h e f a c t t h a t t h e s t r i n g s ** a r e composed on " m u l t i d i m e n s i o n a l l e t t e r s " , t h a t in L
i s , t h e y a r e j o i n t a c t i o n s by a number o f p e r s o n s . Suppose t h a t u = [ ~ ( l ) , . . . , a ( n ) ] i s a s t r i n g of j o i n t a c t i o n s , and l e t
(7.11)
be t h e v e c t o r o f acti?ons performed by v a r i o u s members of t h e group a t t i m e t = i . We may now d e f i n e t h e a c t i o n v o c a b u l a r y A
l a n g u a g e Li o f i - t h
and a c t i o n
i p e r s o n as follows. We p u t f i r s t
A . as t h e s e t o f a l l a c t i o n s a C A s u c h t h a t 1
829
FORMAL THEORY OF ACTIONS
3u
... , a ( n ) 1
= [~(l),
L** s u c h t h a t for some r we
have
DEFINITION. The s e t A . w i l l be c a l l e d t h e a c t i o n vocal
b u l a r y of p e r s o n i .
*
Denote now by A i
t h e monoid o v e r A i ,
that i s , the class
of a l l f i n i t e s t r i n g s of e l e m e n t s of A i ' We s h a l l now d e f i n e t h e language o f a c t i o n s o f i - t h p e r s o n , t o be t d e n o t e d by L i , as t h e c l a s s o f a l l s t r i n g s i n A which may a p p e a r i n m u l t i d i m e n s i o n a l s t r i n g s i n L
**
.
i
DEFINITION. We s h a l l s a y t h a t a s t r i n g ui = [ a i ( l ) ,
...,
or s i m p l y : b e l o n g s t o t h e ** s e t L i , i f t h e r e e x i s t s u = [a(l), a(n)] L s u c h t h a t t h e i - t h component of a ( j ) e q u a l s a i ( j ) f o r
a . ( n ) ] i s Li-admissible,
...,
1
j = 1,
...,n .
We s h a l l now a n a l y s e t h e s t r u c t u r a l c o n s t r a i n t s on mut u a l c o m p a t i b i l . i t y o f p e r f o r m i n g a c t i o n s by v a r i o u s members o f t h e g r o u p . DEFINITION. We s h a l l s a y t h a t t h e s t a t e
s allows
pendence o f a c t i o n s , i f t h e r e e x i s t s s e t s C1,
inde-
...,Ck
CA
such t h a t d(s) =
C1)(
...
Y
ck
where d i s t h e f u n c t i o n g i v e n i n a d m i s s i b i l i t y of j o i n t a c t i o n s .
(7.12)
(7.7),
describing
830
CHAPTER 6
Thus, s t a t e s a l l o w s independence o f a c t i o n s , i f i t g i v e s t o e a c h p e r s o n a c e r t a i n s e t o f o p t i o n s , and t h e r e a r e no c o n s t r a i n t s on t h e c h o i c e : e a c h o p t i o n may be comb i n e d w i t h any o p t i o n s of o t h e r p e r s o n s . S i n c e a l l o w i n g independence o f a c t i o n s i s a p r o p e r t y o f t h e s t a t e , it i s n a t u r a l t o i n t r o d u c e t h e f o l l o w i n g definition.
DEFINITION. A p a i r ( u , ~ ) ,c o n s i s t i n g of a s t r i n g of a c t i o n s and i n i t i a l s t a t e a l l o w s independence o f a c t i o n s
a t t i m e m (where m < ( u l ) , i f t h e s t a t e g p ) ( g ) a l l o w s independence o f a c t i o n s . We have t h e n
...,
THEOREM. Assume t h a t t h e p a i r ( u , ~ )w i t h u = [ ~ ( l ) , a ( n ) ] a l l o w s independence o f a c t i o n s a t some t i m e
...
and l e t d [ g p ) ( s ) ] = C1x 1,. ,k we have a i ( m t l ) E C i .
..
x Ck.
m
< n,
Then f o r a l l i =
T h i s theorem e x p r e s s e s t h e p r o p e r t y t h a t i f a s t r i n g
of a c t i o n s a l l o w s independence a t some moment b e f o r e it t e r m i n a t e s , t h e n t h e next a c t i o n s i n t h i s s t r i n g must b e l o n g t o t h e a p p r o p r i a t e s e t s C i a l l o w e d i n t h e s t a t e a t which independence i s a c h i e v e d . L e t u s now c o n s i d e r a s t a t e
which d o e s n o t a l l o w i n dependence o f a c t i o n s . In t h i s c a s e , t h e s e t d ( 2 ) does n o t have t h e form o f a C a r t e s i a n p r o d u c t o f s e t s o f a c t i o n s allowed f o r p a r t i c u l a r p e r s o n s . L e t us d e n o t e by A!'
t h e p r o j e c t i o n o f t h e s e t d ( g ) on i - t h c o o r d i n a t e , so t h a t 1
FORMAL THEORY OF ACTIONS
a E A; i f
32
= (al,,..,"
k
83 1
) w i t h ai = a .
S i n c e d ( 2 ) i s not empty, we have a l s o A; # 0 f o r a l l i . Clearly , d(2)
<
A'; x
...
:A
(7.13)
and t h e i n c l u s i o n must be s t r i c t , i f t h e s t a t e s does n o t a l l o w independence o f a c t i o n s . T h i s means t h a t t h e r e e x i s t s a v e c t o r of a c t i o n s a z ( a l , . . . , a k ) w i t h aiE A; for a l l i , and s u c h t h a t a & d ( s ) . I n o t h e r words, each a c t i o n i n 2 i s a d m i s s i b l e , i n t h e sense t h a t it may b e performed i n some c i r c u m s t a n c e s ( c o n t e x t o f o t h e r a c t i o n s ) , b u t all o f them j o i n t l y a r e n o t admissible.
P e r f o r m i n g a j o i n t a c t i o n i n s i t u a t i o n s when t h e r e i s no independence r e q u i r e s some s o r t o f communication between members o f t h e group. To s e e why it i s s o , cons i d e r t h e s i m p l e s t c a s e o f two p e r s o n s . Suppose t h a t t h e a c t i o n s e t c o n t a i n t s two e l e m e n t s , x and y , and t h a t o u t o f f o u r p a i r s o f j o i n t a c t i o n s , xx, xy, y x and y y , o n l y t h e f i r s t and t h e l a s t a r e p o s s i b l e . Then i f t h e f i r s t p e r s o n p e r f o r m s x, t h e second must a l s o p e r f o r m x , and s u c h knowledge ( t h a t he cannot p e r f o r m y i n t h e g i v e n c i r c u m s t a n c e s ) must be a c q u i r e d by some communication between t h e two p e r s o n s .
it means simply t h a t e a c h p e r s o n knows enough about t h e i n t e n d e d a c t i o n s o f o t h e r s t o b e a b l e to d e t e r m i n e which a c t i o n s a r e a v a i l a b l e f o r h i m . T h i s communication need n o t be t r e a t e d l i t e r a l l y ;
Since t h e s e t d ( 2 ) i s , i n e f f e c t , a k-place r e l a t i o n
CHAPTER 6
832
i t i s not p o s s i b l e t o l i s t s y s t e m a t i c a l l y a l l c o n t i n g e n c i e s which may o c c u r . However, i t i s i n t h e set Ak,
p o s s i b l e t o d i s t i n g u i s h t h e most i n t e r e s t i n g t y p e s . DEFINITION. An a c t i o n a 6 A w i l l b e c a l l e d c o o p e r a t i v e , i f t h e c o n d i t i o n s 5 = ( a l , . . . , a k ) r2 d ( 2 ) and ai = a f o r some i imply a = a f o r a l l , j = l,...,k. j
T h i s means t h a t e i t h e r a l l members perform a , or nobody
d o e s . A t y p i c a l c a s e would be moving a n o b j e c t which i s s o heavy t h a t i t may be moved o n l y by a j o i n t e f f o r t of a l l persons. T h i s d e f i n i t i o n may b e g e n e r a l i z e d , s o as t o r e t a i n i t s most e s s e n t i a l f e a t u r e s , b u t r e l a x i n g t h e c o n d i t i o n t h a t
a l l p e r s o n s a r e r e q u i r e d t o perform t h e a c t i o n . DEFINITION. A c t i o n a E A w i l l be c a l l e d c o o p e r a t i v e , i f whenever 5 = ( al , . . . , a k ) t: d ( 2 ) and ai = a f o r some i , t h e n t h e r e e x i s t s a t l e a s t one j # i s u c h t h a t a = a . j
I n o t h e r words, a n a d m i s s i b l e j o i n t a c t i o n 5 e i t h e r d o e s n o t i n v o l v e any a c t i o n a , or i t c o n t a i n s a t l e a s t two o f them, performed by d i f f e r e n t p e r s o n s . If a c t i o n a i s c o o p e r a t i v e , we may d e f i n e t h e mapping q : d ( 2 ) +2K,
which t o e v e r y s t r i n g 5 i n d ( 2 ) a s s i g n s t h e s e t q ( 2 ) o f a l l t h o s e p e r s o n s who perform a i n s t r i n g a , s o t h a t if
3
= (al,..
q(g)
. ,ak) , t h e n
= (i
c
K : ai = a]
.
FORMAL THEORY OF ACTIONS
833
L e t us denote
so t h a t Q ( a ) i s t h e c l a s s of a l l s e t s of p e r s o n s who can j o i n t l y perform a . C l e a r l y , i f a i s c o o p e r a t i v e , i n t h e s e n s e of t h e l a s t d e f i n i t i o n , t h e n Q ( a ) d o e s n o t c o n t a i n any s i n g l e t o n s . If a i s c o o p e r a t i v e i n t h e sense of t h e f i r s t d e f i n i t i o n , t h e n Q ( a ) c o n t a i n s o n l y t h e empty s e t and t h e whole s e t K . On t h e o t h e r hand, we may have a c t i o n s which e x c l u d e cooperation, i n t h e following sense. DEFINITION. An a c t i o n a E A w i l l b e c a l l e d e x c l u s i v e , i f t h e c o n d i t i o n s a = ( a l , . . .,a,) G d ( g ) and ai = a for some i , i m p l y a # a for a l l j # i . j Thus, a n e x c l u s i v e a c t i o n i s s u c h i h a t i f one p e r s o n p e r f o r m s i t , nobody e l s e c a n . S i m i l a r l y as i n t h e c a s e of c o o p e r a t i v e a c t i o n s , we may i n t r o d u c e t h e c l a s s of p e r s o n s , s a y M ( s , a ) , who a r e q u a l i f i e d t o p e r f o r m a c t i o n a i n s t a t e 2, namely
A s mentioned, a t y p i c a l c a s e here o c c u r s when a c t i o n
a c o n s i s t s o f u s i n g some equipment which may be used a t any t i m e by one p e r s o n o n l y , and has t o be shared by t h e members o f t h e group. I n t h i s c a s e , M(2,a) i s
t h e set o f all p e r s o n s who can ( a r e q u a l i f i e d , p e r m i t t e d , e t c . ) t o use t h e equipment.
834
CHAPTER 6
8 . AN APPLICATION TO SYSTEM SYNTHESIS The a n a l y s i s o f t h e p r e c e d i n g s e c t i o n may be m o d i f i e d s o as t o p r o v i d e an a c c e s s t o some fundamental problems o f s y s t e m s t h e o r y , namely t o problems o f s y s t e m s synthesis.
The q u e s t i o n may be f o r m u l a t e d as f o l l o w s . Suppose t h a t we have a number o f systems which o p e r a t e s i m u l t a n e o u s l y . Under which c o n d i t i o n s one s h o u l d t r e a t t h e m j o i n t l y as a s y s t e m ? I n t u i t i v e l y , one r e q u i r e s here some s o r t o f i n t e r a c t i o n , s i n c e two o r more systems which o p e r a t e i n d e p e n d e n t l y , and do n o t i n f l u e n c e one a n o t h e r WoUldi u s u a l l y n o t be r e g a r d e d as a l a r g e r s y s t e m . To f o r m a l l y p r e s e n t t h e problem, assume t h a t w e have
k a c t i o n s y s t e m s , a s c o n s i d e r e d i n S e c t i o n 1, namely ( o m i t t i n g f o r s i m p l i c i t y t h e pauses # ) (Ai,
Si, r i , p i ) ,
i = 1,
... , k
where t h e symbols a r e a s b e f o r e , s o t h a t Ai
(8.1)
i s the set
o f a c t i o n s , S. i s t h e set of s t a t e s of i - t h system, ri: Si 4 2 A1i - { P r f i s t h e a d m i s s i b i l i t y mapping, and pi: Si x A i ->S is the transition function. i
One c o u l d now t r y , i n b u i l d i n g a system o u t o f s u b s y s tems ( 8 . 1 ) , t o t a k e as t h e s e t o f a c t i o n s a v a i l a b l e t h e Cartesian product A = A
1x A 2 r
...
Ak
(8.2)
s o t h a t each a c t i o n i s a vector
(8.3)
835
FORMAL THEORY OF ACTIONS
where a ( i ) C Ai,
i n t e r p r e t e d as s i m u l t a n e o u s p e r f o r m i n g
t h e component a c t i o n s . A s it w i l l t u r n o u t , s u c h a n approach i s not s a t i s f a c t o r y . The problem o r s y s t e m a g g r e g a t i o n c o n s i s t s o f d e f i n i n g t h e p r i m i t i v e c o n c e p t s s o as t o o b t a i n a new s y s t e m , o f
t h e form d e s c r i b e d by (8.1), which would d e s c r i b e t h e j o i n t o p e r a t i o n o f a l l subsystems. Thus, as t h e s e t o f s t a t e s one c o u l d t r y , s i m i l a r l y as for a c t i o n s , t o t a k e t h e C a r t e s i a n p r o d u c t
s
=
s 1 x s2
%
... *
Sk.
(8.4)
B e f o r e d e f i n i n g t h e r e m a i n i n g two n o t i o n s , t h o s e o f r and p , f o r t h e a g g r e g a t e d s y s t e m , i t i s w o r t h t o n o t e t h a t we have h e r e c e r t a i n d i f f i c u l t i e s ( a n a l o g o u s t o t h o s e e n c o u n t e r e d i n a c t i o n s o f many p e r s o n s ) which c a n n o t be overcome w i t h i n t h e frameworks o f s y s t e m s (8.1) separately. The p o i n t i s as f o l l o w s : b o t h t h e a c t i o n v e c t o r 2 and
s t a t e v e c t o r 2 may be s u b j e c t t o v a r i o u s i n t e r n a l cons t r a i n t s , which e l i m i n a t e c e r t a i n v e c t o r s o f s t a t e s o r a c t i o n s as i m p o s s i b l e ( e v e n though t h e c o o r d i n a t e s a r e possible separately i n each system). These c o n s t r a i n t s l e a d t o some s e t s AM and SM o f a c t i o n s and s t a t e s f o r t h e a g g r e g a t e d system; t h e s e s e t s may be i n v a r i o u s r e l a t i o n s t o t h e C a r t e s i a n p r o d u c t s ( 8 . 2 ) and ( 8 . 4 ) . One may i n t r o d u c e h e r e t h e f o l l o w i n g d e f i n ition.
DEFINITION. If' AM = A ( = A 1 x . . . Y Ak) , we s a y t h a t systems ( 8 . 1 ) a r e a c t i o n a l l y i n d e p e n d e n t . If SM = S
CHAPTER 6
836
( = S1 x
.. .
r- S k ) , w e s a y t h a t t h e s y s t e m s
state-independent
(8.1) are
.
If AM # A , we s a y t h a t s y s t e m s ( 8 . 1 ) a r e a c t i o n a l l y i n t e r f e r i n g . If A 3 A , t h e i n t e r f e r e n c e i s M p o s i t i v e , or s y n e r g e t i c , w h i l e i f A M C A , t h e i n t e r f e r e n c e i s n e g a t i v e , o r e x c l u d i n g . F i n a l l y , i f AM - A # 0 and A - AM # @, we s p e a k of b i l a t e r a l i n t e r f e r e n c e . DEFINITION.
Analogous n o t i o n s may be i n t r o d u c e d for s t a t e s , t h r o u g h r e l a t i o n s between S and SM. T h u s , we have h e r e s t a t e s y n e r g y and s t a t e e x c l u s i o n . An i m p o r t a n t c a s e o f i n t e r f e r e n c e i s q u a s i - i n d e p e n d e n c e . DEFINITION. We s a y t h a t s y s t e m s ( 8 . 1 ) a r e a c t i o n a l l y quasi-independent, i f AM = A;
'P
...
%
AL
(8.5)
and s t a t e q u a s i - i n d e p e n d e n t , i f SM =
s;
...
n
A' k'
(8.6)
We have t h e n THEOREM. Assume t h a t s y s t e m s ( 8 . 1 ) a r e a c t i o n a l l y q u a s i i n d e p e n d e n t . Then t h e y a r e p o s i t i v e l y i n t e r f e r i n g , i f A! 3 Ai 1
f o r a l l i , w i t h a t l e a s t one i n c l u s i o n b e i n g
strict. S i m i l a r l y , we have THEOREM. Assume t h a t t h e s y s t e m s ( 8 . 1 ) a r e a c t i o n a l l y
FORMAL THEORY OF ACTIONS
837
q u a s i - i n d e p e n d e n t . Then t h e y a r e n e g a t i v e l y i n t e r f e r -
ing, i f A; C Ai f o r a l l i , w i t h a t l e a s t one i n c l u s i o n being s t r i c t . S i m i l a r theorems h o l d f o r t h e c a s e o f s t a t e q u a s i - i n d e pendence. To i l l u s t r a t e t h e s e n o t i o n s , one may u s e t h e examples o f a c t i o n s y s t e m s of many p e r s o n s from t h e p r e c e d i n g s e c t i o n . Imagine namely two s y s t e m s , e a c h c o n s i s t i n g of a b o s s , s e c r e t a r y , d e s k , t y p e w r i t e r , e t c . If we assume t h a t t h e s e systems a r e i n d e p e n d e n t , t h e n t h e r e a r e no r e s t r i c E o n s on t h e p o s s i b i l i t y of performi n g a c t i o n s i n each of t h e s y s t e m s s i m u l t a n e o u s l y , as l o n g as t h e s e a c t i o n s a r e a d m i s s i b l e i n t h e s y s t e m s . However, i f t h e r e i s o n l y one t y p e w r i t e r which must be shared, t h e s y s t e m s are negatively i n t e r f e r i n g ( a c t i o n a l l y ) . S i m i l a r examples may b e found f o r s t a t e i n t e r ference. To i l l u s t r a t e p o s i t i v e i n t e r f e r e n c e , o r s y n e r g y , a s s u -
me t h a t b o t h s e c r e t a r i e s may c a r r y j o i n t l y some o b j e c t , which none o f them can c a r r y a l o n e . We have t h e n a new a c t i o n , a b s e n t i n t h e systems t r e a t e d s e p a r a t e l y , which accounts f o r synergetic e f f e c t . It ought t o be c l e a r t h a t w h e t h e r o r n o t t h e s y s t e m s
a r e p o s i t i v e l y o r n e g a t i v e l y i n t e r f e r i n g i s determined by f a c t o r s o t h e r t h a n t h o s e c o n t a i n e d i n t h e s y s t e m s s e p a r a t e l y ; t h i s i s p r e c i s e l y t h e f e a t u r e which makes a s y s t e m d i s t i n c t from t h e sum o f i t s p a r t s . L e t us now t u r n t o t h e d e f i n i t i o n o f a d m i s s i b i l i t y f u n c t i o n . A s b e f o r e , t h e s t a r t i n g p o i n t w i l l be t h e “natural” d e f i n i t i o n corresponding t o t h e lack of i n t e r ference. Violation of t h i s d e f i n i t i o n causes t h e
838
CHAPTER 6
c o l l e c t i o n of t h e systems t o be a l s o a system.
,. . .
<
Let 2 = (s (1) , s ( ~ ) ) SM be t h e s t a t e o f t h e s y s tem a t t i m e t . L e t Z be t h e c l a s s o f a l l f u n c t i o n s k which a r e monotone w i t h r e s p e c t t o e a c h o f t h e v a r i a b l e s , which map t h e s e t [ O , l l k i n t o [O,l]. DEFINITION. We s a y t h a t s y s t e m s ( 8 . 1 ) a r e n o n - i n t e r f e r i n g , i f t h e r e e x i s t s a f u n c t i o n h C. Z every s t a t e 2
c
SM
A
S and a c t i o n 2
k
such t h a t for
A we have
where I(2) i s t h e c h a r a c t e r i s t i c f u n c t i o n o f t h e s e t D , so t h a t
0
if
2 k DM.
A special case of lack of interference occurs i f the
f u n c t i o n h i s o f t h e form h(xl,
... , x k )
= min [ x l y . . . , x k I .
I n t h i s c a s e , t h e a d m i s s i b i l i t y o f a given j o i n t a c t i o n e q u a l s t h e minimum o f a d m i s s i b i l i t i e s of a c t i o n s i n p a r t i c u l a r systems ( n o t e t h a t w e c o n s i d e r h e r e t h e c a s e o f f u z z y a d m i s s i b i l i t y , b o t h i n systems ( 8 . 1 ) and i n the aggregated system). It i s c l e a r t h a t t h e above d e f i n i t i o n c o n c e r n s o n l y tho-
se a c t i o n s 5 which a r e i n t h e C a r t e s i a n p r o d u c t A , I f t h e s y s t e m s are a c t i o n a l l y i n t e r f e r i n g p o s i t i v e l y or
839
FORMAL THEORY OF ACTIONS
b i l a t e r a l l y , o n e n e e d s t o d e f i n e s e p a r a t e l y t h e admissi b i l i t y f u n c t i o n f o r a c t i o n s 2 which are o u t s i d e t h e
set A . To d e f i n e t h e l a s t n o t i o n o f t h e a g g r e g a t e d system,
namely t h e t r a n s i t i o n law p , w e may i n t r o d u c e t h e f o l l o w ing definition. DEFINITION. W e s a y t h a t s y s t e m s ( 8 . 1 ) h a v e i n d e p e n d e n t , s ( ~ ) )c SM A S t r a n s i t i o n s , if f o r e a c h 2 = ( s (1)
)...
and 2
=
( a( 1 )) .. . , a ( k ) ) E. AM
r\
A
w e have
where
T h u s , i n d e p e n d e n c e o f t r a n s i t i o n s means t h a t t h e n e x t s t a t e i s a v e c t o r , whose components a r e n e x t s t a t e s i n t h e c o r r e s p o n d i n g s u b s y s t e m s , as d e t e r m i n e d by t h e t r a n s i t i o n f u n c t i o n s i n t h e s e subsystems. N a t u r a l l y , t h e most i n t e r e s t i n g c a s e o c c u r s when t h e systems do n o t have independent t r a n s i t i o n s . A t t h i s p o i n t , i t i s w o r t h w h i l e t o s t r e s s t h e f o l l o w i n g asp e c t s of the considerations. E a c h of t h e s y s t e m s ( 8 . 1 ) i s d e s c r i b e d i n terms o f A i , S i , r i and p i ' w h i c h c h a r a c t e r i z e i t s o p e r a t i o n s i n " i s o l a t i o n " . The a g g r e g a t e d s y s t e m i s d e s c r i b e d by t h e c o r r e s p o n d i n g n o t i o n s A MY 'M.3 and P * If t h e l a t t e r c a n be e x p r e s s e d by t h e n o t i o n s f o p t h e
s y s t e m s t r e a t e d s e p a r a t e l y , t h i s means
,
intuitively
CHAPTER 6
840
s p e a k i n g , t h e t h e a g g r e g a t i o n o f ( 8 . 1 ) does n o t form a system i n t h e p r o p e r s e n s e o f t h i s t e r m , i . e . t h e beh a v i o u r o f t h e whole i s c o m p l e t e l y d e t e r m i n e d b y t h e behaviour of t h e p a r t s t r e a t e d s e p a r a t e l y . I n o t h e r words, no new " q u a l i t y " emerges, which would be a n a t t r i b u t e o f t h e whole n o t b e i n g a n a t t r i b u t e o f t h e parts
.
One s h o u l d mention t h a t t h e p r e s e n t c o n s i d e r a t i o n s a r e , i n a s e n s e , "dual" t o t h e t r a d i t i o n a l s y s t e m i c a n a l y s i s . I n d e e d , t y p i c a l l y one c o n s i d e r s a s y s t e m , and a n a l y s e s t h e b e h a v i o u r o f i t s p a r t s , whose p r o p e r t i e s a r e induced by t h e c o r r e s p o n d i n g p r o p e r t i e s o f t h e s y s t e m . The p r e s e n t approach i s d i f f e r e n t : one s t a r t s from t h e component s y s t e m s , and t h e a n a l y s i s c o n c e r n s t h e a g g r e g a t e d s y s t e m , o r r a t h e r , t h e d e v i a t i o n s from "independence" i n form o f s y n e r g y , a d m i s s i b i l i t y i n t e r f e r e n c e , e t c . G e n e r a l l y , one may i n t r o d u c e t h e f o l l o w i n g d e f i n i t i o n . DEFINITION. Subsystems ( 8 . 1 ) form a s y s t e m , i f a t l e a s t one of t h e f o l l o w i n g c o n d i t i o n s i s m e t :
( a ) they a r e not s t a t e and/or a c t i o n a l l y independent; ( b ) t h e y i n t e r f e r e from t h e p o i n t o f view o f a d m i s s i bility;
( c ) t h e y do n o t have i n d e p e n d e n t t r a n s i t i o n s . One can t h e n f o r m u l a t e t h e f o l l o w i n g c r i t e r i a which a l l o w us t o d e t e r m i n e whether ( 8 . 1 ) form a system o r not.
,..., a ( k )
THEOREM. I f t h e r e e x i s t a c t i o n s a (1)
such t h a t
841
FORMAL THEORY OF ACTIONS
,..., a (k))
( a (1)
6 AM, while a ( i )
< Ai
f o r some i , t h e n
( 8 . 1 ) form a s y s t e m .
,...,a (k)
THEOREM. I f t h e r e e x i s t a c t i o n s a (1)
such t h a t
a ( i ) E Ai f o r a l l i , w h i l e
then (8.1) form THEOREM.
(s"),
a system.
,...,
If t h e r e e x i s t s t a t e s s (1) s ( k ) such t h a t i C: SM w h i l e s ( ) 4 Si f o r some i , then
zk))
...,
( 8 . 1 ) form a s y s t e m . THEMREM.
s(i) G
,...,
I f t h e r e e x i s t s t a t e s s (1)
si
s (k) s u c h t h a t
for a l l i , w h i l e
t h e n ( 8 . 1 ) form a s y s t e m . THEOREM.
Suppose t h a t t h e r e e x i s t two a c t i o n s v e c t o r s 1 and two s t a t e v e c t o r s , = (a( ), a ( k) ) , = 1) ( : l ) S ( k ) ) , s ' = and s = ( s (at( -
a
,...,
,...,
...,
,...,
a'
s ' (k)) s u c h t h a t for a l l i w e have
For t h e p r o o f , l e t us observe t h a t i f t h e assumptions of t h e l a s t t h e o r e m a r e s a t i s f i e d , t h e n t h e r e e x i s t s no f u n c t i o n which would s a t i s f y t h e c o n d i t i o n s o f t h e
842
CHAPTER 6
d e f i n i t i o n of t h e non-interference of a d m i s s i b i l i t i e s .
THEOREM
If t h e r e e x i s t s t a t e v e c t o r s
s = -
and a c t on v e c t o r
a = ( a (1) such t h a t
then
p(s,a) =
,..., 2'
a( k ) ) while
( 8 . 1 ) form a s y s t e m .
L e t u s now c o n s i d e r t h e problem o f a t t a i n a b i l i t y i n
c a s e s when t h e a c t i o n s i n one or more subsystems a r e c o n s t r a i n e d i n some way. We assume t h a t t h e g o a l o f t h e s y s t e m , a s a whole i s t o
a t t a i n a t t h e t i m e t one o f t h e s t a t e s i n t h e s e t C C S M , and l e t t h e i n i t i a l s t a t e ( a t t = 0 ) be so. Suppose t h a t t h e s e t C i s a t t a i n a b l e a t t i m e t , s t a r t i n g from so, i . e . t h e r e e x i s t s a sequence o f a c t i o n vectors
such t h a t
FORMAL THEORY OF ACTIONS
843
Let u s assume t h a t i n i - t h s y s t e m one i s c o n s t r a i n e d t o a c t i o n s from a s e t A; C Ai o n l y , I t i s t h e n c l e a r t h a t t h e p o s s i b i l i t y o f a t t a i n i n g a s t a t e from t h e s e t C may o n l y d e c r e a s e .
...
r A i represent the c o n s t r a i n t s , and l e t St (' A 1 , s ) be t h e c l a s s of a l l s t a t e s which a r e a t t a i n a b l e a t t i m e t , s t a r t i n g from 2, and u s i n g o n l y a c t i o n s from A'. Thus, i n t h i s n o t a t i o n , S t ( A , s ) d e n o t e s t h e c l a s s of s t a t e s which a r e a t t a i n a b l e w i t h o u t c o n s t r a i n t s . S i n c e C i s a t t a i n a b l e by a s s u m p t i o n , w e have To d e s c r i b e i t , l e t A '
We have h e r e
= A;
w
t h e f o l l o w i n g theorem.
THEOREM. I f A? C A; C A f o r a l l i , t h e n i
f o r a l l t and a l l i n i t i a l s t a t e s 2. T h i s means t h a t imposing s t r o n g e r c o n s t r a i n t s may only
worsen t h e s i t u a t i o n . C o n s i d e r now t h e v a l u e q ( C , s , A 1 ) ing relation
d e f i n e d by t h e follow-
844
CHAPTER 6
q ( c , g , A ' ) = min [ t : S t ( A ' , g ) n
C #
01.
Thus, q ( C , s , A ' ) i s t h e e a r l i e s t t i m e a t which one can a t t a i n one o f t h e s t a t e s from C , s t a r t i n g from 5 , and under t h e c o n s t r a i n t s A ' . We have h e r e
THEOREM. If A " C A ' , t h e n
and moreover, i f C" C C
I ,
bhen f o r e v e r y A'
c
A
T h i s means t h a t t h e t i m e q ( C , s , A )
increases with the d e c r e a s e o f t h e s e t A , or t a r g e t s e t C .
Thus f a r , t h e g o a l C was n o t f u z z y . Suppose now t h a t C i s a f u z z y s e t o f s t a t e s , d e s c r i b e d by t h e membership f u n c t i o n f c (-s ) , whose v a l u e s e x p r e s s t h e e x t e n t t o which t h e g o a l i s a c h i e v e d i f t h e a c t u a l s t a t e i s 2. Then we can c o n s i d e r t h e f u n c t i o n d e f i n e d as f o l l o w s
f c ( g ) i f g i s a t t a i n a b l e from go a t t i m e t , under c o n s t r a i n t s A '
1
\ft(g,go,A') = *
/. 0
otherwise.
We may t h e n d e f i n e
which r e f l e c t s t h e d e g r e e t o which t h e g o a l C' may be
FORMAL THEORY OF AC'ITONS
845
a t t a i n e d a t t i m e t , s t a r t i n g from s and under con-0 ' s t r a i n t s A ' . C l e a r l y , i f $ t ( C , ~ O , A ' ) = 1, t h e n C i s a t t a i n a b l e i n t h e s e n s e of t h e p r e c e d i n g c o n s i d e r a t i o n s . S i n c e qt(g,gO,A') , r e g a r d e d a s a f u n c t i o n o f 2 , may be t r e a t e d as a membership f u n c t i o n o f a f u z z y s e t , we s e e e a s i l y t h a t t h e s e f u z z y s e t s d e c r e a s e w i t h t h e i . e . we have
constraints A ' , THEOREM.
If
A"
C_
A',
t h e n for a l l s and t
I n a s i m i l a r way as b e f o r e , we may d e f i n e , f o r any A S 1
S O t h a t q ( d ) ( C , s o , A 1 ) i s t h e e a r l i e s t moment a t which a t i t p o s s i b l e t o a t t a i n C i n d e g r e e a t l e a s t o( ( s t a r t i n g from so, and under c o n s t r a i n t s A'), A s b e f o r e , t h e -
se times i n c r e a s e w i t h t h e i n c r e a s e o f s e v e r i t y of c o n s t r a i n t s , i n c r e a s e o f d , and d e c r e a s e o f t h e s e t C . T h i s r e f l e c t s t h e f a c t t h a t i n any c a s e , s t r i c t e r r e q u i r e m e n t s have t o be p a i d b y a p r i c e , i n form of e i t h e r d e c r e a s i n g t h e d e g r e e o f a t t a i n m e n t o f t h e g o a l , or by t h e t i m e of' i t s a t t a i n m e n t .
9 . PREFERENCE SYSTEMS AND RELATIONS TO DECISION THEORY
So f a r t h e c o n c e p t s r e f e r r e d o n l y t o t h e p o s s i b i l i t i e s
o f p e r f o r m i n g c e r t a i n c o m b i n a t i o n s o f a c t i o n s , and a t t a i n i n g c e r t a i n c o m b i n a t i o n s of outcomes. Within t h i s
CHAPTER 6
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framework i t was n o t p o s s i b l e t o e x p r e s s t h e a t t i t u d e s of p a r t i c u l a r p e r s o n s from K t o w a r d s v a r i o u s g o a l s .
L e t us t h e r e f o r e i n t r o d u c e a system o f p r e f e r e n c e s TT
={2j: j c
K!
where e a c h 2 i s a b i n a r y r e l a t i o n on t h e s e t of a l l j g o a l s , s a y S; e a c h of t h e r e l a t i o n s 2 i s assumed j t r a m s i t i v e and c o n n e c t e d . We s h a l l r e f e r t o 2 as t o t h e p r e f e r e n c e system f o r - j p e r s o n j , s o t h a t i f G , G 1 a r e two g o a l s i n S, t h e n
2. GI J
w i l l be i n t e r p r e t e d as t h e f a c t t h a t p e r s o n j p r e f e r s (weakly) g o a l G t o g o a l G I . G
G e n e r a l l y , t h e c o n c e p t of p r e f e r e n c e r e l a t i o n s 2 - j a l l o w s u s t o c h a r a c t e r i z e t h o s e s u b s e t s o f K which c o n s t i t u t e teams, i . e . r o u g h l y s p e a k i n g , g r o u p s o f p e r sons w i t h i d e n t i c a l preferences. L e t S ’ c S be a s u b s e t o f t h e c l a s s o f a l l g o a l s ; i n t u i t i v e l y , S ’ w i l l c o n t a i n t h o s e g o a l s which a r e o f i n t e r e s t i n a g i v e n c o n t e x t . We assume t h a t S 1 c o n t a i n s a t l e a s t two d i s t i n c t e l e m e n t s . We may now i n t r o d u c e t h e f o l l o w i n g d e f i n i t i o n . K a r e s a i d t o be S 1 - e q u i DEFINITION. Two p e r s o n s , i . j v a l e n t , i f f o r every G ’ , G ” f S’ t h e c o n d i t i o n G I >iG1l h o l d s i f and o n l y i f G I 3 . G I 1 .
J
It i s easy t o s e e t h a t t h e r e l a t i o n d e f i n e d above i s i n d e e d an e q u i v a l e n c e , i . e . i t i s r e f l e x i v e , symmetric and t r a n s i t i v e . C o n s e q u e n t l y , for e v e r y s e t S 1 , t h e
FORMAL THEORY OF ACTIONS
847
c l a s s K w i l l s p l i t i n t o c l a s s e s of S'-equivalence. Members o f t h e same c l a s s w i l l a l l have t h e same p r e f e r e n c e s w i t h i n t h e s e t S' o f g o a l s , w h i l e members o f d i f f e r e n t c l a s s e s w i l l have d i f f e r e n t p r e f e r e n c e s ( a t l e a s t f o r some g o a l s ) . Any s u c h S ' - e q u i v a l e n c e c l a s s may be c a l l e d a n S'-team, and f o r any S' t h e cla.ss K may c o n s i s t s o f one or mor e S f - t e a m s . It may a l s o happen t h a t e a c h S1-team cons i s t s o f one p e r s o n o n l y . F o r a d e s c r i p t i o n o f a g i v e n s i t u a t i o n one may now u s e t h e c o n c e p t s o f game t h e o r y , i d e n t i f y i n g s t r i n g s o f actions of a person w i t h h i s " s t r a t e g i e s " . O f p a r t i c u l a r s i g n i f i c a n c e h e r e a r e t h e c o n c e p t s o f e q u i l i b r i u m , and o f c o a l i t i o n of p l a y e r s . DEFINITION. A s t r i n g 2 = (2(1),... >-a ( k ) ) o f j o i n t a c t i o n s i s s a i d t o be i n e q u i l i b r i u m , i f f o r a l l i K and a l l s t r i n g s si, whenever t h e j o i n t s t r i n g v' = (i-1) a( k ) ) i s admissible, ( 2( 1 ) y . - * , s a(itl) t h e n 2 '2i s ' , where 2 and 2' a r e s t a t e s a t t a i n e d by s t r i n g s 1 and 1'.
,...,
J
I n t u i t i v e l y , a j o i n t s t r i n g of a c t i o n s i s i n e q u i l i b r i u m i f t h e f o l l o w i n g i s t r u e f o r every person: i f a pers o n knows t h a t a l l o t h e r s w i l l p e r f o r m a c t i o n s which a r e components o f 2 , t h e n he may o n l y l o s e ( i . e . a t t a i n somelihing l e s s p r e f e r r e d ) i f he c h o s e s any o t h e r s t r i n g o f a c t i o n s t h a n 2( i )
.
E q u i l i b r i u m i n a g i v e n s i t u a t i o n need n o t be u n i q u e . T h i s i s t r u e i n c a s e o f N-person games i n g e n e r a l ; i n o u r c a s e we have a l s o t h e f o l l o w i n g theorem.
CHAPTER 6
848
THEOREM. Suppose t h a t a l l members o f K form as S1-team,
and l e t G be t h e most p r e f e r r e d g o a l i n S ' . Then any s t r i n g of a c t i o n s which b r i n g s about t h i s g o a l i s i n eq u i 1ib r i um
.
It i s w o r t h w h i l e t o d i s c u s s h e r e t h e r e l a t i o n between t h e t h e o r y o f a c t i o n s o f t h i s c h a p t e r , b o t h of a s i n g l e
p e r s o n and o f groups of p e r s o n s , and t h e d e c i s i o n and game t h e o r y . In g r e a t e s t s i m p l i f i c a t i o n , t h e difference l i e s primar i l y i n t h e emphasis, which t h e t h e o r i e s of t h e p r e s e n t c h a p t e r p u t on t h e s t r u c t u r a l a s p e c t s of t h e s i t u a t i o n , as opposed t o t h e more " g l o b a l " a p p r o a c h e s of t h e c l a s s i c a l d e c i s i o n t h e o r y and t h e o r y o f games. L e t us c o n s i d e r f i r s t i n some d e t a i l t h e r e l a t i o n s b e t ween t h e t h e o r i e s i n q u e s t i o n i n c a s e o f a c t i o n s o f one p e r s o n . A l l known d e c i s i o n t h e o r i e s r e d u c e t h e s i t u a t i o n t o
e s s e n t i a l l y the following elements: t h e s e t of availa b l e o p t i o n s , o f which one and o n l y one m u s t be c h o s e n , t h e s e t o f s t a t e s of "Nature", r e p r e s e n t i n g t h e d e c i s i o n maker's i g n o r a n c e as t o t h e e x a c t s i t u a t i o n which he f a c e s ( i n extreme c a s e s , he knows t h e s i t u a t i o n e x a c t l y ) , s e t o f outcomes, and t h e a s s u m p t i o n s which allow us t o r e p r e s e n t t h e p r e f e r e n c e s of t h e d e c i s i o n maker among t h e outcomes. I n t h e s i m p l e s t c a s e , when t h e d e c i s i o n maker knows e x a c t l y t h e s i t u a t i o n , t h e o n l y u n c e r t a i n t y for h i m i s t h e p o s s i b l y s t o c h a s t i c laws t h a t connect t h e o p t i o n s and outcomes. I n t h i s c a s e , t h e problem l i e s i n
FORMAL THEORY OF ACTIONS
849
d e f i n i n g t h e u t i l i t y i n d e x s o t h a t t h e b e s t outcome w i l l be t h a t which y i e l d s t h e h i g h e s t e x p e c t e d u t i l i t y .
V a r i o u s t h e o r i e s s u g g e s t e d t h u s f a r ( o f which S E U , or S u b j e c t i v e Expected U t i l i t y , i s b e s t known) f a i l e d t o p r o v i d e d e s c r i p t i o n which would be a d e q u a t e i n a l l s i t u a t i o n s ; e a c h of them g i v e s good p r e d i c t i o n s i n some c l a s s e s o f s i t u a t i o n s , and f a i l s i n some o t h e r c l a s s e s . It would a p p e a r t h e r e f o r e , t h a t d i f f e r e n t c l a s s e s o f
s i t u a t i o n s m a y r e q u i r e d i f f e r e n t d e c i s i o n models. T h i s makes i t i m p o r t a n t to p r o v i d e a t h e o r y a l l o w i n g a t least a c l a s s i f i c a t i o n of various decision s i t u a t i o n s . The t h e o r y o f t h e p r e s e n t c h a p t e r , b y s t r e s s i n g t h e a n a l y s i s o f s t r u c t u r a l a s p e c t s of t h e s i t u a t i o n , may provide a solution here. In f a c t , t h e action theory p r e s e n t e d here may a l s o b e termed a t h e o r y o f pre-decisional situations. F i r s t l y , i n s t e a d o f j u s t assuming t h e a d e c i s i o n m a k e r f a c e s a set of o p t i o n s , t h e p r e s e n t t h e o r y a t t e m p t s t o e x p l i c a t e t h e s t r u c t u r e of t h i s s e t , by d i s t i n g u i s h i n g d i s c r e t e u n i t s o f b e h a v i o u r . O p t i o n s a r e namely r e p r e s e n t e d as s t r i n g s of elementary a c t i o n s , t h u s s t r e s s i n g t h e dynamic a s p e c t s o f d e c i s i o n making, and t h i s a l l o w s us to i n t r o d u c e and a n a l y s e s u c h n o t i o n s a s d e c i s i v e moments. Here t h e s t r u c t u r e i s i n d u c e d simply b y t h e f a c t t h a t two o p t i o n s , e a c h o f them bei n g a s t r i n g o f a c t i o n s , may have a common b e g i n n i n g , which a l l o w s t h e d e c i s i o n maker t o p o s t p o n e t h e c h o i c e between them. N e x t , t h e p r e s e n t t h e o r y e x p l a i n s i n some d e t a i l t h e
r e l a t i o n s between o p t i o n s ( s t r i n g s o f a c t i o n s ) and outcomes. The l a t t e r have now t h e form o f s t r i n g s of
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t i m e - e v e n t s , and a g o a l may be d e f i n e d as a d i s j u n c t i o n o f some c o n f i g u r a t i o n s of t h e l a t t e r . T h i s leads t o an a l g e b r a i c a n a l y s i s o f g o a l s , and t h e c e n t r a l f a c t o f t h e t h e o r y i s t h e isomorphism between t h e s t r u c t u r e o f t h e g o a l and t h e s t r u c t u r e o f t h e s e t o f s t r i n g s o f a c t i o n s which b r i n g about t h i s g o a l . Again, t h i s approach s t r e s s e s d i f f e r e n t a s p e c t s t h a n t h o s e i n " c l a s s i c a l " d e c i s i o n t h e o r y , by i n t r o d u c i n g s t r u c t u r a l a s p e c t s o f b o t h t h e o p t i o n s ( s t r i n g s of a c t i o n s ) and outcomes ( s t r i n g s o f e v e n t s ) . I n c a s e o f a c t i o n s o f many p e r s o n s , t h e d i f f e r e n c e s between t h e o r y o f games and t h e p r e s e n t a p p r o a c h a r e a g a i n s i m i l a r . What i s o f s p e c i a l i m p o r t a n c e i s t h e d i s t i n c t i o n between t h e n o t i o n s o f c o o p e r a t i o n , blocki n g , e t c . i n game t h e o r y and h e r e (where i n t h e l a t t e r c a s e t h e y a r e d e f i n e d t h r o u g h t h e s i t u a t i o n a l constr-v a i n t s , and n o t t h r o u g h u t i l i t i e s ) . To i l l u s t r a t e t h i s p o i n t , c o n s i d e r t h e game s u c h a s c h e s s . The a c t i o n s h e r e a r e c o n s e c u t i v e moves by White and B l a c k . Whether a p a r t i c u l a r move i n a g i v e n s i t u a t i o n i s a l l o w e d or n o t depends on t h e c o n f i g u r a t i o n of p i e c e s on t h e c h e s s b o a r d , and i n o r d e r f o r B l a c k ( s a y ) t o make a c e r t a i n move, i t i s n e c e s s a r y t h a t White moved b e f o r e i n some s p e c i a l way. The a c t i o n s o f White and B l a c k have t h e r e f o r e forms o f s t r i n g s
w1#w2#w3#.
..
and # b l # b 2 # b 3 . .
.
and are c o o p e r a t i v e ( s o t h a t t h e j o i n t s t r i n g wlblw 2 . . . i s a n a d m i s s i b l e game). T h i s s h o u l d n o t be c o n f u s e d w i t h t h e f a c t t h a t c h e s s i s a s t r i c t l y c o m p e t i t i v e game i n which t h e i n t e r e s t s o f p l a y e r s are opposed.
FORMAL THEORY OF ACTIONS
851
I n o t h e r words, t h e p r e f e r e n t i a l s t r u c t u r e o f t h e dec i s i o n s i t u a t i o n i s t r e a t e d s e p a r a t e l y from t h e a c t i o n a l s t r u c t u r e a s l o g i c a l l y i n d e p e n d e n t from i t , and i n t h i s c h a p t e r , t h e main S t r e s s i s p u t on t h e l a t t e r , b o t h i n t h e c a s e o f a c t i o n s y s t e m s f o r one and f o r many persons.
1 0 . P L A N N I N G ACTIONS AND GENERATIVE GRAMMARS The f o r m a l s y s t e m o f t h i s c h a p t e r may s e r v e as a b a s i s
f o r c o n s t r u c t i o n of f o r m a l t h e o r y o f p l a n n i n g and d e s i g n . The main i d e a c o n s i s t s h e r e of a p p l y i n g t h e n o t i o n s o f g e n e r a t i v e grammar, and g e n e r a t i n g s e n t e n c e s by t h e s e grammars. Suppose t h a t we a r e i n t e r e s t e d i n a t t a i n i n g a c e r t a i n g o a l G , descri.bed as a c o n f i g u r a t i o n o f t i m e - e v e n t s . I n such a c a s e , i f w e d e c i d e t o u s e s t r i n g s o f a c t i o n s w i t h a d m i s s i b i l i t y a t l e a s t b , we must choose a s t r i n g o f a c t i o n s from t h e s e t R i l ( G ) . These s e t s , f o r v a r y i n g G and b y form some l a n g u a g e s , and one may a t t e m p t t o
f i n d t h e g e n e r a t i n g grammars o f t h e s e l a n g u a g e s . To c o n s t r u c t a grammar, one n e e d s t o s p e c i f y i t s metaa l p h a b e t and p r o d u c t i o n r u l e s , s o t h a t t h e c l a s s o f a l l s t r i n g i n t e r m i n a l a l p h a b e t (which i n t h i s c a s e c o i n c i d e s w i t h t h e c l a s s o f a l l a c t i o n s ) which may be g e n e r a t ed u s i n g t h e s e r u l e s o f p r o d u c t i o n e q u a l s t h e c l a s s o f a l l a d m i s s i b l e s t r i n g s , i . e . t h e c o n s i d e r e d language of actions.
I n t h e c a s e o f t h e l a n g u a g e s o f t h e form R b l ( G ) , t h e t e r m i n a l a l p h a b e t c o i n c i d e s simply w i t h t h e c l a s s of
852
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a c t i o n s . The p r o c e s s of g e n e r a t i n g a c t i o n s t r i n g s from
RL1(G) i s , i n t h i s c a s e , t h e p r o c e s s o f s u c c e s s i v e s p e c i f i c a t i o n of t h e p l a n o f a t t a i n i n g G . The r o l e o f m e t a a l p h a b e t i s p l a y e d by t h e l a b e l s o f d e s c r i p t i o n s o f a c t i o n s o r c l a s s e s o f a c t i o n s . The p r o d u c t i o n r u l e s
depend, o f c o u r s e , on s p e c i f i c f e a t u r e s o f t h e s i t u a t i o n which i s t o be a n a l y s e d . On most c a s e s , g e n e r a t h g s t r i n g s of a c t i o n s may b e i n t e r p r e t e d simply as t h e s u c c e s s i v e s t a d i a of e l a b o r a t i o n of t h e p l a n . F o r m a l l y , t o e a c h element o f t h e a c t i o n s p a c e A one may a s s i g n some l a b e l , which c h a r a c t e r i z e s some f e a t u r e s of t h i s a c t i o n ( d i f f e r e n t a c t i o n s may be a s s i g n e d t h e same l a b e l ) . L a b e l s c o r r e s p o n d t o v a r i o u s l e v e l s o f g e n e r a l i t y . F o r i n s t a n c e many o f t h e a c t i o n s which are performed i n t h e k i t c h e n i n c o n n e c t i o n w i t h p r e p a r a t i o n o f meals may b e termed g e n e r a l l y , e . g . " f r y i n g " , "cooki n g " , e t c . The p l a n may be i d e n t i f i e d w i t h a s u b s e t o f t h e s e t o f a l l l a b e l s , c o n s i s t i n g o f t h o s e which a r e a s s i g n e d t o a c t i o n s i n R-'(G). The s m a l l e r i s t h i s b subset, the l e s s general i s t h e plan. In the particular c a s e , when a one-element
s e t i s chosen ( i . e . a s i n g l e
s t r i n g o f a c t i o n s ) , t h e p l a n i s most s p e c i f i c , i . e .
it
p r e s c r i b e s which a c t i o n s a r e t o be t a k e n , and i n which order. The p r o d u c t i o n r u l e s , i.e. t h e r u l e s which a l l o w r e p l a -
cement o f some l a b e l s by l a b e l s o f more s p e c i f i c s t r k n g s o f a c t i o n s , may be o b t a i n e d by u s i n g t h e r u l e s o f mo-
t i v a t i o n a l consistency:
t h e l a b e l c may be r e p l a c e d b y t h e composite l a b e l b l b 2 , . ,b k ' i f i n e a c h c o n t e x t , two s t r i n g s which d i f f e r o n l y by t h e f a c t t h a t one con-
..
t a i n s l a b e l a , and t h e o t h e r - l a b e l b l b 2 . . . b k , a r e both consistent o r both inconsistent motivationally.
FORMAL THEORY OF ACTIONS
853
One may e x p e c t t o o b t a i n some s p e c i f i c r e s u l t s here i n a s p e c i a l c a s e of s t r i n g s which a r e t o produce some o b j e c t s o u t of t h e i r c o n s t i t u e n t p a r t s . The g o a l i s t o c r e a t e a n o b j e c t w i t h some d e s i r e d p r o p e r t i e s , and i n t h i s case w i l l one may e x p e c t t h a t t h e s e t R-'(G) b b e r e l a t e d i n some way t o t h e s t r u c t u r e of t h e o b j e c t . The c r u c i a l p o i n t h e r e i s c o n n e c t e d w i t h l a n g u a g e s of a c t i o n s of many p e r s o n s , and e s p e c i a l l y t h e p o s s i b i l i t y o f p e r f o r m i n g some a c t i o n s i n p a r a l l e l , and n o t s e r i a l l y . The most i n t e r e s t i n g problem h e r e i s t h a t of opera t i o n s on p l a n s , s o as t o o b t a i n a p l a n which would l e a d t o c r e a t i o n of a p a i r o f o b j e c t s c r e a t e d b y separate plans. One may a l s o e x p e c t t h a t s u c h problems a r e c l o s e l y r e l a t e d t o problems of a r t i f i c i a l i n t e l l i g e n c e , and a l s o a r e u n d e r l y i n g t h e usage o f language and t h o u g h t .
11. AN APPLICATION TO O R G A N I Z A T I O N THEORY
The system of a c t i o n s from t h e p r e c e d i n g s e c t i o n s may be e a s i l y a p p l i e d t o a d e s c r i p t i o n of o p e r a t i o n of org a n i z a t i o n s , u n d e r s t o o d as i n s t i t u t i o n a l i z e d s e t s of p e r s o n s . Formally t h e r e f o r e i t i s t h e same as a n a c t i o n system of many p e r s o n s . One m a y , however, make t h e s i t u a t i o n more s p e c i f i c b y e x h i b i t i n g some s p e c i a l f e a t u r e s of a n o r g a n i z a t i o n , as opposed t o an a c t i o n system o f some o t h e r groups o f p e r s o n s . We s h a l l namel y c o n s i d e r a s p e c i a l c a s e , of two o r g a n i z a t i o n s of which one i s d i r e c t i n g t h e o p e r a t i o n s of a n o t h e r . We s h a l l t h e r e f o r e c o n s i d e r a system c o n s i s t i n g o f a pair
854
CHAPTER 6
p2)
(11.1)
where P1 w i l l be i n t e r p r e t e d as t h e d i r e c t i n g o r g a n i z a t i o n , and P2 -- as t h e o r g a n i z a t i o n which i s b e i n g d i rected. Now, under t h i s i n t e r p r e t a t i o n , i t may be t h a t P1 and P a r e d e p a r t m e n t s o f t h e same i n s t i t u t i o n , e t c . 2
Although e a c h o f t h e components P1 and P2 forms t y p i c a l l y a complex m u l t i p e r s o n s y s t e m , t o d e s c r i b e t h e e s s e n t i a l f e a t u r e s o f d i r e c t i n g , some of t h e i n t r i c a c i e s of i n t e r n a l s t r u c t u r e s o f P1 and P 2 may be d i s r e garded. F i r s t l y , we assume t h a t P1 i s a d e c i s i o n making body, which s e t s t h e p o l i c i e s , t o be c a r r i e d o u t b y P Thus, 2' P1 w i l l have a s p e c i a l a c t i o n a l p h a b e t , namely i t s a c t -
ions will be d e c i s i o n s , o r more s p e c i f i c a l l y , d i r e c t i o n s which a r e t o be c a r r i e d o u t b y P 2 ( o t h e r p o s s i b l e a c t i o n s o f P1 may be d i s r e g a r d e d f o r t h e p r e s e n t a n a l y s i s ) . A s r e g a r d s P 2 , it i s t o be i n t e r p r e t e d as a n a c t i o n s y s t e m ( a s a r u l e , o f many p e r s o n s ) , which i s to f u l -
f i l l t h e d i r e c t i o n s of P
1'
T h i s a c t i o n s y s t e m i s cha-
r a c t e r i z e d by t h e u s u a l components, i . e . s e t o f p e r s o n s , s e t o f s t a t e s , s e t o f a c t i o n s , and a d m i s s i b i l i t y and t r a n s i t i o n f u n c t i o n s . Thus, we may summarize i t b y r e p r e s e n t i n g P as 2
P2 =(K,
LA, O i d S l , E, R)
(11.2)
where K i s t h e s e t o f p e r s o n s i n P each L d i s t h e 2' c l a s s o f a l l k - s t r i n g s o f a c t i o n s which a r e a d m i s s i b l e
FORMAL THEORY OF ACTIONS
855
i n d e g r e e a t l e a s t g c , E i s t h e s e t o f r e s u l t s , and R i s t h e r e l a t i o n c o n n e c t i n g k - s t r i n g s from L w i t h outpc comes from E . Now, t h e i n s t r u c t i o n s o f P1 t o P2 may be p a r t i t i o n e d i n t o t h r e e broad c a t e g o r i e s :
--
t h o s e which s p e c i f y t h e g o a l t o be a t t a i n e d by P2, i . e . t h e y s p e c i f y some c o n f i g u r a t i o n s o f t i m e - e v e n t s ;
--
t h o s e which s p e c i f y s t r i n g s o f a c t i o n s t o be perform-
ed by P 2 ;
--
t h o s e which modify t h e a d m i s s i b i l i t i e s o f s t r i n g s o f a c t i o n s from P 2 . N a t u r a l l y , a n i n s t r u c t i o n may be “mixed” and i n v o l v e two or more o f t h e above components. F o r t h e p u r p o s e o f a n a l y s i s , however, i t i s most c o n v e n i e n t t o i n t r o duce t h e c o n c e p t o f ” e l e m e n t a r y ” i n s t r u c t i o n , and a s s u me t h a t i t may be o n l y of; one o f t h e above t y p e s , w i t h mixed i n s t r u c t i o n s r e p r e s e n t a b l e as s t r i n g s o f elementary instructions. A c c o r d i n g l y , we i n t e r p r e t P1 as a system
where D l Y D2 and D a r e some d i s j o i n t s e t s , i n t e r p r e t e d 3 as s e t s o f e l e m e n t a r y i n s t r u c t i o n s o f t h e above t h r e e types. We s h a l l d e n o t e D = D
1
D 2 V D
3
856
CHAPTER 6
*
and d e n o t e by D t h e monoid o v e r t h e a l p h a b e t Du?#), i . e . the c l a s s of a l l f i n i t e s t r i n g s consisting of i n s t r u c t i o n s from D and p a u s e s # . Now, e a c h element o f D s p e c i f i e s a g o a l G t o be a t t i n 1 ed by P2. C o n s e q u e n t l y , we may a s s i g n t o e a c h element d f D1 t h e s e t C ( d l ) d e f i n e d as R - ' ( G ) , where G i s 1 t h e g o a l s p e c i f i e d by d l . I n o t h e r words, t h e i n s t r u c t i o n d l may b e i n t e r p r e t e d as "perform a s t r i n g o f a c t i o n s from t h e s e t R-'(G)". N e x t , t o e a c h element d2 6 D2 we may a s s i g n a s e t C ( d ) 2 2 of s t r i n g s of a c t i o n s , o f which one i s t o be performed. I n extreme c a s e s , C 2 ( d 2 ) c o n t a i n s j u s t one s t r i n g , i f t h e i n s t r u c t i o n d2 i s s p e c i f i c . F i n a l l y , e l e m e n t s o f D are i n s t r u c t i o n s which modify 3 the admissibilities of s t r i n g s of actions of P This 2. means f o r m a l l y t h a t e a c h s u c h i n s t r u c t i o n s changes t h e f a m i l y of s e t s L, i n t o some o t h e r f a m i l y L b , s o t h a t f o r any l e v e l of a d m i s s i b i l i t y o( , t h e c l a s s o f s t r i n g s a d m i s s i b l e w i t h d e g r e e a t l e a s t d changes from L A t o a new s e t L k . To g i v e a n example, a n i n s t r u c t i o n from D may i n v o l v e 3 a new a l l o c a t i o n o f f u n d s , changes o f o r g a n i z a t i o n a l s t r u c t u r e o f P2, e t c . s o t h a t s t r i n g s o f a c t i o n s t h a t were p r e v i o u s l y i n a d m i s s i b l e ( b e c a u s e , s a y , o f b u d g e t a r y c o n s t r a i n t s ) become now a d m i s s i b l e . C o n s i d e r i n g f o r s i m p l i c i t y o n l y t h e i n s t r u c t i o n s from D1 and D 2 ( p e r t a i n i n g t o some s p e c i f i c p e r i o d o f a c t i v i t y o f P2), one may a s s i g n t o a composite i n s t (n) of r u n c t i o n v which i n v o l v e s e l e m e n t s d l(1),..., d 1 D and e l e m e n t s d 2(1),.. , d:m) o f D2, t h e s e t of s t r i n g s 1
.
FORMAL THEORY OF ACTlONS
857
of actions
m
n
(11.4)
If Q ( v ) n Ld # 0
(11.5)
w e s a y t h a t v i s s a t i s f i a b l e on t h e l e v e l o f a d m i s s i b i l i t y o ( . The l a r g e s t o< f o r which ( 1 1 . 5 ) h o l d s , i . e . dV = s u p i d :
Q W n L&
z
m
(11.6)
i s c a l l e d t h e l e v e l a t which v i s s a t i s f i a b l e .
N a t u r a l l y , s i n c e t h e l a n g u a g e s L, a r e m o n o t o n i c a l l y d e c r e a s i n g w i t h i n c r e a s e o f P(, we have THEOREM. I f v i s s a t i s f i a b l e on a l e v e l d , i t i s a l s o s a t i s f i a b l e on any lower l e v e l o f a d m i s s i b i l i t y .
Two composite i n s t r u c t i o n s v and v ' f o r which Q ( v ) = Q ( v l ) a r e e q u i v a l e n t . I f Q ( v ) C Q ( v ' ) , t h e n v ' i s less s p e c i f i c , o r l e s s r e s t r i c t i v e than v. Alternatively, w e may s a y t h a t v imposes a d d i t i o n a l c o n s t r a i n t s , r e l a t i v e t o v ' , or t h a t v ' r e l a x e s t h e c o n s t r a i n t s , r e l a t i v e t o v . F i n a l l y , i f n e i t h e r o f t h e s e t s Q ( v ) and Q ( v f ) i s conta.ined i n t h e o t h e r , t h e n v and v ' a r e c a l l e d independent. I n t h i s c o n c e p t u a l framework one c o u l d f o r m u l a t e v a r i o u s q u e s t i o n s i n o r g a n i z a t i o n t h e o r y , a n a t t e m p t t o answer them, For i n s t a n c e , imposing some c r i t e r i a o f o p t i m a l i t y , which t y p e s o f i n s t r u c t i o n s a r e most e f f e c t i v e ?
858
CHAPTER 6
What s h o u l d be t h e p r o p o r t i o n o f t a c t i c a l i n s t r u c t i o n s
t o a t t a i n a g i v e n g o a l , where a t a c t i c a l i n s t r u c t i o n i s such t h a t modifies t h e c l a s s o f g o a l s , but otherwise d o e s n o t r e s t r i c t t h e c l a s s o f means t o a t t a i n i t ? What s h o u l d b e t h e p r o p o r t i o n of t a c t i c a l i n s t r u c t i o n s i n an i n s t i t u t i o n w i t h given l e v e l o f complexity o f s t r u c t u r e? If we assume t h a t P c o n s i s t s , i n f a c t , o f s e v e r a l 2 o r g a n i z a t i o n s , a l l s u b j e c t t o P1, a n o t h e r c l a s s o f problems c o n c e r n s c o n c e p t s s u c h as t h e i r h i e r a c h i z a t i o n , e q u i v a l e n c e from t h e p o i n t o f view o f some g o a l s , e t c . One c o u l d c o n s i d e r here some o p e r a t i o n s on o r g a n i z a t i o n s , s u c h as j o i n i n g them, e t c .
F i n a l l y , s t i l l a n o t h e r p o s s i b l e i n t e r p r e t a t i o n o f cons i d e r a t i o n s o f t h i s s e c t i o n i s o b t a i n e d as f o l l o w s . The s e t of a d m i s s i b l e s t r i n g s o f i n s t r u c t i o n s forms a * c e r t a n i n language ( a s u b s e t o f t h e monoid D ) ; t h e " a l p h a b e t " of t h i s language a r e s i m p l y i n s t r u c t i o n s . "Words" a r e h e r e s t r i n g s of i n s t r u c t i o n s , and t o e a c h word t h e r e i s a s s i g n e d s e t o f t h e form Q ( v ) d e f i n e d b y (11.4), which may t h e r e f o r e be regarded as "meaning" o f v . These "meanings" have, i n t u r n , a l s o a l i n g u i s t i c s t r u c t u r e , b e i n g s e t s of s t r i n g s ( o f j o i n t a c t i o n s ) . I n t h i s sense, instructions define a c e r t a i n abstract l a n g u a g e , hence t h e y t h e m s e l v e s are e l e m e n t s o f metal a n g u a g e w i t h r e s p e c t t o l a n g u a g e s Q ( v ) . I n some s i m p l i f i c a t i o n one may s a y t h a t t h e language o f i n s t r u c t i o n s i s a meta-language w i t h r e s p e c t t o a c t i o n l a n guages which a r e c o n s i s t e n t w i t h t h e s e i n s t r u c t i o n s . A l t e r n a t i v e l y , l a n g u a g e s Q ( v ) c o n s t t i t u t e t h e semant i c s for t h e language o f i n s t r u c t i o n s .
FORMAL THEORY OF ACTIONS
859
1 2 . ETHICAL VALUATIONS
I n t h i s s e c t i o n , we s h a l l c o n s i d e r t h e problem o f e t h i c a l v a l u a t i o n s o f outcomes o f a c t i o n s , and a l s o t h e a c t i o n s t h e m s e l v e s . T h i s , n a t u r a l l y , r e q u i r e s an e x t e n s i o n o f t h e f o r m a l s y s t e m , s o as t o i n c l u d e t h e n o t i o n o f e t h i c a l v a l u e s o f a c t i o n s a n d / o r outcomes. Incidentally, despite the f a c t that the considerations w i l l c o n c e r n e t h i c a l v a l u e s , t h e same a n a l y s i s may b e
a p p l i e d t o many o t h e r v a l u a t i o n s , e . g . a e s t h e t i c , e t c . E t h i c a l v a l u a t i o n s a r e one o f t h e major c o n s t r a i n t s on a c t i o n s l e a d i n g t o a g i v e n g o a l . These c o n s t r a i n t s a r e o f g r e a t i n t e r e s t , e s p e c i a l l y i n group d e c i s i o n s i n l a r g e s y s t e m s , under c o l l e c t i v e r e s p o n s i b i l i t y . Moreover, they are closely r e l a t e d t o t h e notions dealing w i t h a l l o c a t i o n s o f goods i n t h e s o c i e t y , t o be a n a l y s ed i n n e x t s e c t i o n . One o f t h e main p r o b l e m s , which w i l l be a n a l y s e d i n t h e present section, i s that o f a j o i n t o v e r a l l evaluation of a s t r i n g of actions, i f t h i s s t r i n g brings outcomes o f which some have p o s i t i v e , and some have negative e t h i c a l value. I n a b s t r a c t f o r m u l a t i o n , t h e problem i s as f o l l o w s . The a c t i v i t y ( o f a p e r s o n X , t o be e v a l u a t e d , s a y ) , b r i n g s a c e r t a i n number o f r e s u l t s ; n e g l e c t i n g f o r s i m p l i c i t y t h e t e m p o r a l a s p e c t s , suppose t h a t t h e e t h i c a l v a l u e s o f t h e s e outcomes a r e e l , e 2 , . , en
..
.
Such a f o r m u l a t i o n p r e s u p p o s e s t h a t e t h i c a l v a l u e s a r e r e p r e s e n t a b l e n u m e r i c a l l y ; we s h a l l assume, i n f a c t t h a t t h e s e v a l u e s r e p r e s e n t measurement on a r a t i o
CHAPTER 6
860
s c a l e ( f o r a d i s c u s s i o n o f measurement problems i n c o n n e c t i o n w i t h e t h i c a l v a l u a t i o n s , s e e Nowakowska 1 9 7 3 ) .
.
We assume t h a t some of t h e v a l u e s e l,.. ,en are p o s i t i -
ve and some a r e n e g a t i v e , Moreover, we assume t h a t t h e i n d i c e s correspond t o t h e o r d e r o f occurrences o f t h e r e s p e c t i v e e v e n t s , so t h a t t h e event valued e l i s t h e e a r l i e s t , and t h e e v e n t v a l u e d en i s t h e l a t e s t . If V d e n o t e s t h e s e t o f a l l outcomes which e n t e r t h e
e v a l u a t i o n , t h e n t h e o b j e c t is t o a s s i g n t o V a v a l u e , say E(V), representing the overall ethical valuation o f outcomes i n V . We may r e q u i r e h e r e t h e f o l l o w i n g postulates. POSTULATE 1. The v a l u e E ( V ) depends o n l y on t h e v a l u e s e ) f o r some funce l , . . . , e n’ s o t h a t E ( V ) = f ( e l , n tion f.
...,
POSTULATE 2 . The f u n c t i o n f i s monotone i n c r e a s i n g w i t h r e s p e c t t o each o f i t s arguments, i . e . f o r every i, yei,l,ei+l,. ,en whenever e . e ’ then f o r a l l el,. 1 iy we have
<
.
f ( e l y . . ,eiy..
..
. .en) < f ( e l y . . .
..
..,en).
POSTULATE 3. The f u n c t i o n f i s symmetric w i t h r e s p e c t
t o i t s arguments.
POSTULATE 4 . If e i , . . . , e * = 0 , t h e n m f(el,...,en,ei,...,e~)
= f(el,...,en).
The i n t e r p r e t a t i o n o f t h e s e p o s t u l a t e s i s as f o l l o w s .
FORMAL THEORY OF ACTIONS
86 1
The f i r s t p o s t u l a t e a s s e r t s t h a t i n t h e o v e r a l l e v a l u a t i o n , o n l y t h e e t h i c a l v a l u e s o f t h e outcomes i n t e r v e n e , and n o t any o t h e r a s p e c t s of t h e s e outcomes. I n o t h e r words, two s e t s o f outcomes, which have t h e same e t h i c a l v a l u a t i o n s o f i n d i v i d u a l outcomes, have t h e same o v e r a l l e v a l u a t i o n s , r e g a r d l e s s o f t h e spec i f i c c o n t e n t o f t h e e v e n t s t o be judged. The second p o s t u l a t e r e q u i r e s no comments; f o r t h e t h i r d (symmetry), o b s e r v e t h a t t h i s p o s t u l a t e r u l e s o u t t h e " o r d e r e f f e c t s " : a bad a c t i o n f o l l o w i n g a good one i s e v a l u a t e d i n t h e same way as a good a c t i o n f o l lowing a bad one. One c a n o b j e c t a g a i n s t t h i s p o s t u l a t e , and a t t h e end o f t h e s e c t i o n , we s h a l l show a way of analysing t h e order effects. F i n a l l y , t h e f o u r t h p o s t u l a t e a s s e r t s t h a t what d e t e r mines t h e e v a l u a t i o n i s t h e p a r t o f outcome s e t which c o n s i s t s o f n o n - n e u t r a l e t h i c a l l y outcomes. I n s h o r t , e t h i c a l l y n e u t r a l outcomes do n o t i n t e r v e n e i n t h e evaluation. N a t u r a l l y , t h e s e p o s t u l a t e s do n o t d e t e r m i n e t h e f u n c t i o n f , s o t h a t t h e r e may be many f u n c t i o n s , a l l o f t h e m m e e t i n g all f o u r p o s t u l a t e s . We s h a l l s u g g e s t below one s p e c i a l form o f s u c h a f u n c t i o n f , which has been used i n t h e model o f p r o s o c i a l b e h a v i o u r ( s e e Nowakowska 1978). We assume namely t h a t t h e r e e x i s t a c o n s t a n t k .( 0 and
i n c r e a s i n g Yunctions o f one v a r i a b l e , s a y such t h a t f ( el , . . . , e n )
gl(el+ =
...+e n )
g2(min e i )
g l and g 2 ,
if min e i 2
k
i f min ei 4 k .
862
CHAPTER 6
The f u n c t i o n s gl and g2 may be c a l l e d r e s p e c t i v e l y t h e " l i b e r a l " and " p u r i t a n i c " f u n c t i o n s . To s e e why s u c h a t e r m i n o l o g y i s a d e q u a t e , l e t us o b s e r v e t h a t t h e above e v a l u a t i o n p r o c e d u r e works as f o l l o w s . The number k r e p r e s e n t s a t h r e s h o l d o f some s o r t ; t h e f a c t t h a t k i s assumed n e g a t i v e , means t h a t t h i s t h r e s h o l d b e g i n s t o o p e r a t e o n l y when a n outcome i s j u d g e d n e g a t i v e l y i n t h e d e g r e e below k . NOW, as l o n g as t h e r e are no outcomes v a l u e d below t h e
t h r e s h o l d , s o t h a t min ei i s above k , t h e e v a l u a t i o n and as t h e argument t h e r e a p p e a r s uses t h e function g 1' t h e sum e l t t e n o f v a l u e s . T h i s means t h a t t h e o v e r a l l e v a l u a t i o n i s some s o r t of an a v e r a g e , i n which good and bad c u m u l a t e , p a r t i a l l y c a n c e l one a n o t h e r , e t c . On t h e o t h e r hand, i f some e assumes a v a l u e j below t h e t h r e s h o l d , t h e n f u n c t i o n g2 s t a r t s t o opera t e , w i t h t h e argument b e i n g s i m p l y t h e worst e t h i c a l l y outcome. I n o t h e r words, i n t h i s c a s e , o n l y t h e w o r s t outcome i s c o u n t e d , w h i l e p o s i t i v e outcomes are disregarded.
...
The above j o i n t e v a l u a t i o n may be i n a d e q u a t e i n some c a s e s , i n which e i t h e r t h e t e m p o r a l c h a r a c t e r i s t i c s , or a t l e a s t t h e o r d e r of e v e n t s , p l a y a n i m p o r t a n t r o l e . A c a s e i n p o i n t h e r e may be S t . A u g u s t i n e , whose e a r l y p e r i o d o f l i f e was n o t v e r y commendable, b u t l a t e r way of l i f e made up f o r t h e e a r l y p e r i o d . It may be argued t h a t i f t h e events i n h i s l i f e occurred i n t h e r e v e r s e o r d e r , he would not be h i g h l y e v a l u a t e d . To b u i l d a model which would a c c o u n t f o r changes o f e t h i c a l v a l u a t i o n s i n t i m e , one may assume t h a t t h e e t h i c a l v a l u e a t i o n s a r e s u b j e c t t o a decay, w i t h t h e
863
FORMAL THEORY OF ACTIONS
decay c o n s t a n t b e i n g ( p o s s i b l y ) dependent on t h e i n i t i a l values. Thus, i f t h e e v e n t s w i t h e v a l u a t i o n s e l , e 2 , . . . , e n occurred a t t h e times t l < t < t n ,t h e n t h e 2 e t h i c a l v a l u e , a t some time t ) t n , i s
< ...
n
where q i s some f u n c t i o n which i s i n c r e a s f n g on t h e n e g a t i v e a x i s and d e c r e a s i n g on t h e p o s i t i v e a x i s . T h i s means t h a t t h e decay r a t e becomes s m a l l e r f o r h i g h l y e t h i c a l o r h i g h l y u n e t h i c a l e v e n t s ( o r more p r e c i s e l y : a c t i o n s a s s o c i a t e w i t h t h e m ) . T h i s seems t o be j u s t i f i e d , e s p e c i a l l y i n t h e n e g a t i v e c a s e . Causing som e p a r t i c u l a r l y abominable e v e n t l e a d s t o condemnation, which i s v e r y slowly f o r g o t t e n or f o r g i v e n . Causing l e s s abominable e v e n t s i s f o r g o t t e n f a s t e r .
1 3 . AGGREGATION OF VALUATIONS
Let u s now c o n s i d e r a more g e n e r a l problem, of a g g r e g a t i o n o f v a l u e s , o b t a i n e d from some e v a l u a t i o n s w i t h respect t o different c r i t e r i a . We assume t h a t t h e e v a l u a t i o n s c o n c e r n a sequence o f e v e n t s caused by a p e r s o n ( t h u s , t h e e v a l u a t i o n s conc e r n i n d i r e c t l y t h i s p e r s o n ) . The e v e n t s , moreover, o c c u r i n a changing e n v i r o n m e n t , s o t h a t s l , s 2,'" a r e t h e s t a t e s o f t h e environment between t h e o c c u r r e n c e s o f c o n s e c u t i v e e v e n t s El, E2, ...
.
864
CHAPTER 6
We s h a l l , as mentioned a b o v e , c o n s i d e r s e v e r a l s y s t e m s o f e v a l u a t i o n s a t t h e same t i m e ( e . g . u t i l i t a r i a n , e t h i c a l , e s t h e t i c , e t c . ) . Each o f t h e s e e v a l u a t i o n
systems i s governed by i t s own s e t o f norms, which a l l o w u s t o a s s i g n e v a l u a t i o n s t o e v e n t s from t h e p o i n t of view o f i - t h e v a l u a t i o n s y s t e m . Moreover, e v a l u a t i o n s a r e done by j u d g e s ( a r b i t e r s ) , who e x p r e s s t h e i r e v a l u a t i o n s on some s c a l e s , s p e c i f i c f o r a g i v e n system. I n a d d i t i o n , t h e e v a l u a t i o n s may b e r e p e a t e d ( f o r t h e same e v e n t , or o b j e c t , and f o r t h e same p e r s o n . F o r m a l l y , we s h a l l i n d e x n = l,...,N r e p r e s e n t e v e n t s (or: e v a l u a t e d o b j e c t s ) ; k = l,...,K w i l l r e p r e s e n t t h e c r i t e r i o n w i t h r e s p e c t t o which t h e e v a l u a t i o n i s made; n e x t , p = 1, ,P w i l l s t a n d for p e r s o n , and f i n a l l y , i = 1, ,r w i l l s t a n d f o r o c c a s i o n when person p evaluates o b j e c t n with respect t o k-th c r i terion.
...
...
e v a l u a t i o n by p o f n - t h obj e c t w i t h r e s e p e c t t o k - t h c r i t e p i o n , t h e n t h e system o f e v a l u a t i o n s may be r e p r e s e n t e d as
If' e n k p i s t a n d s f o r
Ie n k p i '
n=1,
...,N;
i-th
k=1,
...,K;
p=1,
...,P ,
i=l,2,..
I t i s assumed t h a t e a c h enkpi i s e x p r e s s e d on t h e s c a -
l e ( l i n g u i s t i c or n o t ) a p p r o p r i a t e for k - t h c r i t e r i o n , s o t h a t enkpi 6 'k> where Vk i s t h e " v o c a b u l a r y " o f
e v a l u a t i o n s on k - t h c r i t e r i o n ( i n a s p e c i a l c a s e , Vk may b e t h e n u m e r i c a l a x i s ) .
865
FORMAL THEORY OF ACTIONS
N e x t , w e assume t h a t w i t h e a c h v Vk t h e r e i s a s s o c i a t e d a f u z z y s u b s e t of t h e s c a l e Rk on t h e k - t h c r i t e r i o n . We s h a l l d e n o t e by m v ( x ) , x Rk t h e membership function of t h e subset o f R corresponding t o t h e k evaluation v. We may now d i s t i n g u i s h s e v e r a l p r o b l e m s of a g g r e g a t i o n of e v a l u a t i o n s . Problem 1. A g g r e g a t i o n w i t h i n p e r s o n , o b j e c t and c r i t e r i o n . T h i s means a g g r e g a t i o n of e v a l u a t i o n s e n k p l ' e nkp2"
' '
' en k p r
of n - t h o b j e c t w i t h r e s p e c t t o k - t h c r i t e r i o n b y p e r s o n p . Here t h e p r o b l e m i s t o a r r i v e a t some " o v e r a l l 1 ' e v a l u a t i o n C h a r a c t e r i s t i c f o r a g i v e n p e r s o n , w h ic h "aqerages o u t " h i s i n d i v i d u a l f l u c t u a t i o n s o f o p i n i o n , d e p e n d i n g on v a r i o u s e x t r a n e o u s f a c t o r s . A t y p i c a l c a s e m i g h t be h e r e as f o l l o w s . S u p p o s e t h a t
a p e r s o n e x p r e s s e s h i s o p i n i o n s a b o u t some p a r t i c u l a r f u t u r e e v e n t , an d h i s o p i n i o n s i n v o l v e s u b j e c t i v e c e r t a i n t y as t o i t s o c c u r r e n c e . On v a r i o u s o c c a s i o n s , t h e p e r s o n was s e v e r a l times " a b s o l u t e l y c e r t a i n " t h a t t h e e v e n t w i l l o c c u r , w h i l e on few o c c a s i o n s he was " q u i t e c e r t a i n " , an d on s t i l l some o t h e r - he had some d o u b t s . The q u e s t i o n t h e n l i e s i n a s s e s s i n g h i s d e g r e e o f c e r t a i n t y , g i v e n t h e d a t a o n numbers o f t i m e s he used p a r t i c u l a r d e s c r i p t o r s . G e n e r a l l y , t h e problem here i s as f o l l o w s . L e t v l , v2' be t h e d e s c r i p t o r s u s e d , and l e t mv , m be t h e membership f u n c t i o n s o f t h e correspdndi:i fuzzy
...
,...
866
CHAPTER 6
s e t s on t h e continuum c o r r e s p o n d i n g t o k - t h c r i t e r i o n ( e . g . of s u b j e c t i v e p r o b a b i l i t y ) . Then t h e a g g r e g a t e d o p i n i o n , s a y e nkp. ' i s r e p r e s e n t a b l e as a f u z z y s e t a c c o r d i n g t o some b u i l t o u t o f s e t s mv l ' mv* a g g r e g a t i o n r u l e F ( m , m ,... ) . Here F i s some funcv1 t i o n , which a s s i g n s a f u z z 3 s e t on t h e continuum R k
,...
t o fuzzy s e t s c o r r e s p o n d i n g t o v l , v 2 ,
... .
The c h o i c e o f f u n c t i o n F depends s t r o n g l y on t h e s c a l e corresponding t o t h e c r i t e r i o n i n question. Several p o s s i b l e f u n c t i o n s have been d i s c u s s e d i n S e c t i o n 2 of t h i s c h a p t e r , where t h e f u n c t i o s F were d e f i n e d t h r o u g h o p e r a t o r s d , some s u i t a b l e f o r a g g r e g a t i n g p o s s i b i l i t y m e a s u r e s , some f o r b e l i e f m e a s u r e s , e t c . Suppose now t h a t t h e problem o f a g g r e g a t i o n as d e f i n e d above was s o l v e d , and l e t e be t h e a g g r e g a t e d o p i nkp n i o n o f person p o f n-th o b j e c t w i t h r e s p e c t t o k-th c r i t e r i o n ( o f c o u r s e , i n some c a s e s we may have j u s t one o p i n i o n , s a y e i n which c a s e we s h a l l have nkpl' the condition e nkpl = enkp.)'
-
Problem 2 . A g g r e g a t i o n a c r o s s p e r s o n s , f o r t h e same o b j e c t and c r i t e r i o n . Here t h e s i t u a t i o n i s as f o l l o w s . Given t h e v a l u e s
one has t o a r r i v e a t a "summary" e v a l u a t i o n o f n - t h object w i t h respect t o t h e k-th c r i t e r i o n . A t y p i c a l c a s e h e r e may be j u d g i n g t h e i c e - s k a t i n g c o m p e t i t i o n by a p a n e l o f j u d g e s , who e v a l u a t e e a c h c o n t e s t a n t w i t h r e s p c t t o some s p e c i a l c r i t e r i o n ( e . g . t e c h n i c a l perfection, esthetic values, e t c . ) .
867
FORMAL THEORY OF ACTIONS
Another t y p i c a l c a s e might be t h e j u r y d e c i s i o n , where t h e i n d i v i d u a l d e c i s i o n s of t h e j u r o r s a r e t o be aggregated i n t o t h e j u r y decision. Here a g a i n , t h e a g g r e g a t i o n f u n c t i o n depends on t h e
t y p e o f q u e s t i . o n . I n c a s e of j u d g i n g c o m p e t i t i o n s , t h e e v a l u a t i o n s enkp. a r e t y p i c a l l y some numbers, and t h e a g g r e g a t i o n f u n c t i o n i s some s o r t o f a v e r a g e , o f t e n w i t h r e j e c t i o n o f extreme v a l u e s , and sometimes with weighting Thus, i n c a s e o f n u m e r i c a l s c o r e s , one may t a k e as t h e a g g r e g a t e d v a l u e
.
P
P
enkp.
-
min e P
nkp
-1
( i f t h e two e x t r e m a l v a l u e s a r e d i s r e g a r d e d ) . I n c a s e
o f j u r y d e c i s i o n , we have two v a l u e s on t h e s c a l e o f g u i l t , namely " g u i l t y " and "not g u i l t y " , and t h e a g g r e g a t i o n f u n c t i o n has t h e form { g u i l t y , i f enkp. = g u i l t y f o r a l l p enk..
not g u i l t y ,
-
= not g u i l t y f o r if e nkz, a t l e a s t one p .
I n c a s e when t h e v a l u e s e n k p . a r e l i n g u i s t i c , one may use a g a i n s e v e r a l methods o f a g g r e g a t i o n , i n p a r t i c u l a r t h o s e based o f d o p e r a t o r s d i s c u s s e d i n S e c t i o n 2 , i n c l u s i n g i n a d d i t i o n t h e importance (weight) of a n o p i n i o n , and moreover, a s s i g n i n g t o t h e r e s u l t a q u a n t i f i e r , s u c h as t h o s e d i s c u s s e d i n t h e p r e c e d i n g sections (with the difference that the quantifiers w i l l now concern n o t t h e f r e q u e n c i e s o f moments of time when some e v e n t s o c c u r , b u t f r e q u e n c i e s o f j u d g e s
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w i t h a given o p i n i o n ) . L e t u s for s i m p l i c i t y omit t h e i n d i c e s r e f e r r i n g t o t h e o b j e c t , c r i t e r i o n and o c c a s i o n s , and w r i t e s i m p l y With e a c h e one may a s s o c i a t e a f u z z y e for e n k p . P P B s e t w i t h membership f u n c t i o n m n , and t h e i m p o r t a n c e B B say w The v a l u e s m ( . ) r e f e r t o t h e of o p i n i o n e P' P P membership o f some v a r i a b l e ( o b j e c t o f i n t e r e s t ) i n
.
.
some f i x e d s e t B . With e a c h judgment e
P
and B one may t h e n a s s o c i a t e t h e
value
t o be c a l l e d i m p o r t a n c e - t r u n c a t e d r e p r e s e n t a t i o n o f opinion of person p. I n t h i s w a y , we d e f i n e d a membership f u n c t i o n o f some fuzzy set of p e r s o n s p .
We may now choose a f u n c t i o n
Q.,
as d e f i n e d i n S e c t i o n
2 , and d e t e r m i n e d - e x t e n s i o n o f t h e s e t of v a l u e s G ( e ) , i . e . put B P
dB =
dB = G (e ), 1 B 1
I: dl, G ~ ( ~ ~ ) I , . . .
up t o t h e v a l u e c(:, which g i v e s t h e d - e x t e n s i o n o f
,..., P.
t h e sequence o f v a l u e s G ( e ) , p = 1 B P
On t h e o t h e r hand, t a k i n g t h e a v e r a g e G (e ) t B 1
:'w
t
... -t GB(ep) = N
...
B
wP
869
FORMAL THEORY OF ACTIONS
w e may a s s o c i a t e w i t h i t a f r e q u e n c y q u a n t i f i e r , s a y Q ( e . g . a l m o s t a11 p e r s o n s , e t c . ) . The a g g r e g a t e d o p i n i o n of p e r s o n s i s now d : ,
giving
t h e & - b i l i t y r e p r e s e n t a t i o n o f t h e o p i n i o n s about t h e r e l a t i o n s between t h e v a r i a b l e o f i n t e r e s t and s e t B . The q u a n t i f i e r a s s o c i a t e d w i t h i t i s Q , s o t h a t t h e a g g r e g a t e d o p i n i o n i s Q dBp . Let us o b s e r v e t h a t t h i s approach g e n e r a l i z e s t h a t o f Yager ( 1 9 8 3 a , b ) , who i n t r o d u c e d t h e n o t i o n s o f concens o r y and c o m p e t i t i v e a g g r e g a t i o n o f o p i n i o n s , t a k i n g some s p e c i a l f u n c t i o n s i n p l a c e o f d . The r e m a i n i n g two problems of a g g r e g a t i o n r e c e i v e d t h u s f a r much l e s s a t t e n t i o n t h a n t h e f i r s t two. To formul a t e t h e s e p r o b l e m s , assume t h a t Problems 1 and 2 have been s o l v e d , i . e . we can o b t a i n doubly a g g r e g a t e d evaluations e which we s h a l l f o r s i m p l i c i t y denonk. * ' t e by enk. Problem 3. Aggregation o f e v a l u a t i o n s e n l ,
en2,
. . . 'enK
of t h e ( a g g r e g a t e d ) e v a l u a t i o n s o f t h e n-th o b j e c t w i t h respect t o a l l c r i t e r i a . A t y p i c a l case here i s
known as m u l t i c r i t e r i a l d e c i s i o n problem: one h a s t o make a c h o i c e among s e v e r a l o b j e c t s , e a c h o f them eval u a t e d on a number o f c r i t e r i a . Here a g a i n , t h e r e i s no u n i v e r s a l l y a c c e p t a b l e s o l u t i o n . Some o f t h e proposed methods i n v o l v e :
-- w e i g h t i n g the c r i t e r i a , and d e f i n i n g t h e j o i n t " s c o r e " as a w e i g h t e d a v e r a g e o f e v a l u a t i o n s ;
--
r a n k i n g t h e c r i t e r i a , and d e f i n i n g t h e j o i n t r a n k i n g
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through lexicographical ordering;
--
s e l e c t i n g some o f t h e c r i t e r i a as c o n s t r a i n t s , and
r e d u c i n g t h e problem t o t h e c o n s t r a i n e d o p t i m i z a t i o n problem.
The main i s s u e h e r e c o n c e r n s c o m p a r a b i l i t y of e v a l u a t i o n s on v a r i o u s c r i t e r i a . I n d e e d , v a r i o u s e need n o t be nk comparable, and b e f o r e any a v e r a g i n g ( w e i g h t e d or n o t ) i s a t t e m p t e d , t h e y have t o be r e d u c e d t o a common s c a l e . T h i s i s o f t e n a d i f f i c u l t , i f not impossible t a s k . To g i v e an example, suppose t h a t t h e o b j e c t s e v a l u a t e d a r e two methods of t r e a t m e n t i n a c e r t a i n d i s e a s e . Imagine t h a t t h e c r i t e r i a are: reduction of s u f f e r i n g , and change of l i f e e x p e c t a t i o n . Suppose t h a t o b j e c t 1 (method 1) r e d u c e s s u f f e r i n g , s a y by 50 %, as measured on some sczile, b u t a l s o d e c r e a s e s t h e l i f e e x p e c t a t i o n by 5 y e a r s , s a y . O b j e c t 2 ( t h e second method) d o e s n o t reduce s u f f e r i n g , but i n c r e a s e s l i f e e x p e c t a t i o n , s a y by 1 0 y e a r s . There does n o t seem t o e x i s t any meaningf u l method o f " w e i g h t i n g " t h e s u f f e r i n g and l i f e e x p e c t a t i o n , and a r r i v e a t a n i n d e x which would a l l o w t o d e c l a r e one method as s u p e r i o r t o a n o t h e r . I n s c i e n c e , t h e problem o f e v a l u a t i o n s a c c o r d i n g t o m u l t i p l e c r i t e r i a o c c u r s q u i t e o f t e n , and f i n d i n g an o v e r a l l e v a l u a t i o n i s of g r e a t importance. A s an example when one e v a l u a t e s a n o b j e c t a c c o r d i n g t o
m u l t i p l e c r i t e r i a i n s c i e n c e one may t a k e t h e p r o c e s s of r e f e r e e i n g . C r i t e r i a a r e sometimes s p e c i f i e d i n t h e " r e f e r e e q u e s t i o n n a i r e " s e n t by t h e j o u r n a l , e . g . o r i g i n a l i t y , c o r r e c t n e s s , completeness of b i b l i o g r a p h i c a l r e f e r e n c e s , s t y l e , c o n t e n t of a b s t r a c t , e t c . C l e a r l y ,
871
FORMAL THEORY OF ACTIONS
t h e s e c r i t e r i a a r e of v a r i o u s i m p o r t a n c e : i f some a r e n o t m e t , t h e p a p e r i s r e j e c t e d , w h i l e some o t h e r s may be n o t met even by an a c c e p t e d p a p e r . Problem 4 . A g g r e g a t i o n w i t h r e s p e c t t o t h e o b j e c t s ( a n d p o s s i b l y a l s o c r i t e r i a ) . T h i s means ( f o r a f i x e d c r i t e r i o n k ) , a g g r e g a t i n g t h e e v a l u a t i o n s e l k , ... 'eNkm
A t y p i c a l c a s e h e r e might b e a n e v a l u a t i o n o f a r e s t a u -
r a n t ( a s s i g n i n g t o i t a c e r t a i n number of s t a r s ) . The o b j e c t s ( e v e n t s ) a r e d i s h e s sampled on v a r i o u s o c c a s i o n s b y one o r more e x p e r t s . The judgments may concern t h e q u a l i t y o f food ( a g i v e n c r i t e r i o n ) , as w e l l as p r i c e , s e t t i n g , q u a l i t y o f s e r v i c e , e t c . Here t h e probl e m l i e s i n a s s i g n i n g an aggregated value t o t h e resta u r a n t , t h e v a l u e b e i n g e x p r e s s e d b y t h e number o f stars. It seems t h a t t h i s t y p e of a g g r e g a t i o n of v a l u e s i s
t h e most d i f f i c u l t t o f o r m a l i z e , and t h a t i t i s a l w a y s done on a s u b j e c t i v e b a s i s . Each of t h e Problems 1 - 4 may be f o r m u l a t e d g e n e r a l l y as f o l l o w s . The e v a l u a t i o n s t o be a g g r e g a t e d are r e p r e s e n t a b l e as some f u z z y s e t s ( p o s s i b l y r e s u l t i n g a l r e a d y from p r e v i o u s a g g r e g a t i o n s ) , w i t h membership f u n c t i o n s Aggregation means r e p r e s e n t i n g t h e s e f u z z y ml, m2,...
.
s e t s as a f u z z y s e t w i t h membership f u n c t i o n
T h i s g e n e r a l s e t u p has t o be s p e c i f i e d d i f f e r e n t l y i n
c a s e o f Problems 1 and 2 ( o f a g g r e g a t i o n s a c r o s s t h e
872
CHAPTER 6
o c c a s i o n s and p e r s o n s ) , t h a n i n t h e c a s e of Problems 3 and 4 ( o f a g g r e g a t i o n s a c r o s s c r i t e r i a and o b j e c t s ) . The p o i n t i s t h a t i n problems 1 and 2 t h e membership f u n c t i o n s a r e a l l d e f i n e d o v e r t h e same s p a c e ( c o r r e s ponding t o a g i v e n c r i t e r i o n ) . C l e a r l y , i t i s n o t s e n s i b l e t o a t t e m p t t o s p e c i f y one method as " t h e " a g g r e g a t i o n method; t h e c h o i c e must depend on t h e s p e c i f i c f e a t u r e s of t h e a g g r e g a t i o n problem d i s c u s s e d . I n c a s e o f Problems 3 and 4 (of a g g r e g a t i o n a c r o s s c r i t e r i a and o b j e c t s ) t h e main i s s u e l i e s i n d e f i n i n g t h e s c a l e on which t h e f i n a l judgment i s t o be made, s i n c e t h e i n d i v i d u a l e v a l u a t i o n s a r e on d i f f e r e n t s c a l e s and a r e , as a r u l e , n o t comparable. Even i f we have c o m p a r a b i l i t y ( e . g . i n j u d g i n g a s e r i e s of e v e n t s w i t h r e s p e c t t o some a t t r i b u t e ) , t h e f u n c t i o n F ( a g g r e g a t i o n ) may sometimes depend on t h e o r d e r ( i . e . F need n o t be symmetric w i t h r e s p e c t t o i t s arguments). I n g e n e r a l , one may d i s t i n g u i s h h e r e two c l a s s e s of such e v a l u a t i o n s , namely l e n i e n t and s e v e r e . To d e f i n e t h e s e c o n c e p t s , suppose t h a t we have a sequ-
... .
We may t h e n a r r i v e ence of e v a l u a t i o n s , e l , e 2 , a t p a r t i a l e v a l u a t i o n s , of t h e f i r s t m t e r m s o f t h e sequence, s o t h a t
Thus, v ( ~ i) s t h e a g g r e g a t e d e v a l u a t i o n of t h e s e t of t h e f i r s t m e v e n t s i n t h e sequence. We may t h e n s a y t h a t t h e system o f a g g r e g a t i o n i s l e n i e n t , i f v ( ~ may ) b o t h i n c r e a s e and d e c r e a s e w i t h m;
FORMAL THEORY OF ACTIONS
873
i t i s c a l l e d s e v e r e , if v ( ~ may ) o n l y d e c r e a s e , once i t becomes lower t h a n a c e r t a i n t h r e s h o l d ( i . e . i t may f l u c t u a t e o n l y above t h e t h r e s h o l d ) . (m) Thus, t h e system i s l e n i e n t , i f w h a t e v e r t h e v a l u e v , t h e r e e x i s t s a p o s s i b l e c o n t i n u a t i o n of t h e sequence beyond t h e t e r m em s u c h t h a t v ( ~ c) v ( m + r ) f o r some r , (m) (m+t) and a l s o a p o s s i b l e c o n t i n u a t i o n s u c h t h a t v v f o r some t . On t h e o t h e r hand, t h e system i s s e v e r e , i f t h e r e e x i s t s a c o n s t a n t D such t h a t whenever v ( ~ <) D , t h e n v ( m t k ) < D f o r a l l k , r e g a r d l e s s of t h e c o n t i n u a t i o n of t h e s q u e n c e . The e t h i c a l v a l u a t i o n s c o n s i d e r e d a t t h e b e g i n n i n g of t h i s s e c t i o n may s e r v e as a n example of a s e v e r e e v a l u a t i o n system. I n c o n n e c t i o n w i t h t h e s e n o t i o n s , one may mention h e r e a n o t h e r problem r e l a t e d t o v a l u a t i o n s , and d e s c r i p t i o n s i n general. Imagine two d e s c r i p t i o n s , A and B , of an o b j e c t ( s i t u a t i o n ) S. F o r m a l l y , e a c h of A , B and S may be r e p r e s e n t e d as a s e t o f p r o p o s i t i o n s ( a s s e r t i o n s about some p r o p e r t i e s ) . * * * L e t A , B and S d e n o t e t h e c l o s u r e s of t h e s e s e t s under t h e r e l a t i o n of l o g i c a l consequence. We may now i n t r o d u c e t h e f o l l o w i n g d e f i n i t i o n :
DEFINITION. The d e s c r i p t i o n A i s more f a i t h f u l t h a n the description A * o (S*)' C B *
€3,
n
if A (S*)'.
* 0 S* 3 B * n S *
and a l s o We w r i t e t h e n A .> B.
The r e l a t i o n * > i s c l e a r l y t r a n s i t i v e , b u t i n g e n e r a l
874
CHAPTER 6
i t need n o t be c o n n e c t e d , s o t h a t two d e s c r i p t i o n s may n o t be comparable. We have h e r e t h e f o l l o w i n g t h e orem THEOREM.
If A * ,> B *
f o r every B
* , then A *
*.
= S
For t h e p r o o f suppose f i r s t t h a t A * $ S*. We t h e n show
*
*.
t h a t one can f i n d B which i s b e t t e r t h a n A Indeed, * * * * * * * l e t p E S - A , and l e t B = ( A + p ) Then A $c B * * * and B c S , s i n c e p k S Consequently, A S $
B
*
0
* , while A*
S
*.
fi
(Sf)'
= B
*
*n
( S )'
.*
= @,
which
*
*
means t h a t B" .> A " .
*
*
Suppose now t h a t A $ S , and l e t q e A - S so that * * * * 2 q (S")'. Define B = ( A - q ) Then B C A , hence * * * c * * * * B 0 (S*)'$ A 0 (S ) , while A A S = B n S , * * which a g a i n means t h a t B .> A
.
.
*
*
'*.
In general, i f A # S t h e n t h e symmetric d i f f e r e n c e * * * * * A Y S = (A - S ) U (S - A ) i s n o t empty, and one I
Y
can combine t h e above two c a s e s t o c o n s t r u c t B" which * i s b e t t e r than A according t o the r e l a t i o n This completes t h e p r o o f .
>.
Thus, t h e theorem a s s e r t s t h a t t h e u n i v e r s a l l y b e s t A must simply c o i n c i d e w i t h S
*
.
S i m i l a r l y , we have t h e p r o p e r t y t h a t whenever B
*
*
*
* c
*
+> A
*
f o r every B , t h e n A = ( S ) , s o t h a t t h e worst desc r i p t i o n must c o n s i s t o n l y o f s e n t e n c e s which a r e i n t h e * complement o f S
.
It i s also w o r t h w h i l e t o mention t h e f o l l o w i n g i n t e r -
p r e t a t i o n o f t h e a g g r e g a t i o n problem. Suppose t h a t we e v a l u a t e N e v e n t s a c c o r d i n g t o K c r i t e r i a , and t h a t
875
FORMAL THEORY OF ACTIONS
t h e s e e v a l u a t i o n s ( d i s r e g a r d i n g t h e p e r s o n s who p e r f o r m e v a l u a t i o n s and t h e o c c a s i o n s ) a l l o w u s t o o r d e r t h e e v e n t s w i t h r e s p e c t t o e a c h c r i t e r i o n . L e t a
d i f f e r e n c e are def ined t h e n i n t h e uaual way. The problem o f a g g r e g a t i o n may now be s t a t e d a s t h e problem of f i n d i n g a common o r d e r i n g of a l l e v e n t s ,
<
which would, i n some s e n s e , r e p r e s e n t a l l t h e i n d i v i d u a l orderings according t o p a r t i c u l a r c r i t e r i a . Depending on t h e r u l e which i s r e q u i r e d t o connect t h e relations
<,
and t h e o v e r a l l r e l a t i o n
< , one
may o r
may n o t f a c e t h e Arrow I m p o s s i b i l i t y Theorem.
The s e t o f c o n d i t i o n s which a r e known t o be j o i n t l y incompatible a r e l a c k of d i c t a t o r , s o v e r e i g n t y , independence o f i r r e l e v a n t a l t e r n a t i v e s , and c o n s i s t e n c y o f i n d i v i d u a l and o v e r a l o r d e r i n g s . One c o u l d , however, f o r m u l a t e t h e problem i n a somewhat d i f f e r e n t w a y , a v o i d i n g t h e Arrow I m p o s s i b i l i t y Theorem. F o r e a c h two e v e n t s a , b and e v e r y c r i t e r i o n k , t h e r e l a t i o n a ( b e i t h e r h o l d s o r d o e s n o t h o l d . We may k t h e r e f o r e t h i n k o f r e l a t i o n <‘k as a f u n c t i o n R which
k
maps t h e s e t E >( E ( w i t h E b e i n g t h e s e t o f a l l e v e n t s under c o n s i d e r a t i o n ) , i n t o t h e s e t {O ,1]. T h i s f u n c t i o n a s s i g n s t h e v a l u e 1 t o t h e p a i r ( a , b ) of e v e n t s i f a k b , and t h e v a l u e 0 o t h e r w i s e .
<
A n a t u r a l g e n e r a l i z a t i o n c o n s i s t s o f u s i n g i n s t e a d of
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t h e s e t 10,l) some o t h e r a p p r o p r i a t e s e t , e . g . i n t e r v a l [ O , l l -- which w i l l g i v e Zadeh's f u z z i f i c a t i o n , o r any other p a r t i a l l y ordered s e t . Thus, i n c a s e o f mappings i n t o [O,l] Rk:
E x E --3
[O,l],
we have f u n c t i o n s
,..., K,
k = 1
where R ( a , b ) means t h a t a p r e c e d e s b i n d e g r e e x , i n k
k-th c r i t e r i o n . N a t u r a l l y , t h e f u n c t i o n Rk must s a t i s f y t h e c o n d i t i o n s which c o r r e s p o n d t o t r a n s i t i v i t y and c o n n e c t e d n e s s . Thus, one may r e q u i r e a v e r s i o n o f s t r o n g s t o c h a s t i c t r a n s i t i v i t y , which i n t h i s c a s e i s If R k ( a , b )
23
and R k ( b , c )
> 3,
then
For c o n n e c t e d n e s s , one may r e q u i r e Rk(a,b)
+
Rk(b,a)
The o r d e r i n g r e l a t i o n
DEFINITION.
a
W
= 1.
( k may now be d e f i n e d a s f o l l o w s :
We s a y t h a t a C k b , if R k ( a , b )
b i f R (a,b) = k k
>
$, and
4.
We can t h e n p r o v e THEOREM.
The r e l a t i o n N k i s a n e q u i v a l e n c e , w h i l e
t h e r e l a t i o n 7 k i s t r a n s i t i v e and c o n n e c t e d .
877
FORMAL THEORY OF ACTIONS
I n d e e d , we have R ( a , a ) = i , which shows t h a t wk i s k r e f l e x i v e . S i m i l a r l y , i f R ( a , b ) = i, t h e n a l s o k
R ( b , a ) = $, which shows s y m m e t r y o f rk.To p r o v e k
t r a n s i t i v i t y , assume t h a t R ( a , b ) = R ( b , c ) = 2. It k k f o l l o w s t h e n t h a t R k ( a , c ) 2 i . On t h e o t h e r hand, we have a l s o R ( c , b ) = R ( b , a ) = $, which i m p l i e s t h a t k
R (c,a) 3 k
3.
k This implies t h a t R ( a , c ) = k
'r, whiqh
means t h a t t h e r e l a t i o n d k i s t r a n s i t i v e . The a s s e r t e d p r o p e r t i e s o f t h e r e l a t i o n C
k
are proved
i s s i m i l a r way.
Thus, we d e f i n e d a nonfuzzy r e l a t i o n
C_
k
on t h e s e t
of a l l events. Next, we may s a y t h a t t h e r e l a t i o n s R k and R a r e conj c o r d a n t , i f for some q > 3 t h e c o n d i t i o n R k ( a , b ) > q h o l d s i f and o n l y i f R . ( a , b ) 7 q . Here q i s a measure J o f t h e " s t r e n g t h " o f t h e concordance r e l a t i o n .
For t h e a g g r e g a t i o n p r o c e s s , we may r e q u i r e t h e f o l l o w ing postulate. POSTULATE. If f o r a l l k , j t h e r e l a t i o n s R k and R
j
are
c o n c o r d a n t , t h e n t h e o v e r a l l r e l a t i o n R must be concordant w i t h every Rk. The c o n s t a n t q i n t h e l a t t e r concordance i s t h e maximum of t h e c o r r e s p o n d i n g c o n s t a n t s o f concordances b e t ween t h e p a i r s o f r e l a t i o n s R k ,
Rj.
This p o s t u l a t e i s a n a t u r a l requirement, corresponding i n some s e n s e t o t h e p o s t u l a t e o f s o v e r e i g n t y o f Arrow. L e t us f i n a l l y c o n s i d e r t h e s i t u a t i o n of e v a l u a t i o n s by a group o f p e r s o n s i n c a s e when n o t a l l c r i t e r i a a r e
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a c c e p t e d by a l l members o f t h e g r o u p . With r e g a r d s t o a f i x e d p e r s o n p , h i s s t a t e w i t h resp e c t t o t h e a c c e p t a n c e o r r e j e c t i o n o f c r i t e r i a may be d e s c r i b e d by a v e c t o r
where r
i s o o r 1 depending whether person p r e j e c t s kp o r a c c e p t s t h e k-th c r i t e r i o n .
T h u s , t h e g r o u p o f p e r s o n s w i l l b e d e s c r i b e d by t h e matrix
r
9
=
r
21 K 1
r r
12
...
22
K2
...
w h e r e e a c h row c o r r e s p o n d s t o a c r i t e r i o n , a n d e a c h column -- t o a p e r s o n . N a t u r a l l y , when o n e s p e a k s o f a c c e p t a n c e o r r e j e c t i o n of a c r i t e r i o n , one e x p e c t s some d e g r e e o f " s i m i l a r i t y " b e t w e e n members o f t h e g r o u p ( t h i s d o e s n o t mean a g r e e m e n t when t h e s e c r i t e r i a
are a p p l i e d t o any p a r t i c u l a r o b j e c t ) . T h i s c o n d i t i o n means t h a t columns o f t h e m a t r i x Q o u g h t t o h e s i m i l a r i n some s u f f i c i e n t l y h i g h d e g r e e . One o f t h e p o s s i b l e i n d i c e s o f s i m i l a r i t y may h e des c r i b e d as f o l l o w s . L e t (
,
d u c t o f two v e c t o r s , s o t h a t
) d e n o t e t h e s c a l a r pro-
819
FORMAL THEORY OF ACTIONS
...,
Moreover, l e t 1 d e n o t e t h e v e c t o r (l,l, 1). I f now z and z a r e two column v e c t o r s of m a t r i x Q, t h e n i j we may p u t
which i s simply e q u a l t o t h e number o f matching comp o n e n t s i n v e c t o r s z and z i
5
.
A s a c o n s i s t e n c y measure one may now t a k e t h e v a l u e
F o r example, for t h e m a t r i x
C(Q)
=
(3+1+2+3+4+2t1+2+3+1+0+1+3+2t3)
~
=
31
4.6-5 ( t h e m a x i m a l v a l u e o f t h e i n d e x i s 1, when a l l columns
are i d e n t i c a l ) . With r e s p e c t t o a g i v e n c r i t e r i o n , t h e s t a t e o f t h e
group i s d e s c r i b e d by t h e row v e c t o r o f m a t r i x Q. We
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m a y i n t r o d u c e a p r o c e s s o f change o f t h i s v e c t o r . One
can namely e x p e c t t h a t t h e group w i l l atteinpt t o a d j u s t e i t h e r t h e c h o i c e o f c r i t e r i a , o r even i t c o m p o s i t i o n ( b y a c c e p t i n g new members o r r e j e c t i n g old o n e s ) , s o
as t o have C(Q) a s c l o s e t o 1 as p o s s i b l e . F o r f o r m a l d e s c r i p t i o n , we s h a l l d e f i n e a Markov c h a i n Xt, t = 0,1, , w i t h s t a t e s 0,1, P , where X = r t means t h a t a t t i m e t , t h e v e c t o r c o r r e s p o n d i n g t o a g i v e n f i x e d row ( s a y k - t h ) has r c o o r d i n a t e s e q u a l
...
...,
1 and K-r c o o r d i n a t e s e q u a l 0 .
The c h o i c e o f t r a n s i t i o n p r o b a b i l i t y w i l l be based on t h e f o l l o w i n g i n t u i t i o n . I n c a s e when t h e s t a t e i s 0 o r P , t h e whole group r e j e c t s o r t h e whole group a c c e p t s t h e k - t h c r i t e r i o n . Then t h e r e i s a ( s m a l l ) chance t h a t one p e r s o n w i l l change h i s mind, s o t h a t
P0 , l =
pP ,P-1
= b,
and o f c o u r s e , P o o = 1-a, P p p = 1-b. For 1
<' j <
P-1,
w e assume t h e mechanism of " a t t r a c t i o n
( P - 1 ) / 2 , i . e . l e s s perby t h e m a j o r i t y " . Thus, i f j sons accept t h e c r i t e r i o n than r e j e c t i t , t h e r e i s more chance t h a t someone who a c c e p t s t h e c r i t e r i o n w i l l change h i s mind, t h a n f o r someone who r e j e c t t h e
c r i t e r i o n t o change h i s mind. Thus, p a s s a g e t o s t a t e j-1 i s more l i k e l y t h a n a p a s s a g e t o t h e s t a t e j t l , and we may assume t h a t f o r j = 1 , 2 , . . P j ,j-1
.,(P-l)/2
we have
'
'j ,j+1'
w i t h t h e d i r e c t i o n o f i n e q u a l i t y r e v e r s e d for j l a r g e r
FORMAL THEORY OF ACTIONS
881
t h a n (P-1)/2, By assuming some s p e c i f i c form o f t h e s e t r a n s i t i o n p r o b a b i l i t i e s , one may d e t e r m i n e t h e e r g o d i c p r o b a b i l i uP ' T h i s d i s t r i b u t i o n w i l l t h e n t i e s , s a y u0 , ul, be U-shaped, i.n view o f t h e a s s u m p t i o n o f t h e f o r c e which t e n d s t o p o l a r i z e t h e group and t h e n win t h e m i n o r i t y t o t h e s i d e o f m a j o r i t y . Thus, u and up w i l l
...,
0 be h i g h e s t , which c o r r e s p o n d s t o t h e f a c t t h a t t h e
most l i k e l y s t a t e s a r e t o t a l r e j e c t i o n o r t o t a l a c c e p t ance. I f , i n a d d i t i o n , we assume t h a t e a c h column i n m a t r i x
Q undergoes changes i n d e p e n d e n t l y , a c c o r d i n g t o t h e
same s t o c h a s t i c mechanism, we may c a l c u l a t e t h e e x p e c t ed v a l u e o f C ( Q ) , j udgment s
.
i . e . t h e expected consistency of
I n c o n n e c t i o n w i t h t h e above a n a l y s i s o f a g g r e g a t i o n of judgments, l e t us observe t h a t t h e a n a l y s i s o f frequency o f e v e n t s , s t u d i e d i n C h a p t e r 5 i n c o n n e c t i o n w i t h t i m e and P o i s s o n p r o c e s s e s , may be r e g a r d e d as
some s p e c i a l c a s e o f e v a l u a t i o n o f f r e q u e n c y o f e v e n t s . Here t h e b a s i s f o r t h e judgment a r e t h e w a i t i n g t i m e s
f o r a n e v e n t of a g i v e n t y p e , or more p r e c i s e l y , t h e t y p e o f t h e e a r l i e s t e v e n t . T h i s i s based on t h e i d e a t h a t e v e n t s from a more f r e q u e n t t y p e w i l l t e n d , i n g e n e r a l , t o o c c u r e a r l i e r t h a n e v e n t s from t h e l e s s frequent type.
1 4 . APPLICATIONS: A THEORY OF SOCIAL CHANGE T h i s s e c t i o n w i l l be d e v o t e d t o t h e c o n s i d e r a t i o n s o f
some g e n e r a l problems o f s o c i a l dynamics, which oppose
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or i n d u c e a change, and t h e r e s u l t i n g phenomenon o f alienation.
14.1. S t r u c t u r e of t h e s o c i e t y I n o r d e r t o be a b l e t o s p e a k o f s o c i a l change, i t i s n e c e s s a r y t o d e v e l o p some s o r t o f t h e d e s c r i p t i o n o f t h e s t a t e o f t h e s o c i e t y a t any g i v e n moment. A complete d e s c r i p t i o n o f a s t a t e o f t h e s o c i e t y would
i n v o l v e an unmanageably l a r g e number of v a r i a b l e s . T h i s s u g g e s t s u s i n g a d e s c r i p t i o n which c o n c e n t r a t e s f i r s t on some most i m p o r t a n t c h a r a c t e r i s t i c s , and t h e n becomes e n r i c h e d as t h e need a r i s e s . Formally, a d e s c r i p t i o n of t h e s t a t e of t h e s o c i e t y i n v o l v e s s p e c i f i c a t i o n o f a t l e a s t t h e f o l l o w i n g objects:
--
t h e s e t o f p e r s o n s under c o n s i d e r a t i o n , or ' l s o c i e t y " , t o b e d e n o t e d by S , w i t h e l e m e n t s d e n o t e d by s ;
--
a s o c i a l group s t r u c t u r e , t h a t i s , a c l a s s o f s e t s o f e l e m e n t s o f S , r e p r e s e n t i n g v a r i o u s groups one may w i s h t o c o n s i d e r . The groups h e r e may r a n g e from "The f a m i l y o f Mr. Brown", t o "Republican P a r t y " , e t c . It i s i m p o r t a n t t h a t t h e groups need n o t be d i s j o i n t , s o t h a t a p e r s o n may b e l o n g t o s e v e r a l g r o u p s , and a l s o t h a t t h e groups may be f u z z y , i . e . w i t h i m p r e c i s e l y d e f i n e d boundaries. Formally, t h e r e f o r e , t h e s o c i a l s t r u c t u r e may be d e s c r i b e d i n terms o f a f a m i l y , s a y G , o f f u z z y s u b s e t s o f S ( d e f i n e d t h r o u g h t h e i r membership functions).
FORMAL THEORY OF ACTIONS
--
883
t h e s e t of goods, a l l o c a t e d among members o f t h e
s o c i e t y . These goods i n c l u d e , e . g . v o t i n g r i g h t s , funds, administrative positions, e t c .
--
t h e a l l o c a t i o n s o f goods i n t h e s o c i e t y . Thus, i f
C s t a n d s for t h e s e t of a l l goods under c o n s i d e r a t i o n , t h e a l l o c a t i o n may be d e s c r i b e d by t h e f u n c t i o n f which maps t h e s e t C X S i n t o t h e r e a l l i n e . The v a l u e f ( c , s ) i n d i c a t e s t h e amount of goods c b l k o c a t e d t o p e r s o n s.
--
N e x t , we have a system o f p r e f e r e n c e r e l a t i o n s , which d e s c r i b e t h e p r e f e r e n c e s o f members o f t h e soci e t y . For f o r m a l r e a s o n s , i t i s n o t s u f f i c i e n t t o cons i d e r t h e p r e f e r e n c e s on goods o n l y ( b e c a u s e a p e r s o n may p r e f e r t h a t h i s enemy d o e s n o t have a n a c c e s s t o some goods, e t c . ) . Thus, t h e p r e f e r e n c e s a r e d e f i n e d on t h e c l a s s o f a l l a l l o c a t i o n s o f goods. The p r e f e r e n c e s , s a y >s, on a l l o c a t i o n s f l , f 2 , a r e assumed t o be t r a n s i t i v e and c o n n e c t e d . The r e l a t i o n s o f s t r i c t p r e f e r e n c e and i n d i f f e r e n c e aae t h e n d e f i n e d i n t h e u s u a l manner. The n o t i o n o f p r e f e r e n c e a l l o w s u s t o d e f i n e a l s o t h e c o n c e p t of a g o a l , as i n t h e p r e c e d i n g s e c t i o n s . Thus, f o r m a l l y a g o a l i s a c l a s s o f a l l o c a t -
...
i o n s which one wants t o a c h i e v e . The g o a l may a l s o be a c l a s s o f a l l o c a t i o n s which a r e l e s s p r e f e r r e d t h a n t h e e x i s t i n g o n e , and t h e r e f o r e t o be a v o i d e d . Naturally, i n practical situations the goals are often e x p r e s s e d v e r b a l l y , w i t h o u t e x p l i c i t mention of a l l o c a t i o n s . The p o i n t i s , however, t h a t any such g o a l i s a l w a y s e x p r e s s i b l e i n terms of a s e t o f a l l o c a t i o n s t o be a t t a i n e d or t o be a v o i d e d . To g i v e a n example, i f someone wants t o g e t a h i g h e r
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CHAPTER 6
share i n some s p e c i f i c goods c
while preserving h i s 0' shares i n o t h e r goods on a t l e a s t t h e p r e s e n t l e v e l , t h e n h i s w i l l a t t e m p t t o g e t any a l l o c a t i o n f such t h a t
where s 0 d e n o t e s t h e goods i n q u e s t i o n , and s o i s t h e p e r s o n whose g o a l i s b e i n g d e s c r i b e d . Of c o u r s e , t h e s i t u a t i o n becomes much more compGicated i f one t r i e s to d e f i n e t h e g o a l of a group, s i n c e t h i s amounts t o a g g r e g a t i n g t h e p r e f e r e n c e s of members of t h e group i n t o one p r e f e r e n c e . The d i f f i c u l t y which a r e e n c o u n t e r e d i n s u c h s i t u a t i o n s , have a l r e a d y been discussed. A more r e a l i s t i c s o l u t i o n here i s t o abandon t h e r e -
q u i r e m e n t o f a common p r e f e r e n c e for a l l members of t h e group, and s p e c i f y t h e g o a l of a group as a s e t of a l l o c a t i o n s o f goods, t h a t a g r e e s " i n e s s e n t i a l r e s p e c t s ' ' w i t h a l l o c a t i o n s d e s i r e d b y members o f t h e g r o u p . T h i s , n a t u r a l l y , i s a f u z z y c o n c e p t of a g o a l . The s t r e n g t h of t h e group w i l l depend, among o t h e r s , on t h e d e g r e e t o which t h e p r e f e r e n c e s o f members a r e c o n s i s t e n t among t h e m s e l v e s , and a l s o on t h e d e g r e e t o which t h e s e p r e f e r e n c e s a g r e e w i t h t h e d e c l a r e d g o a l s o f t h e group. The f i r s t r e q u i r e m e n t minimizes t h e amount o f i n t e r n a l r i v a l r y , w h i l e t h e second w i l l enhance t h e amount of c o o p e r a t i o n .
---
t h e n e x t c o n c e p t i s a s e t o f r u l e s , s a y R , which
885
FORMAL THEORY OF ACTIONS
o p e r a t e w i t h i n p a r t i c u l a r g r o u p s , r e s t r i c t i n g t h e beh a v i o u r of t h e i r members.
---
F i n a l l y , t h e s t a t e of t h e s o c i e t y i s d e s c r i b e d a l s o
by c h a r a c t e r i z h g t h e communication l i n k s w i t h i n t h e
s e t S. To summarize, t h e s o c i e t y may b e d e s c r i b e d i n terms
of t h e formal s y s t e m
(14.1) where S i s t h e s e t o f p e r s o n s , G i s t h e group s t r u c t u r e , C i s t h e s e t o f goods, f i s t h e a l l o c a t i o n a t a g i v e n i s a s e t o f p r e f e r e n c e s o f members of t h e time,{&s s o c i e t y , R i s t h e s e t of r u l e s of b e h a v i o u r i n p a r t i c u l a r g r o u p s , and L i s t h e communicaticn network.
1
14.2.
S o c i a l change
Given a d e s c r i p t i o n o f t h e s t a t e of o f s y s t e m ( 1 4 . 1 ) , one may c o n s i d e r u n d e r s t o o d as t r a n s i t i o n s t o s t a t e s one o f t h e components of t h e system
t h e s o c i e t y i n form s o c i a l changes , such t h a t a t least undergoes a change.
The c o n c e p t o f s o c i a l change i s , o f c o u r s e , a f u z z y o n e :
i n d e e d , i t i s n o t s u f f i c i e n t t h a t some change o c c u r s i n s y s t e m ( 1 4 . 1 ) . T o q u a l i f y as a s o c i a l change, i t must b e i n some way " s u b s t a n t i a l " . The l a t t e r a d j e c t i v e i n d u c e s f u z z i n e s s among changes i n system ( 1 4 . 1 ) , s o t h a t w i t h e a c h change one c o u l d c o n c e i v a b l y a s s o c i a t e t h e d e g r e e t o which t h i s change i s " s o c i a l " . Now, changes may be c a t e g o r i z e d as t o which o f t h e
886
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components of (14.1) a r e changed, and t o which d e g r e e . The d e g r e e s o f change have t o be i n e a c h c a s e s p e c i f i e d by an a p p r o p r i a t e i n d e x . The t o t a l change may be t h e n measured t h r o u g h a g g r e g a t i o n o f t h e s e i n d i c e s ( i f t h e r e s h o u l d be a need of n u m e r i c a l r e p r e s e n t a t i o n of change). The most t r i v i a l change c o n c e r n s t h e s e t S , and i n f a c t , i f a l l o t h e r v a r i a b l e s remain t h e same, t h e n a change i n S would. t y p i c a l l y not be q u a l i f i e d a s s o c i a l .
Next, l e t u s c o n s i d e r changes i n t h e group s t r u c t u r e G . Here t h e r e a r e two b a s i c t y p e s of changes: t h o s e i n membership i n g r o u p s , and t h o s e c o n s i s t i n g o f a n a p p e a r ence o f a new group ( t h e change c o n s i s t i n g o f a d i s a p p e a r e n c e o f an old group may be r e g a r d e d simply as change of membership i n s u c h a way t h a t no p e r s o n remains i n t h e g r o u p ) . The f i r s t t y p e o f change i s exemp l i f i e d by a change i n a p o l i t i c a l s t r u c t u r e , i n t h e s e n s e o f an i n c r e a s e o f p o p u l a r i t y o f some p a r t y , and d e c r e a s e o f p o p u l a r i t y o f some o t h e r p a r t y ( h e r e t h e group s t r u c t u r e i s induced n o t s o much by p a r t y memb e r s h i p , b u t by s u p p o r t to a g i v e n p a r t y by members o f t h e s o c i e t y ) . The second t y p e of change w i l l o c c u r i f a new p a r t y i s formed, e t c . Another t y p e o f change i s a s s o c i a t e d w i t h s e t s o f goods C . A s b e f o r e , one c a n c o n s i d e r changes c o n s i s t i n g o f a p p e a r e n c e o f some new goods, or d i s a p p e a r e n c e o f some o t h e r goods. The f i r s t t y p e o f change o c c u r s when a new i n s t i t u t i o n i s created, thus giving rise t o a certain number o f a d m i n i s t r a t i v e p o s i t i o n s , t h e l a t t e r b e i n g t h e goods i n q u e s t i o n . The second t y p e o f change i s e x e m p l i f i e d by abandoning a program, s o t h a t f u n d s are
FORMAL THEORY OF ACTIONS
887
no l o n g e r a v a i l a b l e . I n t h i s c a s e , one can a l s o c o n s i d e r t h e changes c o n s i s t i n g o f amounts o f goods a v a i l a b l e f o r a l l o c a t i o n t o undergo some v a r i a t i o n s ( e . g . r e s e a r c h f u n d s , e t c . ) . A s r e g a r d s changes i n t h e a l l o c a t i o n f o f goods among
members o f t h e s o c i e t y , t h e number of p o s s i b i l i t i e s i s much l a r g e r . F i r s t l y , i t i s c o n v e n i e n t t o i n t r o d u c e a c l a s s i f i c a t i o n o f goods, i n t o " i n f i n i t e l y d i v i s i b l e " and n o t - i n f i n i t e l y d i v i s i b l e . A s examples o f t h e f i r s t t y p e , one may t a k e v o t i n g r i g h t s : i t i s p o s s i b l e , a t l e a s t i n t h e o r y , t o g i v e v o t i n g r i g h t t o e v e r y o n e . On t h e o t h e r hand, goods which a r e n o t i n f i n i t e l y d i v i s i b l e a r e t h o s e which come i n l i m i t e d q u a n t i t i e s , 3 0 t h a t g i v i n g them t o some p e r s o n s n e c e s s i t a t e s d e n y i n g
them t o o t h e r s . A s a n example, one can t a k e f u n d s , or administrative positions, etc. To p r o c e e d s y s t e m a t i c a l l y , i f t h e change c o n c e r n s t h e
a l l o c a t i o n of goods o f t h e f i r s t t y p e ( i n f i n i t e l y d i v i s i b l e ) , t h e n i t simply means t h a t t h e s e t o f p e r s o n s who have a c c e s s t o t h e goods i n q u e s t i o n c h a n g e s . A s a n example, one may t a k e g i v i n g v o t i n g r i g h t t o some groups t o which i t was d e n i e d b e f o r e . Another t y p e o f change o c c u r s when one c o n s i d e r s a l l o c a t i o n s o f goods o f t h e second t y p e , and a l s o b i n a r y (such as a d m i n i s t r a t i v e p o s i t i o n s ) . I n c a s e , f o r inst a n c e , o f a p a r t y winning e l e c t i o n t h e change amounts simply t o f i l l i n g t h e a v a i l a b l e p o s i t i o n s w i t h c a n d i d a t e s of t h e winning p a r t y . F i n a l l y , for q u a n t i f i a b l e goods-,
such as f i n a n c e s ,
888
CHAPTER 6
t h e change means s i m p l y a m o d i f i c a t i o n ( d r a s t i c o r n o t ) o f a l l o c a t i o n of funds. The changes i n a l l o c a t i o n s a r e t h o s e which a r e most " v i s i b l e " . A s opposed t o t h a t , changes o f p r e f e r e n c e systems b e l o n g r a t h e r t o t h e domain of c o n s c i o u s n e s s . The change o f p r e f e r e n c e s , o r e q u i v a l e n t l y
-- t h e
change i n group and i n d i v i d u a l g o a l s -- may o c c u r as a r e s u l t of c e r t a i n a c t i o n s by o t h e r g r o u p s o f i n dividuals. S i m i l a r l y , t h e r u l e s which o p e r a t e w i t h i n a g r o u p , and changes o f t h e s e r u l e s , b e l o n g t o t h e domain o f cons c i o u s n e s s . Here t h e changes may concern n o t o n l y a d d i t i o n o f some new r u l e s or r e v o k i n g o l d o n e s , b u t a l s o i n t h e change o f o r d e r o f i m p o r t a n c e o f some r u l e s . F i n a l l y , changes i n t h e communication network may res u l t from f a c t o r s r a n g i n g from changes o f o r g a n i z a t i o n a l schemes w i t h i n some i n s t i t u t i o n s , t o changes o f some s o c i a l customs o r l i f e p a t t e r n s , which i n f l u e n c e t h e f r e q u e n c y o f human c o n t a c t s . The main i s s u e i n any t h e o r y o f s o c i a l change l i e s n o t so much i n d e f i n i n g what i s and what i s n o t a change, b u t a n a l y s i n g t h e r e p e r t o i r e o f a c t i o n s which might produce s o c i a l c h a n g e s , and b u i l d i n g a t h e o r y which would a l l o w t o p r e d i c t t h e change ( i t s magnitude, or a t l e a s t d i r e c t i o n ) i f a given a c t i o n i s undertaken.
14.3. A t h e o r y of freedom The a n a l y s i s of t h e p r e c e d i n g s e c t i o n a l l o w s u s t o
889
FORMAL THEORY OF ACTIONS
introduce
c e r t a i n t h e o r e t i c a l p o s t u l a t e s concerning t h e
c o n c e p t of freedom. C o n s i d e r namely t h e goods of t h e f i r s t t y p e ( i n f i n i t e l y d i v i s i b l e ) and t h e i r a l l o c a t i o n between members of S . S i n c e t h e s e goods a r e s u c h t h a t one may e i t h e r have a n a c c e s s t o t h e s e goods o r n o t ( i . e . t h e a c c e s s i s n o t g r a d e d ) , an a l l o c a t i o n may be r e p r e s e n t e d more c o n v e n i e n t l y i n form o f a f a m i l y of
sets
cs
={c
E
c:
f(s,c) = 13,
s o t h a t Cs i s t h e s e t of goods which a r e a l l o c a t e d t o person s. A l t e r n a t i v e l y , w e may a s s o c i a t e w i t h e a c h goods c t C of t h e t y p e under c o n s i d e r a t i o n , t h e s e t
sc
={s(l
s:
f(s,c) = 13
i . e . t h e s e t o f a l l p e r s o n s who have a c c e s s t o goods c . Take now a s goods t h o s e which a r e r e l a t e d t o freedom, such as v o t i n g r i g h t s , c i v i l l i b e r t i e s , a r i s t o c r a t i c p r i v i l e g e s , e t c . Then t h e s e t s Cs , C describe S
,...
t h e a l l o c a t i o n of t h e s e goods i n t h k s o c ? e t y . One can now d e f i n e v a r i o u s c o n c e p t s r e l a t e d t o t h e o r y o f freedom; t h u s , e q u a l i t y w i t h r e s p e c t t o goods C ' c C i s d e f i n e d by t h e r e q u i r e m e n t t h a t C s 7 C'
for a l l s
S,
i . e . e v e r y p e r s o n has a t l e a s t a l l goods i n C ' .
890
CHAPTER 6
Suppose now t h a t goods i n C are o r d e r e d , f o r e a c h p e r s o n s e p a r a t e l y , from t h o s e deemed most i m p o r t a n t t o t h o s e l e s s i m p o r t a n t . T h i s o r d e r i n g may b e d i f f e r e n t f o r d i f f e r e n t p e r s o n s ; moreover, i t may be o n l y a p a r t i a l o r d e r , i . e . some goods may remain i n c o m p a r a b l e . One may d i v i d e t h e goods i n t o some c a t e g o r i e s , s u c h a s p o l i t i c a l , l e g a l , c u l t u r a l , pertaining t o education, e t c . The a n a l y s i s may be c a r r i e d o u t for e a c h o f t h e s e c l a s s e s s e p a r a t e l y , or j o i n t l y for some c l a s s e s . Let
>s
d e n o t e t h e p r e f e r e n c e o f p e r s o n s among goods.
We may t h e n s a y t h a t t h e a l l o c a t i o n f i s p r e f e r e n t i a l l y c o n s i s t e n t f o r p e r s o n s, i f c
c, cs
& c'
>
c
3
c' f
cs.
T h i s means t h a t i f s has some goods c , t h e n h e a l s o
has a l l goods which he p r e f e r s t o c . T h e main n o t i o n h e r e i s t h a t o f a d m i s s i b i l i t y , or Pare-
t o o p t i m a l i t y . T h i s means t h a t t h e r e i s no a l l o c a t i o n which would be b e t t e r (or a t l e a s t as good) f o r a l l p e r s o n s . F o r m a l l y , an a l l o c a t i o n f i s a d m i s s i b l e , i f t h e r e e x i s t s no a l l o c a t i o n f ' s u c h t h a t
where C' a r e s e t s o f goods a s s i g n e d t o p e r s o n s under S allocation f ' . The main problem o f t h e o r y o f freedom l i e s i n d e t e r m i n i n g t h e c l a s s o f a l l o c a t i o n s which are f e a s i b l e . I n t u i t i -
v e l y , f e a s i b l e a l l o c a t i o n i s such
t h a t may be a t t a i n e d
FORMAL THEORY OF ACTIONS
891
by some a l l o w a b l e means, a n d may a l s o be r e t a i n e d i n
t h e s o c i e t y . N a t u r a l l y , t h e s p e c i a l i s t s may d i f f e r w i d e l y as t o f e a s i b i l i t y o f some a l l o c a t i o n s w h i c h a r e only contemplated. Assuming, h o w e v e r , t h a t o n e may d e t e r m i n e ( p o s s i b l y i n
a f u z z y way) t h e c l a s s o f a l l f e a s i b l e a l l o c a t i o n s , o n e c a n t h e n t r y t o f i n d a n optimum among $hem ( s a y , an admissible a l l o c a t i o n , e t c . ) . Here t h e p r o b l e m a p p e a r s t o be c l o s e l y r e l a t e d t o t h e p r o b l e m o f Arrow o f s o c i a l w e l f a r e f u n c t i o n . However, two f a c t o r s d i s t i n g u i s h i t f r o m t h e A r r o w ' s s e t u p , a n d t h e r e f o r e o f f e r some hope o f a v o i d i n g t h e I m p o s s i b i l i t y Theorem. One o f them i s t h e f a c t t h a t o n e a s s u m e s o n l y p a r t i a l o r d e r i n g o f goods ( s a y , c u l t u r a l g o o d s may b e n o t c o m p a r a b l e t o e d u c a t i o n a l g o o d s , e t c . ) ,
and s e c o n d l y , one l o o k s f o r a s o l u t i o n o n ly i n t h e c l a s s o f a l l f e a s i b l e s o l u t i o n s ( t h e a s s e r t i o n of Arrow Theorem i s known t o be f a l s e i n c a s e when t h e p r e f e r e n c e s of i n d i v i d u a l persons a r e a p p r o p r i a t e l y r e s t r i c t e d ) .
1 4 . 4 . A l i e n a t i o n and dynamics o f s o c i a l change I n t h i s s e c t i o n we s h a l l o u t l i n e t h e main i d e a s o f t h e f o r m a l t h e o r y o f a l i e n a t i o n s u g g e s t e d b y Nowakowska ( 1 9 7 7 a ) , a c c o r d i n g t o which a l i e n a t i o n i s a s t a t e o f b e i n g c o n s c i o u s o f t h e f a c t t h a t o n e i s d e n i e d some goods w h i c h o n e e x p e c t s a n d h a s t h e r i g h t t o have bec a u s e o f t h e work o n e p e r f o r m s , or b e c a u s e o f o n e ' s social status, etc. C o n s i d e r a f r a g m e n t o f t h e s y s t e m from t h e p r e c e d i n g
892
CHAPTER 6
s e c t i o n s , c o n s i s t i n g o f s o c i e t y S, goods C , a l l o c a t i o n s f , and p r e f e r e n c e s 7 F o r s i m p l i c i t y , assume t h a t C / S c o n t a i n s o n l y b i n a r y goods, and c o n s e q u e n t l y , e v e r y
.
a l l o c a t i o n ( a c t u a l or h y p o t h e t i c a l ) assumes o n l y t h e v a l u e s 0 and 1. Two b a s i c a d d i t i o n a l c o n c e p t s w i l l be ( a ) a d m i s s i b i l i t y function a ( t , s , c ) , representing the degree, t o which g i v i n g goods c t o p e r s o n s a t t i m e t i s admissi b l e , and ( b ) t h e f o r c e s F ( s , f , f ' ) and G ( s , f , f ' ) which t h e p e r s o n s i s c a p a b l e , given t h e p r e s e n t a l l o c a t i o n f , o f e x e r t i n g r e s p e c t i v e l y t o w a r d s and a g a i n s t t h e change o f a l l o c a t i o n f t o an a l t e r n a t i v e f ' . The two main p o s t u l a t e s a s s e r t t h e f o l l o w i n g : 1) Every a ( t , s , c ) , sa a f u n c t i o n o f t , h a s a t most one
peak; 2 ) The change from f t o f ' w i l l n o t o c c u r , i f t h e
forces against it are stronger than forces f o r i t , i . e .
lE
G(s,f,f')
s €A
>seBUC
F(s,f,f')
where A , B and C a r e r e s p e c t i v e l y t h e s e t s o f t h o s e p e r s o n s s who p r e f e r f t o f' ( f f ' ) , who a r e i n d i f f e r S e n t between f and f' ( f m S f ' ) , and who p r e f e r f ' t o
>
f (f'
> sf).
One may now d e f i n e t h e a d m i s s i b i l i t y o f an a l l o c a t i o n f , a t t i m e t , as f o l l o w s : A(t,f)
= min{a(t,s,c):
f ( s , c ) = 1)
.
Thus, t h e a d m i s s i b i l i t y o f a n a l l o c a t i o n e q u a l s to t h e
FORMAL THEORY OF ACTIONS
893
a d m i s s i b i l i t y o f t h e l e a s t a d m i s s i b l e of i t s a s s i g n m e n t s
o f goods. The f i r s t p o s t u l a t e i m p l i e s t h a t t h e f u n c t i o n A ( t , f )
has
a l s o t h e p r o p e r t y of s i n g l e - p e a k e d n e s s . T h i s means t h a t a d m i s s i b i l i t y o f any a l l o c a t i o n w i l l e v e n t u a l l y s t a r t
diminishing. Next, an a l l o c a t i o n f i s c a l l e d f a i r ( a t time t ) , i f A ( t , f ) 3 A ( t , f ' ) f o r any a l l o c a t i o n f ' . T h i s means t h a t f i s the allocation w i t h highest admissibility a t t . The n o t i o n o f f a i r n e s s c o v e r s t o some e x t e n t t h e i n t u i t i o n of " s o c i a l j u s t i c e " . An a l l o c a t i o n i s c a l l e d s t a b l e , i f t h e i n e q u a l i t y b e t ween f o r c e s from p o s t u l a t e 2 h o l d s f o r any a l l o c a t i o n f ' . Since A ( f , t ) w i l l eventually begin t o d e c l i n e , ther e w i l l a p p e a r a t l e a s t one a l t e r n a t i v e a l l o c a t i o n f ' which w i l l b e more f a i r t h a n f . T h i s l e a d s t o t h e f o l lowing dilemma: i f one wants t o e n s u r e f a i r n e s s , one has t o make f r e q u e n t changes; i f one wants t o p r e s e r v e s t a b i l i t y , one has t o r e s i g n from f a i r n e s s . Any group of p e r s o n s M c a p a b l e of e n s u r i n g t h e i n e q u a lity
1G(s,f,f') > s t;M
F(s,f,f') SdN
f o r some f ' w i t h A ( t , f ' ) 7 A ( t , f ) , i . e . group c a p a b l e o f b l o c k i n g a s o c i a l l y more f a i r a l t e r n a t i v e f ' , w i l l b e c a l l e d a monopole w i t h r e s p e c t t o a l l o c a t i o n f ' . The n o t i o n o f an e n f o r c i n g monopole ( a s opposed t o b l o c k i n g monopole above) i s d e f i n e d a n a s i m i l a r way. S i n c e t h e f o r c e s F and G a r e , i n g e n e r a l , n o t known
CHAPTER 6
894
e x a c t l y , t h e problem o f c h e c k i n g whether a g i v e n group i s a monopole may be d i f f i c u l t . M o r e o v e r , monopoles
are u s u a l l y fuzzy s u b s e t s of S. Given t h e a b o v e c o n c e p t s , o n e c a n s p e c i f y t h e c o n d i t i o n s f o r a l i e n a t i o n of a person s . T h u s , s u p p o s e t h a t t h e a c t u a l a l l o c a t i o n i s f , and t h e r e e x i s t s f ' s u c h t h a t f ( s , c ) = 0 , f ' ( s , c ) = 1 f o r some goods c ; t h a t i s , p e r s o n s d o e s n o t h a v e a c c e s s t o goods c u n d e r f , b u t he would have s u c h a n a c c e s s u n d e r t h e a l l o c a t i o n f'. If t h e f o l l o w i n g c o n d i t i o n s are met:
1) a ( t , s , c ) i s h i g h a n d i n c r e a s i n g ;
2)
f' >s
3) A ( t , f ' )
f (i.e. s prefers f ' t o f ) ;
>
A(t,f);
4) f' i s b l o c k e d by some monopole M; t h e n o n e s a y s t h a t p e r s o n s i s a l i e n a t e d by monopole M . T h u s , a l i e n a t i o n o c c u r s if t h e r e i s a n a l t e r n a t i v e a l l o c a t i o n f ' , which g i v e s goods c t o s , w h i l e he d o e s n o t h a v e t h e s e goods u n d e r f , a n d d e s e r v e s t h e m ; moreo v e r , t h e a l t e r n a t i v e f ' i s a t t h e same t i m e s o c i a l l y b e t t e r t h a n f , p r e f e r r e d by s , a n d b l o c k e d by a mono-
pole. The p a p e r s of Nowakowska ( 1 9 7 7 a , b ) p r e s e n t a number o f
h y p o t h e s e s a b o u t s t r a t e g i e s o f m o n o p o l e s , and t h e e f f e c t o f a l i e n a t i o n on t h e t e n d e n c i e s t o form counter-monop o l e s which t r y t o i n d u c e changes i n t h e a l l o c a t i o n o f g o o d s . These h y p o t h e s e s may t h e r e f o r e be t r e a t e d as
FORMAL THEORY OF ACTIONS
895
as h y p o t h e s e s about t h e dynamics which induce some s p e c i a l k i n d o f s o c i a l c h a n g e s , namely t h o s e which cons i s t o f a t r a n s f o r m a t i o n of a l l o c a t i o n o f goods i n the society. Under c o n d i t i o n s ( a )
-
( d ) , one may c o n s t r u c t a l s o a
measure o f s t r e n g t h o f a l i e n a t i o n . A p e r s o n may be a l i e n a t e d w i t h r e s p e c t t o v a r i o u s goods,
and by v a r i o u s monopoles. T h i s a l l o w s us t o s p l i t t h e a l i e n a t i o n of a g i v e n p e r s o n i n t o v a r i o u s components, and d e f i n e t h e n o t i o n s , s u c h as scope and d e p t h o f a l i e n a t i o n , depending on t h e r e l a t i v e " l o c a t i o n " o f a p e r s o n w i t h r e s p e c t t o a monopole. When a l a r g e number o f p e r s o n s a r e a l i e n a t e d by t h e
same monopole, and t h e i r communication network i s suf f i c i e n t l y "dense", t h e r e i s a high p r o b a b i l i t y t h a t t h a y w i l l form a new s o c i a l g r o u p , w i t h i t s own g o a l s , program f o r a c t i o n , e t c . , aimed p r i m a r i l y a t r e d u c i n g t h e l e v e l o f a l i e n a t i o n from some goods. I n t h i s s e n s e , a l i e n a t i o n may produce a s o c i a l change i n t h e form o f a t r a n s f o r m a t i o n o f t h e s t r u c t u r e o f t h e s o c i e t y and t r a n s f o r m a t i o n o f t h e p r e f e r e n c e system. Such s o c i a l c h a n g e s , as a r u l e , a r e p r e l i m i n a r y t o major changes i n allocations. One may t h e r e f o r e s a y t h a t , g i v e n a p p r o p r i a t e c o n d i t i o n s , t h e phenomenon of a l i e n a t i o n i s a primary s o u r c e o f s o c i a l c h a n g e s , f i r s t i n t h e " c o n s c i o u s n e s s domain", and t h e n -- i f t h e s t r e n g t h o f t h e new group i s s u f f i cient
--
a l s o i n t h e domain of a l l o c a t i o n of goods.
The main r e s u l t o f t h e s u g g e s t e d model, however, i s t h e t h e o r e t i c a l j u s t i f i c a t i o n of t h e u n a v o i d a b i l i t y
896
CHAPTER 6
o f a l i e n a t i o n , due t o t h e dilemma between s t a b i l i t y and f a i r n e s s o f a l l o c a t i o n s o f goods.
1 4 . 5 . S u b j e c t i v e p e r c e p t i o n of o n e ' s l e v e l o f a l i e n a t i o n I n t h i s s e c t i o n we s h a l l s u g g e s t a c e r t a i n model, des i g n e d t o c a p t u r e c e r t a i n dynamic f e a t u r e s o f t h e p e r c e p t i o n of o n e ' s l e v e l of a l i e n a t i o n . The b a s i c assumption i s t h a t t h e p e r c e i v e d l e v e l o f a l i e n a t i o n changes s e q u e n t i a l l y , as one o b s e r v e s t h e l e v e l s o f a c c e s s t o c e r t a i n goods o f o t h e r p e r s o n s . I n t u i t i v e l y , e v e r y time one e n c o u n t e r s a p e r s o n whose l e v e l o f a c c e s s t o some goods i s e x c e e d i n g o n e ' s own a c c e s s t o t h e same goods, t h e p e r c e i v e d a l i e n a t i o n l e v e l i n c r e a s e s . On t h e o t h e r hand, when one e n c o u n t e r s a p e r s o n who i s , s o t o s p e a k , i n worse c o n d i t i o n s , t h e l e v e l of a l i e n a t i o n d e c r e a s e s . These p e r c e p t i o n s a r e " f i l t e r e d " , o r b i a s e d , depending o f o n e ' s p e r c e i v e d l e v e l of a l i e n a t i o n . The l a t t e r e f f e c t , o f a p s y c h o l o g i c a l c h a r a c t e r , i s r e l a t e d t o t h e f a c t t h a t f o r a person who i s h i g h l y a l i e n a t e d ( w i t h something l i k e a " f e e l i n g of b i t t e r n e s s " ) each encounter w i t h a person w i t h h i g h e r a c c e s s t o some goods t e n d s t o have a somewhat e x a g g e r a t e d e f f e c t . On t h e o t h e r hand, i n s u c h a i t u a t i o n , e n c o u n t e r i n g a p e r s o n who i s worse-off t o be n e g l e c t e d .
tends
Thus, t h e model w i l l a t t e m p t t o c a p t u r e t h e dynamic f e a t u r e s o f p e r c e p t i o n o f o n e ' s own l e v e l of a l i e n a t i o n , as a n i n t e r p l a y o f t h e f o l l o w i n g t h r e e f a c t o r s :
---
t h e a c t u a l perceived l e v e l of alienation; t h e a c t u a l a c c e s s t o t h e goods;
FORMAL THEORY OF ACTIONS
--
891
t h e p e r c e i v e d a c c e s s t o t h e goods by o t h e r s .
14.5.1. The model. We c o n s i d e r p e r c e p t i o n of t h e l e v e l o f a l i e n a t i o n o f an a r b i t r a r y f i x e d s u b j e c t , r e s u l t i n g from h i s p e r c e p t i o n of b e i n g d e p r i v e d of c e r t a i n goods s a y , C . The n a t u r e o f t h e s e goods need n o t be made s p e c i f i c i n t h e g e n e r a l model.
For s i m p l i c i t y , l e t u s assume t h a t t h e l e v e l o f a c c e s s t o the goods C i s r e p r e s e n t a b l e i n form o f a c e r t a i n i n d e x ( i t may be a s i m p l e v a r i a b l e , s u c h as s a l a r y l e v e l , or a composite i n d e x , i n v o l v i n g p e r c e p t i o n o f l e s s t a n g i b l e goods, which may i n t e r v e n e i n a l i e n a t i o n level). N e x t , we assume t h a t from t i m e t o t i m e ( i n t h e model,
a t t i m e s t = 1,2, . . . ) t h e s u b j e c t e n c o u n t e r s o t h e r p e r s o n s , whom he s e e s as e q u a l l y d e s e r v i n g t h e a c c e s s t o t h e goods a s h e , and p e r c e i v e s t h e i r a c t u a l a c c e s s t o t h e s e goods.
...
be t h e s u b j e c t ' s a c t u a l l e v e l s F o r m a l l y , l e t zl, z2, of a c c e s s t o goods C a t t i m e s t = 1,2,..., and l e t xl, x 2 , be h i s p e r c e i v e d l e v e l s o f a c c e s s t o C o f t h e s u c c e s s i v e persons whom he e n c o u n t e r s .
...
...
F i n a l l y , l e t Y1, Y2, be t h e p e r c e i v e d by t h e subj e c t l e v e l s o f a l i e n a t i o n from goods C . B e f o r e f o r m u l a t i n g t h e a s s u m p t i o n s about t h e n a t u r e o f i t i s w o r t h w h i l e t o comment b r i e f l y the process on t h e i n t e r p r e t a t i o n o f t h e t h r e e p r o c e s s e s i n v o l v e d , namely {xn\, { z n l and { y n j .
fYnl,
898
CHAPTER 6
F i r s t l y , i t w i l l be assumed t h a t f z 1 form some f i x e d n s e q u e n c e , r e p r e s e n t i n g t h e s u b j e c t Is e v o l u t i o n w i t h r e s p e c t t o goods C . I f I z n l r e p r e s e n t s t h e s a l a r y l e v e l s , t h e n i t may be assumed t h a t z n grows s t e a d i l y a t a c e r t a i n r a t e , o r remains c o n s t a n t , e t c . S e c o n d l y , xn i s t h e a c c e s s t o goods C o f t h e n - t h successive person encountered (it i s not important t o i n t e r p r e t t h e t e r m " e n c o u n t e r " l i t e r a l l y ) . We imagine t h a t a t t5me t = n , t h e s u b j e c t l e a r n s about some o t h e r p e r s o n , who ( a c c o r d i n g t o h i m ) d e s e r v e s t h e acc e s s t o goods C i n t h e same d e g r e e as he ( e . g . has t h e same q u a l i f i c a t i o n s , e t c . ) , and p e r c e i v e s ( r i g h t l y o r n o t ) h i s a c c e s s t o C as xn
.
>
Now, i f xn z n , t h e s u b j e c t w i l l f e e l l a c k of "fairn e s s " , and t h i s w i l l t e n d t o i n c r e a s e h i s p e r c e p t i o n MOo f a l i e n a t i o n , from Y n t o some h i g h e r v a l u e Y n t l . r e o v e r , t h e i n c r e a s e Y 1 - Y n w i l l t e n d t o be h i g h e r nt i f t h e a c t u a l l e v e l of a l i e n a t i o n Y n i s h i g h e r . On t h e o t h e r hand, i f xn < z n , t h e a l i e n a t i o n l e v e l Yn w i l l d e c r e a s e t o a lower l e v e l Yntl. The d e c r e a s e w i l l be l e s s i n a b s o l u t e v a u l u e , i f t h e s u b j e c t f e e l s h i g h l y a l i e n a t e d , i . e . i f Yn i s h i g h . S p e c i f i c a l l y , t h e model w i l l have t h e f o l l o w i n g form 'n+ 1 = f ( Y n )
+ F(xn, zn, Y n ) .
(14.2)
Here f i s a f u n c t i o n which d e s c r i b e s t h e " s p o n t a n e o u s "
changes i n t h e p e r c e i v e d a l i e n a t i o n l e v e l , o c c u r r i n g d u r i n g the p e r i o d between c o n s e c u t i v e e n c o u n t e r s . One o f t h e p o s s i b l e forms of t h e f u n c t i o n f i s
FORMAL THEORY OF ACDONS
899
where 0 < r 4 1. T h i s c o r r e s p o n d s t o t h e c a s e when t h e p e r c e i v e d l e v e l o f a l i e n a t i o n "wears o f f " i n t h e p e r i o d s between t h e e n c o u n t e r s , i n a b s e n c e of s u p p o r t i n g or i n h i b i t i n g f a c t o r s , w i t h r being t h e decrease r a t e . Next, f u n c t i o n F ( x , z , Y ) r e p r e s e n t s t h e change ( p o s i t i ve o r negative) i n one's perceived a l i e n a t i o n l e v e l , r e s u l t i n g from an e n c o u n t e r w i t h a p e r s o n whose a c c e s s t o goods C i s x , i n s i t u a t i o n when t h e s u b j e c t ' s a c c e s s t o goods C i s z , and h i s a l i e n a t i o n l e v e l i s Y . The gen e r a l a s s u m p t i o n s about t h e f u n c t - o n s f and F are as follows. ASSUMPTION 1. The f u n c t i o n f i s s t r i c t l y i n c r e a s i n g . T h i s a s s u m p t i o n means t h a t t h e h i g h e r i s t h e p e r c e i v e d
l e v e l o f a l i e n a t i o n a f t e r t h e n - t h e n c o u n t e r , t h e highe r w i l l be i t s " r e s i d u a l " l e v e l , p r i o r t o ( n + l l - s t encounter. ASSUMPTION 2 . F o r e v e r y f i x e d Y and z , t h e f u n c t i o n F(x,z,Y) i s s t r i c t l y increasing i n z , with F(x,z,Y) f o r x > z and F ( x , z , Y ) < 0 for x < z .
>
9
T h i s assumption means t h a t e a c h e n c o u n t e r w i t h a p e r -
s o n who i s p e r c e i v e d as " b e t t e r o f f " ( x > z ) g i v e s an i n c r e a s e i n a l e v e l o f a l i e n a t i o n , while each enc-unter w i t h a p e r s o n who i s p e r c e i v e d as "worse-off" ( x < z ) g i v e s a d e c r e a s e of l e v e l of a l i e n a t i o n . The v a l u e F ( x , z , Y ) r e p r e s e n t s a change i n t h e p e r c e i v e d l e v e l o f a l i e n a t i o n r e s u l t i n g from an e n c o u n t e r w i t h a p e r s o n who i s i n t h e same s i t u a t i o n as r e g a r d s t h e goods
900
CHAPTER 6
as t h e s u b j e c t . T h i s v a l u e ma.y be p o s i t i v e , n e g a t i v e
or zero. ASSUMPTION 3 . The f u n c t i o n F(x,z,Y)
increases with Y
f o r any x and z. The i n t e r p r e t a t i o n o f t h i s assumption i s as f o l l o w s . Firstly, if x
> z (an encounter
with
a
p e r s o n who i s
“ b e t t e r o f f ” g i v e s an i n c r e a s e i n a l i e n a t i o n l e v e l ( s i n c e F > 0 ) , and t h i s i n c r e a s e i s l a r g e r when t h e l e v e l of a l i e n a t i o n i s h i g h e r . On t h e o t h e r hand, i f x c z ( a n e n c o u n t e r w i t h a p e r s o n who i s w o r s e - o f f ) , t h e l e v e l of a l i e n a t i o n d e c r e a s e s , but t h e decrease i s s m a l l e r when t h e l e v e l o f a l i e n a t i o n i s h i g h e r . T h i s c o r r e s p o n d s t o t h e p s y c h o l o g i c a l e f f e c t mentioned
a t t h e b e g i n n i n g , a c c o r d i n g t o which when t h e r e i s a h i g h e r f e e l i n g o f d e p r i v a t i o n e a c h new i n s t a n c e o f p e r c e i v e d i n j u s t i c e or u n f a i r n e s s i s f e l t as more a c u t e . On t h e o t h e r hand, a n e n c o u n t e r w i t h a l e s s e s i n j u s t i c e t e n d s t o be n e g l e c t e d .
14.5.2.
A n a l y s i s . The above model i s t o o g e n e r a l t o p e r m i t a n a l y s i s , and one has t o make some a d d i t i o n a l s i m p l i f y i n g a s s u m p t i o n s . T h u s , one may assume f i r s t l y t h a t z1 = z = = z. T h i s means t h a t i n t h e p e r i o d under c o n s i d e r a t i o n , t h e a c c e s s t o goods C by t h e s u b j e c t do n o t change i n any a p p r e c i a b l e manner.
...
i t may be r e a s o n a b l y assumed t h a t t h e y a r e i n d e p e n d e n t i d e n t i c a l l y d i s t r i b u t e d random v a r i a b l e s . T h i s c o r r e s p o n d s t o t h e most i m p o r t a n t c a s e when t h e r e i s some i n h e r e n t v a r i a t i o n i n t h e a c c e s s t o goods C i n t h e p o p u l a t i o n o f p e r s o n s whom t h e subA s r e g a r d s xl, x
2’”’
FORMAL THEORY OF ACTIONS
90 1
j e c t p e r c e i v e s as " e q u a l l y d e s e r v i n g " t h e goods C . Under t h e s e a s s u m p t i o n s , d r o p p i n g for s i m p l i c i t y t h e c o n s t a n t z , t h e model becomes
where G i s some f u n c t i o n (f t F ) . One may now impose c o n d i t i o n s under which t h e sequence Yn i s s t a t i o n a r y , o r t e n d s t o a s t a t i o n a r y d i s t r i b u t i o n , depending on t h e d i s t r i b u t i o n of x and c o n s t a n t z. n The e x i s t e n c e of a s t a t i o n a r y d i s t r i b u t i o n means t h a t t h e p e r s o n ' s p e r c e i v e d l e v e l of a l i e n a t i o n f l u c t u a t e s around some a v e r a g e v a l u e ( d e p e n d i n g on z), w i t h o u t tendency t o grow or d e c r e a s e . I n p r a c t i c a l t e r m s , s u c h a s i t u a t i o n means % h a t ( f r o m p s y c h o l o g i c a l p o i n t o f view) a n a c c e p t a b l e e q u i l i b r i u m has been r e a c h e d : t h e s u b j e c t f e e l s t h a t he i s " l o c a t e d " a t some p l a c e o f t h e s c a l e of a c c e s s t o goods C , and a c c e p t s t h e v a r i a t i o n s o f t h i s l e v e l as normal. He u n d e r s t a n d s t h a t some p e r s o n s who a r e e q u a l l y d e s e r v i n g as h i m s e l f a r e w o r s e - o f f , w h i l e some o t h e r s a r e b e t t e r - o f f , b u t t h e numbers of these persons a r e balanced, so t h a t h i s f e e l i n g of a l i e n a t i o n ( f r o m a c c e s s t o goods C ) i s s t a b l e . On t h e o t h e r hand, i f Y n t e n d s t o i n c r e a s e , we have t h e s i t u a t i o n of a p e r s o n who f e e l s i n c r e a s i n g l y f r u s t r a t e d , b y e n c o u n t e r i n g t o o many p e r s o n s who a r e (undes e r v e d l y ) b e t t e r off t h a n h i m s e l f . I n t h i s c a s e , t h e p e r c e i v e d l e v e l of a l i e n a t i o n i s a self-perpetuating process. It i s c l e a r t h a t t h e model s u g g e s t e d h e r e may be e n r i c h -
902
CHAPTER 6
ed and m o d i f i e d , s o as t o t a k e i n t o a c c o u n t v a r i o u s other factors.
14.6.
Some h y p o t h e s e s
There i s no need t o p o i n t o u t t h a t t h e t h e o r y of t h e p r e s e n t s e c t i o n g i v e s only a very b r i e f s k e t c h o f t h e t h e o r y o f s o c i a l change, w i t h many d e t a i l s o m i t t e d f o r s i m p l i c i t y and s h o r t n e s s , and many a s p e c t s o f t h e phenomenon l e f t o u t . To i l l u s t r a t e , however, t h e a p p l i c a b i l i t y o f t h e s y s t e m
and i t s r i c h n e s s , some h y p o t h e s e s w i l l b e f o r m u l a t e d c o n c e r n i n g v a r i o u s a s p e c t s o f t h e phenomenon d e s c r i b e d . F i r s t l y , a s o c i a l change was d e f i n e d i n a f u z z y way as
a " s u b s t a n t i a l " change i n one o r more v a r i a b l e s o f t h e system. Whether a g i v e n change i s a c c e p t e d as s u b s t a n t i a l or n o t ( i . e . whether i t may be c l a s s i f i e d as a soci a l change) depends on many f a c t o r s , mainly on t h e pos i t i o n of a c l a s s i f y i n g p e r s o n i n t h e s o c i e t y , and h i s a l i e n a t i o n l e v e l . We have h e r e t h e f o l l o w i n g hypothesis. HYPOTHESIS 1. Any p e r s o n h a s a t h r e s h o l d , above which he t e n d s t o c l a s s i f y changes a s s u b s t a n t i a l . T h i s
threshold, i n general, i s
-- h i g h e r for changes which a f f e c t p o s i t i v e l y t h e judgi n g p e r s o n , and lower i f t h e y a f f e c t him n e g a t i v e l y ;
--
n e g a t i v e l y r e l a t e d t o t h e a l i e n a t i o n l e v e l of t h e person.
903
FORMAL THEORY OF ACTIONS
Another a s p e c t
o f s o c i a l changes i s c o n n e c t e d w i t h
t h e concepts sketched b r i e f l y i n t h e preceding sections. The p o i n t h e r e i s t h a t t h e t h e o r y c o v e r s what may be termed " t o t a l " c h a n g e s , a l o n g one or more o f t h e d i mensions o f t h e system i n t r o d u c e d . One can a l s o c o n s i d e r " l o c a l " changes, r e s t r i c t e d t o one o r more o f t h e g r o u p s , and c o n s i s t i n g o f changes of t h e s t r a t e g i e s ; changes o f g o a l s , changes o f r u l e s , as w e l l as changes i n t h e s t r u c t u r e o f t h e groups t h e m s e l v e s , even i f r e s t r i c t e d t o one group o n l y , can be c o n s i d e r e d as s o c i a l changes i n t h e s e n s e o f t h e p r e c e d i n g s e c t i o n s , which l e a v e i n v a r i a n t t h e r e l e v a n t a s p e c t s o f a l l b u t one group. The s t r a t e g y means h e r e t h e change o f s t r e s s o f t h e
a l l o c a t i o n of e f f o r t s by a group among t h e t h r e e domains o f i t s a c t i v i t y : (A1) i d e o l o g i c a l , (A2) f a c t u a l , and i n t e r n a l . Here one may f o r m u l a t e t h e f o l l o w i n g hypothesis. (A3)
HYPOTHESIS 2 . During t h e t i m e p r e c e d i n g and immediat e l y f o l l o w i n g a s o c i a l change i f form o f t r a n s i t i o n t o
a new a l l o c a t i o n o f goods, t h e group a c t i v i t i e s a r e c o n c e n t r a t e d a t f i r s t m o s t l y on a c t i v i t i e s o f t h e t y p e A1, t h e n on a c t i v i t i e s o f t h e t y p e A2, and t h e n , i n t h e p e r i o d immediately p r e c e d i n g t h e t r a n s i t i o n and f o l l o w i n g i t , on a c t i v i t i e s o f t y p e s A and A1.
3
HYPOTHESIS 3 . A s t h e a d m i s s i b i l i t y o f t h e a c t u a l a l l o c a t i o n d e c r e a s e s (as it must, a c c o r d i n g t o t h e p o s t u l a t e s of t h e t h e o r y ) , t h e groups i n t e r e s t e d i n p r e s e r v i n g i t w i l l c o n c e n t r a t e mainly on a c t i v i t i e s o f t h e t y p e s w h i l e t h e groups i n t e r e s t e d i n changing i t A1 and A
3'
904
CHAPTER 6
w i l l c o n c e n t r a t e mainly on a c t i v i t i e s o f t h e t y p e A
2
and A1. HYPOTHESIS 4. The e f f e c t i v e n e s s a f a c t i v i t y o f t h e t y p e A1 ( i d e o l o g i c a l ) depends p r i m a r i l y on t h e p r o p e r t i e s o f t h e network o f communication i n S , and i n c r e a s e s w i t h t h e i n c r e a s e of t h e number o f c o n n e c t i o n s p e r node. HYPOTHESIS 5 . An i n c r e a s e o f i n t e n s i t y of a c t i v i t y o f t y p e A1 ( i d e o l o g i c a l ) by two or more groups w i t h oppos i n g p r e f e r e n c e p a t t e r n s , l e a d s t o a d e c r e a s e o f fuzzin e s s o f t h e groups s t r u c t u r e ( " p o l a r i z a t i o n o f t h e society"). HYPOTHESIS 6 . Major changes i n t h e s t r u c t u r e and cont e n t o f t h e s e t of r u l e s associated with various groups a r e u s u a l l y f o l l o w e d by major changes i n t h e groups s t r u c t u r e i t s e l f , by l e a d i n g t o an a p p e a r e n c e o f new g r o u p s . A s a l r e a d y mentioned, t h e main d r i v i n g f o r c e s b e h i n d s o c i a l changes a r e induced by a l i e n a t i o n of
members
o f v a r i o u s g r o u p s . Thus, t h e h y p o t h e s e s about a l i e n a t -
i o n ( s e e Nowakowskal977a) a r e r e l e v a n t h e r e , i n t h e c o n t e x t o f s o c i a l changes. I n p a r t i c u l a r , one s h o u l d menion here t h e f o l l o w i n g h y p o t h e s e s . HYPOTHESIS 7 . A b l o c k i n g monopole, hence a monopole i n t e r e s t e d i n p r e v e n t i n g a s o c i a l change a d v e r s e t o
i t s i n t e r e s t s , w i l l t e n d t o s u p p o r t o t h e r s o c i a l change s , i n p a r t i c u l a r t h o s e which l e a d t o i n c r e a s e of t h e t o t a l a d m i s s i b i l i t y o f a l l o c a t i o n s , hence a l s o l e a d i n g t o a decrease i n alienation l e v e l s .
FORMAL THEORY OF ACTIONS
905
HYPOTHESIS 8 . A s o c i a l change i n form o f an a p p e a r e n c e o f a new group s t r u c t u r e , t h r o u g h t h e c r e a t i o n o f a counter-monopole, i s more l i k e l y i n c a s e s when t h e l e v e l o f a l i e n a t i o n i s h i g h , and i t s v a r i a n c e i s s m a l l , t h a n i n the opposite cases. F i n a l l y , t h e f o l l o w i n g h y p o t h e s i s i s r e l e v a n t f o r pred i c t i o n o f dynamics of monopoles, hence a l s o t h e dynamics of t h e s o c i a l s t r u c t u r e . HYPOTHESIS 9 . I f f ( t ) i s t h e p r o b a b i l i t y t h a t a p e r s o n ' s p a r t i c i p a t i o n i n a monopole i s e x a c t l y t , and F ( t ) i s t h e probability t h a t it i s t o r l e s s , then t h e r a t i o f ( t ) / [ l - F ( t ) l i s an i n c r e a s i n g f u n c t i o n o f t . The l a s t h y p o t h e s i s , a s s e r t i n g t h e " a g i n g p r o p e r t y " o f t h e d i s t r i b u t i o n o f p a r t i c i p a t i o n i n a monopole i s j u s t i f i e d by t h e i n t u i t i o n a c c o r d i n g t o which t h e longe r a p e r s o n i s i n a monopole, t h e more chances t h a t he i s s a t i s f i e d w i t h t h e goods he r e c e i v e d , and t h e r e f o r e , t h e more c h a n c e s he w i l l a t t e m p t t o j o i n a n o t h e r monopole. The p r e c e d i n g h y p o t h e s e s concerned t h e r o l e o f a l i e n -
a t i o n i n i n d u c i n g s o c i a l changes. I n connection w i t h t h i s , one can a l s o s t a t e t h e f o l l o w i n g h y p o t h e s e s of a "negative" character. HYPOTHESIS 1 0 . A s o c i a l change i s l e s s l i k e l y t o o c c u r , i f t h e a l i e n a t e d groups: -- r e c e i v e some " g o o d s - s u b s t i t u t e s " , as l o n g as t h e y a r e approved o r v a l u a b l e , o r -- t h e group a c t i v i t y i s d i r e c t e d towards some " a c t i o n substitutes", o r
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-- t h e g r o u p ’ s p r e s e n t a l l o c a t i o n i s t h r e a t e n e d . HYPOTHESIS 11. If t h e group i s under n e g a t i v e i n f o r m a t i o n p r e s s u r e o r g a n i z e d by competing g r o u p s , i t may dec r e a s e t h e l i k e l i h o o d o f r e a l changes by making a p p a r e n t changes ( e . g . t h r o u g h s e m a n t i c a l l y e q u i v a l e n t p h r a s i n g o f programs, r e a s o n s and r u l e s ) . HYPOTHESIS 1 2 . The t i g h t e r a group i s o r g a n i z e d , and t h e more s u c h a group a p p r o v e s o f i t s r e a s o n s and programs, t h e more r e s i s t a n t i t w i l l be t o new i d e a s , and t h e l e s s p r o n e t o change.
1 4 . 7 . Some o t h e r mechanisms o f s o c i a l change The mechanisms connected w i t h a l i e n a t i o n p r e s e n t e d above a r e g e n e r a l l y t r e a t e d as one o f t h e main c a u s e s o f s o c i a l changes. Another mechanism i s t h a t o f t h e s o c i a l p r e s s s u r e on d e c i s i o n s . These p r e s s u r e s , i n some con-
t e x t s may be towards t h e c h a n g e s , and i n some o t h e r contexts
--
a g a i n s t t h e changes. I n c o n n e c t i o n w i t h
s o c i a l p r e s s u r e s , we s h a l l s u g g e s t a s i m p l e model o f a group d e c i s i o n i n o p p o s i n g or s u p p o r t i n g some i s s u e . The c e n t r a l a s s u m p t i o n w i l l be t h a t t h e p r o b a b i l i t y o f a p o s i t i v e d e c i s i o n by n - t h p e r s o n depends on t h e number o f p o s i t i v e d e c i s i o n s o f t h e p r e c e d i n g n-1 p e r sons.
I n o t h e r words, we imagine t h a t t h e group d e -
c i s i o n i s r e a c h e d s e q u e n t i a l l y , and t h a t e v e r y p e r s o n h a s some knowledge about t h e d e c i s i o n s o f h i s p r e d e c e s -
sors, and a c t s a c c o r d i n g l y . The r e s u l t s o f t h i s model w i l l allow us t o optimally a l l o c a t e t h e resources between v a r i o u s propaganda t a s k s .
907
FORMAL THEORY OF ACTIONS
I n c i d e n t a l l y , i t i s w o r t h n o t i n g t h a t a new d e c i s i o n model o f p r o s o c i a l and a n t i s o c i a l b e h a v i o u r , g i v e n i n C h a p t e r 5 , i s a l s o r e l e v a n t h e r e , namely as a model of p r e s s u r e e x e r t e d by a group, l e a d i n g t o t h e mo d if ic a t i o n o f g r o u p r u l e s , and i n d i r e c t l y , a s o c i a l c h a n g e o f group s t r u c t u r e and g o a l s . T h i s model r e a c h e s d e e p e s t i n t o t h e mechanisms o f g r o u p b e h a v i o u r , by s h o w i n g t h e r e l a t i o n s b e t w e e n a c t i o n s a n d o p i n i o n s , e x p e c t a t i o n s , and t y p e s o f reward and punishment by t h e g r o u p . One o f t h e most i m p o r t a n t f e a t u r e s o f t h i s model i s i t s a b i l i t y o f s u p p l y i n g p r e d i c t i o n s . F i n a l l y , t h e communication network i n system ( 1 4 . 1 ) and t y p e o f s p r e a d of i n f o r m a t i o n i n t h e group, a r e a l s o o n e o f t h e most i m p o r t a n t f a c t o r s d e t e r m i n i n g
t h e group
a n d i t s d y n a m i c s . An o u t l i n e c f s u c h a t h e o r y o f n e t works w i l l be g i v e n i n n e x t s u b s e c t i o n s .
14.7.1. Group p r e s s u r e s a n d g r o u p c h o i c e .
The r e s u l t s
o f t h i s s e c t i o n were f i r s t p u b l i s h e d i n Nowakowska (1982b), and t h e r e a d e r i s r e f e r r e d t h e r e f o r t h e p r o o f s o f t h e t h e o r e m s . The model c o n c e r n s a s i t u a t i o n i n w h i c h a g r o u p , s a y G , o f a f i x e d s i z e N , makes a d e c i s i o n l l f o r l r or " a g a i n s t " a c e r t a i n i s s u e . C l e a r l y , s u c h
a s i t u a t i o n i s a b a s i c i n g r e d i e n t o f more complex s o c i a l s i t u a t i o n s o f i n t e r p l a y b e t w e e n a number o f g r o u p s f i g h t i n g f o r and a g a i n s t some i s s u e .
We s h a l l assume t h a t t h e i s s u e i n q u e s t i o n , s a y I , i s f i x e d , a n d e v e r y member o f t h e g r o u p h a s t o r e s p o n d t o i t p o s i t i v e l y o r n e g a t i v e l y ( v o t e " f o r " or " a g a i n s t f f ) . One may t h i n k here o f s u c h i s s u e s I as j o i n i n g a s t r i k e ,
908
CHAPTER 6
e l e c t i n g a c a n d i d a t e , d o n a t i n g money t o some c a u s e , e t c . Group membership i n f l u e n c e s a p e r s o n l s r e s p o n s e i n a t l e a s t two ways. F i r s t l y , t h e p r i o r p r o b a b i l i t y o f a pos i t i v e r e s p o n s e may be d i f f e r e n t t h a n f o r members i n o t h e r s o c i a l g r o u p s . I n o t h e r w o r d s , g r o u p membership i s some s e n s e c o n d i t i o n s t h i n k i n g and c o n v i c t i o n s , a n d makes members o f t h e g r o u p t h i n k " a l i k e " i n h i g h e r d e g r e e t h a n members o f d i f f e r e n t g r o u p s ( w h i c h d o e s n o t mean t h a t t h e y have t o be o f t h e same o p i n i o n , o f course. ) S e c o n d l y , a p e r s o n ' s d e c i s i o n may be i n f l u e n c e d by h i s knowledge a b o u t t h e d e c i s i o n s o f o t h e r members o f t h e g r o u p , who have a l r e a d y made t h e d e c i s i o n . The model below c o n c e r n s t h i s s e c o n d e f f e c t , w h i c h may be c a l l e d "group p r e s s u r e " . We assume t h e r e f o r e t h a t e a c h member of t h e g r o u p G i s
t o make a p o s i t i v e o r n e g a t i v e d e c i s i o n a b o u t t h e i s s u e I , w i t h no t i e s a l l o w e d . L e t xN b e t h e t o t a l number of p e r s o n s i n g r o u p G who d e c i d e s d r r f o r r r .Then xN may b e c a l l e d t h e g r o u p s u p p o r t f o r I , a n d o u r main o b j e c t i v e w i l l b e t o make i n f e r e n c e and p r e d i c t i o n s a b o u t x
"
as w e l l as t o f i n d ways o f c o n t r o l l i n g E ( x N ) . We c o n c e p t u a l i z e t h e d e c i s i o n p r o c e s s as o c c u r r i n g
s e q u e n t i a l l y ; moreover, w e imagine t h a t a f t e r n-th p e r s o n made h i s d e c i s i o n , t h e ( n + l ) - s t p e r s o n knows how many among t h e f i r s t n made t h e d e c i s i o n i n f a v o u r of I. N a t u r a l l y , such an assumption i s a n i d e a l i z a t i o n : i n r e a l s i t u a t i o n s , a p e r o n s who makes h i s d e c i s i o n d o e s n o t h a v e a c o m p l e t e knowledge a b o u t t h e d e c i s i o n s of
909
FORMAL THEORY OF ACTIONS
o t h e r s . However, he may h av e a f a i r i d e a a b o u t i t , a n d t h i s know l e dg e, v ag u e as i t may b e , c a n i n f l u e n c e t h e p r o b a b i l i t y of h i s p o s i t i v e d e c i s i o n . A s an example, i m a g i n e a v o t i n g a t a c o n v e n t i o n o f some s o r t . A g i v e n v o t e r u s u a l l y has some i d e a ( f r o m t a l k i n g w i t h o t h e r d e l e g a t e s , from s p e e c h e s made, e t c . ) w h e t h e r m a j o r i t y w i l l be r r f o r r lo r " a g a i n s t " , and t h i s knowledge may influence his decision. I n t h e model of t h i s s e c t i o n , t h e a b o v e phenomenon w i l l b e d e s c r i b e d i n form o f d ep en d e n c e o f p r o b a b i l i t y of p o s i t i v e d e c i s i o n of n-th c o n s e c u t i v e p e r s o n on t h e number o f v o t e s ' ' f o r r r among t h e n -1 p e r s o n s who p r e c e ded him.
...,
x l , x2, x N , where x i s t h e number o f v o t e s " f o r r r among t h e f i r s t n n members o f t h e g r o u p . We assume t h a t t h i s p r o c e s s i s a Markov c h a i n , w i t h t h e f o l l o w i n g t r a n s i t i o n p r o b a b i l i t i e s : l e t xo = 0 , an d f o r n = 1,2, We d e a l h e r e w i t h t h e p r o c e s s
...
P(xntl
= ktllx,
= k) = p
P
= k
I xn
= k) = 1
+
k
a(1 -p )--nt1
(14.5)
an d X
where 0
nt1 p
<
1 an d 0 <, a
5
- P ( xn t 1= k t l ( x n 1
= k),
a r e some c o n s t a n t s .
Formu a ( 1 4 . 5 ) g i v e s t h e p r o b a b i l i t y t h a t t h e ( n t 1 ) - s t p e r s o n w i l l make t h e d e c i s i o n i n f a v o u r o f I , g i v e n t h a t among t h e f i r s t n p e r s o n s v o t i n g t h e r e were e x a c t l y k f a v o u r i n g t h e d e c i s i o n . The n e x t f o r m u l a s t a t e s s i m p l y t h a t a p e r s o n must e i t h e r f a v o u r o r
910
CHAPTER 6
oppose t h e i s s u e .
Here p i s t h e " g r o u p v a l u e " o f t h e p r o b a b i l i t y o f fav o u r i n g t h e i s s u e . One may i n t e r p r e t i t a s t h e p r i o r p r o b a b i l i t y o f f a v o u r i n g t h e i s s u e by a member o f t h e g r o u p , w i t h o u t a n y knowledge a b o u t t h e v o t e s o f o t h e r s . On t h e o t h e r h a n d , t h e c o n s t a n t a r e f l e c t s ( l a c k o f ) r e s i s t a n c e t o group p r e s s u r e : h i g h e r v a l u e o f a c o r r e s pond t o s i t u a t i o n s when g r o u p members a r e more a f f e c t e d by t h e knowledge o f t h e d e c i s i o n s by o t h e r s . Thus, t h e p r o b a b i l i t y o f a d e c i s i o n i n f a v o u r i s e q u a l t o t h e group v a l u e p , i n c r e a s e d p r o p o r t i o n a l l y t o t h e number who a l r e a d y v o t e d " y e s " . To f o r m u l a t e t h e main t h e o r e m s a b o u t t h e p r o c e s s d e n o t e for j = 1 , 2 ,
b c
3 j
...
= 1 t 2a(l-p)/j
(14.7)
= 2p t a ( l - p ) / j ,
(14.8)
and p u t f o r i
<
j
R?, = aiai+l.. . a
J
!Q
xn '
= b2bifl..
. bJf .
(14.9) (14.10)
We h a v e now t h e f o l l o w i n g two t h e o r e m s w h i c h e x p r e s s t h e e x p e c t a t i o n a n d v a r i a n c e o f t h e v a r i a b l e x" e q u a l t o t h e t o t a l g r o u p s u p p o r t for t h e i s s u e .
91 1
FORMAL THEORY OF ACTIONS
THEOREM 1.
The e x p e c t e d group s u p p o r t e q u a l s
-
N- 1
P C +~
E(xN) =
2-
R Nj t l l ,
(14.11)
j=l
where RN i s g i v e n by ( 1 4 . 9 ) . j
THEOREM 2 . The v a r i a n c e o f group s u p p o r t e q u a l s V a r (x,)
N-1 N CN 'jQj+1 = E ( x )[-- + ---m--1 "N j = 1 ajRj+l
2:
+
N pQ1
N
2 C7
1
j=1 Q 1 where c
k --:TI c
-
E(xN12,
(14.12)
Y=IQ;R;
and
Ql
and
E ( x N ) is g i v e n by ( 1 4 . 1 1 ) .
j -
(14.10),
-
a r e g i v e n r e s p e c t i v e l y by ( 1 4 . 8 )
and
The p r o o f s i n v o l v e c o n s t r u c t i o n of a n a p p r o p r i a t e mart i n g a l e ; t h e y may be found i n Nowakowska (198213). T a b l e s 1 and 2 show t h e e f f e c t s o f group " c o h e s i v e n e s s "
a on e x p e c t a t i o n and v a r i a n c e ; i t may be s e e n t h a t these e f f e c t s are quite appreciable. The v a l u e s o f t h e e x p e c t a t i o n E ( x N ) = E ( x N ; p , a ) i n c r e a s e w i t h b o t h p and a . When a = 0 , t h e d i f f e r e n c e s xn - x n- 1 a r e i n d e p e n d e n t , and e q u a l 0 or 1 w i t h p r o b a b i l i t i e s 1-p and p r e s p e c t i v e l y . C o n s e q u e n t l y , xN h a s b i n o m i a l distribution P ( x N = k ) = (,)p N k( l - ~ ) ~ k =- ~ 0 , 1,,
...,N
(14.13)
s o t h a t E ( xN ) = E(xN;p,O) = Np, and V a r (x,) = N p ( 1 - p ) . Thus, t h e s e v a l u e s p r o v i d e a c o n v e n i e n t r e f e r e n c e v a l u e s .
912
CHAPTER .6
P
a
100
N 500
1000
25.00 125.00 250.00 153.08 306.78 30.28 (5.2%) (5.6%) (5.6%) 0.50 195.77 394.40 37.86 (14.1%) (14.4%) ( 12.8%) 265.09 540.71 49.11 0.75 ( 24.1%) (28.0%) (29.0%) 66.37 385.80 807.38 1.00 ( 52.1%) (55.7%) (41.3%) _---- .-______----____--------------------------50.00 250.00 500.00 0.00 0.25 56.63 284.97 570.56 (6.6%) (6.9%) (7.0%) 330.52 663.20 0.50 65.00 ( 15.0%) (16.1%) (16.3%) 391.54 788.80 75.72 0.75 (28.3%) (28.8%) (25.7%) 0.00
0.25
1.00 .----,
0.00
0.25 0.50 0.75 1.00 .----. Table
375.00 399.58 79.70 ( (4.6%) (4.9%) 427,45 84.95 (9.9%) (10.4%) 459.25 90.85 (16.8%) ( 15.8%) 495.78 97.51 (22.5%) --------------(24.1%) .- -- ----------
750.00 799.53 (4.9%) 855.84 (10.5%) 920.34 ( 17.0%) 994.79 (24.4%) .--------
1. E x p e c t a t i o n s E ( x N ; p , a ) for s e l e c t e d v a l u e s
N , p and a . The numbers i n p a r e n t h e s e s g i v e t h e p e r c e n t a g e i n c r e a s e o f group s u p p o r t due t o
group cohe-
s i v e n e s s a , i .e . [ E ( x N ; p , a ) - E ( x N ; p , O ) 1100/N.
913
FORMAL THEORY OF ACTIONS
The numbers
i n p a r e n t h e s e s i n Table 1 g i v e t h e r e l a -
t i v e p e r c e n t a g e i n c r e a s e of E ( x ) due t o a . For i n s t a n N c e , i f N = 500 and p = 0.5, for a = 0.75 we may exp e c t o v e r 390 p e r s o n s i n t h e group t o f a v o u r I , which i s an i n c r e a s e o f 28.3% o v e r t h e e x p e c t e d 250 i n cas e when a = 0 .
0.1 0.2
13.05
0.3
13.64
2.61 2.72
0.7 0.8
17.31 18.22
14.29
2.84
0.9
19.21
3.34 3.46 3.57
T a b l e 2. Expected group s u p p o r t E ( x N ) and s t a n d a r d d e v i a t i o n of x for N = 25, p = 0.5 and v a r i o u s a .
N
T a b l e 2 g i v e s t h e e x p e c t a t i o n s and s t a n d a r d d e v i a t i o n s of xN f o r N
=
25 and p = 0.5, under v a r i o u s v a l u e s o f
a . Again, t h e r e f e r e n c e p o i n t i s t h e b i n o m i a l d i s t r i b u t i o n , for a = 0 . A s may be s e e n , t h e v a r i a n c e of xN i n c r e a s e s w i t h a . T h i s means i n p r a c t i c e , t h a t t h e a c t u a l v a l u e s o f xN i n c a s e o f p o s i t i v e group cohes i v e n e s s ( a > 0 ) a r e more d i f f i c u l t t o p r e d i c t t h a n i n t h e binomial c a s e . T o g e t a n i d e a about t h e s h a p e o f t h e d i s t r i b u t i o n o f
xN ’ some s i m u l a t i o n s t u d i e s were c a r r i e d o u t (see Nowakowska 1982b). It a p p e a r s t h a t t h e d i s t r i b u t i o n i s asymmetric, w i t h l o n g e r l e f t t a i l .
914
CHAPTER 6
I f one a c c e p t s t h e model o f t h i s s e c t i o n a s a r e a s o n -
a b l e a p p r o x i m a t i o n of r e a l p r o c e s s e s which o c c u r i n group d e c i s i o n s , t h e n one c o u l d use t h e r e s u l t s t o o b t a i n t h e o p t i m a l r e s o u r c e a l l o c a t i o n s i n propaganda e f f o r t s i n t h e f o l l o w i n g way. Suppose t h a t a t some moment, t h e v a l u e s which c h a r a c t e r i z e a g i v e n group a t some t i m e ( p r i o r t o a c t u a l d e c i s i o n ) a r e p and a o . We may t h e n c a l c u l a t e t h e 0 e x p e c t e d group s u p p o r t E ( x N ; p O , a O ) . Assume f u r t h e r t h a t t h e l a t t e r v a l u e i s n o t s u f f i c i e p t for t h e s o c i a l p u r p o s e under c o n s i d e r a t i o n , and one needs t o s e c u r e t h e e x p e c t e d v a l u e a t l e a s t K , where
T h i s g o a l may be a t t a i n e d by i n c r e a s i n g p a n d / o r a . A l o o k a t F i g . 1 shows t h a t t h e c o n t o u r s E ( x N ; p , a ) =
c o n s t i n t h e ( p , a ) - p l a n e a r e concave. T h i s f a c t w i l l p l a y a n e s s e n t i a l r o l e i n Theorem 3 below. Now, t h e c o s t of i n c r e a s i n g t h e p a r a m e t e r s from ( p O , a O ) t o ( p , a ) w i t h P 2 p O J a 3 a. i s some v a l u e C ( p , a ; p o , a o ) , and t h e problem may i n g e n e r a l be f o r m u l a t e d as t h a t * * o f f i n d i n g ( p , a ) which s a t i s f y t h e g o a l a t minimal cost, that is,
The s o l u t i o n w i l l , i n g e n e r a l depend on t h e form o f t h e f u n c t i o n C ; i f C i s n o t l i n e a r , we have a n o n l i n e a r o p t i m i z a t i o n problem. Assume, however, t h a t C i s lin e a r i n p and a , s o t h a t
FORMAL THEORY OF ACTIONS
915
Fig. 1. Contous E ( x N ; p , a ) = const in the (p,a)-plane, f o r N = 20, and c = 8, 10, 12, 14, 16 and 18.
916
CHAPTER 6
We have i n t h i s c a s e t h e f o l l o w i n g t h e o r e m THEOREM 3. I n t h e l i n e a r c a s e (14.15), t h e s o l u t i o n
t o t h e o p t i m a l a l l o c a t i o n problem i s a l w a y s of one o f the following three types: Type 1.
a
Type 2 .
p
Type 3 .
a
* f
*
=
ao, p
= po,
= 1, p
a
*
* * =
= m i n i p : E ( x N ; p , a o ) >/
K]
= m i n i a : E ( x N ; p O J a )2
K]
minip: E ( x N ; p , l )
2 K].
For t h e p r o o f , s e e s e l f - e x p l a n a t o r y F i g u r e 2 . G e n e r a l l y , Type 1 w i l l be o p t i m a l f o r s u f f i c i e n t l y s m a l l r a t i o C,/C,
o f c o s t o f u n i t i n c r e a s e of p and
a r e s p e c t i v e l y ; Types 2 and 3 w i l l be a p p l i c a b l e f o r h i g h C,/C2.
Whether or n o t t h e s o l u t i o n i s of t y p e 2
or 3 depends on w h e t h e r E ( x N ; p O , l ) e x c e e d s K o r n o t . These r e s u l t s may be o f g r e a t i m p o r t a n c e i n d e s i g n n i n g s o c i a l p o l i c i e s . G e n e r a l l y , an i n c r e a s e o f p i n v o l v e s e f f o r t s aimed a t e x p l a i n i n g t h e m e r i t s of t h e i s s u e I t o members of t h e g r o u p , s t r e s s i n g t h r e a t s c o n n e c t e d w i t h r e j e c t i n g I , and s o f o r t h . On t h e o t h e r h a n d , e f f o r t s aimed a t i n c r e a s i n g a w i l l c o n s i s t o f s t r e s s i n g t h e need f o r group u n i t y , e t c . The c o s t s i n v o l v e d i n t h e i n c r e a s e o f p and a by t h e same amount may b e different. Theorem 3 a s s e r t s t h a t e x c e p t f o r Type 3 , t h e o p t i m a l strategy advocates concentrating t h e e f f o r t s e n t i r e l y on a n i n c r e a s e of one p a r a m e t e r o n l y . Type 3 i n v o l v e s
maximal i n c r e a s e o f a ( t o a
=
l), f o l l o w e d by m i n i m a l
FORMAL THEORY OF ACTIONS
917
n e c e s s a r y i n c r e a s e of p .
a
an =a*
a
1
I
PO
P
3 P
H
I
I
0----
I
I
I
IH
)P
P
PO Type 3
F i g . 2 . I l l u s t r a t i o n o f Theorem 3; t h e c u r v i l i n e a r cont o u r i s E ( x N ; p , 3 ) = K , arid t h e s t r a i g h t l i n e i s t h e c o s t f u n c t i o n c,(p-p0)
t
c,(a-a,)
= const.
918
14.7.2.
CHAPTER 6
Communication n e t w o r k s .
One o f t h e f a c t o r s
which d e t e r m i n e t h e v a l u e o f c o h e s i v e n e s s o f t h e g r o u p , e s s e n t i a l i n t h e model o f t h e p r e c e d i n g s e c t i o n , i s t h e communication network between members o f t h e group. I n t h i s s e c t i o n , t h e r e f o r e , we s h a l l s u g g e s t a model f o r t h e s p e e d and scope o f d i s s e m i n a t i o n o f i n f o r m a t i o n i n a group. The a n a l y s i s w i l l concern t h e communication network.
It i s i m p o r t a n t t o mention t h a t t h e t e r m "communication" need n o t be t r e a t e d l i t e r a l l y . F o r m a l l y , a communicati o n network Lwill be a b i n a r y r e l a t i o n i n group G , r e p r e s e n t i n g some s o r t o f s o c i a l c o n t a c t s , and i t w i l l be assumed t h a t t h e o b j e c t s which a r e b e i n g d i s s e m i n a t e d i n t h e s o c i e t y -- such a s news, i n n o v a t i o n s , d i s e a s e s , o p i n i o n s , e t c , -- are t r a n s m i t t e d from one p e r s o n t o another i f they a r e r e l a t e d by t h e r e l a t i o n
2.
The problem w i l l be t o s t u d y t h e p r o p o r t i o n o f p e r s o n s whom t h e new i d e a e v e n t u a l l y r e a c h e s , t h e r a t e o f d i s semination, e t c . B a s i c a l l y , t h e r e a r e two a p p r o a c h e s p o s s i b l e h e r e . One ( v e r y much e x p l o i t e d , as t h e problem i s one o f t h e o l d e s t i n s o c i o l o g i c a l r e s e a r c h and has a l r e a d y a l o n g t r a d i t i o n ) i s t o make some a s s u m p t i o n s about t h e r e l a t i o n 2 , and t h e n t r y t o deduce t h e p r o p e r t i e s o f i n t e r e s t from t h e s e a s s u m p t i o n s . Such a n approach i s r e a s o n a b l e f o r small g r o u p s . However, f o r a s o c i e t y c o n s i s t i n g o f a l a r g e number o f i n d i v i d u a l s ( m i l l i o n s , i f one c o n s i d e r s s u c h groups as a n a t i o n , i n h a b i t a n t s o f a town, e t c . ) , s u c h a n a p p r o a c h a p p e a r s i m p o s s i b l e . Thus, a n a l t e r n a t i v e i s t o assume t h a t t h e r e l a t i o n i s random and t o a n a l y s e t h e p r o b a b i l i t i e s o f v a r i o u s
c
919
FORMAL THEORY OF ACTIONS
e v e n t s , e s p e c i a l l y t h o s e which have p r o b a b i l i t y c l o s e t o 1 ( s u c h e v e n t s a r e o f primary i n t e r e s t , s i n c e t h e y a l l o w t o i n f e r t h a t a n a l o g o u s e v e n t s must o c c u r i n t h e r e a l s o c i e t y , r e g a r d l e s s o f t h e s p e c i f i c form or' t h e relation
1).
To p r o c e e d f o r m a l l y , i t w i l l be more c o n v e n i e n t t o de-
n o t e t h e f a c t t h a t t h e e l e m e n t s a r e r e l a t e d t h r o u g h relation by a n arrow; t h u s , s 3 t means t h a t t h e r e i s
1
a r e l a t i o n between s and t ( t h o u g h n o t n e c e s s a r i l y b e t ween t and s ) . One can now d e f i n e t h e r e l a t i o n s
+t
if
s ->t;
s
Tt
if
( 3 ~ )s : a x
>- n
and x
as f o l l o w s :
Tt.
F i n a l l y , a d o u b l e arrow w i l l i n d i c a t e t h e sum o f a l l i . e . t h e t r a n s i t i v e c l o s u r e o f --+ : relations
2,
s 3 t
if
( 3 n ) : s *t.
Let now s be f i x e d , and c o n s i d e r t h e s e t s
A(s) = f t : s 3 j t )
(14.16)
i t : t =>.I.
(14.17)
and
B(s) =
Thus, i f t h e r e l a t i o n 4 i s r e p r e s e n t e d as a d i r e c t e d
...
and e d g e s c o r r e s g r a p h , w i t h nodes a t p o i n t s s , t , ponding t o p a i r s connected w i t h t h e r e l a t i o n , t h e n
920
CHAPTER 6
A ( s ) i s t h e s e t of a l l p o i n t s which can be r e a c h e d from
s f o l l o w i n g t h e e d g e s , and B ( s ) i s t h e s e t o f a l l p o i n t s from which s can be r e a c h e d f o l l o w i n g t h e edges ( i . e . which may be r e a c h e d from s by g o i n g a g a i n s t t h e arrows)
.
Moreover, i f D C G , t h e n l e t
(14.18) and
Thus, A ( D ) and B ( D ) a r e t h e s e t s o f p o i n t s which may be r e a c h e d from p o i n t s i n D by g o i n g a l o n g , and a g a i n s t t h e arrows.
NOW, i f t h e r e l a t i o n + i s
random, s o a r e t h e s e t s
A(D) and B ( D ) . One o f t h e i n t e r e s t i n g q u e s t i o n s i s : suppose t h a t t h e s e t D c o n t a i n s k e l e m e n t s , and t h a t t h e s e t G c o n t a i n s N e l e m e n t s . Given t h e a v e r a g e number o f e d g e s l e a v i n g a g i v e n node t o be q , what i s t h e d i s t r i b u t i o n o f t h e s i z e s o f t h e s e t s A ( D ) and B ( D ) , and i n p a r t i c u l a r , what i s t h e p r o b a b i l i t y t h a t A(D) = G?
F o r a p o s s i b l e a p p l i c a t i o n , imagine t h a t D i s t h e s e t o f p e r s o n s who o r i g i n a t e some new i d e a , and t h a t t h i s i d e a s p r e a d s a l o n g t h e a r r o w s o f communication network. Then A ( D ) i s t h e s e t o f p e r s o n s who w i l l e v e n t u a l l y be " i n f e c t e d " w i t h t h e idea. One c a n c o n j e c t u r e h e r e t h a t t h e r e e x i s t s a t h r e s h o l d o f d e n s i t y o f c o n n e c t i o n s ; t h i s c o n j e c t u r e may be for-
FORMAL THEORY OF ACTIONS
921
mulated as f o l l o w s . For a f i x e d N and k ( s i z e s o f G and D ) , and t h e a v e r a g e number o f e d g e s p e r node b e i n g q , ( s o t h a t t h e r e a r e N q edges a l t o g e t h e r ) , l e t
be t h e p r o b a b i l i t y t h a t t h e s i z e of t h e s e t A(D) w i l l exceed A N ( h e r e 0 < d G l ) , i . e . t h a t t h e " i n f e c t i o n " w i l l c o v e r t h e f r a c t i o n d o f t h e group. HYPOTHESIS 1. For s u f f i c i e n t l y l a r g e N , f i x e d k and
M,
there e x i s t s a threshold i n q , i . e . P ( ( A ( D ) l ZdN) N ,k,q w i l l be c l o s e t o 0 i f q i s l e s s t h a n t h i s t h r e s h o l d , and c l o s e t o 1 i f q e x c e e d s t h i s t h r e s h o l d . HYPOTHESIS 2 . For s u f f i c i e n t l y l a t g e N , f i x e d q and d , t h e r e e x i s t s a t h r e s h o l d i n k , s u c h t h a t PN ( WD)I ,q,k ,&N) w i l l be c l o s e t o 0 i f k i s below t h e t h r e s h o l d , and c l o s e t o 1, i f k i s above t h i s t h r e s h o l d . The f i r s t o f t h e s e h y p o t h e s e s p r e d i c t s e s s e n t i a l q u a l i -
t a t i v e changes i n t h e scope and s p r e a d o f t h e i d e a when t h e d e n s i t y o f r e l a t i o n s ( f r e q u e n c y o f c o n t a c t s , " t i g h t n e s s " of t h e g r o u p , e t c . ) e x c e e d s a c e r t a i n lim i t . The second h y p o t h e s i s p r e d i c t s t h e same phenomenon under t h e changes of t h e s i z e o f t h e i n i t i a l s e t . I f t h e s e h y p o t h e s e s a r e t r u e , t h e y would have some
very i n t e r e s t i n g s o c i a l implications concerning t h e o p t i m a l c h o i c e of network d e n s i t y a'nd a l s o t h e o p t i m a l c h o i c e o f t h e s i z e of t h e group which i s t o e f f e c t i v e l y s p r e a d t h e i d e a a c r o s s t h e group. The s u g g e s t e d a p p r o a c h o f f e r s a l s o numerous o t h e r po-
922
ssibilities.
CHAPTER 6
To mention j u s t one of them, suppose t h a t
one wants t o s t u d y t h e r e s i s t a n c e t o a c c e p t i n g new i d e a s b e f o r e p a s s i n g them on ( t h i s may be i n t e r p r e t e d , f o r i n s t a n c e , as i d e o l o g i c a l r e s i s t a n c e , e t c . ) . T h i s may b e a t t a i n e d simply by assuming t h a t t h e i n f o r m a t i o n i s a c c e p t e d b y a p e r s o n o n l y i f he r e c e i v e s i t from a t l e a s t m s o u r c e s , s o t h a t m i s a measure of h i s r e s i s t a n c e . Thus, t h e s p r e a d i n g p r o c e e d s as b e f o r e , e x c e p t t h a t i t l e a v e s a g i v e n node o n l y i f t h e r e a r e a t l e a s t m a r r o w s l e a d i n g t o i t , e a c h of t h e s e arrows o r i g i n a t i n g a g a i n from a node t o which t h e r e a r e a t l e a s t m a r r o w s , e t c . Here one may a l s o e x p e c t some t h r e s h o l d e f f e c t s , under t h e change of r e s i s t a n c e e x p r e s s e d by t h e number m.
1 4 . 8 . Concluding remarks. Some f u r t h e r p e r s p e c t i v e s Thus f a r , system ( 1 4 . 1 ) was a n a l y s e d t h r o u g h c e r t a i n submodels, s u c h as t h a t of a l i e n a t i o n , freedom, s o c i a l p r e s s u r e , o r communication n e t w o r k s . These models were aimed a t a n a n a l y s i s of some i n t e r r e l a t i o n s between t h e components of system ( 1 4 . 1 ) . Thus, for i n s t a n c e , t h e a l i e n a t i o n model was b a s e d on t h e n o t i o n s of p r e f e r e n c e s , g r o u p s , goods, and a l l o c a t i o n s . The t h e o r y was o n l y b r i e f l y s k e t c h e d h e r e ; f o r a more d e t a i l e d a n a l y s i s i n c a s e of a l i e n a t i o n i n s c i e n c e , t h e r e a d e r i s r e f e r r e d t o Nowakowska ( 1 9 7 7 a , b , c , 1 9 8 3 ) .
S i m i l a r l y , t h e model o f s o c i a l p r e s s u r e shows how t h e i n f o r m a t i o n about t h e “ d e c i s i o n s t a t e ” o f t h e group a c t s as a m o d i f i e r of t h e i n d i v i d u a l d e c i s i o n . T h i s model s t r e s s e s t h e c r u c i a l r o l e o f s t r u c t u r e and i n f o r m a t i o n s p r e a d i n u n i f i c a t i o n of a c t i o n i n a c h i e v i n g
FORMAL THEORY OF ACTIONS
923
some g o a l s . It a l s o shows t h e c o n t r o l l i n g r o l e o f propaganda ( u n d e r s t o o d i n i t s w i d e s t s e n s e ) . The problem o f s p r e a d o f i n f o r m a t i o n i s s t u d i e d as an-
o t h e r s e p a r a t e submodel, which t r e a t s t h i s s p r e a d as a macrophenomenon. Let u s a l s o remark t h a t f o r t h e a n a l y s i s of s y s t e m ( 1 4 . 1 ) one may a l s o use t h e n o t i o n s developed i n t h e s e c t i o n on s y s t e m s y n t h e s i s , i n p a r t i c u l a r a l l t h o s e which c o n c e r n t h e mutual i n t e r r e l a t i o n s between two or more a c t i o n s y s t e m s . I n g e n e r a l , one may s a y t h a t f o r m a l a c t i o n t h e o r y o f t h i s chapter provides a unifying character t o a l l t h e
submodels under c o n s i d e r a t i o n . I n p a r t i c u l a r , as a l r e ady e x p l a i n e d , t h e g o a l s o f e a c h p e r s o n and t h e g o a l s o f t h e group as a whole, may be e x p r e s s e d i n terms o f p r e f e r e n c e s o v e r t h e s e t o f a l l o c a t i o n s o f goods i n t h e s e t S. These g o a l s a r e i n g e n e r a l s t r u c t u r e d i n some w ay, t h a t i s , a t t a i n i n g one o f them may f a c i l i t a t e a t t a i n i g some o t h e r g o a l s . T h i s i s due t o t h e f a c t t h a t a t t a i n i n g a g o a l s p e c i f i e d i n t h e form o f an a l l o c a t i o n o f goods f , y i e l d s new powers t o v a r i o u s members
o f t h e group, s o t h a t t h e y may f i n d i t e a s i e r t o change f i n t o some o t h e r a l l o c a t i o n f ' , and s o o n . I n a d d i t i o n , t h e g o a l s ( p r e f e r e n t i a l s t r u c t u r e s ) o f soc i a l groups may be r e l a t e d i n some way a c r o s s g r o u p s . The b a s i c t y p e s o f r e l a t i o n s h e r e a r e c o n c o r d a n c e , o r t h o g o n a l i t y and o p p o s i t i o n . The f i r s t and t h i r d o f t h e s e r e l a t i o n s r e q u i r e no comments, as t h e y simply mean agreement o r d i s a g r e e m e n t o f p r e f e r e n t i a l o r d e r i n g s , w h i l e o r t h o g o n a l i t y means t h a t whenever an a l t e r n a t i v e i s s t r i c t l y p r e f e r r e d t o a n o t h e r by one
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g r o u p , t h e n t h e second group i s i n d i f f e r e n t between t h e s e a l t e r n a t i v e s . There i s no need t o s t r e s s t h e i m p o r t a n c e o f t h e s e n o t i o n s f o r any t h e o r y o f c o a l i t i o n s and c o n f l i c t s between s o c i a l g r o u p s . R e t u r n i n g t o t h e c a s e o f a s i n g l e group, i t s g o a l may t h e r e f o r e be f o r m a l l y i d e n t i f i e d w i t h a sequence o f allocations f , f ’ , f ” , Each a l l o c a t i o n i n s u c h a sequence may be r e g a r d e d as a c o n j u n c t i o n o f v a r i o u s c o n d i t i o n s , s t a t i n g t h a t s u c h and s u c h goods a r e t o be a l l o c a t e d t o s u c h and s u c h p e r s o n s a t a g i v e n s t a g e o f p u r s u i t of t h e g o a l .
...
Any s u c h c l a s s o f sequences may b e r e p r e s e n t e d i n t h e form o f a “ g o a l - t r e e “ , c o n s e c u t i v e b r a n c h e s c o r r e s p o n d i n g t o t h e terms of sequences. To b r i n g about such changes as s p e c i f i e d b y t h e g o a l , i . e . t o achieve successive t r a n s i t i o n s t o f , then t o f ’ , e t c . , as s p e c i f i e d b y t h e g o a l , t h e group as a whole must p l a n and perform some a c t i o n s . Such a p l a n , covering a l l p o s s i b i l i t i e s , i . e . specifying the actions a t e a c h node o f t h e t r e e , i s c a l l e d a s t r a t e g y . G e n e r a l l y , w i t h e a c h p e r s o n s one may a s s o c i a t e h i s a c t i o n r e p e r t o i r e U(s). The a c t i o n s i n v a r i o u s s e t s U(s) a r e , as a r u l e , m u t u a l l y r e l a t e d - i n t h e s e n s e t h a t p e r f o r m i n g some a c t i o n s i n U(s) may i n h i b i t t h e p o s s i b i l i t i e s o f p e r f o r m i n g some o t h e r a c t i o n s from s e t s U(s’). Such c o n s t r a i n t s on a c t i o n s a r e i n d u c e d b y v a r i o u s f a c t o r s , among them t h e s e t R o f r u l e s a c c e p t ed by t h e group. Moreover, s i n c e t h e s e t R i s s t r u c t u r e d , v a r i o u s c o n s t r a i n t s on a c t i o n s have v a r y i n g degr e e s o f “ t i g h t n e s s ” , i . e . b r e a k i n g some o f t h e r u l e s i s c o n s i d e r e d ” w o r s e ” , from t h e p o i n t o f view o f t h e g r o u p ,
FORMAL THEORY OF ACTIONS
than breaking other r u l e s .
925
These f e a t u r e s were, i n a
s e n s e , i n c l u d e d i n t h e model o f p r o s o c i a l and a n t i s o c i a l b e h a v i o u r o f C h a p t e r 5, i n form of h i g h e r or lower changes o f o p i n i o n f o l l o w i n g some s p e c i f i c a c t i o n s . Of c o u r s e , a more s u b t l e t h e o r y s h o u l d a l s o i n c l u d e t h e ways by which a p e r s o n a r r i v e s a t s u c h changes of opinion. G e n e r a l l y , a c t i o n s of a group may be d i v i d e d i n t o t h r e e broad c a t e g o r i e s : i d e o l o g i c a l , f a c t u a l and i n t e r n a l , a s s p e c i f i e d i n t h e p r e c e d i n g s e c t i o n s , where some hyp o t h e s e s were f o r m u l a t e d a b o u t t h e s e t y p e s of a c t i o n s . I d e o l o g i c a l and f a c t u a l a c t i o n s a r e i n t e n d e d t o a t t a i n i n g a n a l l o c a t i o n from a s e t s p e c i f i e d by a g o a l - t r e e . I n t h e f i r s t c a s e , t h e a c t i o n s a r e more " i n d i r e c t " , by p r o v i d i n g t h e r e a s o n s for n e c e s s i t y , j u s t i f i c a t i o n , e t c . o f a g i v e n a l l o c a t i o n o f goods. I n t h i s s e n s e , t h e a c t i o n s a r e i n t e n d e d t o p r o v i d e e f f e c t s i n t h e domain o f consciousness. On t h e o t h e r hand, t h e f a c t u a l a c t i o n s a r e some changes o f s t a t e , which e i t h e r i m p l y , or make more l i k e l y , some i n t e n d e d a l l o c a t i o n s . The t h i r d t y p e o f a c t i o n s may be c a l l e d i n t e r n a l . The p o i n t i s t h a t when a group as a whole r e c e i v e s a c e r t a i n amount o f goods, a s s p e c i f i e d b y i t s g o a l , t h e problem a r i s e s as t o a l l o c a t i n g t h e s e goods w i t h i n t h e g r o u p . F o r m a l l y , t h e ( p a r t i a l ) g o a l o f t h e group may be t o b r i n g about any a l l o c a t i o n which p r o v i d e s t h e t o t a l amount K of goods c , i . e . any a l l o c a t i o n which meets
rscG f(s,c)
a number o f o t h e r c o n s t r a i n t s ) . The problem i s t h e n t o choose a p a r t i c u l a r
the condition
= K (plus
a l l o c a t i o n i n t h e c l a s s o f a l l o c a t i o n s which s a t i s f y
926
t h e above c o n s t r a i n .
CHAPTER 6
The s o l u t i o n d e p e n d s , o f c o u r s e ,
on t h e i n t e r n a l s t r u c t u r e o f t h e g r o u p , a n d t h e f o r c e s w h i c h o p e r a t e w i t h i n i t , t h e powers o f v a r i o u s members, and s o on. F o r a dynamic a n a l y s i s o f s y s t e m ( 1 4 . 1 ) i t would be necessary t o i n t r o d u c e assumptions about changes of t h e components of t h e s y s t e m . F o r i n s t a n c e , t h e number of goods i n t h e s e t C , which t h e g r o u p may h a v e , would i n g e n e r a l i n c r e a s e ( w h i c h d o e s n o t mean t h a t t h e g r o u p h a s t h e s e goods a c c e s s i b l e ) . The r a t e o f t h i s i n c r e a s e may v a r y t o g e t h e r w i t h t h e e c o n o m i c a l l e v e l of t h e s o c i e t y . The c h a r a c t e r o f goods a l s o c h a n g e s : some goods may d i s a p p e a r , w h i l e o t h e r s a p p e a r , w h i c h may be r e l a t e d t o w i t h t h e e x i s t i n g and planned g o a l s t o which t h e goods a r e t o s e r v e . T h i s i s i n t u r n , r e l a t e d t o t h e t e c h n o l o g i c a l development and s e a r c h f o r i n n o v a t i o n s ,
as w e l l a s w i t h c e r t a i n c o g n i t i v e l i m i t a t i o n s imposed on t h e s e t of g o o d s , i . e . t e n d e n c y t o c e r t a i n bounds on t h e s e t o f g o o d s , t h e i r b e t t e r a l l o c a t i o n s a n d r e s i s t i n g t h e s t r e e s o f u s i n g unknown a n d u n t e s t e d g o o d s . Lack o f knowledge and e x p e r i e n c e f a v o u r s s t r o n g e r lim i t a t i o n s on t h e s p e c t r u m o f a v a i l a b l e g o o d s . I n t h i s way, t h e a c t i n g p e r s o n s a r e a d a p t e d t o l o w e r i n g c o g n i t i v e c a p a c i t y , and economical r e s t r i c t i o n s . Freedom goods a r e s u b j e c t t o a c e r t a i n v a r i a b i l i t y , w i t h i n l i m i t s p r e s c r i b e d by c o n s t i t u t i o n . There a r e , however, t e n d e n c i e s t o a l l o c a t e t h e s e goods more a n d more e v e n l y , a l t h o u g h t h i s p r o c e s s may b e s l o w . Temporary l i m i t a t i o n s on t h e a c c e s s t o s u c h g o o d s may i n h i b i t o r d e l a y t h e d e v e l o p m e n t o f some s o c i a l g r o u p s . There a r e a l s o c o n s c i o u s e f f o r t s t o l i m i t t h e growth
FORMAL THEORY OF ACTIONS
927
o f t h e s e t of r u l e s c r e a t e d and i n t r o d u c e d by v a r i o u s
i n s t i t u t i o n s . T h e i r g r a d u a l ( o r sometimes sudden) i n c r e a s e i n d u c e s t e n d e n c i e s t o s i m p l i f i c a t i o n and new f o r m u l a t i o n s , s o as t o make t h e s e r u l e s p e r c e p t u a l l y more manageable. One may a l s o speak about t h e development o f t h e system of p r e f e r e n c e s i n groups: p r e f e r e n c e s a r e d e f i n e d f o r a l a r g e r and l a r g e r s e t o f goods and a l l o c a t i o n s . A l s o , t h e a b i l i t y t o d i s c r i m i n a t e may i n c r e a s e . G e n e r a l l y , however, as i n t h e p r e c e d i n g c a s e s , t h e development o f p r e f e r e n c e systems i s l i m i t e d c o g n i t i v e l y , and i t s s i z e i s an o b j e c t o f p a r t i a l c o n t r o l . An i n c r e a s e i n t h e d e n s i t y o f communication network i s a l s o l i m i t e d by s l o w e r growth o f "communication capac i t y " o f members o f t h e group. I f t h e s i z e o f t h e group i n c r e a s e s , t h e network a l s o i n c r e a s e s , b u t t h e r e may a p p e a r t e n d e n c i e s t o b r e a k t h e network, and form some separate c l u s t e r s . To sum u p , i t i s assumed t h a t a l l e l e m e n t s o f system
( 1 4 . 1 ) , e s p e c i a l l y t h o s e which a r e r e p r e s e n t a b l e a s s e t s , are s u b j e c t t o some c h a n g e s , w i t h new e l e m e n t s a p p e a r i n g , some e l e m e n t s d i s a p p e a r i n g , and w i t h a poss i b i l i t y o f exchange o f e l e m e n t s (some " c o r e " p a r t o f t h e s e t r e m a i n i n g i n v a r i a n t ) . T h i s p r o c e s s may be i n p a r t c o n t r o l l e d . V a r i o u s o u t s i d e e v e n t s , p o l i t i c a l , econ o m i c a l , e n v i r o n m e n t a l , e t c . , i n t r o d u c e randomness i n t h e p r o c e s s , s o t h a t we d e a l i n e f f e c t w i t h a semi-stoc h a s t i c p r o c e s s o f changes o f components o f ( 1 4 . 1 ) . What i s t a k e n as a s o c i a l change i n i n f a c t an i n t e r p l a y and s u p e r p o s i t i o n o f t h e s e p r o c e s s e s . One o f t h e most i n t e r e s t i n g problems i s t o r e c o n s t r u c t t h e s e p r o c e s s e s
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by s i m u l a t i o n , and t h u s e s t i m a t e t h e r e l e v a n t p a r a m e t e r s . A c o m p l e t e d e s c r i p t i o n i s , o f c o u r s e , beyond t h e r a n g e o f p o s s i b i l i t i e s . To o u t l i n e t h e p r o b l e m , one may s a y t h a t t h e s t a t e o f t h e s o c i e t y a t some time t i s d e s c r i bed by a v a r i a b l e S ( t ) , where S ( t ) i s a m u l t i d i m e n s i o n -
a l s p e c i f i c a t i o n o f a l l e l e m e n t s of ( 1 4 . 1 ) . N e x t , l e t al, a2, be t h e c o n t r o l p a r a m e t e r s , a v a i l a b l e f o r
...
v a r i o u s g r o u p s . The problem t h e n r e d u c e s t o s p e c i f i c a t i o n o f laws which g i v e t h e p r o b a b i l i t y t h a t S ( t + T ) w i l l s a t i s f y some c o n d i t i o n s , s a y S ( t + T ) c A , t h e s t a t e S ( t ) and a c t i o n s a l , a * , . . . ,
given
i . e . t h e pro-
bability PCS(t+T) E A I S ( t ) , a l , a 2
,...
1.
U n f o r t u n a t e l y , no c o h e r e n t t h e o r y e x i s t s a t p r e s e n t , which would a l l o w t o d e t e r m i n e s u c h p r o b a b i l i t i e s , even i n t h e s i m p l e r c a s e s , o f p a r t i a l d e s c r i p t i o n s o f some o f t h e components o f s y s t e m ( 1 4 . 1 ) .
1 5 . EXAMPLES OF APPLICATION TO DEVELOPMENTAL PSYCHOLOGY 1 5 . 1 . S t a d i a 1 languages of development. G e n e r a l l y , i n d e v e l o p m e n t a l p s y c h o l o g y , one may d i s t i n g u i s h t h r e e groups of problems:
(1) i n t r o d u c i n g a s y s t e m o f t h e o r -
e t i c a l n o t i o n s which d e s c r i b e t h e development o f a person,
(2) i n t r o d u c i n g t h e s e t o f n o t i o n s which e x p l a i n
t h e p s y c h o l o g i c a l d e v e l o p m e n t , and ( 3 ) i n t r o d u c i n g t h e s e t o f n o t i o n s which d e s c r i b e t h e e f f e c t s o f i n f l u e n c i n g t h e development ( e d u c a t i o n , r e a r i n g ) . Theory of a c t i o n s a l l o w s for some c o n c e p t u a l c l a r i f i c -
FORMAL THEORY OF ACTIONS
929
a t i o n and r e f r e s h i n g t h e t h e o r e t i c a l c o n c e p t s , and a l s o t h e d e s c r i p t i o n of development i n terms o f a c t i o n l a n guages and s t a d i a 1 l a n g u a g e s . L i n g u i s t i c i n t u i t i o n s used i n a c t i o n t h e o r y a l l o w u s t o d e s c r i b e development as a n e x t e n s i o n o f t h e " v o c a b u l a r y " o f a c t i o n s , and a b i l i t y t o b u i l d more complex s t r i n g s of a c t i o n s . T h i s c o r r e s p o n d s t o l e a r n i n g u n i t s o f a c t i o n s and b u i l d i n g c o r r e c t s t r i n g s o u t of them, p r o p e r f o r a g i v e n g o a l . Looking from t h e p o i n t o f view of o b j e c t s t o be o p e r a t ed upon by t h e a c t i n g p e r s o n , and which have some a c t i v a t i n g p r o p e r t i e s , development c o n s i s t s of l e a r n i n g languages o f wider c l a s s e s o f o b j e c t s . U n d e r s t a n d i n g c a u s a l r e l a t i o n s between o b j e c t s assumes t h a t development i s i n t e r p r e t e d i n a c t i o n t h e o r y as t h e knowledge how t o form m u l t i d i m e n s i o n a l s t r i n g s of a c t i o n s , which a r e c a u s a l l y and t e m p o r a l l y o r d e r e d w i t h r e s p e c t t o some c l a s s e s of o b j e c t s . T h i s c o r r e s ponds t o d e v e l o p i n g t h e a b i l i t y o f o p e r a t i n g , p o s s i b l y s i m u l t a n e o u s l y , w i t h c l a s s e s o f o b j e c t s , and a r r a n g i n g t h e i r i n t e r - l i n g u a l o r g a n i z a t i o n s . The l a t t e r i s one of t h e major d e t e r m i n a n t s o f development. D u a l l y , one may look a t t h e p r o c e s s of l e a r n i n g new more and more complex grammars of a c t i o n l a n g u a g e s , which a l l o w t o b u i l d more composite and l o n g e r a c t i o n strings. I n a d d i t i o n , l e a r n i n g how t o form new r u l e s of grammars c o r r e s p o n d s t o development o f c r e a t i v i t y and a b i l i t y of b u i l d i n g new o b j e c t s and a c t i o n s , a s w e l l as new t y p e s of s t r i n g s o f a c t i o n s . The d e s c r i p t i o n i n t e r m s of a t h e o r y , i n which t h e c e n t -
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r a l n o t i o n s are t h o s e o f a u n i t , r u l e s o f grammar, o r l i n g u i s t i c i n t u i t i o n s a l l o w i n g t o d e t e r m i n e whether a g i v e n s t r i n g b e l o n g s t o a language o f a c t i o n s i n q u e s t i o n , may a l l o w a more r i g o r o u s and b e t t e r q u a n t i f i a b l e e x p e r i m e n t a and s t a t i s t i c a l a n a l y s e s . One can a l s o use t h e methods o f d e s c r i p t i o n o f development and e x i s t i n g taxonomies o f d e v e l o p m e n t a l psychol o g y , t o s t u d y some " s t a d i a l l a n g u a g e s " , o f i n t e r e s t i n p a r t i c u l a r for e a r l y s t a g e s o f development, when t h e b e h a v i o u r i s s t i l l r e l a t i v e l y s i m p l e , and t h e s t a d i a ( p h a s e s ) a r e easy t o d i s t i n g u i s h ( e . g . p h a s e when t h e c h i l d u t t e r s o n l y s y l l a b l e s , t h e n words, and t h e n f i r s t s e n t e n c e s , o r -- i n n o n v e r b a l b e h a v i o u r -- p h a s e s d e s c r i b e d by t h e d e g r e e s o f a b i l i t y o f g r a s p i n g and operating with objects). T h e i d e a s u n d e r l y i n g s t a d i a l language i s t h a t some
s t r i n g s of p h a s e s a r e p r o p e r ( g r a m m a t i c a l ) and some are not. According t o Przetacznikowa (1973), t h e r e e x i s t s a t p r e s e n t 18 systems o f d e s c r i p t i o n i n t e r m s o f developm e n t a l s t a d i a , i n v o l v i n g a l t o g e t h e r 6 1 s t a d i a . Thus, t h e development may be d e s c r i b e d i n terms o f 18 p a r a l l e l s t r i n g s o f s t a d i a . The scheme o f such a r e p r e s e n t a t i o n , simplified t o the case of t h r e e , instead of 18 s y s t e m s , may be r e p r e s e n t e d g r a p h i c a l l y a s f o l l o w s .
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Thus, a t e a c h moment t h e development i s d e s c r i b e d i n t e r m s of a s t r i n g o f symbols, r e p r e s e n t i n g t h e s t a d i a , s u c h as AKQ o r CMS ( a s marked on t h e f i g u r e a b o v e ) . Here t h e " a l p h a b e t " c o n s i s t s of 6 1 symbols ( " l e t t e r s " ) and e a c h "word" has 1 8 l e t t e r s , s i n c e t h e r e a r e 18 systems o f s t a d i a . A sequence o f o b s e r v a t i o n s a t v a r i o u s t i m e s may be
t h e r e f o r e r e p r e s e n t e d as a " s e n t e n c e " , 1 . e . s t r i n g o f words , i n t h e a b s t r a c t " s t a d i a l l a n g u a g e " . S i n c e t h e s t a d i a l systems a r e n o t i n d e p e n d e n t , l o g i c a l l y o r s t a t i s t i c a l l y , one may e x p e c t t h a t c e r t a i n words w i l l n o t a p p e a r a t a l l , w h i l e some o t h e r words w i l l a p p e a r l e s s f r e q u e n t l y . The same c o n c e r n s a l s o " s e n t e n c e s " i n s t a d i a l l a n g u a g e . A s a consequence, one may s p e a k o f "grammar" of words and "grammar" o f s e n t e n c e s . The f i r s t of t h e s e grammars i s c o n n e c t e d w i t h t h e f a c t
t h a t some words a r e i m p o s s i b l e , w h i l e some o t h e r l e t t e r
combinations correspond only t o p a t h o l o g i c a l development s . The grammar o f " s e n t e n c e s " i s c o n n e c t e d w i t h t h e f a c t t h a t some s t a d i a a r e c o r r e l a t e d among t h e m s e l v e s , have
some a v e r a g e d u r a t i o n , and f i n a l l y -- i n s i d e one s y s t e m , t h e y a p p e a r i n some more o r l e s s s p e c i f i e d o r d e r . The a n a l y s i s of s t r u c t u r a l p r o p e r t i e s of such "words" and " s e n t e n c e s " of s t a d i a l language may be h e l p f u l i n e x p l i c a t i n g t h e l o g i c a l r e l a t i o n s between t h e s t a d i a l s y s t e m s introduced by various researchers. The t a x o n o m i c a l systems s u g g e s t e d up t o now have numerous a d v a n t a g e s and d i s a d v a n t a g e s , and it may be a l s o
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be d i f f i c u l t t o p a s s from one s y s t e m t o a n o t h e r , i n o r d e r t o a c h i e v e some c o m p a r i s o n s . Here a c t i o n t h e o r y s u p p l i e s not only convenient methodological standards
of a n a l y s i s , but a l s o s t a t i s t i c a l approaches t o study of p r o p e r t i e s of a c t i o n languages. This i s e s p e c i a l l y i m p o r t a n t f o r c h a r a c t e r i z a t i o n o f normal, harmonious development o f a p e r s o n , c a p t u r i n g t h e i d e a o f a v e r a g e , a n d most common t e n d e n c i e s i n t h e p o p u l a t i o n . The n o t i o n o f " n o r m a l " , o r " s t a n d a r d " s t a d i a 1 s t r i n g s e n a b l e s us t o d i s t i n g u i s h sub-normal s t r i n g s , and t o l e r a n c e l i m i t s , t h u s s u g g e s t i n g some r e m e d i a l e d u c ational techniques. N a t u r a l l y , t h i s system i s n o t free of d i f f i c u l t i e s of developmental psychology, i . e . f u z z i n e s s o f d i s t i n g u i s h e d u n i t s a n d s t a t e s . One may s u g g e s t h e r e a n a p p l i c a t i o n o f fuzzy set t h e o r y . I n t h i s c a s e , t h e a p p l i c a t i o n would c o n s i s t o f t r e a t i n g "words" as c e r t a i n i d e a l t y p e s . T h u s , a t any moment, t h e s e t o f a l l c h i l d r e n who b e l o n g t o a g i v e n t y p e i s a f u z z y s e t . I n o t h e r w o r d s , e a c h c h i l d would b e l o n g i n some d e g r e e ( p o s s i b l y zero) t o each of the types.
A f u z z y s e t , s u c h as AKQ, would b e i n t h i s c a s e a p r o d u c t o f three s e t s , each corresponding t o a phase i n one system. I n o t h e r words, t h e d e g r e e t o which a c h i l d would b e l o n g t o t h e t y p e AKQ would e q u a l t h e m i n i m a l o u t o f t h e d e g r e e s t o w h i c h he b e l o n g s t o t y p e s A, K a n d Q.
As r e g a r d s t h e d e f i n i t i o n o f t h e d e g r e e o f membership i n a f u z z y p h a s e o f d e v e l o p m e n t , o n e c o u l d u s e t h e conv e n t i o n a l methods, scaled).
e.g. test scores (appropriately
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One s h o u l d s t r e s s h e r e t h a t s t a d i a l l a n g u a g e s , as d e f i n e d here a r e n o t r e s t r i c t e d t o c h i l d l s d e v e l o p m e n t . The idea may b e u s e f u l l y a p p l i e d i n d e s c r i p t i o n o f o t h e r domains o f b e h a v i o u r a n d a c t i o n , s u c h as f o r b u i l d i n g m o d e l s o f d e v e l o p m e n t o f o r g a n i z a t i o n s , resea r c h teams, d i s c i p l i n e s , e t c . A l l t h a t i s n e e d e d i s t h a t o n e may d i s t i n g u i s h v a r i o u s p h a s e s o f d e v e l o p m e n t , a n d t h a t t h e s e p h a s e s may b e c o n c a t e n a t e d i n a "gramm a t i c a l " way ( p o s s i b l e w i t h d i f f e r e n t d e g r e e s o f a c c e p t a b i l i t y o f terms o f t h e s t r i n g ) . Moreover, an a d v a n t a g e h e r e i s t h a t s t a d i a l la n g u a g e s a l l o w t h e d e s c r i p t i o n of c l a s s e s o f a c t i o n s , a n d n o t o n l y r e l a t i o n s b e t w e e n u n i t s . One c a n a l s o t h i n k a b o u t a d u a l d e s c r i p t i o n , i n terms o f a u t o m a t a t h e o r y , o f stadial languages.
15.2. E d u c a t i o n a n d p s y c h o l o g i c a l d e v e l o p m e n t : modeli n g of t h e process A t t h e e n d , l e t us o u t l i n e a p o s s i b l e a p p l i c a t i o n o f
a c t i o n t h e o r y o f t h i s c h a p t e r , t o e d u c a t i o n a l a n d dev e l o p m e n t a l p s y c h o l o g y . We s h a l l namely p r e s e n t a v e r y g e n e r a l model of e d u c a t i o n p r o c e s s , t r e a t i n g i t as a c e r t a i n process of s t o c h a s t i c c o n t r o l with incomplete information
(for o t h e r m o d e l s o f e d u c a t i o n a l p r o c e s s e s , see e . g . Kochen 1 9 7 5 , Doran 1 9 6 8 , C e t l i n 1 9 6 3 , Menzel
1 9 7 0 , Kokawa e t a l . , 1 9 7 4 , M i t c h e l l 1 9 8 1 , o r B a n a t h y 1 9 8 0 , 1 9 8 0 a , b . F o r l e a r n i n g l a n g u a g e s , s e e Hamburger and W e x l e r , 1 9 7 3 ) . Generally, let X stand f o r the person being taught, and l e t Y be t h e p e r s o n t e a c h i n g . We assume t h a t t h e
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subject being taught i s a c e r t a i n r e l a t i o n a l s t r u c t u r e , c o n s i s t i n g of a s e t of o b j e c t s o f some k i n d , and some r e l a t i o n s between t h e s e o b j e c t s . A s a n example, one may t h i n k h e r e o f s u b j e c t s u c h as geometry, i t s o b j e c t s b e i n g v a r i o u s f a c t s a b o u t p r o p e r t i e s of f i g u r e s , and r e l a t i o n s b e i n g induced b y l o g i c a l d e p e n d e n c i e s between t h e s e f a c t s . A s a n o t h e r example, one may t h i n k o f s u b j e c t s u c h as d r i v i n g , where " o b j e c t s " a r e v a r i o u s t r a f f i c r u l e s and a c t i o n s o f t h e d r i v e r , and r e l a t i o n s are g i v e n by c a u s a l depende i e s between d r i v e r ' s a c t i o n s and motions of t h e v e h i c l e .
7
G e n e r a l l y , l e t S be t h e s u b j e c t t a u g h t , s o t h a t we may r e p r e s e n t i t as
S =(A,
R1, R2,...>
where A i s some s e t , and R1, R2,.. ween e l e m e n t s o f A .
(15.1)
.
are r e l a t i o n s bet-
Now, t h e " s t a t e " o f t h e e d u c a t i o n of X a t any t i m e may be i d e n t i f i e d w i t h h i s knowledge about S . T h i s knowl e d g e c o n c e r n s b o t h t h e f a c t s i n A , and r e l a t i o n s between t h e s e f a c t s . G e n e r a l l y , w e assume t h a t knowledge i s r e p r e s e n t a b l e as a n element of some s e t Q , whose e l e m e n t s a r e p a r t i a l l y o r d e r e d by a r e l a t i o n 9 . Here q < q t means t h a t t h e s t a t e q r e p r e s e n t s l e s s knowledge than s t a t e q . We may now i n t r o d u c e t h e r e m a i n i n g e l e m e n t s o f t h e
system; t h u s , by an e d u c a t i o n system we s h a l l mean a 9-tup l e (15.2)
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where t h e r e m a i n i n g symbols o f ( 1 5 . 2 ) have t h e f o l l o w ing intended i n t e r p r e t a tion. Firstly, Q
*
i s a ( f u z z y ) s u b s e t of Q , r e p r e s e n t i n g t h e
*
g o a l o f e d u c a t i o n . Formally, Q i s r e p r e s e n t a b l e through i t s membership f u n c t i o n , s a y f , s o t h a t f ( q ) i s t h e d e g r e e t o which t h e e d u c a t i o n a l g o a l i s a t t a i n e d , i f X reaches a s t a t e q. Next, D i s t h e s e t o f a l l p o s s i b l e a c t i o n s which may be u n d e r t a k e n by Y i n t h e p r o c e s s o f e d u c a t i o n : some w a y s o f e x p l a i n i n g t h e t o p i c s , arguments u s e d , i l l u s t r a t i v e examples, homwwork p r o b l e m s , t e s t p r o b l e m s , r e w a r d s , e t c . On t h e o t h e r hand, V i s t h e s e t o f X ' s responses t o a c t i o n s o f Y . F i n a l l y , P i s t h e t r a n s i t i o n p r o b a b i l i t y , which t o e v e r y s t a t e q i n Q and a c t i o n d i n D a s s i g n s a p a i r , c o n s i s t i n g o f t h e next s t a t e q ' ( p o s s i b l e w i t h q ' = q ) , and r e s p o n s e v . We s h a l l denot e t h i s p r o b a b i l i t y by (15.3) The main problem l i e s h e r e i n t h e f a c t t h a t s t a t e s q and q' are n o t o b s e r v a b l e d i r e c t l y , and one can o n l y i n f e r about them from o b s e r v i n g r e s p o n s e s v t o s t i m u l i d. I n t u i t i v e l y , a response t o a stimulus v , say a t e s t i t e m , w i l l s t r o n g l y depend on t h e s t a t e o f knowledge q o f X . What i s c r u c i a l h e r e , i s t h a t t h e n e x t s t a t e and r e s p o n s e are n o t i n d e p e n d e n t -- t h i s y i e l d s t h e p o s s i b i l i t y o f i n f e r e n c e a l s o about t h e s t a t e t o which a t r a n s i t i o n o c c u r r e d , on t h e b a s i s o f t h e observed response. One may now u s e t h e p r o b a b i l i t y ( 1 5 . 3 ) t o c l a s s i f y
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v a r i o u s a c t i o n s d . F i r s t l y , an a c t i o n d such t h a t
(15.4) whenever q ' # q , w i l l be c a l l e d p u r e l y t e s t i n g . T h i s s i m p l y means t h a t a c t i o n d i s s u c h t h a t it w i l l n o t change t h e s t a t e q o f p e r s o n X . The o n l y r o l e o f such a n a c t i o n , t h e r e f o r e , i s t o p r o v i d e i n d i r e c t i n f o r m a t i o n about t h e s t a t e q (from o b s e r v i n g v ) . Let now q be f i x e d , and l e t
(15.5) be t h e s e t o f a l l s t a t e s which r e p r e s e n t t h e e d u c a t i o n a l l e v e l a t l e a s t as h i g h as q . We may t h e n s a y t h a t a c t ion d i s instructive at q, i f
2
( q ' , v ) : q ' c B(q)
( q ' , v ) = 1.
P
(15.6)
q'd
T h i s c o n d i t i o n means t h a t i f a c t i o n d i s a p p l i e d a t s t a t e q , then t h e next s t a t e q ' w i l l represent a t least as h i g h l e v e l o f a t t a i n m e n t a s q . T h i s means t h a t a c t i o n d l e a d s t o some p r o g r e s s i n t e a c h i n g ( o r : no regress). The t r o u b l e i s t h a t a n a c t i o n may be i n s t r u c t i v e a t some s t a t e q , and n o t i n s t r u c t i v e a t some o t h e r s t a t e q ' . S i n c e when t h e a c t i o n i s a p p l i e d , one does n o t have complete knowledge about q , i t may happen t h a t a n o t h e r w i s e i n s t r u c t i v e a c t i o n w i l l g i v e some r e g r e s s (be confusing f o r X ) . The t e a c h i n g p r o c e s s p r o c e e d s s e q u e n t i a l l y . P e r s o n Y
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FORMAL THEORY OF ACTIONS
applies stimuli
dl,
d2,...
(testing or instructional)
.
After n successand o b s e r v e s t h e r e s p o n s e s v l , v 2 , . . , ive pairs (dl,vl), ( d n , v n ) , p e r s o n Y may have a
...,
r e a s o n a b l e i d e a about t h e s t a t e q o f X , and t h e n he may s e a r c h f o r a n a c t i o n d which would maximize t h e * nt1 probability of a t t a i n i n g t h e goal Q i n highest possib l e d e g r e e i n some t i m e l i m i t , e t c . W i t h t h i s f o r m u l a t i o n , i n s t r u c t i o n p r o c e s s becomes a n o b j e c t o f p o s s i b l e o p t i m i z a t i o n p r o c e d u r e s , and one c o u l d hope t h a t a t l e a s t i n some c a s e s i t might be p o s s i b l e t o s p e c i f y i t t o t h e l e v e l a l l o w i n g t h e use o f Bellman e q u a t i o n . One s h o u l d remark, however, t h a t t h i s e q u a t i o n would have t o be m o d i f i e d s o as t o t a k e i n t o a c c o u n t t h e f a c t t h a t t h e s t a t e i s n o t o b s e r v a b l e d i r e c t l y , and t h e o p t i m a l s t r a t e g y w i l l depend on t h e o b s e r v e d r e s p o n s e s , and n o t t h e a c t u a l s t a t e s . I n c o n n e c t i o n w i t h t h i s problem o f e d u c a t i o n , one may a l s o c o n s i d e r t h e f o l l o w i n g problems. F i r s t l y , v a r i o u s a c t i o n s o f p e r s o n Y may have v a r i o u s l e v e l s o f admissi b i l i t y , which may be r e l a t e d t o e f f e c t i v e n e s s o f t h e s e a c t i o n s i n a n i n v e r s e way ( e . g . b e a t i n g i s a n i n a d m i s s i b l e method, d e s p i t e t h e f a c t t h a t i n some c a s e s i t may be e f f e c t l v e ) . The c h o i c e o f l e v e l o f adm i s s i b i l i t y depends p r o b a b l y on t h e problem, on a b i l i t i e s o f t h e s t u d e n t and t e a c h e r . S e c o n d l y , t h e p r o c e s s o f e d u c a t i o n may be d e s c r i b e d by d i f f e r e n t i n d i c e s , measuring s u c h f a c t o r s as r e s i s t a n c e , e f f i c i e n c y , e t c . Tne e d u c a t i o n p r o c e s s may t h u s be i n t e r p r e t e d as a f a m i l y o f a c t i o n s t r i n g s , or a c t i o n l a n g u a g e , w i t h v a r i o u s l e v e s o f a d m i s s i b i l i t y . For e a c h g o a l and i t s l e v e l o f a t t a i n m e n t , one can a n a l y s e t h e c l a s s of a l l s t r i n g s w i t h a given a d m i s s i b i l i t y ,
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which a t t a i n t h i s g o a l . S e c o n d l y , g i v e n t h e c l a s s of r e s p o n s e s , one can d i f f e r e n t i a t e t h e p o p u l a t i o n from t h e p o i n t of view o f t h e d e g r e e o f a t t a i n m e n t o f t h e g o a l , and a n a l y s e s u c h d i f f e r e n t i a t i o n s ( t h i s means i n p r a c t i c e , a s e a r c h f o r o p t i m a l , o r e f f i c i e n t , w a y s o f t e s t i n g knowledge of s t u d e n t s ) . A p a r t i c u l a r l y i n t e r e s t i n g s p e c i a l c a s e o f t h e above
scheme i s t h e e r i s t i c d i a l o g u e , t h a t i s , a r g u m e n t a t i o n and c o n v i n c i n g . Here t h e s t i m u l i ( a c t i o n s ) and respons e s a r e u t t e r a n c e s . The d i f f e r e n c e l i e s o n l y i n t h e f a c t t h a t t h e i n t e r a c t i o n i s proceeding both ways, i . e . r e s p o n s e s o f one p e r s o n a r e a t t h e same t i m e s e r v i n g as s t i m u l i f o r t h e o t h e r ( w h i l e i n t h e model of e d u c a t i o n , t h e s i t u a t i o n i s a s y m m e t r i c ) . G e n e r a l i z i n g t h i s model t o t h e c a s e o f t e a c h i n g n o t a s i n g l e p e r s o n but a group ( e . g . a c l a s s ) , one c o u l d i n c o r p o r a t e h e r e -- by a n a p p r o p r i a t e i n t e r p r e t a t i o n o f t h e c o n c e p t o f s t . a t e -- a l s o t e a c h i n g p r o s o c i a l beh a v i o u r . I n t h i s c a s e , one would have t o e n r i c h t h e model by i n c l u d i n g t h e p o s s i b i l i t y o f t r a n s i t i o n s dep e n d i n g on t h e r e w a r d s and punishments. R e t u r n i n g t o t h e terminology o f a c t i o n t h e o r y , t h e p s y c h o l o g i c a l development o f a p e r s o n can be d e f i n e d , i n some s e n s e , i n two dimensions -- s y n t a c t i c and sem a n t i c . I n s y n t a c t i c a s p e c t , development i s l e a r n i n g more and more complex a c t i o n l a n g u a g e s , e s p e c i a l l y o f many p e r s o n s , and " m u l t i d i m e n s i o n a l grammars", of h i g h e r o r d e r s , u n d e r s t o o d as r u l e s o f c r e a t i n g new languages ( c r e a t i v i t y ) . For t h e d e s c r i p t i o n o f t h e l e v e l o f s o c i a l development, t h e knowledge o f r u l e s
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o f m u l t i p e r s o n l a n g u a g e s seems e s s e n t i a l ( s u c h a s r u l e s o f l a n g u a g e s of s o c i a l r o l e s , o r g a n i z a t i o n s , e t c . ) An i m p o r t a n t a s p e c t o f development d e s c r i b e d on a synt a c t i c level i s acquisition of "praxiological" properti e s o f a c t i o n s , s u c h as t h e i r e f f e c t i v e n e s s , e t c . The s e m a n t i c dimension i s d e s c r i b e d by t h e c o n t e n t and c o m p l e x i t y o f g o a l s , as w e l l a s p l a n s o f s t r i n g s o f a c t i o n s l e a d i n g t o t h e s e g o a l s . Semantics understood i n t h i s way ( a s g o a l s , p l a n s and r o l e s ) d e t e r m i n e s t h e c l a s s and l e v e l o f c o m p l e x i t y o f a c t i o n l a n g u a g e s acquired. I n s e m a n t i c a s p e c t , s o c i a l i z a t i o n i s i n t e r p r e t e d as a h i g h v a l u a t i o n of some c l a s s o f g o a l s ( p l a n s , o u t c o m e s ) , g e n e r a t e d by v a r i o u s n o r m a t i v e s t r u c t u r e s . It seems t h a t two b a s i c mechanisms o f s o c i a l i z a t i o n a r e l e a r n i n g m o t i v a t i o n a l c o n s i s t e n c y and l e a r n i n g cooper a t i v e n e s s , i . e . t r a n s a c t i o n s , u n d e r s t o o d as exchange o f some p s y c h o l o g i c a l o r m a t e r i a l goods. A t t h e e n d , l e t u s o b s e r v e t h a t a c t i o n t h e o r y appro-
a c h e s and automata i n t e r p r e t a t i o n o f l a n g u a g e s , may be s u c c e s s f u l l y a p p l i e d a l s o t o o t h e r domains, s u c h as
e . g . d i a l o g u e t h e o r y (Nowakowska 1976a) or t h e o r y o f development (Nowakowska ( 1 9 8 3 a ) . A l s o , S k v o r e t z and Fararo (1980) obtained very i n t e r e s t i n g r e s u l t s i n automata i n t e r p r e t a t i o n o f formal t h e o r y o f a c t i o n s .
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CHAPTER 6
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