Temporal Logic: From Ancient Ideas to Artificial Intelligence

TEMPORAL LOGIC Studies in Linguistics and Philosophy Volume 57 Managing Editors GENNARO CHIERCHIA, University of Mila...

Author: Peter Øhrstrøm | Per Hasle

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TEMPORAL LOGIC

Studies in Linguistics and Philosophy Volume 57

Managing Editors GENNARO CHIERCHIA, University of Milan PAULINE JACOBSON, Brown University FRANCIS J. PELLETIER, University of Alberta

Editorial Board JOHAN VAN BENTHEM, University of Amsterdam GREGORY N. CARLSON, University of Rochester DAVID DOWTY, Ohio State University, Columbus GERALD GAZDAR, University of Sussex, Brighton IRENE HEIM, M./.T., Cambridge EWAN KLEIN, University of Edinburgh BILL LADUSAW, University of California at Santa Cruz TERRENCE PARSONS, University of California, lrvine

The titles published in this series are listed at the end of this volume.

TEMPORAL LOGIC From Ancient Ideas to Artificial Intelligence

by PETER OHRSTROM Department of Communication, Aalborg University,Denmark and

PER F. V. HASLE Department of Information Science, University of Aarhus, Denmark

KLUWER ACADEMIC PUBLISHERS DORDRECHT / BOSTON / LONDON

Library of Congress Cataloging-in-Publication Data

~hrstrem, Peter. Temporal l o g i c : From a n c i e n t i d e a s to a r t l f l c l a l by Peter Ohrstrem and Per F.V. Hasle.

Intelligence

p. cm. - - ( S t u d l e s in l i n g u i s t ~ c s and p h i l o s o p h y Includes bibliographical r e f e r e n c e s and i n d e x . ISBN 0 - 7 9 2 3 - 3 5 8 6 - 4 (hb . a l k . p a p e r ) 1. Tense ( L o g i c ) 2. L o g i c , S y m b o l i c and m a t h e m a t i c a l . I . H a s t e , Per F. V. II. Title. III. Series. BC199.T4037 1995 160--dc20

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57)

3.

Time.

95-22191

ISBN 0-7923-3586-4

Published by Kluwer Academic Publishers, P.O. Box 17, 3300 AA Dordrecht, The Netherlands. Kluwer Academic Publishers incorporates the publishing programmes of D. Reidel, Martinus Nijhoff, Dr W. Junk and MTP Press. Sold and distributed in the U.S.A. and Canada by Kluwer Academic Publishers, 101 Philip Drive, Norwell, MA 02061, U.S.A. In all other countries, sold and distributed by Kluwer Academic Publishers Group, P.O. Box 322, 3300 AH Dordrecht, The Netherlands.

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TABLE

PREFACE

OF CONTENTS

.............................................................................................................. ~ i

I N T R O D U C T I O N ........................................................................................................ 1

P A R T 1. 1.1. 1.2. 1.3. 1.4. 1.5. 1.6. 1.7. 1.8. 1.9. 1.10. 1.11.

TIMEAND I A ~ C ........................................................................... 6 T h e Sea-fight T o m o r r o w ........................................................ 10 T h e M a s t e r A r g u m e n t o f D i o d o r u s C r o n u s ................. 15 T h e S t u d y o f T e n s e s in t h e M i d d l e A g e s ......................... 33 T e m p o r a l A m p l i a t i o n ............................................................... 39 T h e D u r a t i o n o f t h e P r e s e n t ................................................. 4 3 T h e L o g i c o f B e g i n n i n g a n d E n d i n g ................................. 52 T i m e a n d C o n s e q u e n t i a .......................................................... 65 T e m p o r a l i s - t h e Logic of"While'. ....................................... 71 H u m a n F r e e d o m a n d D i v i n e F o r e k n o w l e d g e ............ 87 T h e D o w n f a l l of M e d i e v a l Tense-logic .......................... 109 L o g i c a s a T i m e l e s s S c i e n c e ................................................. 114

PART 2. 2.1. 2.2. 2.3. 2.4. 2.5. 2.6. 2.7. 2.8. 2.9.

T I M E AND LOGIC R E U N I T E D ............................................. 118 T h e 19th c e n t u r y a n d Boolean logic .............................. 122 C.S. P e i r c e o n T i m e a n d M o d a l i t y ................................... 128 L u k a s i e w i c z ' s C o n t r i b u t i o n to T e m p o r a l logic .......... 149 A T h r e e - P o i n t S t r u c t u r e o f T e n s e s ................................. 155 A. N. Prior's Tense-logic ........................................................ 167 T h e I d e a o f B r a n c h i n g T i m e .............................................. 180 T e n s e L o g i c a n d S p e c i a l R e l a t i v i t y .................................. 197 S o m e B a s i c S y s t e m s of T e m p o r a l Logic ........................ 2 0 3 F o u r G r a d e s o f T e n s e - L o g i c a l I n v o l v e m e n t .............. 2 1 6

V

vi 2.10. PART 3. 3.1. 3.2. 3.3. 3.4. 3.5. 3.6. 3.7.

Metric T e n s e Logic ..................................................................

231

MODERN ISSUES IN TEMPORAL L O G I C ........................ 2 4 1 T w o P a r a d i g m s o f T e m p o r a l L o g i c ................................ 2 4 3 I n d e t e r m i n i s t i c T e n s e Logic ............................................... 2 5 7 L e i b n i z i a n T e n s e L o g i c .......................................................... 2 7 0 T e n s e L o g i c a n d C o u n t e r f a c t u a l R e a s o n i n g .............. 2 8 2 Logic o f D u r a t i o n s .................................................................... 3 0 3 G r a p h s f o r T i m e a n d M o d a l i t y ......................................... 3 2 0 T e m p o r a l L o g i c a n d C o m p u t e r S c i e n c e ....................... 3 4 4

4. CONCLUSION ................................................................................................... 3 6 6 A P P E N D I X ............................................................................................................ 3 7 3

BIBLI~

...................................................................................................

INDEY~......................................................................................................................

386 405

SOME IMI~RTANT LOGICIANS................................................................... 4 1 3

PREFACE

Time is ubiquitous. Look to such diverse fields as literature and computers, ethics and physics, logic and rhetoric, philosophy a n d n a t u r a l science. I f you are studying any of these subjects, professionally or c o n a m o r e , you are very likely to come across temporality as a crucial factor to your studies. For this reason, people are led into the study of time from a v a r i e t y of highly different disciplines. For the s a m e reason, the s t u d y of time is useful and enlightening, both for its own sake a n d for a l a r g e n u m b e r of specific purposes. The r a t h e r ambitious goal of this book is to comprehend time in its diversity, a n d yet to do this in a focused manner. O u r study stretches from Antiquity to the p r e s e n t day, and s p a n s the field from l i t e r a t u r e to computer science. It t h u s comprises a historical as well as a systematical dimension. We believe t h a t such a comprehensive approach is n e c e s s a r y in order to achieve a fuller understanding of time. The cost of this approach is t h a t not all aspects can be given a t r e a t m e n t quite as thorough as t h e y deserve. J u s t for example, t h e r e is much more to say about such fields as program verification, trivalent and m a n y - v a l u e d logic, and quantified temporal logic t h a n w h a t we h a v e m a n a g e d to cover here. There are also r e l e v a n t topics w h i c h have been e n t i r e l y left out: for instance, the c u r r e n t discussions on indexicals, and the logic of t r u t h value gaps, to mention two of the most important omissions. With these disclaimers we wish to make it clear t h a t we are ourselves a w a r e of some limitations of this book. But we also believe t h a t it does contain an u n u s u a l l y c o m p r e h e n s i v e exposition of the study of time. We must say a few words about the genesis of this book. Peter O h r s t r ¢ m was the first of us to do research on t h e concept of time, leading to his 1988 dr. scient, thesis on this subject. This thesis put a special emphasis on the relation between the logic of time and the general history of natural science. Per Hasle began studies on the logic of time in 1988, adding to our incipient common project perspectives from linguistics a n d information

vii

viii

PREFACE

science. Differences in our backgrounds notwiths£anding, the contribution contained in this volume is the result of essentiality joint work. The book contains entirely new results as well as previously published material, which has been reworked and put into the wider context of our exposition here. A particularly important source for our book has been several interviews with Dr. Mary Prior, who has graciously provided valuable and interesting information on the work of her late husband Arthur Norman Prior - the founder of m o d e r n temporal logic. Furthermore, Mary Prior has granted us access to/k N. Prior's papers kept at the Bodleian Library in Oxford, also a crucial source for some of the new findings presented here. A very special thanks must go to her. We are indebted to many persons for advice, criticism and inspiring discussions. We especially thank Mogens W e g e n e r and Marta Ujvari for carefully reading and constructively criticising our manuscript. We also want to thank Harmen van den Berg, Knud Capion, Jack Copeland, M. J. Cresswell, Dick Crouch, Sten Ebbesen, Milea Angela Simoes Froes, Claudine Engel-Tiercelin, Antony Galton, Richard Gaskin, Nils Klarlund, Inger Lytje, Claus Myltoft, Jakob M~ller, Stig Andur Pedersen, Amir Pnueli, Anne Rasmussen, Stephen Read, Jan Schmidt, Peter Simons, and Jan Tapdrup. All these persons have in various ways been helpful and inspiring for our work. Peter OhrstrCm Per Hasle Aalborg, January 1995

INTRODUCTION: LOGIC A N D THE STUDY OF TIME What, then, is time? I f no one asks me, I know: if I wish to explain it to one that asketh, I know not. St. Augustine [Confessiones XI, c. XIV, xvii. / Gale p. 40] Every concept of time arises in the context of some (no doubt useful) human purpose and bears, inevitably and essentially the stamp of that human intent. N. Lawrence [1978, p. 24] Philosophers have had much to say about the nature of Time. Mathematicians and Physicists add a lot from their perspective. More recently, linguists are also becoming interested in the temporal constructions of natural language. Can a logician add anything of value to all this wisdom? J.F.A.K. van Benthem [1983, xi] According to the Ancients as well as most of the later European t h i n k e r s in philosophy and science, time is primarily to be understood as strongly related to movement. In addition it is assumed t h a t time can be described by numbers. Time has consequently been thought of as a basic concept for n a t u r a l science, first and foremost physics and astronomy. In m a n y circles physics is still a s s u m e d to be the key science for anyone who wants to study the concept of time, so let us first say a few words about the contribution from physics in this respect. According to Newtonian mechanics time is viewed plainly as a co-ordinate. The bodies in the world are supposed to move according to the laws of dynamics. These movements can be fully predicted in principle. All past a n d future states are implicit in the present state. Predictions and retrodictions can be expressed by m e a n s of spatial and temporal co-ordinates. At this level there is no proper temporal a s y m m e t r y , since the laws of dynamics permit time reversal. A concept of entropy might, however, be defined at this level, and the probability of increasing entropy will be

2

INTRODUCTION

high. In t h e r m o d y n a m i c s the law of ever increasing e n t r o p y has been used as an argument for the so-called 'arrow of time'. Things become even more complicated when q u a n t u m physics is t a k e n into consideration, a n d relativity theory raises some special problems for the logical study of time. The study of physical time is certainly very i m p o r t a n t and useful. In our opinion, however, it is even more i m p o r t a n t to realise the relevance of what N. Lawrence [1979] has pointed out in his study of various levels in the discourse about time: every concept of time bears the s t a m p of h u m a n intent. W h e n hum a n s are t a k e n into consideration the concepts of activity and creativity become very important for the understanding of time. In this connection it is clearly also possible - in fact, necessary to introduce the idea of the 'NOW' and the direction of time. This observation m u s t have general consequences, if it is accepted t h a t every concept, including the concept of time, has to be related to the h u m a n mind. U n d e r this perspective it becomes more n a t u r a l to describe time by m e a n s of tenses: past, present and future, t h a n by means of i n s t a n t s (dates, clock-time, etc.). With tenses, we can express t h a t the past is forever lost and the f u t u r e is not yet here. Without t h e s e ideas we cannot hope to grasp the idea of the passing of time. P h e n o m e n a such as memory, experience, observation, anticipation and hope are all essential for the way time is understood. Notions of past and future time, the interpretation of the past as well as expectations of the f u t u r e are all interwoven in the h u m a n mind. In this qualitative sense the past has not ceased to exist w h e n followed by the next time period. T h e r e are m a n y common expressions for the qualitative and q u a n t i t a t i v e aspects of time, for example 'the sight of time' 'time will tell' - 'old time' - 'I don't have time' - 'to waste time' - 'to buy time' - 'long time' - 'short time'. Apparently, h u m a n beings experience a tension between time as a quantity and time as a quality. We can certainly see t h e numerical or quantitative aspect of time, witness the clock and the calendar. But we also highly value the qualitative aspects of time such as the 'nowness' of events, and the passing of t i m e as expressed by the tenses [Lundmark 1991].

INTRODUCTION However, time should not be seen as an idea merely d e p e n d e n t upon the individual mind, but also as an intersubjective idea. An i n d i v i d u a l c a n n o t u n d e r s t a n d t i m e p r o p e r l y only f r o m t h e viewpoint of his or h e r own m e n t a l life. It is a very i m p o r t a n t fact t h a t t h e tenses past, present, a n d future are not private, b u t at least intersubjective, if not objective. A satisfactory u n d e r s t a n d i n g of time requires a careful s t u d y of temporal relations in h u m a n society. It m u s t be a d m i t t e d t h a t this sociotemporality is very complex and t h a t little h a s been done in order to reach a deeper u n d e r s t a n d i n g of it. B u t it is clear t h a t l a n g u a g e a n d communication are in general essential for an u n d e r s t a n d i n g of social time. The above description of the various notions of time does n o t explicitly say which idea of t e m p o r a l i t y is t h e most f u n d a m e n tal. In our opinion, the answer to t h e question of f u n d a m e n t a l i t y m u s t be t h a t the concepts of past, p r e s e n t a n d future are basic, b u t t h a t t h e y cannot be fully u n d e r s t o o d unless sociotemporal relations a n d especially the preconditions for c o m m u n i c a t i o n are t a k e n into consideration. Therefore a proper study of t i m e m u s t involve an analysis of the general m e a n s and f e a t u r e s of communication. So the study of m e a n i n g a n d language is essential for the u n d e r s t a n d i n g of time. Nobody h a s yet p r e s e n t e d a satisfactory definition of time. Every a t t e m p t to tell w h a t time is can be understood as a n acc e n t u a t i o n of some aspects of t i m e at t h e expense of others. Plato's definition of time as t h e 'moving image of eternity' a n d Aristotle's suggestion t h a t 'time is the n u m b e r of motion w i t h respect to earlier and later' are no exceptions (see e. g. [Whitrow 1972]). In our opinion the a t t e m p t to e s t a b l i s h a conclusive definition of time u l t i m a t e l y leads to confusion. Time is not definable by any other concepts. Time, in its fullness, is u n i q u e a n d sui generis. The Augustinean wisdom that time cannot be satisfactorily described using just one single formula, definition, or explanation, is now generally accepted among philosophers of time. I n order to gain more knowledge about t h e t e m p o r a l aspects of reality, time h a s to be studied within m a n y different strands of science. If such studies are to lead to a deeper u n d e r s t a n d i n g of t i m e it-

4

INTRODUCTION

self, various disciplines have to be brought together in the hope t h a t their findings m a y form a n e w synthesis, even t h o u g h we s h o u l d not expect any u l t i m a t e answer r e g a r d i n g the question of t h e n a t u r e of time! If a synthesis is to succeed, a common lang u a g e for t h e discussion of time h a s to be established. We are convinced t h a t t e m p o r a l logic (or 'the logic of time') is a crucial p a r t of such a language. The use of n u m b e r s in the description of time has made it obvious to see a c o n n e c t i o n b e t w e e n time a n d m a t h e m a t i c s . However, some people m a y be t a k e n aback by t h e claim t h a t the concept of time is a subject for the discipline of logic. This reaction is p r i m a r i l y c a u s e d by t h e idea t h a t logic is essentially timeless. Nevertheless, we will here a t t e m p t to d o c u m e n t t h a t t i m e h a s been relevant in t h e development of logic, and indeed, t h a t its relevance has n e v e r been more acute t h a n today. We shall argue t h a t this relation between time a n d logic is twoways: logical investigations into time are required for a deeper u n d e r s t a n d i n g of the concept, as well as for t h e development of a general language for t h e discussion of time. On t h e other hand, t e m p o r a l notions are required for a richer logic, applicable to a wider scope of problems r a n g i n g from c o m p u t e r science to philosophy. We intend to d e m o n s t r a t e t h a t the concept of time can in fact be studied using temporal logic. According to St. Augustine we all have a tacit knowledge of w h a t time is, even t h o u g h we cannot define time as such. In a sense the endeavour of temporal logic is to study some manifestations of this tacit knowledge. In the first part of this book the question will be discussed from t h e perspective of the history of logic. It will be documented that t h e r e is a rich tradition of t e m p o r a l logic from the ancient and medieval periods. We shall take t h e liberty of presenting some of these old ideas utilising t h e explanatory power of symbolic logic. The application of symbolic logic to ancient and medieval logic is in fact disputed - some researchers claim t h a t such a procedure is a n a c h r o n i s t i c a n d misleading. We shall n o t t a k e up t h a t methodological discussion, except in the form of 'arguing by doing' - showing how concrete examples do lend themselves to a discussion p a r t l y in t e r m s of symbolic logic. The great Polish

INTRODUCTION

5

School of Logic set the example for this approach, and indeed, temporal logic itself partly began from the conviction t h a t some classical logical ideas could and should be so studied. In our opinion, some of the brilliant insights of those ancient and medieval logicians and philosophers can in fact only be fully u n d e r s t o o d and studied f u r t h e r by m o d e r n logicians w h e n recast in a symbolic formalism. The rediscovery in our century of the importance of time and tense is first and foremost due to the works of Arthur Norman Prior, who was deeply inspired by his studies in ancient and m e d i e v a l logic. In the 1950s and 1960s Prior laid out the foundation of temporal logic and showed t h a t this important discipline was i n t i m a t e l y connected with modal logic. Prior revived the medieval a t t e m p t at formulating a temporal logic corresponding to natural language. In doing so, he also used his symbolic formalism for investigating the ideas put forward by these logicians. Prior argued that temporal logic is fundamental for understanding and describing the world in which we live. He r e g a r d e d tense and modal logic as particularly relevant to a n u m b e r of i m p o r t a n t theological as well as philosophical problems. Using his temporal logic Prior analysed the fund a m e n t a l question of d e t e r m i n i s m versus freedom of choice. The second part of the book will describe this rediscovery of the logic of time, focusing on Prior's contribution for the reasons just given. But we shall also describe his most important forerunners in the field of temporal logic in the 19th century and the first decades of the 20th century. In fact, Prior himself preferred the term 'tense logic', but it has since t h e n become commonplace to call the general quest for a logic of time as well as the resulting systems 'temporal logic'. We shall adopt the modern usage in this respect; later, we shall clarify the special m e a n i n g of 'tense logic' within the general picture. The main parts of temporal logic have been developed using mathematical symbolism and calculi, but nevertheless it has first and foremost been a philosophical enterprise. During the last decades it has become clear that temporal logic also has a n u m b e r of practical applications. In part three we intend to outline some modern issues of temporal logic.

1.1. THE SEA-FIGHT TOMORROW C h a p t e r IX of Aristotle's work, On Interpretation, is without doubt the philosophical text which has had the g r e a t e s t impact on t h e debate about the relations between time, t r u t h , and possibility. In this text we fred the famous example of 'the sea-fight t o m o r r o w ' ; the discussion of this example c e r t a i n l y bears witness to the fact t h a t Ancient philosophy was highly conscious of tense-logical problems (see [Gaskin 1995]). Central to the discussion is the question of how to interpret the following two statements: 'Tomorrow there will be a sea-fight'. 'Tomorrow there will not be a sea-fight'. Aristotle makes the following observation: Let us take, for example, a sea-fight. It is requisite on our hypothesis t h a t it should neither take place nor yet fail to t a k e place on the morrow. These and other s t r a n g e consequences follow, provided we assume in the case of a pair of contradictory opposites having universals for subjects and being themselves universal or having an individual subject, t h a t one must be true, the other false, t h a t contingency t h e r e can be none and that all things that are or take place come about in the world by necessity. [On Interpretation, 18 b 23 ff.] It is n a t u r a l to discuss this text with a special view to the tenselogical semantics of operators concerning the f u t u r e . The two s t a t e m e n t s above can be symbolised by

F(1)p F(1) -p w h e r e p stands for the statement 'there is a sea-fight', and F(1) is read 'it will be the case in one time unit' - this is w h a t we would today call a metrical tense operator, since it is combined with an 10

THE SEA-FIGHT TOMORROW

11

explicit m e a s u r e of time. In the present context F(1) simply means 'it will be the case tomorrow', p stands for 'there is a sea-fight going on', and ~p stands for the negation 'there is not a sea-fight going on'. Can s t a t e m e n t s like F(1)p and F(1)~p be said to be t r u e (or false) a l r e a d y today? Alternatively, is the t r u t h value of t h e s t a t e m e n t undetermined, such that it c~nnot be said to have any actual t r u t h value today? The answers to these questions in t u r n bear upon the interpretation of modality. For if we assume t h a t F(1)p is t r u e today, is the statement then not also necessary today? And further, if it t u r n s out that there is no sea-fight tomorrow, can F(1)p t h e n be possible today? Aristotle was clearly aware of these relations, and in the discussion of the example he as well as later t h i n k e r s also examined the r e l a t e d problems concerning the modal concepts of possibility and necessity. On grounds of his basic assumption of indeterminism, Aristotle claimed t h a t neither s t a t e m e n t could be necessary today. However, the same does not apply to statements about the past or the present; t h e y are either necessarily true or necessarily false. Aristotle is apparently a 'past-determinist' and a 'present-determinist', but a 'future-indeterminist'. The Aristotelian logic m u s t therefore be assumed to allow the following proposition within its framework:

P(n)p ~ NP(n)p P(n) stands for 'it was the case n time units ago' and N stands for 'it is necessary that ...'. This implication must as a m i n i m u m be consistent with the general theory, that is, its negation m u s t not be valid. But still more likely, it should in fact itself be a t h e o r e m of the theory. On the other hand, the theory m u s t reject the validity of

NF(n)p v NF(n)-p i.e. it should accept that in some cases it holds t h a t

MF(n)p A MF(n)-p

12

CHAPTER 1.1

where M stands for 'it is possible that ...'. T h e Polish logician J a n Lukasiewicz [1920] has a r g u e d t h a t Aristotle in fact considered propositions about future contingents to be n e i t h e r t r u e nor false. Such an interpretation is not merely a m o d e r n c o n s t r u c t i o n . Richard of L a v e n h a m (c.1380) said s o m e t h i n g very similar, w h e n he f o r m u l a t e d the Aristotelian position in the following way: The third opinion, which was Aristotle's opinion, opposes the Christian faith in so far as this opinion presupposes t h a t God does not k n o w more d e t e r m i n a t e l y t h a t Antichrist will be t h a n t h a t Antichrist will not be; a n d t h a t He does not know more d e t e r m i n a t e l y t h a t the day of j u d g e m e n t will be t h a n t h a t the d a y of j u d g e m e n t will not be; a n d t h a t He does not know more d e t e r m i n a t e l y t h a t t h e resurrection of t h e dead will be t h a n t h a t the resurrection of the dead will not be. And the reason is t h a t there is no d e t e r m i n a t e t r u t h of any of the two propositions about contingent future events. B u t these propositions 'the day of j u d g e m e n t will be' and 'the resurrection of t h e dead will be' are contingent propositions about t h e future, t h e r e f o r e they are not d e t e r m i n a t e d to t r u t h , a n d in consequence not more d e t e r m i n a t e d to t r u t h t h a n to falsity (and also not conversely). The consequence is clear, and the m a j o r p r e m i s e is Aristotle's opinion in 'On Interpretation'. A n d this opinion presupposes t h a t no contingent proposition about the future is true, and t h a t no such proposition is false. This was Aristotle's intention as Ockham says in his book about 'On Interpretation'. [OhrstrCm 1983] L a v e n h a m ' s version of Aristotle's s t a t e m e n t clearly m e a n s t h a t F(n)p as well as F(n)~p are n e i t h e r t r u e nor false. It is, however, u n c l e a r w h e t h e r he h a d in m i n d a third true-value corresponding to 'indeterminate', or simply held t h a t no t r u t h value is defined for such contingent f u t u r e propositions. In any case, L a v e n h a m r e g a r d e d the Aristotelian view as contrary to t h e C h r i s t i a n faith, a n d he preferred a solution suggested by William of O c k h a m .

THE SEA-FIGHT TOMORROW

13

Lukasiewicz has argued t h a t Aristotle's text in chapter IX of

On Interpretation should be read as an a r g u m e n t for a threevalued logic. At least in the early 1950's A. N. Prior shared this view [1957a, p.86]. At this time he thought t h a t this was the only w a y to construct an indeterministic tense-logic [1967, p.128]. Therefore he suggested a three-valued logic of tensed propositions [1953]. L a t e r it became clear to h i m t h a t indeterministic tense-logic with bivalence is possible in at least two i m p o r t a n t ways - k n o w n as 'the Ockhamist system' and 'the P e i r c e a n system'. These systems are Prior's formalisations of ideas by O c k h a m and Peirce; both systems will be e x a m i n e d in detail later on. The i n t e r p r e t a t i v e problems regarding On Interpretation c h a p t e r IX are by no m e a n s simple. N. Rescher [1968] h a s shown in a word-by-word analysis of t h e critical passage of chapter IX how a realistic interpretation, which maintains the principle of bivalence, can be consistently defended, and in fact was defended by Scholastic and Moslem philosophers in the Middle Ages. As we shall see, this medieval interpretation and the tense logic pertaining to it provide an affirmative answer to the question of w h e t h e r statements about the contingent future do h a v e a t r u t h - v a l u e (at the time of utterance). T h e y also confirm t h a t even if it turns out that there is no sea-fight tomorrow, F(1)p can be regarded as possible today. In principle, pastd e t e r m i n i s m was also accepted, but it was observed t h a t it only holds for s t a t e m e n t s which are properly about the past. What was at stake here was to rule out necessitation for sentences of the form P(n)F(m)p, whose g r a m m a t i c a l form is in the past tense, but which are only spuriously about the past when m > n. C o n c e r n i n g the g e n e r a l discussion of t h e r61e of necessity, Rescher referred to Peter Abelard (c. 1079-1142), who stated: No proposition about the contingent future can be either det e r m i n a t e l y true or determinately false in the same sense, but this is not to say that no such proposition can be t r u e or false. On the contrary, any such proposition is true if the outcome is to be true as it states though this is still unknown to us. What Aristotle wished to m a i n t a i n in his De Inter-

14

C H A P T E R 1.1 pretatione was that while a proposition is necessary w h e n it is true, it is not t h e r e f o r e necessarily t r u e s i m p l y a n d always. [Kneale p.214]

As we shall see later the O c k h a m i s t system m a k e s it possible t h a t a proposition about t h e contingent future can be t r u e now, even t h o u g h its truth-value is still u n k n o w n to us. In this crucial sense Abelard's i n t e r p r e t a t i o n is in a g r e e m e n t w i t h Prior's O c k h a m i s t system. H e n n i n g Boje Andersen a n d J a n Faye [1980] have, however, p u t forth a different i n t e r p r e t a t i o n of chapter IX. T h e y claimed t h a t Aristotle would probably reject t h e general validity of w h a t could be called 'the law of excluded middle for s t a t e m e n t s in the future tense', i.e. for allp:

F(n)p v F(n)-p Given that this proposition is not valid, it must be accepted that

~F(n)p A -F(n)-p m a y indeed be t r u e for some proposition p. In fact, according to this i n t e r p r e t a t i o n the latter formula is possible for a n y conting e n t s t a t e m e n t about the future. On t h e other h a n d , it is also clear t h a t F(n)p and F(n)-p cannot both be true. Therefore

-F(n)p v ~F(n)-p m u s t be a t h e o r e m in t h e A r i s t o t e l i a n s y s t e m u n d e r this interpretation. It is w o r t h p o i n t i n g o u t t h a t t h i s i n t e r p r e t a t i o n m a k e s Aristotle's observations c o n s i s t e n t w i t h the a f o r e m e n t i o n e d P e i r c e a n system. Thus, t h e r e is a line from t h e two basic i n t e r p r e t a t i o n s of Aristotle's text p r e s e n t e d here to Prior's two major indeterministic tense logical systems.

1.2. THE MASTER ARGUMENT OF DIODORUS CRONUS Diodorus Cronus (ca. 340-280 B.C.) was a philosopher of the M e g a r i a n school [Sedley 1977]. He achieved wide f a m e as a logician a n d a formulator of philosophical paradoxes. T h e most w e l l - k n o w n of t h e s e p a r a d o x e s is t h e so-called ' M a s t e r A r g u m e n t ' which in Antiquity was u n d e r s t o o d as an a r g u m e n t designed to prove t h e t r u t h of fatalism. Unfortunately, only the p r e m i s e s a n d t h e conclusion of t h e a r g u m e n t are k n o w n . We know almost nothing about t h e way in which Diodorus u s e d his p r e m i s e s in order to reach t h e conclusion. D u r i n g t h e last few decades various philosophers and logicians have tried to reconstruct t h e a r g u m e n t as it m i g h t have been. The reconstruction of t h e M a s t e r A r g u m e n t certainly constitutes a g e n u i n e problem within the history of logic. It should, however, be noted t h a t the a r g u m e n t has been studied for reasons other t h a n historical. First of all, t h e M a s t e r A r g u m e n t h a s been read as a n argum e n t for determinism. Secondly, t h e M a s t e r A r g u m e n t can be r e g a r d e d as an a t t e m p t to clarify the conceptual relations between t i m e and modality. W h e n seen in this perspective any att e m p t e d reconstruction of the a r g u m e n t is i m p o r t a n t also from a s y s t e m a t i c point of view, and this is obviously t r u e for any version of the argument, even if it is historically incorrect. Our approach in this chapter will in t h e first p a r t be m a i n l y historical. We shall c o m m e n t on some of t h e r e c o n s t r u c t i o n s which have been suggested, and p r e s e n t an elaborated version of one of them. At the end of t h e chapter, we shall discuss some of t h e philosophical and c o n c e p t u a l problems r e l a t e d to t h e Master Argument. The M a s t e r A r g u m e n t is a t r i l e m m a . According to Epictetus, Diodorus argued t h a t the following three propositions cannot all be true [Mates 1961, p.38] :

(D1) (D2)

Every proposition true about the past is necessary. An impossible proposition cannot follow from (or after) a possible one. 15

16

CHAPTER 1.2 (D3)

T h e r e is a proposition which is possible, but which n e i t h e r is nor will be true.

Diodorus used this incompatibility combined with t h e plausibility of (D1) and (D2) to justify t h a t (D3) is false. Assuming (D1) and (D2) he went on to define possibility and necessity as follows: (DM) (DN)

The possible is t h a t which either is or will be true. The necessary is t h a t which, being true, will not be false.

In order to reconstruct the Master A r g u m e n t two f u n d a m e n tal questions m u s t be answered: (1) (2)

How should 'proposition' in (D1-3) be understood? How should 'follow' in (D2) be understood?

For the sake of completeness it should be m e n t i o n e d t h a t for some reconstructions it is also relevant w h e t h e r the structure of time is assumed to be discrete or continuous. The first of the above questions can be answered in at least two ways : (1.1) (1.2)

The propositions m e n t i o n e d in (D1-3) are temporally definite statements. The M a s t e r A r g u m e n t refers in fact to s t a t e m e n t s corresponding to propositional functions.

F.S. Michael [1976] has suggested a reconstruction of the Master A r g u m e n t based on (1.1). According to Michael the t r u t h or falsity of such s t a t e m e n t s is entirely unaffected by t h e time of assertion. In his version the first premise of the a r g u m e n t can be formulated in t h e following way: (D1M)

If t h e proposition Po is true at some time t' before t, then t h e t r u t h ofpo is necessary at t. In symbols: (T(t',po) A t' < t ) ~ N(t, po)

THE MASTER ARGUMENT

17

Note t h a t this can only be reasonable if the proposition Po in (DIM) itself takes the form T(t",r). Using (D1M) Michael could in fact construct an a r g u m e n t like the Master A r g u m e n t without using (D2) directly. For his a t t e m p t at a reconstruction, however, Michael had to presuppose t h a t a necessary proposition is true. This principle seems to be uncontroversial, but it is not implied by (D1-3) alone. His proof can be presented in the following way: According to (D3) it is assumed that there is a proposition qo, which is possible, but false now and also at any future time. The proposition qo m u s t in the a r g u m e n t itself be of the form T(t",r) by Michael's assumption of (1.1). This means t h a t the following holds: M(n, qo) A T(n,~qo) A (Vt: t>n ~ T(t,~qo)) Now, qo must be false also before n, since if for some t' T(t',qo) A t'
18

CHAPTER 1.2

sons it seems m o s t probable that Diodorus t h o u g h t of propositions as corresponding to w h a t we today would call functions. His examples include s t a t e m e n t s like 'It is day', 'I am conversing', 'It is light'. As Mates [1961, p.36] has stated, these propositions 'are t r u e at certain times and false at others', or equivalently, 'they b e c o m e t r u e and become false'. F u r t h e r m o r e , Mates could also conclude t h a t Diodorean necessity would in most cases apply to such 'functional propositions', so generally s p e a k i n g we should expect (1.2) to be t h e correct answer as reg a r d s t h e s t a t u s or n a t u r e of p r o p o s i t i o n s in t h e M a s t e r A r g u m e n t . Nevertheless, Mates did not t h i n k t h a t (D1) could m a k e sense if'proposition' is understood in this way [1961, p.39]. Therefore Mates' analysis apparently left us with an enigma: according to this analysis, (1.2) was t h e most probable answer, but Mates could not see how this a s s u m p t i o n could be consistent with t h e context of t h e Master Argument. However, as we shall see in the following, Prior has shown how a reading of(D1) consistent with (1.2) is in fact possible. But first we m u s t examine t h e question regarding the u n d e r s t a n d i n g of (D2). This question can also be answered in at least two different ways: (2.1) (2.2)

'Follows' in (D2) refers to temporal order. 'Follows' in (D2) refers to logical implication.

Like t h e r e c o n s t r u c t i o n s of Zeller [1882] and of Copleston [1962], Rescher's reconstruction [1966] of the Master A r g u m e n t is based on an a s s u m p t i o n like (2.1), i.e. on a temporal version of (D2). Rescher a s s u m e s t h a t the original formulation of this premise can be r e f o r m u l a t e d in the following way: (D2x)

The impossible does not follow after the possible.

(D2x) implies t h a t w h a t has been possible will always be possible. This 'principle of possibility-conservation' is obviously not very plausible. E v e n if some proposition p could once be regarded as possible, consistently with w h a t e v e r else obtained at t h a t time, some of t h e conditions for p m a y change p e r m a n e n t l y

T H E M A S T E R ARGUMENT

19

at a later time such as to m a k e it impossible always thereafter. Moreover, Mates observed t h a t the word u s e d by Epictetus in (D2), which Rescher t r a n s l a t e s into 'follow after', is t h e s a m e word used by Diodorus for 'is a consequent of. It should also be n o t e d t h a t C h r y s i p p u s , who rejected the M a s t e r A r g u m e n t , understood its second premise as referring to logical consequence r a t h e r t h a n t e m p o r a l succession [Mates 1961, p.39]. Finally, a c i r c u m s t a n t i a l b u t i m p o r t a n t piece of evidence t h a t (D2) is concerned w i t h logical consequence is t h e fact t h a t Diodorus studied the n a t u r e of implication very carefully. The famous debate between Diodorus and Philo of Megara precisely concerned t h e relation b e t w e e n t i m e a n d implication. Their views on implication w e r e described in t h e following way by Sextus Empiricus: according to Philo such a conditional as 'If it is day, t h e n I a m conversing' is t r u e w h e n it is day a n d I a m conversing, since in t h a t case its antecedent, 'It is day' is t r u e and its consequent, 'I a m conversing', is true; b u t according to Diodorus it is false, for it is possible for its antecedent, 'It is day', to be t r u e and its consequent 'I a m conversing' to be false at some time, namely, after I h a v e become quiet... [Adv.Math. VIII, ll2ff; Mates, 1961, p. 98] This conflict b e t w e e n Diodorus and Philo was obviously concerned w i t h w h e t h e r one could allow t h e t r u t h values of t h e implication to vary w i t h time or not. As Mates [p.46] has argued, a conditional was proved to be Diodorus-true by showing that it n e v e r has a t r u e a n t e c e d e n t and a false consequent. T h a t is, Diodorus favoured w h a t we today could call temporally strict implication, w h e r e a s Philo argued for material implication. The quotation also bears on the status of propositions, for Diodorus' a r g u m e n t as referred by Sextus Empiricus presupposes t h a t propositions are understood as functions. It appears t h a t Diodorus regarded logical implication as very important. Therefore, it is only natural to a s s u m e t h a t it played an i m p o r t a n t rSle in his M a s t e r A r g u m e n t . We believe t h a t (2.1) should be rejected and t h a t (2.2) should be accepted, a n d

20

CHAPTER 1.2

also t h a t it is n a t u r a l to assume that the implication in question was t h e Diodorus-implication, which is true just in case it never has a t r u e antecedent and a false consequent.

PRIOR'S RECONSTRUCTION Prior's reconstruction [1967, p.32 ft.] of the Master A r g u m e n t follows the line of the interpretations (1.2) and (2.2). Thus it basicaUy adopts the same u n d e r s t a n d i n g of 'proposition' and consequence as we have been arguing for above. Prior uses tenseand m o d a l operators in his reconstruction, and i n t e r p r e t s the logical (Diodorean) consequence involved in (D2) as w h a t is in modal logic usually called 'strict implication', symbolised by -->. On t h e s e assumptions it is possible to restate the reconstruction problem. Using symbols, (D1-3) can be formulated in the following way: (DI') (D2') (D3')

Pq ~ NPq ((p ~ q) A Mp) ~ Mq (3 r) (Mr A - r A -Fr)

w h e r e F is read as 'it will be the case that...', P is read as 'it has been t h e case that ...'., and --~ is the strict implication defined as p -->q =N(p ~ q ) We are now r e a d y to reformulate Prior's reconstruction. In doing so, we shall at first leave aside some of the problematic points about it, in order to make the m a i n thrust of the argum e n t as clear as possible. We shall use the propositional function q: 'Dion is here' as an example. The reconstruction, then, runs as following way. Let us make the following two assumptions: (P1) (P2)

It is possible for Dion to be here. In symbols: Mq Dion is not here and he never will be here. In symbols: -q A -Fq

THE MASTER ARGUMENT

21

Obviously, (P1) and (P2) t o g e t h e r m a k e up an i n s t a n c e of (D3). Now intuitively speaking, if Dion is not here now and from now on n e v e r will be here, t h e n in the 'immediate past' it was true simply t h a t Dion never would be here. Thus, it follows from (P2) t h a t (P3)

It has been the case that Dion never will be here. In symbols: P~Fq

By s u b s t i t u t i o n into ( D r ) we have (P-Fq ~ NP-Fq). Therefore, it follows from (P3) and (DI') t h a t (P4)

It is necessary t h a t it has been the case t h a t Dion never will be here. In symbols: NP-Fq

For t h e sake of exposition, it is useful to subject (P4) to two transformations. First, since N is equivalent with ~M-, we directly obtain (P5)

It is impossible t h a t it has not been the case t h a t Dion never will be here. In symbols: ~M-P-Fq

We can now m a k e use of the common tense-logical symbol H, which is an abbreviation o f - P - , and which m a y be read 'it has always been the case that ...' Using H in (P5), we get (P6)

It is impossible t h a t it has always been the case t h a t Dion will be here. In symbols: -MHFq

If Dion is h e r e now, then at any time in the past it has been true to say 'Dion will be here'. Hence, the following implication is true: (P7)

If Dion is here, then it has always been the case that Dion will be here. In symbols: q ---->HFq

22

CHAPTER 1.2

By conjoining (P1) a n d (P7) we obtain ((q ~ H F q ) A Mq). Using (D2') we can then deduce M H F q . We have now arrived at a contradiction, since on a s s u m i n g (P1) a n d (P2) we h a v e derived ~ M H F q (P6) as well as M H F q . Therefore, the combined assumption of (P1) a n d (P2) m u s t be rejected. Unfortunately, it is clear t h a t Prior is not able to reconstruct t h e a r g u m e n t only u s i n g (D1), (D2) a n d (D3). In addition to these, he needs two extra premises. In order to m a k e sure that t h e a r g u m e n t from (P2) to (P3) is valid, he m u s t a s s u m e t h a t ( - q A -Fq) ~ P ~ F q

or, to p u t it in a general form, that (D4) (p A Gp) ~ P G p w h e r e G - ~F~ ('it will always be the case that...'). F u r t h e r m o r e , he m u s t a s s u m e t h a t (P7) is in fact a valid strict implication such t h a t (D5) N ( p D H F p ) is valid in general. Prior's proof t h a t t h e three Diodorean premises (DI', D2', D3') are inconsistent given (D4) and (D5) can be s u m m a r i s e d as a reductio ad a b s u r d u m proof in the following way: (1) (2) (3) (4) (5) (6) (7) (8) Q.E.D.

M r A - r A ~Fr Mr N(r ~ HFr) MHFr -r A G-r PG-r NPG-r ~MHFr

(from (from (from (from (from (from (from (from

D3') 2) D5) D2, 2 & 3) 1) 5 & D4) 6 & D1) 7; contradicts 4)

THE MASTER ARGUMENT

23

O. Becker [1960] has shown t h a t the extra premises (D4) and (D5) can be found in the writings of Aristotle. For t h a t reason Becker concludes t h a t it seems reasonable to assume t h a t the extra premises were generally accepted in antiquity. However, Prior's addition of (D4) and (D5) is nevertheless problematic (even though the argument t h u s reconstructed is interesting in its own right). (D4) is in fact a r a t h e r complicated s t a t e m e n t and not so innocuous as it may s e e m at first glance observations which will indeed become clear w h e n we are going to discuss the Ockhamist and Peircean systems. It is not v e r y likely t h a t Diodorus would involve such an a r g u m e n t without m a k i n g it a n explicit premise in the M a s t e r Argument. As r e g a r d s (D5), we k n o w t h a t Diodorus u s e d t h e M a s t e r A r g u m e n t as a case for the definitions (DM) and (DN). That is, in t h e a r g u m e n t i t s e l f M (or N) should in a sense be regarded as primitive. It is hard to believe t h a t Diodorus would involve a premise about N without stating it explicitly.

A NEW RECONSTRUCTION OF THE MASTER ARGUMENT As we have argued, Mates in his excellent analysis gave all the essential information needed for a reconstruction of the Master Argument. On the basis of the considerations so far we shall suggest a very simple a r g u m e n t as a possible reconstruction. We shall see that the a r g u m e n t can be formulated without the use of complicated extra premises as it is the case in Prior's reconstruction. We shall assume t h a t in the M a s t e r A r g u m e n t c e r t a i n notions regarding time and propositions are t a k e n for granted:

(a) (b)

Time is discrete. Diodorean propositions are functions of time. Thus, propositions are functions from i n s t a n t s into t r u t h values a n d conversely, s u c h f u n c t i o n s a r e propositions. For the function application of a proposition p to an instant t we write T(t,p).

24

CHAPTER 1.2

(c)

The Diodorean implication involved in (D2) can be defmed in terms of present-day temporal logic as

(p ~ q) if and only if (Vt) (T(t,p) ~ T(t,q)) Ad (a): It is not possible to prove directly t h a t Diodorus took t i m e to be m a d e up of t e m p o r a l atoms, although there is evidence t h a t Diodorus believed in indivisible places and bodies [Adv. Phys. II,142-143]. Richard Sorabji [p.19] has m a i n t a i n e d t h a t a c e r t a i n p a s s a g e in the w o r k s of Sextus E m p i r i c u s [M 10.86-90] indicates t h a t Diodorus was a temporal atomist. But e v e n if Sorabji is w r o n g and Diodorus was not a t e m p o r a l atomist, we m i g h t still u n d e r t a k e a reconstruction along the lines w h i c h we have been suggesting, provided t h a t Diodorus held s o m e t h i n g like (A) No proposition has a first i n s t a n t of truth. If a proposition is true, it has already been true for some time. A l t h o u g h we have no direct information indicating t h a t Diodor u s actually m a d e this assumption, it is indeed very likely t h a t he was aware of Aristotle's point of view: For a change can actually be completed, and there is such a thing as its end, because it is a limit. But with reference to t h e b e g i n n i n g t h e phrase h a s no meaning, for t h e r e is no b e g i n n i n g of a process of change, and no p r i m a r y 'when' in which t h e change was first in progress. [Phys. 236a 12-14] It is n o t u n r e a s o n a b l e to s u r m i s e t h a t Diodorus tried to elaborate t h i s observation, and t h a t this work led h i m to an a s s u m p t i o n like (A). We shall, h o w e v e r , omit a d e t a i l e d reconstruction of the m a s t e r a r g u m e n t on the basis of (A). Ad (b): Diodorus apparently t h o u g h t of propositions as t h o u g h t h e y c o n t a i n e d time-variables. T h e s e propositions are t r u e at certain times a n d false at other times. On the other hand, Mates h a s m a i n t a i n e d t h a t "although Diodorus u s u a l l y predicates

THE MASTER ARGUMENT

25

necessity of w h a t are in effect propositional functions, it seems t h a t in the first of his three incompatibles, necessity is predicated of a proposition" [1961, p. 39]. We shall d e m o n s t r a t e how an u n d e r s t a n d i n g of the Master Argument based on (1.2) as well as (2.2) is possible. Ad (c): According to Mates [1961, p.451 "a conditional holds in the Diodorean sense if and only if it holds at all times in the Philonian sense". (The Philonian implication is simply the material implication). Mates has demonstrated that his conclusion is a clear consequence of a number of passages from the sources. Note t h a t t h e assumptions (a), (b), a n d (c) are all well d o c u m e n t e d in t h e known sources about Diodorus' logic. Moreover, they do not involve the modal concepts which are at stake in the argument. For these reasons (a)-(c) should not be regarded as extra premises like Prior's (D4) and (D5). In (c), we use ' 9 ' instead of ' ~ ' in order to emphasise t h a t our definition is distinct from Prior's definition of Diodorean implication, which was (p -->q) if and only if N(p ~ q) If we did not keep these two definitions apart, (c) might be seen as d e f i n i n g modality in terms of temporality. However, the M a s t e r A r g u m e n t was thought to lead to such a definition, to wit, (DM) and (DN), not to presuppose it. On (c), (D2) m a y be r e n d e r e d as (p ~ q) A M p ) ~ Mq w h e r e the possibility-operator should be understood as a still u n a n a l y s e d concept. We shall assume, however, t h a t Diodorus accepted the u s u a l interdefinability b e t w e e n n e c e s s i t y and possibility (as he indeed most likely did). In symbols, this means M = -N-, N = ~M-.

26

CHAPTER 1.2

Using t h e a s s u m p t i o n s (a) - (c), it is possible to reconstruct t h e argument. It should be noted t h a t a l t h o u g h (c) defines (p ~ q) in terms of temporality, it is very different from the k i n d of temporal definition involved in Rescher's u n d e r s t a n d i n g of the Diodorean 'follows'. O u r u n d e r s t a n d i n g of (p ~ q) refers to a quantification over t e m p o r a l instants r a t h e r t h a n a t e m p o r a l order. Let us a s s u m e (D3) for some s t a t e m e n t q, e.g. 'Dion is here'. In symbols: - q A ~Fq A Mq T h e n t h e s t a t e m e n t is false now and at every future time, alt h o u g h Dion's being here is possible. We intend to show t h a t t h e a s s u m p t i o n of (D3) contradicts t h e premises (D1) and (D2). Let r be a s t a t e m e n t t r u e only at the time just before the present time. A l t h o u g h any a r b i t r a r y s t a t e m e n t fulfilling t h e req u i r e m e n t would do, we m a y choose t h e more i n t u i t i v e l y appealing r: 'The prophet says: Dion will never be here.' From t h e propositional function r, we can construct the propositional function Pr, which is obviously false at any past time, true now a n d always in the future. We can illustrate the situation by the following figure, where t h e i n s t a n t 'now' is represented by the n u m b e r 10:

~Pr

~r

?q

~Pr

~Pr

~r

Pr

Pr

Pr

~r

~r

~r

11

8

9

10

?q

?q

~q

~q

12

~q

THE MASTER A R G U M E N T

27

Clearly r is false at any i n s t a n t other t h a n 9, the i n s t a n t immediately preceding the now. ~Pr is true at any past time, i.e. any instant lesser t h a n 10. On the other hand, Pr is true now, at 10, and always thereafter. Finally, by our assumption of (D3), q is false now and always in the future. However, q might be true or false at any past time. Since Pr is t r u e now, we can by (D1) obtain NPr, which is equivalent with ~M-Pr. It is also evident that

(q ~ - P r ) . This Diodorean implication is valid since if q is true at time t, t h e n t m u s t be a past time; this follows from our assumption of (D3) as illustrated in the figure. Furthermore, -Pr is t r u e at any past time. Therefore the antecedent can never be t r u e w h e n the consequent is false. But the validity of this Diodorean implication contradicts (D2), since the impossible, -Pr, follows from the possible, q. Therefore the assumption of (D3) has to be rejected. In this w a y the Master A r g u m e n t can be reconstructed using discrete time and the Diodorean idea of implication. We t h i n k it very likely t h a t this was the kind of reasoning actually used by Diodorus. It is interesting t h a t the above a r g u m e n t works even if it is assumed t h a t the first premise (D1) of the Master A r g u m e n t is concerned only with propositions which are genuinely about the past. An example of a proposition which is not genuinely about the past would be 'One day ago it was the case that in two days, Dion will be here'. Such propositions should not be necessitated by (D1), although they may be necessitated on other grounds. In Prior's reconstruction, s t a t e m e n t s which are only spuriously about the p a s t are regarded as necessary. In this w a y the validity of implications like PGq ~ NPGq can be derived. In our reconstruction, however, such a questionable use of (D1) is completely unnecessary. The w a y (D2) is used in our reconstruction bears some resemblance to one of the paradoxes of implication, since we can without loss of generality assume t h a t q is not only false in the

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CHAPTER 1.2

present and the future, b u t also in the past - that Dion has n e v e r been here, is not here a n d never will be here. In this case a n y proposition will follow from q in the Diodorean sense. Indeed, it is not required that t h e r e be any semantical relation between q and r in the a r g u m e n t . In general, ifq is any proposition w h i c h is always false, then t h e Dioderean implication (q ~ p) holds for any a r b i t r a r y proposition p; in this case, the implication obviously n e v e r has a t r u e antecedent and a false consequent. But t h e n we m a y choose any possible proposition q in order to show t h a t p m u s t be possible. Hence, any proposition which is always false m u s t be possible on the assumption of (D2). In this connection it should be noted that the ancients w e r e aware of the paradoxes of implication. There can be no doubt that Diodorus, too, realised t h a t any proposition which is always false, implies any other proposition.

LOGICAL DETERMINISM It is very likely that the M a s t e r A r g u m e n t w a s originally designed to prove fatalism or determinism. Because of the a p p a r e n t plausibility o f ( D 1 ) a n d (D2), the a r g u m e n t was understood as a r a t h e r s t r o n g case a g a i n s t (D3). The denial of (D3) is equivalent to the view t h a t if a proposition is possible, t h e n either it is t r u e now or it will be true at some future time. So in a nutshell the a r g u m e n t is that an event which never will h a p p e n and is not h a p p e n i n g now cannot be possible, and hence e v e r y t h i n g h a p p e n i n g now or in the f u t u r e is necessary. It should be clear, then, that the a r g u m e n t is interesting not only for historical reasons. Its systematical content is entirely relevant for a m o dern discussion of d e t e r m i n i s m , too. The p r e s e n t - d a y philosopher w a n t i n g to argue against fatalism and d e t e r m i n i s m m u s t r e l a t e t o all known versions of the Master A r g u m e n t , directly or indirectly. If the fatalistic or deterministic conclusion of the Master A r g u m e n t is to be avoided, at least one of the two premises (D1) and (D2) h a s to be denied - at a n y rate, t h a t is t h e case as long as we accept the tacit assumption that time is a lin-

THE MASTER ARGUMENT

29

e a r structure. Now for a n y version of the M a s t e r A r g u m e n t based on t h a t assumption we believe that it is in fact quite reasonable to deny at least one of (D1) and (D2). Let us consider the versions which have been discussed above. As mentioned above, the second premise in Rescher's version of the Master A r g u m e n t t u r n s out to be equivalent to a 'principle of possibility-conservation'. It would certainly be reasonable to deny the validity of this principle. In Michael's version of the M a s t e r A r g u m e n t the first premise, (DIM), should be denied, since it is not reasonable to view a true proposition about the future as necessary, j u s t because it is formulated as a prophecy stated in the past. Such a proposition is about t h e past only in a spurious sense. Regarding (D1) in Prior's reconstruction we can make a similar observation. The statement 'It has been that Dion never will be here', (in symbols: P-Fq) should not be counted as necessary even if it is true. Even if we accept -q, -Fq, and P~Fq, there is no a priori reason to exclude the conceptual possibility of Dion's being h e r e at some future time, or his 'having always been going to be here', i.e. MFq and MPGq. Therefore, the w a y in which (D1) is u s e d in Prior's version of the argument should certainly be questioned. In our reconstruction, we do not have to a s s u m e any more t h a n the necessity of propositions which are genuinely about the past. When (D1) is seen in this way, it a p p e a r s reasonable, whereas (D2) should be rejected if time is linear. The reason is t h a t if there is a propositional function q which is possible but never true, t h e n our version of (D2) implies t h a t any absurdity (p A -p) also becomes possible. Obviously, it is not acceptable to regard an absurdity as being possible. Given t h a t time is linear it seems entirely reasonable to deny (D2). Prior himself questioned the validity of (D5) i.e. (D5) N(p ~ HFp)

30

CHAPTER 1.2

If we u n d e r s t a n d 'will be' as 'determinately will be', t h e n (D5) can certainly be denied, as in fact it is in the Peircean system, which Prior elaborated and to which he indeed preferred himself. We shall return to this system in part 2.

SOME CONCEPTUAL CONSIDERATIONS The Master A r g u m e n t can also be read as an attempt to relate the modal concepts of possibility and necessity to the concept of time. The various versions of the a r g u m e n t emphasise t h e impact of temporal indices on the operators of possibility a n d necessity. For instance, what is possible now need not be possible in the future. And what is now not necessary but a mere possibility, can become necessary in the future. It is obvious that the notion of modality involved in such assumptions should be linked to the idea of time. A proposition is necessary if it is 'now-unpreventable', and a proposition is possible if its negation is 'now-preventable'. In formulating his argument Diodorus was aiming at a justification of his definitions of possibility and necessity, (DM) and (DN), which were:

(DM) (DN)

The possible is that which either is or will be true. The necessary is that which, being true, will not be false.

But if these definitions are accepted, and if time is understood as a linear structure, then we are led to some kind of fatalism or determinism. As we have seen, we do not have to accept (DM) and (DN) on account of the a r g u m e n t itself, since at least one of the premises (D1) and (D2) should be rejected if time is implicitly or explicitly u n d e r s t o o d to be a linear structure. However, the picture is somewhat different if we avail ourselves of the modern notion of branching time: t h a t is, if time is considered to be a b r a n c h i n g structure, it is not representable as a subset of the real numbers, and both (D1) and (D2) as understood in our reconstruction become plausible. In p a r t 2 we shall e x a m i n e the notion of

THE MASTER ARGUMENT

31

branching time in detail. The basic idea can, however, easily be illustrated by t h e following figure: future a

future b

future c no

future d

The central idea is that for any given 'now' there are a n u m b e r of possible and different futures - sometimes called the 'forking paths into the future'. Just one of these will become actualised in the course of time. In this kind of structure a propositional function cannot be represented by a series of truth-values. Rather, it must be r e p r e s e n t e d as a complex structure of values. It should not be too h a r d to see that if the complex structures of branching time are discrete, t h e n our new version of the Master Argum e n t is still valid. The premises (D1) and (D2) as understood in our version can be accepted within all theories of b r a n c h i n g time, in which case the conclusion of the Master Argument also has to be accepted within these theories. An adequate conception of the notion of 'possibility' can then be captured by the formula M r - ( r v Fr)

Obviously this m e a n s that the definitions (DM) and (DN) should also be adopted in theories of branching time. In fact, t h e very use of the idea of 'possible futures' can be understood as a n acceptance of t h e conclusion of the Master Argument, since it is evident t h a t if time is branching then any possibility must belong to some possible future. So when we investigate the M a s t e r Argument from the perspective of the historical development of the logical s t u d y of time, the argument t u r n s out to be a demon-

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CHAPTER 1.2

s t r a t i o n of a f u n d a m e n t a l relationship between time and modality r a t h e r t h a n a case for fatalism or determinism. T h e relation between t i m e and modality and t h e a t t e m p t to d e f i n e m o d a l i t y in t e r m s of tense were very i m p o r t a n t to the f o u n d e r of m o d e r n symbolic tense logic, A. N. Prior. As we shall see in p a r t 2, Prior elaborated the formula above into a very c o m p l e x a n d c o n c e p t u a l l y refined definition - his so-called f o u r t h grade of tense-logical involvement, wherein t h e concept of m o d a l i t y becomes entirely absorbed by this tense logic. This f o u r t h grade expressed Prior's own conception of time.

1.3. THE STUDY OF TENSES IN THE MmDLE AGES The Diodorean Master A r g u m e n t can be seen as an example of t h a t interest in the logic of statements involving time which is p a r t of a tradition dating back to Aristotle and other A n c i e n t philosophers. T h e Scholastic logicians in particular m a d e a n u m b e r of original contributions to tense-logic. We shall now devote a few chapters to a brief survey of the most i m p o r t a n t of these contributions. Medieval logicians were engaged in an attempt to develop a logic of n a t u r a l language. With this objective they had to t a k e h e e d of the fact t h a t some statements do not have fixed t r u t h values. A proposition like 'Socrates is alive' is true when Socrates is alive, and it is false w h e n he is not alive. Therefore it is an i n t e g r a l p a r t of m e d i e v a l logic t h a t the t r u t h - v a l u e of a proposition can v a r y from time to time. For the same reasons it was n a t u r a l , i n d e e d inevitable, for t h e m to analyse t e n s e d s t a t e m e n t s in t h e i r logical studies. It was an important goal of theirs to be able to describe the logical content of propositions about past and future events. The Scholastic logicians discussed the status of t e n s e d s t a t e m e n t s with a view to theological problems. In the course of t i m e the difference between s t a t e m e n t s such as 'Christ was born', 'Christ is born', and 'Christ will be born' had given rise to a theological and logical problem. On the one hand, a distinction between the t h r e e forms from a purely logical point of view was considered legitimate. On the other hand, some claimed t h a t t h e r e was in principle no difference between w h a t h a d b e e n believed by the prophets (the third form), the contemporaries of J e s u s (the second form), and w h a t has been believed by Christians in all the succeeding centuries (the first form). The object of the faith is therefore the same one. But how can t h e u n i t y of faith and its independence of time be maintained, w h e n its m a i n t e n e t s are described by s t a t e m e n t s whose m e a n i n g s s e e m to v a r y in t i m e in the same m a n n e r as other t e n s e d statements? T h e r e were two different solutions in the Middle Ages, as pointed out by N u c h e l m a n [1980, p.133]. Firstly, t h e r e was a 33

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CHAPTER 1.3

c o m p a r a t i v e l y small g r o u p of thinkers, who d e f e n d e d one or a n o t h e r v a r i a n t of t h e so-called 'res-theory'. T h e s e scholars w a n t e d to consider the object of faith as an unchangeable entity. F o r i n s t a n c e , t h e object m i g h t be certain e v e n t s which were believed to have actually occurred. Secondly, t h e r e was a larger g r o u p of Schoolmen, who a d h e r e d to the 'enuntiabile-theory'. T h e y m a i n t a i n e d that t h e three statements: 'Christ was born', ' C h r i s t is born', and ' C h r i s t will be born' are significantly different, a l t h o u g h a h a r d core of m e a n i n g remains. It is quite conceivable t h a t doctrines (e.g. t h e virgin birth) ought to be expressed without tense, but that a confession (e.g. 'I believe t h a t C h r i s t was born by a virgin') becomes tensed w h e n the faith is to be expressed by an individual. The tense free doctrine is t h a t w h i c h all the tensed creeds refer to. T h o m a s Aquinas [ S u m m a theologiae II 2. q. 1. art. 2 & De v e r i t a t e q. 14, art. 12], a m o n g others, a t t e m p t e d to act as an i n t e r m e d i a r y between t h e two theories. He pointed out t h a t one can consider t h e object of faith either from the point of view of t h e object itself, or from t h e point of view of t h e faith. This c o r r e s p o n d s exactly to t h e difference between t h e 'res-theory' a n d t h e 'enuntiabile-theory'. The duality b e t w e e n t h e s e two perspectives is also evident when it comes to a discussion of the r e l a t i o n b e t w e e n God's a n d m a n ' s possibilities for h a v i n g k n o w l e d g e . A c c o r d i n g to T h o m a s , d i v i n e k n o w l e d g e is p r i m a r i l y a i m e d at t h e object itself (res), while m a n can only k n o w and believe in enuntiabile [Summa Theologia I q. 14. art. 15]. The t e n s i o n between these two Scholastic theories, which t a k e their s t a r t i n g points in enuntiabile and res, respectively, in a h i g h l y s t r i k i n g m a n n e r corresponds to the m o d e r n debate in tense-logic regarding A- and B-theories. We shall discuss those notions in p a r t 2. The debate about the semantic status of tense inflected statem e n t s can be regarded as an example of the scholastic emphasis on w h a t we now call tense-logic. This subject m a t t e r was of g e n e r a l i n t e r e s t d u r i n g t h e entire Middle Ages and covered a b r o a d s p e c t r u m of t h e o r i e s , w h i c h also i n c l u d e d w o r k c o n c e r n i n g q u e s t i o n s b o r d e r i n g on t e n s e - l o g i c . T h e s e i n v e s t i g a t i o n s s o m e t i m e s w e n t to t h e borderline of possible

TENSES IN THE MIDDLE AGES

35

l a n g u a g e use, as in a discussion by Boethius de Dacia (c.1270) about the s t a t e m e n t s 'heri curram' ('I will run yesterday') and 'cras cucurri' ('I r a n tomorrow') [Boethii Daci Opera, IV, I, p.203] - a discussion which was meant to be entirely serious. Medieval logicians in general were also very m u c h a w a r e of the problems related to logical arguments involving tenses. In his famous L o g i c a M a g n a - which is representative for a great deal of medieval logic - Paul of Venice {c. 1369-1429) dealt with a n u m b e r of questions concerning reasoning about time and t e n s e s . For instance, he considered the following a r g u m e n t [Part II Fasc. 8, p.271]: (Arg. 1) Socrates is in Rome at moment A; You are in Rome at some moment; therefore you are in Rome at moment A. This a r g u m e n t is in fact based on two other arguments, which can be stated in the following way: (Arg. 2) Socrates is; therefore Socrates is now (i.e. at the present moment). (Arg. 3) You are; therefore you are now (i.e. at the present moment). P a u l obviously r e a l i s e d why t h e use of t h i s k i n d of a r g u m e n t a t i o n can be critised. We m a y reformulate the m a t t e r in t e r m s of t h e 'res-theory' and t h e ' e n u n t i a b i l e - t h e o r y ' . According to the 'res-theory' (correspoding to t h e modern Bt h e o r y ) (Arg. 2) and (Arg. 3) are invalid. According to the 'enuntiabile-theory' (corresponding to the m o d e r n tense logic) the arguments are all valid, but the premise 'Socrates is in Rome at m o m e n t A' is a contradiction unless the m o m e n t A is a s s u m e d to be the present moment. If A is not identical with the

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CHAPTER 1.3

p r e s e n t m o m e n t , one should according to the 'enuntiabiletheory' perhaps rather formulate the premise as Socrates has been, is or will be in Rome at moment A; but with t h a t prem~.qe the argument is clearly invalid. In his Sophismata, John Buridan (c. 1295-1358) dealt with the problem of self-reference in a setting, which r e l a t e d t h a t problem to the subjects of time and tense. Medieval logicians used the t e r m 'sophism' to describe propositions which were in a given situation considered true by some and false by others. That is, arguments could be made both for and against the t r u t h of the proposition in question. Such was exactly the case in Buridan's discussion of this sophism: ' ~ o u will throw me into the water" [Buridan 1966, p. 219]. For the discussion of that sophism B u r i d a n imagined the following scenario: Socrates w a n t s to cross a river and comes to a bridge g u a r d e d by Plato, who says: "Socrates, if in the first proposition w h i c h you utter, you speak the truth, I will permit you to cross. But surely if you s p e a k falsely, I shall throw you into the water." Buridan assumed t h a t Socrates then replies with the sophism in question. Obviously, it would be very hard for Plato to fred out w h a t he should do. He m u s t a d m i t t h a t he c a n n o t k e e p his promise. B u r i d a n m a i n t a i n e d t h a t the sophism when u t t e r e d by Socrates has a t r u t h value, i.e. it is either true or false. It is, however, "not d e t e r m i n a t e l y true or d e t e r m i n a t e l y false" [Buridan 1966, p. 220]. This m e a n s that we cannot d e t e r m i n a t e l y know w h e t h e r it is true or false, until we have seen how Plato acts w h e n Socrates is crossing. The two implications which can be found in Plato's s t a t e m e n t are invalid, since in both cases the antecedent can be true and the consequence false. And since in this case w h a t he promised is simply false, h e cannot be u n d e r a n y obligation to keep his promise. In order to deal with this sophism one also has to provide an a n s w e r to the very difficult question concerning the status of s t a t e m e n t s about the contingent future. Buridan's solution was t h a t a s t a t e m e n t about t h e contingent future is true or false,

T E N S E S IN THE MIDDLE AGES

37

although its t r u t h value cannot be k n o w n by anybody now. T h a t solution is just one of the possible answers - as we shall see. T h e logicians of the Middle Ages in g e n e r a l took t h e Aristotelian view t h a t a statement can change its t r u t h value w i t h time. A proposition such as 'Socrates runs' is not true at all times. The truth value depends on the actual state of affairs. The idea of 'the truth of a proposition at a given time' thus comes into t h e picture. M a t t e r s got more c o m p l i c a t e d w h e n c e r t a i n logicians introduced propositions such as 'Sortes fuit currens in a' ('Socrates was running at the time a') into the discussion. A s s u m i n g t h a t S o c r a t e s a c t u a l l y r a n at t i m e a , such a proposition w a s regarded as false before and at time a, but t r u e at all times after a [Nuchelman 1980, p.133]. The t r u t h value of the proposition w a s t h u s r e g a r d e d as relative to the time at which it w a s p u t forth - its 'moment of utterance'. Several factors are i m p o r t a n t in determining t h e t r u t h value of a proposition: The 'present time' (understood as the moment of utterance), the time at which the event does or does not take place (tempus significantum), as well as the tense of the verb in the proposition (tempus consignificatum). To maintain t h a t the truth value of a proposition in any given case is to be d e t e r m i n e d relative to t h e 'present time' - t h e m o m e n t of utterance - is not so simple as it may seem. One can question the n a t u r e of that present time relative to which the proposition is to be evaluated. For we might very well consider the present as a duration rather t h a n an atomic instant. J o h n Buridan argued for this conception and noted t h a t the 'present duration' which we have in mind m a y indeed be quite extended, "for we call this y e a r present and this d a y present and this hour present " [Buridan 1966, p. 170]. Hence, Buridan argued t h a t the t r u t h value of a proposition should be understood as v a r y i n g relative to the p r e s e n t regarded as a duration whose length is conventionally determined. The m a j o r i t y of logicians in t h e Middle Ages, however, took the view t h a t the truth value should be discussed as 'truth value corresponding to a certain time'. They believed t h a t w h a t is needed in logic is 'durationless time', not determined as a part of a duration, b u t as a limit for the duration. T r u t h and time were considered as being closely

38

CHAPTER 1.3

i n t e r c o n n e c t e d . Walter B u r l e i g h (c. 1275-1345) e x a m i n e d t h e sophism "nothing is true u n l e s s at this instant" Cnihil est v e r u m nisi in hoc instanti" [Burleigh p.173] ). In his solution to t h a t sophism he concluded t h a t if a proposition is true it m u s t be t r u e now, t h a t is to say at the p r e s e n t time. Even so, m a n y medieval philosophers realised t h a t t h e idea of the t r u t h of a proposition at a d u r a t i o n l e s s time was n o t w i t h o u t problems. We shall later r e t u r n to t h e s e q u e s t i o n s w h i c h m a i n l y have to do w i t h beginning and ending (incipit and desinit). B u r i d a n ' s a n s w e r to the question about the relation between t i m e a n d t r u t h r e p r e s e n t s an i m p o r t a n t a l t e r n a t i v e to t h e majority view. In his view t h e present, as well as t h e p a s t a n d the f u t u r e , were to be c o n s i d e r e d as h a v i n g a certain span. Accordingly, one cannot give a definite answer to the question of t h e t r u t h v a l u e of a p r o p o s i t i o n w i t h o u t k n o w i n g t h e p r e s u p p o s e d convention of t h e d u r a t i o n of the present. The t r u t h of a proposition t h u s depends on the choice of the duration which is considered to be t h e present. We shall analyse some details of Buridan's position in the next chapter. In his article in The Encyclopedia of Philosophy [London 1967, vol.3, p.528], Ernest A. Moody identified t h e four most important c o m p o n e n t s of Medieval logic to be the general theories of 1) suppositio t e r m i n o r u m (theory of terms) and 2) c o n s e q u e n t i a (theory of entailment), 3) modal concepts, and finally, 4) t h e general preoccupation w i t h philosophical problems which were in t h e m a i n related to logic and language. We shall in t h e following c h a p t e r s consider some of t h e most characteristic features of t h e tense-logic of the Middle Ages. We shall see t h a t the concept of time is r e l e v a n t to all four areas m e n t i o n e d by Moody. It s h o u l d be m e n t i o n e d t h a t m a n y of t h e medieval texts still exist only in m a n u s c r i p t form. We have no g u a r a n t e e t h a t the texts w h i c h have been published are generally representative of the m e d i e v a l logicians' views on the relation between time and logic. However, it may well t u r n out that temporal logic received even g r e a t e r attention in t h e Middle Ages t h a n it appears from the Medieval texts which have been published so far.

1.4. TEMPORAL AMPLIATION T h e t e m p o r a l reference of t e r m s is one of t h e p r o b l e m dom a i n s of tense-logic. The basic n a t u r e of the problem involved s h o u l d be clear w h e n c o n s i d e r i n g sentences such as 'Young Socrates was going to argue', a n d 'The king of France was bald'. Obviously, an adequate logical analysis of these propositions requires an analysis of the temporal content of their subject terms. In t h e Middle Ages, this p r o b l e m field was commonly called 'ampliatio', a n d great e n e r g y w a s invested into its solution. Indeed, t h e work of the Medieval logicians on 'ampliatio' is perh a p s t h e clearest example of t h e great importance which t h e y attributed to the logical study of temporal aspects of propositions. One can h a r d l y t h i n k of a Scholastic a u t h o r of a major logical w o r k from t h e 14th century onwards, who would not also be concerned w i t h t h e temporal reference of terms. However, this does n o t m e a n t h a t all logicians called t h e problem d o m a i n 'ampliatio': to our knowledge O c k h a m did not use this particular t e r m at all in his analysis, p e r h a p s because he h a d his very own solution to this problem. Most Medieval logicians nevertheless did use t h e t e r m 'ampliatio' w h e n discussing how to determ i n e the t e m p o r a l reference of t h e subject. The three rules p u t forth by W a l t e r Burleigh in his De Puritate Artis Logicae are probably typical: The first rule is t h a t a c o m m o n t e r m s t a n d i n g (in a sentence) with a non-ampliating verb about the present stands only for p r e s e n t things. The second rule is t h a t a c o m m o n t e r m s t a n d i n g (in a sentence) with a verb about t h e past is able to s t a n d indifferently for present and past things. The t h i r d rule is t h a t a c o m m o n t e r m standing (in a sentence) w i t h a verb about the future is able to stand indifferently for p r e s e n t and future things. [Normore 1975 p.51] One of the crucial problems motivating the work on 'ampliatio' was t h e p r o b l e m s r e g a r d i n g t h e naive conception of t e n s e d s t a t e m e n t s . According to t h a t conception, a proposition of the type 'A will be B' is equivalent to t h e claim of the existence of a 39

40

C H A P T E R 1.4

future in which ~ is B', and similarly a proposition of the t y p e 'A has b e e n B' is r e g a r d e d as t r u e if and only if t h e proposition ~4 is B' w a s t r u e at s o m e p a s t time. But this n a i v e conception c a n n o t be u p h e l d in all cases. Consider for instance t h e s t a t e m e n t 'The little boy will become a famous man'. This proposition c a n certainly be true, e v e n though t h e statem e n t 'the little b o y is a f a m o u s man' c a n n o t be fulfilled at a n y time. T h e s o l u t i o n w a s to i n t e r p r e t t h e s t a t e m e n t as b e i n g equivalent to: 'For a given p e r s o n x, x is now a little b o y and x will b e c o m e a f a m o u s man' 'The little boy' t h u s refers to something in t h e present although the v e r b is r e f e r r i n g to t h e future. B u t e v e n this more refined t r e a t m e n t cannot e n c o m p a s s all cases, as w e can see from t h e sentence 'Antichrist will be an orator'. Crucial to this e x a m p l e is the theological observation t h a t Antichrist does not yet exist. The s t a t e m e n t could c o n s e q u e n t l y not be p a r a p h r a s e d in t h e s a m e w a y as t h e s t a t e m e n t a b o u t t h e little boy, b u t w a s u n d e r s t o o d as being equivalent to: 'For some p e r s o n x: it is true that x will be Antichrist, a n d x will be an orator'. In an a n a l o g o u s m a n n e r t h e proposition, ' S o m e t h i n g w h i t e w a s black', might be t r u e b e c a u s e t h e following s t a t e m e n t is true: 'Something w h i c h h a s b e e n white, w a s black', or it m i g h t be t r u e b e c a u s e of t h e t r u t h of, ' S o m e t h i n g w h i c h is w h i t e w a s black'. Actually, even t h i s fairly innocuous f o r m u l a t i o n is slightly biased, for it implicitly favours Ockham's solution over t h e traditional one. Burleigh's rules for ampliatio q u o t e d above are repres e n t a t i v e of the t r a d i t i o n a l solution, w h i c h t r e a t s s e n t e n c e s of the form 'A has b e e n B' as equivalent to 'For some x: x is or h a s been A, and x h a s b e e n B', and analogously for future t e n s e sen-

TEMPORAL AMPLIATION

41

t e n c e s . It is possible t h a t O c k h a m h a d Burleigh's logic at h a n d w h e n h e w r o t e his Summa Logicae, w h i c h states: F o r e x a m p l e if this proposition is t r u e 'a w h i t e t h i n g w a s Socrates', and if'white' supposits for t h a t which is white, it is n o t required t h a t this will have sometime b e e n t r u e 'a w h i t e s o m e t h i n g is Socrates', b u t it is required t h a t this will h a v e b e e n t r u e 'this is Socrates' d e m o n s t r a t i n g t h a t for w h i c h t h e s u b j e c t s t a n d s in 'a w h i t e t h i n g w a s Socrates'.' [Normore 1975 p.48] W i t h this explanation O c k h a m s h o w e d how t h e proposition 'a w h i t e t h i n g w a s Socrates' can be t r u e in t h e v e r y first m o m e n t in w h i c h Socrates w a s w h i t e for t h e first time. W e h a v e indicat i o n s t h a t O c k h a m did n o t a c c e p t B u r l e i g h ' s r u l e s s i n c e O c k h a m t h o u g h t t h a t ',4 w a s B' is either to be i n t e r p r e t e d as - in m o d e r n t e r m s - 'for some x: x is A and w a s B', or as 'for some x: x w a s A a n d w a s B', b u t not as t h e disjunction of t h e t w o possibilities. T h a t is to say, O c k h a m considered such s e n t e n c e s as inher e n t l y a m b i g u o u s . S u c h an i n t e r p r e t a t i o n of O c k h a m h a s b e e n d e f e n d e d by G r a h a m Priest and S t e p h e n R e a d [1981]. T h e difference b e t w e e n t h e two solutions is m o r e significant t h a n one might t h i n k at first glance. One of t h e m o s t p e r s u a s i v e a r g u m e n t s in favour of t h e last kind of t r e a t m e n t - as opposed to t h e t r a d i t i o n a l 'ampliatio-theory' - e m e r g e s w h e n we a n a l y s e p r o p o s i t i o n s which require t h e u s e of a rule defining t h e p a s t as w e l l a s a rule defining t h e f u t u r e . L e t u s u s e a n e x a m p l e of B u r i d a n ' s [1966, p.150]: 'Young Socrates w a s going to argue'. In t h e c a s e of this sentence, it does not s e e m acceptable to let t h e t e m p o r a l reference of t h e subject t e r m 'young Socrates' e x t e n d to t h e f u t u r e j u s t b e c a u s e a n e l e m e n t of f u t u r e occurs in t h e v e r b p h r a s e . More specifically, the 'disjunctive t r e a t m e n t ' forces u p o n u s a reading equivalent to 'for s o m e x : x w a s going to be y o u n g Socrates, a n d x w a s going to argue, or x w a s y o u n g Socrates, and x w a s going to argue'.

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CHAPTER 1.4

The first p a r t of this reading does not seem plausible, and is not necessarily forced upon us by the 'ambiguity treatment'. One of the m o s t famous analyses of 'ampliatio' comes from Albert of Saxony, who was the rector of the University of Paris in 1353. Albert defined 'ampliatio' with a p a r t i c u l a r view to terms which do not refer to actually existing entities. That is, his rules accepted t h a t statements can be made about thing which do not exist now or at any other time. He defined a n u m b e r of r a t h e r precise r u l e s for 'ampliatio' in a w a y similar to Burleigh's. The study of 'ampliatio' was made a central part of logic during the later Scholastic period. The problem was studied as late as in the 17th c e n t u r y by the Portuguese logician J o h n of St. Thomas (1589-1644), who was defending and still w o r k i n g within the tradition of Scholastic logic. He wrote two short passages directly concerning 'ampliatio', as well as a third passage, in which t h a t p h e n o m e n o n forms part of the problem. In t h e first of these passages he defined 'ampliatio', and in the second passage he p r e s e n t e d four rules related to it. Two of these rules are similar to t h e above with respect to t e m p o r a l and modal propositions, while the others are formulated in the following way: A term signifying a beginning amplifies all t e r m s before and after to w h a t is or w h a t will be; a t e r m signifying cessation, to what is or was... [John of St. Thomas, p.73] The term 'imaginatively' and the verb 'imagine' amplify all a n t e c e d e n t and s u b s e q u e n t t e r m s to the imaginable... Similarly, signifying an interior act of the soul, as I wish, I understand, etc., can amplify to the imaginable the term on which it hits as its object. [John of St. Thomas, p.74] The second of these two rules can naturally be looked upon as an extension of 'ampliatio'- n a m e l y a rule for modal propositions. The first rule demonstrates yet another connection between time a n d 'ampliatio'. Here the study of ampliatio is related to the extensive Scholastic debate on the logic of 'incipit' and 'desinit', to which we shall return.

1.5. T H E D U R A T I O N OF THE P R E S E N T The investigations in the last chapter clearly d e m o n s t r a t e d t h a t the naive conception of t e n s e d s t a t e m e n t s should be rejected as a general solution. On the other hand, it is also clear t h a t Buridan defined tensed s t a t e m e n t s in a recursive way, reducing them to s t a t e m e n t s in the present tense. It t u r n s out, however, t h a t even t h e s t a t e m e n t s in t h e present are r a t h e r complicated. The r e a s o n for this is t h a t Buridan regarded the present as a d u r a t i o n and not as a point in time. He explained the problems regarding the present in the following way: Also I say t h a t it is not determined for us how much is the present time w h i c h we ought to use as the present. But we are allowed to use as much as we wish, for we call this year present and this day present and this hour present. [Buridan 1966 p. 170] Obviously, B u r i d a n ' s notion of the p r e s e n t was t h a t of a duration. There w a s clearly an element of convention involved in this notion, since we are allowed to use as much of the present time as we wish as the present. Buridan's concept of t r u t h is relative to a choice of the present. That is, it only m a k e s sense to talk about t h e t r u t h of a contingent proposition if t h e present is specified. Buridan introduced his idea of the t r u t h of a proposition in the following way: Thus, if in one part of the present time, Socrates stands or is white or is dead, it is simply true to say t h a t he stands or is white or is dead. [Buridan 1966 p. 173] According to Buridan's defmition a proposition p is true during the present if and only if there exists a part of the present time during which the t r u t h ofp is given. The scope of t h a t definition should p e r h a p s be r e s t r i c t e d by observing t h a t B u r i d a n ' s examples in this context are all concerned with what we would call 'stative propositions'. We may illustrate one of his examples by the following figure. 43

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CHAPTER 1.5

The present (Now) I

I

I

I

Time

Socrates is alive Figure 1 The t r u t h of the proposition 'Socrates is alive' is given with respect to the interval I, which is a subinterval of the present assumed to be specified in the context. Thus, according to Buridan's ideas the proposition is to be regarded as true with respect to the entire present. Or in other words, if we have in mind the situation above as well as the present specified above, the presenttense proposition 'Socrates is alive' should be evaluated as true. This also m e a n s t h a t there is a distinction between t h e general notion of being true with respect to an interval, and the notion of being given for as certain interval. The latter is the stronger notion of the two, and it reflects the intuition that the proposition in question is true throughout the interval.

A MODERN REPRESENTATION OF BURIDAN'S IDEAS In order to give a symbolic representation of Buridan's ideas, we shall use the following conventions: - variables p, q, ... stand for atomic propositions; variables/, I',... denote intervals (durations); In is understood to denote the present (as specified by some choice); - included(I,I') means t h a t the interval I is included in the interval I'; the formula T(I,p) m e a n s 'p is true with respect to the interval I'; - t h e formula given(I,p) m e a n s 'the t r u t h o f p is given for the interval I'. -

-

-

T H E DURATION OF THE P R E S E N T

45

T h e relation 'included' is the u s u a l inclusion relation a m o n g intervals, a n d it is t h u s both reflexive (I1) and transitive (I2): (I1) (I2)

V I: included(I,I) V LI',I": (included(I,I') /, included(I',I")) included(I,I")

The intuitions for 'given' are not stated explicitly in B u r i d a n ' s text, b u t it seems reasonable to assume that the following theses m u s t hold: (I3) (I4)

(I5) (I6)

given(I,A) ~ V I'. (included(I',D D given(I',A)) V I. (given(I,A) ~ -given(L-A)) -given(In, A) ~ (3 L included(I,I,) A given(I,-A)) (given(In,A) /x given(In, A ~ B)) ~ given(In, B)

We observed above t h a t given(I,A) is a strong notion of t r u t h , i m p l y i n g t h a t t h e state of affairs d e n o t e d by A o b t a i n s t h r o u g h o u t the interval I. This intuition is what is formalised by (I3)-(I5). Specifically, (I3) reflects the intuition that i f A is given for some interval, t h e n it is also given for any of its subintervals. (I4) c a p t u r e s t h e intuition that A and -A cannot be given for t h e same interval. (I5) ensures t h a t if given does not obtain for some predicate w i t h respect to an interval, t h e n the negated predicate is given for at least one of its subintervals. Finally, (I6) s t a t e s t h a t t h e given-relation is closed u n d e r a m o d u s - p o n e n s - l i k e operation. We suggest t h e following symbolic representation of B u r i d a n ' s definition of t r u t h with respect to an interval (understood to be 'the present'): (B1)

T(In, A) --def ~ I: included(I, In) A given(I,A)

A consequence of (B1) is the following one: (B2)

T(In,A) - ~I: included(I, In) A T(I,A)

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CHAPTER 1.5

(B1-2) can of course be f u r t h e r generalised to cover all wellformed propositions, but we do not currently wish to state the rules for expressions like T(In,p A q). Now let us consider a situation which gives rise to some intuitive problems. Suppose t h a t it is given that in one p a r t of the present time, Socrates is alive, and in another part of the present time, he is dead: The present (Now) I

I

I

Socrates is alive

I

I

I

Time

Socrates is dead Figure 2

It follows from the definition (B1) t h a t Buridan is obliged to accept the t r u t h of t h e conjunction 'Socrates is alive and he is dead'. This s e e m s to be a violation of t h e principle of contradiction, and so it would be if the following formula were valid:

(B3)

T(In,p /, q) = T(In,p) A T(In, q)

Consequently, Buridan had to reject the principle embodied by (B3). But how could he establish a consistent framework such t h a t (B3) would be invalid? In order to solve this problem Buridan h a d to m a k e a distinction between, 'Socrates is alive and he is dead' and 'Socrates is alive and dead' . The latter can never be true, w h e r e a s the former can in fact in some cases be accepted as true. In order to analyse this problem in further detail it was very important for Buridan to distinguish between affirmative and negative propositions. Affirmative propositions are s t a t e m e n t s of the f o r m ' S is P', which are not negated. Conjunctions like 'Socrates is alive and he is not alive' i.e.

T(In,p) A -T(In, p)

THE DURATION OF THE P R E S E N T

47

can never be true. B u t if this is so, how can t h e r e be a duration for w h i c h the conjunction 'Socrates is alive a n d he is dead' is t r u e ? B u r i d a n solves t h e problem by p o i n t i n g out t h a t while 'Socrates is dead' is an affirmative proposition, 'Socrates is not alive' is a negative proposition. According to B u r i d a n , t h e conjunction T(In,p) /, T ( I n , - p )

can in fact be t r u e in some cases, n a m e l y in s u c h situations w h e r e p is true in some p a r t of the present d u r a t i o n In, a n d -p is t r u e in some other p a r t of In (see figure 2). Obviously t h e r e are two kinds of negation involved in the temporal logic of Buridan: (i) negation of predicates, e.g. 'non-alive' (='dead') is t h e negation of the predicate 'alive' ; (ii) negation of propositions, e.g. 'Socrates is not alive' is the negation of the proposition 'Socrates is alive'. We'll m a k e use of t h e notation (1) (2) (3) (4)

T(In,A) for 'Socrates is alive', T(In,-A) for 'Socrates is dead', -T(In,A) for 'Socrates is not alive', ~T(In,-.4) for 'Socrates is not dead'.

So t h e negation involved in (2) is a predicate negation, whereas (3) is t h e usual sentential negation. In (4) we see both kinds of n e g a t i o n occurring. Before p r o c e e d i n g it should be noted t h a t in our logical l a n g u a g e , A and -.4 are well-formed propositions in their own right. W h e n they are not preceded by t h e T-operator, t h e y are u n d e r s t o o d to refer to t h e present. U s i n g (B2) we find t h e following t r u t h conditions: (B4) (B5) (B6)

T ( I n , - A ) --- 3 I: included(I, In) .~, T(I,-.4) -T(In, A ) - V I: included(I, In) ~ ~ T ( I , A ) - T ( I n , ~ A ) - V I: included(I, In) ~ - T ( I , - A )

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CHAPTER 1.5

Intuitively, it is obvious that an analogue of the traditional rule for s e n t e n t i a l double negation should also hold for predicate negation. Using (I1) - (I6) it can in fact be verified that (B7)

-T(In,A r, B) - -T(In,A) A -T(In, B)

Or equivalently, (B8)

T(In,A v B) - T ( I ~ A ) v T ( I ~ B )

From (I5) and (B4) we can now deduce t h a t (B9) (B10)

- given(In,A) ~ T ( I n , - A ) ~ given(In,~A) ~ T(In,A)

By contraposition we fmd (Bll) (B12)

- T ( I n , - A ) ~ given(In, A ) ~T(In,A) ~ given(In, - A )

It now follows from (B1) and (I1)-(I5) t h a t (B13) (B14)

- T ( I n , - A ) ~ T(In, A) ~T(In,A) ~ T(In,-A)

But obviously the opposite implications do not hold, for as we have seen it m a y very well be true t h a t T(In,-A) A T(In,A)

is the case, as in figure 2 - in which case neither of -T(In,A) and -T(In, ~A) are true! And in general, the consequences of (B12) and (B13) m a y well be true, without the respective antecedents being true. One consequence of all these observations is of course t h a t the n a t u r a l language inference

THE DURATION OF THE PRESENT

49

'if Socrates is alive, then he is not dead' is invalid on Buridan's account of the d u r a t i o n of the present. This last result seems to us to be in conflict w i t h the logic of n a t u r a l language. If t h a t is so, t h e n B u r i d a n ' s ingenious investigations raise some problems of which a n y a t t e m p t e d interval semantics for n a t u r a l language should take heed. It may seem by now t h a t this logic is suspiciously complicated. But we do not t h i n k t h a t this observation by itself m a k e s Buridan's ideas dubitable; interval semantics is in general a more complicated business than instant semantics. If we wish to study t r u t h relative to durations, we must be prepared to accept a complicated framework.

TWO KINDS OF TENSES As we h a v e seen B u r i d a n took it for g r a n t e d t h a t tensedistinctions are important to logical reflection. But he was also aware of the fact t h a t a logic of tenses which pays due regard to a logic of d u r a t i o n s is v e r y complicated. For this reason, probably, he was content to sketch his ideas of tense logic. Buridan suggested two alternative ideas for the construction of the logic of tenses. The first one leads to the fairly natural kind of semantics, which we have discussed above. The tenses, past and future, are t a k e n absolutely, in the sense t h a t no part of t h e present time is said to be past or future. Buridan made no attempt at formulating a detailed semantics for the tense operators, but he maintained t h a t if the tenses are t a k e n in a n absolute sense, the Aristotelian proposition 'All which is moved was moved previously' cannot be valid [Buridan 1966, p. 177]. Generally speaking, the implication

moving(X) ~ P(moving(X)) is not a valid thesis in Buridan's temporal logic. In the same way he would also reject the validity of the implication

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CHAPTER 1.5

moving(X) ~ F(moving(X)) On the other hand, these two propositions become valid if the tenses are taken in the relative sense, which Buridan explained in the following way: But in another way, 'past' and 'future' are t a k e n relatively, so that the earlier part of the present time is called past with respect to the later, and the later part is called future with respect to the earlier. This w a y of t a k i n g the t e r m s is customary. [Buridan 1966, p. 175] B u r i d a n pointed out that if some t h i n g is moving now, t h e n t h e r e is a part of the present d u r i n g which it is moving, and hence, it is moving in some part of the present which is earlier t h a n some other part of the present. Therefore, if the t h i n g is moving, t h e n it was moving (if the past is taken in the relative sense). For this reason, t h e A r i s t o t e l i a n sophism m u s t be conceded if the past is understood relatively. It seems plausible to represent Buridan's idea of a relative past in the following way:

T(In, PrelA) --clef 3I': included(I',In) A HI". before(I",I') A given(I",A) whereas the absolute past can be defined as

T(In, PabsA) ~ def 3I': before(I',In) A given(I',A) Let us assume t h a t it is true for some 'now' In that some thing X is moving. According to the definition (B1) this m e a n s t h a t there is an interval I' for which it holds that

included(I', In) A given(I', moving(X)) If I1 is included in I ' and 12 is included in In s u c h t h a t before(Ii, I2), then by (I2) and (I3) we have

THE DURATION OF THE PRESENT

51

included(It,In) A before(Ii,I2) /~ given(Ii, moving(X)) (Of course, our scenario here does not preclude t h a t

given(I2, mo ving(X)) m a y also hold - indeed, it might be t r u e t h a t X is moving throughout the present - but on the other hand, this is clearly not a necessary condition, i.e. it is not entailed.) These observations fit nicely into the above definition of the relative past; it follows directly from our assumptions that

312: included(I2,In) A 3II:(before(Ii,I2) /, given(I1, moving(X)) which is t h e definition of T(In, Prel(moving(X))). Therefore, it follows t h a t

T(In, moving(X)) ~ Prel(moving(X)) Obviously, an analogous thesis cannot be proved for Pabs instead of Prel. It t u r n s out t h a t the two intuitively different k i n d s of tenses are also very different from a formal point of view.

1 . 6 . T H E L O G I C O F B E G I N N I N G AND E N D I N G A very special chapter of Medieval logic was opened when philosophers of t h a t time took up the analysis of the verbs 'incipit' (it begins) and 'desinit' (it ends). The starting point was found in Aristotle's Physics, books 6 and 8, so it was no coincidence t h a t their deliberations proved to be relevant not only to logic but also to physics. The questions concerning beginning and ending n a t u r a l l y led to the consideration of temporal limits. The n u m b e r of Medieval logicians who worked on these questions was very large [Kretzmann, 1976, pp.101ffi. As pointed out by William and Martha Kneale [1962, p.233-34], the very fact t h a t so m u c h attention was given to this type of problem constitutes an excellent proof of the formal character of Medieval logic. The general problem had to do with the correct understanding of 'incipit-statements' such as: (1) 'Socrates begins to be white', (2) 'Socrates begins to run', and analogously for statements containing the verb 'desinit'. The task of the logician was to give clear semantic definitions of 'incipit' ('begins') a n d 'desinit' ('ends'). The most common definition given in order to clarify the m e a n i n g of the above examples was the following: (1') 'Socrates is w h i t e and was not white immediately before' (2') 'Socrates does not run, but will run immediately after' This interpretation was for example defended by Peter of Spain (d. 1277). Obviously, the treatment offered by (1') and (2') does not fit into the same pattern, or paradigm; 'whiteness' and 'running' are treated differently. This difference - inspired by Aristotle's t r e a t m e n t in the Physics - originates in a distinction b e t w e e n p e r m a n e n t things or states (whose parts appear simultaneously), and successive things or states (whose parts appear one after another). Medieval logicians considered the 52

THE LOGIC OF B E G I N N I N G AND E N D I N G

53

p r o p e r t y of 'running' to be successive a n d 'whiteness' to be p e r m a n e n t . Hence the two kinds of predicates h a d to be t r e a t e d differently. In addition to t h e two types of p h e n o m e n a , t h e p e r m a n e n t a n d t h e successive, a t h i r d object type, t h e i n s t a n t a n e o u s , was observed and discussed by Burleigh a n d T h o m a s Bradwardine (c.1295-1349) [Nielsen, 1982, p.29]. T h e discussion of b e g i n n i n g and e n d i n g is in our opinion a s t r i k i n g example of t h e m a n n e r in w h i c h m e d i e v a l logic is r e l e v a n t even today for s e m a n t i c discussions. W h e n u s i n g symbolic language for t h e discussion, we shall use t h e following convention: p is a variable ranging over p e r m a n e n t state propositions, s is a variable ranging over successive state propositions, and q is a meta-variable r a n g i n g over both types of propositions. T h e questions concerning 'incipit'/'desinit' were a m o n g s t the m o s t discussed problems in t h e Middle Ages, as Simo K n u u t t i l a [1985, p.165fI] has pointed out. The analysis of s t a t e m e n t s was central to t h e Medieval approach to scientific questions in general, a n d this particular p r o b l e m was regarded as i m p o r t a n t in scientific and physical t h i n k i n g as early as the 12th century (at w h i c h time 'the present' w a s regarded as a primitive concept). T h u s , Scholastic n a t u r a l philosophers who were i n t e r e s t e d in k i n e m a t i c s t u r n e d t h e i r a t t e n t i o n to propositions r e g a r d i n g b e g i n n i n g and ending [Murdoch p.ll7ffi. In t h e medieval treatises on the question t h e r e is evidently a connection between the tense-logical analysis of t h e problems of 'incipit' a n d 'desinit', a n d t h e e m e r g i n g a w a r e n e s s of t h e p r o b l e m of c o n t i n u i t y in connection w i t h e s t a b l i s h i n g t h e ' m a t h e m a t i c a l m o m e n t ' . T h i s is clearly t h e case w i t h for i n s t a n c e Richard of L a v e n h a m ' s analysis in his t r e a t i s e s De Natura Instantium and De Primo Instanti [OhrstrCm 1985b]. T h e s t u d y of 'incipit' a n d 'desinit' is an e x t r e m e l y difficult m a t t e r . Two c o m p l i c a t i n g factors o u g h t to be m e n t i o n e d . Firstly, a special challenge is constituted by s t a t e m e n t s m a k i n g i t e r a t i v e use of 'incipit' a n d 'desinit'. Secondly, t h e use of ' i m m e d i a t e l y before' a n d ' i m m e d i a t e l y after' calls for very

54

CHAPTER 1.6

specific tense-logical constructions, since it is not obvious how one is to p r e c i s e l y u n d e r s t a n d t h e s e expressions. It is t h e question of t h e continuity of time which is at stake here. Richard Kilvington (died 1361) amongst others analysed these problems thoroughly in his Sophismata [Kretzmann 1982, p.270ffi. Two characteristic features of Medieval logic was t h a t it dealt w i t h propositions whose t r u t h - v a l u e s could vary from t i m e to time, and t h a t it took tensed s t a t e m e n t s into serious consideration. On t h a t basis medieval logicians p u t forth some very interesting ideas of temporal logic, also w i t h respect to t h e problems of 'incipit'Pdesinit'. In the following we will concentrate on some findings of William of Sherwood, w h i c h he f o r m u l a t e d in his Syncategoremata [Kretzmann 1968]. According to Sherwood t h e t e r m s 'incipit' ('begins') a n d 'desinit' ('ceases') can be used c a t e g o r e m a t i c a l l y as well as s y n c a t e g o r e m a t i c a l l y . This d i s t i n c t i o n was well k n o w n in Medieval logic, as described by [Kretzmarm et al.]: Medieval logicians regularly classified m e a n i n g f u l w o r d s into s u c h as have m e a n i n g in t h e i r own right (termini significativi ...), and such as are m e a n i n g f u l only w h e n joined to words of the first kind (termini consignificativi...). The f o r m e r are also called categorematic terms..., the latter, syncategorematic terms... [p. 162] Typical s y n c a t e g o r e m a t i c words are q u a n t i f i e r s s u c h as 'every', 'all', 'some', etc. These r e m a r k s can be s u p p l e m e n t e d by the observation t h a t in a syncategorematic use of an expression, the expression is considered to be 'incomplete'. For instance, the verb 'to begin' m a y be combined w i t h an infinitive c o m p l e m e n t as in 'begin to run' in order to form a complete predicate (i.e. an intransitive verb phrase). In this use of 'begin', it is considered incomplete u n t i l adjoined with its complement. A categorematic expression, on t h e other hand, is complete by itself- in t h e w a y a verb such as 'walk' m a y m a k e up a full verb p h r a s e by itself. Here, we will c o n c e n t r a t e on t h e sync a t e g o r e m a t i c use of t h e verbs. In this syntactic role t h e y indicate h o w t h i n g s are qualified a n d how t h e y are to be

THE LOGIC OF BEGINNING AND ENDING

55

interrelated, but t h e y c~nnot by themselves be used as the predicate of a sentence. In medieval logic some of the syncategorematic terms w e r e classed as exponible terms, i.e. t e r m s having an obscure sense which has to be explained or clarified. Sherwood stated t h a t 'incipit' and 'desinit' are such exponible terms. He offered the following explication of 'desinit': Therefore, if I say 'he ceases to be sick, or unhealthy', t h e n 'to cease' indicates t h a t the thing is at the end of the time in which it was such and such, (in termino temporis in quo fuit talis). [Kretzmann 1968 p.109]

A MODERN REPRESENTATIONOF THE IDEAS Let the statement variable p stand for an arbitrary proposition, e.g. 'Socrates is alive', and let Cp r e p r e s e n t the proposition stating that p ceases to be true, i.e. 'Socrates ceases to be alive'. Sherwood's description of the time at which 'he ceases to be sick' as 'terminus temporis in quo fuit talis', i.e. 'the end of the time d u r i n g which t h e person was sick', gives rise to the following explication: (1) The proposition Cp is true at the time t only i f t is a limit between times at which p is t r u e and times at which p is false.

IfBp represents the proposition stating that p begins to be true, the following condition seems to be n a t u r a l in addition to (1): (2) The proposition Bp is true at the time t only if t is a limit between t i m e s at which p is false and times at which p is true.

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CHAPTER 1.6

It m u s t be noted that (1) and (2) are not complete definitions, since we have only stated some necessary conditions for Cp and Bp to be true at t. If time is continuous, and the distribution of the t r u t h values of propositions corresponds to t e m p o r a l intervals, the limits mentioned in (1) and (2) are well-defined. Then the definitions are in fact complete - so in that case, we m a y substit u t e 'only if by 'if and only if. If on the other h a n d time is discrete, the conditions in (1) and (2) are not sufficient. For let the truth-values (T: true, F: false) forp be as follows:

time 1 truth-value F

2 T

3 T

4 T

5 T

6 F

7 F

... ...

w h e r e t h e integers are used to indicate t h e succession of discrete i n s t a n t s (or discrete periods). Now t h e r e is a basic intuition according to which it is reasonable to say t h a t Cp is true at t = 6 and that Bp is true at t = 2. But on the other hand, it also might be intuitively plausible to say that Cp is true at t = 5 and t h a t Bp is true at t = 1. In fact, when operating on the basis of d i s c r e t e time, (1) a n d (2) give rise to four possible combinations. It must be concluded that t h e y are not yet full definitions. In fact, what the precise 'limits' of (1) and (2) should be d e t e r m i n e d to be also depends upon w h e t h e r we are talking about successive or p e r m a n e n t states. So far, we have been using only the example of the p e r m a n e n t state p, but actually (1) and (2) in their general formulation are m e a n t to apply to successive states also. A fully precise version of (1) and (2) will have to be differentiated according to the type of propositions in question. Sherwood realised all of this and did indeed arrive at a clear definition based on his concept of time. In the notes to his translation of William of Sherwood's Syncategoremata Norman K r e t z m a n n stated: "Sherwood's analysis is evidently based on a view of time as a sequence of discrete ins t a n t s or periods..." [Kretzmann 1968, p.109]. We agree with K r e t z m a n n ' s observation. Some of the phrases and expressions used by Sherwood obviously presuppose that it is possible to iden-

THE LOGIC OF BEGINNING AND ENDING

57

tify an i m m e d i a t e successor and an immediate predecessor for every instant in time. His use, for instance, of the expression 'in penultimo i n s t a n t i vitae suae' ('in the next to the last instant of his life') clearly indicates t h a t according to Sherwood there is a n instant immediately preceding 'the last instant of his life'.

PERMANENTIA AND SUCCESSIVA Since Sherwood used a discrete p a r a m e t e r of time, further explanations of 'incipit' and 'desinit' were required. In his expositions he was r e l y i n g on the distinction between ' p e r m a n e n t states' (permanentia) and 'successive states' (successiva), w h i c h we discussed above. The parts of a p e r m a n e n t state are at one and the s a m e time, whereas the parts of a successive state are not at one a n d t h e same time. The property of 'being white' r e p r e s e n t s a p e r m a n e n t state, a n d 'running' r e p r e s e n t s a successive state. As we also mentioned the distinction goes back to Aristotle. T h e philosophical s t a r t i n g point for Sherwood's discussion is to be sought in Aristotle's Physics book VI and book VIII. Aristotle stated that For a change can actually be completed, and there is such a thing as its end, because it is a limit. But with reference to the beginning the phrase has no meaning, for there is no beginning of a process of change, and no p r i m a r y 'when' in which t h e change was first in progress. [Phys. 236a 12-14] Obviously 'a process of change' is a successive state. Hence, according to Aristotle there is an end to, but no beginning of a successive state. The latter observation m a y seem counterintuitive, but what is m e a n t is simply that a thing is not in the state w h e n the thing begins to be in the state. Aristotle's view also m e a n s t h a t a thing is in the state when the thing ceases to be in t h e state. In symbols: (3.1) (3.2)

(Bs ~ ~s) (Cs ~ s)

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C H A P T E R 1.6

w h e r e s is an a r b i t r a r y proposition a b o u t a s u c c e s s i v e state. S h e r w o o d m a i n t a i n e d t h a t (3) is valid for such propositions. L e t t h e t r u t h - v a l u e s for s be as follows:

time truth-value

1 F

2 T

3 T

4 T

5 T

6 F

7 F

o~

ooo

W e discussed above t h e same series of t r u t h v a l u e s for t h e perm a n e n t s t a t e proposition p. It was a n d observed t h a t from an int u i t i v e p o i n t of view, it could be a r g u e d t h a t B p w a s t r u e at either t = i or t = 2, a n d analogously for Cp at t = 5 a n d t = 6. At a p r e - t h e o r e t i c a l level, a similar a r g u m e n t m i g h t be m a d e a b o u t B s a n d Cs. H o w e v e r , if w e combine t h e i n s i g h t of (3) w i t h (1) a n d (2), w e m u s t arrive at the conclusion t h a t Bs is t r u e if and only if t = 1 and that Cs is true if and only if t = 5. On t h e o t h e r h a n d , if p is a proposition c o r r e s p o n d i n g to a p e r m a n e n t state w i t h t h e above v a r i a t i o n of t r u t h - v a l u e s , Sherwood's v i e w can be f o r m u l a t e d in t h e following way:

(4.1)

(Bp ~p)

(4.2)

(Cp ~ ~p)

T h a t condition is also in good accordance with t h e observations b y Aristotle in his Physics, book VIII: It is also evident that, when speaking of the subject of motion or change, unless w e assign the instant t h a t divides p a s t and f u t u r e time to t h e s t a t e into w h i c h it will be for t h e f u t u r e r a t h e r t h a n to t h a t w h i c h it t u r n s out of and in w h i c h it w a s in t h e past, we shall have to s a y t h a t t h e s a m e t h i n g b o t h exists and does not exist at the s a m e instant, a n d w h e n it has b e c o m e s o m e t h i n g it is not t h a t s o m e t h i n g w h i c h it h a s become. (263 b 10-14). Aristotle examined t h e proposition: 'The object D is white'. H e ass u m e d t h a t D is w h i t e before the i n s t a n t t, and n o t w h i t e after t. Obviously, t is a limit: D ceases to be white at t a n d begins to be n o t - w h i t e at t. B u t since the change from w h i t e to n o t - w h i t e h a s

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59

been completed at t, D must be not-white at t. Hence, according to Aristotle (4) is obviously valid for the proposition p: 'The object D is not white'. We can now generalise the results of the discussion into the following equivalences: (5.1) (5.2) (6.1) (6.2)

Bs Cs Bp Cp

- ( - s A F(1)s) ~- (s A F(1)-s) - ( p A P(1)~p) - ( ~ p A P(1)p)

P(n) is r e a d 'n time units ago it was the case t h a t ...' and F(n) is read 'in n time units it will be the case that...'. In t h e following we intend to investigate Sherwood's solutions to some sophisms concerning 'incipit' and 'desinit'.

(A) W H A T

BEGINS TO BE CEASES N O T TO BE

Using the s a m e kind of symbolic language as above, w e m a y render this sophism as:

(7)

Bq ~ C - q

w h e r e q r a n g e s over a r b i t r a r y propositions. Sherwood, of course, held this statement to be true. But since it is a sophism, t h e r e also exists an a r g u m e n t implying t h a t t h e s t a t e m e n t is false, n a m e l y this one: 'But w h a t begins to be is, and what ceases not to be is not; therefore what is is not'. [Kretzmann 1968 p. 113]. In symbols: (Bq ~ q) and (C-q ~ -q) But this combination of statements is neither in a g r e e m e n t with (5.1-2) nor with (6.1-2), i.e. q can neither be a p e r m a n e n t state nor a successive state. On the other hand, if we accept (5)

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and (6), it t u r n s out that (7) is a valid thesis, and hence afortiori t h e sophism is true. Accordingly, the premise of the a r g u m e n t above should be rejected (and so should its conclusion). Using (5) a n d (6) it can be verified t h a t (7) holds in general. We shall show (7) for permanent propositions: (i) (ii) (iii) (iv)

Bp p A P(1)-p - - p A P(1)-p C-p

(assumption) (by 5.1) (double negation) (by 5.1 used on ~p)

Here it is assumed that ifp is a permanent state, then -p is also a p e r m a n e n t state. - The proof of (Bs ~ C-s) is analogous, and c o m b i n i n g t h e two we of course h a v e (7). In fact, t h e implications in (7) can be substituted by equivalences, as should be obvious from the proof above.

(B) CEASING TO BE NOT CEASING Sherwood formulates another sophism in the following way: 'Suppose t h a t Socrates is in the next to the last instant of his life. T h e n S o c r a t e s ceases to be not ceasing to be'. [Kretzmann 1968 p.l14] In the following, let p represent the proposition 'Socrates is alive'. - According to Sherwood the sophism is not valid, if'ceases to be' and 'not ceasing to be' are distributed as in 'Socrates ceases to be and Socrates doesn't cease to be'. This is obvious, for if the sophism is r e a d in this way, its conclusion will be Cp A -Cp, which clearly cannot be accepted. But if 'ceases to be' and 'not ceasing to be' are iterated as in C~Cp, the sophism should be regarded as true, indeed as a valid thesis. Let the truth-values forp be as follows: time truth-value

... 3 4 5 6 7... ... ? ? T F F ...

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61

Before the validity of the sophism can be demonstrated, it m u s t be clarified how 'in the next to the last instant of his life' ('in penultimo vitae suae') should be understood. Is it the instant immediately before the d e a t h of Socrates, i.e. t = 5, or is it the instant immediately preceding that instant, i.e. t = 4? It turns out t h a t in the latter case, the sophism will not be valid. For this reason we shall assume t h a t the instant in question is the instant immediately preceding the death of Socrates, t = 5. On the assumptions made so far the sophism can be symbolised in the following way: (8)

F(1)Cp ~ C~Cp

Note t h a t the antecedent is true exactly at t = 5. For the proposition p corresponds to a p e r m a n e n t state, and consequently Cp has to be understood according to the exposition in (6). But it does not n e c e s s a r i l y follow t h a t its n e g a t i o n ~Cp is also a p e r m a n e n t state. To u n d e r s t a n d (8) fully we m u s t k n o w w h e t h e r the proposition -Cp corresponds to a p e r m a n e n t state or to a successive state. It can be shown t h a t (8) is not valid if the exposition in (6.2) is used for both occurrences of C in C~Cp. Hence, it seems to be natural to assume t h a t -Cp corresponds to a successive state. In general, n e g a t i n g a predicate of one aspectual type m a y t u r n the predicate t h u s formed into another type (a fact which is also realised within modern linguistics). On this observation, we can use the definition given in (5.2), and this makes (8) equivalent to (9)

F(1)Cp ~ (-Cp A F(1)Cp)

which is valid if and only if the following is a valid thesis:

(lo)

F(1)Cp ~ ~Cp

Since p is a p e r m a n e n t state proposition, we m u s t now use (6.2), which makes (10) equivalent to (11)

F(1)(-p ~ P(1)p) D (p vP(1)-p)

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C H A P T E R 1.6

L e t u s t a k e a semi-formal look at the antecedent of (11): on our a s s u m p t i o n s , F(1)~p is clearly t r u e (~p is true at t = 6), so in this case t h e a n t e c e d e n t of (11) is t r u e if a n d only ifF(1)P(1)p is true. As for t h e consequent, t h e t r u t h v a l u e of P(1)-p c a n n o t be det e r m i n e d by t h e a s s u m p t i o n s l e g i t i m a t e in connection w i t h t h e s o p h i s m in question. Nevertheless, if we can show t h a t p m u s t be true, t h e entire c o n s e q u e n t is of course true, and w e can neglect P(1)-p. In short, if (12)

F(1)P(1)p ~ p

is a v a l i d t h e s i s , t h e n (11) is also a valid t h e s i s .

Since

F(1)P(1)p ~ p is intuitively valid, it follows t h a t t h e v a l i d i t y of (8) is a c o n s e q u e n c e of n a t u r a l a n d obvious reasoning. W e m a y a d d t h a t in m o d e r n a x i o m a t i s a t i o n s of metrical t e n s e logic, (12) is also valid, t h a t is, a theorem. In the context of t h e discussion of this sophism, S h e r w o o d m a k e s an i n t e r e s t i n g o b s e r v a t i o n . H e considers the proposition (13)

Cp ~ C~Cp ( w h e r e p is as above).

This proposition is a valid thesis if time is dense. To see this, ass u m e t h a t Cp is t r u e at t h e t i m e t and t h a t time is dense. According to (1) it follows t h a t t is a limit b e t w e e n times at w h i c h p is t r u e a n d times at which p is false. Then t is also a limit b e t w e e n t i m e s at which ~Cp is true, a n d times - in this case only one time t - at w h i c h -Cp is false. F o r this r e a s o n C-Cp is t r u e at t. Hence, (13) is a valid thesis. Obviously (13) is not v a l i d if time is discrete, w h e r e a s (8) is n o t v a l i d if time is dense. T h e s e results a r e r a t h e r r e m a r k a b l e . S h e r w o o d did in fact d e m o n s t r a t e t h a t t h e difference b e t w e e n a c c e p t i n g (8) as valid, or a l t e r n a t i v e l y , a c c e p t i n g (13) as valid, c o r r e s p o n d s to t h e distinction b e t w e e n discrete and d e n s e time. F o r a m o d e r n logician, (8) a n d (13) will be n a t u r a l c a n d i d a t e s for a n a x i o m a t i c d e s c r i p t i o n of this distinction.

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63

(C) CEASING TO BE BEFORE SOMETHING Sherwood presents another sophism p e r t a i n i n g to some of t h e same ideas in the following way: 'Suppose that Socrates is in the next to the last instant before Plato's death. T h e n Socrates ceases to be before Plato's death'. [Kretzmann 1968 p.l14] Let p represent the proposition 'Plato is alive'. As with the sophism in the above section we will assume the instant in question to be t h e i n s t a n t immediately preceding Plato's death. Hence t h e a n t e c e d e n t in the sophism is t r u e if and only if F(1)Cp is true, w h e r e a s t h e consequent is t r u e if and only if CF(1)p is true. Therefore the whole sophism ca_u be represented as follows (14)

F(1)Cp~ CF(1)p

According to Sherwood this sophism is valid, which is easily verflied if F(1)p is considered to correspond to a p e r m a n e n t state and t i m e is discrete. SOME FUTHER REMARKS The above results should serve to d e m o n s t r a t e t h a t Sherwood was c o m m i t t e d to the view t h a t t h e concept of t i m e is n e e d e d w i t h i n logical analysis. He considered the logical operators corresponding to 'beginning' and 'ceasing' to be interesting w i t h i n a t e m p o r a l logic, and formulated the semantics of these operators. He did so by giving some basic theses for each of t h e m and arguing for t h e validity of those theses. By this work William of S h e r w o o d provided a valuable c o n t r i b u t i o n to t h e m e d i e v a l s t u d y of temporal logic. It is r e m a r k a b l e t h a t he was a w a r e of the possibility of distinguishing between discrete and dense time by m e a n s of theses from temporal logic. Another debate, which is to some extent related to the problems concerning 'incipit' and 'desinit', is the debate of the concepts of

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CHAPTER 1.6

' m o m e n t of change' a n d ' m o m e n t of creation', issues which bear on the ideas of'creatio ex nihilo' and related questions. The analysis of 'creation', 'beginning', 'end', and similar concepts also p l a y e d an i m p o r t a n t role after t h e Middle Ages. Moses M e n d e l s s o h n (1729-1786) was convinced t h a t an analysis of t i m e a n d logical relations in connection w i t h change would cons t i t u t e an a r g u m e n t for the immortality of the soul, an argum e n t which K a n t a n d Brentano resumed [Chisholm 1980]. It is also worth pointing out the distinction between successiva a n d p e r m a n e n t i a clearly anticipate those syntactic a n d t e n s e logical distinctions a m o n g types of verb phrases discussed in m o d e r n linguistics. In fact it m a y be easier to u n d e r s t a n d t h e m w h e n c o m p a r i n g t h e m w i t h p r e s e n t - d a y terminology. Zeno Vendler [1967] divided verb phrases into four major types: (a) states, corresponding to p e r m a n e n t properties - such as 'is white', (b) achievements, roughly corresponding to i n s t a n t a n e o u s events/properties, such as 'the deer was hit by an arrow', (c) accomplishments, which m a y be described as noni n s t a n t a n e o u s events, such as 'to draw a circle', (d) activities, approximately the same as successive properties, such as 'to run'. In m o d e r n formal semantics, it is realised that these different types of verb p h r a s e s call for different tense-logical t r e a t m e n t s [Galton 1987, p. 13]. Also in Artificial Intelligence we fred similar distinctions, as for instance in J. F. Allen's distinction between states, events, a n d processes [Allen 83, 84] - the last concept being comparable to t h a t of successive states.

1.7. TIME AND

CONSEQUENTIA

In the introduction, we emphasised the fact t h a t the subject m a t t e r of logic has not been constant throughout t h e history of logic, and t h a t the focus of interest has changed several times. Even so, the very notion of logical consequence is an almost definitional property of logical studies. The Middle Ages are no exception in this respect. The s t u d y of logical consequence, known as consequentia, constituted one of the most central fields within Medieval logic. Since we h a v e been stressing t h e importance attributed to time and tense, we should now balance our account by observing t h a t the concept of time was not of crucial importance to the formulation of most of theories on consequentia. On the other hand, in some cases time did in fact play a r61e in such theories. Some medieval texts on consequentia appears to be about c o n d i t i o n a l s ('if A t h e n B'), but in most of t h e texts on consequentia it seems t h a t the a u t h o r is in fact dealing with w h a t we would now call inference. But as A l e x a n d e r Broadie has pointed out [1987, p.51] it is plain that medieval logicians in general were aware of the difference between w h a t we would call respectively conditionals and inferences, although they used the same term for the two relations. In this chapter we shall also use the term in this ambiguos way. In medieval logic the study of syllogisms was considered to be one of the key parts. Sometimes other syllogisms t h a t the ones of the classical figures were studied. Given the medieval awareness of the importance of temporal logic it is not surprising t h a t t h e y introduced a syllogistic tense logic. According to Broadie the first detailed discussion of the topic was given by William of Ockham. In his Summa Logicae he stated for instance: When both premisses are past-tensed in the second figure and the subject of each of these supposits for things which are, there always follows a present-tensed conclusion, and not a past-tensed conclusion. [Summa Logicae III, 1,18]

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This m e a n s that although the syllogism No (present) A is B Some (present) C is B Ergo: Some C is not A is in fact valid, the tensed syllogism, No (present) A was B Some (present) C was B Ergo: Some C was not A has to be rejected in general, whereas the No (present) A was B Some (present) C was B Ergo: Some C is not A should be accepted as valid syllogism. - In this way William of O c k h a m p r e s e n t e d some r a t h e r precise components for t h e formulation of a formal (but of course not symbolic) tense logic. Time was also involved in the medieval study of consequentia in a n o t h e r way. The conception of s t a t e m e n t s as units w i t h temporally variable t r u t h values led the medieval logicians to the notion of c o n s e q u e n t i a ut nunc, which was the medieval t e r m for t h a t form of logical consequence whose t r u t h value varies w i t h time. T h a t is, this type of logical consequence is capable of being t r u e at one time and false at another. P e t e r King, for instance, has described an u t nunc consequence as a s t a t e m e n t with an antecedent and a consequent 'such t h a t it is not the case that the antecedent obtains and the consequent fails to obtain', and he has stressed that we must take the tense of the verb in 'it is not the case' seriously [King p.62-63]. So in medieval logic we once again fred a distinction between an implication which can be valid at one time (or some times, or some period of time), but invalid at other times, and a n implication which m u s t be valid at all times, if it is to be valid at all. That distinction had already been discussed in Ancient times

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(Diodorus, Philo of Megara, the Stoics), an issue which we touched upon in our discussion of the Master Argument. There is an exact correspondence between the distinction underlying t h e Ancient discussion and the later Medieval distinction between 'as-of-now consequence' ( c o n s e q u e n t i a u t n u n c ) and 'absolute consequence' ( c o n s e q u e n t i a s i m p l e x ) - as pointed out by L. N. Roberts [1967]. As we have seen, the famous debate between Diodorus and Philo of Megara was precisely concerned with the relation between time and implication. The question was w h e t h e r to allow the t r u t h values of the implication to vary with time or not. The Medieval logicians were aware of t h e problem, and solved it by allowing both kinds of implications, whilst duly distinguishing between t h e m in any concrete case. Burleigh, for example, presented one implication of the same type as Diodorean implication, that is, it had to be valid at any time, and another - c o n s e q u e n t i a u t n u n c - in the style of Philo, that is, it only h a d to be valid at one time. Walter B u r l e i g h put forth the following example of a consequentia ut nunc:

(1) 'Every m a n is running, so Socrates is running' This is clearly a consequence, which is correct at certain times and not at others. It is obviously only valid during a period of time, in w h i c h S o c r a t e s is alive. On t h e o t h e r hand, a consequentia such as: (2) 'All living beings are running, so all men are running' is valid at all times, since the set of living beings must at a n y t i m e i n c l u d e t h e set of h u m a n beings. However, t h e consequentia of (1) will prove to be false at times where no m a n by the name of Socrates exists. While consequentia was in general a field of much interest for medieval logicians, t h e y did not pay m u c h attention to t h e specific c o n s e q u e n t i a u t n u n c , nor did t h e y develop any real theory about it. It appears t h a t only a few leading medieval

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logicians a t t e m p t e d to define rules for this kind of consequentia. Burleigh, for one, stated this rule: In an as-of-now consequence, however, the antecedent cannot be true without the consequent as of now. [Kretzmann 1988, p.285] This m e a n s t h a t at a time at which the consequence holds good, t h e antecedent cannot be true without the consequent. J o h n B u r i d a n formulated another rule: F r o m any false sentence a n y other sentence follows as a c o n s e q u e n c e ut nunc, and also any true sentence follows from a n y other as a consequence ut nunc. [King p. 196] W h a t this rule states is obviously close to what we would call the p a r a d o x e s of implication. That should come as no surprise, for it is evident that consequentia ut nunc is essentially the same as m a t e r i a l implication - w h e r e a s c o n s e q u e n t i a s i m p l e x corresponds to strict implication. According to Buridan common people often use as-of-nowconsequences [King, p.185]. As an example he mentioned the consequence: 'Cardinal White has been elected Pope; therefore, a Master of Theology has been elected Pope.' Clearly, this consequence can only be true if the proposition 'Cardinal White is a Master of Theology' is true. In general, an ut-nunc-consequentia is true only w h e n some tacit a s s u m p t i o n related to the consequence is also true. As Alexander Broadie [1987, p.61] h a s pointed out a valid ut-nuncc o n s e q u e n t i a can be t r a n s f o r m e d in to a valid inference (or conditional) by the addition of a relevant proposition, which is true now.

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Buridan even claimed that the consequence: 'Socrates is running, Plato is running, Robert is running; therefore, every man is running.' is t r u e if it is perfected with the t r u t h of the proposition 'Any man is Socrates, Plato or Robert'. Buridan also pointed out t h a t promissory consequences are asof-now-consequences. He considered an example, w h e r e Plato says to Socrates: 'If you come to me, I shall give you a horse' This consequence can be t r u e as-of-now, if it is perfected with the following conditions, which m u s t be assumed to be t r u e [King p.186]: (a) 'Plato wills to give a horse to Socrates', and (b) 'whatever Plato wills to do in the future, (i) he will be able to do by holding to t h a t volition (and holding in any circumstances to that what he wills); and (ii) w h e n he is not prevented he does t h a t thing w h e n and how h e wills'. Attempts to involve past and future statements directly in the studies of the consequence-as-of-now were rare. One of the few examples is this one: If Antichrist will never be generated, Aristotle never existed. [King p. 196] Buridan's accepted this consequence as being true as-of-now on the basis of his Christian belief that Antichrist is in fact going to be. Buridan did acknowledge that it is logically possible for the antecedent to be true, but he also asserted that its truth would be incompatible with other facts which should be accepted. Hence, the a n t e c e d e n t m u s t be r e g a r d e d as false, and therefore the consequence is to be accepted as true as-of-now. However, after

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the generation of Antichrist it must be n a t u r a l to reject the consequence. Obviously, B u r i d a n in this m a n n e r b r o u g h t t e m p o r a l c o n s i d e r a t i o n s into t h e s t u d y of c o n c r e t e e x a m p l e s of consequentia ut nunc. However, to do so in connection with the c o n s e q u e n c e - a s - o f - n o w was a fairly r a r e t h i n g a m o n g s t medieval logicians, according to Bochenski [1961, p. 193]. On the other h a n d it was common-place in the late Middle Ages to base a distinction among different types of consequentia on criteria relating to the concept of time. But around the time of 1500 A.D., t h a t kind of investigation was clearly fading a w a y a m o n g logicians. Consequentia ut nunc was occasionally mentioned as consequentiae vulgares, since it was regarded as primarily conc e r n e d w i t h e v e r y d a y language r a t h e r t h a n with scientific r e a s o n i n g [Kneale p. 289]. This view was probably also the reason for the a b a n d o n m e n t of consequentia ut nunc: as logic g r a d u a l l y d i s t a n c e d i t s e l f from e v e r y d a y l a n g u a g e , consequentia ut n u n c came to be seen as u n i n t e r e s t i n g from a scientific point of view.

1 . 8 . T E M P O R A H S - THE LOGIC OF ' W H I L E ' Medieval logicians generally accepted a distinction between atomic and molecular propositions. Molecular propositions are formed from atomic (or simple) propositions by m e a n s of propositional connectives. Propositions thus formed were also known as hypotheticals, a t e r m which was applied not only to implicational s t a t e m e n t s , but also to the other k i n d s of molecular propositions such as conjunctions, disjunctions etc. In the Middle Ages, however, there was little agreement as to which connectives should be t a k e n into consideration w i t h i n logical studies. Thus the number of propositional connectives was not fixed in general. William of Ockham [EL, p.198] suggested t h a t there w e r e at least five: conditionalis, i.e. implication (in a broad sense), copulativa, i.e. conjunction, disiunctiva, i.e. disjunction, and temporalis and causalis. Temporalis will be discussed below. With respect to causalis, we m a y mention two of the m a n y examples put forth by Paul of Venice (c. 1369-1429) in his Logica

Magna: Because you are a m a n you are not a donkey [II, 3, 27e] Because the sun is this light is [II, 3,29e] The use of causalis is not presently a concern of ours, and we shall leave it aside for now. It m a y be mentioned, though, t h a t causalis roughly corresponds to 'because', a n d t h u s has to do with causation only in a very broad sense. O c k h a m did not claim, though, t h a t t h e corresponding molecular propositions are the only possible ones, or t h a t they were mutually independent in a strong logical sense. He merely considered the molecular propositions listed above to be the most i n t e r e s t i n g and important ones. Walter Burleigh [1955, p.107] agreed with Ockham on the number of propositional connectives. He admitted t h a t other types of molecular propositions m i g h t be thought of, but he also maintained t h a t such f u r t h e r types would prove to be reducible into the five f u n d a m e n t a l ones m e n t i o n e d above. Ockham and Burleigh t h u s agreed on which molecular propositions were to be regarded as important or ba71

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sic. Writing a few decades later, J o h n Buridan listed six molecular propositions [MS Krakow BJ 662 f.7r.]. He accepted the five molecular propositions of Ockham a n d Burleigh, but in addition he considered one more type, which he called localis, to be a connective in its own right. Richard L a v e n h a m [Spade 1973, p.57] later in the 14th century suggested the addition of even one more type of molecular proposition, called rationalis. Ockham and Burleigh, as well as Buridan, had probably considered rationalis to be nothing more t h a n a yet another variant of conditionalis. It appears t h a t these medieval writers all agreed on the import a n c e of 'temporal propositions' (temporalis), i.e. molecular propositions composed from two or more atomic propositions conjoined by some adverb of time. Temporal connectives such as 'cum', 'dum', and 'quando' appeared to these logicians to form an i m p o r t a n t class of logical constructions, just like conjunction, disjunction, and implication. This idea is in fact very old. The s t u d y of temporalis can be traced back to Boethius (c. 480-524), who discussed for instance the s t a t e m e n t 'when a m a n is, an animal is' - 'when' being the temporal connective, of course. In the Middle Ages, one of the first philosophers to discuss temporal propositions was the Islamic logician Ibn Sina (980-1037), in Christian Europe known as Avicenna. In his logic he discussed s u c h Arabic temporalis s t a t e m e n t s as can be seen in t h e following English counterparts: 'Whenever the sun is out, then it is day', 'It is never the case that if the sun is out, then it is night', 'It is never the case that either the sun is out or it is day', 'If, whenever the sun is out, it is day, then either the sun is out, or it is not day' One cannot say t h a t Avicenna developed a real theory of such temporal propositions, but he did t r y to work out the relationship between temporalis and the implication, as it can be seen from the following quotation: Take the word 'if. You do not say 'If the day of resurrection comes, t h e n people will be judged' because the consequent is

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not implied by the posited antecedent. For [the consequent] is not something necessary but d e p e n d s on God's will. R a t h e r you say 'When the day of r e s u r r e c t i o n comes, people will be judged'. Moreover, you do not say 'If m a n exists, then two is even' or 'the void is non-existent'. You say 'whenever m a n exists, then two is also even' or 'the void is non-existent'. It seems that the word 'if is very strong in showing implication, while 'whenever' is w e a k in this respect and 'when' is in between. [Shehaby 1973, p.38] According to Avicenna temporalis is an implication t h a t is slightly weaker t h a n the implication corresponding to 'if-then'. H e r e it must be observed that Avicenna h a d a sort of 'relevant implication' in mind. As he saw it, 'if p, t h e n q' presupposes a s t r o n g semantical relation between p and q. Since Avicenna considered 'p while q' to be weaker than this kind of implication, h e accepted that it could be true also in cases where the implication 'if q, then p' does not obtain. Following Avicenna, however, w h e n e v e r q' should be interpreted as the m a t e r i a l implication, t h a t is, p and q need not be semantically relevant to one another. Most medieval logicians in the following centuries appear to have agreed that a necessary condition for the t r u t h of 'p while q' is t h a t both ofp and q are true at some time in the past, the p r e s e n t or the future. It is a fact that in ordinary language use temporalis is often conflated with implication. Thus, propositions such as 'When the sun shines, it is day', and 'Wood becomes w a r m , w h e n fire is brought near to it', are often supposed to be equivalent to the corresponding conditionals 'If the sun shines, t h e n it is day', and 'If fire is brought near to wood then it becomes warm'. Similar intuitions must have been at work in the a t t e m p t s of various logicians, who tried to describe temporalis in t e r m s of the conditional. Ockham was obviously a c q u a i n t e d with some of those attempts, but he nevertheless rejected t h e i r underlying idea of reducing temporalis into a conditional. I n s t e a d , he r e l a t e d the semantics of temporalis to the conjunction, as we shall see in detail below. According to Ockham, the following

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propositions are all instances of temporalis: (1) 'Socrates is running, while Plato is debating'. (2) 'Socrates was running, before Plato was debating'. (3) 'Socrates was running, after Plato was debating'. However, on Buridan's definitions only (1) is a proper instance of temporalis, whereas (2) and (3) are not. Buridan seems to have a s s u m e d t h a t molecular propositions constructed by means of 'after' and 'before' can be reduced into propositions with the connective 'while'. In the following we shall concentrate on temporal propositions formulated by means of 'while' (or some equivalent word); we shall use the connective w for 'while'. Given Buridan's interest in durational logic, it would only seem natural if h e h a d also tried to account for the logic of temporalis within a d u r a t i o n a l framework. To our knowledge, however, Buridan n e v e r formulated such an explication. Now w h a t is the truth-condition for the temporal proposition (p w q)? Here all 14th c e n t u r y logicians gave almost the same answer: The truth-condition for the temporal proposition (p w q), where both p and q a r e in the p r e s e n t tense, can be formulated in t h e following way: (C)

(p w q) is true i f f b o t h p and q are true (now).

If (C) is accepted without any constraints, the following thesis will be valid: (T1)

(p wq) ~(p Aq)

The same of course goes for the converse of (T1). Thus it would be a consequence of a completely general adoption of (C) t h a t temporalis would simply be equivalent with the usual conjunction. However, according to Buridan (C) should not be accepted in general, but only for propositions in the present tense. Now let p and q represent atomic propositions (with the verbs in the present tense), and let Pp, Pq, Fp, and Fq stand for the corresponding propositions with verbs in the past and in the future tense,

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r e s p e c t i v e l y . W i t h r e s p e c t to t h e t e m p o r a l p r o p o s i t i o n s (Pp w Pq) a n d (Fp w Fq),,the truth-conditions will be different from (C). As a precondition for giving t h e proper definitions, Dorp [1499, p.41] suggested t h a t a n y adequate definition would have to comply w i t h the following observations: (B1) T h e r e is a temporal proposition (p w q), which is false, although p and q are both true. (B2) T h e r e is a temporal proposition (p w q), which is true, although one of its parts is false. In order to prove (B1) Dorp considered t h e temporal proposition 'My f a t h e r was while Adam was'. The c o n s t i t u e n t s of this proposition, 'My f a t h e r was' a n d 'Adam was', are both true, w h e r e a s t h e temporally combined proposition is false. This is indeed sufficient to prove (B1), and as a consequence of t h a t principle t h e following proposition is not a valid thesis in Dorp's (and Buridan's) system: (N1)

(Pp A Pq) ~ (Pp w Pq)

For this r e a s o n t h e temporalis a n d t h e conjunction are obviously not equivalent. (T1) can be valid only for propositions in t h e present tense. T h u s restricted (T1) can on the other h a n d be s t r e n g t h e n e d into (TI') (p w q) --(p A q) where p a n d q are atomic a n d present-tense propositions. -

In order to prove (B2) Dorp fielded t h e following t e m p o r a l proposition: 'My f a t h e r was not while A d a m was' ('Pater m e u s non fuit quando A d a m fuit').

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L e t p s t a n d for 'My f a t h e r is', and q stand for 'Adam is'. It appears t h a t t h e first p a r t of this proposition, 'My father was not', is false, whereas t h e o t h e r part, 'Adam was', is true. It also appears t h a t t h e combined temporal proposition is itself t r u e - as already observed by B u r i d a n . Then the proposition would s e e m to constitute a s t r a i g h t f o r w a r d example of (B2). However, t h e r e obviously is a problem here. In general, the whole proposition is three-ways ambiguous, as signified by (a) (b) (c)

-(Pp w Pq) (P-p w Pq) (-Pp w Pq)

The (a)-case is not i n t e r e s t i n g here, though, for clearly Pp a n d Pq are both true, a n d t h u s it could never be m a d e into a case for (B2). In fact the (a)-case is a n o t h e r example of the principle (B1). So let us turn o u r attention to the (b)- a n d (c)-cases, which together reflect the fact t h a t the proposition 'My father was not' is itself ambiguous. It can be u n d e r s t o o d as P~p or as ~pp. Obviously it is ~Pp t h a t is false, while (P~p w Pq), P-p, a n d Pq are all true. It seems t h a t Dorp mixed up his readings of 'My f a t h e r w a s not'; in a r g u i n g t h a t the c o n s t i t u e n t was false he a d o p t e d t h e r e a d i n g - P p , b u t in a r g u i n g t h a t t h e whole temporalis was true h e w a s referring to the reading P - p , i.e. the (b)-case. (It m u s t be a d m i t t e d , however, t h a t t h e original Latin formulation of the e x a m p l e ('pater meus non fuit') takes a form t h a t makes it tempting to u n d e r s t a n d it is as -Pp.) We conclude that a l t h o u g h it m a y be questioned w h e t h e r Dorp m a n a g e d to show (B2), b u t he did show t h a t t h e following formula is not a thesis: (N2)

(P-p w Pq) ~ -Pp

What he did not m a n a g e to show was that this formula is not a thesis: (N3)

(P-p w Pq) ~ P-p

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although he apparently thought he had done so. In fact, it m a y well be the case that (N3) is a thesis, and that (B2) is not correct. Ockham in his S u m m a Logicae actually denied the validity of (B2): It is also clear from w h a t has been said t h a t t h e r e is a valid consequence from a temporal proposition to one of its parts but not conversely. Similarly, a conjunctive proposition follows from a t e m p o r a l proposition - but not conversely. [Ockham 1980, p.192] T h a t is, Ockham claimed t h a t (T2)

(Pp w Pq) ~ (Pp A Pq)

is a valid thesis, whereas he (like Dorp) rejected (N1), i.e. he denied t h a t t h e r e is a valid consequence from t h e conjunction of t h e parts of a temporal proposition to the temporal proposition itself. The validity of (T2) clearly implies the rejection of (B2). Since (N1) is not a thesis, there is no straightforward reduction of temporal propositions into conjunctions, a n d (C) therefore h a s to be amplified in a m a n n e r , which will m a k e it applicable also to tensed propositions such as (Pp w Pq). It appears that the a p p r o p r i a t e truth-condition according to both B u r i d a n and Dorp can be formulated as follows: (CP)

(Pp w Pq) is t r u e if and only if there is some time in the past at which both p and q are true.

It is evident t h a t this truth-condition is identical with the truthcondition for the proposition P(p A q). Therefore we have now arrived at the following equivalence, which must be adopted as a valid thesis within Buridan's and Dorp's logic for temporalis: (T3)

(Pp w Pq) - P ( p A q)

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It is reasonable to assume that the similar thesis for w i t h respect to the future operator is also valid: (T4)

(Fp w Fq) -=F(p A q)

Therefore the temporalis can in some sense be reduced to t h e conjunction, although it is no straight-forward reduction. T h e r e l a t i o n b e t w e e n t h e t e m p o r a l i s and the conjunction is explained in (C), (T3) and (T4). It is clear, however, t h a t t h e s e theses do not apply to all possible syntactical constructions. For instance, one might ask for the truth-conditions of propositions like (Fp w Pq). To the best of our knowledge no medieval logician took such propositions into serious consideration. The r e a s o n for this m u s t have been the view t h a t such hybrid propositions simply do not m a k e sense. What meaning could be attributed to e.g. 'Socrates will be running while Plato was debating'? The acceptance of such propositions as meaningful presupposes t h e assumption t h a t some future time is also past, that is, t h a t t h e structure of time is circular (or possibly cyclical). We shall not rule out t h a t such an assumption might be consistent at a n ontological level, but it is not consistent with the semantics of Latin, or English, for t h a t matter. For then consequences of the form

Fp Pp would have to be counted as intuitively valid - which t h e y are not. It should be noted, however, that we do not rule out propositions like 'Socrates will be running while Plato will have been debating'. B u t t h e s t r u c t u r e of such propositions is n o t (Fp w Pq), w h e r e p and q are simple propositions with the verbs in the present tense. The structure of such propositions is r a t h e r to be represented as (Fp w FPq). The complete truth-condition for meaningful temporal propositions can be stated in the following way: I f p and q are atomic propositions in the present tense, t h e n (p w q) --(p A q)

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I f p and q are arbitrary propositions (atomic or molecular, in any tense), then (Pp w Pq) - P(p A q) and (Fp w Fq) - F(p A q). This definition reflects the views of the logicians whose opinions have been discussed above, and on those views it covers all non-modal, t e m p o r a l propositions which are considered to be meaningful. In our opinion, this t r e a t m e n t does go a long way in getting us an appropriate semantics for temporalis, but as we suggested above there also are limitations to it. It seems to us that in certain cases temporalis should perhaps be related to implication r a t h e r t h a n conjunction. Let us give just one example of this. Consider the sentence (S) 'Socrates will be running, while Plato will be debating'. Now the t r e a t m e n t offered above clearly implies t h a t both of the events of S will occur. Let p represent 'Socrates is running', and q represent 'Plato is debating'. Then S will be represented as (Fp w Fq), which is equivalent to F(p A q), which clearly entails the future 'occurrence' ofp as well as q. But it seems that there is another reading which does not strictly foresee that either event will occur; it m e r e l y says t h a t i f p will ever be the case, t h e n so will q.. Let G s t a n d for 'it will always be the case that', i.e. G = - F - . Then the latter reading could tentatively be represented as G(p ~ q). This reading is close to what we above called a 'generic reading'. However, it can be argued t h a t this reading is the one that should be expressed as (S') 'Whenever Plato will be debating, Socrates will be r u n ning', and hence, the reading G(p ~ q) for S might be viewed as m e r e l y a consequence of an imprecise use of language. But in fact, it seems t h a t t h e r e is an even weaker reading of (S), which might be represented as

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Fq D F(p A q) Loosely, this reading says that if Plato is ever going to debate at some future time, and possibly at several future times, t h e n Socr a t e s is going to be r u n n i n g at least one of those times. In other words, we h e r e need to define 'while' in terms of implication r a t h e r t h a n conjunction. We admit t h a t this r e a d i n g m a y be quite rare, but we believe t h a t there is a systematical a r g u m e n t for it. The t r e a t m e n t of (Fp w Fq) as equivalent w i t h F(p A q) m a k e s no difference between (a) forming (S) by first forming p and q, t h e n forming (p w q), and finally putting the whole thing into the future tense; in this case (S) is seen to be equivalent w i t h 'it will be t h e case t h a t Socrates is running, while Plato is debating'. This is appropriately r e p r e s e n t e d as F(p w q), a n d adequately treated as equivalent to F(p A q); (b) forming (S) by first forming two future proposition, Fp and Fq, and then conjoining t h e m by means of the whileconnective. This s t r u c t u r e is i m m e d i a t e l y r e f l e c t e d by (Fp w Fq). We see no prima facie evidence t h a t this reading m u s t also entail the actual occurrence ofp and q. Now we admit that no difference between the two s t r u c t u r e s can be seen in the syntactic surface structures of either Latin or English. But it is clear t h a t (S) could be built in both ways suggested, and moreover, it is a fact t h a t there has also been a persistent tradition of understanding temporalis in t e r m s of the implication. From these observations we conclude t h a t (i) it is problematic to define temporalis exclusively in t e r m s of conjunction (and t e n s e operations) - implication should be granted a rSle, too; (ii) temporalis in some cases invites a kind of generic reading, which can be pleasantly represented by the G-operator; it m a y be noted t h a t this kind of generic r e a d i n g m a y be relaxed somewhat if we would introduce intervals.

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The logicians of t h e 14th century also considered t e m p o r a l propositions involving modal operators. Walter Burleigh s t u d i e d t e m p o r a l propositions involving modal operators, and in t h i s connection he also considered the Aristotelian principle " O m n e quod est q u a n d o est necesse est esse" [Burleigh 1955, p. 130], which Aristotle proposed in De Interpretatione IX. According to Burleigh this principle can be understood in two different ways, which m a y be representable as follows (as usual, we shall let Mp and Np stand for 'it is possible that p', and 'it is necessary t h a t p', respectively): (N4) (T5)

N(p w p) p ~ (Np w p)

(N4) corresponds to the translation 'it is necessary, t h a t everyt h i n g is, when it is', where as (T5) can be read 'everything t h a t is true, is necessary, w h e n it is true'. Burleigh m a i n t a i n e d t h a t (T5) is a valid thesis, whereas (N4) is not. B u t how does t h e notion of possibility more precisely a t t a c h to t h e temporalis operator? Burleigh pointed out t h a t a precondition for t h e possibility of (p w q), t h a t is, for the t r u t h of M(p w q), is - r a t h e r of course - t h a t both of p and q m u s t be possible in their own right. So we have the following valid thesis : (T6)

M(p w q) ~ (Mp A Mq)

However, the converse of (T6) is not valid: (N5)

(Mp A Mq) ~ M(p w q)

In his S u m m a Logicae Ockham i n v e s t i g a t e d several d e t a i l s r e g a r d i n g propositions containing both modal and temporalis operators. He first investigated which conditions would h a v e to obtain in order for a t e m p o r a l proposition to be n e c e s s a r y . O c k h a m claimed t h a t ... in order for a temporal proposition to be necessary it is required t h a t each p a r t be necessary. [Ockham 1980, p.180]

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This condition can be formulated by means of t h e following thesis: (T7)

N(p w q) ~ (Np z, Nq)

However, t h e converse of (T7) is not valid, t h a t is, no temporal proposition (p w q) is n e c e s s i t a t e d simply because each of its c o m p o n e n t propositions is necessary by itself. In fact, even temporal propositions such as 'a m a n is a bachelor, while he is not married', 'Socrates exists, while he exists', and 'Socrates is moving, while he is running', are not necessary according to Ockham. This is so, in spite of the fact t h a t t h e s e t e m p o r a l propositions are composed of atomic propositions which could form necessary conditionals: 'Necessarily, if a m a n is not married, then he is a bachelor', and similarly for the other two sentences. The fact t h a t p necessarily implies q does not, however, imply t h a t t h e t e m p o r a l p r o p o s i t i o n of p and q is necessary, so according to Ockham's ideas (N6)

N(p D q) ~ N(p w q)

is not a valid thesis. Not even (N7)

N(p ~ p) ~ N(p w p)

is a valid thesis in Ockham's logic. The reason is of course t h a t t e m p o r a l propositions according to Ockham are modified conj u n c t i o n s a n d not modified conditionals. Hence, (p w p) will be equivalent to p, and consequently (T8)

N(p w p) -=Np

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is a valid thesis in Ockham's logic. In his own words: Hence, the proposition 'Socrates exists while he exists' or 'Socrates is moving while he is running' is not necessary, b u t can be false. [Ockham 1980, p.192] T h e r e is a n o t h e r q u e s t i o n r e g a r d i n g m o d a l t e m p o r a l propositions to which O c k h a m gave an answer; t h a t question is concerned w i t h impossibility. He m a d e it clear t h a t a t e m p o r a l proposition can be impossible, even t h o u g h none of its p a r t s is impossible, so we do not have the following formula as a thesis: (N8)

- M ( p w q) ~ ( - M p v - Mq)

I n fact, O c k h a m did not state a necessary condition for t h e impossibility of (p w q), but he did formulate a sufficient condition: ... for a temporal proposition to be impossible it is n o t req u i r e d t h a t some p a r t be impossible. Rather, it is sufficient t h a t the parts be incompossible. Thus, this is impossible: God creates while he does not create. [Ockham 1980, p.192] Since p and - p are incompossible, the statement implies t h a t (T9)

~M(p w -p)

is a thesis. It should be m e n t i o n e d t h a t (T9) can be shown as a c o n s e q u e n c e of two basic principles of modal logic: (i) t h e fact t h a t t h e contradiction is impossible, i.e. the thesis (TIO)

- M ( p A -p),

a n d (ii), t h e well-known modal principle t h a t no impossible proposition follows with necessity from a possible one, i.e. the thesis (Tll)

N(p ~ q) ~ ( - M q ~ -Mp)

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Because of (TI'), (T3), (T4), and s t a n d a r d modal logic (T12)

N((p w ~p) ~ (p A -p))

is also a thesis. It is easy to verify t h a t t h e validity of (T9) follows from (T10-12). In the 15th century temporalis b e c a m e more a n d more neglected. The medieval discussion r e g a r d i n g the n u m b e r of basic kinds of molecular proposition has b e e n s u m m e d up by P a u l of Venice in his Logica Magna: Some posit five kinds of molecular proposition, some six, others seven, others ten, others fourteen, and so on. B u t p u t t i n g all these opinions to one side, I say t h a t of k i n d s of m o l e c u l a r proposition which a r e n o t identical in t h e i r signification there are three a n d no more ... [Logica Magna, II, Fasc. 3, 2el. According to Paul himself, t h e r e are only t h r e e species of molecular proposition: conditionalis, copulativa and disiunctiva. Nevertheless, he made some very careful studies of the logic temporal propositions such as 'when 'while 'when 'when

I was awake I did not sleep' [13e} I shall not be Antichrist will not be' [15e] every m a n disputed every m a n was white' [15e] one single m a n will die every m a n will die' [18e]

and other temporal propositions like 'you will be a priest before you will be a bishop' [11e] 'you will begin to be after A will be' [16e] However, although Paul of Venice considered such t e m p o r a l propositions to be relevant for the logical reflection, he did n o t accept temporalis as one of the f u n d a m e n t a l kinds of molecular propositions. He simply did not accept 'dum', 'ubi', 'quia' etc. as proposition-forming functors. The opinion expressed h e r e by

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P a u l of Venice came to be the usual one in logical studies during t h e Renaissance. I n post-medieval logic temporalis eventually disappeared a n d can only exceptionally be found in 16th c e n t u r y logic. A s h w o r t h m e n t i o n s the P o r t u g u e s e logician P e t r u s de F o n s e c a (15281599) as one of t h e exceptions. B u t how can t h e fact t h a t the temporalis s a n k into oblivion be explained? One factor seems to have been t h e growing h u m a nistic criticism of scholastic logic. According to t h e h u m a n i s t s t h e language used by the logicians of the scholastic tradition was perverted. It is i n d e e d likely t h a t the h u m a n i s t s saw t h e med i e v a l discussion of the temporaIis as a clear e x a m p l e of scholastic linguistic perversions. On t h a t basis, t h e temporalis should of course be rejected as an i m p o r t a n t e l e m e n t of logic. It must, however, be admitted that this can only be a part of the e x p l a n a t i o n . For t h e very idea t h a t t e m p o r a l p r o p o s i t i o n s f o r m e d a class of basic molecular propositions was rejected before t h e general downfall of scholastic logic - in fact, it was rej e c t e d within t h e scholastic tradition itself. As we have ment i o n e d , Paul of Venice in his very p o p u l a r Logica Magna claimed that there are only three species of molecular propositions. Obviously the majority of logicians of t h e late scholastic period agreed with Paul. They devoted significantly more interest to t h e conjunction, t h e disjunction, and t h e conditional, t h a n to t h e other putative molecular propositions. One reason for this preference may have been the simple fact t h a t it is relatively easy to formulate t h e truth-conditions for these three molecular propositions in terms of truth, falsity, and modality, whereas the truth-conditions for the other molecular propositions are more complicated. As t h e above discussion s h o u l d have shown, t h e t r u t h - v a l u e of a t e m p o r a l proposition (p w q) is n o t a simple function of the truth-values of its components. On t h e contrary, it comes out as a r a t h e r complicated combination of conjunctions a n d tense-operators. These properties do not seem to be adequate for a f u n d a m e n t a l notion. One further partial explanation should be mentioned. The rejection of the temporalis might be seen in t h e light of the general features of medieval logic. We believe t h a t medieval logic can in

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a general sense be characterised as a temporal logic. That is, it was a logic of propositions whose truth-values can vary from time to time, and a logic in which temporal expressions were considered to be important. It is only natural, then, t h a t the logicians of the Middle Ages, who were working within t h a t f r a m e w o r k , and who took t h e temporalis into serious consideration, were also to put forth some of clearest statements ever of the basic assumptions of temporal logic. Late-scholastic and h u m a n i s t logicians paid less attention to the temporal structures of logic t h a n their predecessors, and so it seems understandable t h a t the temporalis became increasingly neglected, and was ignored as an important propositional connective. Even so, the reasons for the rejection of the temporalis which we have m e n t i o n e d here do not seem sufficient for fully explaining this development. There is certainly still much to be done with respect to reconstructing the medieval use of the temporalis as well as the final rejection of it as a connective. But we hope to have a r g u e d convincingly t h a t the temporalis is an interesting construction of medieval logic, and t h a t it deserves f u r t h e r

study.

1.9. HUMAN FREEDOM AND DIVINE FOREKNOWLEDGE During t h e Middle Ages logicians as a m a t t e r of course related their science to theology. Clearly t h e y felt that t h e y h a d somet h i n g i m p o r t a n t to offer with r e g a r d to solving f u n d a m e n t a l logical q u e s t i o n s in theology. T h e most i m p o r t a n t question of t h a t k i n d w a s t h e problem of t h e contingent future. This problem h a s since come to be r e g a r d e d as one of t h e most central p r o b l e m s in t h e logic of time, t o g e t h e r w i t h t h e concomitant question of t h e relation between time and modality. In our day, it is not primarily seen as a theological problem, but intellectuals of t h e Middle Ages saw the problem as intimately connected w i t h t h e relation between two f u n d a m e n t a l Christian dogmas. T h e s e a r e t h e d o g m a s of h u m a n f r e e d o m a n d God's omniscience, respectively. God's omniscience is a s s u m e d to also comprise k n o w l e d g e of t h e f u t u r e choices to be m a d e by men. B u t t h e n t h e l a t t e r d o g m a a p p a r e n t l y gives rise to a s t r a i g h t f o r w a r d a r g u m e n t f r o m d i v i n e f o r e k n o w l e d g e to necessity of t h e future: if God already now knows the decision I will m a k e tomorrow, ~t h e n an inevitable t r u t h about m y choice tomorrow is already given now! Hence, there seems to be no basis for t h e claim t h a t I have a free choice, a conclusion which violates t h e d o g m a of h u m a n freedom. To s u m it up, t h e a r g u m e n t p r o c e e d s in two p h a s e s : first f r o m d i v i n e f o r e k n o w l e d g e to n e c e s s i t y of t h e f u t u r e , a n d from t h a t a r g u m e n t to t h e s u b s e q u e n t conclusion t h a t t h e r e can be no real h u m a n freedom of choice. A m o n g m a n y others t h e great Danish 12th century philosopher, Boethius de Dacia [Sajo, vol.V, p.241] tried to solve this difficult problem. According to h i m the m a i n question is w h e t h e r the s t a t u s of the contingent future is compatible w i t h the certainty of divine knowledge, t h a t is, the belief t h a t God h a s certain knowledge of arbitrary contingent events in t h e future. Boethius in his analysis insisted t h a t God fully k n o w s f u t u r e events, w h i c h a m o n g other t h i n g s m e a n s t h a t he k n o w s events, which in a n u m b e r of cases are not necessary b u t contingent.

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The approach to t h e problem was to regard it as a consistency problem, which h a d to be solved within logic. It was p r i m a r i l y s t u d i e d in connection w i t h Aristotle's text from De Interpretatione I X (the s e a - b a t t l e tomorrow etc.). A n o t h e r piece of classical text which w a s occasionally t a k e n into consideration was Cicero's De Fato, w h i c h among other m a t t e r s describes t h e Diodorean Master A r g u m e n t . The problem obviously bears on the theological t a s k of clarifying questions such as 'In which way can God know t h e future?' or 'What is to be understood by 'free-will' and 'freedom of choice'?' Extensive literature about this subject, p r i m a r y as well as secondary, can be found, a n d any a t t e m p t to produce a detailed exposition of this subject seems hopeless at t h e outset. On the other h a n d , it is possible to get a systematical overview of basic approaches to the problem. We shall accordingly restrict ourselves to an exposition of the four possible solutions to the appar e n t conflict b e t w e e n t h e two d o g m a s , w h i c h R i c h a r d L a v e n h a m (c.1380) e n u m e r a t e d in his t r e a t i s e De eventu f u t u r o r u m . L a v e n h a m ' s central idea is quite clear: If two dogmas are s e e m i n g l y contradictory, t h e n one can solve t h e problem by denying one of t h e dogmas, or by showing t h a t the a p p a r e n t contradiction is not real. Denial of the d o g m a of h u m a n freedom leads to fatalism ( l s t possibility). Denial of t h e dogma of God's foreknowledge can eit h e r be based on t h e claim t h a t God does not know the t r u t h about t h e future (2nd possibility), or t h e a s s u m p t i o n t h a t no t r u t h about the c o n t i n g e n t future h a s yet been decided (3rd possibility). One can alternatively formulate a system, which shows t h a t the two dogmas, r i g h t l y understood, can be u n i t e d in a consistent w a y (4th possibility). Lavenham h i m s e l f preferred the last approach which he called 'opinio m o d e r n o r u m ' , and which can justly be called the typical 'medieval solution' to the problem reg a r d i n g h u m a n f r e e d o m a n d divine f o r e k n o w l e d g e . T h e central feature of t h a t solution was its use of the notion of a 'true future' a m o n g a n u m b e r of possible futures. It was originally formulated by William of Ockham (d. 1349), although some of its e l e m e n t s can a l r e a d y be found in A n s e l m of C a n t e r b u r y

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(d.1109). It is also i n t e r e s t i n g t h a t Leibniz (1646-1711) m u c h l a t e r w o r k e d w i t h a s i m i l a r s y s t e m as a p a r t of h i s m e t a p h y s i c a l considerations. In t h e following we shall follow this line from Anselm to Leibniz. It seems t h a t Lavenham, like Ockham, regarded t h e Aristotelian a p p r o a c h to propositions concerning t h e contingent f u t u r e as being equivalent with t h e 3rd possibility. However, this interp r e t a t i o n of Aristotle is, as shown by Nicholas Rescher, by no m e a n s t h e only one. There is also a medieval interpretation of Aristotle, according to w h i c h his solution was t h o u g h t to be identical w i t h w h a t we have called t h e 'medieval solution', i.e. the 4 t h possibility. On t h e o t h e r h a n d , B o e h n e r [1945] h a s clearly d e m o n s t r a t e d t h a t a n u m b e r of Ockham's contemporaries f a v o u r e d t h e 3rd possibility. Peter Aureole (c.1280-1322), for instance, claimed t h a t n e i t h e r the s t a t e m e n t 'Antichrist will come' n o r t h e s t a t e m e n t 'Antichrist will not come' is true, w h e r e a s t h e disjunction of t h e two s t a t e m e n t s is actually true. F r o m t h a t point of view, one can n a t u r a l l y claim t h a t t h e dogma of God's omniscience is still tenable, even if God does not know if A n t i c h r i s t will come or not. God knows all the t r u t h s given, a n d c a n n o t know if A n t i c h r i s t will come due to t h e simple reason t h a t no t r u t h value for t h e s t a t e m e n t 'Antichrist will come' yet exists. It nevertheless appears quite sensible t h a t L a v e n h a m rejected t h e 3rd possibility as c o n t r a r y to t h e C h r i s t i a n faith, since the u n d e r s t a n d i n g of the dogma of God's foreknowledge does seem s o m e w h a t clobbered. T h e m o s t c h a r a c t e r i s t i c f e a t u r e of L a v e n h a m ' s a n d O c k h a m ' s t h e o r y is its theoretical concept of 'the true future'. The C h r i s t i a n faith says t h a t God' possesses certain knowledge not only of t h e n e c e s s a r y future, b u t also of t h e c o n t i n g e n t future. T h i s m e a n s t h a t a m o n g t h e possible contingent f u t u r e s t h e r e m u s t be one which has a special status, simply because it c o r r e s p o n d s to t h e actual course of events in the future. We h a v e v e n t u r e d to call t h i s line of t h i n k i n g 'the m e d i e v a l solution', even t h o u g h other approaches existed as described in the foregoing. The justification for this is partly t h a t the notion of 'the t r u e future' is the specifically medieval contribution to this p r o b l e m , a n d p a r t l y t h a t l e a d i n g m e d i e v a l logicians

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regarded this solution as the best one ('opinio modernorum'). We shall now t r y follow the development of the medieval solution from Anselm to Leibniz.

SCT. ANSELM Anselm t r e a t e d t h e problem concerning divine foreknowledge and h u m a n freedom in his work De Concordia Praescientiae et Praedestinationationis et Gratiae Dei cum Libero Arbitrio [Hopkins 1967]. In this work Anselm undertook to answer t h r e e questions, of w h i c h the first one directly concerns the problem of divine foreknowledge and h u m a n freedom. The central idea in Anselm's solution to the problem is his distinction b e t w e e n two kinds of modality. In chapter III of De Concordia he considers two propositions: 'There will be a revolution tomorrow', and 'The sun will rise tomorrow'. The first of t h e s e sentences can be regarded as a contingent sentence, w h e r e a s the second one can be regarded as necessary. U s i n g the d a y as the time u n i t these propositions can be symbolised as F(1)p and F(1)q respectively. If F(1)p and F(1)q are true, they a r e necessary on the basis of w h a t Anselm calls subsequent necessity (necessitas sequens) - in symbols:

(1)

F(1)p ~ NsF(1)p

(and similarly for q). But according to Anselm there is another kind of necessity. He calls it antecedent necessity (necessitas praecedens). In t e r m s of antecedent necessity the proposition F(1)p is not necessary, so we have (2)

F(1)p A ~NpF(1)p

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or equivalently (3)

F(1)p A Mp ~F(1)p

(where Mp - - Np ~)

according to which it is possible t h a t there will not be any revolution tomorrow even t h o u g h it is in fact t r u e t h a t there will be a revolution tomorrow. On the other h a n d t h e proposition F(1)q is n e c e s s a r y on t h e basis of a n t e c e d e n t necessity. T h a t is: NpF(1) q. B u t w h a t is t h e difference between the two kinds of necessity? According to A n s e l m s u b s e q u e n t necessity follows from t r u e propositions about t h e state of affairs, while t h e a n t e c e d e n t necessity of a proposition m e a n s that it is compelled to be true. Obviously subsequent necessity is 'factual necessity' - t h a t is to say, it is necessity in t e r m s of simply b e i n g true. Following Anselm, a proposition is necessary on t h e basis of s u b s e q u e n t n e c e s s i t y if a n d only if a c o n t r a d i c t i o n follows from a conjunction of its negation and a n u m b e r of t r u e propositions. Now in the a r g u m e n t from divine foreknowledge to necessity of the future one m a y interpret 'necessity' (N) as ' s u b s e q u e n t necessity' (Ns). T h e n the a r g u m e n t and its conclusion are fully acceptable to A n s e l m . Likewise, in t e r m s of s u b s e q u e n t necessity Anselm did not have any misgivings about t h e thesis: 'What will be, necessarily will be', that is (4)

V x: F(x)p ~ NsF(x)p

Anselm formulated his view as follows: For when I say, 'If a thing will be, t h e n necessarily it will be', this necessity follows, r a t h e r t h a n precedes, the p r e s u m e d existence of t h e thing. [Hopkins, p.51] This acceptance, however, does not imply any reduction of h u m a n freedom. To Anselm, t h e n e c e s s i t y involved is only verbal a n d factual, b u t it does not cause a n y t h i n g to be t r u e concerning the future.

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A n t e c e d e n t necessity is stronger t h a n subsequent necessity. If t h e o c c u r r e n c e of a certain e v e n t is necessary in t e r m s of a n t e c e d e n t necessity, t h e n t h e necessity causes t h e e v e n t to occur. A n t e c e d e n t necessity can be described as a c a u s a l necessity. This distinction between two k i n d s of necessity is originally Aristotelian. In his Prior Analytics Aristotle clearly d r e w a distinction between absolute and relative necessity: F u r t h e r , it can be shown by t a k i n g examples of t e r m s t h a t the conclusion is necessary, not absolutely, but given certain conditions. [30b 32] In De Interpretatione the distinction between the two kinds of necessity is also expressed. It is very likely t h a t A n s e l m k n e w the Aristotelian distinction. In fact a Latin version of Aristotle's De Interpretatione along w i t h Boethius' c o m m e n t a r i e s was certainly at his disposal. Let us again consider the argument from divine foreknowledge to necessity of t h e future, and t h e s u b s e q u e n t conclusion t h a t there can be no h u m a n freedom. Now, w h a t is the A n s e l m i a n reaction to t h a t argument, when Np is u s e d as N in the a r g u m e n t ? It is obvious t h a t Anselm rejects the conclusion of t h e a r g u m e n t . According to h i m t h e r e is no i n s o l u b l e conflict b e t w e e n t h e doctrines of divine foreknowledge a n d h u m a n freedom. He says: It is clear from these considerations that there is no inconsistency in m a i n t a i n i n g both t h a t God foreknows all things a n d t h a t t h e r e are m a n y t h i n g s which, t h o u g h h a v i n g before t h e y occur the possibility of never occurring, do actually occur through free will. [Hopkins p.55] Therefore, according to Anselm there exists true propositions about t h e future such that their negations are also possible. The p r o p o s i t i o n F(1)p about t o m o r r o w ' s revolution is s u c h a proposition, as expressed in (2). It is clear that if t h e r e will be a

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revolution tomorrow, it cannot be possible - on the basis of subsequent necessity - that there is no revolution tomorrow. If it is possible t h a t t h e r e is no revolution tomorrow, it has to be on the basis of antecedent necessity, as we can see in (2). The acceptance of (2) clearly indicates t h a t Anselm rejected t h e classical a r g u m e n t from divine foreknowledge to necessity of the future. This being so Anselm had to reject at least is one of its premises. It seems clear that he in fact denied t h a t a n y t r u e s t a t e m e n t about the past is antecedently necessary. In Cur Deus Homo II.1 Anselm was discussing the Virgin's belief t h a t Christ was going to die of his own will: It is in accordance with this consequent and non-creative n e c e s s i t y t h a t since the belief or prophecy c o n c e r n i n g Christ, and according to which he was to die voluntarily, and not from necessity, was t r u e it was necessary t h a t these things should be. [Henry p.176] Here Anselm admits the truth of the proposition 'It was t r u e to say: God k n o w s t h a t Christ is going to die v o l u n t a r i l y ' According to Anselm, however, this proposition is necessary on t h e basis of s u b s e q u e n t necessity, but not on the basis of a n t e c e d e n t necessity. Let us clarify this position by u s i n g symbolic language. Let p stand for the proposition 'Christ dies voluntarily', D for the operator 'God knows that', and let x and y be suitable time units (mlmbers signifying for instance days or years). Consider now the statement

P(y)DF(x+y)p which can be read 'y years ago God knew that Christ was going to die voluntarily x+y years later'. In this case Anselm rejected

NpP(y)DF(x+y) p. It should be noted t h a t this position implies the rejection of the first of t h e p r e m i s e s in the so-called M a s t e r A r g u m e n t of

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Diodorus, wherein necessity (N) is understood in a completely general sense: (D1)

P(x)A ~ N P ( x ) A , where A is an arbitrary proposition.

Anselm obviously would reject (D1), i f N is interpreted as Np and A is the proposition DF(x+y)p. Nevertheless, it seems t h a t Anselm was willing to accept (D1) in some limited sense. In De Concordia he said: Now, the past event has a characteristic which neither the p r e s e n t nor the future event has. For w h a t is past can never become not-past as w h a t is present can become notpresent and as w h a t is going to occur without necessity can be not going to occur. [Hopkins p.52] Let us ponder this s t a t e m e n t carefully. The phrase 'what is going to occur .. can be not going to occur' shows us t h a t Anselm m u s t be talking about events, of which it is possible t h a t t h e y would not occur, even though t h e y actually do occur. This in t u r n shows us that the kind of possibility, respectively necessity, in question must be antecedent possibility. For the occurrence of an event entails its subsequent necessity, a n d hence, in t h a t sense it cannot be going not to occur. We r e p e a t the formula used earlier on to capture the kind of possibility at stake:

(2)

F(1)p A -NpF(1)p

Now let us apply these observations to the s t a t e m e n t 'what is past can never become not-past'. Given t h a t we are talking about a n t e c e d e n t necessity, it must be i n t e r p r e t e d in the following way: (DI')

P(x)A ~ NpP(x)A

Since Anselm rejected the general version of (D1), he m u s t have presupposed some constraints on the type of propositions

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w h i c h can be accepted as A in (DI'). In order to explain which c o n s t r a i n t is n a t u r a l from the A n s e l m i a n p o i n t of view one s h o u l d n o t e t h a t A n s e l m did not say t h a t a n y p a s t t e n s e proposition is necessary, but rather he made his assertion about p a s t events. We do not t h i n k t h a t he would accept God's forek n o w l e d g e in t h e p a s t as a past event, for according to h i m d i v i n e k n o w l e d g e is different from h u m a n knowledge. In De Concordia he says: We s h o u l d also u n d e r s t a n d t h a t like f o r e k n o w l e d g e , predestination is not properly attributed to God. For there is no before or after in God, but all things are p r e s e n t to Him at once. [Hopkins p.68] So according to A n s e l m t h e fact t h a t God k n e w s o m e t h i n g in t h e p a s t c a n n o t be p r o p e r l y characterised as a p a s t event. Following A n s e l m God's knowledge should be u n d e r s t o o d as timeless knowledge, but it is also true t h a t he a s s u m e d t h a t this divine knowledge can be transformed into the t e m p o r a l dimension. This seems to be how prophecy works.

THOMAS AQUINAS T h e idea of v i e w i n g God's k n o w l e d g e as t i m e l e s s w a s s u g g e s t e d by Boethius (480 - 524), and since t h e n it h a s been discussed m a n y times (see e.g. [Lucas 1989, p. 209 ff.]). D u r i n g t h e Middle ages it became common to appeal to this idea in a t t e m p t s at solving t h e problem of the logical t e n s i o n between t h e doctrines of h u m a n freedom and divine foreknowledge. The m e d i e v a l p h i l o s o p h e r who c o n t r i b u t e d t h e m o s t to t h e elaboration of this solution was T h o m a s Aquinas (1225-1274). I n Aquinas' opinion, God's eternity is timelessly s i m u l t a n e o u s w i t h all parts of time. He compared this view w i t h t h e relation b e t w e e n t h e c e n t e r a n d t h e circumference of a circle. T h e relation between t h e center and the circumference is t h e same

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all the w a y round; in a similar m a n n e r , God relates in the s a m e w a y to all times. In his own words: F u r t h e r m o r e , since the being of w h a t is eternal does n o t pass away, eternity is present in its presentiality to any time or i n s t a n t of time. We m a y see an example of sorts in t h e case of a circle. Although it is indivisible, it does not co-exist simultaneously with any other point as to position, since it is the order of position t h a t p r o d u c e s t h e continuity of t h e circumference. On the other h a n d , t h e center of t h e circle, which is no part of the circumference, is directly opposed to any given d e t e r m i n a t e point on t h e circumference. Hence, w h a t e v e r is found in any p a r t of time coexists with w h a t is eternal as being present to it, although with respect to some other t i m e it be past or future. [Summa contra gentiles I, c. 66] As Marilyn McCord Adams [1987 II, p.1121] has pointed out, A q u i n a s a p p a r e n t l y a s s u m e d n o t only t h a t God a n d His knowledge are timeless, but also t h a t time should be r e g a r d e d as a s y s t e m in which the basic r e l a t i o n s of succession a n d s i m u l t a n e i t y are given in a timeless way - owing to the fact t h a t t i m e is given to God in a t i m e l e s s way. B u t Aquinas also m a i n t a i n e d t h a t the divine knowledge can be transformed into t h e t e m p o r a l d i m e n s i o n by m e a n s of prophecies. In [Summa contra gentiles I, c. 67] he emphasised this possibility quoting t h e biblical s t a t e m e n t "I foretold thee of old, before they came to pass I told thee" [Isaias 48:5]. So the conceptual difference b e t w e e n past, p r e s e n t , and f u t u r e is r e l e v a n t only w h e n h u m a n s a r e involved, either as the subjects of cognition or as participants in c o m m u n i c a t i o n . F u r t h e r m o r e , A q u i n a s clearly stated t h a t a t e m p o r a l being cannot have any c e r t a i n knowledge of f u t u r e contingents at all. T h u s Aquinas w a s suggesting a distinction between t i m e as it is for temporal beings such as h u m a n s , a n d time as it is for God, who is eternal.

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WILLIAM OF OCKHAM O c k h a m discussed t h e problem of divine foreknowledge and h u m a n freedom in his w o r k Tractatus de praedestinatione et de futuris contingentibus [Ockham 1969]. He asserted t h a t God k n o w s all f u t u r e c o n t i n g e n t s , but he also m a i n t a i n e d t h a t h u m a n beings can choose between alternative possibilities. In his T r a c t a t u s he a r g u e d t h a t the doctrines of divine foreknowledge a n d h u m a n freedom are compatible. O c k h a m was aware t h a t the concept of c o m m u n i c a t i o n w a s essential to this discussion - especially, of course, the communication coming from God to h u m a n beings. He claimed t h a t God c a n c o m m u n i c a t e t h e t r u t h about t h e f u t u r e to us. N e v e r t h e l e s s , according to Ockham divine k n o w l e d g e regarding f u t u r e contingents does not imply t h a t t h e y are necessary. As an example O c k h a m considered the prophecy of Jonah: "Yet forty days, a n d N i n e v e h shall be overthrown" (Jonah ch. 3 v. 4). This prophecy is a communication from God about t h e future. Therefore, it might seem to follow that w h e n this prophecy h a s been proclaimed, t h e n t h e future d e s t r u c t i o n of N i n e v e h is necessary. B u t O c k h a m did not accept that. Instead, he m a d e room for h u m a n f r e e d o m in the face of t r u e prophecies by a s s u m i n g t h a t "all prophecies about f u t u r e c o n t i n g e n t s were conditionals" [Ockham 1969, p.44]. So according to O c k h a m we m u s t u n d e r s t a n d t h e prophecy of J o n a h as p r e s u p p o s i n g t h e condition 'unless the citizens of Nineveh repent'. Obviously, this is in fact exactly how t h e citizens of N i n e v e h u n d e r s t o o d the st a t e m e n t of Jonah! O c k h a m realised t h a t t h e revelation of t h e f u t u r e by m e a n s of an u n c o n d i t i o n a l s t a t e m e n t , c o m m u n i c a t e d from God to t h e prophet, is incompatible with the contingency of t h e prophecy. If God reveals the f u t u r e by means of unconditional statements, t h e n t h e future is inevitable, since the divine revelation m u s t be true. T h e concept of divine communication (revelation) m u s t be t a k e n into consideration, if t h e belief in divine foreknowledge is to be compatible w i t h t h e belief in t h e f r e e d o m of h u m a n actions. So Ockham u n d e r s t o o d that the compatibility can only

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be established w i t h i n a f r a m e w o r k which duly considers w h a t we in t h e introduction called sociotemporal notions. O c k h a m a t t e m p t e d to clarify t h e issue as much as possible. About the divine foreknowledge, he stated: ... t h e divine essence is an perfect, so clear, t h a t it is an p a s t a n d f u t u r e , so t h a t contradiction [involving such is false. [Ockham, 1969, p.50]

i n t u i t i v e cognition t h a t is so evident cognition of all t h i n g s it k n o w s w h i c h p a r t of a things] is true and w h i c h p a r t

However, he h a d to admit t h a t this is not very clear. In fact, he m a i n t a i n e d t h a t it is impossible to express clearly t h e w a y in which God knows f u t u r e contingents. He also had to conclude t h a t in g e n e r a l t h e divine k n o w l e d g e about the c o n t i n g e n t f u t u r e is inaccessible. God is able to communicate t h e t r u t h about t h e f u t u r e to us, but if God reveals t h e t r u t h about t h e f u t u r e by m e a n s of u n c o n d i t i o n a l s t a t e m e n t s , t h e f u t u r e s t a t e m e n t s c a n n o t be c o n t i n g e n t a n y m o r e . H e n c e , God's unconditional foreknowledge r e g a r d i n g future contingents is in principle n o t revealed, whereas conditionals can be c o m m u n i cated to the prophets. Even so, t h a t p a r t of divine foreknowledge about f u t u r e contingents which is not revealed m u s t also be considered as true according to Ockham. Richard of L a v e n h a m m a d e a r e m a r k a b l e effort to c a p t u r e and in a clear way to present t h e logical features of O c k h a m ' s system as opposed to Aristotle's solution. L a v e n h a m described some examples. In his view the propositions 'Antichrist will be', 'The Day of J u d g e m e n t will be', and 'The resurrection will be' are all about f u t u r e contingent facts. Then, L a v e n h a m maintained, t h e y are n e i t h e r d e t e r m i n a t e l y true nor d e t e r m i n a t e l y false on Aristotle's account. To s u b s t a n t i a t e that i n t e r p r e t a t i o n

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L a v e n h a m referred to the following consequentia, w h i c h he held to be the crucial claim of Aristotle's theory: 'If a proposition is about a future contingent fact, t h e n the proposition is not determinately true.' The Christian faith, however, goes against the acceptance of the consequentia, since a Christian person must believe t h a t God foreknows all future contingent facts. If Antichrist will indeed be, t h e n God knows that Antichrist will be, and for this reason it is determinately true that Antichrist will be. It will be recalled that Richard of L a v e n h a m enumerated four possible approaches to solving the a p p a r e n t conflict b e t w e e n God's foreknowledge and h u m a n freedom. He rejected the three classical opinions corresponding to the first, second, and third solutions, and t h e n formulated his own answer to t h e problem - w h i c h was also t h e opinion of m a n y of his contemporaries. L a v e n h a m held t h a t the doctrines of divine f o r e k n o w l e d g e and h u m a n freedom are compatible. H e considered two versions of the inference from God's prescience to the necessity of the future, and he explained why they should be rejected. Let us with L a v e n h a m consider the first version. The starting point is this example: q: 'The Day of Judgement will be.' The proposition q is regarded as being about a future contingent fact. The following consequentia can now be formed: (C) 'God knew from eternity t h a t q; therefore q'. This consequentia is obviously valid. The a r g u m e n t now proceeds by utihsing the principle: (P) 'A t r u e proposition about the past, the truth of w h i c h does not depend on the future, is necessary'.

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T h e principle is f o r m u l a t e d s u c h as to deal only w i t h sentences, which are g e n u i n e l y about the past. (P) seems to be equivalent to the idea t h a t w h a t has already been (or is now) the case cannot be undone. It m i g h t appear to follow from (P) that t h e antecedent of (C) is necessary, t h a t is, that we should have (A) 'Necessarily, God k n e w from eternity t h a t q'. (As you shall see, this is t h e step which L a v e n h a m rejected, but w h i c h was crucial to t h e argument.) We can now apply a well k n o w n principle of modal logic (medieval as well as modern): (M) 'If 'q follows from p' is a valid consequentia, and p is necessary, then q will also be necessary'. Hence, the consequent of (C) will also be necessary. In short, the a r g u m e n t goes as follows: (1) (2) (3) (4)

'The Day of J u d g m e n t will be'. 'God knew t h a t the Day of J u d g m e n t will be'. 'It is necessary t h a t God knew that the Day of J u d g m e n t will be'. 'It is necessary t h a t the Day of J u d g e m e n t will be'.

B u t L a v e n h a m rejected the inference from (2) to (3). He c l a i m e d t h a t (P) c a n n o t be u s e d in order to j u s t i f y this inference, precisely b e c a u s e t h e t r u t h of (2) d e p e n d s on the future. For if the Day of J u d g e m e n t will not be, t h e n (2) m u s t also be false! L a v e n h a m ' s answer to this a r g u m e n t obviously d e p e n d s on O c k h a m ' s view in De praedestinatione et de praescientia Dei. In considering the other version of the a r g u m e n t , L a v e n h a m u s e d the examples: p:

g:

'Antichrist will be'. 'God wills t h a t Antichrist will be'.

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W i t h i n t h e f r a m e w o r k of t h e dogmas of the Christian faith, t h e consequentia from g to p is clearly valid So i f g can be proved to be necessary, t h e n p will also be necessary in virtue of (M). To see h o w one m i g h t go on to e s t a b l i s h t h e necessity of g, L a v e n h a m investigated t h e following syllogism:

Major premise: g is u n c h a n g e a b l y known to God. Minor premise: W h a t is unchangeably k n o w n to God is necessarily known to GOd. Conclusion: g is necessarily known to God. In this syllogism the m i n o r premise is valid on grounds of t h e principle t h a t w h a t e v e r is unchangeable is also necessary. T h e m a j o r p r e m i s e is s h o w n by a proof ad a b s u r d u m : if g w e r e k n o w n to God, b u t n o t u n c h a n g e a b l y k n o w n to him, t h e n g would be changeably known to God. But this is absurd. T h u s it is p r o v e d t h a t g is necessarily known to God. But more t h a n t h a t is n e e d e d , since it s h o u l d be d e m o n s t r a t e d t h a t g by itself is n e c e s s a r y . It s e e m s t h a t L a v e n h a m forgot to m e n t i o n t h e following premise: 'What is necessarily k n o w n to God is necessary'. However, there is no doubt that this premise is presupposed in his r e c o n s t r u c t i o n of t h e a r g u m e n t . By m e a n s of this extra p r e m i s e it is easily shown t h a t g is itself necessary, and conseque n t l y p is also necessary. T h u s goes the second version of t h e a r g u m e n t as r e n d e r e d by L a v e n h a m . But L a v e n h a m h i m s e l f of course rejected the a r g u m e n t . He pointed out t h a t the m i n o r p r e m i s e of the syllogism above is not valid. For we might j u s t as well a s s u m e t h a t w h a t is u n c h a n g e a b l y k n o w n to God could after all have been different, and therefore it does not have to be necessary! It can t h e r e f o r e be said t h a t in a p p e a l i n g to this m i n o r premise, the a r g u m e n t was in a sense presupposing t h a t w h i c h it w a s going to d e m o n s t r a t e , a fact w h i c h L a v e n h a m a p p a r e n t l y reahsed. As we h a v e s e e n L a v e n h a m i d e n t i f i e d four possible a p p r o a c h e s to solving t h e tension between our two a p p a r e n t l y

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conflicting dogmas. However, these approaches were not t h e only ones to be considered by medieval philosophers. At least one i m p o r t a n t position seems to have been left out. That is the opinion of St. Thomas Aquinas and others who claimed that the knowledge of God abstracts from t h e difference between past, present and future. According to this view it might be said t h a t all events are 'always' present to God - in an atemporal sense of 'always'! It was m e n t i o n e d e a r l i e r t h a t Leibniz worked out a metaphysics of time, which from a systematical point of view is v e r y similar to the thoughts of Anselm and Ockham. We shall now for a passage leave the Middle Ages in order to examine his system.

LEIBNIZ Leibniz accepted the doctrine of divine foreknowledge as well as t h a t of h u m a n freedom. He of course knew the standard argum e n t s that can be constructed in order to prove the incompatibility of the two doctrines, but he claimed that those arguments were invalid: Nor does t h e foreknowledge or preordination of God impose necessity even though it is also infallible. For God has seen things in a n ideal series of possibles, such as they were to be, and a m o n g t h e m m a n freely sinning. By seeing the existence of this series He did not change the n a t u r e of things, nor did h e m a k e w h a t is contingent necessary. [Rescher 1967 p.39] Leibniz's central idea was that God had chosen the best of all possible worlds and made it actual. But in actualising the creatures of t h a t world He did not change their free natures. So it is not necessary for a m a n to do that which he will in fact be doing according to t h e foreknowledge of God. It would have been

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possible for h i m to m a k e different decisions leading to different acts. But if t h i s is so, how can t h e foreknowledge of God be infallible? Leibniz' solution to this p r o b l e m is very close to Anselm's solution. Like Anselm, Leibniz introduced a distinction between two kinds of necessity: For we m u s t distinguish between an absolute and a hypothetical necessity. [Alexander p.56] These two concepts of necessity correspond exactly to Anselm's a n t e c e d e n t a n d subsequent necessity, respectively. W i t h respect to t h e a r g u m e n t from God's foreknowledge to necessity of the future, Leibniz would have no objection so long as t h e necessity in q u e s t i o n is t h e hypothetical necessity - j u s t like Anselm accepted the a r g u m e n t w h e n i n t e r p r e t e d as r e f e r r i n g to succedent necessity. Leibniz observed that Hypothetical necessity is that, w h i c h the supposition or hypothesis of God's foresight and pre-ordination lays u p o n future contingents. [Alexander p.56] This s t a t e m e n t is equivalent to Anselm's 'What will be, necessarily will be', i.e.

F(x)p ~ NsF(x)p - w h e r e Ns m a y be i n t e r p r e t e d as s u b s e q u e n t as well as h y p o t h e t i c a l necessity. It is i m p o r t a n t to realise t h a t t h i s s t a t e m e n t does not provide any i n f o r m a t i o n at all about t h e n u m b e r of f u t u r e possibilities. Perhaps this is more easily seen, if we once a g a i n allow ourselves to i l l u s t r a t e t h e t h o u g h t s involved by m e a n s of t h e modern notion of 'branching time'. For ease of reference, we here repeat t h e illustration w h i c h we used in the previous discussion of the Master Argument:

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future b

future c no

future d

Suppose that, say, future b is the course of events t h a t is actually going to be. T h e n to assert hypothetical necessity of some f u t u r e event E can be interpreted as saying simply t h a t t h e event E 'is on t h e b-branch'. But t h a t is exactly t h e same as simply saying t h a t E is going to occur, which m a k e s t h e f o r m u l a above a s i m p l e tautology. And in fact it is very likely t h a t Leibniz considered this type of s t a t e m e n t as outright tautological, t h a t is, as stating w h a t we would now express with t h e formula

N(F(x)p ~ F(x)p) It should be obvious, then, t h a t this kind of s t a t e m e n t does not convey any information as to the n u m b e r of possible futures. On t h e other hand, Leibniz would certainly reject the validity of t h e formula

F(x)p ~ NF(x)p, w h e r e N represents necessity in general, t h a t is, it also includes absolute necessity. In terms of the branching time model above, a b s o l u t e n e c e s s i t y in effect q u a n t i f i e s over all b r a n c h e s e x p a n d i n g from t h e given 'now'. T h a t rejection is a consequence of his refusal to accept 'pastnecessitation'- (D1) - for arbitrary statements. A l t h o u g h Leibniz rejected t h e generalised version of (D1), he w a s willing to accept a limited version of t h a t principle. In his Theodicy [II §

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170] he explained that there is a difference in modality between the past a n d the future. For while it is not possible to cause a past event, it is now possible to cause some of the future events. Therefore, i f p is a s t a t e m e n t variable such t h a t P(x)p is genuinely about the past, it follows t h a t P(x)p is also necessary in the general sense, i.e.

P(x)p ~ NP(x)p Regarding t h e contingent future there are statement variables, e.g. the variable q, such that it is possible to make F(x)q false, in spite of the fact that it will be true - that is

F(x)q A M-F(x)q This m e a n s t h a t while there is no alternative to the actual past, t h e r e a r e alternatives to the future. These a l t e r n a t i v e futures correspond to the Leibnizian concept of possible worlds. The connection which Leibniz established between modality and the m u l t i t u d e of possible futures is the one which is also commonly u s e d within present-day modal logic and possible world semantics: w h a t is necessary is t h a t which holds in all possible futures, and what is possible is t h a t which holds in at least one possible future. The concept of modality involved here is clearly of a temporal nature. This m e a n s t h a t a proposition which describes some event is necessary (in the absolute sense) if and only if t h e proposition follows from a proposition about the past or the present. It seems that the implication in question is a kind of causal implication. Thus, an alternative formulation would be to say t h a t a future event is necessary if and only it is unchangeably caused by present or past events. It follows t h a t in terms of our branching time model a necessary event m u s t be true in all future branches. The possible worlds of Leibniz represent the ways in which the entire history might have been different from what it is. Therefore it seems to be reasonable to identify a possible world with a possible history.

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Leibniz claimed t h a t among all the possible worlds God has chosen the best one: It follows from the supreme perfection of God t h a t he has chosen the best possible plan in producing t h e universe, a plan which combines the greatest variety together with the g r e a t e s t order ... For as all possible things have a claim to existence in God's understanding in proportion to their perfections, the result of all these claims must be t h e most perfect actual world which is possible. Without this it would be impossible to give a reason why things have gone as t h e y have rather than otherwise. [Leibniz 1969, p.639] Leibniz' idea of possible worlds can in the context of temporal logic be viewed as a n u m b e r of sequences of events. In each of t h e s e chronicles the f u t u r e events follow logically from the present. In this connection it should be noted t h a t all relevant information about the present also includes information about all past events. Leibniz formulated his position as follows: For everything has been regulated in things, once and for all, with as much order and agreement as possible; the sup r e m e wisdom and goodness cannot act except w i t h perfect h a r m o n y . The present is great with the future; the future could be read in the past; the distant is expressed in the near. One could learn the beauty of the u n i v e r s e in each soul if one could unravel all that is rolled up in it but t h a t develops perceptibly only with time. [Leibniz 1969, p.640] It m a y seem that Leibniz in this way left no room for the idea of f r e e choice. T h a t would, however, be a n e r r o n e o u s conclusion. In dealing with the question of h u m a n freedom he stated: Since the individual concept of every person includes once and for all everything which can ever happen to him, one sees in it a priori proofs or reasons for the t r u t h s of each event and why one has happened rather t h a n another, but

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t h e s e t r u t h s , however certain, are nevertheless contingent, being based on t h e free will of God and of creatures. It is t r u e t h a t their choice always has its reasons, b u t these incline without necessitating. [p.310] Leibniz took t h e person of Julius Caesar as an example. The concept of this person involves already at his birth (and in fact, ever before) a f u t u r e crossing of t h e Rubicon, a f u t u r e dictatorship etc. Nevertheless, the assumption t h a t t h e person who w a s crossing the Rubicon on a certain day in 52 B.C., a n d who h a s also done exactly e v e r y t h i n g Caesar did before t h e crossing, will choose not to be a dictator, does not imply a contradiction. Therefore Caesar's becoming a dictator is not necessary, b u t merely certain as foreseen by God.

T H E ANSELM-OCKHAM-LEIBNIZ S O L U T I O N It is evident t h a t the solutions presented by Anselm, Ockham, a n d Leibniz, h a v e very m u c h in common. In spite of m i n o r d i f f e r e n c e s it is m e a n i n g f u l to s p e a k about ' t h e A n s e l m Ockham-Leibniz solution'. T h e analysis of the relation between the d o g m a s of h u m a n freedom a n d God's omniscience, which led to t h e 'the AnselmOckham-Leibniz solution', has proved to be very i m p o r t a n t also for t h e d e v e l o p m e n t of m o d e r n tense-logic. W i t h i n the m o d e r n discipline t h e problems concerning d e t e r m i n i s m a n d t h e s t a t u s of t h e c o n t i n g e n t f u t u r e are n o r m a l l y n o t t h o u g h t of in theological t e r m s , but r a t h e r it is discussed at a purely tempomodal level: W h a t does it m e a n for an event E to t a k e place? H o w s h a l l we solve t h e p r o b l e m s of d e t e r m i n i s m v e r s u s i n d e t e r m i n i s m ? W h a t is t h e relation b e t w e e n t i m e a n d modality in general? In p a r t 2 a n d 3 we intend to examine some different theories for t h e f u t u r e operator in an i n d e t e r m i n i s t i c tense-logic, t h e o r i e s w h i c h form m o d e r n c o u n t e r p a r t s of t h e v a r i o u s m e d i e v a l approaches we have been discussing. In this m o d e r n context, we shall a r g u e t h a t 'the A n s e l m - O c k h a m - L e i b n i z

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solution' is consistent as well as plausible. We shall also qualify our own h e a d i n g of 'the Anselm-Ockham-Leibniz solution' by showing t h a t from a strictly formal point of view, the solution can be differentiated into two slightly different systems. This is so because Leibniz' ideas can give rise to a model of time which differs slightly from w h a t could be called an Ockhamistic model of time.

1.10. THE DOWNF ,LT OF 1V[EDIEVAL TENSE-LOGIC In a short but thought-provoking sketch of the history of logic w i t h a special view to tense-logic, A. N. Prior has argued t h a t the central t e n e t s of Medieval logic with respect to time a n d tense can be summarised in the following way: (i) t e n s e distinctions are a proper subject of logical reflection, and (ii) w h a t is true at one time is in many cases false at another time, and vice versa. [1957a, p.104] Prior a d m i t t e d t h a t he had not actually documented t h e s e claims in his sketch. However, as we have seen in the preceding chapters t h e r e are m a n y concrete examples in support of his claims. Prior's s t a t e m e n t can be made more precise, though, by mentioning its two main points in the reverse order, since (ii) can be seen as a n a t u r a l presupposition for (i). One can h a r d l y imagine a logical system based on the first claim which rejects the second. That is to say: if, in accordance with a rejection of (ii), logic is to t r e a t timeless truths only, then it seems rather futile to establish theories for tensed propositions. On t h e other h a n d there is no inconsistency in recognising t h a t t h e t r u t h value of propositions can in principle vary with time, but finding w o r k on this subject uninteresting for logic. And in fact the waning of t e n s e logic began with a gradual loss of i n t e r e s t in t e m p o r a l structures, t h a t is, it was (i) which was first abandoned by the different schools of logic, and (ii) came to be rejected only afterwards. We shall now sketch a few m a j o r points of this gradual transformation of logic as a discipline. The downfall of Scholasticism was a process unfolding w i t h the rise of the Renaissance Humanism. One of the losses was the logical studies practised within the Scholastic discipline of dialectics. The Scholastic disputation, which can be seen as a method of unravelling logical intricacies, came to be particularly despised. It was perceived as expressive of an a b s t r a c t philosophy, which could not lead to anything constructive, a n d which did not have any worthwhile qualities in its own right. 109

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The struggle against the Scholastic tradition initiated by the dawning Renaissance lasted for at least a century, although E. J. Ashworth [1982] is probably correct in suggesting t h a t the most significant phase of the battle took place in the years around 1530. In t h e e a r l y period some of the major critics of Scholasticism w e r e Laurentius Valla (1407-1457) and Rudolf Agricola (1443-1485). In the footsteps of Cicero and Quintilian, Valla and Agricola wanted to study and define logic within the discipline of rhetoric. For instance Agricola defined logic as "the art of expressing yourself convincingly of anything" [Dumitriu p.232]. This considerable change in the conception of logic was to a large extent a reaction against the perceived m a l t r e a t m e n t of the Latin language by Scholastic logicians. In the decisive phase of the strife the most i m p o r t a n t h u m a n i s t s were J u a n Luis Vives (1493-1540) and Peter Ramus (1515-1572), who also introduced the new ' h u m a n i s t logic', based on the same ideas as those of Valla and Agricola. Vives w e n t to the university of Paris in 1509 to study w i t h i n the Scholastic tradition, but w h e n he left it again in 1512, he was totally convinced that Scholastic logic had very little going for it, if indeed a n y t h i n g at all. He especially reacted against the sophisticated, almost artificial, l a n g u a g e of the Scholastic logicians. That language was actually semi-artificial in m u c h the same m a n n e r as the verbiage of present-day philosophical logic, as seen also in the preceding pages - think of phrases such as 'it will always have been the case that...', etc. However, modern logicians do not have to rely on this kind of language, because we have actual symbolic logic at our disposal. But the Scholastics h a d no other m e a n s t h a n this q u a s i - f o r m a l language in order to make their ideas precise, and for t h a t reason it became an i m p o r t a n t and pervasive p a r t of their logical tradition. Against t h a t tradition Vives maintained, like Valla and Agricola before him, t h a t contemporary logicians ought to stick to ordinary language. In this connection they fielded the extra argument t h a t it had been possible for Aristotle and Cicero to describe their logical rules in everyday Greek and Latin. In passing it is w o r t h noting t h a t there are striking similarities between the H u m a n i s t criticism of Scholastic logic,

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and t h e m o d e m 'pragmatical' opposition against t h e project of formal semantics for n a t u r a l language. In both cases t h e r e is a r e a c t i o n a g a i n s t t h e (real or i m a g i n e d ) r e g i m e n t a t i o n of n a t u r a l l a n g u a g e , a n d a t u r n to p r a g m a t i c a l p h e n o m e n a instead. Especially the downfall of Logical Atomism in t h e face of O r d i n a r y L a n g u a g e P h i l o s o p h y in t h e F o u r t i e s , w h i c h is recorded in [Urmson 1967], m a k e s out a striking parallel. T h e R e n a i s s a n c e perception of t h e Scholastic t r a d i t i o n is reflected in strong terms in Vives' Adversus pseudodialecticos of 1520, in which passages like this one can be found: One is n o w a d a y s a s h a m e d of s p e a k i n g of 'incipit' a n d 'desinit'. Who, by chance, passed on this subtle rigourism, these futile examples, these inane [examples]. Wives p.59] In this arrogant way Vives dismissed t h e a t t e m p t s of t h e previous centuries to build a conceptual apparatus, which a m o n g s t other t h i n g s should provide a n account of t e m p o r a l c o n t i n u i t y and limits. He also ridiculed Scholastic distinctions b e t w e e n propositions such as: (I) 'Antichristus qui fuit erit' (Antichrist who was, will be) and (II) 'Antichristus erit, qui fuit' (Antichrist will be t h e one, who was). According to the Scholastic analysis t h e first proposition is false, b e c a u s e it implies t h a t A n t i c h r i s t h a s a l r e a d y lived, whereas the second one is considered to be true, since Antichrist - according to the Bible - will come, whereafter he will be t h e one who was! For a m o d e r n tense-logician this is familiar as t h e d i s t i n c t i o n b e t w e e n (Fq A Pq) and FPq, respectively. Vives r e g a r d e d this discussion primarily as an example of bad Latin, and did not realise t h a t t h e r e was indeed a significant logical difference underlying the discussion. Vives w a s greatly a p p l a u d e d for his endeavours. E r a s m u s of Rotterdam, for example, wrote t h a t Vives was more suited t h a n

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any other person for the task of refuting Scholastic logic, due to his previous service for several years within t h a t tradition. It is possible, however, that the Scholastic tradition was also to some e x t e n t d i s s o l v i n g from within. A s h w o r t h [1982] h a s d i s c u s s e d w h e t h e r t h e r e w a s a general recognition of t h e inadequacy of t h e i r logic a m o n g the better k n o w n logicians at t h e University of Paris. T h a t question m u s t still be said to be open, a l t h o u g h it is obvious t h a t in their work t h e s e logicians were g e t t i n g close to the limit of w h a t linguistic formulations could bear in order to gain more insight by logical analysis. In an ironical m a n n e r , this is the very same problem which also gave rise to Vives' type of accusation, n a m e l y t h a t of u n n e c e s s a r y sophistry and m a l t r e a t m e n t of Latin. It is h a r d l y possible to find any real progress, or a n y real novelties, in t h e modified logic of the Renaissance, as Robert A d a m s o n [1911] h a s r e m a r k e d . The same is t r u e of t h e only logician of 'the C r u n c h period', Peter Ramus, whose works were very p o p u l a r in t h e 16th and 17th centuries. He became t h e m a i n p r o p o n e n t of t h e so-called h u m a n i s t logic. The result of t h e leading Renaissance logician's work was not a recreation of logic, b u t an a m p u t a t i o n . The e m p h a s i s on rhetoric and t h e s i g n i f i c a n c e a t t r i b u t e d to R o m a n logic, w h i c h m a i n l y accentuated elegance and simplicity, t u r n e d logic into a science of t h e art of a r g u m e n t a t i o n , or an 'ars docendi' as seen by M e l a n c h t h o n (1497-1560). As a consequence of Humanistic logic the temporal dimensions of logic became progressively more neglected. D u r i n g t h e 16th c e n t u r y i n t e r e s t in t e m p o r a l c o n s t r u c t i o n s s u c h as those discussed in the previous sections nearly disappeared, although a few logicians c o n t i n u e d to w o r k along t h e lines of t h e Scholastic t r a d i t i o n (see [ T r e n t m a n 1982]). 'Ampliatio' was among the temporal constructions which attracted the a t t e n t i o n of l o g i c i a n s for t h e longest period of t i m e (see [Ashworth 1982]). By t h e 1 7 t h c e n t u r y , t h e i n t e r e s t in s u c h t e m p o r a l c o n s t r u c t i o n s h a d n e a r l y d i s a p p e a r e d a m o n g logicians. Nevertheless t h e r e were a n u m b e r of logicians who felt t h a t the t r u t h value of propositions m u s t in principle be looked upon as

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varying with time, as N u c h e l m a n [1980 p.131 if] has shown in his thorough analysis. But in the famous and representative work La Logique ou l'art de penser, which was first published in 1662, Antoine Arnauld (1612-1694) and Pierre Nicole (16251695) presented a persuasive and coherent logical theory, in which little room was left for the medieval approach to the logic of tensed propositions. The idea of temporally varying t r u t h was not categorically rejected in the text, but on the other h a n d it just did not play any r61e. In the following chapter we shall briefly consider the new u n d e r s t a n d i n g of logic developed by Leibniz and others.

1.11. L O G I C AS A T I M E I . ~ S S S C I E N C E In his thorough history of logic Anton Dumitriu [1977 p. 11 ff] puts m u c h e m p h a s i s on the significance of Francis Bacon (1561-1626). In Bacon's attempt to establish e x p e r i m e n t a l science and define its methods, he presented logic as a tool to be applied within the respective scientific disciplines, as well as a more general tool for analysing the conditions of each discipline. Thus Bacon emphasised the rSle of logic as methodology. This emphasis would eventually lead to the dissociation of logic from language, that very connection which in the Scholastic times had inter alia legitimised the study of propositions with time reference. Dumitriu attributes almost the same importance to the rSle of Ren~ Descartes (1596-1650) within post-Scholastic logic. In Descartes' methodology, mathematics becomes a model for all of science. Since mathematical t r u t h s are in general considered to be independent of and without reference to time, Descartes' point of view also seemed to motivate t h a t time be neglected in logic. One of the great Cartesians, Malebranche (1638-1715) wrote, in his Recherche de la v~rit~: La v~rit~ est i n c r ~ e , immuable, ~ternelle, au-dessus de toutes choses. [Risse 1970, p.ll0] Let us now consider Gottfried Wilhelm Leibniz (1646-1716), who was of p a r a m o u n t importance in the history of logic. Leibniz must be considered to be the founder of symbolic logic. He is also one of the logicians responsible for the definitive a b a n d o n m e n t of tense-logic. In 1679 he presented a subjectpredicate logic, in which the study of the copula (English 'be', or Latin 'esse') was not significant (see [Leibniz 1969 p.235]). In so doing he effectively distanced himself from a considerable n u m b e r of the subjects with which the Scholastic logicians h a d been concerned. Leibniz was clearly influenced by P e t e r R a m u s a n d Melanchthon (see [Leibniz 1969 p.464 & p.471]), and followed them in finding Scholastic logic inadequate. He also mentioned the logicians J a c o b u s Zabrella (1533-1599) and J o a c h i m 114

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J u n g i u s (1587-1657) as persons who revealed t h e inadequacy of Scholastic logic. Nevertheless, Leibniz' approach to logic w a s very different from t h e methods k n o w n within ' h u m a n i s t logic'. He w a n t e d to m a t h e m a t i c i z e logic, and to construct a calculus w h i c h could be u s e d as a m a t h e m a t i c a l description of logical s t r u c t u r e s a n d inference. In this endeavour, he left out t h e copula and its conjugation, as well as other auxiliaries, for instance m o d a l ones. In Leibniz' symbolic logic it is implicitly understood t h a t a copula has to appear in the present tense (and an o m n i t e m p o r a l sense). An i m p o r t a n t person in the development was Gabriel Wagner, who in 1696 settled in Hamburg,. Here he began to publish t h e journal 'Vernunfttibungen', in which he led a bitter fight against c o n t e m p o r a r y Scholasticism, r o u n d l y a t t a c k i n g logic. This p r o m p t e d Leibniz, in a letter to Wagner in t h e s a m e year, to defend t h e discipline as extremely valuable (see [Leibniz 1969 p.462 fi]). In his l e t t e r Leibniz tried to d e t e r m i n e w h a t exactly logic was. He established that logic ought to be looked upon as an art, which can m a k e t h e knowledgeable m o r e secure. T h i s h a p p e n s n o t only by e v a l u a t i n g t h e t r u t h v a l u e s of given propositions, b u t also by logical investigations a n d m e t h o d s l e a d i n g to n e w a n d h i t h e r t o h i d d e n t r u t h s . Leibniz t h u s regarded logic as a science of t h o u g h t and method. In his a t t e m p t to d e t e r m i n e w h a t logic is, Leibniz pointed to it as a science i m p o r t a n t for all kinds of intellectual work. In his opinion, it h a d to be considered t h e key to all i n t e l l e c t u a l e v a l u a t i o n s , a n d h e n c e to all of science. Since W a g n e r w a s unwilling to draw this conclusion he m u s t either have disagreed w i t h Leibniz' definition of logic, or else he m u s t have considered t h e state of logic as well as its results so far to be p r e t t y poor. Some of Wagner's r e m a r k s indicate t h a t he did indeed hold t h e latter, and actually, so did Leibniz himself, at least to some extent. He recognised t h a t logic at present was 'but a shadow' of w h a t he w a n t e d it to be. But even t h o u g h he t h u s partly agreed w i t h Wagner, he t h o u g h t it w a s wrong to reject t h e entire logical t r a d i t i o n o u t of h a n d , since he c o n s i d e r e d m u c h traditional logic as both thought-provoking and useful.

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T h e logic which Leibniz h i m s e l f w a n t e d to p r o m o t e w a s a timeless logic. But t h a t is not to say t h a t Leibniz did not t a k e an interest in questions involving time, and in fact, questions which we would call tense logical. Indeed, we have argued t h a t Leibniz' philosophy regarding the relation between God's foreknowledge and m a n ' s free will is in t u n e w i t h Medieval as well as m o d e r n tense-logic. N e v e r t h e l e s s , in his actual logical works Leibniz f o r m u l a t e d his logic in such a way t h a t it did not take time into account. We shall h e r e suggest w h a t we believe to be the main r e a s o n s for Leibniz' deliberate neglect of time within his logic. Firstly, it was Leibniz' ambition to bring m a t h e m a t i c s a n d logic close t o g e t h e r . Leibniz a d m i t t e d t h a t m a t h e m a t i c s is identical w i t h logic, but he m a i n t a i n e d t h a t it is 'one of t h e eldest sons' of logic. In particular, he e m p h a s i s e d what he considered to be a n i m p o r t a n t discovery, n a m e l y t h e insight t h a t m a n y of the a d v a n t a g e o u s features of algebra can be traced back to t h e a u g u s t science of logic! (See [Leibniz 1969 p.469 ffl) Given Leibniz' clear interest in t h e relation between m a t h e m a t i c s and logic, a n d especially in the use of logic w i t h i n mathematics, it is easy to u n d e r s t a n d t h a t he would favour the timeless v a r i e t y of truth. A s e c o n d a n d m o r e p h i l o s o p h i c a l reason for Leibniz' preference for a tenseless logic can be found in his concept of the i n d i v i d u a l substance. The complete or perfect n o t i o n of an individual substance on his view includes everything w h i c h can be said of the substance with respect to past, present, a n d future (see [Leibniz 1969 p.268 f/I). We have already d i s c u s s e d t h e example of the concept of J u l i u s Caesar. Leibniz also m e n t i o n e d the Apostles Peter and J u d a s as examples: it is i n h e r e n t in t h e complete notion or concept of P e t e r t h a t he was going to d e n y Jesus, a n d likewise is i n h e r e n t in the complete notion of J u d a s t h a t h e w a s lost. T h e r e f o r e , a c c o r d i n g to L e i b n i z a n y a r g u m e n t a t i o n or i n v e s t i g a t i o n concerning complete n o t i o n s does n o t need to m a k e any reference to time. Predicates belong, or do n o t belong, to the complete notion irrespective of t e m p o r a l relations. It should however be noted t h a t the temporal aspects

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are clearly incorporated into the very formulation of t h e concepts! Leibniz' conception of logic was continued by people like Wolff and the Dane Jens Krai~, who in their books on logic referred to m a t h e m a t i c a l examples, a l t h o u g h they did not use a c t u a l symbolic logic. Examples of actual symbolic logic were rare in the 18th century and the first part of 19th century. I m p o r t a n t exceptions w e r e J o h a n Heinrich Lambert, Gottfried Ploucquet, Leonard Euler, and J. D. Gergonne [Kneale 1962 p.348 ft.]. But the typical attitude of the time was that there was no need for a further development of logic. Kant, notably, had in the preface to Critique of Pure Reason stated t h a t logic had been unable to m a k e a n y s u b s t a n t i a l advances since Aristotle, and logic appeared to h i m as "allem A n s e h e n nach geschlossen u n d vollendet zu sein" [2. ed. p.VIII]. K a n t defined logic as the discipline concerned w i t h the formal rules for any k i n d of thinking ("die formalen Regeln alles Denken"). In his opinion, the study of these rules, i.e. logic as such, had already been completed by Aristotle. This of course did not exclude t h a t there could still be work to be done on the foundation of logic, in order to be able to formulate the conditions for 'pure reason' - this was exactly w h a t Kant himself was doing. But the set of formal rules of thought, t h a t is, logic itself, was considered to have been already e x h a u s t i v e l y determined. Therefore a t t e n t i o n was turned a w a y from a supposedly futile study of actual logic, and instead directed towards a discussion of the application of logic, primarily w i t h i n reasoning and scientific methodology. Thus logical studies were concerned with general truths, and logic became timeless.

2.1. THE 19TH CENTURY AND B O O L E A N LOGIC In t h e 19th c e n t u r y t e m p o r a l distinctions were u s u a l l y considered to be irrelevant to logic. The timeless character of logic was often a r g u e d for by reference to philosophy of science: the p r i m a r y i n t e r e s t s of science should be timeless (or omni-temporal); a n d since logic was t h o u g h t to be a tool for sciences, it too had to follow suit. T h u s Alexander P f ' ~ d e r , who worked within the phenomenological tradition, asserted t h a t the non-historical sciences h a d exclusively to aim at t r u e propositions described w i t h "eine d u r c h alle Zeiten h i n d u r c h g e h e n d e G e g e n w a r t " [1921, p.269] (app. "a p r e s e n t s t r e t c h i n g t h r o u g h all times"). Pf~inder w a s by no m e a n s t h e first person to state such views. On t h e c o n t r a r y , t h e y m u s t be said to have been p r e v a l e n t d u r i n g t h e previous two centuries. The perception of logic as t i m e l e s s w a s still a c o m m o n - p l a c e a r o u n d the t u r n of t h e century. In the following we shall as one example show how this conception was expressed by one Danish logician of t h a t period. K. K r o m a n was one of the most p r o m i n e n t logicians of his day in D e n m a r k . His work on logic entitled T~enke og Sj~elel~ere (app. " T e x t b o o k on T h o u g h t a n d t h e Soul") [1899] w a s concerned w i t h the n a t u r e of scientific statements: [To say] t h a t N. N. has a horse, which h a d a fall yesterday, is n o t [to make] a scientific s t a t e m e n t . But the observation t h a t t h e horse is a solid u n g u l a t e hoofed m a m m a l , which is n o r m a l l y used as a domestic animal, is on the other h a n d a scientific statement. [1899, p.5] (our translation) K r o m a n ' s u n d e r s t a n d i n g of t h e r61e of logic as a tool for t h e other sciences is apparent in the following quotes: ...it is w i t h the aid of logic, t h a t we build any other science... Logic is ... the science of correct thinking. [1899, p.5] (our translation)

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T h u s logic is, according to K r o m a n as well as m o s t of his c o n t e m p o r a r i e s , a tool to be u s e d in connection w i t h t h e f o r m u l a t i o n of any science. The u n d e r l y i n g a s s u m p t i o n is t h a t scientific theories are essentially s y s t e m s of propositions. Logic i t s e l f c o n s i s t s of t h r e e p a r t s since it s t u d i e s c o n c e p t s , propositions, and inferences: The basic items of logic are.., concepts, and w i t h concepts the first p a r t of logic is thus concerned [p.15]; Words refined into concepts are not sufficient as a basis for logic, b u t logic must, whilst following everyday language, also seek a refined expression for t h e life and m o v e m e n t of c o n c e p t u a l c o n t e n t , c o r r e s p o n d i n g to t h e [ e v e r y d a y l a n g u a g e ] s e n t e n c e ; this e x p r e s s i o n it f i n d s in t h e proposition. [p.22]; •..it (is) the t a s k of logic to teach us how we form inferences, or how we should correctly derive n e w propositions from given ones. [p.23] (our translation) This trichotomy is historical and obvious in Scholastic logic, too. There, however, temporal considerations were interwoven w i t h all t h r e e elements. But Kroman only discussed t h e r61e of time explicitly in connection with the formulation of propositions: S c i e n c e is f u r t h e r m o r e c o n c e r n e d e s s e n t i a l l y w i t h conditions and activities of a general character, conditions a n d activities which are not d e p e n d i n g on a n y specific p r e s e n t m o m e n t of time, but w h i c h are of a lasting validity; hence t h e different 'tenses' of t h e verb would as a rule also be superfluous, and we could in general abide by the present tense only. [p.31-32] (our translation) T h e r e is t h u s no room for c o n s i d e r a t i o n s c o n c e r n i n g t e n s e inflected propositions in Kroman's logic. The reason for this is clearly t h a t logic is to serve science as K r o m a n a n d his contemporaries saw it. Here time comes into the picture, partly in c o n n e c t i o n w i t h t h e basis for prediction, a n d p a r t l y in connection w i t h his ideas on the development of knowledge.

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Kroman's u n d e r s t a n d i n g of logic as a timeless discipline was continued in Hoffding's logic, in which the copula of propositions is seen to be always in t h e present tense: 'is'. H¢ffding did allow t h a t t i m e relations c a n be considered, b u t only w h e n t h e expression indicating t i m e occurs either as t h e logical subject or as a predicate. Not u n t i l Jorgen Jorgensen, who e m p h a s i s e d Boole's i n c l u s i o n of t i m e and L u k a s i e w i c z ' s a n a l y s i s of propositions relating to t h e future, did time-logic fred a m o d e s t place w i t h i n Danish logic. Kroman's view on logic was certainly typical of his period, not only in D e n m a r k b u t in Western logic as a whole. The most i m p o r t a n t logician of t h e n i n e t e e n t h century w a s Gottlob Frege (1848-1925). For Frege, t r u t h in logic was completely timeless: t h e time at which an u t t e r a n c e is m a d e is considered as p a r t of t h e t h o u g h t which is b e i n g expressed [Klemke, p.361]. If somebody w a n t s to s a y t h e same today, as he said y e s t e r d a y t h e n he m u s t s u b s t i t u t e t h e e x p r e s s i o n 'today' w i t h t h e expression 'yesterday' [Prior 1957]. In Frege's logical s y s t e m there is no room for t h e conception of a proposition as a function w i t h t i m e as a v a r i a b l e . Likewise, t h e s t u d y of t e n s e d propositions is not c o n s i d e r e d to be i n t e r e s t i n g . While this w i t h o u t doubt holds for his symbolic logic a n d its concomitant conception of what logic is about, it is paradoxical t h a t Frege at the same t i m e was k e e n l y aware of intensionality in language. In his famous article "Uber Sinn u n d Bedeutung" [Frege 1969] (On Sense and Reference), he actually did invite a conception of propositions as functions. His observations in t h a t paper played a crucial r61e for the later development of i n t e n s i o n a l logic (and possible world semantics), w h e r e propositions are ordinarily c o n s t r u e d as f u n c t i o n s w i t h t i m e as one of t h e i r crucial p a r a m e t e r s [Montague 1976a]. In Prior's overview of t h e history of logic it is described how m a n y 19th century logicians - for instance R. Whately, H.L. Mansel, Francis Bowen, a n d T h o m a s F o w l e r - d e n i e d t h a t tensed propositions w e r e i m p o r t a n t or at all relevant in logical analysis. There are s o m e notable exceptions, though; these include J. S. Mill (1806-73), George Boole (1815-64) and C. S. Peirce (1839-1914). In t h e n e x t section we shall e x a m i n e

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Peirce's r61e, but for now we shall concentrate on George Boole, w h o constructed a logical formalism modelled on algebra. This formalism proved vital for t h e development of m o d e m symbolic logic. But Boole is also p e r h a p s t h e first 19th c e n t u r y logician to h a v e included t h e concept of t i m e explicitly in his theories ( a l t h o u g h only in a few passages). N a t u r a l l y , we shall here concentrate on this e l e m e n t of Boole's logic a n d leave aside a more exhaustive exposition. Some i n t e r e s t i n g c o n s i d e r a t i o n s r e g a r d i n g t h e relation between time and logic can be found in the m a n u s c r i p t entitled Sketch of a Theory and Method of Probabilities Founded upon the Calculus of Logic, w h i c h Boole m u s t have w r i t t e n between 1848 and 1854. Boole h e r e discussed e l e m e n t a r y propositions such as "The T h e r m o m e t e r falls" and "It will rain". Boole made this observation: Accordingly I have ... i n t e r p r e t e d t h e symbols x, y, z, as e x p r e s s i n g t h e c a s e s in w h i c h t h o s e e l e m e n t a r y propositions are t r u e Ix corresponding to 'it rains' and y corresponding to 'it hails']. This is in a g r e e m e n t w i t h the ordinary doctrine of t h e 'Reduction of Hypotheticals'. But more exact analysis h a s led m e to another conclusion. And without stopping here to assign the reason u p o n which that interpretation is founded, I shall simply state t h a t it consists in r e g a r d i n g t h e symbols as r e p r e s e n t i n g t h e t i m e s in which t h e e l e m e n t a r y propositions to w h i c h t h e y refer are true. [LP, p.146] F r o m this passage it is n o t clear w h a t convinced Boole t h a t a t e m p o r a l approach is preferable to t h e approach based on an a n a l y s i s of t h e given s i t u a t i o n . B u t an a p p e n d i x to t h e m a n u s c r i p t shows t h a t h e r e c o g n i s e d t h e i m p o r t a n c e of o r d i n a r y l a n g u a g e as a guide for - or p e r h a p s a test-bed o f logical considerations (especially w i t h respect to t h e s t u d y of universals a n d particulars): T h e l a n g u a g e of c o m m o n discourse, w h i c h in m a n y respects outstrips t h e limits w i t h i n w h i c h t h e logicians

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C H A P T E R 2.1 would fain h a v e restricted it, recognizes, however, t h e p a r t i c u l a r as well as the u n i v e r s a l in h y p o t h e t i c a l j u d g e m e n t s , - and it distinguishes t h e m by the particle 'sometimes'. The s y s t e m which I have e n d e a v o u r e d to establish introduces the same element, Time, and in t h e same m a n n e r . This I was not aware of w h e n I was led to form t h a t system, and I accordingly esteem it an interesting verification. [LP, p.162-63]

Boole obviously held t h a t a proposition refers to one or more durations. If two propositions refer to the duration x and t h e d u r a t i o n y, respectively, t h e n t h e conjunction between two propositions equals xy (i.e. the intersection between t h e two durations). It thus becomes especially interesting to examine the numerical constants 0 and 1: .. the numerical values 0 and 1 will be equally admissible w i t h this s y s t e m of interpretation, the former as t h e representative of t h e nothing of time or never: the l a t t e r as the Universal of time, which w h e n unlimited is Eternity, w h e n limited the duration to which our discourse refers. [LP, p.146] For example, Boole represented propositions such as: 'If it rains, it hails' by an equation of the type:

x=vy Let us r e n d e r the ideas involved in this example in m o d e r n terms: y is to be understood as the function defined on the set of times, y i e l d i n g t h e v a l u e 1 when 'it hails' is true, a n d 0 otherwise; x is the analogous function corresponding to 'it rains'. The third function of the equation v, is according to Boole "the representative of time partially indefinite and is a symbol of the same kind as x and y". The set theoretical r61e, which v plays in

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the equation is to ensure that the set of t r u t h s for x is a subset of t h e set of t r u t h s for y. Similarly, one could say t h a t 'it hails' a n d 'it rains' can respectively be represented by t h e equations y = l a n d x = l . H e r e t h e implication 'if it rains, it hails', has b e e n captured by an equation. Alternatively, one could represent it by t h e function (1 - x + vy). In his major w o r k A n Investigation of the Laws of Thought, on which are f o u n d e d the Mathematical Theories of Logic a n d Probability [1854], Boole presented some of the same t h o u g h t s w h i c h a p p e a r in t h e earlier manuscript. He considered t h e socalled s e c o n d a r y propositions [p.159], verdict functions w h i c h relate to other verdicts. It is characteristic of secondary verdicts t h a t they involve a time relation. This means t h a t logic m u s t be about relations b e t w e e n 'valid times'. J o r g e n J ¢ r g e n s e n h a s p e r t i n e n t l y c h a r a c t e r i s e d Boole's theory for secondary verdicts as a "time interval calculus" [1937, p.48]. This kind of theory as was unique in n i n e t e e n t h century logic. It m u s t be a d m i t t e d t h a t the introduction of time into a logic for secondary verdicts does not seem to have been one of Boole's chief concerns (judged on the basis of the n u m b e r of pages in his w o r k s d e v o t e d to t h e t r e a t m e n t of this question). B u t h i s r e m a r k s on t h e m a t t e r are on the other hand clear enough. I n any case they c a u s e d John Venn [1894, p.451-52] to realise as a consequence of Boole's theory t h a t tense inflected propositions m u s t be considered in logic, even t h o u g h Venn himself did n o t like the idea much. Half a century later, Boole's inclusion of t i m e in logic became one of the inspiring factors for t h e founder of m o d e r n tense-logic, A.N. Priori1957], who strongly e m p h a s i s e d Boole's suggestions.

2.2. C.S. PEIRCE ON TIME AND MODALITY Time has usually been considered by logicians to be what is called 'extra-logical' matter. I have never shared this opinion. But I have thought that logic had not yet reached the state of development at which the introduction of temporal modifications of its forms would not result in great confusion; and I am much of that way of thinking yet. C.S. Peirce [CP 4.523] To Charles Sanders Peirce (1839-1914) semiotics (or to use his own expression: 'semeiotic') gradually became identical with logic in a broad sense. In defining this relation between semiotics and logic, Peirce was no doubt highly influenced by the w a y the scholastic realists understood science, as for instance demons t r a t e d by Emily Michael [1977]. He got his inspiration first and foremost from the medieval juxtapositon of three of the seven free a r t s into the so-called trivium. The trivium consisted of the disciplines G r a m m a r , Dialectics (or: Logic), and Rhetoric. As d e m o n s t r a t e d by Max H. Fisch [1978], Peirce's work from 1865 to 1903 shows a constant development of reflections on the content and application of this tripartition. In the Spring of 1865 he s u b d i v i d e d the g e n e r a l science of r e p r e s e n t a t i o n s into 'General G r a m m a r ' , 'Logic' and 'Universal Rhetorics'. In May t h e s a m e y e a r he called this division 'General G r a m m a r ' , ' G e n e r a l Logic', and 'General Rhetorics', and in 1867 it was p r e s e n t e d as ' F o r m a l G r a m m a r ' , 'Logic' a n d ' F o r m a l Rhetorics'. T w e n t y y e a r s later, in 1897, it had become 'Pure G r a m m a r ' , 'Logic Proper' and 'Pure Rhetorics'. In 1903 Peirce w i t h i n his own now more m a t u r e d framework - d e t e r m i n e d t h e t r i p a r t i t i o n as 'Speculative G r a m m a r ' , 'Critic', a n d 'Methodeutic'. By t h e n it was also clear to him t h a t semiotics subdivided in that way - can in fact be understood as logic in the broad sense. Altogether Peirce's semiotics can be looked upon as a m o d e r n i s a t i o n of the u n d e r s t a n d i n g of logic from t h e late Middle Ages. 128

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Not only t h e t r i p a r t i o n , but also several other e l e m e n t s of medieval logic h a d an impact on Peirce's analyses and his develo p m e n t of semiotics. One example is t h e tripartition of t h e subjects of logic into t e r m s , propositions a n d arguments - a division, which can be found in almost every mediaeval introduction to logic. It was clear to Peirce t h a t this classification was r e l e v a n t not only w i t h i n logic (in t h e narrow sense), but also w i t h i n b o t h g r a m m a r a n d rhetoric, a fact which h a d also been recognised by t h e ancients a n d t h e medievals. It should be mentioned, however, t h a t Peirce rejected t h e idea of completely non-assertoric terms. In his opinion even terms are in general assertoric [CP 2.341]. One of t h e very obvious differences between mediaeval logic and t h e logic of later centuries is the rble of time in logic. I n m e d i a e v a l logic t i m e w a s t a k e n very seriously. Words a n d t e r m s w i t h a t e m p o r a l content such as 'begin', 'end', 'while' w e r e analysed, a n d t h e t e n s e s of the verbs were made t h e object of endless logical/semantical analyses. Peirce was certainly a w a r e of this, and t h e r e are m a n y indications t h a t he realised as one of the earliest m o d e r n philosophers and logicians t h a t t i m e could and even should be gradually included in logic. As t h e introductory quote above m a k e s evident, Peirce m a d e h i m s e l f a s p o k e s m a n for an open and undogmatic u n d e r s t a n d ing of logic. This openness, which was obviously due to his extensive knowledge of classic and scholastic logic, also m e a n t t h a t he would not accept logic as an u n t e m p o r a l science. He could well i m a g i n e a n e w development of a logic, which would t a k e time seriously. Peirce, however, held t h a t logicians a r o u n d t h e t u r n of t h e c e n t u r y were not ready to (re)introduce t i m e into logic w i t h o u t creating great confusion; not until later would it be possible to introduce the logic of time. Peirce's prophetical vision of a temporal logic proved to be correct. In the 1950's a n d 60's A. N. Prior succeeded in re-establishing the logic of time as a proper part of logic. It is obvious t h a t t h e s t u d y of Peirce's p h i l o s o p h y m e a n t a great deal to Prior. I n Prior's first g r e a t time logical work Time and Modality [1957], he gave a brief p r e s e n t a t i o n of the history of the modern logic of t i m e in an a p p e n d i x ; about one f o u r t h of this exposition is

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d e v o t e d to t h e i m p o r t a n c e of Peirce with r e s p e c t to t h e development of the new logic of time. Peirce's philosophy contains features, which could well be i n t e r p r e t e d as an e m e r g e n t logic for events. For example, he defined the notion of a 'Token' as applying to "A Single event w h i c h h a p p e n s once and whose identity is limited to t h a t one h a p p e n i n g or a Single object or thing which is in one single place a t a n y one i n s t a n t of time" [CP 4.537]. As we have already seen P e i r c e was h e s i t a n t about advancing a formal logic of time bSm~elf, but nevertheless it is relatively easy in his authorship to find clear ideas which can be used in a presentation of a formal time logic. In t h e following we shall first discuss Peirce's conception of time. Then we shall examine those r u d i m e n t a r y elements of a t i m e logic, which can be found in Peirce's work a f t e r all. This e x a m i n a t i o n will be followed by a preliminary discussion of Prior's formalisation of those elements. In a l a t e r chapter we shall compare these Peircean answers regarding future contingents with a formal version of the Ockham a n s w e r discussed in P a r t One.

PEIRCE'S UNDERSTANDING OF TIME It is reasonable first to discuss Peirce's u n d e r s t a n d i n g of time w i t h i n mathematics. Peirce was fully aware of the fact t h a t one of his g r e a t e s t sources of inspiration, the philosophy of Imman u e l Kant, h a d in Anglo-Saxon thinking given rise to an extrao r d i n a r y linkage of the concept of time with mathematics: However, Sir William Rowan Hamilton and De Morgan influenced (the latter only indirectly) by K a n t defined mat h e m a t i c s as the science of time and space. This definition never h a d very wide vogue. It is one of the v e r y worst any science ever received. [NEM, p.594].

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I n s p i r e d by Kantian thinking, Hamilton had felt t h a t if geomet r y could be understood as a pure m a t h e m a t i c a l s t u d y of space, t h e n a similar pure m a t h e m a t i c a l s t u d y of time o u g h t to exist. T h e r e s e a r c h p r o g r a m e m e r g i n g from this conviction can be described as an a t t e m p t to establish algebra as t h e 'science of p u r e time'. H a m i l t o n e n c o u n t e r e d m a n y difficulties in t h a t endeavour. In fact, there are several indications t h a t he actually gave u p t h e f u n d a m e n t a l idea himself [Ohrstrom 1985a]. Peirce w a s v e r y categoric in his rejection of Hamilton's p r o g r a m . He even stated that "it m u s t be an unclear head that cannot see that n u m b e r a n d c o u n t i n g h a v e n o t h i n g in p a r t i c u l a r to do w i t h time." [NEM, p.594] However, the validity of t h a t rejection is doubtful. Aristotle h a d already determined time as "the n u m b e r of m o t i o n with respect to earlier and later" IPhysica IV, 220b]. In fact, t h e r e seems to be an etymological connection b e t w e e n the G r e e k w o r d s for r h y t h m a n d n u m b e r , respectively. This c o n n e c t i o n a p p a r e n t l y s t r e n g t h e n e d t h e belief w i t h i n Greek p h i l o s o p h y t h a t t h e s e two concepts are i n t e r d e p e n d e n t , or at least semantically related. At any rate, there is ample historical proof t h a t time a n d n u m b e r s are closely interwoven. I m m e d i a t e e x a m p l e s are the calendar and the clock. As a m a t t e r of fact, in Peirce's own work t h e r e are enough examples to s u p p o r t t h e v i e w p o i n t t h a t time is r e l e v a n t to m a t h e m a t i c s . Peirce's first rejection of Hamilton's p r o g r a m was r a t h e r injudicious, b u t as we shall see below the following observation is more convincing: H a m i l t o n called algebra the science of Time. B u t t h e most r e m a r k a b l e characteristic of time, namely, t h a t t h e passage from t h e past to the f u t u r e is qualitatively different from t h e passage from the future to the past is not r e p r e s e n t e d in algebra. [NEM, p.9] According to Peirce, Hamilton's p r o g r a m failed because it did n o t in its algebra incorporate the temporal a s y m m e t r y between t h e p a s t and the future. I n his c o m p r e h e n s i v e w o r k New Elements of Mathematics [NEM] (which was a rewriting, or p e r h a p s a p a r a p h r a s e , of his f a t h e r Benjamin's manuscript), Peirce included a brief c h a p t e r

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concerned w i t h time. In a letter to William E. Story (dated t h e 2 2 n d of M a r c h 1896), Peirce's s t a t e d his m o t i v a t i o n for including this chapter as follows: "The science of Time receives a brief chapter, chiefly because it affords an o p p o r t u n i t y for s t u d y i n g t r u e continuity" [NEM, p.vi]. T h e m a t h e m a t i c a l character of time is defined in this chapter in the following way: Time is t h a t by the variations of w h i c h individual t h i n g s have inconsistent characters. T h u s , to be alive and to be dead are inconsistent states; but at different times the s a m e body m a y be alive and dead. [NEM, p.248] Obviously, this definition is not merely mathematical, but also s u b s t a n t i a l l y logical; this can be seen from t h e way it uses t h e notion of (in)consistency, a n d also from t h e implicit reference to assertions. Moreover, the kind of logic implicit in the definition is a time logic, since it involves assertions which are true at some times, b u t false at other times. In general, it is clear t h a t to Peirce time was to be understood in relation to events, and it is unlikely t h a t his framework should leave any room for repres e n t a t i o n s of an 'empty time', w h e r e i n no change at all would t a k e place. T h e s e observations are s u p p o r t e d by t h e following quotation from 1892: Time, as t h e universal form of change, cannot exist u n l e s s t h e r e is s o m e t h i n g to undergo c h a n g e a n d to u n d e r g o change continuous in time there m u s t be a continuity of changeable qualities. [CP 6.132] F u r t h e r Peirce defines the past, the p r e s e n t and t h e f u t u r e in the following way: The present is the existing state of things .... The past is t h a t p a r t of t i m e w i t h which the m e m o r y is concerned ... T h e future is t h a t part of time with w h i c h t h e will is concerned. [NEM, p.248-49]

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Obviously, these definitions are not themselves exactly w h a t we would call p u r e mathematics. Nonetheless, t h e y invite a mathematical discussion of the concept of time, as witnessed by the appeal to the notion of continuity; evidently, such a discussion should on the other hand be influenced by cognition and psychology. The chapter on time is mainly concerned with considerations about temporal continuity. The crucial point is here the gradual change in the course of time. In t h a t connection it becomes very important to distinguish between 'instant' corresponding to the mathematical time and 'moment' which is an infinitesimal duration and which can be used in a m a t h e m a t i c a l description of the gradual change. In Peirce's philosophy, experience was a crucial notion, and in t h a t connection he naturally had to discuss time. Any realisation process, as for instance the change from doubt to belief, m u s t involve something temporal, he stressed [CP 7.346]. As S a n d r a B. Rosenthal [1987] noted, Peirce was a w a r e of the fact that no experience is so limited as not to contain a flow of continuity. Peirce put it like this: There is no span of present time so short as not to contain ... something for the confirmation of which we are waiting. [CP 7.675] In 'The Law of Mind' (1892) he tried to determine the salient features of how we as h u m a n beings u n d e r s t a n d time: One of the most marked features about the law of mind is t h a t it makes time to have a definite direction of flow from past to future. The relation of past to future is different from the relation of future to past. [CP 6.127] This temporal a s y m m e t r y is clearly in opposition to the laws of mechanics, which are fully s y m m e t r i c a l with respect to the time co-ordinate - t h e two temporal directions being no more different in relation to mechanics t h a n two spatial directions. Nevertheless, Peirce maintained that our experience of time is

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asymmetrical. Our cognitive relation to the past is certainly different from our relation to the future. The fact t h a t "future conduct is the only conduct that is subject to self-control" [CP 5.427] was very important for him with respect to a theory of meaning: The rational meaning of every proposition lies in the future. How so? The m e a n i n g of a proposition is itself a proposition.., it must be simply the general description of all the e x p e r i m e n t a l p h e n o m e n a w h i c h t h e assertion of t h e proposition virtually predicts. [CP 5.427] Peirce believed that the power or principle shaping the history of n a t u r e is neither coincidence nor necessity, but r a t h e r it is love, agape: the divine love which the Creator expresses towards creation in the course of time. In this way nature can be viewed as a continuous flow. But it should also be clear that according to Peirce m a n is not only living in this progressive time. H u m a n time is also tense-oriented, t h a t is, the concepts of past, present and f u t u r e are essential to the h u m a n mind. The past can be characterised as 'facts'. According to Peirce a fact should be understood as a "fait accompli; its esse is in praeterito" [CP 2.84]. Such facts should be viewed as 'now-unpreventable'. But with the future it is a different matter: Being in futuro appears in mental forms, intentions and expectations. Memory supplies us a knowledge of the past by a sort of brute force, a quite binary action without reasoning. But all our knowledge of the future is obtained through the m e d i u m of something else. [CP 2.86] The m e d i u m mentioned h e r e could for instance be the laws of physics, or n a t u r e in general. That is, in some cases the future can be p r e s e n t in its causes, and in these cases we can have knowledge of the future. But in other cases we must confine ourselves to other kinds of law-like statements. It should, however, be mentioned that Peirce did not consider n a t u r a l laws to be quite as compelling as logical laws. Natural laws he saw as

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'habits of nature', and he even accepted the possibility of a "sudden stoppage of everything" [CP 4.547]. He did not consider the possibility of a law of nature against such a sudden stoppage as an a r g u m e n t which should be t a k e n very seriously! To which extent would Peirce allow t h a t a scientific law can be a m e d i u m t h r o u g h which aspects of the future can be known? This is, as far as we can see, an open question. Peirce's views on the relation between time and cognition as well as his idea of time in general are very complex, and we must admit t h a t his s t a t e m e n t s do not unambiguously point in any single direction. A few quotations should illustrate that: For time is itself an organized something, having its law or regularity; so that time itself is a part of that universe whose origin is to be considered. We have therefore to suppose a s t a t e of things before t i m e was organized. Accordingly, w h e n we speak of the universe as "arising", we do not mean that literally. [CP 6.214] The idea of time must be employed in arriving at the conception of logical consecution; but the idea once obtained, the time element m a y be omitted, this leaving the logical sequence free from time. That done, time appears as an existential analogue of the logical flow. [CP 1.491] S t a t e m e n t s like these are typical for Peirce's philosophy. They form a good inspiration for f u r t h e r speculation r e g a r d i n g the concept of time. It must be admitted, however, that his ideas of t i m e become very complicated w h e n it is added t h a t Peirce a p p a r e n t l y believed in w h a t Milic Capek [1991, p.265] has t e r m e d a 'self-returning n a t u r e of time'. Peirce stated: The other question is w h e t h e r time is infinite in duration or not. If it has no flaw in its continuity, it must, as we shall see in Chapter 4 return to itself. This m a y happen after a finite time, as Pythagoras is said to have supposed, or in infinite time, which would be a doctrine of consistent pessimism. [CP 1.498]

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Peirce formulated similar views in [CP 1.498] and in [CP 6.210]. It is hard to see how the idea of eternal recurrence can fit with the rest of the Peircean thinking. Peirce m a y in fact have had problems with this question himself, since the 'Chapter 4' to which h e refers in t h e above quotation was apparently n e v e r written. We shall leave this issue here.

TIME AND MODALITY The concept of possibility has always played a great r61e in philosophy, and Peirce is no exception to this rule - on the contrary, it was one of his essential goals to find a suitable definition of'possibility'. Early in his authorship his attempts at a definition were characterised by semantically negative expressions, but later he emphasised the positive character of the notion. On the 18th of March 1897 Peirce wrote: ... m y old definition of the possible as t h a t which we do not know not to be t r u e (in some state of information real or feigned) is an anacoluthon. The possible is a positive universe, and the two negations happen to fit in, but t h a t is all ... I found myself a r r e s t e d until I could form a whole logic of possibility, - a very difficult and laborious task. [CP 8.308] Later he formulated the positive character of possibility in still stronger terms:

Potentiality is t h e absence of Determination (in the u s u a l broad sense) not of a mere negative kind, but a positive capacity to be Yea and to be Nay; not ignorance but a state of being ... Actuality is the Act which d e t e r m i n e s the merely possible... Necessitation is the support of Actuality by reason ... [Ms 277, 1908; quoted from Fisch and Turquette 1966:78] It is a n a t u r a l consequence of Peircean t h o u g h t t h a t he in some contexts related modality, including the definition of the

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possible, to time or temporality. In order to examine this relation closer we m u s t first scrutinise Peirce's view on the relation b e t w e e n time and reality, which m e a n s t h a t we have to s t a r t with t h e ontological foundation for his ideas about the logic of time. Peirce distinguished between three modes of being, which can be understood from the following quotations: M y view is that there are three modes of being. I hold t h a t we can directly observe t h e m in elements of w h a t e v e r is at a n y time before the mind in any way. They are the being of positive qualitative possibility, the being of actual fact, and t h e being of law t h a t will govern facts in the future. [CP 1.21-1.23] T h u s the t h r e e modes of being in Peirce's philosophy are: actuality, possibility and necessity. In 'temporal terms', 'actuality' (understood as the 'given') will cover both the past and the present. The future is thought of as a possibility sphere with certain p r e d e t e r m i n e d incidents (logically necessary or d e t e r m i n e d by n a t u r a l law). In this way possibility as well as necessity are both r e l a t e d to the future; and conversely, future events are subdivided into either necessary ones or merely possible ones. In accordance with this binary subdivision, Peirce rejected the idea t h a t t h e t r u t h about the contingent (and undecided) f u t u r e could be known beforehand - or indeed, t h a t assertions about the c o n t i n g e n t f u t u r e could at all be m e a n i n g f u l l y r e g a r d e d as having a truth-value. The following s t a t e m e n t sums up essential features of Peirce's views: T h a t time is a particular variety of objective Modality is too obvious for argumentation. The Past consists of the sum of faits accomplis, and this Accomplishment is the Existential Mode of Time .... the Mode of the Past is t h a t of Actuality. N o t h i n g of the sort is t r u e of the F u t u r e .... (The future) is not Actual, since it does not act except through the idea of it, t h a t is as a law acts; but is either Necessary or Possible... [CP 5.459]

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Peirce did not define the past as 'necessary', but reserved this definition for 'the preordained'. However, he maintained t h a t at a cognitive level t h e relation of the present to the past is decisively different from the relation of the present to the future. Whilst in principle any past event belongs to the domain of the memory, we have no possibilities of obtaining a similar insight into future events. On the contrary, the future lies open before us, t h u s e n a b l i n g us to influence the forming of the future within certain limits. A similar possibility of influencing the past does not exist: I remember t h e past, but I have absolutely no slightest approach to such knowledge of the future. On the other hand I have considerable power over the future, but nobody except the Parisian mob imagines that he can change the past by much or by little. [CP 6.70] Peirce f u r t h e r m o r e wrote: A certain e v e n t either will happen or will not. There is nothing now in existence to constitute the t r u t h of its being about to happen, or of its being not about to happen, unless it be certain circumstances to which only a law or uniformity can lend efficacy. But t h a t law or uniformity, the nominalists say, h a s no real being, it is only a mental representation. If so, n e i t h e r the being about to happen nor the being about not to happen has any reality at present ... If, however, we admit t h a t the law has a real being, and of the mode of being an individual, but even more real, t h e n the future necessary consequent of a present state of things is as real and true as the present state of things itself. [CP 6.368] Peirce saw h i m s e l f as a realist. T r u t h and reality were for him objective, albeit in a sense differing from classical 'naive realism'. As pointed out by for instance H a r r y R. Klocker [1968, p.80 fi], t r u t h is according to Peirce t h a t viewpoint upon which everybody e x a m i n i n g the state of things eventually agree, and reality is the object represented by this viewpoint. (In fact, this

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should probably be modified even further, into approximately 'that viewpoint u p o n w h i c h everybody e x a m i n i n g the state of things by scientific method eventually agree'.)

THE DETERMINISM PROBLEM The d e t e r m i n i s m problem was one of t h e problems of p r i m a r y importance to Prior, and in this problem is included the question of free choice or free will. Strictly speaking Peirce did not leave m u c h latitude for t h e will itself. On the 18th of March 1897 he stressed in a letter to William J a m e s t h a t t h e will as such is not free to any i m p o r t a n t extent. The freedom r a t h e r antecedes t h e will and is being established in a state of unstable equilibrium: T h e freedom lies in t h e choice w h i c h long antecedes t h e will. There a state of nearly unstable equilibrium is found. [CP 8.311] In the history of logic the problems concerning the freedom of m a n have often b e e n discussed in theological t e r m s as a tension between man's p u t a t i v e freedom of choice in t h e face of divine foreknowledge. N o w and t h e n Peirce discussed these questions in that context, too. In 1893 he wrote: T h a t is to say, t h e y suppose that a m a n is perfectly free to do or not to do a given act; and yet t h a t God already knows w h e t h e r he will or will not do it. This seems to most persons flatly self-contradictorary; and so it is, if we conceive God's knowledge to be a m o n g t h e things w h i c h exist at t h e present time. B u t it is a degraded conception to conceive God as subject to Time, which is rather one of his creatures. [CP 4.68]

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In this w a y Peirce accepted t h e view of the church fathers t h a t t h e world is n o t created in time, but t h a t God in t h e b e g i n n i n g created t h e world as well as time. Peirce continued: Literal fore-knowledge is certainly contradictory to literal freedom. B u t if we say t h a t t h o u g h God knows (using t h e word k n o w s in a trans-temporal sense), he never did know, does n o t know, and never will know, t h e n his knowledge in no wise interferes with freedom. [CP 4.68] The y e a r before Peirce had in the Monist published a survey of w h a t he called 'The Doctrine of Necessity', which he described as "the c o m m o n belief t h a t every single fact in the u n i v e r s e is precisely d e t e r m i n e d by law" [CP 6.36], or alternatively: The proposition in question is t h a t the state of t h i n g s existing at a n y time, t o g e t h e r w i t h certain i m m u t a b l e laws, completely determine the state of things at every other time [CP 6.37] This doctrine he traced back to the Stoics, who according to Peirce linked t h e doctrine with materialism. He pointed out t h a t the later advances in mechanical physics gave an i m p e t u s to the doctrine. On t h e other hand, it was not generally accepted, exactly b e c a u s e it appeared irreconcilable w i t h the belief in the freedom of t h e will and the possibility of miracles. Peirce himself a r g u e d a g a i n s t mechanical d e t e r m i n i s m a n d insisted t h a t in the description of courses of events there would h a v e to be a decisive e l e m e n t of probability, s p o n t a n e o u s n e s s a n d real possibility. As indicated by J o h n E. Smith [1987] this position fits very well w i t h t h e viewpoints published by William J a m e s some years earlier. Peirce rejected t h a t conception of science upon which t h e doctrine of n e c e s s i t y rests by s t r e s s i n g the observation t h a t conclusions d r a w n from science are never more t h a n probable. He a r g u e d t h a t t h e doctrine p r e s u p p o s e d the idea t h a t physical q u a n t i t i e s do in fact have m a t h e m a t i c a l values, an idea which just like t h e doctrine of necessity itself could not be established by

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m e a n s of observation. Peirce could not come to t e r m s w i t h a u n i v e r s a l m e c h a n i c a l determinism. He felt t h a t especially t h e origin of life in its infinite complex forms fitted very badly w i t h the doctrine of necessity. Moreover, reflections on consciousness also u n d e r m i n e d t h e doctrine. If t h e doctrine is a c c e p t e d w i t h o u t modification, t h e n one m u s t reject the i d e a of consciousness as a source or ground of real choices and decisions. For all functions of the mind m u s t t h e n be understood as p a r t s of t h e physical universe, and t h u s t h e perception t h a t we are free to do a given act will be reduced to an illusion. In rejecting t h e doctrine Peirce believed to have made room for consciousness, or in his own words: "room to insert mind into our scheme, a n d p u t it into t h e place where it is needed, into t h e position, which as t h e sole self-intelligible thing, it is entitled to occupy, t h a t of t h e fountain of existence ..." [CP 6.61] It is w o r t h m e n t i o n i n g t h a t Peirce's refutation of t h e doctrine of necessity a n d his accentuation of indei'miteness, coincidence and s p o n t a n e o u s n e s s can be seen as a forerunner of t h e philosophy p r e v a l e n t today in the interpretation of q u a n t u m physics, as pointed out by Peder Voetmann Christiansen [1988 p.38 •. Peirce rejected the notion that indefiniteness should be seen as a d e g e n e r a t i o n from definiteness. To him, indefiniteness was of p r i m a r y importance: Get rid, t h o u g h t f u l Reader, of the Ockhamistic prejudice of political p a r t i z e n s h i p t h a t in t h o u g h t , in being, a n d in d e v e l o p m e n t the indefinite is due to a degeneration from a p r i m a l state of perfect definiteness. The t r u t h is r a t h e r on t h e side of t h e scholastic realists t h a t the u n s e t t l e d is t h e p r i m a l state, and that defmiteness and determinateness, the two poles of settledness, are, in the large, approximations, d e v e l o p m e n t a l l y , epistemologically, and m e t a p h y s i c a l l y [CP 6.348] T h e s e observations are, in fact, an ontological c o u n t e r p a r t of the position t h a t s t a t e m e n t s r e g a r d i n g the c o n t i n g e n t f u t u r e cannot be t r u e now (perhaps an additional reason w h y O c k h a m is m e n t i o n e d explicitly in the quote). In a Peircean system, t r u t h

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cannot be the only basic concept - if indeed, it can be basic at all. Some kind of theoretical vagueness also h a s to be involved. Peirce gave this example: Again, statisticians can tell us pretty accurately how m a n y people in the city of New York will commit suicide in the y e a r after the next. None of these persons have at present a n y idea of doing such a thing, and it is v e r y doubtful w h e t h e r it can properly be said to be d e t e r m i n a t e now who they will be, although their n u m b e r is approximately fLxed. [CP 4.172] Even though statisticans can predict the number of suicides in New York pretty accurately, they cannot tell which persons will commit suicide in t h e y e a r after the next. It is sufficiently evident t h a t in Peirce's opinion, a proposition like 'Mr. Smith is going to commit suicide in the y e a r after t h e next' cannot be t r u e now, since Mr. Smith has not yet made up his mind - or, if h e had, he might change it. (The only possible exception to this rule would be for us to establish some kind of n a t u r a l law, which would in the fLxed amount of time lead inevitably to Mr. Smith's suicide. But apart from the extreme unlikelihood of finding such a law, if we found it we would not be dealing with the contingent future any more, but with the necessary future.)

THE FORMALISATION OF THE PEIRCEAN IDEAS Peirce did not m a k e any a t t e m p t to formalise his ideas on t e m p o r a l and modal logic in t e r m s of an operator calculus in t h e m o d e r n sense. However, he made s o m e i n t e r e s t i n g attempts of relevance in this field, using his so-called existential graphs. This however was not recognised, or at a n y rate did not h a d a n y r61e to play in the development of tempo-modal logic d u r i n g the 1950s and 1960s. But later it h a s become widely recognised t h a t Peirce in fact established a general calculating

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t e c h n i q u e , w h i c h c o m p r e h e n d s w h a t we t o d a y call predicate logic. I n p a r t 3 we shall outline t h e ideas involved, a n d also discuss how they can be utilised in a computational context. In t h e following we shall however concentrate on those Peircean i d e a s , w h i c h c a m e to play a rSle for A. N. P r i o r in his d e v e l o p m e n t of t e m p o - m o d a l logic. Peirce m a i n t a i n e d t h a t logical assertions become t r u e by rep r e s e n t i n g facts. If 'a fact' is understood as 'a portion of reality' s m a l l e n o u g h to be represented in one single (atomic) assertion, t h e n t h e assertions of logic will s t a n d as figures r e p r e s e n t i n g features of reality. B u t how does this conception of logic s t a n d in relation to a c h a n g i n g world? If assertions represent features of reality, t h e n t h e p e r c e p t i o n of t h e single fact m u s t be l i n k e d w i t h time. P e i r c e ' s i n s i s t e n c e on a correspondence b e t w e e n logic a n d r e a l i t y p r e s u p p o s e s a conception of logic w h i c h t a k e s t i m e seriously. In his i n t r o d u c t i o n to the logic of t i m e A. N. Prior a n a l y s e d Peirce's position, especially with r e s p e c t to f u t u r e s t a t e m e n t s . The analysis showed t h a t in Peirce's framework, a p r o p o s i t i o n like (i) ' T o m o r r o w J a n e c h o o s e s to go to C o p e n h a g e n ' is e q u i v a l e n t w i t h e i t h e r (ii) ' T o m o r r o w J a n e n e c e s s a r i l y chooses to go to Copenhagen', or (iii) 'Tomorrow J a n e possibly chooses to go to Copenhagen'. T h e s y s t e m t h u s m a k e s no room for a n y concept of plain t r u t h 'in between' 'necessarily' and 'possibly'. On t h e basis of his studies of Peirce's philosophy Prior p u t f o r w a r d a tense logical system, w i t h which he, by t h e way, declared himself to be very satisfied. In his own words: .. C.S. Peirce's description of t h e past (with, of course the present) as the region of the 'actual', the area of 'brute fact', a n d the future as t h e region of t h e necessary a n d t h e possible. T h a t is w h y I call this system 'Peircean'. [Prior 1967, p.132] T h e r e is h a r d l y any doubt t h a t Prior's r e n d i t i o n of Peirce's a m b i t i o n s as regards t h e logic of time and m o d a l i t y is correct.

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We are even able to support this w i t h Peirce quotations, w h i c h Prior, in all likelihood, did not have at his disposal: A simply assertory Proposition differs j u s t half as m u c h as the assertion of a Possibility, or t h a t of a Necessity, as these two differ from each other. For as we have seen above, t h a t which characterises and defines an assertion of Possibility is its emancipation from the Principle of Contradiction, while it r e m a i n s subject to the Principle of Excluded Third; while t h a t w h i c h c h a r a c t e r i z e s a n d defines an a s s e r t i o n of N e c e s s i t y is t h a t it r e m a i n s subject to t h e P r i n c i p l e of Contradiction, b u t throws of t h e yoke of the Principle of Excluded Third; and w h a t characterizes and defines an assertion of Actuality or simple existence, is t h a t it acknowledges allegiance to both formulae, and is j u s t m i d w a y bet w e e n t h e two rational 'Modals' as t h e modified forms are called by all the old logicians. [The Art of Reasoning elucidated, 1910; quoted from Fisch a n d Turquette 1966, p. 78] Obviously, this s t a t e m e n t m a k e s a distinction between t h r e e types of assertions. For ordinary assertions both 'the Principle of Excluded Third': p v~p

and 'the Principle of Contradiction'

-(p A-p) are valid. T h a t is also rather to be expected, for by the ordinary laws of (bivalent) logic the two principles are equivalent. B u t in modal contexts m a t t e r s are not t h a t simple. For instance, as a rule

~(Np A N - p ) applies for assertions c o n t a i n i n g t h e n e c e s s i t y - o p e r a t o r N. However, t h e following is not valid:

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Np v N - p As for the possibility-operator M it is the other way round:

Mp v M~p is valid, but

-(Mp A M - p ) in not valid in general. - By analysing Peirce's way of thinking and transferring this into the modern logic of time, we arrive at t h e conclusion t h a t the following formula m u s t hold for a n y proposition p:

~(F(x)p A F(x)~p), whereas its 'excluded middle' analogue

F(x)p v F(x)-p does not hold in general. - This is due to the fact t h a t both assertions, F(x)p and F(x)~p, can be false, if t h e y represent a pair of s t a t e m e n t s about the contingent future. On the other hand, if t h e y are t a k e n to represent statements about the necessary future, involved precisely one of them is t r u e - t h a t is, the law of excluded middle holds in that case. It seems, however, t h a t this theory gives offence to the intuition on which everyday language is based. We normally accept a concept of future which is logically b e t w e e n possible future and necessary future, and which certainly m a k e s no distinction between F(x)~p and -F(x)p. For instance, we m a y well wish to a s s e r t t h a t 'Tomorrow J a n e chooses to go to Copenhagen' without saying that this choice is necessary. And if J a n e the next day does choose to go to Copenhagen, we shall feel justified in h a v i n g m a d e the assertion y e s t e r d a y - t h a t is, we would be inclined to consider it as having been true, when we made it.

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In any case it is clear that Peirce did not depart from t h e classical logic, but r a t h e r added a g r e a t deal to it. This a t t i t u d e is shown clearly in the following r e m a r k s : I have long felt t h a t it is a serious defect in existing logic that it takes no heed of the limit between two realms. I do not say t h a t t h e Principle of Excluded Middle is d o w n r i g h t false; b u t I do say t h a t in every field of t h o u g h t whatsoever t h e r e is an i n t e r m e d i a t e ground b e t w e e n positive a s s e r t i o n a n d positive negation which is j u s t as Real as they. M a t h e m a ticians always recognize this, a n d seek for the limit as t h e presumable lair of powerful concepts, while m e t a p h y s i c i a n s a n d old fashioned logicians, t h e sheep and goat s e p a r a t o r s never recognize this. The recognition does not involve any denial of existing logic, but it involves a great deal of addition to it. [Letter to William James, dated Feb. 26, 1906] It would have been interesting to learn more about w h a t k i n d of 'additions' to 'the existing logic' Peirce h a d in mind. F r o m w h a t he said about t h e principle of excluded middle a n d t h e law of contradiction it seems very likely t h a t he had in m i n d some k i n d of operator logic, as the one we have presented here. One m i g h t even with some j u s t i f i c a t i o n conjecture t h a t Peirce realised how a distinction between -F(x)p and F(x)~p would become necessary, w h e n it comes to f o r m u l a t i n g a logic of time. P e r h a p s he was t h i n k i n g of the need for this kind of distinction, w h e n he stated t h a t t h e introduction of temporal modifications of t h e forms of logic would r e s u l t in great confusion, a n d t h a t logic h a d to be developed f u r t h e r before it could be done [CP 4.523].

SOME FORMALITIES OF THE PEIRCEAN SOLUTION We shall now finally sketch Prior's 'Peircean system' (it will also be described in chapter 2.5 a n d s u b s e q u e n t c h a p t e r s in increasing detail). The essential f e a t u r e of this s y s t e m can be

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explained in t e r m s of the old Aristotelian example about t h e possible sea-fight tomorrow. According to Peirce's ideas w e can model the future in the following way: sea-fight

_J no sea-fight In the Peirce-model it makes no sense to speak about 'the t r u e future' as one of the possible futures. There is no future yet. Let s stand for 'there is a sea-battle going on', and let us m a k e t h e reasonable assumption that 'tomorrow it will be the case t h a t s' is contingent. Then in this model F(1)s as well as F ( 1 ) - s are false, w h e r e a s -F(1)s and - F ( 1 ) ~ s are both true. There is certainly a tension between this hall-mark of the system a n d t h e intuition normally involved in everyday reasoning. Prior realised that according to the Peirce-solution we cannot infer t h a t there was going to be a sea-battle from the fact t h a t there is a sea-battle going on, although it certainly does follow that there will have been one. [Prior 1957a, p.95] That is, in the Peirce-model one m u s t accept that even if s, 'there is a sea-battle going on', is given (true), we cannot infer P(1)F(1)s. Therefore, q ~P(x)F(x)q , is not a thesis in the Peirce system. On the other hand, it should be obvious t h a t in this system, the proposition schemas F(x)P(x)q ~ q P(x)F(x)q ~ q

are generally valid. We m a y sum up these features by noting with Prior t h a t in the Peirce model F(x)q is understood in the strong way, i.e. as "it is bound to be the case after x time units that q" [Prior 1969, p.329]. Moreover, f u t u r e contingents cannot be known 'now', a n d

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hence there cannot be any true s t a t e m e n t s about the contingent future. We h a v e seen t h a t in this system the s t a t e m e n t 'there will be a sea-battle tomorrow' cannot be true today, for there is no u n i q u e future, but r a t h e r a n u m b e r of alternative, possible futures. The basic question concerns the interpretation of expressions r e g a r d i n g the future: can it be m a i n t a i n e d with conceptual and logical consistency t h a t '(some event) E will happen', whilst dist i n g u i s h i n g this s t a t e m e n t from 'E could happen', and 'E will n e c e s s a r i l y h a p p e n ' ? We have shown how in t h e Peircean S y s t e m the plain 'tomorrow' becomes equivalent to 'necessarily tomorrow', or 'possibly tomorrow'. But we have also previously seen how the O c k h a m view on s t a t e m e n t s permits a differentiation between actual, possible and necessary future, and hence a d i f f e r e n t i a t i o n b e t w e e n 'tomorrow', 'possibly tomorrow' and 'necessarily tomorrow'. This f u n d a m e n t a l question of the status of t h e c o n t i n g e n t f u t u r e h a s certainly not been definitively s e t t l e d (and p e r h a p s can n e v e r be). The d e b a t e b e t w e e n O c k h a m i s t s a n d Peirceans goes on. Many t h i n g s indicate t h a t t h e discussion about which position to prefer is actually a question about the very u n d e r s t a n d i n g of logic. Later we shall also c o m p a r e Peircean answers r e g a r d i n g future contingents with t h e Ockhamistic answers.

2.3. ~UKASIEW~CZ'S CONTRIBUTION TO T E M P O R A L LOGIC In a series of articles d u r i n g t h e 1920's a n d 30's t h e famous P o l i s h logician J a n L u k a s i e w i c z a d v o c a t e d a p a r t i c u l a r i n t e r p r e t a t i o n of Aristotle's discussion of the s t a t u s of sentences a b o u t t h e c o n t i n g e n t f u t u r e , as developed in his 'sea-fight' e x a m p l e (from De I n t e r p r e t a t i o n e c h a p t e r IX). L u k a s i e w i c z ' i n t e r p r e t a t i o n crucially rests on a rejection of t h e principle of bivalence. In fact, this k i n d of interpretation w a s not new, but h a d b e e n formulated as early as by the Epicureans. However, Lukasiewicz presented this position more clearly t h a n h a d ever b e e n done before, a n d developed it w i t h t h e aid of m o d e r n symbolic logic. Now it is clear t h a t philosophical determinism goes nicely with some tempo-modal logical systems, and conversely; but on t h e o t h e r h a n d , a t e m p o - m o d a l system can be constructed so as to allow for indeterminism. Lukasiewicz used his interpretation of Aristotle and the status of sentences about the contingent future as a n a r g u m e n t against logical d e t e r m i n i s m a n d in favour of logical indeterminism, for which he declared his w h o l e h e a r t e d s u p p o r t . He defined (logical) d e t e r m i n i s m as t h e a s s u m p t i o n that I f A is B at time t; t h e n it is true at any time before t, that A is B at t. [McCall 1967, p. 22] Generally speaking, d e t e r m i n i s m t h u s becomes equivalent to a thesis of omnitemporal t r u t h , since 'A is B at time t' is identified w i t h 'it is true for any time t~, t h a t A is B at time t'; the restriction in t h e above quote that tl be earlier t h a n t disappears in a fuller development, as will be shown below. Let p stand for the s t a t e m e n t 'A is B', the expression T(t,p) for is t r u e at time t', and (tl < t) for 't is earlier t h a n (before) t'; t h e n Lukasiewicz's rendition of d e t e r m i n i s m can be symbolised as (D) (T(t,p) A (tI < t)) ~ T(tl, T(t,p)) 149

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CHAPTER 2.3

On Lukasiewicz' interpretation, Aristotle's considerations in the sea-fight e x a m p l e were intended to show that (D) follows from two common (logical) presuppositions. Firstly, the principle of bivalence w h i c h can be expressed in the following way: (B) For a n y time t and any proposition p: either T(t,p) or T(t, ~p), but not both. If (B) holds p and ~p cannot both be true at the same time, but one of t h e m h a s to be true and the other false. This means t h a t the following two formulae (B 1) (B2)

- (T(t,p) A T(t,-p)) T(t,p) v T(t, ~p)

hold for any p and any t. Consequently (C 1) (C2)

- T(t,p) - T(t, -p) -T(t,T(Q,p)) - T(t,T(Q,-p))

Lukasiewicz f u r t h e r m o r e considered the principle expressed in the following Aristotelian statement: ... if [a certain thing] was white or was not white, t h e n it is true to confirm or deny it. [18a39]. The Aristotelian assumption that if it was true that X is Y t h e n it is true t h a t is t r u e that X was Y, can according to Lukasiewicz be translated as the following principle: (P)

(T(t,p) A t< tP ~ T( tl, T(t,p))

The difference between (P) and (D) is thus merely rooted in the before/after relation between t and tl. The proof t h a t (B) and (P)

LUKASIEWICZ'S CONTRIBUTION TO TEMPORAL LOGIC 151 implies (D) can be done indirectly. Assume t h a t (D) is invalid i.e. that T(t,p) A t~ < t A - T ( tl, T(t,p))

holds for a proposition p and for two times t and tl. By (C2) this is equivalent to T(t,p) A tl < t A T ( tl, T(t, ~p)).

By applying (P) we get: T(t,p) A tl < t A T ( t, T ( t , - p ) )

which assuming T(t, T(t,p)) - T ( t , p ) m u s t be equivalent to T(t,p) A t l < t A T(t, ~p).

This clearly contradicts (B1), so we conclude t h a t it is possible to infer (D) from (P) a n d (B1-2)! Lukasiewicz s u g g e s t e d t h a t the principles embodied by t h e above t h e o r e m s (D, B, a n d P) were t h e u n d e r l y i n g t e n e t s , respectively t h e implications of the Aristotelian text. This m e a n s t h a t Aristotle in order to avoid d e t e r m i n i s m h a d to restrict t h e validity of bivalence. Nevertheless, Lukasiewicz had to a d m i t t h a t t h e putative limitations of this sacred principle are by no m e a n s self-evident in Aristotle's discussion; t h e very need for i n t e r p r e t a t i o n a n d ' r e c o n s t r u c t i o n ' b e a r s w i t n e s s to t h i s observation! I n d e e d , Aristotle does n o t s e e m to have b e e n definitive in his a t t i t u d e towards the principle of bivalence. W h i c h e v e r w a y Aristotle h i m s e l f is to be u n d e r s t o o d , Lukasiewicz's solution to t h e problem of sentences about t h e contingent future and the associated problems with d e t e r m i n i s m w a s v e r y m u c h i n s p i r e d by t h e A r i s t o t e l i a n analysis. His solution, t h e n , was to consistently reject t h e principle of bivalence by introducing a t h i r d t r u t h value. This

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CHAPTER 2.3

t r u t h value, 'undetermined', is applied to contingent propositions r e g a r d i n g t h e f u t u r e [McCall 1967, p. 64]. F o r i n s t a n c e , a proposition s t a t i n g t h a t t h e r e will be a sea-fight tomorrow can be a s s i g n e d t h e t r u t h v a l u e ' u n d e t e r m i n e d ' today. This is b e c a u s e today it is not given or definitely d e t e r m i n e d w h e t h e r t h e sea-fight is actually going to take place tomorrow or not. Lukasiewicz' interpretation was disputed by Prior [1962, p. 240 ff.], w h o p o i n t e d o u t a s i g n i f i c a n t d i f f e r e n c e b e t w e e n Lukasiewicz's trivalent logic and Aristotle's text: according to Aristotle it is t r u e already today, t h a t there either t h e r e will or t h e r e will n o t be a sea-fight tomorrow, even t h o u g h t h e t r u t h v a l u e of t h e two constituents of the disjunction are separately u n k n o w n , or possibly ' u n d e t e r m i n e d ' . This c o n t r a d i c t s one i m p o r t a n t and presumably inevitable property of Lukasiewicz' t h r e e - v a l u e d logic: n a m e l y t h e fact t h a t the t r u t h value of t h e d i s j u n c t i o n of two (separately u n d e t e r m i n e d ) propositions is a c c o r d i n g to L u k a s i e w i c z ' u n d e t e r m i n e d ' , i.e. (p v q ) i s u n d e t e r m i n e d for p u n d e t e r m i n e d and q u n d e t e r m i n e d . For t h i s r e a s o n it m u s t be concluded t h a t Lukasiewicz' t r i v a l e n t logic does not provide a convincing basis for the interpretation of Aristotle's sea-fight e x a m p l e and t h e associated logical a n d philosophical problems. As we have seen in p a r t I, Nicholas Rescher's [1968] so-called 'medieval i n t e r p r e t a t i o n ' seems far more promising, although it has to be admitted t h a t a distinction b e t w e e n t h e principles of bivalence and 'tertium n o n datur' can be r e a d into t h e Aristotelian text (see e.g. [Andersen & Faye 1980]). We have to follow Prior in his refutation of L u k a s i e w i c z ' i n t e r p r e t a t i o n , and we m a y add that on quite different grounds we ourselves - like m a n y others - are uncomfortable w i t h t h e idea of a trivalent logic. B u t we could not possibly m e n t i o n t h e n a m e a n d p a r t of the work of J a n Lukasiewicz w i t h o u t saying a word in praise of this great logician - and wise philosopher, too. F i r s t of all, L u k a s i e w i c z m a d e a n u m b e r of specific c o n t r i b u t i o n s to the d e v e l o p m e n t of formal a n d m a t h e m a t i c a l logic - t h e m o s t well-known one being his 'Polish Notation', w h i c h is superior to s t a n d a r d logical (infix) n o t a t i o n by being syntactically unambiguous without the aid of parentheses. Even

LUKASIEWICZ'S CONTRIBUTION TO TEMPORAL LOGIC 153 t h o u g h t h i s n o t a t i o n (which Prior also used) h a s failed to establish itself as a standard, it has proved its practical w o r t h at least within c o m p u t e r science. Secondly, a l t h o u g h Lukasiewicz can not be said to h a v e developed a g e n u i n e temporal logic, he was the first logician to actually w o r k out a symbolic calculus sensitive to some of t h e logical a n d philosophical problems associated w i t h t i m e a n d tense; to t h a t e x t e n t he anticipated Prior's work by 30-40 years. T h i r d l y , a n d in o u r opinion m o s t i m p o r t a n t l y of all, Lukasiewicz w a s one of t h e first significant logicians to relate m o d e m symbolic logic to Classical a n d Medieval discussions of logic. T o g e t h e r w i t h a n u m b e r of o t h e r p r o m i n e n t logicians, including A. Tarski, he m a n a g e d to establish a fruitful Polish e n v i r o n m e n t w h i c h took a particular interest in t h e history of logic, especially Ancient and Scholastic. It is worth n o t i n g t h e m a n n e r w h e r e i n m o d e m symbolic logic was related to t h e historical sources: •ukasiewicz and his associates did not m e r e l y apply the f o r m e r to t h e latter, they also took direct i n s p i r a t i o n from the l a t t e r for t h e development of t h e former. The w a y in which Lukasiewicz related Aritotle's text in De Interpretatione, C h a p t e r 9, to a trivalent logic is a first class example of this. T h u s t h e s e P o l i s h logicians e s t a b l i s h e d the very p a r a d i g m , w i t h i n w h i c h Prior h i m s e l f obviously worked - and on which, we may add, this book is also based. P e r h a p s [.ukasiewicz's most r e m a r k a b l e achievement w i t h i n t h i s p a r a d i g m w a s his i n v e s t i g a t i o n of t h e h i s t o r y of t h e propositional logic [1935]. One of Lukasiewicz's students d u r i n g t h e 1930s w a s J. Salamucha, who in 1935 published a book on Ockham's propositional logic. We m a y conclude this section by quoting a p a s s a g e from the G e r m a n t r a n s l a t i o n of t h a t book, w h i c h n e a t l y typifies t h e way in w h i c h this group of P o l i s h logicians t h o u g h t and worked: Wir h a b e n bei Ockham, wie bei fast allen mittelalterlichen A u t o r e n , n e b e n d e r k a t a g o r i s c h e n Syllogistik, in d e r A u s s a g e n u n d A u s s a g e n f u n k t i o n e n von Typos: A ist B, auftreten, noch andere Syllogistiken; vor allem entwicklet sich n a c h Aristoteles die Syllogistik der m o d a l i s i e r t e n

154

CHAPTER 2.3 Aussagen, ferner die Syllogistik der Aussagen, in d e n e n das Zeitwort die F o r m der Z u k u n f t oder der V e r g a n g e n h e i t h a t .... [Salamucha 1950]

T h e quote is typical for this group of pre-war Polish logicians. T h e r e was an awareness of the fact t h a t logicians in the Ancient t i m e s and in the Middle Ages had analysed t e n s e d propositions a n d a r g u m e n t s , and t h a t such analyses were still relevant; at t h e time, however, t h e s e were given less priority t h a n o t h e r k i n d s of logical analysis.

2.4. A TIHtEE-POINT STRUCTURE OF TENSES I n his Elements of Symbolic Logic [1947], H a n s Reichenbach p u t forth a description of tenses which was to h a v e a significant i m p a c t u p o n t h e linguistic community. Reichenbach suggested t h a t in order to u n d e r s t a n d how tenses work we m u s t consider not only t h e t i m e of u t t e r a n c e , and the time of t h e event in question, but also a 'point of reference'. To u n d e r s t a n d the idea of this three-fold distinction, it is probably best first to consider the future perfect, as in 'I shall have s e e n J o h n ' . This sentence clearly speaks of a c e r t a i n event, n a m e l y 'my seeing John'; but it is also clear t h a t it directs us to a f u t u r e t i m e different from t h e time of the (expected) event n a m e l y a t i m e prior to which the event has a l r e a d y occurred. Thus, we m u s t distinguish between the time of t h e event and the t i m e to w h i c h t h e sentence refers. Reichenbach called t h e form e r 'point of t h e event' and the latter 'point of reference', symbolised by E a n d R, respectively. F u r t h e r m o r e , b o t h m u s t of course be d e t e r m i n e d with respect to the time of utterance, t h e 'point of speech' S. A r m e d w i t h these distinctions Reichenbach could give t h e following d i a g r a m for the future perfect:

Future Perfect I shall have seen John

S

E

R

A quite similar analysis can be given for the p a s t perfect 'I h a d seen John'. These two tenses, t h e n - the p a s t perfect and t h e f u t u r e perfect - establish t h e prima facie case for d i s t i n g u i s h i n g b e t w e e n E, S, and R in the description of tenses. However, if t h e difference b e t w e e n E and R is crucial in e x p l a i n i n g t h e p a s t perfect a n d t h e future perfect, it is precisely t h e coincidence b e t w e e n one or more of E, R, a n d S, w h i c h is crucial in e x p l a i n i n g some of the other tenses. Indeed, w h a t particularly i m p r e s s e d linguists was t h e elegant and concise account of t h e 155

156

CHAPTER 2.4

difference b e t w e e n the simple past and the p r e s e n t perfect w h i c h Reichenbach could give on t h e basis of t h e three-fold distinction. In g r a m m a r s of English, six tenses are standardly recognised; the d i a g r a m for each of these can be seen in this figure (cf. [Reichenbach 1947, p. 290]):

Future Perfect

Simple Future

I shall have seen John

I shall see John

I

I

i

S

E

R

~

I

i

S,R

E

Past Perfect

Simple Past

I had seen J o h n

I saw John

I

I

l

E

R

S

,,,_

I

I

R, E

S

Present Perfect

Present

I have seen John

I see John

I

t

E

S, R

~

..-

I

S, R, E

On this account, the crucial difference between the simple past and the present perfect is determined by the relative 'position' of the reference point. In the case of the simple past, the diagram clearly suggests t h a t the point of reference coincides with the point of the event. Thus the sentence 'I saw John' clearly refers to the past, but it makes no discernible distinction between the time of the event - E - and the time from which this event is seen, i.e. the reference time R. In the case of the present perfect, the event is also situated in the past, but here, the point of reference coincides with the point of speech. Reichenbach's system makes a r a t h e r strong prediction about the notion of tenses, logically as well as grammatically. If tenses are in general to be construed as a three-point structure, the

A THREE-POINT STRUCTURE OF TENSES

157

p o s s i b l e a r r a n g e m e n t s of this kind of s t r u c t u r e m u s t e x h a u s t t h e s e t of p o s s i b l e t e n s e s . In p r i n c i p l e , R e i c h e n b a c h ' s s y s t e m a t i s a t i o n allows for 13 different t e n s e s ; he only r e g a r d e d nine of t h e s e as significantly different, though: I f w e w i s h to s y s t e m a t i z e t h e p o s s i b l e t e n s e s w e c a n p r o c e e d as follows. We choose t h e p o i n t of speech as t h e s t a r t i n g point; relative to it the point of reference can be in t h e past, at t h e s a m e time, or in t h e future. This f u r n i s h e s t h r e e possibilities. N e x t we consider t h e point of the event; it can be before, s i m u l t a n e o u s with, or a f t e r t h e reference point. We t h u s arrive at 3 • 3 = 9 possible forms, which w e call fundamental forms. F u r t h e r differences of form r e s u l t only w h e n t h e position of the e v e n t r e l a t i v e to t h e point of s p e e c h is c o n s i d e r e d ; this position, however, is usually irrelevant [our italics] [p. 296] T h e fact t h a t R e i c h e n b a c h considered t h e relative positions of E a n d S as basically irrelevant explains a slight oddity a b o u t his d i a g r a m for t h e f u t u r e perfect. The s e n t e n c e 'I shall h a v e s e e n J o h n ' w o u l d also s e e m to be t r u e even if t h e s p e a k e r h a s in m i n d an e v e n t w h i c h h a s a l r e a d y occurred - t h a t is, t h e s t r u c t u r e w o u l d be E .... S .... R (this is p e r h a p s a less n a t u r a l reading, b u t quite possible). However, t h e above quotation m a k e s it clear t h a t a c c o r d i n g to R e i c h e n b a c h , t h e r e is no i m p o r t a n t d i f f e r e n c e b e t w e e n E .... S .... R a n d S .... E .... R. I n d e e d , in s u m m i n g u p t h e possible tenses he explicitly aligns S . . . . E .... R S, E .... R E .... S .... R u n d e r t h e c o m m o n h e a d i n g of ' f u t u r e p e r f e c t ' . A s i m i l a r a c c o u n t is g i v e n for R .... E .... S, R .... S . . . . E, a n d R .... S,E, w h i c h h e collects u n d e r t h e h e a d i n g 'posterior past'. None of t h e six t r a d i t i o n a l t e n s e s corresponds to posterior past, b u t it can be s t a t e d b y some transcription, as in 'I w a s to see J o h n once more' or 'the letter w a s to cause h e r great anxiety'.

158

C H A P T E R 2.4

OTTO J E S P E R S E N ON TIME AND TENSE According to Reichenbach, t h e idea of a three-point s t r u c t u r e for t e n s e s h a d already b e e n s u g g e s t e d by the g r e a t D a n i s h linguist Otto Jespersen (1860-1943): In J. 0. H. Jespersen's excellent analysis of G r a m m a r (The Philosophy of Grammar, H. Holt, New York, 1924) I find t h e three-point s t r u c t u r e indicated for such t e n s e s as the past perfect and the f u t u r e perfect (p. 256), b u t not applied to the interpretation of t h e other tenses. This explains the difficulties which even J e s p e r s e n has in distinguishing the p r e s e n t perfect from t h e simple p a s t (p. 269). He sees correctly t h e close connection between the p r e s e n t tense a n d the p r e s e n t perfect, recognizable in such sentences as 'now I have e a t e n enough'. B u t he gives a r a t h e r vague defmition of t h e present perfect and calls it 'a retrospective variety of the present'. [Reichenbach 1947, p. 290] I n fact, w h e n looking into J e s p e r s e n ' s text it t a k e s some consideration to see how t h e three-point structure can be said to be suggested here. Jespersen's book deals with time a n d tense in two chapters. In these chapters there are no explicit s t a t e m e n t s t h a t s u c h a three-fold distinction h a s been made, n o r do t h e y m a k e any clear q u a l i t a t i v e - let alone t e r m i n o l o g i c a l distinction between speech t i m e a n d reference time. However, J e s p e r s e n first suggests t h a t we should basically consider seven different possible tenses. For these he introduces this diagram, which m u s t be w h a t Reichenbach sees as the first suggestion of a three-point structure: present

I

I past

I

before-past

r

I

future

I

after-past

i

before-future

i

I

after-future

A THREE-POINT STRUCTURE OF TENSES

159

However, the diagram is immediately reworked into the one below, since "it is much better to arrange the seven "times" in one straight line... For there can be no doubt t h a t we are obliged (by the essence of time itself, or at any rate by a necessity of our thinking) to figure to ourselves time as something h a v i n g one d i m e n s i o n only, t h u s capable of being r e p r e s e n t e d by one straight line" [Jespersen, p. 256]. Such a representation - in which an indication of a three-point structure is easier to see - is given on p. 257:

post-future

Cc

after-future

future

Cb

future

ante-future

Ca

before-futureJ

present

C) B

~

C future

/

present

post-preterit

Ac

after-past

preterit

Ab

past

ante-preterit

Aa

before-past

~ A past

J e s p e r s e n here uses the terms before-past, past, etc., in an ontological sense, i.e. concerning the 'essence of time', whereas the t e r m s ante-preterit, preterit, etc., are the corresponding grammatical terms. The four 'subordinate times' can be briefly described as follows: a) before-past (ante-preterit): corresponds to the past perfect; b) after-past: is described by periphrastic forms such as 'The letter was to cause anxiety'; c) before-future: corresponds to the future perfect;

160

CHAPTER 2.4 d) after-future: is described by periphrastic forms such as 'I shall be going to see John'.

The closest J e s p e r s e n gets anywhere in his text to describing a Reichenbach-like three-point structure is in his explanation of the figure above: This figure, a n d the letters indicating the various divisions, show the relative value of the seven points, the subordinate "times" being orientated with regard to some point in the p a s t (Ab) a n d in t h e future (Cb) exactly as t h e m a i n times (A and C) a r e o r i e n t a t e d w i t h r e g a r d to t h e p r e s e n t m o m e n t (B). [p. 257] Clearly, in addition to the present (the time of utterance), two m o r e 'points of o r i e n t a t i o n ' are b r o u g h t into t h e picture. However, R e i c h e n b a c h ' s i m p l e m e n t a t i o n of this idea differs from Jespersen in t h r e e important respects: 1) Obviously, J e s p e r s e n considers three points - r a t h e r t h a n just two - to be relevant only for the subordinate tenses. This is crucial, of course, for "the difficulties which even J e s p e r s e n has" in explaining the difference between t h e s i m p l e p a s t a n d t h e f u t u r e perfect - in t h e m a n n e r suggested by Reichenbach. 2) J e s p e r s e n m a k e s no q u a l i t a t i v e or t e r m i n o l o g i c a l distinction between the two points besides the present. 3) An e x h a u s t i v e s y s t e m of tenses, respectively, times, cannot be constructed on the basis of t h e above distinctions. J e s p e r s e n says: "The system t h u s a t t a i n e d s e e m s to be logically impregnable, but, as we shall see, it does not claim to comprise all possible time-categories nor all those tenses t h a t are a c t u a l l y found in languages" [p. 257]. This is obviously in c o n t r a s t to Reichenbach, who, as we have a l r e a d y seen, p r o p o s e s his t h r e e - p o i n t s y s t e m as a comprehensive account of all possible tenses.

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161

It is a very interesting fact t h a t one tense which J e s p e r s e n considers to be 'beyond' his seven-tense system is exactly the present perfect: The system of tenses given above will probably have to meet the objection t h a t it assigns no place to the perfect, have written... This, however, is really no defect of the system, for the perfect cannot be fitted into t h e simple series, because besides the purely temporal e l e m e n t it contains the element of result.., it represents the present state as the outcome of past events, and may therefore be called a retrospective variety of the present. [p. 269] F u r t h e r m o r e , J e s p e r s e n points out another significant difference between the simple past and the p r e s e n t perfect, namely t h a t the former is about some definite point in the past, as opposed to the latter. Indeed, in English this difference is taken so strictly that it "does not allow the use of the perfect if a definite point in the past is meant, w h e t h e r this be expressly m e n t i o n e d or not" [p. 270]. This is in contrast to some other languages, e.g. German and Danish, the latter of which tolerates combinations like "jeg h a r set h a m igor (I have seen him yesterday)" [p. 271]. It appears, then, that Jespersen considered the present perfect to be a tense which could not be fitted into his general 'structure of time' with corresponding tenses (the diagram on p. 257). Or perhaps we should rather say t h a t in this structure it would not be wrong to place the present perfect under Ab, together with the simple past (preterit) - but this would not be sufficient to describe it. The reason for this is that the present perfect bring in an element which is not strictly temporal (the element of result). Now Reichenbach, on the other h a n d , in his g e n e r a l i s e d f r a m e w o r k did m a n a g e to give a clear formal distinction between the simple past and the present perfect. Therefore, his system seems to be an improvement of Jespersen's ideas. But of course, this only holds provided t h a t his generalisation is also otherwise logically and linguistically tenable. We shall now try to assess these questions.

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PRIOR, J E S P E R S E N AND REICHENBACH For all its intuitive elegance, it is clear that Reichenbach's formalism is very limited. It is certainly not a complete calculus, but at best it could be seen as a suggestion of some guidelines along which such a system could be constructed. However, even w h e n m e a s u r e d on its own terms the system harbours severe difficulties. As we have seen t h e r e is a difference between Jespersen a n d Reichenbach in t h a t the latter makes a sharp distinction between 'point of reference' and 'point of event'. This is the v e r y move on which the general viability of Reichenbach's systematisation rests - as well as its accounts of the individual tenses. One who clearly saw this was Prior, who in [1967] discussed the precursors of tense logic. Herein he gave Reichenbach some credit for his o b s e r v a t i o n s , b u t t h e n w e n t on to s t a t e t h a t "Reichenbach's scheme, however, will not do as it stands; it is at once too simple and too complicated" [1967, p. 13]. The main target of Prior's a t t a c k was exactly the sharp distinction between 'point of reference' and 'point of event'. Consider a complicated future tense like this one: 'I shall have been going to see John'. T h i s s e n t e n c e is p e r h a p s not v e r y n a t u r a l , but it is g r a m m a t i c a l l y correct, and it does express a tense-relation for which we must be able to account. It is not too hard to see t h a t to describe this tense, we in fact need two points of reference. Prior's 'Reichenbachian' diagram for this case looks like this:

I

I

I

I

S

R2

E

R1

So, for such a t e n s e the Reichenbachian framework would have to be extended to allow for two points of reference; and in

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general, an a r b i t r a r y n u m b e r of 'reference points' m i g h t be needed. Prior could therefore observe t h a t ... once this possibility is seen, it becomes unnecessary a n d misleading to m a k e such a sharp distinction between t h e point or points of reference and t h e point of speech; t h e point of speech is j u s t the first point of reference. (This, no doubt, destroys Reichenbach's way of d i s t i n g u i s h i n g t h e simple p a s t a n d t h e p r e s e n t perfect; b u t t h a t distinction needs more subtle machinery in any case.) [1967, p. 13] It is crucial for Reichenbach's system t h a t three points of t i m e should always be t a k e n into consideration. But we have just seen t h a t this m a y sometimes be too little; and, as t h e quotation also suggests, it is sometimes too much. For in the account of, say, t h e simple past - in t e r m s of an R,E .... S diagram, w h e r e R = E w h y should we accept t h a t t h e r e is really m o r e t h a n two temporal indicators involved? And even more so, w h y should we accept such a t h i n g for t h e present S,R,E (where S=R=E)? Only cogent logico-linguistic reasons should m a k e one accept t h a t t h e r e are three t e m p o r a l indicators at play in these cases. B u t r e f e r r i n g to t h e fact t h a t Reichenbach's account a p p a r e n t l y explains the difference between the simple p a s t and the p r e s e n t perfect is at best c i r c u m s t a n t i a l evidence; for it e x p l a i n s this difference only if the distinctions are valid beforehand. I n c i d e n t a l l y , t h e s e o b s e r v a t i o n s also show t h a t t h e Reichenbach f r a m e w o r k really ought to distinguish between on one h a n d t h e t e m p o r a l indicators - or c o n c e p t s - of 'event', 'reference' and 'speech', and on the other h a n d t h e points of time w h i c h they 'indicate'. T h u s for instance, if the event E occurs at t, we might say t h a t z(E)=t. Only thus can a diagram like T(R), z(E) .... ~(S)

m a k e a m e a n i n g f u l d i s t i n c t i o n b e t w e e n m o r e t h a n two indicators. Here, R a n d E are co-extensive w i t h respect to their t i m e - p a r a m e t e r , b u t they m u s t be a s s u m e d to be intensionaUy different (i.e. v(R)=~(E), but E ¢R).

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As we have already seen Jespersen did not assume that t h r e e points of time were relevant for the tenses in general; rather, in his s y s t e m such a structure is applicable only to the subordinate times. For the past and for the future, only one point of 'orientation' besides the present is taken into consideration - a n d for the present, only the present is relevant. Moreover, even w h e n a three-point structure becomes relevant, there is no suggestion that the two points besides the present qualitatively differ from each other. In all these respects, it is Prior - r a t h e r t h a n Reichenbach - who is in agreement with Jespersen. To be true, Jespersen's system does not foresee a multiple-point s t r u c t u r e as Prior does; but then again, Jespersen explicitly stated t h a t his a r r a n g e m e n t of the seven-tense system was not exhaustive. The divisions which he does make are, however, more n a t u r a l l y expressed in terms of tense logic than in terms of the three-point structure: for instance, we have the following correspondences (assuming linear discrete time): Aa: before-past (past perfect) Ab: past (simple past) Ac: after-past

PPq Pq PFq

These tense-logical forms are really closer to Jespersen's system t h a n the three-point structures. In each case, the n u m b e r of tense-operators clearly agrees with the n u m b e r of 'points of orientation' considered relevant by Jespersen. F o r m s where still more 'points of orientation' are needed, as in 'I shall have been going to see John', can be represented by tense-logical formulae with a corresponding n u m b e r of operators, e.g. FPFq. Obviously, these tense-logical forms also agree with Jespersen in m a k i n g no qualitative differences between the corresponding 'points of orientation'. One m i n o r discrepancy should be mentioned: the tense-logical form PFq_differs slightly from the category Ac, which in Jespersen's diagram seems unambiguously situated in the past. PFq, on the other hand, may also be true if q takes place at the present m o m e n t or even at a future moment. However, this does not contradict Ac, but is simply more general. As far as we can

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see, PFq actually describes tense-constructions of the f o r m 'it was to be t h a t q' better t h a n Ac. For it is clear that such linguistic forms can also refer to an event q which is in the 'absolute future' (even t h o u g h this m a y be rare in actual language use). But if somebody insists that this category Ac m u s t refer strictly to the past, we could obtain this by a metrical tense-logical form, as in P(n)F(m)q, w h e r e n > m. We m e n t i o n e d earlier three differences between J e s p e r s e n a n d Reichenbach h a v i n g to do with (a) whether t h r e e - p o i n t s t r u c t u r e s should always be used, (b) w h e t h e r there are a n y qualitative differences between the different points involved, a n d (c) w h e t h e r the respective systems are exhaustive or not. T h e previous p a r a g r a p h s should have m a d e it clear that on the first two points Prior is obviously m u c h closer to Jespersen - t h e y b o t h a n s w e r in the negative - t h a n is Reichenbach, who c o n f i r m s both of these points. As for (c), it is at least clear that J e s p e r s e n did not consider his seven tense system to be linguistically complete, b u t there are some indications that he considered it to be logically, or conceptually, exhaustive. Nevertheless, he clearly did not t h i n k that all linguistically realised tenses could be uniquely c a p t u r e d by his system, as opposed to R e i c h e n b a c h ' s belief in his 9 tenses. But neither Jespersen nor Reichenbach h a d available formal tense logic. In this discipline, it has been m a d e clear t h a t t h e n u m b e r of tenses depends on several a s s u m p t i o n s about the s t r u c t u r e of time (one result by Prior and H a m b l i n , yielding 30 different tenses on certain given assumptions, is m e n t i o n e d in the next chapter). Jespersen's openness in this r e spect, however, goes better with Prior's findings t h a n Reichenbach's strong prediction of just 9 (or 13) possible tenses. Reichenbach was a brilliant mind, and m a n y of his results also on the philosophy of time - have h a d lasting value. F a i r n e s s d e m a n d s t h a t this be acknowledged, and in the case of his 'threepoint structure' it m u s t at least be admitted that for its day it w a s a n e l e g a n t a n d advanced proposal. But its real deficiencies together w i t h its very success m a d e it counter-productive Prior considered Reichenbach's work in this respect as an i m pediment r a t h e r t h a n a help in the development of tense logic.

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Apparently, Prior did not question t h a t Reichenbach's ideas h a d essentially originated in Jespersen's work. In fact, he even accepted that "Jespersen only used this 'three-point structure' to explain these two tenses [future perfect and past perfect]..." [1967, p. 12]. However, the discussion of Jespersen should have m a d e it clear that this is oversimplifying m a t t e r s somewhat, since all four "subordinate times" depend on some kind of threepoint structure. It is true, however, that only two of these times correspond directly to traditionally recognised tenses. Nevertheless, the fact of the matter is that Jespersen's ideas in many w a y s seem more compatible w i t h t e n s e logic t h a n w i t h Reichenbach's system, a fact which Prior could well have put to good use.

2.5. A.N. PRIOR'S TENSE-LOGIC The history of tensed logic proper began with Prior's insight: Tensed propositions are propositional functions, with times as arguments. [Bas C. van F r a s s e n 1980] A. N. Prior m u s t be said to have laid the foundation for m o d e r n tense-logic. He revived t h e medieval a t t e m p t at f o r m u l a t i n g a t e m p o r a l logic for n a t u r a l language. T h e r e f o r e his w o r k also established a paradigm applicable to the exact s t u d y of t h e logic of n a t u r a l language. Prior held t h a t logic s h o u l d be related as closely as possible to intuitions embodied in everyday discourse, a n d his tense logic can indeed account for a large n u m b e r of linguistic inferences. In t h e 1950's and 1960's he laid out t h e found a t i o n of tense-logic and showed that this i m p o r t a n t discipline w a s i n t i m a t e l y connected w i t h modal logic. P r i o r also a r g u e d t h a t t e m p o r a l logic is f u n d a m e n t a l for u n d e r s t a n d i n g and describing t h e world in which we live. He r e g a r d e d tense a n d m o d a l logic as particularly relevant to a n u m b e r of i m p o r t a n t theological problems. U s i n g his temporal logic Prior a n a l y s e d t h e f u n d a m e n t a l question of d e t e r m i n i s m v e r s u s freedom of choice. A r t h u r N o r m a n Prior was born in Masterton, New Zealand, on December 4th., 1914. His m o t h e r died a f o r t n i g h t after his birth. His father was a doctor and a medical officer d u r i n g t h e First World War, and Prior was brought u p by his a u n t s a n d g r a n d p a r e n t s . Both of his grandfathers w e r e M e t h o d i s t ministem. Prior went to Otago University at Dunedin in 1932. He set out to s t u d y medicine, but after a short time he i n s t e a d w e n t into p h i l o s o p h y a n d psychology. In 1934 he a t t e n d e d F i n d l a y ' s courses on ethics and logic. Through Findlay Prior became interested in the history of logic and was introduced to Prantl's textbooks. His M.A. thesis was devoted to this subject. In 1949 Prior wrote about Findlay: "I owe to his teaching, directly or indirectly, all t h a t I know of either Logic or Ethics" [Kenny p. 323]. Prior was brought up as a Methodist, but while he was a stud e n t he came to consider Methodistic theology too unsystematic, 167

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a n d he became a Presbyterian. He also became a very active m e m b e r of t h e S t u d e n t C h r i s t i a n M o v e m e n t (SCM). In t h e years about 1940 he found himself in a crisis of belief. D u r i n g t h e s e y e a r s h e w r o t e t h e a r t i c l e 'Can religion be discussed?'(1942), in which he advocated an almost atheistic position. This view, however, does not s e e m to have lasted very long. He c o n t i n u e d to t r e a s u r e his theological library and to join t h e w o r k of t h e SCM [Kenny p. 326]. L a t e r in his life, however, he became an agnostic. It is very likely t h a t Prior's abandoning of Christianity a n d his becoming an agnostic was related to t h e problems concerning f r e e d o m a n d time. He was acutely a w a r e of the fact t h a t a n u m b e r of significant Christian t h i n k e r s in the course of history h a d attacked or criticised the idea of free will. In a paper entitled ' D e t e r m i n i s m in Philosophy a n d Theology' [DPT] (probably w r i t t e n in his Calvinist period), he formulated this in t h e follow i n g way: It is extremely rare for philosophers to pay any great attention to t h e fact t h a t a whole line of Christian thinkers, runn i n g from A u g u s t i n e (to trace it no further back) t h r o u g h L u t h e r and Calvin and Pascal to B a r t h and B r u n n e r in our own day, have attacked free will in the n a m e of religion. [DPT, p. 1] Prior added t h a t for instance J o n a t h a n Edwards, who produced a novel defence of Calvinism in 18th-century New England, did it by " d e m o n s t r a t i n g t h e absurdity of free will itself' [DPT p. 1]. However, even if we accept that t h e idea of free will is illusory (and at the t i m e of writing DPT, Prior seems to have accepted this, in contrast to his later convictions), the ordinary perception of freedom and of guilt has to be explained: E v e n t h o s e of us who accept a s t r a i g h t f o r w a r d d e t e r m i nism have to give some account of men's feeling of freedom, and their feeling of guilt; ... [DPT, p. 2-3]

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This state of inner conflict between two p a r t s of the self, in which we feel both responsible and enslaved, is also one to which no one can be a stranger... [DPT, p. 4] Even so, Prior felt t h a t a Christian had to be a determinist, and t h a t the believer m u s t accept that we are guilty of t h a t which we are totally helpless to alter [DPT, p. 2]. It appears t h a t Prior accepted Edward's and others' a r g u m e n t t h a t the doctrine of God's infallible and complete foreknowledge is incompatible w i t h the contingency of future events. "I m u s t confess I can't see t h a t foreknowledge is compatible with preventability", he said [IWB, p. 12]. Prior clearly understood t h a t foreknowledge should not itself be seen as the cause of that which is foreknown, but r a t h e r as an effect. But what has got so far as to have effects is surely "beyond stopping", he pointed out [IWB, p. 12]. The only w a y out of this for anyone who wants to accept the doctrine of divine foreknowledge appeared to be Thomas Aquinas' idea of atemporal knowledge. Thomas "taught t h a t God doesn't experience time as passing, but has it present all at once. In other words, God sees time as tapestry" [SFTT, p. 2]. This solution was not at all attractive to Prior, since it seems to be in conflict with the reality of tenses. Moreover, atemporal knowledge cannot be foreknowledge in the strict sense. Prior's own view was t h a t God "cannot know the answer to the question "How will t h a t person choose?" because there isn't any answer to it until he has chosen" [SFTT, p. 3]. This position of course suggests t h a t Prior at some s t a g e h a d adopted an indeterministic position. T h a t in t u r n would m e a n t h a t a full Christian faith could no longer be held, provided that Christianity implies a full forekonowledge by God, and t h a t such foreknowledge is incompatible with the notion of free will. In 1943 he married Mary. From 1946 to 1958 he taught philosophy at C a n t e r b u r y University College in New Zealand. In 1953 he became a professor of philosophy. In 1949 his book Logic and the Basis of Ethics had been published. After t h a t time he became even more interested in logical problems. During 1950 and 1951 he wrote a manuscript for a book with the work i n g title The Craft of Logic. This book was, however, never

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published as a whole, but in 1976 P. T. Geach and A. J. P. Kenny edited p a r t s of it. In the first chapter of the book, Propositions and Sentences, the a u t h o r a m o n g o t h e r things a n a l y s e d Aristotle's view on some of the problems concerning time and tense. Prior found t h a t according to the ancient as well as the medieval view a proposition may be true at one time and false at another. He described this view in the following way: ... t h e s t a t e m e n t or opinion t h a t someone is sitting will be true so long as the person in question is in fact seated, and will become false - if it is persisted in - as soon as he rises. [Prior 1976b, p. 38] In the following years Prior worked mainly on questions in the history of logic. From 1952 to 1955 he had seven articles on the history of logic published. Four of these were concerned with Medieval logic and one with Diodorean logic. His interest in the history of logic is also evident in his Formal Logic, published in 1955. According to M a r y Prior his resurging i n t e r e s t in the history of logic was very much due to the fact that the university library bought Bochenski's Prdcis de Logique Mathdmatique (1948). It seems t h a t a short article by Benson Mates [1949] made Prior even more aware of the interesting relation between time and logic. The paper was concerned with Diodorean logic, primarily Diodorus' definition of implication. Prior seemed to realise t h a t it might be possible to relate Diodorus' ideas to contemporary works on modality by developing a calculus which included t e m p o r a l operators analogous to the operators of modal logic. M a r y .Prior has described the first occurrence of this idea: "I r e m e m b e r his waking me one night, coming and sitting on m y bed, a n d reading a footnote from John Findlay's article on Time, and saying he thought one could make a formalised tense logic." This m u s t have been some time in 1953 [Kenny p. 336]. The footnote which Prior studied that night was the following: And o u r conventions with regard to tenses are so well worked out that we have practically the materials in t h e m

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for a formal calculus... The calculus of tenses should have been included in the m o d e m development of modal logics. It includes such obvious propositions as t h a t x p r e s e n t = (x present) p r e s e n t ; x f u t u r e = (x future) present = (x present) future; also such comparatively recondite propositions as t h a t (x).(x past) future; i.e. all events, past and future will be past. [Gale p. 159-60] To be sure, Findlay's considerations on the relation b e t w e e n t i m e a n d logic in this footnote were n o t very elaborated, b u t it gave Prior t h e idea of developing a formal calculus which would c a p t u r e t h i s relation in detail. For this reason Prior called Findlay "the founding father of m o d e r n tense logic" [Prior 1967, p. 1]. But t h e r e are, in our opinion, certainly not sufficient reasons for viewing Findlay as the founder of tense logic. The hon o u r of being t h e founder m u s t w i t h o u t doubt be a t t r i b u t e d to Prior himself. W i t h his m a n y articles and books on questions in t e n s e logic he p r e s e n t e d a very extensive and t h o r o u g h corpus, which still forms the basis of tense logic as a discipline. Findlay's major merit in tense logic is, as Jean-Louis Gardies [1975, p. 40] h a s r e m a r k e d , to have h a d the luck of inspiring Prior to initiate t h e development of formal tense logic. In fact, Findlay's footnote was certainly not the only source of inspiration for Prior's incipient formal study of the logic of time. Prior highly v a l u e d various parts of Polish logic like L u k a s i e wicz's three-valued logic. And of course, from the previous stages of his career he was well acquainted with a huge historical material on questions related to t e m p o r a l logic. A persistent feat u r e t h r o u g h o u t Prior's works is a clear interest in the history of logic. Indeed, Prior took an interest in t h e history of logic not only as a subject in its own right, but he also saw the works of ancient and medieval logicians as a significant contribution to the c o n t e m p o r a r y development of logic. He was particularly interested in Aristotle, Diodorus, and t h e Scholastics, but his interest also extended to more recent logicians such as Boole and Peirce, w h o m he called "the greatest of all symbolic logicians" [1957c].

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It is likely that Prior w a s already in the early fifties acquainted w i t h McTaggart's considerations on time [1908], and Reichenbach's examinations of t h e tenses of verbs [1947]. However, he m a d e no reference to t h o s e ideas in his introductions to tense logic d u r i n g the 1950's. T h e reason m a y be t h a t he t h o u g h t of these philosophers as adversaries. At least, he himself declared t h a t at first he considered McTaggart an 'enemy', until P e t e r Geach m a d e him a w a r e of t h e i m p o r t a n c e a n d relevance of McTaggart's distinction between the so-called A- and B-series conceptions of time [1967, p. vii. The A-series conception is based on the notions of past, p r e s e n t , and future, as opposed to a 'tapestry' view on time, as embodied by t h e B-series conception of time. Prior later formally elaborated McTaggart's distinction, and showed that we c a n discuss time using either a tense logic, corresponding to t h e A-series conception, or u s i n g an earlierlater calculus, corresponding to the B-series conception; we shall show in detail how he r e l a t e d the two to one a n o t h e r in chapter 2.8. Prior's i n t e r e s t in M c T a g g a r t s o b s e r v a t i o n s was first a r o u s e d w h e n he r e a l i s e d t h a t M c T a g g a r t h a d offered an a r g u m e n t to the effect t h a t t h e B-series p r e s u p p o s e s t h e Aseries r a t h e r t h a n vice versa [1967, p.2]. Prior was particularly concerned w i t h M c T a g g a r t ' s a r g u m e n t a g a i n s t t h e reality of tenses. He pointed out t h a t the a r g u m e n t is in fact based on one crucial assumption, n a m e l y t h a t tenses should be explicated in t e r m s of a n o n - t e m p o r a l 'is', a t t a c h i n g e i t h e r an event or a ' m o m e n t ' to a ' m o m e n t ' . T h a t a s s u m p t i o n is certainly very controversial. N e v e r t h e l e s s , since Prior's s t u d i e s b r o u g h t r e n e w e d f a m e to M c T a g g a r t ' s a r g u m e n t , t h i s so-called McTaggart's paradox h a s been very i m p o r t a n t in the debate a b o u t v a r i o u s k i n d s of t e m p o r a l logic a n d t h e i r m u t u a l relations. In the next p a r t of this book, we shall discuss t h e kind of reasoning involved in McTaggart's paradox. With r e g a r d to R e i c h e n b a c h ' s ideas, however, he did n o t change his mind. As we h a v e seen in chapter 2.4, Reichenbach m a d e some elegant observations, but t h e f o r m a l i s m he constructed w a s very limited. Indeed, its crucial idea of a threepoint s t r u c t u r e directly resists some required logical generalisations. It is also d u b i t a b l e w h e t h e r its capacity for linguistic

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generalisation really goes very far - in spite of considerable initial success in the linguistic community. In consequence, Prior r e g a r d e d Reichenbach's analysis as in some w a y s "a hindrance r a t h e r t h a n a help to the construction of a logic of tenses" [1967, p. 13]. Prior shared the medieval view on statements. He presented this view in Past, Present and Future, quoting P e t e r Geach, who h a d formulated it as early as 1949: Such expressions as 'at time t' are out of place in expounding scholastic views of time and motion. For a scholastic, 'Socrates is sitting' is a complete proposition, enuntiabile, which is sometimes true, sometimes false; not an incomplete expression requiring a further phrase like 'at time t' to make it into an assertion. [Prior 1967, p. 15] Prior continued to examine the Scholastic sources himself, and in his writings he clearly d e m o n s t r a t e d the validity of Geach's formulation of the Scholastics' view on propositions. Prior was invited to Oxford as 'John Locke Lecturer' in Philosophy in 1955-56. This led on to the Prior family moving in 1959 to M a n c h e s t e r and a few y e a r s l a t e r to Oxford, w h e r e Prior worked at BaUiol College. The J o h n Locke lectures gave Prior an excellent opportunity to present his new findings regarding time and modality. The lectures were held on Mondays. Among the participants were J o h n Lemmon, Ivo Thomas, and P e t e r Geach [Kenny p. 337]. T h e lectures were l a t e r published as the book Time and Modality (1957). It was this work which made Prior internationally known. After the publication of Time and Modality he received a n u m b e r of important and interesting letters from various logicians. One of the logicians who wrote to Prior was Saul Kripke. In two letters to Prior in September and October 1958 Kripke put forth some very stimulating ideas r e g a r d i n g temporal logic. In the next section we shall examine Kripke's ideas and t h e i r impact on Prior's work. According to Peter Geach, Prior regarded his own research into the logic of ordinary language constructions as a continua-

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tion of the medieval tradition [Geach p. 188]. His attitude was congenial to t h a t of the young Russell in Principles of Mathematics: ordinary language is not a logician's master, but it m u s t be his guide [Geach p. 187]. After all logic in Prior's opinion "is not primarily about language, but about the real world" [TR, p. 1]. For this reason he strongly opposed the formalistic view on logic: Formalism, i.e. the theory that logic is j u s t about symbols and not about things, is false. [TR p. 1] I cannot see how any statement whatever can be m a d e t r u e simply by using language in a particular way... [WL, p. 2] Prior's own answer to the question about the nature of logic ran as follows: Logic deals, at bottom, with s t a t e m e n t s - it enquires into what s t a t e m e n t s follow from what - but logicians aren't entirely agreed as to w h a t a statement is. Ancient and medieval logicians thought of a statement as something t h a t can be true at one time and false at another. [SFTT, p. 1] It is an obvious consequence of the ancient and medieval view t h a t time should not be ignored in logic. Following this view Prior stressed t h a t "the tense of a statement must be t a k e n seriously" [SFTT, p. 2]. To Prior, all logic was in a sense tense logic: "... tenseless statements of modern logic are just a special case of statements in the old sense ..." [SFTT, p. 2]. He argued t h a t tense logic is based on two fundamental assumptions [Prior 1957a, p. 104]: 1) tense-distinctions are a proper subject of logical reflection, 2) what is true at one time is in m a n y cases false at a n o t h e r time, and vice versa. Prior observed t h a t ancient and medieval logicians took t h e s e assumptions for granted, but that they were eventually denied

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175

(or simply ignored) after the Renaissance. Prior h i m s e l f can be said to have realised the possibility of formulating a logic based on these old assumptions. In fact, he took the assumptions even further and also claimed the reality of tenses: So far, then, as I have anything t h a t you could call a philosophical creed, its first article is this: ! believe in the reality of the distinction between past, present, and future. I believe t h a t w h a t we see as a progress of events is a progress of events, a coming to pass of one t h i n g after another, and n o t j u s t a t i m e l e s s t a p e s t r y w i t h e v e r y t h i n g stuck there for good and all. [SFTT, p. 1] It was Prior's conviction t h a t tense logic was not merely a form a l language t o g e t h e r with rules for purely syntactic m a n i p u lations. It also embodied a crucial ontological and epistemological point of view according to which "the tenses (it will be, it w a s the case) are primitive; only present objects exist." [Prior & Fine, 1977, p. 116] To Prior, the present and the real were one and t h e s a m e concept. Shortly before he died he formulated his view in the following way: ...the p r e s e n t simply is the real considered in relation to two p a r t i c u l a r species of unreality, n a m e l y p a s t and f u t u r e . [Prior 1972] It is obvious t h a t Prior was strongly attracted by questions concerning t h e relation between time and existence. In Time and Modality he proposed a s y s t e m called 'Q' which w a s specifically m e a n t to be a 'logic for contingent beings' [1957a, 41 ff.]. System Q can deal with a certain kind of propositions, w h i c h are n o t always 'statable'. Such propositions are p a r t i c u l a r l y interesting from a tense logical point of view. Consider t h e proposition r: 'x exists'. If t h e object x is a c o n t i n g e n t being, t h e n we m a y a s s u m e t h a t it h a s come into existence at s o m e p a s t t i m e (or p e r h a p s it is coming i n t o existence r i g h t now). Before its coming into existence t h e proposition r was not statable. One consequence is t h a t in a tense

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logic w h i c h is sensitive to this problem, such as Q, we cannot in g e n e r a l a s s u m e t h a t Pr and - H ~ r are equivalent. This m a k e s t h i n g s v e r y complicated, so it is u n d e r s t a n d a b l e t h a t Prior s u b s e q u e n t l y chose to leave aside t h a t problem in his first p i o n e e r i n g development of symbolic tense logic. However, late in h i s w o r k h e again took u p this challenge and f o r m u l a t e d a tense-logic for n o n - p e r m a n e n t existents [1968, p. 145 ff.]. I n 1958 Prior entered into a very interesting correspondence w i t h Charles H a m b l i n of T h e New South Wales U n i v e r s i t y of Technology in Australia. T h e i r correspondence led to i m p o r t a n t r e s u l t s , especially on implication relations among t e n s e d propositions. Prior and Hamblin discussed two central issues in tense logic: t h e n u m b e r of non-equivalent tenses, and t h e implicative s t r u c t u r e of the (non-metric) tense operators. In a letter to Prior d a t e d 18th April 1958 H a m b l i n suggested a set of axioms w i t h P a n d F as m o n a d i c o p e r a t o r s , c o r r e s p o n d i n g to "a s i m p l e i n t e r p r e t a t i o n in terms of a two-way infinite c o n t i n u o u s timescale". Hamblin's axioms are: Axl: Ax2: Ax3: Ax4: Ax5:

F(p v q ) - ( F p vFq) - F - p ~ Fp FFp - Fp FPp --- (p v Fp v Pp) - F - P q - ( q v Pq)

H a m b l i n also assumed 3 rules of inference: RI: R2: R3:

IfA is a thesis, t h e n -F~A is also a thesis. IfA -=B is a thesis, then FA - F B is also a thesis. IfA is a thesis, and A' is the result of simultaneously replacing each occurrence of F in A by P a n d each occurrence of P in A by F, then A' is also a thesis. (A' is the socalled mirror-image of A.)

W h e n these axioms and rules are added to the u s u a l proposit i o n a l calculus a n u m b e r of interesting theorems can be proved. In fact, H a m b l i n could prove t h a t "there are j u s t 30 distinct t e n s e s " , w h i c h can be f o r m e d using only P, F a n d n e g a t i o n .

177

A. N. PRIOR'S TENSE-LOGIC

H a m b l i n also s u g g e s t e d a certain implicative structure for t h e tenses. His suggestion can be illustrated like this:

/

P- ~

P-P~~

-P~P ~

P

"x

P-

-F-P

\

/ -F- ~

F-F- ~

-F-F ~

F

These results b e c a m e even more appealing w h e n Prior s t a r t e d to u s e the o p e r a t o r s G (= ~F~) and H (= -P-). (We h a v e n o t been able to fired any explicit explanation as to why Prior chose exactly those two letters. However, M. J. Cresswell has by personal communication suggested to us t h a t G was inspired by t h e p h r a s e 'is always going to be', and H by the phrase 'has always been' .) Using G a n d H, H a m b l i n could s u m m a r i s e the reduction of tenses with more t h a n two adjacent tense-operators into t h e following diagram: GH

FH

PH

HP

GP

FP

HF

P

GH

PH

PH

HP

P

FP

FP

GF

FG

PG

H

GH

H

PH

HP

HP

FP

HF

GF

FG

HG

F

GH

FH

PH

HP

FP

FP

F

GF

FG

FG

G

GH

GH

PH

HP

GP

FP

GF

GF

GF

FG

FG

PG

G

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Finally, in 1965 H a m b l i n a n d Prior e n d e d u p w i t h t h e following nice implicative s t r u c t u r e for the non-metrical tenseoperators, which according to Hamblin is "a bit like a bird's nest" [Hamblin, letter of 6th J u l y 1965]:

/

H~

PH~

HP~

P,,~

In 1967 Prior p u b l i s h e d his major work, Past, Present and Future, in which his approach to tense logic h a d reached a very convincing form. The decade of intense work in t h e field since t h e J o h n Locke lectures h a d brought him a lot further. Also he h a d been able to benefit greatly from the correspondence with logicians like Kripke and Hamblin. As a teacher Prior was very inspiring. He was always able to find nice and u n d e r s t a n d a b l e illustrations of t h e logical systems he w a n t e d to introduce. F o r instance, he would illustrate the fact t h a t LMp cannot be deduced from Mp in t h e following way (where Mp in this context m e a n s ~ either is or will be true', and Lp stands for 'p is and always will be true'): ...it is or will be that Uncle Joe's car is running, but it will not always be true t h a t this is or will be true; so in this sense Mp does not imply LMp [1957c]

A. N. PRIOR'S TENSE-LOGIC

179

It seems clear t h a t he very much liked teaching and lecturing. He w a s not 'the Oxford type', but it appears that he almost imm e d i a t e l y build up a reputation as one of the best lecturers in Oxford. Prior died on October 6th., 1969, whilst on a lecture t o u r in Scandinavia. On the day of his death he was visiting Trondheim in Norway. Prior had by t h e n accomplished an impressive production. The bibliographical overview of Prior's philosophical works comprises more t h a n 150 titles IFlo 1970]. In this overview one can follow how Prior's interests developed in t h e course of his work. Summarising the main trends it can be said t h a t his work until the middle of the 1950's was c h a r a c t e r i s e d by a preoccupation with ethics and the history of logic. From the mid-fifties and onwards he devoted himself mainly to the study of t h e relation between time, modality, and logic. That should be seen as a n a t u r a l consequence of his endeavour to develop the formal calculus of tense logic, a task which he took up a r o u n d 1953 (at the time of being inspired by Findlay's footnote). Nevertheless, we hope to have also made clear that there is no sharp distinction between Prior's philosophical and historical concerns on one h a n d and his work as a formal logician on the other.

2.6. THE IDEA OF BRANCHING TIME I f the determinist sees T i m e as a line, the indeterminist sees it as a s y s t e m o f forking paths... J o h n P. Burgess [1978, p.157]

T h e s t r a i g h t line and the circle, respectively, are the traditional geometrical r e p r e s e n t a t i o n s of time. According to t h e linear conception time is progressive. Strictly speaking, n o t h i n g will stay as it was, everything will change. Even if a p h e n o m e non appears to be stable, say, the whiteness of an object, it is still n o t seen to be identical w i t h 'same' p h e n o m e n o n one m o m e n t ago - since t h a t p h e n o m e n o n does not really exist as opposed to t h e p h e n o m e n o n we are c o n t e m p l a t i n g 'now', and which does exist. According to the circular conception of t i m e n o t h i n g is really new. Any event is a repetition of previous events, and will be repeated indefinitely in the future. T h e s e two geometrical i m a g e s of t i m e have been d o m i n a n t w i t h i n the philosophy of n a t u r e and o t h e r strands of systematic t h i n k i n g from t h e antiquity and up to this century. However, d u r i n g the last decades a n u m b e r of intellectuals have s u g g e s t e d a new k i n d of t i m e models. According to these models t i m e is viewed as a branching system - a tree-structure. Since b r a n c h i n g time models are very i m p o r t a n t in t h e modern analysis of temporality, it is w o r t h t r y i n g to u n d e r s t a n d this new image of time in relation to t h e history of ideas. Consider this figure:

X Y One of the first philosophers of t i m e to formulate the idea of b r a n c h i n g t i m e in a precise m a n n e r w a s Henri Bergson (1859180

THE IDEA OF BRANCHING TIME

181

1941) in his book from 1889 Essai sur les donn~es immddiates de la conscience. In this book Bergson considered t h e problems r e g a r d i n g time a n d free will. As a possible i l l u s t r a t i o n of t h e process of deliberation he discussed t h e above figure [Bergson 1950, p. 176]. B e r g s o n considered a i n t e r p r e t a t i o n of t h i s i l l u s t r a t i o n like t h e following: The p e r s o n in q u e s t i o n h a s t r a v e r s e d a series, MO, of conscious states. At t h e state O he finds t h e two directions, OX and OY, equally open for him. However, Bergson argued t h a t this geometrical r e p r e s e n t a t i o n of the process of coming to a decision is deceptive: This figure does not show me the deed in t h e doing b u t t h e deed already done. Do not ask m e t h e n w h e t h e r t h e self, h a v i n g t r a v e r s e d t h e p a t h MO a n d decided in favour X, could or could not choose Y: I should a n s w e r t h a t t h e question is meaningless, because there is no line MO, no point O, no p a t h OX, no direction OY. To ask such a question is to a d m i t t h e possibility of adequatel~ r e p r e s e n t i n g t i m e by space and a succession by a s i m u l t a n e i t y . [Bergson 1950, p.180] In our century t h e idea of branching t i m e h a s become more acceptable t h a n it was in the 19th century. In this connection t h e a u t h o r s h i p of Borges s t a n d s out p r o m i n e n t l y . A p p a r e n t l y , Borges was the first intellectual to give a detailed description of the new model of time, namely in his short story from 1941 The Garden of Forking Paths [in Borges 1962] (which in some of its e l e m e n t s appears almost a thriller). In t h e following we shall account for the n e w u n d e r s t a n d i n g of t i m e a n t i c i p a t e d a n d compellingly unfolded by this story. Borges' story is set during World War I. T h e Chinese Yu T s u n is a spy for t h e G e r m a n s in England. B u t a c e r t a i n E n g l i s h counter-intelligence officer, captain R i c h a r d M a d d e n , h a s j u s t m a n a g e d to q u a s h t h e spying network to w h i c h Yu T s u n belongs. Yu T s u n h i m s e l f has not yet b e e n t a k e n , b u t c a p t a i n M a d d e n is right on his heels. The Chinese spy, however, still has one i m p o r t a n t t a s k to accomplish for his G e r m a n superiors in Berlin. He m u s t point out to t h e m the t o w n of Albert, where the

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E n g l i s h are building a n e w strategically i m p o r t a n t artillery park. In t h e circumstances he sees no other w a y of achieving this goal t h a n to kill some person w i t h the name of 'Albert'; this incident, then, should reach the headlines of the English newspapers, w h e r e the Berlin office regularly looks for clues from its spies. Yu T s u n searches for possible victims in a telephone directory, a n d t h e only possible victim t u r n s out to be one S t e p h e n Albert, w h o lives about h a l f an hour's travel by t r a i n from Yu Tsun's home. Yu Tsun plans t h e m u r d e r and leaves his place; he s t a r t s c a r r y i n g out his plan, w h i l s t t r y i n g to observe t h e following maxim: W h o s o e v e r would u n d e r t a k e some atrocious e n t e r p r i s e should act as if it were already accomplished, should impose u p o n himself a future as irrevocable as the past. [p. 92] Yu T s u n now sees the m u r d e r of Albert as something inevitable. Albert is already dead in this p l a n n e d future, which has been given t h e glow of necessity by Yu Tsun. And yet the m u r d e r is no necessity. This is strongly s u g g e s t e d by the fact t h a t captain R i c h a r d M a d d e n was right on t h e trail of Yu T s u n and w e n t after h i m on his way to the train, but "by an accident of fate" he does n o t r e a c h h i m [Borges, p. 92]. M a d d e n is only a few m i n u t e s late for the train, b u t this "small victory" [Borges, p. 92] is t h e difference between life and d e a t h for Stephen Albert - and for t h e fate of Yu T s u n himself. Already in this introductory s e q u e n c e t h e b r a n c h i n g b e t w e e n different f u t u r e courses of events is evident to the attentive reader. The past, on the other hand, is irrevocable, necessary a n d unchangeable. P l a n n i n g is an a t t e m p t to assign to the f u t u r e the same characteristics as those of t h e past, even t h o u g h on grounds of principle this m a y only succeed to a certain extent. And Yu Tsun's maxim, it m a y be added, is a recipe for soothing one's conscience by projecting the properties of the past onto the future. References to bifurcations in time pervade the story. W h e n Yu T s u n leaves t h e train at Ashgrove and has to w a l k t h e last stretch to S t e p h e n Albert's house, some local children give h i m the following piece of guidance:

THE IDEA OF BRANCHING TIME

183

The house is a good distance away, but you won't get lost if you t a k e the road to the left and bear to the left at every crossroad. [p. 93] The children's' instruction about turning always to the left reminds Yu Tsun t h a t "such was t h e common formula for finding the central courtyard of certain labyrinths" [p. 93]. His thoughts are t h u s led on to his g r e a t - g r a n d f a t h e r Ts'ui P~n, who for t h i r t e e n years worked on the construction of a maze, "in which all m e n would lose themselves". [p. 93]. The analogy between a labyrinth and time now becomes explicit to Yu Tsun's mind: ... I thought of a maze of mazes, of a sinuous, ever growing maze which would take in both past and future and would somehow involve the stars. [p. 94] These thoughts are being mirrored by nature itself." "... overhead the branches of trees intermingled..." [p. 93]. For a moment Yu T s u n feels as if he is allied with eternity - as a spectator to the totality of temporal courses of events: For an u n d e t e r m i n e d period of time I felt myself cut off from the world, an abstract spectator. [p. 94] W h e n approaching Albert's home, Yu Tsun to his surprise hears Chinese music coming from the garden. Stephen Albert at first m i s t a k e s Yu Tsun for a Chinese consul, who was apparently expected to come round some time to see Albert's garden. Thus a conversation is started, and Yu Tsun eventually learns that his victim-to-be is a sinologist, who holds a profound knowledge about his forefather's Ts'ui P~n's universe of ideas. For this reason Yu Tsun decides to postpone the execution of his otherwise irrevocable decision to kill Albert for about an hour. He lets Albert know t h a t he is a descendant of Ts'ui P~n. The two tog e t h e r t a k e a stroll through the garden. Also here the suggestions of the concept of branching time are clear:

184

CHAPTER 2.6 The damp path zigzagged like those of m y childhood. [p. 95]

A philosophical - a n d at the same time highly existential - conversation then takes place in Albert's library. Stephen Albert is sitting with his back to a large circular clock. Yu Tsun is seated facing t h e clock. The conversation is about Ts'ui P~n, who for t h i r t e e n years lived remote from the world in order to write a book and construct a labyrinth. After the d e a t h of Ts'ui P~n his heirs found only a m e s s of chaotic manuscripts, which were published only because the executor of his will insisted. Yu Tsun himself never u n d e r s t o o d the book. T h a t is evident from his r e m a r k s about it: Such a publication was madness. The book is a shapeless mass of contradictory rough drafts. I examined it once upon a time: the hero dies in the third chapter, while in the fourth he is alive. [p. 96] Yu T s u n holds t h a t the book is by no m e a n s characterised by the logical rules w h i c h in his view every a u t h o r should obey. Thus for instance h e t h i n k s that the t h i r d chapter of the book should respect c h a p t e r two as a p h a s e w h i c h has been concluded. It just h a s not occurred to him t h a t the logic of the book could be quite new. Due to his close studies Stephen Albert, however, has been able to see through the mystery. A fragment of a letter from Ts'ui P~n has proved to hold the decisive key to the right understanding of the book: I leave to various future times, but not to all, m y garden of forking paths. [p. 97] The book and the l a b y r i n t h were not to be considered as two separate pieces of w o r k to be carried out by Ts'ui P~n. They were in essence the same thing. The book was to be constructed as a l a b y r i n t h of time. The a p p a r e n t conflict b e t w e e n the different parts of the book is simply due to the fact that Ts'ui P~n w a n t e d to describe all the possible f u t u r e s concurrently. The

THE IDEA OF BRANCHING TIME

185

book therefore does not respect the usual logic but defines its own logic - a new kind of temporal logic: This is the cause of the contradictions in the novel. Fang, let us say, has a secret. A s t r a n g e r knocks at his door. F a n g m a k e s up his mind to kill him. Naturally there are various possible outcomes. Fang can kill the intruder, the i n t r u d e r can kill Fang, both can be saved, both can die and so on and so on. In Ts'ui P~n's work, all the possible solutions occur, each one being the point of departure for other bifurcations. Sometimes the pathways of this labyrinth converge ..... [p. 98] In other words the novel depicts time as an infinite branching system. Thus Ts'ui P~n has h a n d e d over his proposal for a solution of t h e enigma of time to posteriority. That is to say, he has h a n d e d it over to the different futures after his d e a t h - even t h o u g h it will in fact not be received in some of them. For instance, the solution is not received in the case of those possible futures, in which the executor of the will has the manuscripts burnt in order to prevent their publication. T h r o u g h o u t Borges' short story the description of time as a gigantic branching system gets still more precise. Towards the end of the short story he lets Stephen Albert say: The explanation is obvious. The Garden of Forking P a t h s is a picture, incomplete yet not false, of the universe such as Ts'ui P~n conceived it to be. Differing from Newton and S c h o p e n h a u e r , your a n c e s t o r did not t h i n k of time as absolute and uniform. He believed in an infinite series of times, in a dizzily growing, ever spreading n e t w o r k of diverging, converging and parallel times. This web of time t h e s t r a n d s of which approach one another, bifurcate, i n t e r s e c t or ignore each o t h e r t h r o u g h the c e n t u r i e s embraces every possibility. [p. 100] Borges' conception of time bears m a n y similarities to Leibniz' idea of possible worlds. The different futures represent different

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CHAPTER 2.6

possibilities, a n d this aspect assumes a particular importance with respect to t h e existence of persons. Even though a person exists in one series of time, it cannot at all be taken for granted t h a t he or she exists in a n o t h e r series of time. Borges lets Stephen Albert emphasise the fact that in most times - we might say, courses of events - neither Yu Tsun or he himself (Albert) exist. Moreover, t h e question about the existence of persons in the different series of time gives rise to some considerations on t h e e x t r e m e l y difficult philosophical problems concerning temporal and counterfactual identity: Once again I sensed the population of which I have already spoken. It s e e m e d to me t h a t t h e d e w - d a m p g a r d e n s u r r o u n d i n g t h e house was i n f i n i t e l y s a t u r a t e d w i t h invisible people. All were Albert and myself, secretive, busy and multiform in other dimensions of time. [p. 100] Yu Tsun experiences the counterfactual, yet in a sense real and s i m u l t a n e o u s existence of other 'editions' of himself and Albert - v a r i e d i n f i n i t e l y as in a n i g h t m a r e . And it is understandable t h a t this should appear like a nightmare, for if the n u m b e r of Yu Tsuns is infinite, who t h e n is the real Yu Tsun? A c o m p l e m e n t a r y question to this philosophical and existential problem is this one: what exactly does it m e a n that a n u m b e r of different possible persons are in some sense all Yu Tsun? P e r h a p s in his short story Borges presupposed the a n s w e r later to become prevalent in philosophy, n a m e l y t h a t two possible persons are identical in a 'simultaneous' sense if they have a common history (indistinguishable histories) up to a certain point of time. Borges increases the dramatic effect of the idea of simultaneous identity by letting Albert say: Time is forever dividing itself toward innumerable futures, and in one of t h e m I am your enemy. [p. 100] This u t t e r a n c e answers a warped r e m a r k by Yu Tsun, who claims that in all possible times he would appreciate and be grateful to Albert for his reconstruction of the garden of forking

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paths. Albert's a n s w e r s t a n d s out in d r a m a t i c contrast to Yu T s u n ' s proclaimed gratefulness, since Yu T s u n shortly after fires t h e k i l l i n g s h o t at Albert in a c c o r d a n c e w i t h t h e 'irrevocable decision' m a d e before he even m e t Albert. Borges n e w idea about time is represented in a literary figure, and therefore it is no wonder t h a t a n u m b e r of philosophical and logical problems r e m a i n unanswered. In particular, t h e question about the b r a n c h i n g towards the p a s t is conspicuous. How can Borges accept an idea about a b r a n c h i n g past? W h a t does it m e a n w h e n "the web of time - t h e s t r a n d s of w h i c h approach one another ..., intersect" [p. 100], and "Sometimes the pathways converge" [p. 98]? Does Borges actually m e a n t h a t it m a k e s sense to talk about alternative possibilities of t h e past in the same way as one m a y operate w i t h alternative possibilities of t h e future? Clearly, it is m e a n i n g f u l to talk about an alternative p a s t in an epistemological sense, since we do n o t have a full or definite knowledge about t h e vast majority of questions about t h e past. This epistemological limitation is different from an ontological a s s u m p t i o n t h a t t h e r e be several different courses of events in t h e past, w h i c h are equally real. However, t h e r e is hardly a n y evidence of such a distinction being m a d e in Borges' story. On t h e other h a n d it is difficult to believe t h a t Borges would really m a k e room for a liberty of choice r e g a r d i n g t h e past. The story repeatedly stresses the observation t h a t t h e past is irrevocable. The ethical tension arises exactly out of the wilful and forced projection of this property of t h e p a s t onto t h e future. Neither Yu T s u n nor anybody else can r e p e a t or alter the past, and this fact in t h e end influences Yu T s u n ' s attitude towards t h e m u r d e r which he h a s committed. I m m e d i a t e l y after t h e deed, Yu T s u n is a p p r e h e n d e d by R i c h a r d M a d d e n , who has somehow m a n a g e d to trace h i m to Albert's home. Afterwards, writing in his cell, w h e r e he is awaiting execution, Yu T s u n expresses his anguish at his deed: W h a t r e m a i n s is u n r e a l and u n i m p o r t a n t . M a d d e n in a n d a r r e s t e d me. I have been c o n d e m n e d to Abominably, I have yet triumphed! T h e secret n a m e city to be attacked got t h r o u g h to Berlin. Yesterday

broke hang. of t h e it was

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CHAPTER 2.6 b o m b e d . . . H e [the G e r m a n s u p e r i o r in t h e B e r l i n h e a d q u a r t e r s ] does not know, for no one can, of m y infinite penitence a n d sickness of t h e heart. [p. 101]

Those last words conclude the story. The feeling of r e p e n t a n c e and fatigue expressed in t h e m seems to be the n e a r e s t a h u m a n being can come in an attempt to change the past. - A solution to the question about alternative pasts in Borges short story can be based on t h e observation t h a t the text seems to contain two concepts of eventuality (possible courses of events). One is connected w i t h t h e s i t u a t i o n of the h u m a n being. Our a l t e r n a t i v e s (of choice) w i t h respect to eventuality regards the future only, since t h e past h a s already been settled. The other concept of eventuality is r e l a t e d to the conceivable or the consistent. It is a very c o m p r e h e n s i v e concept, since everything t h a t does n o t directly involve a logical self-contradiction is regarded as possible. (Specific causal restrictions m i g h t be superimposed on this notion.) A p p a r e n t l y , Borges is relying on t h e latter concept of e v e n t u a l i t y in his depiction of a temporal b r a n c h i n g system. From t h e viewpoint of the p r e s e n t state of things it is possible to imagine different past courses of events, which have in various ways led to t h e present situation. These different pasts would be p o s s i b l e in so far as t h e y a m o n g t h e m s e l v e s m a k e no (recognisable) contrast to the p r e s e n t state of affairs, for if t h e y did we could rule some of t h e m out. It is a quite striking fact t h a t Borges wrote his s h o r t story already in 1941, t h e very same year w h e n J. Findlay's article Time: A Treatment of Some Puzzles was p u b l i s h e d in t h e A u s t r a l a s i a n J o u r n a l of Psychology and Philosophy. This article is n o r m a l l y considered to be t h e starting point for t h e m o d e r n logic of t i m e (although Lukasiewicz trivalent logic m i g h t be seen as an e v e n earlier forerunner of t e m p o r a l logic, as we have shown in a c h a p t e r 2.3). There can be no doubt t h a t The Garden of Forking Paths is a compelling picture of the very s a m e basic intuitions w h i c h also underlie t h e later formal d e v e l o p m e n t of b r a n c h i n g time. Nevertheless, it is hard to establish a n y direct impact of Borges' ideas in t h e development of the formal logic of t i m e - in spite of the fact t h a t m a n y leading logicians and

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189

philosophers within the study of time have evidently been aware of Borges' short story. In order to u n d e r s t a n d the more or less s i m u l t a n e o u s a p p e a r a n c e of Borges' short stories about t i m e and t h e incipient study of temporal logic, we should p e r h a p s r a t h e r focus on a general desire of understanding the n a t u r e of time in a more satisfactory way t h a n the classical models could provide. There appears to have been a widespread concern w i t h f u n d a m e n t a l questions about time among intellectuals in the 1940's and the 1950's. Both the early logic of time and Borges' literary description of time can be said to have had the purpose of stressing the reality of time. Time is seen as an aspect of the real world and not an illusion. But what does this mean and how do we work out these ideas in detail? The idea of branching time is a f r a m e w o r k within which we can begin to answer some of those questions. At least as an experiment, we can w i t h Borges take on the r61e of an 'abstract spectator' of the world and t r y to u n d e r s t a n d t h e infinite t e m p o r a l b r a n c h i n g s t r u c t u r e of possible events. The idea of branching time was not reahsed in early w o r k on temporal logic. Indeed it had not yet been formulated in Prior's Time and Modality (1957), which otherwise marked the major b r e a k t h r o u g h of the new logic of time. As an explicit (or formalised) idea, branching time was first suggested to Prior in a letter from Saul Kripke in September 1958. This letter contains an initial version of t h e idea and a system of branching time, a l t h o u g h it was of course not worked out in details. Kripke suggested t h a t we m a y consider the present as a point of 'rank 0', and possible 'events' or 'states' at the next moment as points of ' r a n k 1'; for every such possible state in turn, there would be various possible future states at the next moment from ' r a n k l ' , t h e set of which could be labelled 'rank2', and so forth. In this way it is possible to form a tree structure representing the entire set of possible futures expanding from the present (rank0) - indeed a set of possible futures can be said to be identified for any state, or node in the tree. In this structure every point determines a subtree consisting of its own present and future.

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CHAPTER 2.6 Branching Time according to Saul Kripke, 1 •

Rank 0

Rank 1

Rank 2

Prior clearly found this view of time highly interesting, and in the following years he substantially developed it. He worked out the formal details of several different systems, which constitute different a n d even competing interpretations of this idea, as you shall see below. Eventually, he incorporated t h e idea of branching into the concept of time itself. We m a y refine t h e intuitive picture of b r a n c h i n g time by the figure below. In this picture, it makes sense to say t h a t for every event there is one u n a m b i g u o u s past. For instance, in relation to the event Es, the past contains the linear a r r a n g e m e n t of events represented by Eo, El, and E2. In relation to E~ considered as the p r e s e n t t i m e the events E9 and Elo are alternative future possibilities. Relative to Es, the events E4, E6 and Ev will be counterfactual; t h a t is, if E~ is ever 'realised', E4, E6 a n d E7 are indeed 'by now' (E~) beyond possible realisation. Each E-node really represents a set of events and facts; if two facts both 'belong to' one and t h e s a m e node, say Es, they are of course g e n u i n e l y simultaneous at Es. E4, E6 and ET, on the other hand, r e p r e s e n t a pseudos i m u l t a n e i t y w i t h E~ for w h a t would h a v e been real u n d e r different a n d counterfactual conditions.

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THE IDEA OF BRANCHING TIME

E4 E2 ~

s

E8

E9 E 5 ~ E 1 0

E0

~E1

It is, however, still possible to interpret this general idea in various ways. Prior himself worked out two different interpretations, inspired respectively by Ockham and Peirce [Prior 1968, p.122 ff.]. This fundamental work has led to a large number of articles in various journals. A significant n u m b e r of these papers are concerned with the problem of determinism versus indeterminism, and we shall in part 3 examine in detail how indeterministic tense logics based on the idea of branching time can be worked out. In order to shed light on the concept of time, Prior's procedure basically was to work out different temporal systems and t h e n to examine their logical consequences. Other researchers have t a k e n a more 'ontological approach', focusing on the concept of t i m e itself; from an analysis of that concept, one can then construct the corresponding logic. (Needless to say, the two procedures cannot be kept strictly apart, but they do differ somewhat in their methodological consequences.) Nicholas Rescher [1968], for one, has reacted against Prior's rendition of branching time,

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arguing t h a t time itself is not really branching, in spite of t h e fact t h a t a w e a l t h of possibilities for the future course of events can be found (as seen from the present). To Rescher, we h a v e a "branching in time", but not "branching of time" [1971, p.173]. Storrs McCall [1976], on the other hand, h a s argued t h a t t h e passing of t i m e is genuinely related to the u n d e r s t a n d i n g of time as a branching system: the passing of time is equivalent to a loss of possibilities! This observation emphasises how the b r a n c h i n g of time is directed towards the future only, t h a t is, for any point in the s y s t e m there exists only one possible past. Of course, t h e problem of t h e ontological status of t h e possible futures is a very difficult one. Should we consent to w h a t Borges lets Yu T s u n say i m m e d i a t e l y before he kills S t e p h e n Albert: "The future exists now 2, [p. 101]? Prior would certainly disagree; he r e p e a t e d l y stated t h e conviction t h a t only t h e p r e s e n t exists. The t e n s i o n between t h e s e two creeds is in fact also manifest in The Garden of Forking Paths. Before Yu T s u n p l a n s t h e m u r d e r a n d embarks on his chosen mission, he ponders his probable fate in t h e near future, n a m e l y the ordinary p u n i s h m e n t meted out to spies: execution. But reflecting on t h e importance of the p r e s e n t as constituting reality, he finds some solace: T h e n I reflected t h a t all t h i n g s h a p p e n , h a p p e n to one, precisely now. Century follows century, and things h a p p e n only in the present. [p. 90] T h u s Yu T s u n comforts himself w i t h an observation exactly opposed to t h e m a x i m w i t h which he later tries to justify his deed. The above words bear a striking resemblance to some of Prior's r e m a r k s on t h e present as reality. In tense logic, t h e picture of b r a n c h i n g time unfolded in t h e story is a c t u a l l y compatible w i t h the identification of t h e p r e s e n t with t h e real. Nevertheless, while Borges' story certainly depicts and apparently advocates the branching view of time, it is not quite so clear w h e t h e r it also agrees with t h e notion t h a t 'only p r e s e n t objects exist' [Prior & Fine 1977, p. 116]. Even so, the fact t h a t t h e story also pays attention to t h e special rSle of the p r e s e n t bears yet more witness to its profundity.

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193

Finally, it may be w o r t h considering the fact t h a t the whole course of t h e story is itself w h a t we would ordinarily consider as e x t r e m e l y unlikely. It is quite a bit of a coincidence t h a t Yu Tsun's only possible victim should turn out to be a sinologist, indeed a sinologist who happens to have studied intensely t h e work of Yu Tsun's great-grandfather. In this circumstance one might seek evidence to the effect that aider all, courses of events are seen as governed by Fate, or Providence. But on the other hand, it might also be seen as a suggestion t h a t no future possibility should be ruled out or considered 'too unlikely' (excepting those which would violate the laws of logic or physics). The latter interpretation would be in good accordance with general features in Borges' work, to the best of our knowledge. Several models of branching time have been proposed. The m a i n difference between these models has to do with the status of t h e future. The models fall into a small n u m b e r of groups, w h e r e the basic ideas can be shown in a very intuitive way: consider once again t h e old Aristotelian example about t h e possible sea-fight tomorrow. How should we define t r u t h for s t a t e m e n t s like F(1)p? One particular line of answer to this question can be based on a simple but radical assumption, namely the rejection of the principle of bivalence. As we have seen J a n Lukasiewicz maintained t h a t we should view the logic of time as three-valued, attaching a third truth-value: 'indeterminate' to s t a t e m e n t s about the contingent future. A comparable line has been t a k e n by Richmond H. T h o m a s o n [1970], according to which the t r u t h - v a l u e of s t a t e m e n t s about the contingent future are in general undefined. Thomason's theory is certainly consistent, and it is also interesting t h a t he has been able to use it in the context of deontic logic (i.e. the logic of moral obligation) [Thomason 1981, pp. 165 ff.]. The crucial problem with this approach as well as t h a t of L u k a s i e w i c z is t h e c i r c u m s t a n c e t h a t t h e u s u a l t r u t h functional technique breaks down for these theories. This condition is a source of serious formal problems as well as highly counter-intuitive features. For instance, if F(1)p and ~F(1)p are both 'indeterminate' (or 'undefined'), it is very h a r d to explain how s t a t e m e n t s like the conjunction F(1)p A -F(1)p and the dis-

194

C H A P T E R 2.6

j u n c t i o n F(1)p v ~F(1)p can be a n y t h i n g else t h a n ' i n d e t e r m i n a t e ' (or 'undeiVmed ') [Prior 1967, p. 135]. We t h i n k t h a t t h e i n t r o d u c t i o n of 'indeterminate' or 'undefined' s t a t e m e n t s is a n u n n e c e s s a r y complication. For t h i s r e a s o n we shall l e a v e a s i d e f u r t h e r discussion o f solutions b a s e d on t h e rejection of bivalence. L e t u s consider t h e four b i v a l e n t a n s w e r s which h a v e b e e n given in t h e l i t e r a t u r e . For t h e s a k e of simplicity, w e s h a l l u s e m e t r i c a l t i m e in o u r examples; b u t t h e r e s u l t s can be g e n e r a lised into n o n - m e t r i c a l time, if e a c h b r a n c h defines an e q u i v a lence class of futures. 1) The first a n s w e r is that the two possibilities, sea-fight a n d no sea-fight, a r e b o t h future, and t h a t none of t h e m h a s a n y superior s t a t u s relative to the other. This a n s w e r can be r e p r e s e n t e d graphically in t h e following way: sea-fight

no sea-fight The a r r o w s on end of t h e two f u t u r e b r a n c h e s indicate t h a t t h e s t a t e m e n t s 'there is going to be a s e a - b a t t l e (tomorrow)' a n d 'there is not going to be a sea-battle (tomorrow)' are b o t h t r u e in this picture of b r a n c h i n g time. T h a t is, if we let p s t a n d for 'there is a s e a - b a t t l e going on', and F(1)p s t a n d for 'there is going to be a sea-battle tomorrow', then

F(1)p A F(1)-p is true. T h e corresponding tense-logical s y s t e m is called Kb after Saul Kripke. 2) According to t h e O c k h a m - m o d e l only one possible f u t u r e is t h e t r u e one, a l t h o u g h we as h u m a n beings do not k n o w w h i c h of t h e m it is. Let u s a s s u m e t h a t t h e r e is in fact going to b e no

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195

s e a - f i g h t tomorrow. I n t h i s ease t h e f u t u r e s h o u l d be r e p r e sented g r a p h i c a l l y in t h e following way, w h e r e a line not e n d i n g in a n a r r o w indicates t h a t it will be false to a s s e r t t h a t t h e corresponding state-of-affairs will be t h e case tomorrow: sea-fight

J

H1

-~H2 no sea-fight So, -F(1)p A F(1)~p is t h e t r u e description of t h i s situation, even t h o u g h we m a y be u n a b l e to k n o w this at t h e p r e s e n t m o m e n t (p etc. being defined as above). 3) A c c o r d i n g to t h e P e i r c e - m o d e l - w h i c h Prior h i m s e l f adopted as covering his own view - it m a k e s no sense to s p e a k about t h e t r u e future as one of t h e possible f u t u r e s . T h e r e is no f u t u r e yet, j u s t a n u m b e r of possibilities. Hence, the f u t u r e - or p e r h a p s r a t h e r , the 'hypothetical future' - s h o u l d be r e p r e s e n t e d graphically in this way: sea-fight H1

- ~H2 no sea-fight N e i t h e r F(1)p nor F(1)~p are t r u e on t h i s picture. H o w e v e r , if some proposition q holds tomorrow in all possible f u t u r e s - t h a t is, if t h e t r u t h of q t o m o r r o w is r e g a r d e d as n e c e s s a r y - t h e n F(1)q is true. 4) The possibility of t h e first t h r e e a n s w e r s m e n t i o n e d above were realised by A. N. Prior. However, l a t e r H i r o k a z u N i s h i m u r a [1979] f o r m u l a t e d a new t e m p o r a l model w h i c h t u r n e d o u t to

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CHAPTER 2.6

be slightly different from the Ockham-model which Prior had considered. Nishimura's model involved not only times, but also histories defined as linear subsets of the set of times. In fact, it is n a t u r a l to view Nishimura's model of time as a union of disjoint histories. According to t h e model the tenses (past, present, future) are always relative to a history. Relative to one possible history there is going to be a sea-fight tomorrow, and relative to a n o t h e r history there is not going to be a sea-fight tomorrow. Graphically, this model can be presented in the following way: sea-fight

_j.-z

no sea-fight Here, F(1)p is true with respect to H1, whilst F(1)-p is true with respect to H2. Clearly Nishimura's model has to involve some relation of i d e n t i t y of histories before certain events. H1 and H2 m a y be identical in all past times except for the fact t h a t F(x)p is t r u e at all such times in H1 (for some x), while it is false in H2. Therefore, in order to achieve such an identity relation future s t a t e m e n t s must be disregarded. In dealing with the model, it is n a t u r a l to consider the full set of histories as pre-defined. As we have seen, this view is similar to Leibniz' concept of creation of a t e m p o r a l world. In general, it is interesting t h a t the constructions in Nishimura's model come very close to ideas that can be found in Leibniz' philosophy, in spite of the fact t h a t Leibniz h i m s e l f ruled out time from his endeavour to establish a symbolic logic. Nishimura's ideas can be incorporated into a formal branching time model, which we shall call the Leibniz System, to be presented in due course. This system seems to be very close to t h e Ockhamist one, but it t u r n s out that t h e r e are certain s t a t e m e n t s which are t r u e from an Ockhamistic point of view, but false within the Leibniz System - as we shall see later.

2.7. TENSE LOGIC AND SPECIAL RELATIVITY A c c o r d i n g to P r i o r m a n y p h i l o s o p h e r s a n d s c i e n t i s t s w h o a c c e p t t h e t a p e s t r y v i e w of time h a v e claimed t h a t "they h a v e on t h e i r side a v e r y a u g u s t scientific theory, t h e t h e o r y of relativity, a n d of course it w o u l d n ' t do for m e r e p h i l o s o p h e r s to q u e s t i o n a u g u s t scientific theories" [SFTT, p. 3]. Prior early bec a m e a w a r e of t h e conflict b e t w e e n t e n s e logic a n d special relativity. It w a s m e n t i o n e d b y Saul K r i p k e in a l e t t e r to P r i o r as e a r l y a s 1958. Prior described the conflict in a v e r y clear way: T h e t r o u b l e arises w h e n we come to c o m p a r e another's exp e r i e n c e s , w h e n , for e x a m p l e , I w a n t to k n o w w h e t h e r I s a w a certain flash of light before you did, or y o u s a w it before I did .... It could h a p p e n t h a t if I a s s u m e d m y s e l f to be s t a t i o n a r y and y o u moving, I'd get one r e s u l t - s a y t h a t I s a w t h e flash first - a n d if you a s s u m e d t h a t y o u were statio n a r y a n d I moving, you'd get a different r e s u l t ... A n d t h e c o n c l u s i o n d r a w n in t h e t h e o r y of r e l a t i v i t y is t h a t t h i s q u e s t i o n - the question as to which of us is right, which of us r e a l l y s a w it first - is a m e a n i n g l e s s question ... Now I don't w a n t to be d i s r e s p e c t f u l to people w h o s e r e s e a r c h e s lie in o t h e r fields t h a n m y own, b u t I feel compelled to s a y t h a t this j u s t won't do. [SFTT, p. 3-4] It is e a s y to u n d e r s t a n d w h a t Prior m e a n s . S u p p o s e t h a t two o b s e r v e r s , A a n d B, are moving w i t h velocities v a n d -v, from an e m i t t e r E, b o t h leaving E w h e n the E-clock r e a d s t=O. A -V

0

E

0

B

O

V

According to special r e l a t i v i t y the following t r a n s f o r m a t i o n s for t h e t i m e co-ordinates hold: tA = L (tE + VXF_) tB = L ( t ) - VXF_)

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w h e r e L= (1 - v2).112 and the speed of light is taken as unity (c = 1). A flash is e m i t t e d from E and received simultaneously by A and B, yielding same readings, tE, on the E-clocks. The time coordinates for seeing the flash on A (XE = -vtE) a n d B (XE= VtE) can be calculated in A's system in the following way: tAA = L ( t E + v x ~ = L(1-v2)tE tA, B ---- L ( t E - VXE) = L ( I +v2)tE

Clearly according to this A is the first to see the flash. The arrivals of the light signals can also be calculated in the B-system: tBA = L ( t E - VXF_) = L ( I +v2)tE tB,B = L ( t E - VXE) = L(1-v2)tE

According to this calculation B sees the flash before A. For this reason some physicists would say t h a t the question as to which of the two observers really saw a certain flash first can only make sense if a n inertial frame is specified relative to which the calculation should be carried out. However, Prior thought t h a t the question as to which of the two observers really saw a certain flash first is indeed a meaningful one. H e stated that what it means is simply this: "When I w a s seeing t h e flash, had you already seen it, or had you not?" [SFTT, p. 5] Of course, it might be doubted t h a t a physicist committed to t h e ordinary interpretation of special relativity would be convinced by t h a t definition. He would probably s a y t h a t this is begging the question. As a precondition for accepting t h e question as a meaningful one he would instead d e m a n d some experimental procedure, by m e a n s of which the question can be settled. Prior a d m i t t e d t h a t we cannot in all cases know w h e t h e r a given event is present or not, i.e. whether it is really taking place 'now' or not, b u t he m a i n t a i n e d t h a t this epistemological question is v e r y different from t h e corresponding ontological question. H e w a n t e d to make it clear t h a t all what physics could show would be t h a t "in some cases we can never know, we can

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never physically find out [our italics], w h e t h e r something is actually h a p p e n i n g or merely has h a p p e n e d or will happen" [Prior 1972, p. 323]. Nevertheless, m a n y modern physicists want to go even further, and claim with Albert Einstein: There is no irreversibility in the basic laws of physics. You have to accept t h e idea t h a t subjective time w i t h its emphasis on the now has no objective meaning. [Letter to Michele Besso; quoted from Prigogine 1980, p. 203] On the other hand, Prior could also note - without doubt w i t h some pleasure - t h a t not even Einstein was quite content w i t h this view. Einstein once said to Carnap that the problem of t h e Now worried h i m seriously, explaining t h a t "the experience of the Now means something special for men, something different from the past and the future, but t h a t this important difference does not and cannot occur within physics" [Prior 1968, pp. 133134]. Following this kind of reasoning, Prior m a i n t a i n e d t h a t questions concerning the h u m a n Now make sense, even t h o u g h we cannot be sure t h a t such questions can ever be decided by physical m e a n s . On logical and philosophical grounds P r i o r maintained t h a t w h e n an event X is happening, another e v e n t Y either has h a p p e n e d or has not happened. He strongly rejected the idea of treating 'having happened' as a property t h a t can attach to an event from one point of view whilst not from some other point of view: So it seems to me t h a t there's a strong case for just digging our heels in here and saying that, relativity or no relativity, if I say I saw a certain flash before you, and you say you saw it first, one of us is just wrong - is misled it may be, by the effect of speed on his i n s t r u m e n t s - even if there is j u s t no physical m e a n s w h a t e v e r of deciding which of us it is. [SFTT, p. 51 There seems to be two different ways of solving the conflict between tense logic and special relativity. We can either reject (or adjust) the f u n d a m e n t a l beliefs underlying tense logic, or we

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can reject (or adjust) the basic assumptions of special relativity. In t h e following we shall prefer to adjust t h e p h i l o s o p h i c a l a s s u m p t i o n s of special relativity in such a way t h a t no empirical (or measurable) consequence of the theory is denied. T h e p a p e r [0hrstrOm 1990] analyses a n u m b e r of conceptual possibilities for upholding at the s a m e time the a s s u m p t i o n s of t h e Special T h e o r y of Relativity a n d Prior's e q u a t i n g reality w i t h t h e present. The analysis shows t h a t this c a n be done in various ways. One of the most obvious ways presupposes t h e selection of a privileged inertial system, to whose time-coordinates special m e a n i n g s are attributed. If such a selection is not to be m a d e ad hoc, t h e n it m u s t be possible to list t h e r e a s o n s (preferably cosmological ones) for it. It should be p o i n t e d o u t t h a t t h e principle of relativity does not exclude a cosmological time (that is, a 'natural' inertial system, which d i s t i n g u i s h e s itself t h r o u g h t h e distribution and m o v e m e n t of m a t t e r in t h e universe). However, even on the assumption of a h o m o g e n e o u s u n i v e r s e it can be doubted t h a t cosmic time c a n actually be v i e w e d as an ontological f e a t u r e of t h e u n i v e r s e ; Whitrow, s h a r i n g the assumption of a homogeneous universe, stated: It is doubtful whether there exists a precise definition which h a s so great merits t h a t there would be sufficient reason to consider t h e time t h u s obtained as the true one. [Whitrow 1980, p. 304] This point of view is not shared by all researchers. As Mogens W e g e n e r h a s pointed out in his Simultaneity and Weak Relativity [1992, pp. 10 ff.] some scientists t h i n k t h a t t h e cosmological evidence supports t h e existence of a u n i v e r s a l substrat u m relative to which a cosmic and absolute s i m u l t a n e i t y can be introduced. At least, it is clear t h a t it is possible to hold Prior's v e r y strong tense-logical position w i t h o u t violating any of t h e e m p i r i c a l consequences of special relativity, as long as we conceive t h e t e n s e s as relative to one privileged observer. A r g u i n g from a theological point of view, J. R. Lucas [1989, p. 220] has come to the same conclusion. Lucas points out t h a t "the canon of s i m u l t a n e i t y implicit in the i n s t a n t a n e o u s acquisition

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of knowledge by an omniscient being" is not incompatible w i t h t h e special t h e o r y of relativity, since t h e r e m a y be "a divinely preferred f r a m e of reference". If t h e r e is s o m e privileged f r a m e of reference, t h e n t h e temporal co-ordinates in other systems do not strictly s p e a k i n g represent p r o p e r time. For this reason Prior claimed: we m a y say t h a t the theory of relativity isn't about real space a n d time.., t h e time which e n t e r s into t h e so-called space-time of relativity theory ... is j u s t p a r t of an artificial f r a m e w o r k w h i c h t h e scientists h a v e constructed to l i n k together observed facts in the simplest way possible... [SFTT, p. 5] Prior did n o t m i n d playing that p a r l o u r game, too. He realised t h a t the non-linear structure of space-time points, ordered w i t h absolute before-after relations, possibly of a causal nature, constitutes an i n t e r e s t i n g object of study for the tense logician. The s t r u c t u r e b r a n c h e s both forwards a n d backwards, so it is n o t immediately clear how the corresponding tense logic is to be axiomatised. He a r g u e d [Prior 1967, p. 203ff.] t h a t t h e characteristic axioms for relativistic space-time are:

FGq ~ GFq PHq ~ HPq His a r g u m e n t a t i o n was t h o r o u g h a n d detailed, a l t h o u g h a more s y s t e m a t i c investigation of t h e relation b e t w e e n special relativity a n d t e n s e logic was not carried out u n t i l 1980 (see [Goldblatt 1980]). A decade earlier on, Professor Gerald Masseyfrom Michigan State University h a d directed a frontal attack on t e n s e logic as a new discipline. He h a d specifically referred to results from the Special Theory of Relativity, accusing Prior of p r o m o t i n g "bad physics and indefensible metaphysics" [Massey 1969]. However, in the light of a m o n g s t other t h i n g s Goldblatt's results, Massey's attack was s o m e w h a t u n r e a s o n a ble.

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R e g a r d i n g a tense logical approach to relativity, Prior also pointed out t h a t there is a logic of such functors as 'It appears from a certain point of view that 2. Hence, it is possible to make good sense out of talk about an infinity of different 'apparent' time-series. Prior suspected t h a t the infinity of 'local proper times', which figure in relativistic physics, a m o u n t s simply to w h a t appears from various points of view, or w h a t appears to be the course of events in various 'frames of reference'. If the physicist w a n t s to obtain a more general picture, he can "indicate w h a t features of the course of events (what t e m p o r a l orderings of those events) will be common to all points of view, and one can w o r k out a tense logic for t h a t too" [Prior 1968 p. 133]. Prior h i m s e l f made some important contributions to the development of such a relativistic tense logic [Prior 1967, p. 203 ff.] even t h o u g h he felt that the project of a relativistic tense logic was on the whole a bit strange. Although some results regarding relativistic t e n s e logic have been obtained by Prior and his followers, J. P. Burgess [1984] in his overview of tense logic had to observe t h a t a tense logic for special relativity had not yet been worked out fully - indeed t h a t t h e results which had been produced so far had been sparse. In our opinion this is still the case.

2.8. SOME BASIC SYSTEMS OF TEMPORAL LOGIC Temporal reasoning is captured in one m a n n e r by tense logic, and in a n o t h e r m a n n e r by the logic of instants; the tension between the two approaches was reflected in relation to the traditional interpretation of the special theory of relativity, w h i c h w a s a n a l y s e d in the previous chapter. In t e r m s of McTaggart's time-series we can say that tense logic is A-logical, whereas the logic of instants (or dates) is B-logical. Thus, we can speak about two kinds of temporal logic (A and B). In this c h a p t e r we shall study the relation between these kinds of temporal logic from a formal point of view. Unlike most other disciplines in m o d e m logic, temporal logic and its symbolic calculi were first developed entirely outside of the field of mathematics. This stands out in contrast to the comparable discipline of, say, modal logic, which also has clear philosophical motivations and implications, but in whose development regular mathematicians played an important rSle from t h e beginning. A. N. Prior, however, who was himself a philosopher by training, established temporal logic as a part of philosophical logic. In consequence, the emphasis was put on c o n c e p t u a l i n v e s t i g a t i o n s r a t h e r t h a n s t u d i e s of p u r e l y m a t h e m a t i c a l aspects of temporal logic. This should not be m i s c o n s t r u e d as a failure in m a t h e m a t i c a l competence, for P r i o r w a s clearly a w a r e of the i m p o r t a n c e of m e t a m a t h e m a t i c a l questions concerning general properties and m u t u a l relations of logical systems, and contributed to these issues in developing t e n s e logical systems. However, in constructing the systems conceptual considerations would t a k e p r i o r i t y over m a t h e m a t i c a l n e a t n e s s . The s t r e n g t h of philosophical logic lies in its self-imposed obligation to take the logical intuitions embodied in everyday language into serious consideration. On the other hand, it is also clear t h a t until recently, temporal logic has lacked the kind of m a t h e m a t i c a l glamour exhibited in many other fields of symbolic logic. In most presentations of temporal logic there is a very clear distinction between axiomatics and proof theory on one h a n d 203

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a n d semantics a n d model-theory on t h e other. A-logic is viewed as axiomatics, a n d B-theory m a i n l y as a k i n d of semantics, d e a l i n g w i t h t r u t h - c o n d i t i o n s a n d t e m p o r a l models. Prior's a p p r o a c h to t e m p o r a l logic was different. E l a b o r a t i n g some observations by McTaggart, he m a i n t a i n e d t h a t A-logic is basic a n d t h a t B-logic can be derived from it. In this chapter and t h e next one we shall expound some of the basic systems of temporal logics (i.e. A- a n d B-logics), largely following Prior's ideas. We shall p r e s e n t s o m e of his most i m p o r t a n t results r e g a r d i n g t e m p o r a l logic. A n y A-logic, i.e. tense logic, is based on t h e primitive tenseo p e r a t o r s P a n d F; its axiomatisation is often f o r m u l a t e d in t e r m s of the derived operators H and G (as we have pointed out earlier, H and G are inter-definable w i t h P and F, respectively, so either pair of operators can in fact be chosen as primitives). A v e r y f u n d a m e n t a l s y s t e m has been n a m e d Kt (where t h e 'K' is p r o b a b l y in h o n o u r of Saul Kripke). T h i s tense logic can be p r e s e n t e d as a n axiomatic system w i t h t h e following axiom schemes [Prior 1967 p. 176; McArthur 1976, p. 17 ff.]: (A1) (A2) (A3) (A4) (A5)

p, where p is a tautology of t h e propositional calculus G(p ~ q) ~ (Gp ~ Gq) H(p ~ q) ~ (Hp ~ Hq) p ~ HFp p ~GPp

In (A2) - (A5), p and q are arbitrary, well-formed formulas. All axioms are said to be immediately provable, while other theses can be proved by inference. In Kt, Modus Ponens is the basic rule of inference: (RMP)

I f ~ p and l-p ~ q , then l-q.

In addition we h a v e two rules, which introduce tense-operators: (RG) (RH)

If ~p, t h e n ~ Gp. I f ~ p , then I-Hp.

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F r o m Kt, o t h e r t e n s e logical systems can be defined by a d d i n g m o r e axioms to t h e above list, (A1) - (A5), as we shall see in t h e following. In order to introduce a logic of instants or dates, i.e. a B-logic, we n e e d a set T I M E of i n s t a n t s (or dates) w i t h a relation, < , w h i c h a t t r i b u t e s to T I M E some s t r u c t u r e . T h e r e l a t i o n '<' is called t h e before-after-relation. For any t e m p o r a l i n s t a n t t a n d a n y s t a t e m e n t p, T(t,p) is a n e w s t a t e m e n t , w h i c h can be r e a d 'p is t r u e at t'. In most B-logics it is assumed t h a t (T1) (T2)

T(t,p A q) = (IYt, p) A T(t,q)) T(t,-p) - ~ T(t,p)

Note t h a t in principle we should m a k e a distinction b e t w e e n two k i n d s of c o n j u n c t i o n ( a n d also b e t w e e n two k i n d s of n e g a t i o n ) in (T1)-(T2). The reason is t h a t in m o s t B-logics, p a n d q a r e t r e a t e d as propositional functions r a t h e r t h a n fullfledged propositions such as T(t,p). This m e a n s t h a t t h e two k i n d s of expressions would be of different types. On t h e o t h e r h a n d , it is also possible in a B-logic to p u t b o t h t y p e s of expressions syntactically on a par, as you shall see in t h e n e x t chapter. So we shall neglect this complication, since it is after all r a t h e r clear how t h e conjunctions, negations etc. should be r e a d in each case. Now, t h e definitions (DF) (DP)

T(t, Fp) ~def 3Q: (t
would allow us to evaluate a n y tense logical f o r m u l a p, in t e r m s of T(t,p). F r o m t h e definitions Hp ~def ~P-P and Gp -~def - F - p it i m m e d i a t e l y follows (DG) (DH)

T(t, Gp) - Vt1: ( t < t l ~ T ( t l , p ) ) T(t, Hp) - Vt1: (t1
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We shall say t h a t a s t r u c t u r e ( T I M E , < , T ) is a B - l o g i c a l structure , if T satisfies (T1-2) and the definitions (DF), (DP), (DG), and (DH). T is called the T-operator (or t h e valuation operator) of the structure. It is easy to see that the axioms (A1) - (A5) are all true at any t in a n y B-logical structure. Let us consider the case of (A4). In this case the following proof can be given: (1) (2) (3) (4) (5) (6) (7) (8)

(t
Using standard quantification theory etc. it is also easy to see t h a t the rules of inference all preserve t r u t h by a n y T-operator in a B-logical structure. S u m m i n g up these observations, we have the following result: Theorem. If a tense-logical s t a t e m e n t p is provable in Kt, t h e n T(t,p) (i.e. p is t r u e at t) for any t in a n y B-logical structure (TIME, <, T). Kt makes less assumptions on the 'structure of time' t h a n any other tense-logical system; t h a t is, no restriction on the beforeafter-relation is required in the corresponding B-logic. This is why the system Kt is said to be minimal. If we add the axiom (A6)

FFp ~ Fp

we get a new tense-logical system corresponding to a transitive before-after relation, i.e. (B1)

(tl < t2 A t2 < t3) ~ tl < t3

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If to Kt we add the axioms (A6) and (A7)

FPp ~ (Pp v p v Fp)

we get t h e system known as Kb. The subscript 'b' indicates t h a t this s y s t e m allows for b r a n c h i n g time. Provable t h e o r e m s in Kb are t r u e for any T-operator w i t h (T1-3) and (B2)

(tl
(B2) can be called 'backwards linearity'. A.N. Prior [1967, p.205 ft.] has d e m o n s t r a t e d t h a t the following s t a t e m e n t is provable in Kt if (A7) is accepted as an ~xiom: (A7x)

(Pp A Pq) ~ (P(p A q) v P(p A Pq) v P(Pp A q))

(A7x) is less elegant t h a n (A7), b u t at the intuitive level the former more directly expresses t h e idea of backwards linearity in a b r a n c h i n g time model t h a n does t h e latter. It is in fact also easy to see t h a t (A7) is provable from Kt w i t h (A7x). T h u s t h e r e is a free choice between (A7) a n d (A7x), if one wishes to enlarge Kt into Kb. We shall present a version of Prior's proof in order to give an e x a m p l e of t h e kind of very powerful r e a s o n i n g w h i c h can be carried out in tense logic. So, we are going to prove t h e following meta-theorem: In Kt enlarged with (A7), (A7x) is a theorem. Similarly, in Kt enlarged w i t h (A7x), (A7) is a theorem. - Prior proves t h e following lemmas: L e m m a 1. In a tense logical s y s t e m with the axioms (A1)(A5) a n d (A7) and the rules (RMP), (RG), and (RH) H(p ~ (Hp ~q)) v H ( H q ~ p ) ) is provable, w h e r e p a n d q are a r b i t r a r y well f o r m e d formulas. Proof." The proof is carried out by reductio ad a b s u r d u m i.e.

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CHAPTER 2.8 (1) (2) (3) (4) (5) (6) (7) (8)

-(H(p ~ (Hp ~ q)) vH(HqDp))) ~H(p ~ (Hp ~ q)) -H(Hq~p)) P(p A Itp A -q) P(Hq A -p) HFP(Hq A -p) P(p A Hp A -q A FP(Hq A ~p)) P((p A Hp A ~q A P(Hq A ~p)) V (p A Hp A ~q A Hq A ~p) v (p A Hp A -q A F(nq A -p)) ) But (8) is clearly impossible since all the disjunction are impossible.

(assumption) (from 1) (from 1) (from 2) (from 3) (from 5 and A4) (from 4 and 6) (from 7 and A7) components in the

L e m m a 2. In a tense logical system with the axioms (A1)(A5) and (A7) and the rules (RMP), (RG), and (RH) (H(p ~ q) A H(p ~ Hq) A H(Pp ~ q) A Pp) ~ Hq is provable, w h e r e p a n d q are a r b i t r a r y well formed formulas. Proof: By substitution in Lemma 1 we fred H(q ~ ( H q ~ - p ) ) vH(H~p ~ q ) ) Therefore, the problem can be split into two cases: In the first case H(q ~ (Hq ~ ~p)) is assumed and in the second H(H~p ~ q)) is assumed.

1) In the first case we can argue in the following way: (1) H(p ~ q) (assumption) (2) H(p ~ Hq) (assumption) (3) H(Pp ~ q) (assumption) (4) Pp (assumption) (5) H(q ~ (Hq ~ - p ) ) (assumption) (6) H(p ~ (Hq ~ -p)) (from 1 and 5) (7) H(p ~ ~p) (from 2 and 6) (8) ~P(p Ap) (from 7) (9) -Pp (from 8) - contradicts (4) This means t h a t the assumptions in the antecedent rule out the first case.

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2) In the second case, in which we can a r g u e in the following way: (1) H(p ~ q) (assumption) (2) H(p ~ Hq) (assumption) (3) H(Pp ~ q) (assumption) (4) Pp (assumption) (5) H ( H - p ~ q)) (assumption) (6) H ( ~ q ~ Pp)) (from 5) (7) H(~q ~ q)) (from 6 and 3) (8) ~P(~q A .-q)) (from 7) (9) Hq (from 8) Q.E.D. Now, (A7x) can be proved from l e m m a 2 in the following way: (H(p ~ q) A H(p ~ Hq) A H ( P p ~ q) A Pp) ~ Hq (H(p ~ q) A H ( p ~ Hq) A H ( P p ~ q) ) ~ (Pp ~ Hq) ~(Pp ~ Hq) ~ - ( H ( p ~ q) A H(p ~ Hq) A H(Pp ~ q) ) (Pp A P - q ) ~ (P(p A -q) V P(p A P - q ) v P(Pp A -q))

From this (A7x) can be obtained by substitution. Moreover, Kt together with the axiom (A8)

PFp ~ ( P p v p v F p )

makes it possible by a proof similar to the above proof to deduce (A8x)

(Fp A Fq) ~ (F(p A q) V F(p A Fq) v F(Fp A q))

The axiom (A8) corresponds to the r e q u i r e m e n t of forward linearity for the temporal ordering i.e. (B3)

(t2 < tl A t2 < t3) ~ (tl < ta v t l = t3 vt3 < tl)

We can also to Kt - or any of the suggested enlarged systems add the axioms corresponding to non-ending time

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(A9) (A10)

Gp ~ F p H p ~ Pp

(B4) (B5)

Vt13t2: tl < t2 Vt13t2 : t2 < tl

i.e.

and dense time (All)

Fp ~ F F p

(B6)

V'tl V't2~3 : tl < t2 ~ (tl < tz A t3 < t2)

i.e.

Kt t o g e t h e r w i t h all of t h e axioms (A7)-(A11) yields Prior's linear tense logic K1, for which all the H a m b l i n implications can be proved. In tense logics like Kl, based on just two primitive operators P a n d F , we can by d e f i n i t i o n i n t r o d u c e a n u m b e r of n e w operators, for instance A p = (p A Gp A H p ) (- a l w a y s p ) Ip - (p v F p v Pp) (- s o m e t i m e p )

However, H a n s K a m p [1968] has d e m o n s t r a t e d t h a t s o m e t e m p o r a l operators expressible in t e r m s of a T-operator c a n n o t be defined in this way. One of these operators can be verbalised as 'is going to be u n i n t e r r u p t e d l y t h e case for some time' [Burgess 1984, p.l17]; if we symbolise this operator as 'X', t h e relevant defmition is T(t, X p ) -~-def 3t2,t3: (t < t2 < t3 A Vtl: t2 < tl < t3 ~ T ( t i , p ) )

Kamp, however, m a n a g e d to prove t h a t this operator, a n d indeed, every t e m p o r a l operator in a linear, dense, n o n - e n d i n g instant-logic, can be defined in terms of his two operators U a n d S, u n t i l and s i n c e , p r o v i d e d t h a t t i m e is a s s u m e d to be continuous [Burgess 1984, p.l17]. Kamp's two operators can be defined in the following way (where Upq m a y be read as ~o u n t i l q', and S p q can be read as ~ since q '):

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T(t, Upq) ~def ~t2: (t < t2 A T(t2,q) A Vtl: t < tl < t2 ~ (/2(Q,p) A T ( tl,-q))) T(t, S p q ) - de/' ~t2: (t2 < t A T(t2,q) A Vtl: t2 < tl < t ~ (T( tl,p) A T ( t l , - q ) ) )

Almost from the v e r y beginning of his development of tense logic, Prior [1967 p . l l l ] was a w a r e of problems concerning limitations to t h e expressive power of tense logic. But his approach to a solution to this problem was very different from t h a t of Hans Kamp. Inspired by some observations due to Peter Geach, Prior pointed out that we can in fact define U and S in terms of the tense-operators P and F, if we allow ourselves (i) (ii)

the use of propositional quantifiers, and the assumption that at each instant there is some proposition true at that instant only.

A tense-logical system based on these conditions has in fact very interesting and far-reaching implications. We shall study this issue in the next chapter.

TEMPO-MODAL SYSTEMS Above, we h a v e discussed purely tense-logical systems. Matters obviously become more complex, when besides the two primitive t e n s e o p e r a t o r s a primitive modal o p e r a t o r is i n t r o d u c e d - as r e q u i r e d in the cases of Prior's so-called Ockhamistic, respectively Peircean system. In order to describe the semantics for these tempo-modal systems Prior [1967, p. 126 ff.] needs a notion of temporal 'routes' or ' t e m p o r a l branches' i.e. m a x i m a l l y ordered (i.e. linear) subsets in (TIME,<). We prefer the term 'chronicle'. The set of all such chronicles will be called C. We shall also need the concept of a chronicle-section (c,t), where c ~ C and t e c and a relation, = on the set of chronicle-sections according to which

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(c,t) --(c',t) m e a n s t h a t the two sections a r e identical u p to t i.e. {t'~ c ] t ' _< t} = {t' E c ' ] t' _< t}.

~

C

C~

An o p e r a t o r Ock is an O c k h a m i s t i c v a l u a t i o n o p e r a t o r in a given O c k h a m i s t i c s t r u c t u r e , if for a n y t e m p o r a l i n s t a n t t in any chronicle c a n d a n y tense-logical s t a t e m e n t p, Ock(t,c,p) is a m e t a - s t a t e m e n t w h i c h can be r e a d 'p is t r u e at t in t h e chronicle C I

(a) (b) (c) (d) (e)

Ock(t,c, p/~ q) iff both Ock(t,c,p) and Ock(t,c,q) Ock(t,c,-p) iff not Ock(t,c,p) Ock(t,c, Fp) iffOck(t',c,p) for some t' e c w i t h t < t' Ock(t,c, Pp) iffOck(t',c,p) for some t' e c w i t h t' < t Ock(t,c, Np) iff Ock(t,c ',p) for all (c',t) w i t h (c,t) =(c',t)

(TIME,<,C,--,Ock)is said to be a n Ockhamistic structure. - A f o r m u l a p is said to be O c k h a m valid i f a n d only if Ock(t,c,p) for a n y t and c (with t e c) a n d a n y If t h e s e c o n d i t i o n s hold

Ockhamistic structure. It m a y be d o u b t e d w h e t h e r Prior's Ockhamistic s y s t e m is in fact a n a d e q u a t e r e p r e s e n t a t i o n of t h e t e n s e logical i d e a s p r o p a g a t e d b y W i l l i a m of O c k h a m . According to O c k h a m God k n o w s t h e c o n t i n g e n t future, so it s e e m s t h a t he w o u l d accept an i d e a of a b s o l u t e truth, also w h e n regarding a s t a t e m e n t Fq about t h e contingent future - a n d not only w h a t Prior h a s called "prima-facie assignments" [1967, p.126] like Ock(t,c,Fq). T h a t is, s u c h a p r o p o s i t i o n can be m a d e t r u e 'by fiat' s i m p l y b y c o n s t r u c t i n g a c o n c r e t e s t r u c t u r e w h i c h s a t i s f i e s it. B u t O c k h a m w o u l d accept t h a t F q could be t r u e at t w i t h o u t b e i n g r e l a t i v i s e d to a n y chronicle. And t h a t actually brings u s b a c k to a t w o - p l a c e T - o p e r a t o r , like t h e ones w e h a v e p r e v i o u s l y d i s c u s s e d . In t h e n e x t p a r t of t h e book w e shall show t h a t it is

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possible to establish a system which seems to be a bit closer to Ockham's original ideas. On the other hand, it should be noted t h a t t h e q u e s t i o n concerning t h e notion of t r u t h is m a i n l y p h i l o s o p h i c a l . P r i o r ' s O c k h a m i s t i c s y s t e m a p p e a r s to comprehend at least all the theorems which should be accepted according to Ockham's original ideas. Let us, for i n s t a n c e , consider one tense logical formula:

q ~ HFq It is obvious from the above definitions that Ock(t,c,q ~ HFq) for any t and a n y c with t e c. Therefore q ~ HFq is a t h e o r e m in Prior's Ockhamistic system. Likewise Pq ~ N P q (where F does not occur in q ) is obviously a theorem, w h e r e a s the formula PFq ~ NPFq is not a t h e o r e m in the system. This difference corresponds exactly to the difference between proper past and pseudo-past (see chapter 1.9). It h a s p r o v e d quite difficult to find a s a t i s f a c t o r y axiomatisation of Prior's Ockhamistic system. One p r o m i n e n t a t t e m p t was m a d e by Robert P. McArthur [1976, p. 47]. He introduced a primitive operator L, for which he s t a t e d t h e following axioms: (L1) (L3) (LG) (LP)

L(p ~ q) D (Lp D Lq) Lp ~ L L p

Lp p ~ LPp, where p contains no occurrences o f F

and the rule (RL)

If ~p, then ~ Lp.

L is, of course, intuitively related to 'necessity', but McArthur's L-operator is clearly more t h a n j u s t a modal operator: from (LG) it is obvious t h a t L is also temporal. It is n a t u r a l to u n d e r s t a n d L as equivalent to N G , where N is a pure necessity operator. Conceived in this way, all the axioms in the system are

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certainly Ockham-valid. But as far as we know, t h e question of completeness for Prior's Ockhamistic system is still open. Now, let us t u r n to the other tempo-modal s y s t e m t h a t Prior s t u d i e d carefully, t h e so-called Peirce system. In this s y s t e m four different operators, F, G, f, and g, regarding the f u t u r e can be considered. T h e s e operators can be t r a n s l a t e d i n t o an Ockhamistic formulation in the following way: F --> N F f ---> M F G --->NG g-.--> M G

This process of translation is well-defined, b u t it should be noted t h a t t h e r e is no Peircean expression which t r a n s l a t e s into the O c k h a m i s t i c F. The great a c h i e v e m e n t of t h e O c k h a m i s t i c s y s t e m could arguably be said to be its p r o p e r t y of m a k i n g a g e n u i n e d i s t i n c t i o n b e t w e e n t h e following t h r e e t y p e s of statement: (i) (ii) (iii)

Necessarily, Mr. S m i t h will commit suicide. Possibly, Mr. Smith will commit suicide. Mr. S m i t h will commit suicide.

However, in the Peirce-system t h e type of f u t u r e s t a t e m e n t seen in (iii) will have to be interpreted as m e a n i n g e i t h e r (i) or (ii). There is no 'plain future' in this system. Of course, t h a t is not a consequence of sloppiness on Peirce's side, b u t r a t h e r it is a deliberate and philosophically motivated choice, as explained in c h a p t e r 2.2. Therefore, the Ockhamistic s y s t e m c a n n o t a priori be preferred on philosophical grounds; b u t on linguistic grounds at least, it seems clear that (iii) should be distinguished from (i) a n d (ii). Given the above translation rules, t r u t h and validity within the Peirce s y s t e m can clearly be defined in t e r m s of t r u t h and validity in Ockhamistic structures:

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A f o r m u l a p is s a i d to be P e i r c e - t r u e a t a t i m e t in (TIME, <, C, _Peirce) if and only if the t r a n s l a t i o n o f p into t h e Ockhamistic l a n g u a g e is true at t w i t h t ~ c. A f o r m u l a is P e i r c e - v a l i d iff its t r a n s l a t i o n is O c k h a m valid. L e t u s consider t h e formula: q ~ H F q . W h e n this f o r m u l a is t r a n s l a t e d into the Ockhamistic language, we g e t t h e formula:

q ~ HNFq It is obvious from t h e above definitions t h a t Ock(t,c,q ~ HNFq) c a n be m a d e false for s o m e t and some c w i t h t E c in s o m e s t r u c t u r e . Therefore, t h e formula q ~ HFq, is n o t a t h e o r e m in Prior's P e i r c e a n system. Now, t h e t r u t h - o p e r a t o r in the Peircean s y s t e m does not h a v e to be d e f i n e d in t e r m s of t h e Ockhamistic o p e r a t o r . It w o u l d h a v e been possible to p r e s e n t it quite independently. B u t since w e w a n t to c o m p a r e the two systems, the above definitions are v e r y useful. W e can i m m e d i a t e l y verify the most i n t e r e s t i n g f e a t u r e of Prior's definition of P e i r c e a n truth:

Peirce(t, Fp) if and only if for all (c',t) w i t h (c,t) = (c',t) Peirce(t',p) for some t' e c' w i t h t < t' This a p p e a r s to be in v e r y good accordance w i t h t h e ideas of C. S. Peirce, since he as we h a v e seen in chapter 2.2 rejected t h e v e r y i d e a t h a t s t a t e m e n t s r e g a r d i n g the contingent f u t u r e could be true.

2.9. FOUR GRADES OF TENSE-LOGICAL INVOLVEMENT In order to construct a tempo-modal logic, which is intuitively satisfactory, we m a y proceed semantically; it is not d e m a n d e d t h a t a full axiomatic system t o g e t h e r w i t h proofs of s o u n d n e s s a n d completeness be given. Needless to say, such results would certainly be desirable, where they can be achieved. However, in t h e general project of formal s e m a n t i c s for n a t u r a l languages it s e e m s to be c o m m o n l y accepted t h a t from some p o i n t of complexity, we m u s t necessarily d e p a r t from deductive systems in favour of model-theory [Dowty et al. 1979, p. 50 ft.]. T h a t is, a general and satisfactory formal s e m a n t i c s for n a t u r a l language probably c a n n o t be finitely and completely axiomatised. On t h e o t h e r h a n d , t h e s e observations do n o t necessarily apply to t h e restricted case of tense logic. Thus, in t h e discussion of issues in philosophical logic in relation to everyday l a n g u a g e we can in principle confine ourselves to t h e semantics of t h e systems. This is exactly w h a t we i n t e n d to do w h e n dealing with the problem of finding an indeterministic t e n s e logic, w h i c h is intuitively satisfactory. Nevertheless, the tension between a proof-theoretical approach to tense logic and a s e m a n t i c a l approach should not be exaggerated. As we shall see in this chapter, A.N. Prior has shown t h a t for tense logic the two approaches can (and should) be e m b e d d e d in an approach of a higher order. I n c h a p t e r 2.4 we briefly m e n t i o n e d t h a t tense logic corresp o n d s to McTaggart's A-series conception, which sees t i m e in t e r m s of past, present, and future, w h e r e a s an earlier-later calculus corresponds to his B-series conception, which sees time as a set of objectively existing instants. Prior clearly considered the A-conception to be t h e f u n d a m e n t a l one: Time is not an object, but w h a t e v e r is real exists and acts in time... B u t this earlier-later calculus is only a convenient b u t indirect w a y of expressing t r u t h s t h a t are not really about 'events' but about things ... [TR p. 2-3]

216

FOUR GRADES OF TENSE-LOGICAL INVOLVEMENT

217

Prior introduced four grades of 'tense logical involvement'. The first grade defines tenses entirely in t e r m s of objective instants a n d an earlier-later relation. For instance, a sentence such as Fp, 'it will be the case thatp', is defined as a short-hand for 'there exists some instant t which is later than now, a n d p is true at t', a n d similarly for the past tense; these definitions are, of course, the same ones as those we already stated in chapter 2.8, namely (DF) (DP)

T(t, Fp) ~lef 3 t l : t
Tenses, then, can be considered as mere meta-linguistic abbreviations, so this is the lowest grade of tense logical involvement. Prior succinctly described the first grade as follows: ...there is a nice economy about it ... it reduces the minimal tense logic to a by-product of the introduction of four definitions into an ordinary first-order theory, and richer [tense logical] systems to by-products of conditions imposed on a relation in that theory. [Prior 1968, p. 118] In t h e first grade, tense operators are simply a h a n d y w a y of s u m m a r i z i n g the properties of the before-after relations, which constitute the B-theory of McTaggart. Hence, in the first grade B - t h e o r y concepts are seen to be d e t e r m i n i n g for a proper u n d e r s t a n d i n g of time and reality; tenses are deemed to have no independent epistemological status. The basic idea is a definition of t r u t h relative to temporal instants (this definition is in fact a l r e a d y incorporated into the notion of a B-logical s t r u c t u r e defined in chapter 2.8): (T1) (T2)

T(t,p A q) - (T(t,p) A T(t,q)) T(t,-p) - -T(t,p)

In addition, there may be some specified properties of the beforeafter relation, like for instance transitivity: (B1)

(tl
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In this way, instants acquire an independent ontological status. As we h a v e seen, Prior rejected the idea of temporal instants as something primitive and objective. In the second grade of tense logical involvement, tenses are not reduced into B-series notions. Rather, t h e y are treated on a par with the earlier-later relation. Specifically, a bare proposition p is t r e a t e d as a syntactically full-fledged proposition, on a par with w h a t Rescher and U r q u h a r t [1971] called 'chronologically definite' propositions such as T(t,p) ('it is true at time t t h a t p'). The point of the second grade is t h a t a bare proposition with no explicit temporal reference is not to be viewed as an incomplete proposition. One consequence of this is t h a t an expression such as T(t,T(t',p)) is also well-formed, and of t h e same type as T(t,p) and p. Prior showed how such a system leads to a n u m b e r of theses, which relates tense logic to the earlier-later calculus and vice v e r s a [Prior 1968, p. 119]. The following crucial rule of inference m a k e s this relation within the second grade especially obvious: (RT) If ~p, then e T(t,p) for any t and any truth-operator T. He also stated the following basic assumptions regarding the truth-operator: (TX1) (TX2) (TX3)

(Vt: T(t,p)) ~ p (Vt1: T(tl,p)) ~ T(t2, Vt3: T(t3,p)) T(tl,p) ~ T(t2, T(t~,p))

The philosophical implication of this second grade of t e n s e logical involvement is t h a t one must regard the basic A- and Btheory concepts as being on the same conceptual level. Neither set of concepts is conditioned by the other. The B-theory is sometimes considered as the semantics of the corresponding A-theory. This is not surprising if we again consider the first-grade formulation of Fp, 'it will be the case t h a t p', as a s h o r t - h a n d for 'there exists some instant t which is later than now, and p is true at t' (cf. (DF)).

FOUR GRADES OF TENSE-LOGICAL INVOLVEMENT

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This is t a n t a m o u n t to stating a t r u t h condition for F p . On this view of the relationship between the A- and B-theories, it m a y be a bit puzzling t h a t p and T ( t , p ) can be t r e a t e d as being on the same logical level - the former apparently belonging to the logical language (or object language) and the latter to the semantics (or meta-language). In Prior's opinion, however, this is not at all surprising. In a paper on some problems of self-reference he stated: In other words, a language can contain its own semantics, t h a t is to say its own theory of meaning, provided t h a t this semantics contains the law t h a t for any sentence x, x means t h a t x is true. [Prior 1976a, p. 141] It seems t h a t this statement is exemplified exactly by the relation of the logic of tenses (the A-theory) to the logic of earlier and l a t e r (the B-theory), provided t h a t we are willing to t a k e the step of the second grade: syntactically conflating 'bare' p w i t h T ( t , p ) . The relation becomes even clearer in the third grade, a system which has crucial implications for the status of the indication of time. Prior introduced the third grade in the following way: W h a t I shall call the t h i r d g r a d e of t e n s e logical involvement consists in treating the instant-variables a, b, c, etc. as representing propositions. [Prior 1968, p. 122-23] Such instant-propositions describe the world uniquely at any given i n s t a n t , and are for this reason also called world-state propositions. Like Prior we shall use a, b, c ... as i n s t a n t propositions instead of tl, t2, ... In fact, Prior assumed t h a t such propositions are what ought to be m e a n t by 'instants': A world-state proposition in the tense-logical sense is simply an index of an instant; indeed, I would like to say that it is an instant, in the only sense in which 'instants' are not highly fictitious entities. [Prior 1967, p. 188-89]

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The traditional distinction between t h e description of t h e c o n t e n t a n d the i n d i c a t i o n of t i m e for an e v e n t is t h e r e b y dissolved. F r o m the p r o p e r t i e s of the logical l a n g u a g e w h i c h embodies t h e third grade of tense logical involvement Prior also showed t h a t T(a,p) c a n be defined in t e r m s of a p r i m i t i v e necessity-operator. T h e n tense logic, and indeed, all of t e m p o r a l logic can be developed from t h e purely 'modal notions' of past, present, future, and necessity. In our opinion this idea of t r e a t i n g i n s t a n t s as some k i n d of world propositions was one of his most interesting constructions. We believe t h a t the full s t r e n g t h of this view h a s not yet been displayed. It is very likely that this notion will t u r n out to be very useful in t h e part of c o m p u t e r science called n a t u r a l l a n g u a g e u n d e r s t a n d i n g [Hasle 1991]. The f o u r t h grade consists in a tense logical definition of t h e necessity-operator such t h a t t h e only primitive operators in t h e t h e o r y are the two t e n s e logical ones: P and F. Prior h i m s e l f favoured t h i s fourth grade. It a p p e a r s t h a t his r e a s o n s for w a n t i n g to r e d u c e m o d a l i t y to t e n s e s w e r e m a i n l y metaphysical, since it h a s to do with his rejection of the concept of the (one) true (but still u n k n o w n ) future. If one accepts t h e fourth grade of tense-logical involvement, it will t u r n out t h a t s o m e t h i n g like the Peirce solution will be natural, and t h a t we have to reject solutions which involve crucially t h e idea of a t r u e or simple future - like t h e Ockhamistic theory. A tense-logical a p p r o a c h to t h e concept of t i m e involves a c o m m i t m e n t to t h e t h i r d or the fourth grade. We ourselves prefer a tense-logical approach, essentially for t h e same reasons as Prior. However, s y s t e m s based on t h e t h i r d grade are obviously more general t h a n systems based on the fourth grade. If the fourth grade is accepted, interesting s y s t e m s such as t h e Ockhamistic have to be ruled out. Therefore, we are inclined to consider t h e t h i r d g r a d e to be t h e d e s i r a b l e basis for a conceptually adequate logic of time. In t h e following we shall present some of Prior's most i m p o r t a n t results with respect to the third grade - that is, the theorems in question are valid, if the third grade is accepted. Our presentation will differ s o m e w h a t from Prior's, though. Before proceeding it m a y be noted t h a t

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[Hasle 91] gives an exposition w h i c h follows Prior's own presentation more closely.

THE LOGIC OF INSTANT-PROPOSITIONS With t h e s t a n d a r d set of well formed f o r m u l a e (wff) of propositional tense logic we assume Kt, i.e. the axioms (A1)

p, where p is a tautology of the propositional calculus

(A2) (A3) (A4) (A5)

G(p ~ q) ~ (Gp ~ Gq) H(p ~q) ~ (Hp ~ H q ) p ~GPp p ~ HFp

and the rules of inference (RMP) (RG)

(RH)

If ~p and ~p ~ q, then e q. If e p , then e Gp. If e p , then e Hp.

w h e r e '~' m e a n s 'it is provable in the system that'. It is sometimes useful to mention the system explicitly, as in Kt t-p. In chapter 2.8 we demonstrated t h a t if Kt ep t h e n T(t,p) holds for any t, and any T-operator which satisfies (T1-2) and the definitions (DP), (DF), (DG) and (DH). In other words, Kt is sound. We shall now argue that if the system is 'interpreted' as in Prior's third grade, it is also complete. More precisely, if T(t,p) holds for a n y t and for any T satisfying (T1-2), (DP), (DF), (DG) and (DH) t h e n p is provable in Kt, provided t h a t we adopt the assumptions on which the third grade is based. We are not going to d e m o n s t r a t e completeness in a traditional m a t h e m a t i c a l way, but we intend to show t h a t a result very similar to completeness can be obtained in the context of Prior's t h i r d grade. In order to do t h a t we need a set of i n s t a n t propositions {a, b, c, ...]. We shall define an instant proposition as

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C H A P T E R 2.9

a m a x i m a l c o n s i s t e n t s e t of Kt-wffs. We e x t e n d the n o t i o n of well f o r m e d f o r m u l a e in t h e following way, calling the e n l a r g e d s y s t e m Prt:

(1) (2) (3) (4)

A n y Kt-wff is a Prt-wff. A n y i n s t a n t proposition a is a Prt-wff. I f a and fl are Prt-wff, and a is an instant proposition, then -a, aAfl, Va:a, Pa, and Fa are all Prt-wffs. T h e r e are no other Prt-wffs.

In addition, w e a s s u m e t h e s t a n d a r d definitions of propositional a n d p r e d i c a t e logic, including t h e definition of 3a: a a s ~ Va:~a. In t h e following, 'p' s t a n d s for an a r b i t r a r y P r t - w f f , w h e r e a s 'a' s t a n d s for an arbitrary i n s t a n t proposition. The axioms of Prt are the axioms of Kt together with the a x i o m (I1)

3a: a

and the rule:

(RI)

F o r a n y i n s t a n t proposition a and a n y wffp: Exactly one of ~ a ~ p and ~ a ~ - p

together w i t h t h e rules included in Prior's quantification t h e o r y [Prior 1955, p.76 ft.]:

(HD (n2)

I f ~ ~(x)~fl, t h e n ~ (Vx:~(x))~fl. I f ~ a ~ ( x ) , t h e n ~ a~Vx:O(x), for x not free in a.

F r o m (H1-2) it is easy to deduce [Prior 1955, p. 82] that (El) (Z2)

I f ~ O(x)~fl t h e n ~ 3x:~x)~fl, for x not free in ft. I f ~ a~O(x) t h e n ~ otD3x:¢(x).

It should b e noted t h a t (RI) is n a t u r a l in t h e light of w h a t it m e a n s to b e a m a x i m a l consistent set. Intuitively, an i n s t a n t proposition a m a y be v i e w e d as the conjunction of the e l e m e n t s

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in the maximal consistent set. (I1) is also r a t h e r natural since it simply states t h a t some instant proposition holds now. It is not difficult to prove that the system Prt defined in this way is sound a s s u m i n g (T1-2) and - quantification as in (H1-2) within the scope of T; T(t,a) defines a one to one correspondence between t i m e s and instant propositions; -

for the T-operators in question. In order to obtain a result similar to 'ordinary completeness', we need a l a n g u a g e in which we can express a n o t i o n corresponding to Prt-provability, i.e. a meta-language relative to the Prt-language. For this reason we again extend the notion of well-formed formulae and call the resulting language Priort: (1) (2) (3)

A n y Prt-wff is a Priort-wff. If a and fl are Priort-wffs, then -a, a A fl, La, and Va:a are all Priort-wtTs. T h e r e are no other Priort-wffs.

In addition, we assume the standard definitions from propositional and predicate modal logic, especially the definition of M as ~ L - . The axiomatic system of Priort consists of Prt and t h e axioms (L1) (L2) (L3) (I2) (I3) (BF) (LG) (LH)

L(p Dq) D(Lp ~ L q ) Lp ~ p Lp ~ LLp -L-a L(a D p) v L ( a ~ ~p) L(Va: ~(a)) --- Va: L(O(a)) Lp ~ Gp Lp ~ Hp

along with the rule (RL)

If ~p, then ~Lp.

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It is obvious t h a t (RG) and (RH) follow from (RL), (LG), and (LH). It is also worth noting t h a t Lp intuitively m a y be r e a d as 'provable in Prt' (for a n y Prt-wffp). When read in this w a y it s e e m s reasonable t h a t L satisfies (RL) and the axioms above, provided t h a t these are restricted to Prt-wffs. (L1) m u s t hold for a n y notion of provability. (I2) holds, since i f - a is provable, then a c a n n o t be consistent. (I3) is in fact a consequence of (RI) t o g e t h e r with the consistency and the maximality of a. (BF) is known as the Barcan formula after Ruth C. B a r c a n [1946], who was able to d e m o n s t r a t e it for modal logics which satisfy a few basic conditions. I f L p is understood as 'provable in Prt', t h e n (BF) is r a t h e r n a t u r a l since it m a y be read as stating w h a t it m e a n s to prove Va: ¢~(a). Now we w a n t to construct a T-operator based on the full logic of i n s t a n t propositions i.e. Priort. That is, we wish show how an e n t i r e earlier-later-calculus can be developed - one m i g h t say boot-strapped - from definitions in the tense-logical theory. Let ~ denote the set of i n s t a n t propositions. For a r b i t r a r y elem e n t s a and b in £~ we introduce the following definitions: (DB)

a < b ~ef L (a D F b)

corresponding to 'the instant a is earlier t h a n the instant b', and (DT)

T(a,p) -~-def L(a ~ p )

corresponding to 'it is true at time a that p'. U s i n g these assumptions and definitions we can prove the theo r e m s (T1-2), as well as (DG) and (DH). In turn, this m e a n s t h a t (~,<,T) is a B-logical structure (with T defined as above). The proofs can be carried out in the following way: (TI.1) Proof." (1) (2) (3)

T(a,p Aq) ~(T(a,p) A T(a,q)) T(a,p A q) L(a ~ (p A q)) L((a ~ p ) A (a ~q))

(assumption) (1, using DT) (2)

FOUR GRADES OF TENSE-LOGICALINVOLVEMENT 225 (4) L(a ~ p ) AL(a ~ q ) (5) T(a, p) A T(a,q) Q.E.D.

(3) (4, using DT)

(T1.2) (T(a,p) A T(a,q)) ~ T(a,p A q) Proof." (assumption) T(a, p) A T(a,q) (1) L(a Dp) A L(a :=xl) (1, using DT) (2) (3) L((a ~ p) A (a ==rl)) (2) (3) L(a ~ (p A q)) (4) (4, using DT) T(a,p A q) (5) Q.E.D. Obviously, (T1) follows from (TI.1) and (T1.2). (T2.1) T(a,-p) ~ -T(a,p) Proof." This is proved by reductio ad absurdum. (1) T(a, -p) (assumption) (2) T(a,p) (assumption) (3) L(a ~ -p) (1) (4) L(p ~ -a) (3, using L1) (5) L(a mp) (2) (6) L~a (4 & 5; Contradicts I2) Q.E.D. (T2.2) -T(a,p) ~ T(a,-p) Proof." (1) -T(a,p) (2) L(a ~ -p) Q.E.D.

(assumption) (1, using I3)

Obviously, (T2) follows from (T2.1) and (T2.2). (DL.1) ~L-p ~ _=Vo:T(b,p) Proof." (1) -L-p (2) L(rVo: b)

(assumption) (using I1 and RL)

226

CHAPTER 2.9 (3) (4) (5) (6) (7) Q.E.D.

-L~(._'Tb:b A p) -~: ~L~(b rip) ::t-b: -L(b m -p) ._Wo:L(b rap) 9'o: T(b,p)

(1 and 2, using L1) (3, using BF) (4) (5, using I3) (6, using DT)

(DL.2) Lp m Vb: T(b,p) Proof."

(1) Lp (2) p m (b ~ p ) (3) L(b rap) (4) T(b,p) (5) Lp ~ T(b,p) (6) Lp ~ Vb: T(b,p) Q.E.D.

(assumption) (A1) (1 and 2, using L1) (3, using DT) (1 and 4) (H2)

It follows from (DL.1) and (DL.2) that (DL) Va: T(a,p) - L p In order to prove the remaining theorems, we need the following lemma about the ordering relation: (DB.1)

a < b mL(b ~ P a )

Proof."

This proved by reductio ad absurdum: (1) a
FOUR GRADES OF TENSE-LOGICALINVOLVEMENT 227 Similarly, it can be proved that (DB.2) L(b ~ Pa) ~ a < b This means that (DB.3)

a < b - L ( b ~ Pa )

The remaining task is to prove (DG) and IDH) for the structure (~-2,T,<) defined here. This can be done in the following way: (DG.1) T(a, Gp) ~ (a
This is proved by reductio ad (1) T(a, Gp) (2) a
absurdum: (assumption) (assumption) (assumption) (3, using DT and I3) (4 by LG) (1, using DT) (5 and 6) (7) (2) (8 and 9) (10).

(DG.2) T(a, Gp) ~ ~zb:(a
This follows immediately from the (H2) and the fact that (DG.1) is proved for an arbitrary b. (DG.3) Vb:(a
This is proved by reductio ad absurdum: (1) ~zb:(a
228

CHAPTER 2.9 (4) _~b: T(b, Pa /~ -p) (5) __~b:T(b, Pa) A T(b,-p) (6) __~b:a
(3 a n d DL.1) (4, using T1) (5, using DB.3) (6 and 1)

(7) (8)

(assumption) (1 and DG.3) (2 and LG) (A4) (3 and 4)

(5)

(DG) T(a, Gp) - Vb:(a
FOUR GRADES OF TENSE-LOGICAL INVOLVEMENT

229

A n u m b e r of other interesting theorems may be proved, such as the following ones: (TX1) (Va: T(a,p)) ~ p Proof" L(a ~ p ) ~ (a ~ p ) (1) (Va: L(a ~p)) ~ (a ~p) (2) a ~ ((Va: L(a Dp)) Dp) (3) (3a: a) ~ ((Va: L(a ~p)) ~ p ) (4) (Va: L(a ~p)) ~ p (5) (Va: T(a,p)) ~ p (6) Q.E.D. (TX2) (Va: T(a,p)) ~ T(b, Vc: T(c,p)) Proof" (1) L(a ~ p) ~ (b ~ L(a ~ p)) L(a ~ p) ~ L(b ~L(a ~ p)) (2) Va: L(a D p) ~ Va: L(b DL(a D p)) (3) (4) Va: L(a D p) D L(b ~ Va: L(a ~ p)) Va: T(a,p) ~ T(b, Va: T(a,p)) (5) (6) Va: T(a,p) ~ T(b, Vc: T(c,p)) Q.E.D. (TX3) T(a,p) ~ T(b,T(a,p)) Proof" (1) L(a ~ p) ~ (b ~ L(a ~ p)) (2) L(a ~ p) D L(b DL(a D p)) T(a,p) ~ T(b,T(a,p)) (3) Q.E.D.

(L2) (1, H1) (2) (3, H2) (4, I1) (5)

(1, L1, L3) (2) (3 and BF)

(1, L1, L3)

In this way formulae of the T-calculus are mixed w i t h wffs from Prt. In Priort, everything is included in one single language comprising the T-calculus as well as ordinary tense logic. This extended language is simply Prt with the addition of the logic of i n s t a n t propositions - since the elements of the T-calculus are introduced by definitions based on Prt. This way of seeing things is far from the 'main-stream' t r a d i t i o n within formal logic, w h e r e proof theory (in this case the axiomatics of Prt) is kept

230

C H A P T E R 2.9

strictly s e p a r a t e d from s e m a n t i c s (in this case the T-calculus). But as Prior pointed out t h e r e is n o t h i n g semantically wrong with it, if t h e T-calculus is given an interpretation within tense logic. He also pointed out t h a t such an interpretation could be 'metalogically useful', since in m a n y cases T(a,p) t u r n s out to be easier to prove t h a n t h e 'bare' tense-logical f o r m u l a p i t s e l f [1967, p.89]. Prior h a s t h u s shown t h a t we can in fact i n t e r p r e t B-logic w i t h i n A-logic, n a m e l y in a given modal context in which we can i n t e r p r e t instants as propositions a n d quantify over them. In this sense B-logical s e m a n t i c s is absorbed within an entirely A-logical axiomatics. In Prior's own words, this m e a n s "to t r e a t the first order theory of t h e earlier-later relation as a m e r e byproduct of tense logic" [1968, p.160].

2.10. METRIC TENSE LOGIC In the discussion of problems like 'the sea-fight tomorrow' we need more t h a n modal logics and tense-logical systems like K~ and Kl. We also need metric t e n s e logic, in which n u m e r i c a l durations are taken into consideration. In metric tense logic it is a s s u m e d t h a t some metrical systems for duration (including a relevant time unit) are given. Of course, we have already used metric tense logic in earlier discussions, precisely because this is w h a t is required for cases like 'the sea-fight tomorrow'. In this c h a p t e r we shall examine this subject more systematically. Prior h i m s e l f dealt with t h e problems of metric t e n s e logic several times. We shall now study his basic ideas and present his version of the so-called minimal system for metric tense logic [Prior 1968, pp. 88-97]. We shall call the system MT. The language of MT is based on a set of propositional variables: p, q, r..., and the following definition of well-formed formulae:

(1) (2)

Propositional variables are wff. If a and fl are wff, and x is a positive number, then

(3)

are all wff. There are no other wff.

-a, a_~fl, aAfl, a v fl, Vx:v¢ 3x: a, P(x)~, and F(x)a

So t h e essentially new element in metric tense logic are the expressions P(x)a and F(x)a, stating respectively 'x time units ago it was the case t h a t d and 'in x time units it will be the case t h a t a'. In the following x and y s t a n d for a r b i t r a r y positive n u m b e r s . N. Rescher [1966] has suggested the use of not only positive, but also negative n u m b e r s along with the definition P(x)a =defF(-x)a. Prior, on the other hand, argued t h a t things can become very complicated, if we want such a definition in its full generality [1967 p.98]. For this reason he suggested t h a t no negative n u m b e r s should occur in metric tense logic. Prior did discuss the use of the n u m b e r zero in the forms P(O)a and F(O)a, but towards the end of his work he decided to leave such

231

232

C H A P T E R 2.10

possibilities out. - W e shall define

G (x) a --def ~F (x)~ a a n d H (x) a -~def- P ( x ) - oc. A few w o r d s on t h e s e m a n t i c a l i n t u i t i o n involved in t h e s e definitions might be profitable. In relation to a b r a n c h i n g t i m e model, G(x)a is t r u e iff a is t r u e in x t i m e u n i t s from 'now' in a n y f u t u r e branch:

~llOW~i!i!il I

I x time

units

The 'quantification' implicit in t h e G - o p e r a t o r is over t h e set of all b r a n c h e s rooted in t h e 'now'. On t h e o t h e r h a n d it is n o t required t h a t a a f t e r x t i m e units should be t r u e 'forever' on a n y branch. We m e n t i o n t h i s explicitly, b e c a u s e w e are o t h e r w i s e used to read G as 'it is always going to be t h e case that'. In a model b a s e d on linear time, t h e r e w o u l d be no difference b e t w e e n G(x)a a n d F(x)a, for in such a m o d e l -F(x)a - F ( x ) - a . H o w e v e r , since w e a r e p r e s e n t l y c o n c e r n e d w i t h a m i n i m a l m e t r i c t e n s e logic, w e s h a l l m a k e no a s s u m p t i o n s on t h e s t r u c t u r e of time. S i m i l a r observations go for H(x)a and P(x)a. The axioms of t h e non-modal p a r t of M T a r e t h e following: (MT1) (MT2) (MT3)

G(x)(p ~ q) ~ (G(x)p ~ G(x)q) F(x)H(x)p ~ p F(y+x)p ~ F(y)F(x)p

METRIC TENSE LOGIC

233

In addition to these axioms Prior also stated some axioms for the past-operator. However, t h e s e axioms can be omitted if we introduce the so called 'mirror-image-rule': (RM) The 'mirror image' of any t h e o r e m (in which all occurrences of P are replaced by F and vice versa) is also a theorem. For instance, (MT2') P(x)G(x)p ~ p is t h e mirror-image of (MT2). In the following, references in proofs to axioms and t h e o r e m s may also refer to their mirrorimage 'versions'. The other rules of MT are the following: ( R M P ) If~p and ~p D q , then ~q. If e p , then ~ G(x)p (for any x). (RF) (RP) If e p , then ~ H(x)p (for any x). (Hi) If ~ e)(x)~fl, t h e n ~ Vx:C)(x)~fl. (n2) If ~ a ~ x ) , t h e n e a~Vx:C)(x), for x not free in a. (ZI) If e ~(x)~fl, t h e n e _~:¢(x)~fl, for x not free in ft. If e a~O(x), t h e n e a~3x:O(x). (E2) As in chapter 2.9, we are using Prior's theory of quantification, which presupposes t h a t quantification does not take place over e m p t y sets. Since the quantification in MT is over the set of positive numbers, this is not in practice a restriction. In MT it is possible to prove a number of interesting theorems like (MT4) (MT5) (MT6)

H(x)(p ~ q) ~ (P(x)p ~ P(x)q) p ~ G(x)P(x)p P(x)G(x)p ~ p

234

CHAPTER 2.10

Prior actually s u g g e s t e d two more axioms t h a n (MT1-3): (MT7) (MT8)

Vx:G(y)G(x)p ~ G(y)Vx:G(x)p Vx:G(y)H(x)p ~ G(y) Vx:H(x)p

However, these formulae can be proved in the (1) G(y)q ~ G(y)q (2) Y'x:G(y)q~ G(y)q (3) H(y)(Vx:G(y)q ~ G(y)q) (4) P(y) Vx:G(y)q ~ P(y)G(y)q (5) P(y) Y'x:G(y)q ~ q (6) P(y)Vx:G(y)q ~Vx:q (7) G(y)(P(y) Vx:G(y)q ~ Vx:q) (8) G(y)P(y) Vx:G(y)q ~ G(y) Vx:q (9) Vx:G(y)q ~ G(y) Vx:q

following way: (1 (2 (3 (4 (5 (6 (7 (8

and and and and and and and and

H1) RP) MT4) MT6) 1-I2) RF) MT1) MT5)

In order to get (MT7) and (MT8) we should replace q by G(x)p and H(x)p respectively. Observe that (MT7-8) are very close to t h e Barcan Formula(e). On the basis of t h e metric tense logic MT we can build a nonmetric tense logic. We introduce the definitions ( D U F ) Fp --~def3x:F(x)p (DUG) Gp -~-defVx:G(x)p ( D U P ) Pp ~-def3x:P(x)p (DUH) Hp --~defVx:H(x)p (We are presently not a s s u m i n g any special s t r u c t u r e of time.) Note t h a t in order to make these definitions plausible we have to assume that x is not free Lap. As pointed out by Prior [1967, p.95] the reason is t h a t two propositions like

F(q A F(x)p) a n d 3x: F(x)(q A F(x)p) are obviously n o t equivalent.) - From t h e above definitions it is possible to prove t h e axioms of K~ as theorems of MT. Because of the 'mirror-image-rule' it is sufficient to prove (A2) and (A4).

METRIC TENSE LOGIC

(A2) Proof."

G(q ~ p) ~ (Gq ~ Gp)

(1)

G(q ~ p) Vx:G(x)(q ~ p) Vx: (G(x)q ~ G(x)p) Vx:G(x)q ~ Vx:G(x)p Gq ~Gp

(2)

(3) (4)

(5)

235

(assumption)

(1) (2, u s i n g MT1) (3) (4)

Q.E.D. (A4) Proof." (1) (2) (3) (4) (5) (6) (7) Q.E.D.

q ~ HFq F(x)q ~ F(x)q F(x)q ~ Ex: F(x)q F(x)q ~ Fq H(x)F(x)q ~ H(x)Fq q ~ H(x)Fq q ~ Vx: H(x)Fq q ~HFq

(1 by Z2)

(2) (3 by RP) (4 a n d MT5) (5, u s i n g H2)

(6)

In some systems w i t h more axioms t h a n MT the propositions F(x)q a n d G(x)q will be equivalent - for instance, in s y s t e m s e n l a r g e d w i t h an axiom for forward l i n e a r i t y , as we have already suggested earlier. However, t h e difference in s y s t e m s like MT between F(x) a n d G(x) is interesting. It t u r n s out t h a t in s u c h s y s t e m s G(x) comes very close to w h a t we h a v e presented as the Peircean notion of 'in x t i m e units it is going to be that', whereas F(x) corresponds to 'in x t i m e units it is possibly going to be'. - It should also be noted t h a t it is possible to define new non-metric tense-operators corresponding to the following four expressions: Vx: F(x)p, Vx: P(x)p, :Tx: G(x)p, and !~x: H(x)p. So MT could in fact give rise to a richer t e n s e logic t h a n Kt.

236

CHAPTER 2.10

I N S T A N T - P R O P O S I T I O N S AND METRIC T E N S E - L O G I C MT can be extended so that the extended system, MT*, includes instants-propositions a, b, c... as well as a modal operator L for which the Barcan formulae

(BF1) (BF2)

Vx:L¢~(x) ~ L(Vx: ~(x)) 3x:L¢~(x) ~ L(gx: ~x))

hold along with the axioms

(LHX) Lq DH(x)q ( L G X ) Lq ~G(x)q Note t h a t MT* involves quantification over positive n u m b e r s as well as q u a n t i f i c a t i o n over i n s t a n t propositions. B a r c a n ' s formulae are a s s u m e d for both kinds of quantification. We also use the same quantifier symbols, since it is obvious in each case which kind of quantification is at play. It is k n o w n from modal logic t h a t the Barcan formulae are provable in a n y 'L-calculus' satisfying a few basic conditions. However, in the same way as in chapter 2.9 we m a y intuitively assume L to be at least as strong as 'provability' in MT. For these reasons the results demonstrated in the following w i t h respect to MT* could also serve as an a r g u m e n t for t h e k i n d of completeness mentioned in chapter 2.9. It is obvious t h a t the following theorems can be proved from the axioms mentioned above

(LH)

Lq ~Hq

(LG)

Lq ~ G q

The a s s u m p t i o n s (I1)-(I3) are also added to the system. This m e a n s t h a t we now have a s y s t e m stronger t h a n t h e one we d e v e l o p e d in c h a p t e r 2.8-9. Hence, t h e proofs of (T1-2) regarding t h e 'truth-operator' T can also be carried out in the present system. - In addition, we defme

METRIC TENSE LOGIC (DB1)

237

before(a,b,x) -~-defL(a ~F(x)b)

We can now prove a number of further theorems: (T8) before(a,b,x) -=L(b ~P(x)a) Proof." This is proved by reductio ad absurdum: (1) L(a ~F(x)b) (assumption) (2) ~L(b wP(x)a) (assumption) (3) L(b ~-P(x)a) 12, using I3) (4) L(F(x)b ~ F ( x ) - P ( x ) a ) (3 and MT4) (5) L(a ~F(x)-P(x)a) (1 and 4) (6) L(a ~F(x)H(x)~a) (5) (7) L(a ~ ~a) (6 and MT2) (8) L-a (7) (8) is contradicting II!- The opposite implication is similar. Q.E.D. (T9) a
3 x: before(a,b,x) ~ a < b

Proof."

(1) (2) (3) (4) (5) (6) Q.E.D.

3 x: before(a,b,x) 3 x: L(a ~ F(x)b) L ( 3 x: (a ~ F(x)b)) L(a ~ 3 x: F(x)b) L(a ~ Fb) a
(assumption) (1, using DB1) (2 and BF2) (3) (4, using DMF) (5, using DB)

CHAPTER 2.10

238

It follows from (T9) and (T10) that (Tll) a
(T12)

3 b: (before(a,b,x) A T(b,p)) ~ T(a,F(x)p)

Proof"

(1) (2) (3) (4) Q.E.D.

L(a ~ F(x)b) L(b ~ p) L(F(x)b ~ F(x)p) L(a ~ F(x)p)

(assumption) (assumption) (2)

(1 and 3)

(T13) L(P(x)a ~ -p) ~ L(a ~ ~F(x)p) Proof"

(1) (2) (3) (4) Q.E.D.

L(P(x)a ~ ~p) L(G(x)P(x)a ~ G(x)-p) L(a ~ G(x)~p) L(a ~ ~F(x)p)

(assumption) (1) (2 and MT5) (3)

(T14) T(a,F(x)p) ~ (3 b: before(a,b,x) A T(b,p)) Proof"

(1) (2) (3) (4) (5) (6) (7) (8) Q.E.D.

T(a,F(x)p) L(a ~ F(x)p) ~L(a ~ ~F(x)p) -L(P(x)a ~ -p) -L~(P(x)a A p) _~b: T(b,P(x)a A p) Fo: (T(b,P(x)a) A T(b,p)) _~-b:(before(a,b,x) A T(b,p))

(assumption) (1) (2) (3 and T13) (4) (5 and DL) (6) (7)

METRIC TENSE LOGIC

239

It follows immediately from (T12) and (T14) that (T15) T(a,F(x)p) - ( 3 b: before(a,b,x) ,~ T(b,p)). Moreover, by using the definition of before(a,b,x) (DB1) and the 'mirror-image rule' RM on (B19), we obtain (T16)

T(a,P(x)p) - ( 3 b: before(b,a,x) A T(b,p))

Prior suggested that M T corresponds to any T-calculus for which we have (T1-2), (T15), (T16), and (T17) (T18)

T(a,3x:q) - 3x:T(a,q) before(a,b,x+y) ~ 3c:(before(a,c,x) A before(c,b,y))

These theses can also be proved in the instant-calculus. It can be done in the following way: (T 17.1 )T(a, =Tx:q) ~ ~x: T(a, q) Proof." This is proved by reductio ad absurdum. (1) T(a, gx:q) (assumption) (2) Vx:~T(a,q) (assumption) (3) Vx:L(a ~-q) (2) (4) L(Vx:(a ~-q)) (3 and BF1) (5) L(a ~Vx:-q) (4) (6) L(a ~-(3x:q)) (5, contradicting 1) Q.E.D. (T17.2) Proof." (1) (2) (3) (4) Q.E.D.

_~x:T(a,q) ~ T(a,3x:q) _~x:T(a,q) 3x:L(a ~ q) L(3x:(a ~ q)) L(a ~ ~x:q)

(assumption) (1) (2 and BF2) (3)

240

CHAPTER 2.10 (T18)

before(a,b,x+y) ~ 3c:(before(a,c,x) A before(c,b,y))

Proof." (1)

(2) (3) (4) (5)

before(a,b,x+y) T(a,F(x +y)b) T(a,F(x)F(y)b ) 3c:(before(a,c,x) A T(c,F(y)b)) 3c:(before(a,c,x) A before(b,c,y))

Q.E.D. The MT axioms are valid in each T-calculus in which (T1-2) and (T15-18) hold. Such a T-calculus can be said to be minimal, since no f u r t h e r conditions on t h e before-relation have to be assumed. Given a set of instant-propositions for which (I1-3) hold, a n d a modal operator L for which (BF1-2) and (LHX + LGX) hold, we have also d e m o n s t r a t e d t h a t it is possible to define a T-calculus within MT. In t h a t sense MT corresponds to a m i n i m a l T-calculus. Given t h e e x i s t e n c e of i n s t a n t propositions, any theorem that can be proved in MT will also be valid in any T-calculus, and vice versa.

3.1. Two

PARADIGMS OF TEMPORAL LOGIC

Not only do we measure the movement by the time, but also the time by the movement, because they define each other. The time marks the movement, since it is its number; and the movement the time. [Aristotle, Physics, IV 220 b] Since A n t i q u i t y two images of t i m e have been discussed: t h e flow of t h e river and the line made up of stationary points. T h e tension b e t w e e n the two pictures of time, t h e dynamic a n d t h e static view, has for instance been expressed by t h e Aristotelian idea of t i m e as the n u m b e r of motion with respect to earlier a n d later - an idea, which comprises both pictures. On the one h a n d time is l i n k e d to motion, i.e. changes in t h e world, and on t h e other h a n d time can be conceived as a stationary order of e v e n t s represented by numbers. The basic set of concepts for the d y n a m i c u n d e r s t a n d i n g of time are past, present, and future. After McTaggart's a n a l y s i s of time, t h e s e concepts are called t h e A-concepts. They are well suited for describing the flow of time, since t h e present t i m e will become past, i.e. flow into past. The basic set of concepts for t h e s t a t i o n a r y u n d e r s t a n d i n g of time are before, simultaneously, a n d after. F o l l o w i n g M c T a g g a r t , t h e s e a r e called t h e B-concepts, a n d they seem especially a p t for describing t h e p e r m a n e n t a n d temporal order of events. Philosophers and others still discuss intensively which of t h e two conceptions is the more f u n d a m e n t a l one for the philosophical description of time. The situation can well be characterised as a d e b a t e b e t w e e n two K u h n i a n p a r a d i g m s - t h e i d e a s embodied by t h e well-established B-theory, which were for some centuries p r e d o m i n a n t in philosophical and scientific theories of time, a n d t h e rising A-theory, w h i c h in t h e 1950s received a fresh i m p e t u s due to the advent of Prior's tense logic. Still, m a n y researchers do not w a n t to embrace t h e A-conception. The m a i n reason for this reluctance is the perception t h a t t h e A - t h e o r y does not have t h e 'precedence' or priority, which can a p p a r e n t l y

243

244

CHAPTER 3.2

be a t t r i b u t e d to t h e B-theory. Some a r g u m e n t s often put forward in favour of the B-theory are these: 1) A-concepts can only be defined in relation to B-concepts. 2) B-concepts cannot be reduced into A-concepts. 3) The future is just as real as the past. 4) A-concepts are not objective, but depend upon emotions. 5) B-concepts are objective. 6) The physical theories about time support the B-theory. In the following we shall ourselves come down quite openly on one side in the conflict: we shall argue that t h e r e is indeed a stronger case for t h e A-conception than for the B-conception. S u r e l y the B-concepts do play a r61e in thought and language; but t h e y should in our opinion be seen as secondary to or derived from the A-concepts. In our argumentation, we shall go into detail with the theses (1)-(6). Towards the end of the chapter we shall be concerned with McTaggart's paradox and its relevance for the debate; we shall also outline the general contours of the two paradigms.

REDUCTION OF A-CONCEPTS M a n y B-theorists h a v e m a i n t a i n e d t h a t every s t a t e m e n t couched in A-concepts can be reformulated into a s t a t e m e n t in t e r m s of B-concepts - or at least t h a t its truth conditions can and should be defined by B-concepts. Take for instance the A-statem e n t 'It will rain in London', and assume it to be uttered at 1:51 p.m. on the 10th of November in the year 1994. According to B-theorists this s t a t e m e n t is completely equivalent w i t h the B-statement 'It rains in London at a moment of time after 1:51 p.m. on the 10th of November in the year 1994'. It should be noted t h a t it is only possible to reduce the A-statem e n t if the time of u t t e r a n c e is known. Already for this reason the suggested procedure for reducing the A - s t a t e m e n t is not satisfactory. Indeed, it is not t r u l y possible to express past,

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p r e s e n t and future in t e r m s of the B-theory. How, for instance, could it be possible to express 'E has happened' in such a Blanguage? One might t r y to go for something like 'E is before the u t t e r a n c e of this very statement'. However, such a formulation is really based on a concealed reference to the present. In other words, t h a t which should be explained away (tenses) is really presupposed - the definition is circular. In addition, there are examples of A-statements that cannot be r e d u c e d in t h a t way w i t h o u t demonstrable loss of meaning. Consider, for example, this statement, u t t e r e d at 1 p.m. on a given date: 'Fortunately, m y consultation w i t h the dentist is over'. According to the B-theory the s t a t e m e n t should be reduced into this: 'Fortunately, the consultation with the dentist is before 1 p.m. (on the given date)'. This B-statement, however, exhibits no semantic equivalence at all with the A-statement which is u t t e r e d at 1 p.m. While it is possible for a person by m e a n s of the A-statement to express the reason for his or her joy and relief, the B-statement merely states t h a t the temporal a r r a n g e m e n t between two instants is fortunate: 'Fortunately, tl is before t2[' In the B-reduction of the A-statement valuable semantic cont e n t pertinent to the reason for one's joy and relief- the fact t h a t the pain i s o v e r - has been lost. Similarly, the B-theoretical reduction of an A-statement about the future would fail to capture such semantic content w h i c h could give one reason to experience fear, expectation or excitement. On this background it m u s t be reasonable to reject the B-theoretical procedure of reduction.

REDUCTION OF B-CONCEPTS It is quite common among B-theorists to maintain that B-concepts cannot be reduced into A-concepts. If one would try to do that, the a r g u m e n t goes, it would be necessary to operate with new a n d extra concepts like 'more past than' and 'more future than'. Such an idea a p p a r e n t l y stems from the perception t h a t

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in A-theory, one can merely express t h a t an event is future, but not that one event is in a nearer future t h a n some other event and analogously for the past. In other words, t h e y assume t h a t in A-theory a temporal ordering among events cannot be expressed in general. This statement, however, is plainly wrong. Of course, it is immediately disproved if we take metric tense logic; but even if we take tense logic in its 'pure' form, it is possible to reduce B-concepts into A-concepts without any loss of meaning. T a k e t h e s t a t e m e n t "The event E1 will take place prior to the event E2'. This B-statement can be reduced into the following A-statement: '(Sometime) in the future, it true t h a t E1 is present and E2 is future'. In general, B-statements of the form 'The event E1 is prior to the event E2' can be reduced into Astatements of the form: The following is either past, present or future (true): 'E2 is present and E1 is past'. Naturally, w h e n giving such a defmition in terms of tense logic one m u s t ensure t h a t this relation between events actually satisfies all d e m a n d s imposed on the before/after-relation of the B-theory. The reduction of the B-theoretical concept of an instant into Aconcepts can only be carried out by employing Prior's deirmition of instants as a special kind of propositions. How this can be done formally we h a v e a l r e a d y seen in c h a p t e r 2.9 and 2.10. However, Robin Le Poidevin [1991, p. 36 ffi has argued t h a t Prior's propositional theory of instants is in tension with another basic tenet of Prior's, namely what Poidevin has called the antirealist construal of past and future tensed statements. He h a s also m a i n t a i n e d t h a t the theory is based upon an incoherent view of propositions, namely the idea that different tokens of t h e same tensed type (e.g. 'Socrates is sitting') uttered at different times express t h e same proposition. In both cases Poidevin's criticism is based on the assumption t h a t according to Prior's view t e n s e d propositions have "present fact as t h e i r t r u t h conditions" [1991, p. 37]. In our opinion, Poidevin's analysis is sound and interesting on its own terms, but we believe t h a t it based on a w r o n g assumption. In Prior's theory the notion of truth-conditions is not basic. It is in fact - as we have argued in part 2 - a derived idea. The semantical concept of being t r u e at an instant is defined in terms of present truth. But as Richard

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Swinburne has pointed out 'there is more to be known about t h e world t h a n you can know by knowing the t r u t h - v a l u e s of sentences at t h e i r time of utterance. You need to know w h i c h ones are true now, which of the ones which are were, or will be t r u e when u t t e r e d are t r u e now. And for such t r u t h timeless t r u t h conditions cannot be given' [1990, p. 121]. It is an essential element of Prior's t h e o r y t h a t the very common assumption of t r u t h as something timeless has to be rejected.

THE REALITYAND THE POSSIBLE FUTURE Some B-theorists have argued that the future is just as real as the past. They h a v e claimed that s t a t e m e n t s about the f u t u r e are true or false today, exactly in the same m a n n e r as statements about the past. Naturally this realism with respect to t h e future is n e c e s s a r y if one advocates a symmetrical concept of time - as a n u m b e r of B-theorists do. With this s y m m e t r i c a l concept it is possible to be safeguarded against a certain line of A-theoretical a r g u m e n t s , namely a r g u m e n t s based on the assumption of t r u t h - v a l u e gaps. For example, the idea t h a t statements about the future (seen in contrast to statements about t h e past) have no t r u t h - v a l u e today calls for a truth-value gap. But as we have seen it is possible to formulate a tense logic (i.e. an A-theory) w i t h o u t having to give up the notion of the reality of the future, whilst preserving the n a t u r a l asymmetry b e t w e e n past and f u t u r e (the Ockham system). Some of these theories can be traced back to a number of medieval considerations regarding h u m a n freedom and divine foreknowledge. While the assumption about the reality of the future can v e r y well be consistent with both B-theories and A-theories, the case is different w h e n looking at the assumptions about alternative future possibilities. Tentatively, t h a t assumption might be expressed as the following maxim: 'Whereas it is impossible to change the past, it is possible to change the future'. But t h a t formulation oversimplifies the matter. First, it excludes the reality of the future, and second, it is really self-contradictory to

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say t h a t one can change the future. The principle in question should r a t h e r be put in this way: 'Whereas no (real) alternative possibilities of the past are available, alternative possibilities of t h e future are (at least sometimes) available'. An essential feat u r e in our daffy lives is reflected in this formulation. It is a fund a m e n t a l p a r t of our experience of reality t h a t for instance today, one can choose to travel to Copenhagen tomorrow, and one can also choose to stay at home. Furthermore it is clearly also a p a r t of our experience t h a t one today cannot choose to travel to Copenhagen yesterday. If you were not in Copenhagen yesterday, you have actually lost a n y possibility you m a y have had of going there yesterday - and you shall have to try to bear the loss. On the other hand, in some sense a certain kind of alternative possibilities of the past are in fact available for us today: one can choose to ' m a k e it true yesterday' t h a t one would arrive in Copenhagen within two days - this may be seen as a side-effect of choosing today to go to Copenhagen tomorrow (provided t h a t this choice is also effectuated). However, this kind of influence is certainly felt as somewhat spurious, and it is a t a s k for tense logic (i.e. A-logic) to determine the difference between 'genuine' and 'spurious' events of the past. In our opinion, such considerations build a crucial part of an a r g u m e n t a t i o n in favour of t h e A-theory. It is in fact possible to describe the a s y m m e t r y between past and future in t e r m s of Atheory, and in such a m a n n e r as to leave open the room for g e n u i n e choice (with respect to the future). Moreover, it is possible by reference to the loss of possibilities to define the contents of our perception of the passing of time - a notion which does not fit into the basic framework of the B-theory. We believe t h a t our experience of some freedom of choice - even though it is limited as well as the passing of time are not illusions, but that these p h e n o m e n a are properties of reality; and on t h a t premise we h a v e every reason to claim t h a t A-theory reflects this reality better t h a n does B-theory. -

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THE OBJECTIVITY OF THE A-CONCEPTS It is sometimes argued that A-concepts are not objective, b u t on t h e c o n t r a r y p u r e l y subjective concepts w h i c h are h i g h l y dependent on our consciousness or mind. And it is certainly t r u e t h a t t h e A - c o n c e p t s account well for significant p a r t s of subjective h u m a n experience - they are indeed also m e a n t to do so. Nevertheless, it is very problematic to assert t h a t A-concepts are entirely relative to individual h u m a n minds; for t h e n it becomes v e r y difficult to explain t h e kind of intersubjectivity, which forms t h e basis for the practical a g r e e m e n t concerning t h e Now. T h i s a g r e e m e n t in everyday experience is strong evidence to t h e intersubjectivity of t h e A-concepts, and we e m p h a sise t h a t t h e s e concepts are in any case not 'private'. W i t h i n a group of observers it will not be difficult to establish some agreem e n t r e g a r d i n g t h e A-concepts. Now w h a t a b o u t t h e Btheoretical claim t h a t B-concepts are objective? A-theorists have no quarrel w i t h t h a t contention, for t h e obvious reason t h a t Atheory sees B-concepts as being derived from t h e intersubjective (objective) A-concepts. In an i n t e r e s t i n g way the controversy between A-theory a n d B-theory reflects w h y a n d how t i m e was relegated from logic for centuries. It is t r u e t h a t the B-theory incorporates s o m e notion of t i m e into logic, but essentially it is based on the longcherished idea of tenseless propositions, into which it also strives to reduce t e n s e d propositions. T h e idea of logic as a science concerned w i t h tenseless propositions w a s really t h e m a j o r obstacle to t h e (re)introduction of time into logic. It must, be said, therefore, t h a t B-theory only admits time into logic in t h e m o s t restricted sense possible. To the contrary, A-theory in its fullest consequence actually denies the existence of tense-less facts at all, holding t h a t in principle all facts m u s t be said to be t e n s e formed. T h e A-theorist will m a i n t a i n t h a t our i m m e d i a t e perception is g i v e n in a tensed-formed way. The m e m o r y of a perception is obviously described by m e a n s of t h e past tense, while hope, expectation and choice are described by m e a n s of t h e fut u r e tense. O u r communication, indeed our social lives w i t h each other, are d e p e n d e n t upon the use of modal and tense-logi-

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cal concepts. Therefore t h e t e m p o r a l description of reality is i n s e p a r a b l e from t h e a s s u m p t i o n of t h e possibility of h u m a n experience and communication.

TIME AND EXPERIENCE M a n y B-theorists m a i n t a i n that the B-concepts (in contrast to t h e A-concepts) can be experienced directly. B u t it quickly t u r n s o u t t h a t this position is very difficult to support. W h e n a B-theorist claims to be able to experience directly that for instance 'El is before E2', t h e A-theorists can refer to the fairly obvious fact t h a t in reality this experience has been gained from at least two 'Aexperiences', n a m e l y t h e experiences of the respective events. T h a t is, there is no direct experience corresponding to 'El is before E2'; t h e experience expressed by such a s t a t e m e n t presupposes t h e experience of E1 and E2, respectively. It seems to be clear t h a t in principle all experience m u s t be 'now-experience'. This does not necessarily m e a n that all s t a t e m e n t s of experience m u s t refer to events h a p p e n i n g now, but only t h a t experience is actually being gained now. For instance, we m a y have the 'nowexperience' t h a t E2 is h a p p e n i n g a n d the 'now-experience' t h a t E1 h a s happened. T h e 'nowness' of experience notwithstanding, it is desirable also from t h e point of view of an A-theorist - to reformulate as m u c h as possible of the experience s t a t e m e n t s into earlier-later s t a t e m e n t s like 'El is before E2'. In this w a y it is possible to achieve a formulation of the experience in question such t h a t t h e t r u t h - v a l u e does not vary w i t h time. T h e A-theory in no w a y h a s to forego expressive advantages, w h i c h m a y for some c a s e s be a t t a c h e d to B-theoretical f o r m u l a t i o n s ; A-theory m e r e l y m a k e s it clear w h e r e t h e B - f o r m u l a t e d experience originally comes from - t h a t is, how we should u n d e r s t a n d the s t a t e m e n t 'El is before E2'. It is worth noting t h a t not all 'nowexperience' refers directly to the present. When for instance one sees (the light from) a star, one m a y be well aware of t h e fact t h a t the emission of the light is a past event, even t h o u g h the ex-

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perience of the light is a present one. Strictly speaking, one cannot experience the pastness of the emission of the light, but this is a limitation which would also occur in any B-theoretical epistemology. T h e r e is, in fact, nothing to support the view t h a t it is possible for u s to observe B-relations between events directly. We fmd it very h a r d to see how it can be possible to observe t h a t one event is later t h a n another, without presupposing that this observation is composed of some A-observations - i.e. now-experiences of the respective events. It seems t h a t an immediate observation of Br e l a t i o n s could only be a t t r i b u t e d to God, as indeed it is in T h o m a s Aquinas' philosophy. According to Thomas, all events are p r e s e n t to God at once. Thus God has a timeless knowledge about events and their mutual relations, which m e a n s t h a t God k n o w s and u n d e r s t a n d s B-relations directly. One might perhaps say t h a t the B-theory belongs in a description of reality-asit-is-to-God. B-theory ignores specifically h u m a n limitations and conditions. When, however, the aim is to discuss temporal relations in reality-as-it-is-to-us, we have no doubt t h a t the At h e o r y is the obvious solution. Moreover, to the extent t h a t our experience of reality is actually true and not an illusion, it seems t h a t A-concepts are indispensable. In fact, as pointed out by N. Lawrence [1978, p. 24] it is very hard for us to imagine any kind of temporal notion not related to h u m a n experience.

A- A N D B-CONCEPTS IN PHYSICS

We have ourselves come out quite openly as supporters of the A-theory. Where h u m a n experience, communication and language are involved, B-theory can be criticised on m a n y counts. But A-theory, as we have emphasised, is not m e a n t to be concerned with these matters as opposed to a mind-independent physical reality. To some degree, it is actually a defence of t h e r e a l i t y of some basic h u m a n experiences. Therefore, evidence from h a r d science such as Physics must be t a k e n into account -

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as a m i n i m u m , it m u s t be shown t h a t A-theory is not in conflict w i t h empirical science. N o w m a n y B-theorists in fact m a i n t a i n t h a t evidence especially from Physics directly supports t h e i r theory, as opposed to t h e A-theory. This is supposed to be of p a r t i c u l a r importance, since it is a s s u m e d that physics has a very special r61e to play in t h e study of time. This view has been very clearly stated by H a n s Reichenbach: There is no other way to solve t h e problem of time t h a n t h e w a y t h r o u g h physics. More t h a n a n y other science, physics h a s been concerned w i t h t h e n a t u r e of time. If t i m e is objective, t h e physicist m u s t h a v e discovered t h a t fact, if there is Becoming the physicist m u s t know it; b u t if t i m e is merely subjective and Being is timeless, the physicist m u s t h a v e been able to ignore time in his construction of reality a n d d e s c r i b e t h e world w i t h o u t t h e h e l p of t i m e . Parmenides' claim that time is an illusion, Kant's claim t h a t t i m e is subjective, and Bergson's a n d Heraclitus' claim t h a t flux is everything, are all insufficiently g r o u n d e d theories. [Reichenbach, 1956, p. 16] One a r g u m e n t fielded by B-theorists is the contention t h a t w i t h i n physics t i m e is treated as a p a r a m e t e r associated w i t h a r e l a t i o n , w h i c h is to a very large e x t e n t s i m i l a r to t h e before/after-relation. This is true, b u t t h e r e is a fairly simple r e p l y to it: t h e t i m e p a r a m e t e r w i t h i n physics m u s t be u n d e r s t o o d as a theoretical c o n s t r u c t , w h i c h c o n c e p t u a l l y should really be traced back to the A-concepts. Another a r g u m e n t is that B-concepts fit nicely with the special t h e o r y of relativity, specifically, w i t h t h e relativity of simultaneity. According to Minkowski's rendition of relativity, t i m e is u n d e r s t o o d in t e r m s of geometry. T h a t is, t i m e exists in t h e s a m e sense as space, in an a t e m p o r a l way. This c a n be illustrated by statements like the following by H. Weyl: T h e objective world simply is, it does not happen.. Only to t h e gaze of m y consciousness, crawling u p w a r d along t h e

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life line of m y body, does a section of this world come to life as a fleeting image in space which continuously changes in time. [Weyl, 1949, p. 116] According to this interpretation, it s e e m s t h a t relativity is in conflict w i t h f u n d a m e n t a l A-theoretical concepts about becoming a n d h a p p e n i n g , i.e. 'the passage of time'. However, as we have seen in c h a p t e r 2.7, the tense-logical position does not contradict a n y p a r t of t h e empirical basis of t h e theory. Not all questions r e g a r d i n g relativistic tense logic h a v e been satisfactorily answered, b u t t h e A-theory can by no m e a n s be rejected out of h a n d by appealing to the special t h e o r y of relativity. Indeed, w i t h i n t h e n a t u r a l sciences t h e r e is ample proof of ass u m p t i o n s and ideas which are more in h a r m o n y w i t h t h e At h e o r y t h a n w i t h t h e B-theory. The theories of prediction a n d beforehand calculations illustrate this fact. H a n s Reichenbach h a s argued t h a t The concept of 'becoming' acquires significance in physics: the present, w h i c h separates the f u t u r e from the past, is t h e m o m e n t at w h i c h t h a t which was u n d e t e r m i n e d becomes d e t e r m i n e d , a n d 'becoming' h a s t h e s a m e m e a n i n g as 'becoming determined'. [Grfinbaum 1973, p. 320] In this way Reichenbach has pointed out t h a t there is a crucial difference b e t w e e n p a s t and future, which t h e physicist h a s to take into serious consideration. The difference is t h a t there are f u t u r e facts w h i c h cannot possibly be predicted, w h e r e a s in principle any past fact can be recorded. This m a k e s it possible to establish an epistemological basis for a tense-logical approach w i t h i n t h e physical sciences. In particular, this seems to be i m p o r t a n t with respect to ' q u a n t u m logic', w h i c h is the conceptual foundation of q u a n t u m physics. C. F. von Weizs~icker has stated this in the following way: The most general presupposition of experience is time. Its structure, as expressed by the words present, past and fut u r e is analysed in a logic of t e m p o r a l propositions (tense-

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MCTAGGART'S PARADOX It w a s McTaggart who explicitly identified t h e dichotomy b e t w e e n two major conceptions of time and labelled t h e m 'A' a n d 'B', respectively. M c T a g g a r t h i m s e l f a r r i v e d at t h e conclusion t h a t A-concepts are more f u n d a m e n t a l t h a n Bconcepts. He did not, however, use this analysis as an argument in f a v o u r of A-theory. On the contrary, he u s e d it for a r e f u t a t i o n of the reality of time! McTaggart a r g u e d t h a t Aconcepts give rise to a contradiction - which has become known as 'McTaggart's Paradox'. Due to this p u t a t i v e contradiction within the fundamental conceptualisation of time, he w e n t on to claim that time is not real. The core of McTaggart's a r g u m e n t is t h a t the notions of 'past', 'present' and 'future' are predicates applicable to events. The t h r e e predicates are supposed to be mutually exclusive - any concrete event happens j u s t once (even though a type of event m a y be repeated). On the other hand, any of the three predicates can be applied to any event. In a book on history, it makes sense to speak of 'the death of Queen Anne' as a p a s t event - call it el - but in a document written in the lifetime of Queen Anne, it could well make sense to s p e a k about her d e a t h as a future event. Apparently this gives rise to an inconsistency, since how can el be both past and future - and present as well, by a similar argument? The answer m u s t be t h a t there is a n o t h e r event e2, relative to which for instance el has been p r e s e n t and future, a n d is going to be past. Now, the s a m e kind of a p p a r e n t inconsistency can be e s t a b l i s h e d w i t h respect to e2, and the problem can only be solved by introducing a n e w event e3, for w h i c h a new a p p a r e n t inconsistency will arise etc. - which seems to mean t h a t we have to go ad infinitum in order to solve the inconsistency. The consequence appears to be t h a t the inconsistency can never be solved.

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Prior, however, pointed out a basic flaw in McTaggarts argument: the contradictions arise from an a t t e m p t at forcing t h e Aseries notions into a B-series f r a m e w o r k [1967, p. 6]. Events m a y be described in t e r m s of Prior's instant-propositions, of which it also holds that they 'happen', i.e. are true, exactly once. The condition t h a t the t h r e e predicates are mutually exclusive can be formulated as: a ~ (~Pa A ~Fa) P a ~ (~a A - F a ) F a ~ ( - a A ~Pa)

The fact t h a t any event can be past, present, and future, can be expressed in the following way, where the I-operator stands for 'the present': I a ~ (PFa A FPa) P a ~ (PIa A PFa) F a ~ (FPa A FIa).

But no contradiction follows from these 6 theses. It is t h u s r e v e a l e d t h a t McTaggart's p a r a d o x is in no w a y a cogent a r g u m e n t against the A-series notions, let alone the reality of time.

THE TWO PARADIGMS Both the A-theory and the B-theory are internally consistent theories. They can both profitably be used for describing a range of t e m p o r a l phenomena, and indeed, from a formal point of view each of the theories can be 'absorbed' within the other, under certain premises. So why would philosophers (and we, too) present t h e m as competing paradigms? What is at stake here is a question of two different ways of u n d e r s t a n d i n g reality, and consequently also two different languages for description. One might refer to Henri Bergson who discussed the use of space

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metaphors for the description and analysis of temporal phenomena. Such a language is clearly B-like. According to Bergson this language is obviously unsatisfactory. He denied t h a t time can be adequately represented by space. In this way we can only deal with "time flown" and not with "time flowing". [Bergson 1950, p. 221]. However, Bergson did not say very m u c h about how to express more precisely "time flowing", that is, he did not suggest any A-like language. The two theories oppose each other as two general frameworks, as two different answers to the question of how we should fully u n d e r s t a n d the temporal relations in the world. Hence, we fred it n a t u r a l to compare them w i t h K u h n i a n paradigms. The advocates of the A-paradigm and the Bparadigm, respectively, form two scientific communities with a discussion going on between them. Neither group can present logically cogent arguments. Of course, considerations of logical properties such as expressive power, elegance etc. are not i r r e l e v a n t - a n d t h e s a m e t h i n g obviously goes for ' c i r c u m s t a n t i a l evidence' from e m p i r i c a l sciences. But essentially, the a r g u m e n t a t i o n offered is of a metaphysical nature. From our own A-theoretical position, we think t h a t the following two issues should be weighted highly in the debate between the paradigms: firstly, it seems that the processes of perception, observation, and cognition can only be described satisfactorily by m e a n s of the A-concepts; secondly, the temporal a s y m m e t r y between past and future, and the passage of time can only be described satisfactorily by means of the A-theory. Both of these points, together w i t h the importance of communication, also suggest that A-concepts are closely related to natural language.

3.2. INDETERMINISTIC TENSE LOGIC It is sometimes argued t h a t classical physics establishes a convincing case for determinism, and against h u m a n freedom of choice. In its simplest - a n d original - form, this a r g u m e n t is based on the assumption t h a t all the individual e l e m e n t s of the h u m a n b r a i n and body i n t e r a c t according to N e w t o n i a n mechanics. This was clearly the assumption in t h e famous a r g u m e n t for t h e idea of L'Homme-Machine, 'The M a n Machine', by La Mettrie (1709-51). According to J o h n Cohen, La Mettrie "seems to have been the first to state t h e problem of the m i n d in terms of physics" [Cohen 1966, p. 70]. In fact, there was also a clearly temporal aspect to La Mettrie's idea: La Mettrie was no doubt encouraged to m a k e his g r a n d extrapolation by the ingenious successes of c o n t e m p o r a r y horologists. [Cohen 1966, p. 70] T h a t very same aspect of temporality has also been a basis for criticising La Mettrie's 'Horloge Model', since t h i s model exhibits a "paradoxical timelessness, t h a t is, its insensitivity to duration, which is so vital a feature to h u m a n experience" [1966, p. 70]. Such a criticism could in fact be c a r r i e d out w i t h reference to the philosophy of Henri Bergson who p r e s e n t e d a v e r y interesting analysis of the problem of h u m a n freedom in relation to the concept of time: Freedom is ... a fact, a n d among the facts which we observe t h e r e is none clearer. All the difficulties of t h e problem ... arise from the desire to endow duration w i t h t h e same a t t r i b u t e s as extensity, to i n t e r p r e t a succession by a s i m u l t a n e i t y , and to express the idea of f r e e d o m in a language into which it is obviously untranslatable. [Bergson 1950, p. 221] According to Bergson t h e idea of h u m a n freedom is in fact indefinable. He argued t h a t "we can analyse a t h i n g but not a process; we can break up extensity, but not duration" [Bergson 257

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1950, p. 219]. This seems to be completely correct if all we h a v e got are the conceptualisations w h i c h constitute a B-series conception of time. In the following we shall, however, a s s u m e t h a t a n A-series conception of time is possible. We s h a l l concentrate on an a r g u m e n t concerned with the r61e of h u m a n communication w h e n confronted w i t h the idea of 'the m a n machine'. According to Newtonian mechanics all past and future s t a t e s of a closed system are implicit in the present state. If m a n is such a system, t h e n all future decisions of a h u m a n being can in principle be predicted by an observer, who is fully informed of every r e l e v a n t aspect of the present state. Therefore, t h e r e seems to be no freedom of choice, and there seems to be no w a y to argue t h a t the h u m a n feeling of freedom is more than a mental illusion. The basic idea in the above argument is that the state of a n y individual is in principle a conjunction of statements q l A q2 A .. . A q N

all being true 'now' and therefore also now-unpreventable (i.e. necessary). Moreover, for any possible act a, which the person in question m a y perform in the future, the laws of classical physics make it necessary that either the person will or will not perform t h a t act (for any given a m o u n t of time). Let A be t h e statement 'this person performs d , so that F ( 1 ) A stands for 'this person is going to perform a tomorrow' (taking 'tomorrow as our example). It t h e n follows from the assumptions discussed so far t h a t either (1) or (2) must be true: (1) N ( ( q l A q 2 A . . . A q N ) ~ F ( 1 ) A ) (2) N ( ( q l A q2 A ... A qN) ~ F ( 1 ) - A ) This means t h a t it is already now given whether the person is going to perform a or not. Hence, the person has no freedom of choice, and it cannot consistently be maintained that it is possible that he will perform ~, but also possible that he will not perform a. That is, the following formula is ruled out:

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(3) M F ( 1 ) A A M F ( 1 ) - A It is well k n o w n today t h a t the premises of this a r g u m e n t do n o t hold in general. After all classical m e c h a n i c s has b e e n replaced by q u a n t u m mechanics, for which t h e deterministic ideal does not hold at t h e microscopic level. N e v e r t h e l e s s , v a r i o u s ' m o d e r n i s e d ' v e r s i o n s of t h e a r g u m e n t can be constructed, or it m i g h t simply be m a i n t a i n e d t h a t N e w t o n i a n mechanics still holds for the relevant parts of the brain and body system. However, D o n a l d M. MacKay [1971,1973,1974] h a s shown t h a t even if e v e r y t h i n g were mechanistic, t h e classical a r g u m e n t h a s to be r e j e c t e d , w h e n t h e p o s s i b i l i t y of c o m m u n i c a t i o n is t a k e n into consideration. We shall p r e s e n t t h e m a i n line of MacKay's argument: Let AG be an agent and let P be a predictor. A s s u m e that P is fully informed of AG's state at the time tl. M a k i n g use of this knowledge, P predicts w h a t AG will do at some later time t2. T h a t is, P states t h e prediction F ( 1 ) A (where tl corresponds to t h e p r e s e n t i n s t a n t , a n d t2 is an i n s t a n t one t i m e unit later). Now, is F ( 1 ) A t r u e at tl? MacKay h a d his own definition of t r u t h , which deserves to be mentioned here. He was by t r a i n i n g a physicist, and his analysis of the epistemology of prediction led h i m to the following notion of truth: I do, as a m a t t e r of fact, prefer to reserve t h e word 'true' for propositions t h a t anyone and everyone would be correct to believe and in error to disbelieve. [1974, p. 108] In this sense t h e prediction F ( 1 ) A is not t r u e at tl. For if t h e prediction is c o m m u n i c a t e d to AG at tl, the state of AG's m i n d (or brain and body system) will be changed, and t h e prediction will not be valid in general. The very premises of t h e prediction are changed, if it is communicated to AG. This m e a n s that t h e logical s t r u c t u r e of t h e necessity based on the classical laws is not (1), but r a t h e r (4) N ( ( ql A q2 A ... A qg) ~ ( - C F ( 1 ) A ~ F ( 1 ) A ) ) ,

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where C is an operator m e a n i n g 'it is communicated to AG t h a t ...'. So P might predict AG's future decision, but since h e has to a s s u m e t h a t the prediction is not c o m m u n i c a t e d to AG ~CF(1)A - there does not exist an unconditional prediction of AG's f u t u r e actions. Thus, even on the (really v e r y strong) assumption t h a t h u m a n behaviour is completely d e t e r m i n e d by classical mechanical laws, P can deduce no more t h a n (5) N(~CF(1)A ~ F(1)A). Along the same lines J.W.N. Watkins [1971] has pointed out t h a t t h e r e is a ceteris p a r i b u s clause involved in predictions of future decisions of cognitive agents. According to Watkins, predictions can be valid only on the assumption that t h e y are not communicated to the agents in question, exactly as stated in (4). In this way, even silent predictions concerning cognitive agents are conditional. This m e e t s an obvious - a l t h o u g h r a t h e r superficial - rejoinder to MacKay's argument, namely t h a t ACT's actions are still deterministic, if P simply chooses not to communicate his prediction to AG. Watkins makes it clear t h a t even t h o u g h one may choose to keep silent, one's predictions are still conditioned by exactly t h a t choice! Therefore, it m u s t be concluded t h a t the possibility of silent prediction of AG's future decision does not imply t h a t AG is unfree, since there is no w a y to d e m o n s t r a t e t h a t a predicted decision is necessary. T h e r e is no fully d e t e r m i n a t e specification of AG's future decision, which he would be correct to accept as inevitable, and would be unable to falsify, if it was communicated to him. For these reasons it is not possible to m a i n t a i n N F ( 1 ) A v N F ( 1 ) - A even w i t h i n the classical framework. Therefore, freedom of choice as expressed in (3) can still be consistently upheld. M a c K a y has claimed t h a t even if an agent's brain and body system was as mechanical as a clockwork, it cannot be proved that such a system is unfree, as long as it can receive information a n d react correspondingly. This seems to raise a question about t h e scope of MacKay's argument. Thus L a n d s b e r g and Evans [1970] have argued t h a t even a computer could be free according to the argument. That is in a sense correct. MacKay's

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a r g u m e n t certainly does not refute or rule out t h a t computers can be free. On the other hand, MacKay has neither argued nor proved s u c h a n idea. His a r g u m e n t does not pretend to prove that h u m a n beings (or possibly comparable systems) do have a free will, b u t r a t h e r it is concerned with consistency. He has shown t h a t the existence of a s t a t e m e n t A for which (3) holds is consistent w i t h f u n d a m e n t a l deterministic assumptions - by showing h o w (1-2) must be m a d e conditional. The q u e s t i o n about t h e reality of freedom of choice has to be discussed w i t h reference to o t h e r lines of reasoning, for instance metaphysical or theological. MacKay himself maintained that it is not computers, nor brains, but conscious agents who m a y or m a y not be free. According to him, the crucial property of free agents is their capability of believing - correctly or incorrectly - w h a t t h e y are told [1974, p. 111]. MacKay's definition of t r u t h as well as the observations so far imply t h a t a 'silent' prediction like F(1)A can not be classified as true, even i f A in fact turns out to be true after one time unit. It is i n t e r e s t i n g t h a t MacKay's notion of t r u t h for f u t u r e - t e n s e propositions is very similar to Peirce's position with respect to the contingent future - a position also adopted by Prior. MacKay did not develop a n y proper logical system corresponding to his ideas. But as we shall see in the following Prior certainly did t h a t for a very similar set of ideas. Prior like MacKay had a strong c o m m i t m e n t to w h a t h e called 'a belief in real freedom'. In his opinion, one of t h e most important differences between the past and the future is t h a t ...once something has become past, it is, as it were, out of our r e a c h - once a thing has happened, nothing we can do can m a k e it not to have happened. But the future is to some extent, even though it is only a very small extent, something we can m a k e for ourselves... [SFTT, p. 2]. Prior w a n t e d to develop an indeterministic tense-logic. As he developed Kripke's ideas from 1958 (cf. c h a p t e r 2.5), P r i o r related his belief in real freedom to the concept of b r a n c h i n g time:

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Genuine determinism would be the belief t h a t there is only one possible future, and to express this you really do need to go beyond Kt and add a postulate for nonbranching of the future. [Prior 1969, p. 329] The postulate he has in mind is this one:

(6) PFq ~ (q v Pq v Fq) i.e. "Whatever has been 'on the cards' either is the case or has been the case or is 'on the cards' still" [Prior 1969, p. 329]. As J o h n P. Burgess [1978, p. 157] later explained, Prior would agree t h a t the determinist sees time as a line, and the indeterminist sees it as a system of 'forking paths'. As we have seen he found it highly important to examine the notion of branching in detail and formally elaborated two models for it, n a m e l y those by Ockham and Peirce, respectively [Prior 1968, p. 122 ft.]. Prior himself adopted the so-called Peirce-solution to the problem of the contingent future, according to which the following holds: ... from the fact t h a t there is a sea-battle going on it does not follow that there was going to be one, though it does follow that there will have been one. [Prior 1957a, p. 95] Let q be the statement 'there is a sea-battle going on'. Then it is a thesis that (7)

q~F~P~q

The validity of this thesis is illustrated by the diagram below: if q is t r u e today, then P(n)q will be true in any possible situation a f t e r n days. Therefore F(n)P(n)q m u s t be t r u e today. However, the mirror image of (7):

(8)

q ~ P(n)F(n)q

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does not hold, ifF(n)q is understood in the strong way, i.e. as "it is bound to be the case after n time units t h a t q" [Prior 1969, p. 329]. P(n)q

q F(n)P(n)q

P(n)q

n units

,r

The counter-argument against viewing (8) as a thesis is t h a t it is possible to imagine t h a t q is true at some time, but that F(n)q has not been true n days ago, for which reason P(n)F(n)q is not t r u e now. The following d i a g r a m d e s c r i b e s for i n s t a n c e situations, where the t r u t h of q is brought about by, say, pure coincidence, or by some act of free choice: q

~P(n)F(n)q

~F(n)q

-q

n units

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Similarly, Prior m a i n t a i n e d that a l t h o u g h (9)

q ~ GPq

is a thesis, its mirror image (10) q ~ H F q is n o t valid [SFTT, p. 2], i f F is u n d e r s t o o d in the s t r o n g way. (Prior's considerations here are based on the Peircean notions discussed in chapter 2.8. However, it should be m e n t i o n e d t h a t (10) is t r u e in other b r a n c h i n g t i m e s y s t e m s w i t h d i f f e r e n t definitions, notably Kb and the Ockhamistic theory.) In Prior's opinion, since the t r u t h of future contingents cannot be k n o w n now, t h e r e cannot be a n y t r u e s t a t e m e n t s a b o u t f u t u r e contingents. On this view, t h e s t a t e m e n t 'there will be a sea-battle tomorrow' cannot be t r u e today, since t h e r e is no u n i q u e f u t u r e but r a t h e r a n u m b e r of different possible f u t u r e s (unless, of course, that sentence h a p p e n s to be deterministically entailed by facts of today). T h e basic question concerns the i n t e r p r e t a t i o n of expressions r e g a r d i n g t h e future: Can it be m a i n t a i n e d with conceptual a n d logical consistency of some event E t h a t 'E will happen', this being t a k e n as different from 'E could happen', a n d 'E will necessarily happen'? In chapter 2.8, we pointed out t h a t t h e O c k h a m - s y s t e m m a k e s a g e n u i n e distinction b e t w e e n t h r e e types of s t a t e m e n t , repeated here for ease of reference: (i) Necessarily, Mr. Smith will commit suicide. (ii) Possibly, Mr. Smith will c o m m i t suicide. (iii) Mr. S m i t h will commit suicide. Of course, this means that Ockham would answer the question positively. However, Prior like Peirce took the stance t h a t 'E will h a p p e n ' c a n n o t m a k e sense, u n l e s s it is i n t e r p r e t e d as e q u i v a l e n t to one of t h e two o t h e r types of s t a t e m e n t . T h e

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difference to the OckbAmistic (as well as t h e general medieval) solution clearly h a s to do with the definition of t r u t h . Prior suggested the following condition of t r u t h w i t h respect to future statements: ... n o t h i n g can be said to be t r u l y ' g o i n g - t o - h a p p e n ' (futurum) until it is so 'present in its causes' as to be beyond stopping; until t h a t happens neither 'It will be t h e case t h a t p' nor 'It will not be the case that p' is strictly speaking true. [1968, p. 38] In other words we have the following principle (P): (P) The proposition F(n)p is true now if and only if t h e r e exist now facts which m a k e it true (i.e., will m a k e p true in due course). This definition is very similar to the one MacKay suggested, a n d it is also quite essential in Peirce's philosophy. W h a t m a y a p p e a r more s u r p r i s i n g is Prior's conviction t h a t St. T h o m a s A q u i n a s also accepted these ideas. To be true, Prior a r g u e d a g a i n s t Thomas' view t h a t God's k n o w l e d g e is in some way b e y o n d time, but o t h e r w i s e he c o n s e n t e d to m o s t of w h a t T h o m a s h a d said about tense-logical reasoning. According to Prior's i n t e r p r e t a t i o n of Thomas' philosophy, T h o m a s would even agree on the rejection of (10). (P) implies that the proposition F(n)p can only be true, if it is in principle possible to verify it from facts k n o w n at t h e t i m e of u t t e r a n c e . F u t u r e t e n s e propositions w h i c h c a n n o t be verified a r e false or not well-formed. William of O c k h a m , of course, could not have accepted an idea of t r u t h corresponding to (P). O c k h a m held that it is possible for God to know something, even t h o u g h t h a t which He knows can in no w a y - short of divine revelation - be verified by us. With special regard to propositions about t h e future, O c k h a m claimed t h a t God h a s a complete knowledge: "It m u s t be held beyond q u e s t i o n t h a t God knows w i t h c e r t a i n t y all f u t u r e contingents, i.e., He k n o w s w i t h c e r t a i n t y which p a r t of t h e contradiction is t r u e and which is

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false" [1969, p. 48]. (Here, Ockham obviously had in mind the e m b e d d i n g of contingent sentences into contradictions of the form F(n)p v F(n) ~p.) On this analysis, a future contingent can be true now. The Ockhamist definition of t r u t h with respect to future statements can be formulated in the following way: (O) The proposition F(n)p is true now if and only if God knows t h a t p will be in n days. In Ockham's logic, any proposition is known by God if and only if it is true. Therefore, the above definition can also be expressed in the following way: (O') The proposition t r u e in n days.

F(n)p is true now if and only ifp will be

We h a v e a l r e a d y i n d i c a t e d how t h e formal d e t a i l s c o r r e s p o n d i n g to this definition can be w o r k e d out in an Ockhamistic structure. If (O') is accepted, it m u s t be admitted that we are not in general able to establish whether a proposition is true or false, in spite of the fact t h a t it is assumed to have a definite truth-value at any time - even if it is contingent. We cannot be sure that already now, there exist facts t h a t m a k e F(n)p true. Hence the t r u t h value of F(n)p might be unknown or even unknowable - to us. Thus future contingents can be true now, a l t h o u g h we cannot know w i t h certainty t h a t t h e y are true. In t h e Ockhamistic framework F(n)p is either t r u e or false. Let us a s s u m e that F(n)p is in fact true. Even so, if F(n)p is about the contingent future a rational person does not have to accept its truth; he could deny it w i t h o u t compromising his rationality. On the other hand, while a person would certainly in the a s s u m e d case be correct in believing t h a t F(n)p is true, he would be mistaken in regarding F(n)p as false. Ockham's theory of the future is realistic, since the t r u t h value of F(n)p is well-defined, even if F(n)p is about the contingent future. F u r t h e r m o r e , there exist t r u e propositions about all

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future events and states-of-affairs. In our opinion (0') is a more n a t u r a l definition of t r u t h for future statements t h a n is (P). In fact, we t h i n k t h a t few would be r e a d y to accept t h e full consequences of (P). If (P) is accepted, it follows t h a t any proposition about future contingents is false. As m e n t i o n e d earlier, this also means t h a t 'plain' future s t a t e m e n t s such as 'Mr. S m i t h will commit suicide' m u s t be r e g a r d e d as (a) i n h e r e n t l y false, or (b) ill-formed (or elliptical, o m i t t i n g a required modal expression). Relating this to everyday life, for instance guesses will in general be false: if we are playing the pools and we believe t h a t we are going to win, this belief will according to (P) be false, even if it t u r n s out t h a t we do indeed win. On this basis, it will be almost impossible to state the difference b e t w e e n a true and a false prophet! In order to state such a difference, we need a truth-definition like (O'). It should also be noted that 'excluded middle', (11)

F(n)p v F(n)~p

is not a thesis according to the Peirce-system. His theory makes it conceivable t h a t the proposition 'In n days it will be the case t h a t p', and the proposition 'In n days it will be the case t h a t not p', are both false. The understanding of the concept of the future within the Peirce theory is realistic to the extent t h a t it regards t h e t r u t h v a l u e of the proposition F(n)p as a m e a n i n g f u l concept. If F(n)p is interpreted as either NF(n)p or MF(n)p, it is e i t h e r t r u e or false; but otherwise, it is inherently false. The latter observation makes it clear t h a t the theory is not realistic in the sense t h a t we can form t r u e propositions about future contingents. In general, bare F(n)p and F(n)-p are both false according to the Peirce theory. We t h i n k we have exemplified by now t h a t this theory is not congenial w i t h logical and linguistic intuitions evident in e v e r y d a y communication. Following this investigation, one m i g h t conclude t h a t the Ockham t h e o r y is a fairly accurate r e p r e s e n t a t i o n of our intuitions concerning valid t e m p o r a l reasoning (with regard to the problems with which it deals; m a n y o t h e r aspects of t e m p o r a l r e a s o n i n g , for i n s t a n c e

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durations, have not been dealt with here). In our opinion, Prior's Ockhamist theory is indeed satisfactory for most cases. But as d e m o n s t r a t e d by H i r o k a z u N i s h i m u r a [1979], there are some rare examples in which t h e Ockham theory is not sufficient. We shall state such an example. Let (TIME,<,C,=,Ock) be an Ockhamistic structure, w h e r e C is the set of all m a x i m a l l y ordered (i.e. linear) subsets in (TIME,<). Let us for t h e sake of simplicity assume that T I M E is discrete and each chronicle isomorphic to the set of integers. We may think of the elements in T I M E as possible days. Consider these two statements: AI: 'Inevitably, if today there is life on earth, then either this is the last day (of life on earth), or the last day will come.' Semantically: Ock(t,c, q ~ (G~q v F ( H q A q A G~q))) for a n y t, c (q standing for the statement 'there is life on earth'). -A1 implies t h a t life cannot go on forever, so t h a t on a n y possible day, it is t r u e either that life has become extinct, or that this is the last day of life on earth, or t h a t life will later become extinct. A2: 'At any possible day on which there is life on earth, it is possible that there will be life on earth the following day.' Semantically: Ock(t,c, q ~ MF(1,q)) for any t, c. -A2 implies t h a t t h e r e is hope (and possibilities) for tomorrow as long as there is life!

We assert that a person might hold both AI and A2 w i t h o u t contradicting himself. But we shall show t h a t the assumption t h a t A1 and A2 can b o t h be t r u e is in conflict w i t h t h e Ockhamistic theory. - Assume t h a t A1 and A2 a r e both t r u e and let to be some possible day in T I M E on some chronicle cl. Because of A1, this m e a n s t h a t there is another possible day tl for which Ock(tl, cI, Hq A q A G-q). Because of A2, it follows that Ock(tl, cl, q ~ MF(1,q)).

In consequence, Ock(t l, cl, MF(1,q)).

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This m e a n s t h a t there is a possible day t'2 immediately after tl, on some chronicle c2, w h e r e (t'2, c2) ~ (tl, cp and Ock(t'2,c2,q). B e c a u s e of A1 t h e r e is a possible d a y t2, for w h i c h Ock(t2,c2,q A G-q) and Ock(t, c2,q) for all t b e t w e e n t'2 and t2. Now because of A2 it follows t h a t there is a possible day t'3, i m m e d i a t e l y after t2, on some chronicle C3 where (t'3, c3) ~ (t2, c2) a n d Ock(t'3,cz, q). Because of A1 there is a possible day t3, for which Ock(t3,c3,q A G-q) and Ock(t, c3,q) for all t between t'3 and t3. This procedure can be carried out ad infinitum. - Obviously, t h e series of possible days, c: t1< t2< t3< ... following t h e part of cl before tl, defines a maximally ordered subset in (TIME,<), i.e. a chronicle. It is easy to see t h a t Ock(t,c,q) for all t. But this is clearly a violation of At. Q.E.D. The construction procedure can be illustrated by t h e following figure:

t2 ~

3

c

v

cl

If, on the other hand, we a s s u m e another tense logic different from the Ockhamistic system, in which the construction of t h e chronicle c from the series of cl, c2, ... is forbidden, t h e n t h e c o n j u n c t i o n of A1 and A2 m i g h t be accepted w i t h o u t a n y inconsistency. In such a tense logic the set of possible chronicles cannot be closed under t h e k i n d of construction j u s t mentioned. We m u s t a s s u m e t h a t n o t all linear subsets in (TIME,<) are possible chronicles. An O c h a m i s t i c s y s t e m r e v i s e d in t h i s m a n n e r has an interesting affinity to Leibniz' philosophy. For this reason shall call s u c h a modified Ockhamistic system the Leibniz system. We shall deal with the formalities of this system in the next chapter.

3.3. LEIBNIZIAN TENSE LOGIC In this c h a p t e r we shall present an indeterministic tense logic similar to Prior's Ockhamistic theory, but modified in t h e light of t h e observations made by Hirokazu N i s h i m u r a (mentioned in the p r e c e d i n g chapter). Since the s y s t e m is based on a k i n d of temporal r e a s o n i n g very similar to Leibniz' philosophy, we shall call the s y s t e m 'Leibnizian tempo-modal logic', LT for short. The definition of well-formed formulae w i t h i n LT is: (1) (2)

(3)

Propositional variables are wff. If a and fl are wff and x is a positive n u m b e r , t h e n - a , a_.~fl, aAfl, a v fl, Vx:a, ~x: a, N a , Pa, and Fve are all wff. There are no other wff.

The axioms of LT are: (A1)

A, where A is a tautology of.the propositional calculus

(A2) (A3) (A4) (A5) (A6) (A7) (AS) (A9) (A10) (All) (A12) (A13)

G(A ~ B) ~ (GA ~ GB) H ( A ~ B) ~ (HA ~ H B ) A ~ HFA A ~GPA FFA ~ FA F P A ~ ( P A v A v FA) P F A ~ ( P A v A v FA) GA ~ FA HA ~ PA FA ~ FFA NGA ~ GNA P A ~ N P A , where A contains no occurrences of F

(N1) (N2) (N3) (N4)

N ( A ~ B) ~ ( N A ~ N B ) NA DA NA ~ NNA M N A ~ N A , where M -~def - N -

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(N1)-(N4) a r e t h e $5 axioms for N. The rules of inference a r e t h e s a m e as for t h e Ockhamistic system, i.e.

(RMP) (RG) (RH) (RN)

If ~ A and ~ A ~ B, then I- B.

(nl)

I f ~ ~(x)~fl t h e n I- Vx:~(x)~fl. If/" a ~ x ) t h e n I- a ~ V x : ~ x ) , for x not free in a.

(l-I2)

If ~ A, then I- G A . If I- A, t h e n / - HA. If I- A t h e n I- N A .

w h e r e '1-p' m e a n s 'p is provable'. T h e e a r l i e r - l a t e r logic (B-logic) we h a v e in m i n d can b e p r e s e n t e d b y t h e following definitions. Definition: A L e i b n i z i a n s t r u c t u r e is a q u a d r u p l e ( T I M E , <, _ T ) , w h e r e TIME is a n o n - e m p t y set w i t h t w o

relations < a n d = such t h a t (B1) (B2) (B3) (B4) (BS) (B6) (B7) (B8) (B9) (B10)

(tl < t2 A t2 < t3) ~ tl < t3 (tl < t2 A t3 < t2) ~ (tl < t3 V tl = t3 V t3 < tl) (t2 < tl A t2 < t3) ~ (tl < t3 v t l = t3 v t 3 < t P Vtl~t2 : tl < t2 V Q ~ 2 : t2 < tl Vt I Vt2Jt3 : tl < t2 ~ (tI < t3 • t3 < t2) t=t tl = t 2 ~ t 2 = t l (tl = t 2 A t 2 --t3) ~ t l --t3 (t~ = t e a t3 < re) ~ 3 t 4 : ( t 3 --t4 A t4 < tz)

and an o p e r a t o r T such that (T1) (T2) (T3) (T4) (T5) (T6)

T ( t , A A B ) =-(T(t,A) A T ( t , B ) ) T(t,-A) ~--T(t,A) Vx:T(t,A) =--T(t, Vx:A) w h e r e x is foreign to t T(t, F A ) -= 3Q: (t < tl A T ( t I , A ) ) T(t, P A ) - 3t1: (tl < t A T ( t l , A ) ) T(t, N A ) -~ Vtl: (tl -- t ~ T ( t I , A ) )

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Y'tl Vt2: (tl = t2 A T(tl, PA)) ~ T(t2,PA), where A contains no occurrences of F.

You m a y read T(t~A) as 'A is true at t', but bear in m i n d t h a t we a r e n o t i n t r o d u c i n g a s e p a r a t e s e m a n t i c s following u s u a l m o d e l - t h e o r e t i c p r o c e d u r e s . R a t h e r , we are e n l a r g i n g t h e logical l a n g u a g e such t h e e x t e n d e d s y s t e m ' c o n t a i n s t h e s e m a n t i c s of the original system', following Prior's ideas in this respect. Definition: A s t a t e m e n t is Leibniz-valid if and only if T(t,A) for every Leibnizian s t r u c t u r e (TIME, <, , T) a n d every t in TIME. It is easy to see t h a t t h e axioms (A1) - (A13) a n d (N1) - (N4) a r e all Leibniz-valid. S i n c e t h e inference r u l e s c a r r y t h e Leibniz-validity over, it follows that the following t h e o r e m holds: Theorem 1: IfA is provable in LT, t h e n A is Leibniz-valid. T h i s t h e o r e m expresses t h e s o u n d n e s s of LT. U s i n g Prior's i d e a of i n s t a n t propositions we shall also argue t h a t LT is complete relative to Leibniz-validity. First we shall enlarge t h e logical l a n g u a g e in the m a n n e r already shown in c h a p t e r 2.9. T h e e n l a r g e d system m u s t include i n s t a n t p r o p o s i t i o n s (i.e. m a x i m a l consistent sets from LT) and a modal o p e r a t o r L, for w h i c h we a s s u m e (BF) (I1) (I2) (I3) (L1) (L2) (L3) (LG) (LH) (LN)

L(Va: ¢(a)) ----Va: L(~(a)) :~a: a ~L-a L(a ~ p ) v L ( a ~ -p) L(p ~ q ) ~ (Lp ~ L q ) Lp ~ p Lp ~ LLp Lp ~ Gp Lp ~ Hp Lp ~ Np

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Intuitively, we m a y t h i n k of L as an operator corresponding to 'provability w i t h i n LT'. U s i n g t h e following definitions, we can now d e m o n s t r a t e t h a t t h e set of i n s t a n t propositions forms a Leibnizian structure: a < b --~-defL(a ~ Fb) a -- b -~efL(a ~ M b ) T(a,A) ~-d~fL ( a ~ A )

Note t h a t (A1) - ( A l l ) are simply t h e axioms for Kl. Conseq u e n t l y t h e p r o p e r t i e s (B1) - (B6) a n d (TL1) - (TLS) w h i c h correspond to t h e semantics of K1 follow from t h e completeness of Ki. The r e m a i n i n g properties of t h e s t r u c t u r e can be p r o v e d in t h e following way: Theorem 2: The relation --is an equivalence relation. Proof." Reflexivity is trivial.

S y m m e t r y is proved by reductio ad absurdum: (1) a --b (assumption) (2) -(b =a) (assumption) (3) L(a ~ Mb) (from 1) (4) L(a ~ -Mb) (from 2 and I3) (5) L ( a ~ ~a) (from 3 and 4) (6) L-a (from 5). Contradicts (I2). Transitivity is (1) a =b (2) b =c (3) L(a ~ (4) L(b ~ (5) L(Mb (6) L(a ~ (7) a =c Q.E.D. T h e o r e m 2 implies

proved in a straightforward way: (assumption) (assumption) Mb) (from 1) Mc) (from 2) ~ Mc) (from 4, L1, L3 LN) Mc) (from 3 and 5) (from 6) t h a t (B7) - (B9) hold.

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CHAPTER 3.3 Theorem 3 : T(a, N A ) ~ (a --b ~ T(b,A)) Proof."

Let us assume that in some case this implication is violated. Then we may argue as follows: (1) L(a ~ NA) (assumption) (2) L(a ~ Mb) (assumption) (3) L(b ~ - A ) (assumption and I3) (4) L ( A ~ -b) (from 3) (5) L(NA ~N-b) (from 4) (6) L(a ~ N-b) (from 1 and 5) (7) L(Mb ~-a) (from 6) (8) L ( a ~ ~a) (from 7) (9) L~a (from 8) This obviously contradicts (I1). Therefore the implication in question cannot be violated. Q.E.D. Theorem 4: T(a, N A ) ~ Vb: (a -- b ~ T(b,A)) Proof." This follows from theorem 3. Theorem 5: ~ L - A ~ To: T(b,A) Proof: From chapter 2.9. Theorem 6: Vb: (a = b ~ T(b,A)) ~ L(Ma ~ A ) Proof." This is proved by reductio ad a b s u r d u m (1) Vb: (a = b ~ T(b,A)) (assumption) (2) - L ( M a ~ A) (assumption) (3) -L-(Ma A -A) (assumption) (4) TO: T(b,Ma A -,4) (3, theorem 5) (5) To: T(b, M a ) A T(b,~A) (from 4) (6) To: a --b A T(b,-A) (from 5) (7) To: L(b ~A) A L ( b ~ ~ A ) (from 1 and 6) (8) To: L(b ~ - b ) (from 7) (9) -(Vb: - L - b ) (8)

(9) contradicts (I2). Q.E.D.

LEIBNIZIAN TENSE LOGIC Theorem 7: Proof."

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~o: (a -- b ~ T(b,A)) ~ T(a, NA)

T h e proof is straight forward: (1) Vb: (a -- b ~ T(b,A)) (2) L ( M a ~ A) (3) L ( N M a ~ NA) (4) L(Ma ~ N A ) (5) L(a ~ M a )

(6)

L ( a ~ NA)

(7) Q.E.D.

T(a, NA)

(assumption) (1 by t h e o r e m 6) (2) (3) (from N2) (from 4 a n d 5) (from 6)

T h e o r e m 8: Vb: (a = b ~ T(b,A)) -~ T(a, NA) Proof." F r o m t h e o r e m 4 and theorem 7. T h e o r e m 9: (T(a, PA) A a ~ b) ~ T(b,PA) Proof." Follows from t h e o r e m 2, (A13) and the t h e o r e m 8.

According to t h e o r e m 9, a--b m e a n s t h a t any f o r m u l a of t h e form P A (where A contains no occurrences of F a n d no i n s t a n t p r o p o s i t i o n s ) w h i c h follows w i t h L - n e c e s s i t y f r o m a (i.e. L(a ~ PA))also follows w i t h L-necessity from b (and vice versa). In consequence, equivalent instants have the same g e n u i n e past. Theorem 10: (a
Q.E.D. T h u s we have demonstrated that the set of instant propositions w i t h t h e relations and t h e T-operator defined above is in fact a Leibnizian structure. In consequence, ifA is Leibniz-valid, it will also be valid in this structure, i.e. L(a ~ A) for any a. If we do i n t e r p r e t L as 'LT-provability', it follows that A can be proved in

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LT from any instant proposition. Then by (DL) from chapter 2.9 it follows t h a t LA, i.e. 'A is provable in LT'. When this is t a k e n t o g e t h e r w i t h t h e o r e m 1, we m a y conclude t h a t LT is also complete, i.e. it holds t h a t A is Leibniz-valid if and only i f A is provable in LT. (We t h i n k t h a t for practical purposes this actually does demonstrate completeness, but we are aware t h a t a full m a t h e m a t i c a l proof requires more t h a n w h a t is given here.) However, the above notion of Leibniz-validity does not exclude a cyclical model. In order to make t h e models non-cyclical we would need an additional assumption like ( B l l ) tl --t2 ~ ~(tl < t2) In the enlarged system this would correspond to the axiom a ~ ~MFa

where a stands for an instant proposition. There is, however, no obvious axiom in LT which can do exactly the same job (that is, if instant propositions are not at our disposal).

METRIC TENSE LOGIC We shall now present a metric tense logic MLT, which is a n extension of LT. The language of MLT is based on a set of propositional variables: p, q, r... and the following definition of well-formed formulae (1) Propositional variables are wff. (2) If a and 13 are wff and x is a positive real number, t h e n - a , a~13, gaff, av13, V x:a, 3 x: a, N a , P ( x ) a , and F ( x ) a are all wff. (3) There are no other wff.

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We shall assume $5 for N and in addition the following axioms: (LT1) (LT2) (LT3) (LT4) (LT5) (LT6) (LT7) (LT8) (LT9) (LT10)

G(x)(p ~ q) ~ (G(x)p ~ G(x)q) F(x)H(x)p ~ p F(y+x)p ~ F(y)F(x)p H(x)(p ~ q) ~ (H(x)p ~ H(x)q) P(x)G(x)p ~ p P(y+x)p ~ P(y)P(x)p F(x)-p -~F(x)p P(x)-p -~P(x)p NG(x)p ~ G(x)Np P(x)p ~ NP(x)p, where p contains no occurrences o f F .

In addition we assume standard n u m b e r theory for positive numbers. - The rules of MLT are the following: (RMP) (RF) (RP) (HI) (H2)

I f ~ p and ~p ~ q , then ~q. If e p, then e G(x)p. If ~ p, then ~ H(x)p. If ~ ¢~(x)~flthen ~ Vx:C~x)~fl. If ~ a ~ x ) , then ~ aDVx:C(x) for x not free in a.

Note that in LT the operators F(x) and H(x) are e q u i v a l e n t and similarly for P(x) and H(x). Nevertheless, we use all four operators in order to make the comparison with other s y s t e m s easier. On this basis, all the axioms of KI can obviously be proved. A s s u m i n g s t a n d a r d n u m b e r theory it is e a s y to prove (A8-11). (A12) and (A13) can be proved from (LT9) a n d (LT10), respectively. - Hence, we can conclude t h a t MLT is a proper extension of LT. The semantics we have in mind for MLT can be presented by the following definitions.

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Definition: A metric L e i b n i z i a n s t r u c t u r e is a q u a d r u p l e ( T I M E , before,_T), w h e r e T I M E is a n o n - e m p t y set w i t h t w o relations < and = s u c h t h a t (BI') (before(t l, t2 x) A before(t2,t3,y) ) ~ before(t l, t3,x +y) (B2') (before(tl, t2x) A before(tl, t3,x)) ~ t2 = t3 (B3') (before(tl, t2x) A before(t3,t2,x)) ~ tl = t3 (B4') Vt l Vx3t2 : before(t l, t2 x) (BS') Vt2 Y'x3tl : before(Q, t2x) (B6') t=t (B7') tl =t2~t2 =tl (B8') (tl =t2 A t2 ~t3) ~ tl ~t3 (B9') (tl --t2 A before(t3,t2,x)) ~ 3t4: (t3 •t4 A before(t4,tl, x)) a n d an operator T such t h a t (T1) (T2) (T3) (T4) (TS) (T6) (T7)

T(t,A A B) - (T(t,A) A T(t,B)) T(t, ~A) - ~ T ( t , A ) Vx:T(t,A) - T(t, Vx:A) w h e r e x is foreign to t. T(t,F(x)A) - 3 t 1 : (before(t, tl, x) A T(tl,A)) T(t,P(x)A) - 3 t 1 : (before(t l,t,x) A T(t l,A)) T(t, NA) - VQ: (Q = t ~ T(tl,A)) (t] --t2 A T(tl, P(x)A)) ~ T(t2,P(x)A), w h e r e A contains no occurrences of F.

S t a n d a r d n u m b e r t h e o r y for positive n u m b e r s is also a s s u m e d as a b a c k g r o u n d for the s e m a n t i c a l reasoning r e g a r d i n g metric Leibnizian structures. W e shall s a y t h a t a s t a t e m e n t A is metrically Leibniz-valid if and o n l y i f for a n y m e t r i c Leibnizian structure (TIME, before, , T ) and any t in T I M E , it holds t h a t T(t,A). Now, it is n o t difficult to verify t h a t a n y s t a t e m e n t w h i c h is provable in M L T is also metrically Leibniz-valid. In order to show t h a t this is t h e case we have to a r g u e t h a t the validity in question is c a r r i e d over by t h e MLT r u l e s and t h a t all the M L T axioms are

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metrically Leibniz-valid. Most of these proofs are trivial or very easy. Let us consider for example (LT7). We prove t h a t T(t,F(x)~p ~ ~F(x)p) Proof." This is proved by reductio ad absurdum. (1) T(t,F(x)-p) (assumption) (2) T(t,F(x)p) (assumption) (3) 31: before(t, tI, x) A T(Q, ~p) (4) 32: before(t, t2,x) A T(t2,p) (5) ~ 1 ~ 2 : before(t, Q,x) A before(t, t2,x) A T(t2,p) A T(tl,~p) (6) ~1: before(t, tl,x) A T(tl,p) A T(tl, ~p) (7) 3Q: T(tl,p) A -T(tl,p) - i.e. a contradiction. Q.E.D. It m a y be concluded that ifA is provable in MLT, t h e n A is also m e t r i c a l l y Leibniz-valid. In o r d e r to d e m o n s t r a t e t h a t t h e converse also holds, we shall once again m a k e u s e of an enlarged system. This system includes Prior's instant propositions and t h e modal operator L earlier mentioned in this c h a p t e r , w i t h t h e a d d i t i o n t h a t any two different i n s t a n t propositions are non-equivalent, i.e. L(a = b) if and only if a=b. We w a n t to prove t h a t the set of i n s t a n t propositions forms a metric Leibnizian structure, if we m a k e use of t h e definitions we have used in this chapter and in chapter 2.10: before(a, b,x) =def L(a ~ F(x)b) a -- b -~-defL(a ~ Mb) T(a,A) -~defL(a ~ A)

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Most of the proofs needed for this purpose can be copied from earlier; a few of t h e m are new, though, but fairly easy. Let us take some examples: Va Vx3b : before(a,b,x)

Proof: (1) G(x)(p v ~p) (2) F(x)(p v - p ) (3) L(a ~ F(x)(p v ~p)) (4) 3b: (L(a ~ F(x)b) A L(b ~ (p v - p ) ) (5) 3b: before(a,b,x) (before(a,b,x) A before(a,c,x)) ~ L(b ~ c)

Proof: This is (1) (2) (3) (4) (5) (6) (7) (8) Q.E.D.

proved by reductio ad absurdum. L(a ~ F(x)b) (assumption) L ( a ~ F(x)c) (assumption) ~L(b ~ c) (assumption) L(b ~ -c) L(F(x)b ~ F ( x ) ~ c ) L(a ~ F ( x ) - c ) L(a ~ -F(x)c) L(a ~ -a).

(Contradicts I2).

By a similar a r g u m e n t it can be proved t h a t (before(a,b,x) A before(a,c,x)) ~ L(c ~ b)

In consequence we have (before(a,b,x) A before(a,c,x)) ~ L(b - c)

Since any two instant propositions are non-equivalent, it can be concluded that (before(a,b,x) A before(a,c,x)) ~ b=c

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W i t h respect to the problem of cyclicity we have the same problem as in LT, and again there is an obvious solution within the enlarged system. MLT s e e m s to be v e r y useful in m a n y cases. However, counterfactual implications in n a t u r a l language like (S1) A is not the case, but ifA were the case, t h e n B would also be the case, or w i t h a more n a t u r a l wording ($2) ifA were the case, then B would be the case cannot be satisfactorily expressed in MLT. The obvious solution in a branching time logic would seem to be - A A 3k: P(x)NF(x)(A ~ B )

w h i c h b e c a u s e of (LT9) t u r n s out to imply t h e following formula: -A A N(A ~ B)

This corresponds to statement: ($2) A is not the case, but necessarily, i f A were the case, t h e n B would also be the case. Now, since ($2) is normally considered to be semantically stronger t h a n (S1), we want a logic that can reflect this relation. In t h e n e x t c h a p t e r we shall see how such a logic can be constructed.

3.3. LEIBNIZIAN TENSE LOGIC In this c h a p t e r we shall present an indeterministic tense logic similar to Prior's Ockhamistic theory, but modified in t h e light of t h e observations made by Hirokazu N i s h i m u r a (mentioned in the p r e c e d i n g chapter). Since the s y s t e m is based on a k i n d of temporal r e a s o n i n g very similar to Leibniz' philosophy, we shall call the s y s t e m 'Leibnizian tempo-modal logic', LT for short. The definition of well-formed formulae w i t h i n LT is: (1) (2)

(3)

Propositional variables are wff. If a and fl are wff and x is a positive n u m b e r , t h e n - a , a_.~fl, aAfl, a v fl, Vx:a, ~x: a, N a , Pa, and Fve are all wff. There are no other wff.

The axioms of LT are: (A1)

A, where A is a tautology of.the propositional calculus

(A2) (A3) (A4) (A5) (A6) (A7) (AS) (A9) (A10) (All) (A12) (A13)

G(A ~ B) ~ (GA ~ GB) H ( A ~ B) ~ (HA ~ H B ) A ~ HFA A ~GPA FFA ~ FA F P A ~ ( P A v A v FA) P F A ~ ( P A v A v FA) GA ~ FA HA ~ PA FA ~ FFA NGA ~ GNA P A ~ N P A , where A contains no occurrences of F

(N1) (N2) (N3) (N4)

N ( A ~ B) ~ ( N A ~ N B ) NA DA NA ~ NNA M N A ~ N A , where M -~def - N -

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(N1)-(N4) a r e t h e $5 axioms for N. The rules of inference a r e t h e s a m e as for t h e Ockhamistic system, i.e.

(RMP) (RG) (RH) (RN)

If ~ A and ~ A ~ B, then I- B.

(nl)

I f ~ ~(x)~fl t h e n I- Vx:~(x)~fl. If/" a ~ x ) t h e n I- a ~ V x : ~ x ) , for x not free in a.

(l-I2)

If ~ A, then I- G A . If I- A, t h e n / - HA. If I- A t h e n I- N A .

w h e r e '1-p' m e a n s 'p is provable'. T h e e a r l i e r - l a t e r logic (B-logic) we h a v e in m i n d can b e p r e s e n t e d b y t h e following definitions. Definition: A L e i b n i z i a n s t r u c t u r e is a q u a d r u p l e ( T I M E , <, _ T ) , w h e r e TIME is a n o n - e m p t y set w i t h t w o

relations < a n d = such t h a t (B1) (B2) (B3) (B4) (BS) (B6) (B7) (B8) (B9) (B10)

(tl < t2 A t2 < t3) ~ tl < t3 (tl < t2 A t3 < t2) ~ (tl < t3 V tl = t3 V t3 < tl) (t2 < tl A t2 < t3) ~ (tl < t3 v t l = t3 v t 3 < t P Vtl~t2 : tl < t2 V Q ~ 2 : t2 < tl Vt I Vt2Jt3 : tl < t2 ~ (tI < t3 • t3 < t2) t=t tl = t 2 ~ t 2 = t l (tl = t 2 A t 2 --t3) ~ t l --t3 (t~ = t e a t3 < re) ~ 3 t 4 : ( t 3 --t4 A t4 < tz)

and an o p e r a t o r T such that (T1) (T2) (T3) (T4) (T5) (T6)

T ( t , A A B ) =-(T(t,A) A T ( t , B ) ) T(t,-A) ~--T(t,A) Vx:T(t,A) =--T(t, Vx:A) w h e r e x is foreign to t T(t, F A ) -= 3Q: (t < tl A T ( t I , A ) ) T(t, P A ) - 3t1: (tl < t A T ( t l , A ) ) T(t, N A ) -~ Vtl: (tl -- t ~ T ( t I , A ) )

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Y'tl Vt2: (tl = t2 A T(tl, PA)) ~ T(t2,PA), where A contains no occurrences of F.

You m a y read T(t~A) as 'A is true at t', but bear in m i n d t h a t we a r e n o t i n t r o d u c i n g a s e p a r a t e s e m a n t i c s following u s u a l m o d e l - t h e o r e t i c p r o c e d u r e s . R a t h e r , we are e n l a r g i n g t h e logical l a n g u a g e such t h e e x t e n d e d s y s t e m ' c o n t a i n s t h e s e m a n t i c s of the original system', following Prior's ideas in this respect. Definition: A s t a t e m e n t is Leibniz-valid if and only if T(t,A) for every Leibnizian s t r u c t u r e (TIME, <, , T) a n d every t in TIME. It is easy to see t h a t t h e axioms (A1) - (A13) a n d (N1) - (N4) a r e all Leibniz-valid. S i n c e t h e inference r u l e s c a r r y t h e Leibniz-validity over, it follows that the following t h e o r e m holds: Theorem 1: IfA is provable in LT, t h e n A is Leibniz-valid. T h i s t h e o r e m expresses t h e s o u n d n e s s of LT. U s i n g Prior's i d e a of i n s t a n t propositions we shall also argue t h a t LT is complete relative to Leibniz-validity. First we shall enlarge t h e logical l a n g u a g e in the m a n n e r already shown in c h a p t e r 2.9. T h e e n l a r g e d system m u s t include i n s t a n t p r o p o s i t i o n s (i.e. m a x i m a l consistent sets from LT) and a modal o p e r a t o r L, for w h i c h we a s s u m e (BF) (I1) (I2) (I3) (L1) (L2) (L3) (LG) (LH) (LN)

L(Va: ¢(a)) ----Va: L(~(a)) :~a: a ~L-a L(a ~ p ) v L ( a ~ -p) L(p ~ q ) ~ (Lp ~ L q ) Lp ~ p Lp ~ LLp Lp ~ Gp Lp ~ Hp Lp ~ Np

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Intuitively, we m a y t h i n k of L as an operator corresponding to 'provability w i t h i n LT'. U s i n g t h e following definitions, we can now d e m o n s t r a t e t h a t t h e set of i n s t a n t propositions forms a Leibnizian structure: a < b --~-defL(a ~ Fb) a -- b -~efL(a ~ M b ) T(a,A) ~-d~fL ( a ~ A )

Note t h a t (A1) - ( A l l ) are simply t h e axioms for Kl. Conseq u e n t l y t h e p r o p e r t i e s (B1) - (B6) a n d (TL1) - (TLS) w h i c h correspond to t h e semantics of K1 follow from t h e completeness of Ki. The r e m a i n i n g properties of t h e s t r u c t u r e can be p r o v e d in t h e following way: Theorem 2: The relation --is an equivalence relation. Proof." Reflexivity is trivial.

S y m m e t r y is proved by reductio ad absurdum: (1) a --b (assumption) (2) -(b =a) (assumption) (3) L(a ~ Mb) (from 1) (4) L(a ~ -Mb) (from 2 and I3) (5) L ( a ~ ~a) (from 3 and 4) (6) L-a (from 5). Contradicts (I2). Transitivity is (1) a =b (2) b =c (3) L(a ~ (4) L(b ~ (5) L(Mb (6) L(a ~ (7) a =c Q.E.D. T h e o r e m 2 implies

proved in a straightforward way: (assumption) (assumption) Mb) (from 1) Mc) (from 2) ~ Mc) (from 4, L1, L3 LN) Mc) (from 3 and 5) (from 6) t h a t (B7) - (B9) hold.

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CHAPTER 3.3 Theorem 3 : T(a, N A ) ~ (a --b ~ T(b,A)) Proof."

Let us assume that in some case this implication is violated. Then we may argue as follows: (1) L(a ~ NA) (assumption) (2) L(a ~ Mb) (assumption) (3) L(b ~ - A ) (assumption and I3) (4) L ( A ~ -b) (from 3) (5) L(NA ~N-b) (from 4) (6) L(a ~ N-b) (from 1 and 5) (7) L(Mb ~-a) (from 6) (8) L ( a ~ ~a) (from 7) (9) L~a (from 8) This obviously contradicts (I1). Therefore the implication in question cannot be violated. Q.E.D. Theorem 4: T(a, N A ) ~ Vb: (a -- b ~ T(b,A)) Proof." This follows from theorem 3. Theorem 5: ~ L - A ~ To: T(b,A) Proof: From chapter 2.9. Theorem 6: Vb: (a = b ~ T(b,A)) ~ L(Ma ~ A ) Proof." This is proved by reductio ad a b s u r d u m (1) Vb: (a = b ~ T(b,A)) (assumption) (2) - L ( M a ~ A) (assumption) (3) -L-(Ma A -A) (assumption) (4) TO: T(b,Ma A -,4) (3, theorem 5) (5) To: T(b, M a ) A T(b,~A) (from 4) (6) To: a --b A T(b,-A) (from 5) (7) To: L(b ~A) A L ( b ~ ~ A ) (from 1 and 6) (8) To: L(b ~ - b ) (from 7) (9) -(Vb: - L - b ) (8)

(9) contradicts (I2). Q.E.D.

LEIBNIZIAN TENSE LOGIC Theorem 7: Proof."

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~o: (a -- b ~ T(b,A)) ~ T(a, NA)

T h e proof is straight forward: (1) Vb: (a -- b ~ T(b,A)) (2) L ( M a ~ A) (3) L ( N M a ~ NA) (4) L(Ma ~ N A ) (5) L(a ~ M a )

(6)

L ( a ~ NA)

(7) Q.E.D.

T(a, NA)

(assumption) (1 by t h e o r e m 6) (2) (3) (from N2) (from 4 a n d 5) (from 6)

T h e o r e m 8: Vb: (a = b ~ T(b,A)) -~ T(a, NA) Proof." F r o m t h e o r e m 4 and theorem 7. T h e o r e m 9: (T(a, PA) A a ~ b) ~ T(b,PA) Proof." Follows from t h e o r e m 2, (A13) and the t h e o r e m 8.

According to t h e o r e m 9, a--b m e a n s t h a t any f o r m u l a of t h e form P A (where A contains no occurrences of F a n d no i n s t a n t p r o p o s i t i o n s ) w h i c h follows w i t h L - n e c e s s i t y f r o m a (i.e. L(a ~ PA))also follows w i t h L-necessity from b (and vice versa). In consequence, equivalent instants have the same g e n u i n e past. Theorem 10: (a
Q.E.D. T h u s we have demonstrated that the set of instant propositions w i t h t h e relations and t h e T-operator defined above is in fact a Leibnizian structure. In consequence, ifA is Leibniz-valid, it will also be valid in this structure, i.e. L(a ~ A) for any a. If we do i n t e r p r e t L as 'LT-provability', it follows that A can be proved in

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LT from any instant proposition. Then by (DL) from chapter 2.9 it follows t h a t LA, i.e. 'A is provable in LT'. When this is t a k e n t o g e t h e r w i t h t h e o r e m 1, we m a y conclude t h a t LT is also complete, i.e. it holds t h a t A is Leibniz-valid if and only i f A is provable in LT. (We t h i n k t h a t for practical purposes this actually does demonstrate completeness, but we are aware t h a t a full m a t h e m a t i c a l proof requires more t h a n w h a t is given here.) However, the above notion of Leibniz-validity does not exclude a cyclical model. In order to make t h e models non-cyclical we would need an additional assumption like ( B l l ) tl --t2 ~ ~(tl < t2) In the enlarged system this would correspond to the axiom a ~ ~MFa

where a stands for an instant proposition. There is, however, no obvious axiom in LT which can do exactly the same job (that is, if instant propositions are not at our disposal).

METRIC TENSE LOGIC We shall now present a metric tense logic MLT, which is a n extension of LT. The language of MLT is based on a set of propositional variables: p, q, r... and the following definition of well-formed formulae (1) Propositional variables are wff. (2) If a and 13 are wff and x is a positive real number, t h e n - a , a~13, gaff, av13, V x:a, 3 x: a, N a , P ( x ) a , and F ( x ) a are all wff. (3) There are no other wff.

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We shall assume $5 for N and in addition the following axioms: (LT1) (LT2) (LT3) (LT4) (LT5) (LT6) (LT7) (LT8) (LT9) (LT10)

G(x)(p ~ q) ~ (G(x)p ~ G(x)q) F(x)H(x)p ~ p F(y+x)p ~ F(y)F(x)p H(x)(p ~ q) ~ (H(x)p ~ H(x)q) P(x)G(x)p ~ p P(y+x)p ~ P(y)P(x)p F(x)-p -~F(x)p P(x)-p -~P(x)p NG(x)p ~ G(x)Np P(x)p ~ NP(x)p, where p contains no occurrences o f F .

In addition we assume standard n u m b e r theory for positive numbers. - The rules of MLT are the following: (RMP) (RF) (RP) (HI) (H2)

I f ~ p and ~p ~ q , then ~q. If e p, then e G(x)p. If ~ p, then ~ H(x)p. If ~ ¢~(x)~flthen ~ Vx:C~x)~fl. If ~ a ~ x ) , then ~ aDVx:C(x) for x not free in a.

Note that in LT the operators F(x) and H(x) are e q u i v a l e n t and similarly for P(x) and H(x). Nevertheless, we use all four operators in order to make the comparison with other s y s t e m s easier. On this basis, all the axioms of KI can obviously be proved. A s s u m i n g s t a n d a r d n u m b e r theory it is e a s y to prove (A8-11). (A12) and (A13) can be proved from (LT9) a n d (LT10), respectively. - Hence, we can conclude t h a t MLT is a proper extension of LT. The semantics we have in mind for MLT can be presented by the following definitions.

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Definition: A metric L e i b n i z i a n s t r u c t u r e is a q u a d r u p l e ( T I M E , before,_T), w h e r e T I M E is a n o n - e m p t y set w i t h t w o relations < and = s u c h t h a t (BI') (before(t l, t2 x) A before(t2,t3,y) ) ~ before(t l, t3,x +y) (B2') (before(tl, t2x) A before(tl, t3,x)) ~ t2 = t3 (B3') (before(tl, t2x) A before(t3,t2,x)) ~ tl = t3 (B4') Vt l Vx3t2 : before(t l, t2 x) (BS') Vt2 Y'x3tl : before(Q, t2x) (B6') t=t (B7') tl =t2~t2 =tl (B8') (tl =t2 A t2 ~t3) ~ tl ~t3 (B9') (tl --t2 A before(t3,t2,x)) ~ 3t4: (t3 •t4 A before(t4,tl, x)) a n d an operator T such t h a t (T1) (T2) (T3) (T4) (TS) (T6) (T7)

T(t,A A B) - (T(t,A) A T(t,B)) T(t, ~A) - ~ T ( t , A ) Vx:T(t,A) - T(t, Vx:A) w h e r e x is foreign to t. T(t,F(x)A) - 3 t 1 : (before(t, tl, x) A T(tl,A)) T(t,P(x)A) - 3 t 1 : (before(t l,t,x) A T(t l,A)) T(t, NA) - VQ: (Q = t ~ T(tl,A)) (t] --t2 A T(tl, P(x)A)) ~ T(t2,P(x)A), w h e r e A contains no occurrences of F.

S t a n d a r d n u m b e r t h e o r y for positive n u m b e r s is also a s s u m e d as a b a c k g r o u n d for the s e m a n t i c a l reasoning r e g a r d i n g metric Leibnizian structures. W e shall s a y t h a t a s t a t e m e n t A is metrically Leibniz-valid if and o n l y i f for a n y m e t r i c Leibnizian structure (TIME, before, , T ) and any t in T I M E , it holds t h a t T(t,A). Now, it is n o t difficult to verify t h a t a n y s t a t e m e n t w h i c h is provable in M L T is also metrically Leibniz-valid. In order to show t h a t this is t h e case we have to a r g u e t h a t the validity in question is c a r r i e d over by t h e MLT r u l e s and t h a t all the M L T axioms are

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metrically Leibniz-valid. Most of these proofs are trivial or very easy. Let us consider for example (LT7). We prove t h a t T(t,F(x)~p ~ ~F(x)p) Proof." This is proved by reductio ad absurdum. (1) T(t,F(x)-p) (assumption) (2) T(t,F(x)p) (assumption) (3) 31: before(t, tI, x) A T(Q, ~p) (4) 32: before(t, t2,x) A T(t2,p) (5) ~ 1 ~ 2 : before(t, Q,x) A before(t, t2,x) A T(t2,p) A T(tl,~p) (6) ~1: before(t, tl,x) A T(tl,p) A T(tl, ~p) (7) 3Q: T(tl,p) A -T(tl,p) - i.e. a contradiction. Q.E.D. It m a y be concluded that ifA is provable in MLT, t h e n A is also m e t r i c a l l y Leibniz-valid. In o r d e r to d e m o n s t r a t e t h a t t h e converse also holds, we shall once again m a k e u s e of an enlarged system. This system includes Prior's instant propositions and t h e modal operator L earlier mentioned in this c h a p t e r , w i t h t h e a d d i t i o n t h a t any two different i n s t a n t propositions are non-equivalent, i.e. L(a = b) if and only if a=b. We w a n t to prove t h a t the set of i n s t a n t propositions forms a metric Leibnizian structure, if we m a k e use of t h e definitions we have used in this chapter and in chapter 2.10: before(a, b,x) =def L(a ~ F(x)b) a -- b -~-defL(a ~ Mb) T(a,A) -~defL(a ~ A)

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Most of the proofs needed for this purpose can be copied from earlier; a few of t h e m are new, though, but fairly easy. Let us take some examples: Va Vx3b : before(a,b,x)

Proof: (1) G(x)(p v ~p) (2) F(x)(p v - p ) (3) L(a ~ F(x)(p v ~p)) (4) 3b: (L(a ~ F(x)b) A L(b ~ (p v - p ) ) (5) 3b: before(a,b,x) (before(a,b,x) A before(a,c,x)) ~ L(b ~ c)

Proof: This is (1) (2) (3) (4) (5) (6) (7) (8) Q.E.D.

proved by reductio ad absurdum. L(a ~ F(x)b) (assumption) L ( a ~ F(x)c) (assumption) ~L(b ~ c) (assumption) L(b ~ -c) L(F(x)b ~ F ( x ) ~ c ) L(a ~ F ( x ) - c ) L(a ~ -F(x)c) L(a ~ -a).

(Contradicts I2).

By a similar a r g u m e n t it can be proved t h a t (before(a,b,x) A before(a,c,x)) ~ L(c ~ b)

In consequence we have (before(a,b,x) A before(a,c,x)) ~ L(b - c)

Since any two instant propositions are non-equivalent, it can be concluded that (before(a,b,x) A before(a,c,x)) ~ b=c

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W i t h respect to the problem of cyclicity we have the same problem as in LT, and again there is an obvious solution within the enlarged system. MLT s e e m s to be v e r y useful in m a n y cases. However, counterfactual implications in n a t u r a l language like (S1) A is not the case, but ifA were the case, t h e n B would also be the case, or w i t h a more n a t u r a l wording ($2) ifA were the case, then B would be the case cannot be satisfactorily expressed in MLT. The obvious solution in a branching time logic would seem to be - A A 3k: P(x)NF(x)(A ~ B )

w h i c h b e c a u s e of (LT9) t u r n s out to imply t h e following formula: -A A N(A ~ B)

This corresponds to statement: ($2) A is not the case, but necessarily, i f A were the case, t h e n B would also be the case. Now, since ($2) is normally considered to be semantically stronger t h a n (S1), we want a logic that can reflect this relation. In t h e n e x t c h a p t e r we shall see how such a logic can be constructed.

3 . 4 . T E N S E L O G I C AND COUNTERFACTUAL REASONING Temporal r e a s o n i n g is i n t i m a t e l y related to certain other kinds of reasoning. Some particularly important examples are causal, counterfactual, and diagnostic reasoning. In these kinds of reasoning we fred a type of conditionals, which are crucially interwoven w i t h temporality. (A recent P h . D . Thesis [Crouch 1993] has s h o w n how time and t e n s e are in fact pervasive features of English conditionals in general.) Since time proves to be relevant for counterfactual as well as diagnostic reasoning, we fred it essential to examine the formal logical s t r u c t u r e of such r e a s o n i n g - as well as its t a c i t dimension, w h i c h for these cases plays a somewhat special r61e. We are going to a r g u e t h a t a suitable f r a m e w o r k can be constructed in t h e form of a non-monotonic and tempo-modal logic, which is p a r t l y based on J.L. Mackie's suggestions regarding causality [Mackie 1974]. Moreover, the features of this framework - especially non-monotonicity - broadens the perspectives studied so far on time and tense. Let us consider an example of everyday reasoning about time and causes, based on the following story: Joe wants to prepare a meal for some guests. He goes to the local shop in order to buy the ingredients, and he hires a cook, who is supposed to prepare the meal. In due time the cook arrives at Joe's place, just to find out t h a t there is an electrical failure, which will m a k e it impossible to have the dinner r e a d y on time. In the following a r g u m e n t between Joe and his friend Jim, we see a n u m b e r of (somewhat artificial) counterfactual statements: Joe: If it h a d not been for the failure, we would have h a d dinner on time. Jim:You're wrong. If the cook left, you might still not have had dinner on time, even in the absence of the failure.

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Joe: Well, I was just assuming that the cook would stay. But w h a t I m e a n is t h a t if there were no failure and the cook agreed to stay, we would have had dinner on time. Jim: You're wrong again. If you had not been to the shop to buy the ingredients, you would not have had dinner on time, even if the cook stayed and there were no electrical failure. Joe: I don't agree. Somebody else might have brought the ingredients. As long as the cook has got the ingredients and t h e r e is no electrical failure, we shall w i t h necessity have dinner on time. Jim: Well, the cook might change his m i n d because of all your quarrelling. Again, in that case t h e r e would not be dinner on time, even if there were no electrical failure. Joe: Well, we m i g h t have had dinner on time, if I had ordered some r e a d y - m a d e dinner. The electrical failure could not have prevented the dinner in t h a t case. Jim: I think you're wrong again. The cook is a union man. He will bar the door to the delivery of the ready-made dinner, since the r e a d y - m a d e dinner company takes away jobs from proper cooks. But if the cook leaves, the ready-made dinner can be delivered. What happens in this kind of debate is t h a t still new elements from the scenario are introduced as relevant. At each point, it can be argued t h a t t h e speaker in question is right, if his statem e n t is evaluated from his point of view. On the other hand, it t u r n s out that w h e n e.g. the first statem e n t above is evaluated from a new and broader perspective in w h i c h a new c a u s a l factor (i.e. t h e cook) is t a k e n into consideration, it is false. What is at stake here is the very notion of causation. Each of t h e s t a t e m e n t s can be i n t e r p r e t e d as assertions regarding causes. In order to account for the non-monotonicity, i.e. the apparent change in the truth-values of statements, we m u s t introduce a new representation of causality. Whereas causality is normally conceived as a relation between a cause and its effect, the idea is to represent causality as a predicate which t a k e s three arguments. The point is t h a t there is always a ceteris paribus clause

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involved in a s t a t e m e n t regarding causality. Causal s t a t e m e n t s such as 'if it rains, the grass will become wet', or 'rain causes the grass to become wet', are asserted under the tacit assumption of 'all other things being equal'. For instance, the s t a t e m e n t 'rain causes the grass to become wet' presupposes 'the grass has not been covered' (say, by a large tent for our garden p a r t y tonight). In general, the ceteris paribus clause determines which possible 'chronicles', i.e. possible courses of events, should be t a k e n into consideration in order to evaluate the statement in question. In other words, this extra a r g u m e n t of the causality predicate, which we shall call the 'scope', defines the set of entities relevant for the evaluation of the statement. The minimal scope can be constructed directly from the statement in question as t h e set of all atomic propositions (with their tenses) involved in the statement. We s h a h call this the natural scope of the statement. Also it would be wrong to see the cause as just one single entity. The above conversation shows t h a t it would be more correct to represent the cause as a conjunction of a number of statements. In t h e following we i n t e n d to show how c o u n t e r f a c t u a l implications can be introduced semantically as an extension of MLT discussed in the last chapter. The crucial f e a t u r e s of this extension come from the idea t h a t counterfactual s t a t e m e n t s are e v a l u a t e d u n d e r c e t e r i s p a r i b u s a s s u m p t i o n s , a n d f u r t h e r m o r e , t h a t in any such evaluation one m u s t a s s u m e what we have called a 'scope' for the statement in question. First, we are going to introduce a number of fairly technical definitions, without too m u c h motivation; then we shall come back to the dinner scenario to show how the s y s t e m s work in practice.

CIMP - MODELLING COUNTERFACTUALIMPLICATIONS David Lewis [1973] formulated a very elegant logic of counterfactuals, w h i c h covers m a n y i m p o r t a n t intuitions. Its m a i n defect is, however, that possible worlds appear only as semantic indices - r a t h e r t h a n as concrete conceptual a l t e r n a t i v e s . Instead, we shall propose a construction of possible worlds as

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finite sets of propositions. These constructions will be based on Mackie causal complexes, which are arguably relevant for t h e evaluation of most counterfactuals. We shall describe our ideas partly with reference to a computer implementation called 'the CIMP system'. (The system is implemented in PROLOG and can be used interactively as a tool for modelling and e v a l u a t i n g counterfactuals, cf. [Hasle & Ohrstr~m 1992], [OhrstrCm et al. 1992], and [OhrstrCm and Hasle 1995].) The c e n t r a l idea of CIMP is to consider c o n c e p t u a l alternatives as partial descriptions of a situation. Consider a case of meningitis. If we want to evaluate the conditional if the patient had been vaccinated, he would not have developed meningitis, we m u s t construct a situation in which the patient was vaccin a t e d and exposed to the same pneumococci infection. This construction m u s t respect known singular causal complexes a n d facts about the case. Furthermore, there is a causal field within which the construction takes place. To capture all this, we need formal definitions of causal s t a t e m e n t s and causal models. Moreover, we w a n t to construct possible alternatives as possible ways in which the world might have developed, t h a t is, as possible courses of events. This means t h a t there will be a temporal aspect in our analysis. In CIMP, counterfactual implications are evaluated with respect to 'possible chronicles' r a t h e r t h a n possible worlds. In a sum, CIMP is a semantical system based on the ideas in metric tense logic combined with an u n d e r s t a n d i n g of causality based on Mackie's ideas. David Lewis has stressed the same relation between branching time and counterfactual reasoning in the following way: I suggest t h a t the mysterious a s y m m e t r y between open fut u r e and fixed past is nothing else t h a n the a s y m m e t r y of counterfactual dependence. The forking paths into t h e future - the actual one and all the rest - are the m a n y alternative futures t h a t would come about u n d e r various counterfactual suppositions about the present. [Lewis 1979, p. 462]

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The idea seems to be t h a t given the actual development up to the present state, it does not make sense in addition to assume a n o t h e r past. But even given the actual development up to the p r e s e n t state, we actually have the choice between a number of m u t u a l l y exclusive assumptions about the future. In this w a y Lewis w a n t s to see t h e a s y m m e t r i c a l s t r u c t u r e of time as n o t h i n g but a consequence of the n a t u r e of counterfactual dependence. We do not accept t h a t counterfactual dependence should have conceptual priority over the very concept of time. We suggest t h a t the a s y m m e t r y between open future and fixed past is taken for granted and used for the purpose of defining counterfactual implication. The language of CIMP is based on a finite set of propositional expressions, called the maximal scope: M S C O P E = S.v u . . . ~S.1 ~J So ~ $1 ~ . . . u Sw, where S-i = {P(i,q.il), P(i,q.~, ..., P(i,q.~(-i))}, i = l . . . v So = {qol, qoz ...,qog(o)] Si = {F(i, qil), F(i, qi2), ..., F(i, qiN(i))}, i=l...w and all the qjk are propositional constants and the N(j) are positive integers. F u r t h e r m o r e , we define an operator d u a l in the following way: d u a l ( P ( x , - q p ) = P(x,q'~ dual(P(x, qd) = P(x,~qp dual(~q~ = qi dual(q~ = -qi d u a l ( F ( x , - q p ) = F(x, qp dual(F(x,q.J) = F ( x , - q p We need some additional definitions: Definition: Two sets, S and S', are said to relate to each other if for each element A in S exactly one of A and dual(A) is in S' and vice versa.

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Definition: A m s ~ i m a l set is a set t h a t relates to M S C O P E .

Obviously, related sets have the s a m e n u m b e r of elements. It is also e v i d e n t t h a t a n y m a x i m a l set can be w r i t t e n as a u n i o n of subsets: E-o ~ ... u E.I ~ Eo ~ E1 ~ . . . ~ Ew), w h e r e for each i, Ei is r e l a t e d to t h e Si. The Eis are called events, a n d t h e e l e m e n t s of e a c h Ei are called basic propositions. Note t h a t all e l e m e n t s in an e v e n t h a v e t h e s a m e t e n s e and t h a t t h e events c o n s e q u e n t l y can b e o r d e r e d in an obvious way. E v e r y m a x i m a l s e t corresponds to a p r o p o s i t i o n w h i c h is f o r m e d as t h e c o n j u n c t i o n of all t h e propositions in t h e set. F o r this reason we shall w r i t e a m a x i m a l s e t as Ua(= E.v u ... ~ E.1 u; Eo ~ E1 ~ ... u Ew), w h e r e a is t h e c o r r e s p o n d i n g m a x i m a l proposition. - It is obvious t h a t Theorem 1: I f B c Ua, then dual(B) ~; Ua.

T h i s m e a n s t h a t Ua is m a d e consistent by its construction. It is also e a s y to prove t h a t Theorem 2: If S C O P E is a s u b s e t of M S C O P E a n d Ua is a m a x i m a l set, t h e n t h e r e is a u n i q u e s u b s e t of Ua w h i c h relates to SCOPE.

T h e u n i q u e s u b s e t d e s c r i b e d in t h e t h e o r e m w e s h a l l h e n c e f o r t h call sub(SCOPE, Ua). W e shall m a k e u s e of J o h n Mackie's [1974] conception of s i n g u l a r c a u s a t i o n . M a c k i e defined s i n g u l a r c a u s a t i o n in t h e following w a y (the f o r m u l a t i o n h e r e is t a k e n from [Marsden et al. 1990, p. 66]): c is a c a u s e o f e w h e n t h e occurrence o f c is, b y itself, an insufficient b u t n e c e s s a r y p a r t of a set of conditions which, w h e n c o m b i n e d , a r e u n n e c e s s a r y y e t s u f f i c i e n t for e (in short: c is an I N U S condition for e). L e t E (: t h e effect) be a basic proposition a n d C (: t h e c a u s a l complex) be a set of basic propositions. We shall also a s s u m e t h a t t h e t e n s e s in C are exactly one time unit earlier t h a n E. W e m a y

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c o n s i d e r C as a conjunction of basic s t a t e m e n t s , w h i c h necessarily leads to E.

Definition: An expression of t h e form causal(C,E) is said to be a causal s t a t e m e n t if for some i, C (: the causal complex) relates to a subset of Si a n d E (: t h e effect) relates to a m e m b e r of Si+l. In CIMP, Mackie's ideas are formalised in t e r m s of c a u s a l statements. The elements of C are INUS conditions.

Definition: E is said to be causally d e t e r m i n e d by the set of Ua a n d t h e causal s t a t e m e n t s CS, if there is an e l e m e n t in CS causal(C,E) with C c_Ua. Formally, 3CseCS: Cs=causal(C,E) A C ~ Ua Definition: A maximal set Ua is a permissible chronicle (or history) w.r.t, a set of causal s t a t e m e n t s CS if and only if each basic proposition B which is d e t e r m i n e d by Ua and CS is a m e m b e r of Ua. The set of permissible chronicles is w r i t t e n perm(CS). We shall say t h a t CS is c o n s i s t e n t if perm(CS) is non-empty. Note t h a t if t h e basic proposition B is causally d e t e r m i n e d by a permissible chronicle Uc and the e l e m e n t s in CS, t h e n dual(B) is not causally determined by Uc and t h e elements in CS.

Definition: EeMSCOPE is said to be causally supported by Ua in CS, if there is a causal s t a t e m e n t causal(C, dual(E)) in CS, and for any such statement C is not a subset of Ua. Formally,

3CseCS 3C: Cs=causal(C, dual(E)) A VCseCS VC: (Cs=causal(C, dual(E)) ~-(Co_ Ua)) Definition: E is said to be uniquely supported by Ua in CS, if E is causally supported by Ua a n d CS, but dual(E) is n o t causally supported by Ua and CS.

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It is obvious that

Theorem 3: I f E is supported by Ua in CS t h e n dual(E) is not d e t e r m i n e d by Ua. Theorem 4: If t h e r e is a causal(C,E)eCS and E is uniquely supported by Ua, t h e n E is determined by Ua. T h e set of p e r m i s s i b l e chronicles can be o r g a n i s e d as a b r a n c h i n g time system. It is i m p o r t a n t for t h e evaluation of c o u n t e r f a c t u a l s t h a t this system can be given an additional s t r u c t u r e by t h e reference to causal models introduced in t h e following way:

D e f i n i t i o n : C M ( C S , Uc) is a causal model, if C S is a consistent set of causal statements and Uc is a maximal set, such t h a t ff causal(C,E) e CS and C is a subset of Uc, t h e n E e Uc. - The m a x i m a l proposition c will be called the true chronicle. In t h e following we shall refer to the causal model CM(CS, Uc). It obviously follows from t h e above definitions t h a t if C S is consistent, then it can give rise to at least one causal model. We introduce t h e following algorithm, according to w h i c h a selected event after any given event can be constructed: A s s u m e t h a t Ek is t h e event just before Ek+l in t h e t r u e chronicle Uc, a n d t h a t E'k is a possible e v e n t t h a t relates to Ek. Suppose t h a t Ek+l=[qj,...,qN}. The p r o b l e m is how to construct E'k÷~= {q 'j.... q'g} • The algorithm can be described as follows, where qi is an arbitrary e l e m e n t in Ek+l: a) If q~ is causally determined by E'k in CS, t h e n put q 'i=qi. b) If dual(qp is causally determined by E'k in CS, t h e n p u t q 'i--dual(q~ c) If a)-b) do not apply and dual(q~ is uniquely supported by E'k and CS, t h e n p u t q 'i--dual(q'J. d) If a)-c) do not apply, t h e n put q'i=qi.

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By this algorithm 'the future part' of a permissible chronicle can be constructed from any event. This means t h a t t h e set of permissible chronicles forms a branching time system with the very special property that for each event there is a selected next event in the system, that is, 'a preferred branch'. For each tense t h e r e is a relation between m a x i m a l propositions, w h i c h m a y be w r i t t e n a --~ b, meaning t h a t the future of Ub relative to the tense ~ is constructed from Ua by the above algorithm. The latest tense in A we shall denote by z(A). Before we can present the semantics for the system, we need a proper definition of the full CIMP language. From M S C O P E t h e well formed formulae (wff) of the CIMP language can be defined by the following rules: (a) I f A e MSCOPE, t h e n A is a wff. (b) I f A e MSCOPE, then dual(A) is a wff. (c) I f A and B are wffs and S C c_MSCOPE, then -,4, (A A B), (SC: A > B) are wffs. (d) Nothing else is a wff.

(A v B ) and (A D B) are defined as usual by negations and conjunctions. The t r u t h operator can be p r e s e n t e d in the following way, w h e r e we shall let the maximal propositions play t h e r61e of instants as originally suggested by A. N. Prior: T(a,P(i,q)) iff P(i,q) • Ua T(a,F(i,q)) iff F(i,q) • Ua T(a,-A) iff not T(a,A). T(a,A A B) iff T(a,A) and T(a,B) T(a,q) iff q ~ Ua Let S e l ( C S , c,A) be the set of m a x i m a l propositions and aeperm(CS), where T(a,A) and c =~(A)a. Sc(A, CS, c, S C O P E ) is the set of propositions in Sel(CS, c,A), which are m a x i m a l with respect to an ordering relation <SCOPE, defined in t h e following way:

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a <scoPE b ~def (Uc m (Ua \ sub(SCOPE, U~)) c Ub It is easy to see that if a<SCOPE b then

(Uc ~ (Ua \ sub(SCOPE, Ua)) ~_ (Uc n (Ub \ sub(SCOPE, Ub)) The s t a t e m e n t a<scoPE b clearly m e a n s t h a t everything which is true according to a, but outside SCOPE, is also true according to b. We now define w h a t it means in CIMP for the counterfactual implication (SCOPE: a>b) to be true relative to the set of causal statements CS and the true chronicle c:

Definition: If the evaluation of B d e p e n d s on the f u t u r e course of events relative to ~(A), then T(c, SCOPE: A>B) is defined as: VaeSc(A, CS, c,SCOPE): T(a,B). Three things should be noted about this definition: (a) If no S C O P E is mentioned explicitly, the scope has to be constructed from the basic elements in A and B, i.e. the natural

scope for A > B. (b) If we w a n t to draw counterfactual consequences regarding the f u t u r e relative to ~(A), we have to u s e the algorithm for constructing the selected futures. (c) The definition does not cover cases where the counterfactual consequence B depends only on p a s t or present events relative to ~(A). Such cases are indeed difficult to express in a linguistically natural manner. One example could be (i) If the Germans had not lost the battle of the Marne, t h e n t h e y would not have t r a n s f e r r e d troops to the R u s s i a n Front (before that battle). It would seem, however, that conceptually we are inclined to u n d e r s t a n d such cases as meaning the same thing as (ii) I f the G e r m a n s had not t r a n s f e r r e d troops to t h e R u s s i a n Front, then they would not have lost the b a t t l e of the Marne.

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B u t such a contra-position, that is, the view t h a t (i) and (ii) are equivalent, can be questioned. On the other hand, if (i) is considered as different from (ii), it does express an odd proposition, w h i c h suggests t h a t t h e past relative to t h e a n t e c e d e n t is not settled. This issue is discussed in [Pedersen et al. 1994]. However, h e r e we choose to h a n d l e only t h e ' s t a n d a r d case', w h i c h is covered by the above defmition. CIMP differs from Lewis' classical ideas in several ways. First of all CIMP systematically takes t i m e and tense into account, w h e r e a s Lewis [1973] m a k e s no explicit reference to time in his definitions. Secondly, CIMP in general incorporates the notion of a scope as in (SCOPE: A>B). Lewis' logic, in contrast, only deals with t h e logic of A > B (corresponding to t h e cases, in w h i c h the 'natural scope' is used in CIMP). Thirdly, some of the a x i o m s of Lewis' s y s t e m are not valid in CIMP. Let us for example consider two axioms from [Lewis 1973/1986, p. 132]: (Lewis 1) (A > B) ~ (A ~ B) (Lewis 2) (A A B) ~ (A > B) It is not difficult to verify t h a t (Lewis 1) holds in CIMP: Theorem: (A > B) ~ (A ~ B) is valid in CIMP Proof." We have to prove t h a t T(c,(A > B) ~ (A ~ B)) for any c. This is proved by reductio ad absurdum. (assumption) (1) -T(c,(A > B) ~ (A ~ B)) (2) T(c~A > B) (from 1) (3) -T(c,A ~ B) (from 1) (4) T(c,A) (from 3) (5) -T(c,B) (from 3) (6) VaeSc(A, CS, c, SCOPE): T(a,B)) (from 2, 5 & def.) (from 4) (7) ceSc(A, CS, c, SCOPE) (from 6 and 7). (8) T(c,B) This contradicts (5). - Q.E.D.

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This means that counterfactual implication is in fact stronger t h a n m a t e r i a l implication (given t h a t the converse does not hold). - (Lewis 2), however, is not valid in CIMP. Let us as an illustration consider the causal statement: CS= {causal({q},F(1,-r))] a n d t h e t r u e chronicle: Uc = {p,-q} cJ {F(1,r)}. This basis obviously gives rise to t h e following permissible chronicles: ua = [p,q] • [F(1,-r)] Ub = [-p,q] ~ [g(S,-r)] Ud = {~p,~q} ~ {F(1,r)} Ue = [-p,-q] ~ IF(l, ~r)] Uf= {p,-q} cJ {F(1,-r)} Uc = {p,-q} ~ {F(1,r)}

The following statement is an instance of(Lewis 2): ((p v q) A F(1,r)) ~ ((p v q) > F(1,r)) In the CIMP evaluation of this statement it is easy to see t h a t T(c, (p vq) ~ ( 1 , r ) ) , b e Sc(p vq, CS, c,{p,q,F(1,r)]) and ~ T(b,F(1,r)). So, (Lewis 2) is clearly not valid in CIMP. It should also be emphasised that the logic of counterfactual implication is non-monotonic, t h a t is, (A A C) >B cannot be deduced from A > B. Similarly, we cannot in general deduce the proposition A > (B v C) from A > B. David Lewis has provided several important results regarding the completeness and decidability of his system(s). CIMP is not so well developed. We h a v e no genuine axiomatics for the system. On the other hand, the obvious computability of the logic should m a k e CIMP a proper candidate for t h e modelling of counterfactual reasoning in artificial intelligence and n a t u r a l language understanding. We intend to illustrate this by a closer s t u d y of our 'dinner scenario'. Before doing this, it will be useful to extend the system such as to deal with modalities also. We add these definitions to the system:

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T(a, NA) -~def ~o: same_past(a,b) ~ T(b,A) T(a, MA) -%lef-Tb: same_past(a,b) A T(b,A) where same_past(a,b) means t h a t Va,-v

Ua, o = Ub,-o u . . .

Ub, o

With these definitions it is obviously possible to distinguish b e t w e e n A > B , A > MB, andA > NB.

ANALYSIS OF THE DINNER EXAMPLE Let us consider the dinner scenario example again. Clearly Joe and Jim both t a k e the following facts for granted:

P(1,shopping) : 'Joe has been to the shop one time unit ago to buy the ingredients'

ingredients: 'The ingredients are available' cook: 'The cook is present' el-failure: 'The electrical failure occurs' F(1,- dinner): 'Dinner will not be served during the next period of time'. The list of t h e s e 5 basic statements constitutes the true history in this example. Joe and Jim also both assume a number of causal statements, which can be presented graphically in t h e following way:

P(1 ,shopping)

ingredients cook - el_failure

F(1 ,dinner)

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This model could also be presented as the following database:

causal([ingredients,cook, ~el-failure},F(1,dinner)). causal({P(1,shopping)},ingredients). Now, how should the first statement Joe: If it h a d not been for the failure, we would have h a d dinner on time be e v a l u a t e d w i t h i n this causal model? In symbolic form t h e statement can be formulated as:

~ el-failure > F(1,dinner) w h e r e el-failure and F(1,dinner) are propositional c o n s t a n t s with the obvious meanings. Clearly, in this case only two factors are relevant. The scope of the above s t a t e m e n t can be represented by t h e set {el-failure, F(1,dinner)}. which gives rise to three permissible chronicles: H I . {el-failure, F(1,dinner)} H2. {~el-failure, F(1,dinner)} H3. {el-failure, F(1,-dinner)} (to each of these chronicles one m u s t add the invariant set {P(1,shopping), ingredients, cook}, to obtain t h e full permissible history). -

It is easy to see that the following counterfactual holds:

~el-failure > F(1,dinner) The next s t a t e m e n t is Jim: You're wrong. If the cook left, you might still not h a v e had dinner on time, even in the absence of the failure.

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In symbols: (~ c o o k A - e l - f a i l u r e ) > M F ( 1 , - d i n n e r )

This obviously holds in t h e model, and indeed, t h e following stronger statements also holds: If t h e cook left, you would not have had d i n n e r on time, even in t h e absence of the failure. In symbols: (- cook A ~ el-failure) > F(1, ~dinner)

T h e t r u t h of this counterfactual implication can be seen by inspection into this picture: [ingredients, cook, el_failure]

/4

~~.~'-"'""~

[ingredients, cook, ~ el_failure]

~

dinner)] [F(1, [F(1, dinner)]

[F(f, dinner)]

[P(1, shopping)] ~\

\ ~ [ingredients, ~ cook, el_failure] ~ . . . , , . . \

IF(l, ~ dinner)] [F(1, dinner)]

\

[ingredients, - cook, ~ el failure] ~ -

-'-~ ~"=~

[F(t, - dinner)] [F(1, dinner)]

Joe's next s t a t e m e n t in t h e conversation, "what I m e a n is t h a t if t h e r e were no failure a n d the cook agreed to stay, we would have h a d d i n n e r on time", is simply the counterfactual (cook A - e l - f a i l u r e ) > F ( 1 , d i n n e r )

w h i c h clearly holds.

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However, consider the following answer: Jim: You're wrong again. If you had not been to the shop to buy the ingredients, you would not have had dinner on time, even if the cook stayed and there were no electrical failure. In symbols: (cook A - ingredients A - el-failure A -P(1,shopping)) > F(1,-dinner))

Here the scope is changed such t h a t in principle, 15 permissible c h r o n i c l e s are t a k e n into c o n s i d e r a t i o n . The o n l y chronicles corresponding to a t r u e a n t e c e d e n t of the counterfactual in question are: HI: {P(1,-shopping), -ingredients, cook, -el-failure, F(1,dinner)] H2: P ( 1 , - s h o p p i n g ) , - i n g r e d i e n t s , cook,-el-failure, F(1, - d i n n e r ) ] By our definitions, H2 is selected in this case. The validity of this counterfactual can be verified by inspection into the above diagram. - The next statement is Joe: I don't agree. Somebody else might have brought the ingredients. As long as the cook has got the ingredients and there is no electrical failure, we shall with necessity h a v e dinner on time. In symbols: (cook A - el-failure A P(1,somebody) > N F ( 1 , d i n n e r )

The evaluation of this statement involves an expansion of t h e list of facts with the fact - P ( 1 , s o m e b o d y ) . Obviously, this also means that the scope is expanded. The causal model is expanded with the statement:

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causal([P(1,somebody)},ingredients). O n c e more, we can c o n s t r u c t all t h e p e r m i s s i b l e chronicles, a n d it c a n t h e n be v e r i f i e d t h a t t h e s t a t e m e n t is t r u e (F(1,dinner) h o l d s in all t h o s e c h r o n i c l e s in w h i c h t h e a n t e c e n d e n t is true, so this consequence is true w i t h necessity). T h e n e w causal model can be p r e s e n t e d g r a p h i c a l l y in t h e foll o w i n g way:

I P(1,somebody) I P(1.shopping) Ingredients ~ D . Cook ~ . -el-failure

~i F(1,dinner)!

T h e n e x t move in t h e d e b a t e is similar from a formal point of view; it involves a n e w model with t h e fact unchanged-mind, a n d a n extended causality relation: J i m : Well, t h e cook m i g h t change his m i n d b e c a u s e of all y o u r quarrelling. Again, in that case t h e r e w o u l d not be dinn e r on time, even if t h e r e were no electrical failure.

I P(1,somebody)~ I P(1,shopping)

w

Ingredients unchanged-mind . ~ Cook ;~el-failure

F(1,dinner)I

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With the new scope and the new model it can easily be verified that ( - el-failure A - u n c h a n g e d - m i n d ) > ~ F ( 1 , d i n n e r )

In the next part of the conversation yet a n o t h e r new complex is introduced: Joe: Well, we might have had dinner on time, if I had ordered some ready-made dinner. The electrical failure could not have prevented the dinner in that case.

P(1,somebody)

I readymaded.

I P(1,shopping)

r

Ingredients ~nchanged-mind ~ 3ook -el-failure

F(1,dinner)]

With this new complex and the new fact: - r e a d y - m a d e _ d . , can be shown t h a t

it

(el-failure A r e a d y - m a d e _ d ) > N F ( 1 , d i n n e r )

holds for the scope in question (we take 'could not...prevent' as indicating necessity). The conversation is concluded by the statement Jim: I t h i n k you're wrong again. The cook is a union man. He will bar the door to the delivery of the r e a d y - m a d e dinner, since the ready-made dinner company t a k e s away jobs

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CHAPTER 3.4 from proper cooks. But if the cook leaves, the ready-made dinner can be delivered.

Here, the causal model becomes even more complicated:

I P(1,somebody) I P(1,shopping)

ready made

d.

~ cook ingredients

unchanged-mind ~D=,,-

F(1,dinner) I

;ook -el-failure

As the definition of an INUS-condition suggests, there can be more causal complexes for an effect. The sets of conditions of two causal complexes for some effect E m a y be disjoint, but t h e y m a y also have a non-empty intersection. It is interesting t h a t in the present model the proposition cook as well as its negation -cook appear in two parallel complexes. With the a s s u m e d scope and the above model the counterfactual

(-cook A ready_made_d) > NF(1,dinner) is true. - One might now fear that it holds t h a t

(cook > F(1,dinner)) A (-cook > F(1,dinner)) since cook and -cook are both INUS-conditions in complexes leading to the s a m e effect F(1,dinner). However, with the true history

[P(1,shopping),ingredients, cook, el-failure,-ready_made_d, F(1,- dinner)}

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and t h e described causality relation, t h e counterfactual

-cook > F(1,dinner) does n o t hold. This can be seen by considering t h e permissible history

{P(1,shopping),ingredients,-cook, el-failure,-ready_made_d, F(1,- dinner)}.

PERSPECTIVES As we have already suggested we do n o t consider the CIMP system to be in any sense finished. On the other hand, we believe t h a t ideas like t h e ones proposed here should be included into a g e n e r a l a c c o u n t of c a u s a l and c o u n t e r f a c t u a l r e a s o n i n g . M o r e o v e r , we t h i n k t h a t t h e s e i d e a s h a v e f a r - r e a c h i n g consequences. The dinner scenario is clearly a toy example, b u t it exhibits some p a t t e r n s of r e a s o n i n g w h i c h are crucial for m a n y ( p a r t l y o v e r l a p p i n g ) issues w i t h i n real i n f o r m a t i o n systems:

• artificial intelligence: it is clear that CIMP implements several kinds of reasoning, and moreover, t h a t it m a i n t a i n s an affinity to n a t u r a l l a n g u a g e , w h i c h also m a k e s it a c a n d i d a t e for n a t u r a l language u n d e r s t a n d i n g ([Hasle a n d OhrstrCm 1992]); • updating databases: the non-monotonic g r o w t h of information in databases is in fact a very general problem, having to do w i t h how to m a i n t a i n consistency. The n o n - m o n o t o n i c features of C I M P i l l u s t r a t e t h i s k i n d of problems, a n d some w a y s of h a n d l i n g t h e m are suggested; • planning: the construction of a n u m b e r of futures with respect to a notion of relevance (scope) and a set of u n d e r l y i n g causal

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assumptions is essential in m a n y planning systems (in chapter 3.7, we take up the issue of time in planning systems at a more general level); • diagnosis: the fact that the techniques of CIMP are directly applicable to diagnosis has been investigated at some length, e.g. in [Pedersen et al. 1994]. Here it is shown that the CIMP-logic can be used with considerable generality for error diagnosis as well as medical diagnosis; • decision support and judicial systems: the relevance in decision support is an obvious consequence of the points above. - One i n t e r e s t i n g application of CIMP shows its m o r e specific r e l e v a n c e in judicial systems. Anne R a s m u s s e n [1993] has a n a l y s e d the Danish authorities' official report on t h e fire catastrophe on the ferry boat Scandinavian S t a r (which took place on the 7th of April 1990). In h e r analysis of t h e report, R a s m u s s e n applied the CIMP notions and t h u s obtained a logical - as well as computable - model of the reasoning used. This k i n d of r e a s o n i n g s t u d i e d is f u r t h e r m o r e u s e d for d e t e r m i n i n g t h e responsibility for w h a t w e n t wrong. The conclusions of the report are in t h e real world u s e d both for deciding how to avoid similar events in the f u t u r e , and for deciding upon verdicts in the case. Since CIMP can model such reasoning, it could be used for similar decision-making purposes (although we do not recommend t h a t verdicts be based upon it).

Thus, we are suggesting t h a t the CIMP-ideas are highly versatile for information systems. If t h a t is so, the reason for this is really to be found in its temporal nature r a t h e r t h a n in any special other merits which it might have. Conceptually, CIMP formalises and unifies a n u m b e r of features of reasoning within a temporal framework - specifically, a metric b r a n c h i n g time model. Therefore, the specialised study of this c h a p t e r in fact also holds more general implications for the concept of time and its importance.

3.5. LOGIC OF DURATIONS T h e logics studied so far h a v e mainly been based on some notion of t e m p o r a l i n s t a n t s - r a t h e r t h a n d u r a t i o n s . Of course, we have also seen some exceptions to this rule, for instance J o h n Buridan's t h o u g h t s p r e s e n t e d in chapter 1.5, or Zeno Vendler's distinctions m e n t i o n e d in c h a p t e r 1.6. The p r e v a l e n c e of instant-based logics has not been left unchallenged, though: from a fairly early stage of t h e d e v e l o p m e n t of formal s e m a n t i c s for n a t u r a l languages it has been a r g u e d almost v e h e m e n t l y t h a t h u m a n l a n g u a g e and r e a s o n i n g call for an i n t e r v a l - b a s e d r a t h e r t h a n point-based semantics. This was e m p h a s i s e d a n d formally elaborated in [Dowty 1979], which forms a milestone in this respect. Similar discussions, albeit from r a t h e r different perspectives, have been going on w i t h i n philosophical logic as well as c o m p u t e r science. F o r instance, in philosophical logic P e t e r S i m o n s [1987] h a s carried out some careful s t u d i e s of 'temporal parts' and Antony Galton h a s - with reference to conceptual structures in n a t u r a l l a n g u a g e - argued t h a t t e m p o r a l logic should take heed of durations, and worked out proposals for m e e t i n g t h i s r e q u i r e m e n t [Galton 1984]. S i m i l a r c o n s i d e r a t i o n s have b e e n p u t f o r w a r d w i t h i n a r t i f i c i a l intelligence research, notably by Allen [1983, 1984], Allen a n d Hayes [1985, 1989], a n d P e t e r L a d k i n [1987]. In theoretical c o m p u t e r science, Ben Moszkowski [1983], Roger Hale [1987] a n d o t h e r s have s h o w n t h a t at least for some p u r p o s e s durational logic offers more t h a n instant logic. T h e r e can be no doubt t h a t a fully a d e q u a t e t e m p o r a l logic m u s t be able to account also for durations; and the idea t h a t durations should take conceptual priority over i n s t a n t s is w o r t h considering. But t h a t does n o t m e a n t h a t i n s t a n t - b a s e d approaches m u s t be discarded altogether. On the contrary, m a n y aspects of temporal concepts a n d p h e n o m e n a are best s t u d i e d w i t h i n such logics - if for no other reason, t h e n simply because t h e y are generally s p e a k i n g less complex. The fact t h a t mean i n g f u l and fruitful studies can be conducted w i t h i n i n s t a n t -

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based frameworks should have become quite clear from previous discussions. In fact, the logic of durations was studied even some years before Prior's rediscovery of tense logic. The first modern logician to formulate a calculus in this field was A. G. Walker [1947]. Walker was, however, not concerned with temporal logic in any general sense. Walker considered a structure (S,<), where S is a non-empty set of periods. This set is ordered by a partial ordering relation '<', analogous to the before-after-relation among instants. Two interesting and related aspects of this model should be mentioned right away: first, it does not seem counterintuitive to call one period 'earlier' t h a n another one, even if they 'overlap'. Thus 'Mary opened the door before John rushed in' seems quite right, even if John begins his rushing in before Mary concludes h e r opening the door. Nevertheless, the 'a
a/b ~ef ~(a
(Wl) (W2)

ala (a
In relation to these axioms Walker was able to construct a settheoretic structure of triplets (A,B, C), where A, B, and C are all sets of durations such that 1) A and B are non-empty 2) the union of A, B and C is the set of all durations 3) every element in A is before every element in B

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4) every e l e m e n t in C is overlapping some e l e m e n t in A as well as some element in B. Walker d e m o n s t r a t e d t h a t t h e structure of these triplets has all t h e algebraic properties which we would intuitively expect t h e s t r u c t u r e of t e m p o r a l i n s t a n t s to have. For t h i s reason it m a y be r e a s o n a b l e to view a t e m p o r a l i n s t a n t as s u c h a 'secondary' construct from the logic of durations. As we have seen C. L. H a m b l i n contributed significantly to the d e v e l o p m e n t of t e m p o r a l logic in its very early period. More t h a n two decades later t h a n Walker, H a m b l i n [1972] also p u t forth a theory of t h e logic of durations. Hamblin w a s not aware of Walker's work w h e n he developed his theory [Hamblin 1972, p. 331], but he achieved some similar results u s i n g a different technique. H a m b l i n also considered a f u n d a m e n t a l s t r u c t u r e consisting of a set of d u r a t i o n s w i t h a partial ordering relation (S,<). In addition he defined t h e following relations for arbitrary d u r a t i o n s , w h e r e (al b) m a y be read 'b follows i m m e d i a t e l y aider a', and (a-/b) may be read 'a is contained in b'.: a$ b -%lef(a
azb ~f

Vc: (c/a ~ c / b )

U s i n g the definition of as b, H a m b l i n could also offer a derived notion of an instant: Any pair of d u r a t i o n s (a,b) uniquely defines an i n s t a n t if a n d only if (a$ b). We shall use expressions like as b~ c for the conjunction of a l b and b$ c. Hamblin's axioms can be formulated in t h e following w a y u s i n g our notation (and omitting external u n i v e r s a l quantification): (Hamblin 1): -(a
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CHAPTER 3.5 (Hamblin (Hamblin (Hamblin (Hamblin (Hamblin (Hamblin (Hamblin

4): (a$ c A as d • bs c) ~ b$ d 5): (as bs d A as cs d) ~ b=c 6): _Wo:a
It is interesting t h a t Hamblin 9-10 express two features, w h i c h are also known from lattice-based theories of mass t e r m s [Link 83] a n d e v e n t s t r u c t u r e s [e.g. B a c h 1986, L i n k 1987]. Specifically, (Hamblin 9) states a kind of dissectiveness: if some proposition p 'is t r u e with respect to' some interval a, a n d b is contained in a, t h e n p is t r u e also w i t h respect to b. We m i g h t also say that this expresses 'downwards inheritance'. In a d u a l m a n n e r , ( H a m b l i n 10) expresses a sort of c u m u l a t i v i t y . However, it is well known, at least from later literature on durations, t h a t not all 'properties' of d u r a t i o n s behave like this: t h u s for instance, an 'accomplishment' like 'Mary baked a cake' (say, from 1 p.m. to 4 p.m.) does not entail that Mary b a k e d a cake d u r i n g t h e sub-periods, say, from 2 p.m. to 3 p.m. (Note t h a t even t h o u g h it m a y be t e m p t i n g to say t h a t Mary w a s 'engaged in the process' also during all sub-periods, she certainly did not accomplish it d u r i n g any of those). It is therefore clear t h a t H a m b l i n ' s t h e o r y is confined to c e r t a i n s u b s e t s of (properties of) durations. D u r i n g the last decade various kinds of durational logic h a v e been studied and applied within artificial intelligence r e s e a r c h and n a t u r a l l a n g u a g e u n d e r s t a n d i n g (usually u n d e r t h e heading 'interval semantics', which seems more popular in this scientific community). Two researchers in this field, who h a v e contributed significantly to the development of durational logic, are J a m e s Allen and Patrick J. Hayes [1985, 1989]. Like Walker and Hamblin, Allen a n d Hayes have t a k e n as their s t a r t i n g point t h e study of the structure of t h e partially ordered set of durations. They have suggested an axiomatic system, w h i c h we

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reformulate as (AH 1-5). We shall use ~ for a generalised k i n d of 'exclusive disjunction', i.e. (Pl v--P2 ) ~--(Pl --P2).

A generalised definition can be given as (191 ~ ... Y-PN) -~-defA i = I . . N P i -~ (-P l A ... A--pi. I A --Pi+1 A ... A--pN)

The axioms are: (AH1) (AH2) (AH3) (AH4)

(AH5)

(as C A a S d a b s c) D b s d (as b A Ca d) ~ (as d v 3e:a$ es d v ~.'cs fs b) :Yo, c: bs as c (as bs d A aS ca d) ~ b=c as b ~ 3e Vc, d: (cs a s bs d ~ ca es d)

This axiomatic s y s t e m obviously takes t h e s-relation as t h e primitive. However, this does not constitute any essential step away from Hamblin's system, in which the opposite implication of (Hamblin 3) can easily be proved. We therefore have as a theorem (Hamblin 3'): a
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shall call this resulting duration the s u m of a and b, i.e. e=a+b. However, we point out t h a t this sum-operator is not commutative a n d is in effect a kind of concatenation rather t h a n a 'usual' sum-operator. Allen and Hayes have shown that two arbitrary durations can be r e l a t e d in exactly 13 different ways, which can all be expressed solely in terms of the $ -operator and equality: a a a a a a a a a a a a a

meets b -~-defas b is met by b - ~ f bl a is before b -%~f3c: a$ c$ b is after b -~lef ~c: b$ ci a starts b -~-def_~c: b=a+c is started by b -~lef 3c: a=b+c finishes b -~-def:=~c:b=c+a is f i n i s h e d by b -~ef =:~c:a=c+b overlaps b ~def 3c, d,e: a=c+d A b--d+e is overlapped by b -~-def-~c,d,e: (b=c+d A a=d+e) d u r i n g b ~---def--~C,~b=c+a+d contains b -~lef 3c, d: a=c+b+d equals b -~defa=b

It is v e r y illuminating to study various combinations among these 13 relations. Using Allen's and Hayes' axiomatisation, it is possible to implement a reasoning system, by m e a n s of which s t a t e m e n t s like I f a overlaps b and b is started byc, then a overlaps c; I f a finishes b and b starts c, then a during c; I f a during b and b overlaps c, then a is not met by c; can be proved. This kind of reasoning will be important in any system, which should be able to perform or simulate commonsense reasoning involving time periods. Consider for instance this situation: M a r y was reading during the postman's visit. J o h n finished his beer just when the postman left.

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It is clear t h a t J o h n ' s a n d Mary's respective activities are n o t explicitly r e l a t e d by the above statements. On t h e other h a n d , it is also clear t h a t a certain t e m p o r a l relation m u s t o b t a i n between t h e m . T h e information in this scenario can be c a p t u r e d in durational logic in terms of the following two statements: The p o s t m a n ' s visit takes place during Mary's reading. The p o s t m a n ' s visit finishes or is finished by John's d r i n k i n g the beer. In t h e logic of durations it can be formally d e m o n s t r a t e d from t h e s e s t a t e m e n t s t h a t t h e d u r a t i o n d u r i n g w h i c h ' J o h n is drinking t h e beer' is during, overlaps, or starts 'Mary's reading'. - The t a s k of carrying out all such kinds of reasoning about durations is by no means simple (see also [Knight & Jixin 1992]). We h a v e a l r e a d y pointed out t h a t for some d u r a t i o n s , or p e r h a p s r a t h e r , certain types of events, there are no sub-parts; for instance, if 'John opened t h e door' d u r i n g some period a, it will not be t r u e to say t h a t J o h n opened the door during a n y subinterval b contained in a. In this case, dissectiveness does not obtain (cf. H a m b l i n 9). W h e n r e a s o n i n g about d u r a t i o n s we often come across durations w i t h o u t parts corresponding to for example o p e n i n g a door. Allen's and Hayes' reason for excluding in general t h e axiom (Hamblin 8) is precisely t h a t they w a n t to study t h e s e so-called 'moments', which can be u n d e r s t o o d as d u r a t i o n s w i t h o u t any i n t e r n a l s t r u c t u r e (not to be confused w i t h ' i n s t a n t s ' ) . It a p p e a r s t h a t n o t h i n g is c o n t a i n e d in a m o m e n t , a n d t h a t two m o m e n t s cannot overlap each other. H a m b l i n ' s (as well as Allen's and Hayes') d u r a t i o n a l logic is based on a conception of durations as something similar to real intervals. A n u m b e r of interesting theorems can be proved from Hamblin's axioms, but t h e s y s t e m is not sufficient to e s t a b l i s h t h a t linear intuition about time on which it is obviously based. The reason for this is t h a t there is nothing in (Hamblin 1-8) to exclude a g e n u i n e b r a n c h i n g time model. On t h e other h a n d , if t i m e s h o u l d in fact be conceived as b r a n c h i n g , t h e n t h e

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'containment'-relation z in the above axioms will yield some very strange results, and will be r a t h e r far from the inclusion relation that Hamblin probably had in mind. P e t e r RSper [1980] has developed a more free-grained logic from very much the same intuition as Hamblin's. RSper starts from a non-empty set S of durations and a relation c defined on S, w h i c h should express the 'inclusion' relation among durations. RSper defines a P-frame as a structure (S,~<) satisfying: (A1) (A2) (A3) (A4.1) (A4.2) (A5.1) (A5.2)

Ifx
Obviously, (A5.1) corresponds to forwards linearity, whereas (A5.2) ensures backwards linearity. On the other hand, there is nothing in RSper's system to ensure the irreflexivity of the ordering relation. Some of the further details of RSper's system are mainly concerned with that distinction between dissective and non-dissective 'events', which we have already suggested. We shall recapitulate the main problem by considering the following two propositions: P:

q:

'Percival drinks a pint of bitter' 'Araminta is in Oxford'.

Let us assume t h a t both propositions are true for a duration a, and let b be an arbitrary sub-duration, i.e. b _ca. Then a proposition such as q will also be true for the d u r a t i o n b. Following

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311

RSper, we shall say t h a t q is persistent (i.e. dissective). This can be symbolically expressed as: (T(a,q) A b c_a) m T(b,q)

A p e r s i s t e n t proposition denotes 'a property' in Allen's terminology. On t h e other hand, a proposition such as p may be false for some or all sub-durations. That is, it is in general conceivable t h a t for some sub-duration b, the following formula holds: T(a,p) A b c a A ~T(b,p)

W i t h o u t doubt, this is t r u e for our present example. Suppose t h a t Percival d r a n k one pint of bitter, b e g i n n i n g at 11:30 a.m. and finishing at 11:40 a.m. Then it is false t h a t he dr~nk one pint of b i t t e r d u r i n g t h e s u b i n t e r v a l from 11:35 to 11:36. - Allen reserves t h e t e r m 'an event' for propositions of this type. T h e distinction between these two types of propositions is central for any a t t e m p t at establishing an adequate durational logic. It is evident t h a t Hamblin's theory (cf. H a m b l i n 9-10) is about w h a t Allen has called properties, that is, persistent propositions. RSper, however, m a k e s a distinction between t h e logic of w h a t he h a s called h o m o g e n e o u s sentences a n d t h e logic of 'other sentences'. According to RSper a sentence p is h o m o g e n e o u s if and only if it is 1) persistent (dissective) a n d 2) cumulative (i.e. for any a, i f p is true for all sub-durations of a, t h e n p is t r u e for a). To avoid terminological confusion, let u s r e c a p i t u l a t e t h e major distinctions: in chapter 1.6, we briefly described Zeno Vendler's famous four-fold distinction between verb phrases. It seems fair to say, though, that the main distinction is between (a) predicates, which are dissective and cumulative, and (b) predicates, which describe 'non-divisible' p h e n o m e n a . The former we have called persistent; Allen calls t h e m properties, a n d RSper collects t h e m u n d e r t h e h e a d i n g homogeneous. T h e latter are (somewhat misleadingly) called events by

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CHAPTER 3.5

Allen. F u r t h e r refinements of these distinctions may be found in the work of Allen as well as other authors. RSper's w a y of assigning t r u t h - v a l u e s to homogeneous sentences closely follows the intuitions embodied by (Hamblin 9-10). RSper introduces a valuation function V from pairs consisting of a propositional variable and a time period into the t r u t h values {0,1], in such a w a y t h a t his V(p,a)=l corresponds to Hamblin's T(a,p). RSper defines a P-model b a s e d on the P-Frame (S,c,<) as a structure (S,~<,V), where the V-function satisfies (i) (ii)

If V(p,x)=l and y c_x, then V(p,y)=l If for e v e r y y c x there is a z such that z c y and V(p,z)=l, then V(p,x)=l

The truth of a wff relative to the P-model is defined as

(P1) (P2) (P3) (P4) (P5) (P6) (P7)

I f p is a propositional variable, then P~=xPiff

V(p,x)=l P~x~A ifffor ally c x , not P~yA P /:=xAA B iff P~xA and P]=xB P/=xA v B ifffor e v e r y y c x , there exists a z ~ y such t h a t P~z A or P/=zB Pt=xA ~ B ifffor every y _~x, i f P ~ y A , then P#:y B P~x GA ifffor every y ~ x and all z withy
It is possible to derive truth conditions for F and P:

P ~ x F A ifffor z<w such t h a t (P9) Pt=xPA ifffor w
(P8)

ally c x , there exists z c y and w with

P~wA ally _cx, there exists z _cy and w with

P~wA

RSper h a s formulated an axiomatic system corresponding to the P-models. One of the most interesting features is the fact t h a t the formulae

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A~PA A ~FA t u r n out to be P-valid (true in any P-model), and provable in the corresponding axiomatic system. RSper points out t h a t the presence of these theorems is a "perhaps unexpected feature of t h e system" [p. 459]. It is interesting t h a t J o h n Buridan in his durational logic m a d e a similar observation, as we have seen in c h a p t e r 1.5. He introduced a distinction between relative and a b s o l u t e tenses. The above theses would also be valid in Buridan's logic, provided t h a t the past and the future are understood as relative in this context. However, RSper's logic of h o m o g e n e o u s s e n t e n c e s contains no s i m i l a r distinction. Nevertheless, we shall show that Buridan's idea can also be incorporated into RSper's system. In order to construct a semantical model for the logic of nonh o m o g e n e o u s sentences, RSper has i n t r o d u c e d an E - f r a m e (S,~<) as a structure that satisfies the following conditions: (D1) (D2) (D3) (D4)

_c is reflexive. c is transitive. < is transitive. For any x, y, x',y', ifx
In addition, an E-model based on the E-frame (S,~<) is defined as a structure (S,c,<,V). V is a function from w f f s into {0,1}, which satisfies this condition: For any x in S, there exists a y c x such t h a t either V(p,z)=l for every z c y , or V(p,z)=O for every z _~y. The t r u t h of a wff relative to this E-model is defined as follows:

(El)

Ifp is a propositional variable, then

(E2)

E ~xp iff V(p,x)=l E~-x -A iffnot E/=xA

314

C H A P T E R 3.5 (E3) (E4)

(E5) (E6) (E7)

EI=xA A B iff Eb:xA and Et=xB Et=xA v B iffE]=xA or E~:~B E~:xA ~ B iff: (not E~xA) or E~=xB E]::~ GA iff for e v e r y y w i t h x
So far t h e s e m a n t i c s is exactly as it would be for an 'instantoriented' t e n s e logic. H o w e v e r , in order to express a d u r a t i o n a l e q u i v a l e n t of linearity and d e n s i t y w e need two e x t r a operators, called L a n d W. T h e s e o p e r a t o r s a r e defined by t h e following s e m a n t i c properties:

(E8) (E9)

E ~ x L A ifffor e v e r y y w i t h y ~_x, E/=yA E/=x WA iff for s o m e y w i t h x _cy, E ~ y A

C o n s i d e r n o w forwards and b a c k w a r d s linearity: F o r a n y d u r a t i o n s x, y, z, i f x < y and x
(Fp AFq) ~(F(p AFq) vF(Fp Aq) vF(Wp A Wq)) (Pp A Pq) ~ (P(p A Pq) v P(Pp A q) v P(Wp A Wq))

D e n s i t y is expressed as follows: F o r a n y d u r a t i o n x t h e r e a r e d u r a t i o n s y and z, such t h a t

y
Lp ~ MFp

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Lp ~ MPp

The t r u t h conditions for F and P can easily be derived: (El0) (Ell)

E#:xFA ifffor some y with x
(El0) and ( E l i ) obviously give rise to w h a t Buridan called an absolute understanding of the tenses. The relative tenses may be expressed as follows: (El2) (El3)

El=xFretA ifffor some z c x there is a y with z
It is easy to verify that Fret -=MF and Prel =-MP. - RSper has d e m o n s t r a t e d t h a t the axiomatic system for a minimal non-homogeneous durational tense logic, corresponding to a logic based on (D1-4), would be the axioms of Kt with the addition of these axioms: (MD1) (MD2) (MD3) (MD4) (MD5) (MD6) (MD7) (MD8) (MD9) (MD10) (MD 11) (MD12) and the rules

L(p ~ q) ~ (Lp ~ Lq) ~W~(p ~ q) ~ (Wp ~ Wq) WLp ~ p p ~ LWp Lp ~ p p ~ Wp Lp ~ LLp WWp ~ Wp Gp ~GGp Hp ~ H H p MGp ~ GLp MHp ~ HLp

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CHAPTER 3.5 (MDR1) (MDR2)

I f ~ p , t h e n J'Lp If I-p, t h e n I- ~ W - p

We may call this m i n i m a l system for durational logic DKt. In chapter 2.8 we have shown t h a t for i n s t a n t logic the u s u a l axioms for backwards and forwards linearity can in fact be proved as theorems, if we instead adopt some considerably simpler axioms. Similarly, we shall show t h a t (Axl-2) can be proved on the basis of DKt and some simpler axioms of linearity: (Axl') (Ax2')

(PFp ~ (Fp v P p v M W p ) (FPp ~ (Fp v Pp v M W p )

In order to do that we need some lemmas. L e m m a 1. DKt i- H(L~W~p ~ (Hp ~ q)) v H(Hq~p)) Proof." The proof is carried out by reductio ad absurdum. (1) ~(H(L~W~p ~ (Hp ~ q)) vH(Hq~p))) (ass.) (2) ~ H ( L - W - p ~ (Hp ~ q ) ) (1) (3) ~H(Hq~p)) (1) (4) P ( L - W ~ p A Hp A -q) (2) P(Hq A --p) (3) (5) HFP(Hq A -p) (5, A4) (6) P ( L - W - p A Hp A -q AFP(Hq A -p)) (4,6) (7) P((L-W~p A lip A -q A P(Hq A -p)) v (8) ( L - W - p /~ lip A -q A MW(Hq A - p ) ) V ( L - W - p A Hp A -q A F(Hq A -p)) ) (7, A7)

But (8) is clearly impossible since all the components in the disjunction are impossible. L e m m a 2. DKt t- (H(Wp ~ - W - q ) A H(p ~ Hq) A H(Pp ~ q) A Pp) ~Hq Proof." By substitution in Lemma 1 we fmd

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H(L-W~q ~ (Hq ~ ~p)) v H(H-p ~ q)) Therefore, the problem can be split into two cases: in the first case H(L~W~q ~ (Hq ~ ~p)) is assumed, and in the second H(H~p ~ q)) is assumed. In the first case we make the following derivation: (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12)

H(Wp ~ - W ~ q ) H(p ~ Hq) H(Pp ~ q) Pp H(L-W-q ~ (Hq ~ - p ) ) HL(Wp ~ ~W~q) H(LWp ~ L - W - q ) H(LWp ~ (Hq ~ ~p)) H(p ~ (Hq ~ ~p)) H(p ~ -p) ~P(p Ap) -Pp

(ass.) (ass.) (ass.) (ass.) (ass.) (1, MD12) (1, MD1) (5,7) (8, MD5) (2, 9) (10) (11)

(11) contradicts c o n t r a d i c t s (4). This m e a n s t h a t the assumptions in the antecedent rule out the first case. In the second case, we can argue in the following way: (1) (2) (3) (4) (5) (6) (7) (8) (9) Q.E.D.

H(p ~ q) H(p ~ Hq) H(Pp ~ q) Pp H(H~p ~ q)) H(~q ~ Pp)) H(~q ~ q)) ~P(-q A --q)) Hq

(ass.) (ass.) (ass.) (ass.) (ass.) (5) (6 and 3) (7) (8)

Now, (Axl) can be proved from lemma 2 in the following way:

318

CHAPTER 3.5 ( H ( W p ~ - W - q ) A H(p ~ Hq) A H ( P p ~ q) A Pp) ~ Hq ( H ( W p ~ - W - q ) A H(p ~ Hq) /, H ( P p ~ q) ) ~ (Pp ~ Hq) ~(Pp ~ Hq) ~ ~(H(Wp ~ - W - q ) A H(p ~ Hq) A H(Pp ~ q)) (Pp A P - q ) ~ (P(Wp A W - q ) v P(p A P - q ) v P(Pp A - q ) )

From this (Axl) can be obtained by substitution. In a similar w a y it is possible to prove (Ax2) in a system enlarged w i t h (Ax2'). - In this way we have demonstrated that DKt ~ { A x l ' } I- A x l , and DKt ~ { A x 2 ' } I- A x 2

It can also be demonstrated that DKt u [Axl] I- A x l ' , and DKt ~ {Ax2} !- A x 2 '

This m e a n s t h a t it is in fact a m a t t e r of choice w h e t h e r we w a n t to use {Axl,Ax2} or {Axl',Ax2'} to express linearity. As we have seen RSper uses { A x l ~ 2 } , but Axl' and Ax2' are obviously simpler t h a n Axl and Ax2. It is worth considering whether t h e r e is a durational parallel to Prior's idea of instant propositions. In fact, (E2) corresponds to this Priorian postulate within the third grade: T(x, ~p) - ~T(x,p)

and t h a t (E3) corresponds to T(x,p A q) --(T(x,p) A T(x,q)).

For t h a t reason this logic can be t r e a t e d along the lines of Prior's third grade, now using 'duration propositions' instead of i n s t a n t propositions. Moreover, this can in principle be done not only w i t h a linear conception of time, but also for a logic of durations based on a branching time. On the other hand, it m u s t be recalled t h a t we are currently dealing only with a n o n homogeneous durational logic.

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As we mentioned at the beginning of this chapter, a n u m b e r of researchers have argued that representations of time based on intervals are more n a t u r a l than instant-based systems. Even if the a r g u m e n t s for the conceptual priority of durations over m a t h e m a t i c a l i n s t a n t s seem convincing, it m u s t also be admitted t h a t the notion of durationless instants is very useful in m a n y cases - especially for the description of change. For this reason, the idea of constructing instants from durations - as proposed by Walker and Hamblin - is of considerable interest. Such constructions m a y turn out to be very important, if t h e y can be shown to give rise to a full instant-logic. But it should also be mentioned t h a t the project of durational logic, respectively interval semantics, has been attacked. T h e r e can be no doubt t h a t for n a t u r a l language some idea of durations is required - and hence, t h a t conceptually one m u s t (sometimes) speak of durations. But t h a t certainly does not prove t h a t t h e desired notions of duration could not be built within an instant-logic, a point which has been put forcefully and elegantly by P. Tichy [1985] as well as A. Galton [1990]. This issue cannot be expected to be settled soon, but regardless of the priority b e t w e e n these two conceptions, both approaches are useful for the general study of time.

3 . 6 . GRAPHS FOR TIME AND MODALITY As we have mentioned in part 2, C.S. Peirce established a calculatory technique of logical graphs. These so-called existential g r a p h s have been studied carefully by computer scientists and others for some years. Since the beginning of the 1980's, J o h n Sowa [1984] and others have tried to systematise a m o d e m version of Peirce's existential graphs - and indeed, to implement it c o m p u t a t i o n a l l y . (A s o m e w h a t different but also h i g h l y i n t e r e s t i n g m o d e m i s a t i o n is H a r m e n van d e n Berg's 'Knowledge Graphs' [1993a].) It seems t h a t the m o d e m version of t h e Peircean graphs k n o w n as 'conceptual graphs' is useful w i t h i n artificial intelligence in a broad sense - i n c l u d i n g interfaces to databases, deductive databases, and the like. In this c h a p t e r we shall study t h e Peircean ideas and some of the problems Peirce left open. We shall focus on the 'graphical' representation of tempo-modal problems. The graphs which C.S. Peirce introduced in his logic are divided into 3 classes: the Alpha, Beta, and Gamma graphs. In all of t h e m the statements in question are written on the so-called 'sheet of assertion', SA. The most simple statement is the empty statement, which is supposed to hold according to the only axiom in the Peircean Alpha-system. Propositions on the SA may be enclosed using so-called 'cuts', which in fact correspond to negations (we also speak of 'negated contexts'). That is, the following combination of graphs means t h a t P is the case, Q is not the case, and the conjunction of S and R is not the case.

P

Q Q 320

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321

H e r e the box represents the SA, whereas the curved closures symbolise the cuts, i.e. the negated contexts. In terms of t h e established formalism of propositional logic, the above g r a p h is equivalent to P A - Q A - ( S A R)

It is obvious t h a t such conjunctional forms are rather easy to r e p r e s e n t in P e i r c e a n graphs. Disjunctions and implications are, however, slightly more complicated. The implication (P:::~Q) is represented by the following graph

In s t a n d a r d formalism this g r a p h can be expressed as - ( P A - Q ) , which is exactly the definition of material implication in terms of negation and conjunction. It is a remarkable theoretical result that Peirce's Alpha graphs correspond exactly to standard propositional calculus (cf. J o h n Sowa [1992b]). His Beta graphs, in turn, correspond to first order predicate calculus with a 'non-empty' q u a n t i f i c a t i o n t h e o r y (see below). With that restriction, this m e a n s t h a t theorems w h i c h can be proved in first order logic can also be proved in t e r m s of existential graphs. To prove a t h e o r e m corresponding to a certain graph one m u s t t r a n s f o r m the empty proposition on SA into the graph in question. A n u m b e r of rules are available for this procedure, and they will be stated in the following. J o h n Sowa [1992b] has argued that it is in m a n y cases significantly easier to prove a theorem by using the graphs r a t h e r t h a n the established logical procedures. H e h a s s u b s t a n t i a t e d this view by giving some r a t h e r convincing examples.

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In the Beta graphs Peirce introduced a predicate calculus with a quantification theory formulated in terms of w h a t he called 'lines of identity' (ligatures). These graphs a r e i m m e d i a t e l y designed for existential statements. The statement which is now normally formalised as .=Vx:q(x)is represented by the graph:

I

q

Universal statements have to be represented in a slightly more c o m p l i c a t e d w a y u s i n g two cuts (i.e. two n e g a t i o n s ) corresponding to the formula -(3x:~q(x)):

Roberts [1973, p. 138] enumerates the rules for the Alpha and Beta graphs as follows: R1. The rule of erasure. Any evenly enclosed g r a p h and any evenly enclosed portion of a line of identity m a y be erased. R2. The rule of insertion. Any graph may be scribed on any oddly enclosed area, and two lines of identity (or portions of lines) oddly enclosed on the same area may be joined. R3. The rule of iteration. If a graph P occurs in the SA or in a nest of cuts, it m a y be scribed on any a r e a not part of P, which is contained by the place of P. Consequently, (a) a

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b r a n c h with a loose end may be added to any line of identity, provided t h a t no crossing of cuts results from this addition; (b) any loose end of a ligature m a y be e x t e n d e d i n w a r d s t h r o u g h cuts; (c) a n y ligature t h u s extended m a y be joined to t h e corresponding ligature of an iterated i n s t a n c e of a graph; and (d) a cycle m a y be formed, by joining by i n w a r d extensions t h e two loose ends t h a t are the i n n e r m o s t p a r t s of a ligature. R4. The rule of deiteration. Any graph whose occurrence could be the result of iteration may be erased. Consequently, (a) a branch w i t h a loose end may be retracted into any line of identity, provided t h a t no crossing of cuts occurs in t h e retraction; (b) a n y loose end of a ligature m a y be r e t r a c t e d o u t w a r d s t h r o u g h cuts; and (c) any cyclical p a r t of a ligature m a y be cut at its inmost part. R5. The rule of t h e double cut. The double c u t m a y be i n s e r t e d a r o u n d or removed (where it occurs) from a n y g r a p h on a n y area. A n d these t r a n s f o r m a t i o n s will not be p r e v e n t e d by the presence of ligatures passing from outside t h e outer cut to inside the inner cut. I n addition to t h e s e rules there are two axioms: t h e e m p t y g r a p h (SA) and t h e u n a t t a c h e d line of identity. F r o m these two axioms it is possible to derive a n u m b e r of t h e o r e m s a n d rules u s i n g (R1-5). For i n s t a n c e , the g r a p h c o r r e s p o n d i n g to t h e following implication gx: q(x) ~ ::?x: q(x) t u r n s out to be provable, as demonstrated by Roberts [1992]. The axiom of the u n a t t a c h e d line of identity has a crucial rSle to play in t h i s proof. This m e a n s t h a t quantification cannot be e m p t y in t h e logic of the Beta g r a p h s (which is in fact also t h e case w i t h Prior's quantification theory).

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THE NEED FOR MORE THAN ALPHA AND BETA GRAPHS The logic of Beta graphs is clearly useful in m a n y cases. Peirce realised, however, t h a t t h e Alpha a n d Beta g r a p h s a r e n o t satisfactory in all cases. For i n s t a n c e , he c o n s i d e r e d t h e following two propositions (see [CP 4.546]): (1) Some m a r r i e d w o m a n will c o m m i t suicide, if h e r husband fails in business. (2) Some m a r r i e d w o m a n will c o m m i t suicide, if e v e r y married woman's husband fails in business. Peirce a r g u e d t h a t these two conditionals are equivalent if we analyse t h e m in a merely classical a n d n o n - m o d a l logic - i.e. in t e r m s of Beta g r a p h s within his o w n logical system. F o r t h e s a k e of simplicity we r e f o r m u l a t e t h e p r o b l e m u s i n g o n l y predicates w i t h one argument. According to Peirce's rules for B e t a g r a p h s a n d their lines of identity, t h e g r a p h s corresponding to (1) a n d (2) can be p r o v e d to be equivalent, i.e.

- where fail(x) m e a n s 'x is married to a b u s i n e s s m a n who fails in business', a n d suicide(x) m e a n s 'x c o m m i t s suicide'. T h i s equivalence can be established by t h e rules of transformation for Beta graphs. T h e two graphs respectively correspond to t h e following two expressions of s t a n d a r d predicate notation (where

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quantification is understood to be over the set of women married to businessmen): (la) (2a)

(3x)(fail(x) ~ suicide(x)) (3x)((Vy)fail(y) ~ suicide(x))

Both of these expressions are equivalent with

(3 x)-fail(x) v (3x) suicide(x) The inference from (2a) to (la) a p p e a r s r a t h e r n a t u r a l , w h e r e a s t h e opposite inference is clearly c o u n t e r i n t u i t i v e . Nevertheless, (la) and (2a) turn out to be logically equivalent, as long as we are moving strictly within classical predicate logic, respectively the Beta graphs. Therefore, as long as we are trying to represent our case within those systems, we are obliged to accept the counterintuitive inference. However, it m a y be more natural to formulate the problem in t e r m s of three predicates, so let wife(x) stand for 'x is the wife of a b u s i n e s s m a n ' . W h e n t h e s t a t e m e n t s (1) a n d (2) a r e represented with three predicates, the graphs in question will be:

wife

wife

Again, these graphs can be shown to be equivalent. Essentially, their equivalence is due to the fact t h a t the t e r m wife is outside the scope of the negations. Therefore, the rules of iteration and deiteration for Beta graphs can be applied to the inner copies. The proof using Beta graphs could run as indicated below.

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wife

d---b

Iwife

(1)

wife

41--1

(2)

wife

(3)

Step (1) in the above deduction is the introduction of a double cut in t h e first graph. In step (2) iteration is used, and in step (3) the rules of erasure and deiteration are used; a few other rules are also used, but the details are omitted here. - In every step the opposite operation is also allowed. The only counterintuitive step seems to be the implication from right to left in (3). In t e r m s of standard formalism (1) and (2) are represented by

(lb) (2b)

(3 x)(wife(x) A (fail(x) ~ suicide(x))) (3 x)(wife(x) A ((Vy)(wife(y) ~ (fail(y)) ~ suicide(x)))

U s i n g f u n d a m e n t a l equivalences from first order logic, (lb) and (2b) can be written as disjunctions

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(lb')

(3x)(wife(x) A-fail(x)) v (3x) (wife(x) A suicide(x)))

(2b')

(3x)((wife(x) A (3y)(wife(y) A -fail(y)) v (~x)((wife(x) A suicide(x)))

By the 'omission' of wife(x) in the first part of the disjunction in (2b'), it becomes evident t h a t (lb') follows from (2b'). T h a t is, they are both equivalent to (3x)(wife(x) A -fail(x)) v (3x)(wife(x) A suicide(x)) Peirce stated t h a t the equivalence of these two propositions is "the absurd result of admitting no reality but existence" [CP 4.546]. As Stephen Read [1992] has pointed out, Peirce's analysis is a strong a r g u m e n t against anybody inclined to assert t h a t conditionals in n a t u r a l language are always truth-functional. But t h e Peircean analysis is also an argument for the need of a new tempo-modal logic. Peirce formulated his own solution in the following way: If, however, we suppose t h a t to say t h a t a w o m a n will suicide if h e r h u s b a n d fails, m e a n s t h a t e v e r y possible course of events would either be one in which the husband would not fail or one in which the wife will commit suicide, t h e n , to make t h a t false it will not be requisite for the h u s b a n d actually to fail, but it will suffice t h a t t h e r e are possible circumstances u n d e r which he would fail, while yet his wife would not commit suicide. [CP 4.546] This m e a n s t h a t we have to quantify over 'every possible course of events'. Prior's tense-logical notation systems provide the m e a n s for doing just t h a t . The operator suited for the problem at h a n d is G, corresponding to 'it is always going to be the case that'. As we have seen Prior established a system d e s i g n e d to c a p t u r e Peirce's ideas on t e m p o r a l logic appropriately called 'the Peircean solution' (see chapter 2.2 and 2.6). In the Peircean system, G m e a n s 'always going to be in

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every course of events'. Using the operator in this way, we can express (1) and (2) as respectively (lc) (3 x)G(husband__fail(x) ~ suicide(x)) (2c) (3 x)G((V y)husband_.fail(y) ~ suicide(x)) (It should be mentioned t h a t a linguistically more appropriate r e p r e s e n t a t i o n p e r h a p s should t a k e t h e form N(p ~ Fq). However, (lc) a n d (2c) are sufficient for the c o n c e p t u a l considerations which are important here.) (lc) clearly m e a n s t h a t there is some m a r r i e d woman w for whom (ld) (-husband_fail(w) v suicide(w)) holds at any time in any possible future course of events. (2c) means t h a t there is a married woman w for whom (2d) (3 y) -husband__fail(y) v suicide(w) holds at any time in any possible future course of events. For this reason it is formally clear that (lc) entails (2c), but not conversely. And this corresponds exactly to intuition with respect to the two s t a t e m e n t s (1) and (2): t h e inference from (1) to (2) is valid, but Peirce was justified in maintaining t h a t the inference from (2) to (1) m u s t be rejected. Generally speaking, some kind of tempo-modal logic is r e q u i r e d for d e s c r i b i n g conditionals in n a t u r a l l a n g u a g e reasoning in a satisfactory way - a fact which has quite recently been more systematically expounded [Crouch 1993]. Peirce's considerations on t h e example discussed in this section clearly demonstrate t h a t he realised this.

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THE GAMMA GRAPHS Peirce himself made some attempts at solving the problems of modality by introducing a new kind of graphs. In w h a t he called 'The G a m m a Part of Existential Graphs' [CP 4.510 ft.], he put forth some interesting suggestions regarding modal logic. Some of his considerations on this topic were linked to w h a t is now called epistemic logic, i.e. the logic of knowledge. In the following we shall describe his ideas. In epistemic logic, the idea is that relative to a given state of information a number of propositions are known to be true. In Peirce's graph theory, propositions describing the information in question should be written on the 'sheet of assertion' SA, using just Alpha and Beta graphs. Other propositions, however, are to be r e g a r d e d as m e r e l y possible in the p r e s e n t state of i n f o r m a t i o n . Peirce r e p r e s e n t e d such propositions u s i n g 'broken cuts', combined w i t h the 'unbroken cuts' which we already know from the Alpha and Beta graphs. A broken cut should be interpreted as corresponding to 'it is possible that not ...'. This means that 'it is possible that ...' must be represented as a combination of a broken and an unbroken cut: f

I!

X \ ~

"-.

\A /I

Now consider a contingent proposition, i.e. a proposition, w h i c h is possible, but not necessary according to the present state of information.

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In this state the sheet will include at least two propositions:

I/(f~ \ / \ \X"-"// j

\\

\ j

l

Now suppose t h a t p is contingent relative to some state of information, a n d t h a t we t h e n learn t h a t p is in fact true. This would mean t h a t t h e SA should be changed according to t h e n e w state of i n f o r m a t i o n . The g r a p h c o r r e s p o n d i n g to Mp ('possibly p') s h o u l d be changed into t h e graph for 'it is k n o w n w i t h certainty t h a t p', i.e. -M-p. Obviously, this m e a n s t h a t t h e g r a p h for M~p s h o u l d be dropped, which results in a new (and simpler) g r a p h r e p r e s e n t i n g t h e u p d a t e d state of information. In this way Peirce in effect pointed out t h a t the passage of time does not only lead to new knowledge, b u t also to a loss of possibility. W i t h respect to this epistemic logic Don D. Roberts [1973, p. 85] h a s observed t h a t the notions of necessity a n d possibility both m a y seem to collapse into the notion of t r u t h . Roberts h i m s e l f gave an i m p o r t a n t p a r t of the a n s w e r to this w o r r y by e m p h a s i s i n g how "possibility and n e c e s s i t y a r e relative to t h e state of information" [CP 4.517], a n d t h a t t h e r e will only be a total collapse in case of omniscience. In the context of existential g r a p h s Peirce in fact established an equivalence b e t w e e n 'p is k n o w n to be true' a n d 'p is n e c e s s a r y ' . In consequence, 'p is not known to be false' and ~ is possible' should also be e q u i v a l e n t in a Peircean logic. Therefore, t h e k i n d of m o d a l logic w h i c h Peirce w a s a i m i n g at w a s in fact a n epistemic logic, which should be sensitive to the impact of time.

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He furthermore realised t h a t it would be useful not only to have a logic for knowledge in a strong sense, but also a logic for confidence: The idea of time really is involved in the very idea of an argument. But the gravest complications of logic would be involved, if we took account of time so as to distinguish between w h a t one knows and w h a t one has sufficient reason to be entirely confident of. [CP 4.523]

TEMPO-MODAL PREDICATE LOGIC AND EXISTENTIAL GRAPHS Peirce was concerned with the epistemic aspect of modality, but he also w a n t e d to apply his logical graphs to modality in g e n e r a l - t h a t is, to use t h e m for r e p r e s e n t i n g any kind of modality. However, he was aware of t h e great complexity in which a full-fledged logic involving t e m p o r a l modifications would result. T h i s is probably t h e r e a s o n w h y Peirce's presentations of t h e G a m m a graphs r e m a i n e d tentative and unfinished. In the following we intend to explain some of the problems he was facing, and suggest some ideas regarding the possible continuation of his project. Our analysis of the problem from [CP 4.546] suggests t h a t the two statements should in fact i)e understood as follows: (1') Some m a r r i e d w o m a n will (in every possible future) commit suicide if her husband fails in business. (2') Some m a r r i e d w o m a n will (in every possible future) commit suicide if every married woman's husband fails in business. We intend to f o r m u l a t e a graph-theoretical version of t h e tense-logical solution. So, we have to m a k e sure that there are proper graphical representations of (1') and (2') such t h a t t h e graphs are non-equivalent. In fact, it is not difficult to create graphs corresponding to the modal expressions in (1') and (2').

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Obviously, a g r a p h with a b r o k e n cut inside an u n b r o k e n cut w i t h q would clearly correspond to t h e s t a t e m e n t 'in every possible f u t u r e q'. A representation of ( r ) and (2') could t h e n be (omitting the SA):

wfe i~

+wfe i

In order to treat problems like the one we have been discussing we m u s t be able to handle graphs involving the two kinds of cuts ( b r o k e n a n d u n b r o k e n ) as well as lines of i d e n t i t y . In consequence, we have to establish rules for modal conceptual g r a p h s , specifically such t h a t (1') a n d (2') w o u l d be nonequivalent. H a r m e n v a n den Berg [1993b] has shown how it is possible to f o r m u l a t e propositional modal rules of inference corresponding to c o n c e p t u a l g r a p h s with b r o k e n as well as u n b r o k e n cuts. However, t h e rule of iteration has to be slightly changed, and a few e x t r a rules have to be a d d e d in order to o b t a i n t h e propositional modal logic T (sometimes called M), for which the so-called rule of necessitation and the following axioms hold: (Nec) I f p is provable, t h e n N p is provable N(p ~ q) ~ (Np ~ Nq) Np Dp The question of a similar account for modal predicate logic is left open by H a r m e n van den Berg. In order to obtain a predicate m o d a l logic we have to accept (R1-R5) for g r a p h s only in-

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volving broken cuts in subgraphs which are untouched by the operations of t h e rules. Since classical quantification rules should be valid in general, it follows that any loose end of a ligat u r e m a y be extended inwards through broken as well as unbroken cuts (cf. R3). The resulting minimal change of t h e Rrules and van den Berg's rules then leads to the following rules: RI'. The rule of erasure. Any evenly enclosed g r a p h and a n y portion of a line of identity evenly enclosed by u n b r o k e n cuts m a y be erased. R2'. The rule of insertion. Any graph may be scribed on any oddly enclosed area, and two lines of identity (or portions of lines) oddly enclosed by unbroken cuts on the s a m e a r e a m a y be joined. R3'. The rule of iteration. If a graph P occurs in the SA or in a nest of unbroken cuts, it may be scribed on a n y a r e a not p a r t of P, w h i c h is c o n t a i n e d by the p l a c e of P. Consequently, (a) a branch with a loose end may be added to a n y line of identity, provided that no crossing of cuts results from this addition; (b) any loose end of a ligature m a y be e x t e n d e d i n w a r d s t h r o u g h cuts; (c) any l i g a t u r e t h u s extended m a y be joined to the corresponding ligature of an i t e r a t e d instance of a graph; and (d) a cycle may be formed, by joining by inward extensions through unbroken cuts the two loose ends that are the innermost parts of a ligature. R4'. The rule of deiteration. Any graph whose occurrence could be the result of iteration may be erased. Consequently, (a) a branch with a loose end may be retracted into a n y line of identity, provided that no crossing of cuts occurs in the retraction; (b) any loose end of a ligature may be r e t r a c t e d o u t w a r d s through unbroken cuts; and (c) any cyclical p a r t of a ligature crossing no modal cut may be cut at its inmost part. R5 °. The rule of the double cut. The double cut m a y be i n s e r t e d around or removed from any graph on a n y area. And t h e s e transformations will not be p r e v e n t e d by the presence of ligatures passing from outside the outer cut to inside the inner cut through unbroken cuts.

334

CHAPTER 3.6 R6'. T h e rule of m o d a l conversion. An evenly enclosed u n b r o k e n cut m a y be replaced by a b r o k e n cut. An oddly enclosed broken cut m a y be replaced by an unbroken cut. R7'. The necessitation rule. If the g r a p h corresponding to p is provable, t h e n t h e g r a p h corresponding to Np is also provable. R8'. T h e d i s t r i b u t i o n rule. The g r a p h c o r r e s p o n d i n g to N(p A q) is provable if and only if the g r a p h corresponding to (Np/, Nq) is provable.

The rule of necessitation implies the following rule: (Nec N): If the implication (p ~ q) is provable, t h e n t h e implication (Np ~ Nq) is also provable. This derived rule could also be stated directly in t e r m s of existential graphs. If we want the modal logic $4, we have to add this rule: The duplication rule. A combination of two cuts a r o u n d a g r a p h (broken or u n b r o k e n cuts) m a y be duplicated. The inverse operation is also allowed. U s i n g this rule we can prove t h e graph corresponding to t h e axiom:

Np ~ NNp If we w a n t t h e modal logic $5, we have to add the rule: The generalised duplication rule. A combination of two cuts a r o u n d a g r a p h ( b r o k e n or u n b r o k e n cuts) m a y be duplicated in r a n d o m order. The inverse operation is also allowed.

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With the generalised duplication rule, we can prove the graph corresponding to the axiom: Mp ~ NMp

The change from (R1-5) to (RI'-R8') seems to be minimal if t h e system should include T. For t h a t reason one may suspect t h a t the system w i t h the two Beta axioms together with (RI'R8') is something r a t h e r close to the modal predicate logic corresponding to T. But we shall suggest t h a t something more seems to be needed. As we shall see this idea can in fact be traced back to the Peircean analysis. The question regarding the relation between modal operators a n d quantifiers is crucial for any modal predicate logic. Peirce was aware of this problem. He stated: Now, you will observe t h a t t h e r e is a g r e a t difference between the two following propositions: First, There is some one m a r r i e d w o m a n who u n d e r all possible conditions would commit suicide or else h e r husband would not have failed. Second, U n d e r all possible c i r c u m s t a n c e s t h e r e is some married w o m a n or other who would commit suicide or else her husband would not have failed. [CP 4.546] It is very likely t h a t w h a t Peirce had in mind was the insight t h a t we cannot w i t h complete generality derive 3x: Ns(x) from N(=Tx: s(x)). - that is, not without making some restrictions. This is in fact a very old wisdom which was also k n o w n to the medieval logicians. One cannot deduce 'there is a m a n who will live forever' from 'it will forever be t r u e t h a t t h e r e is a man'. However, the opposite deduction is clearly reasonable. In fact t h e implication (NEX) ~x: Ns(x) ~ N(3x: s(x)) or equivalently: (NEX') M(Vx: s(x)) ~ Vx: Ms(x)

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turns out to be provable in predicate modal logic corresponding to T. This theorem can be represented by the following graph:

) Let an N-double cut be a broken cut inside an unbroken cut, with n o t h i n g enclosed in the outer area except for portions of lines of identity which pass from inside the inner area to outside the outer area. We can then formulate a rule for modal graphs, from which the graph version of (NEX) can be proved: (R-NEX) Any oddly enclosed loose end of a ligature m a y be extended outwards through an N-double cut. - A n y evenly enclosed loose end of a l i g a t u r e m a y be retracted inwards through an N-double cut. It a p p e a r s t h a t we need a rule like (R-NEX) in order to establish the graph-theoretical equivalent to a modal predicate logic including T. We suspect t h a t this rule can be made nicer and more general. (How t h a t should be done in detail will depend on f u r t h e r logical investigations.) As far as we know, Peirce did not investigate the corresponding logical relation between the necessity operator and the universal quantifier. In chapters 2.9 and 2.10 we discussed a certain relation between

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q u a n t i f i e r s and m o d a l operators, n a m e l y Barcan's f o r m u l a [Prior 1967, p. 137 iT.]:

Vx: Ns(x) ~ N(Vx: s(x)) or equivalently M(3x: s(x)) ~ 3x: Ms(x) This formula corresponds to the following graph:

The validity of Barcan's formula within a modal system would m e a n t h a t for instance the following implication holds: if 'at some future time there will be a suiciding wife', t h e n 'there is some wife who at some f u t u r e time will commit suicide'. Such an implication seems not to be acceptable in the context of Peirce's example, since it excludes the possibility t h a t the suiciding wife could come into being at some future time. As we have seen, Barcan's formula is provable in $5, but not so in $4. Therefore $4 could be a reasonable candidate for a modal logic capable of 'solving' t h e Peircean example. But we have to be cautious in trying to 'embed' $4 within the G a m m a graphs.

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The rule of duplication for p r e d i c a t e modal logic m u s t be a d a p t e d very carefully such t h a t Barcan's formula can n o t be derived: R9'. The duplication rule. A combination of two cuts a r o u n d a g r a p h (broken or unbroken cuts) m a y be duplicated. T h e s e t r a n s f o r m a t i o n s will n o t be p r e v e n t e d by t h e presence of ligatures passing from outside the o u t e r cut to inside the inner cut through u n b r o k e n cuts. T h u s revised, the rule of duplication will obviously not validate the Barcan formula. A generalised duplication rule, w h i c h can yield a modal g r a p h logic as s t r o n g as $5, should be f o r m u l a t e d in an analogous manner: RI0'. The generalised duplication rule. A c o m b i n a t i o n of two cuts a r o u n d a graph (broken or unbroken cuts) m a y be duplicated in random order. These transformations will not be p r e v e n t e d by the p r e s e n c e of ligatures p a s s i n g from o u t s i d e t h e o u t e r cut t o i n s i d e t h e inner cut t h r o u g h u n b r o k e n cuts. W i t h this rule we can in fact prove the t h e o r e m s of $5 including Barcan's formula. However, as we have a r g u e d the logic Peirce needed to solve his problem should not allow for the validity of Barcan's formula, since t h a t would m a k e it difficult to give an account of the logical aspect of 'coming into being'. In consequence it seems t h a t t h e logic Peirce was looking for should be weaker t h a n $5. We shall suggest the modal predicate logic corresponding to the s y s t e m w i t h t h e Beta axioms a n d the rules (RI'- R9' + R-NEX).

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CONCEPTUAL GRAPHS AND TEMPO-MODAL LOGIC Peirce clearly realised the need for more modal operators t h a n j u s t one. Peirce's c e n t r a l ideas w e r e in fact a m e n a b l e to s y s t e m a t i s a t i o n in the form of t e m p o - m o d a l calculi, as i n d e e d Prior showed, but these logics have so far not been formulated in t e r m s of Peirce's existential graphs. However, d u r i n g t h e last d e c a d e t h e s t u d y of so-called c o n c e p t u a l g r a p h s h a s b e e n developed as a field w i t h i n artificial intelligence a n d logic. Conceptual graphs constitute a formal logical system based on the ideas laid out in Peirce's existential graphs. The field w a s first e s t a b l i s h e d by J o h n Sowa [1984]. T h e r e are a few differences between conceptual g r a p h s a n d existential g r a p h s , first a n d foremost that (i) in c o n c e p t u a l g r a p h s 'contexts' a r e not in g e n e r a l n e g a t e d as in existential graphs, and (ii) it is possible to treat variables, individuals and types in a m o r e direct and elegant m a n n e r in conceptual graphs. For instance, Sowa [1992a, p. 22] r e p r e s e n t s the s t a t e m e n t 'some dog does not eat meat' in the following way: [DOG: *x] -~[[*x] +- (AGENT) e- [EAT] --~ (PATIENT) --> [MEAT]] In t h e following, we shall use Sowa's notation for conceptual graphs for our discussion. D u r i n g t h e last years some r e s e a r c h e r s have t r i e d to f o r m u l a t e v a r i o u s s y s t e m s of t e m p o r a l logic in t e r m s of conceptual g r a p h s [Esch and Nagle 1992], [Moulin a n d Cot6 1992]. There is, however, still a lot to be done in this area. In order to formulate a graph-theoretical equivalent of Prior's 'Peircean system' (and other relevant tempo-modal systems, for

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t h a t m a t t e r ) we shall need at least two t e n s e operators corresponding to past and future: ( P ) - > [SITUATION: GRAPH] ( F ) - > [SITUATION: GRAPH] m e a n i n g respectively 'the situation GRAPH has been' and 'the situation GRAPH will be'. We can define graphs corresponding to Prior's H ('has always been') and G ('will always be'), i.e. (H) -> [SITUATION: GRAPH] (G) -> [SITUATION: GRAPH] as

-~[(P) -> [SITUATION: -, [GRAPH]]] -,[(F)-> [SITUATION:-~[GRAPH]]]. With such graphs we can also formulate f u n d a m e n t a l tenselogical t h e o r e m s like, say HFq ~ q, in t e r m s of conceptual graphs. We m a y n e e d m e t r i c t e n s e operators, which can be represented graphically in the following way: [TIME: t] -> (P') -> [SITUATION: GRAPH] [TIME: t] -> (F') -> [SITUATION: GRAPH] w h e r e t is any n u m b e r (or integer if time is supposed to be discrete). If (P') a n d (F') are t a k e n to be primitive, we may define (P) and (F) as [TIME: *] -> (P') -> [SITUATION: GRAPH] [TIME: *] -> (F') -> [SITUATION: GRAPH] It is an open problem how a full t e n s e logic should be incorporated into t h e theory of conceptual graphs - in other words, how the G a m m a rules corresponding to Prior's tempomodal systems should be formulated. This problem seems to be a r a t h e r complicated one.

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T h e analysis of t h e Peirce problem can also be formulated in t e r m s of conceptual g r a p h s . Within this theory, t h e crucial s t a t e m e n t s could be expressed in the following way: (1) Some m a r r i e d w o m a n will commit suicide if her h u s b a n d fails in business [WOMAN: *x] [ [WOMAN: *x] -> (MARRIED) -> [MAN] <-(FAILING)

]

[ [WOMAN: *x] <- (AGNT) <- [SUICIDE]

]

(2) Some m a r r i e d w o m a n will commit suicide if every m a r r i e d woman's h u s b a n d fails in business [WOMAN: *x]

[

[WOMAN: @every=*y] [WOMAN: *y] -> ( M A R R I E D ) - > [MAN] <(FAILING) ]

[ [WOMAN: *x] <- (AGNT) <- [SUICIDE]

]

w h e r e we define the g r a p h [WOMAN: @every=*y] as an abbreviation of A[ [WOMAN: *y] A[ [*y] ]]. T h e g r a p h s for (1) a n d (2) t u r n out to be logically equivalent.

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This is, h o w e v e r , not t h e case for t h e c o r r e s p o n d i n g m o d a l statement: (1') S o m e m a r r i e d w o m a n will commit suicide if in every possible future her husband fails in business [WOMAN: *x] (N)->

[

[ [WOMAN: *x] -> ( M A R R I E D ) - > [MAN] <]

(FAILING)

[ ]

[WOMAN: *x] <- (AGNT) <- [SUICIDE]

(2') S o m e m a r r i e d w o m a n will commit suicide if in every possible f u t u r e every m a r r i e d woman's h u s b a n d fails in business [WOMAN: *x] (N)-> [

[ [WOMAN: @every=*y] [WOMAN: *y] -> (MARRIED) -> [MAN] <- (FAILING)

] _-->

[ [WOMAN: *x] <- (AGNT) <- [SUICIDE] ]

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In this case t h e r e is only one modal operator and consequently only two k i n d s of cut, and the rules (RI'-R9') and (R-NEX) m a y still give u s a very convincing logic. However, if we introduce a s e p a r a t e future operator F along with t h e m o d a l operator N, t h i n g s become more complicated. And of course, if we wish to e m b e d a tense logic w i t h i n conceptual graphs, we m u s t include n o t only a future-operator, but also some pasto p e r a t o r P . H o w e v e r , t h e r e is y e t no g r a p h - t h e o r e t i c a l equivalent to Prior's tense logics. It is a question for f u r t h e r investigation how this should be done. J o h n Sowa [1992a, p. 26] h a s h i m s e l f defined t h e t e n s e operators in t e r m s of PTIM (point in time}. For instance, relation PAST(x) is [SITUATION: *x] (PTIM) -~ [TIME] -~ (SUCC) -~ [TIME: #now] T h a t would allow us to refer to situational contexts. The relation b e t w e e n t h i s a p p r o a c h a n d the o p e r a t o r approach is r a t h e r unexplored. Nevertheless, the way in which we dealt w i t h t h e G a m m a g r a p h s should provide useful clues as to how we m a y proceed w i t h t h e t a s k of constructing tense-logical c o n c e p t u a l graph theory.

3.7. TEMPORAL LOGIC AND COMPUTER SCIENCE The usefulness of systems of this sort [on discrete time] does not depend on any serious metaphysical assumption that time is discrete; they are applicable in limited fields of discourse in which we are concerned only with what happens next in a sequence of discrete states, e.g. in the working of a digital computer, tk N. Prior [1967, p. 67] The relevance of temporal logic within computer science was realised in the course of the 1970s. Temporal logic has by now become an established discipline within this science, but the first researchers to take up the connection were not acquainted with Prior's tense logic. The initial studies in the field w e r e based on Temporal Logic by N. Rescher and A. U r q u h a r t [1971]. This book was in fact dedicated to the memory of A r t h u r Prior, and at a n y rate computer scientists would in due course also begin to s t u d y Prior's own works on tense logic. The above quotation m a k e s it clear t h a t one decade earlier it had occurred to Prior himself that his tense logic might be useful in computer science. One of the first computer scientists to realise the relevance of t e m p o r a l logic for the purposes of computer science was Amir P n u e l i . P n u e l i h a s h i m s e l f described [1994, p e r s o n a l communication] how he was working on problems p e r t i n e n t to t h e logic of time, when in late 1975 or early 1976 Saul Gorn of t h e University of Pennsylvania made him a w a r e of Rescher's book Logics of Commands. This book, however, t u r n e d out to be of little relevance for Pnueli's purposes. But on t h e back of the d u s t cover t h e r e was a reference to another book by the same a u t h o r , n a m e l y Rescher's and U r q u h a r t ' s Temporal Logic. Pnueli went on to study this book, where a firm basis for dealing with temporal logic could be found. Pnueli's pioneering work on temporal logic within computer science, as well as subsequent work in this area, has been 344

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concerned not only with the application of temporal logic to specific problems, but also with general theory development. (This fact is highly evident in the contributions to the first international conference on temporal logic in Bonn, 1994 [Gabbay and Ohlbach 1994]). Together with formal linguistics, computer science is today a chief contributor to the continued development in this field. Temporal logic has become an important formalism for various purposes within computer science, ranging from fundamental theoretical issues to special types of application software. One significant example of the latter type is natural language understanding. In the context of this book we have for obvious reasons been paying special attention, directly and indirectly, to this problem domain. In n a t u r a l language understanding we have to deal with the problem of giving a semantical representation of time, as it is manifest in linguistic expressions (tenses, aspect, temporal connectives and adverbs) [Hasle 1991]. An important goal of this kind of work is, of course, to give a formal account of intuitively valid inferences. Broadly speaking, this is tantamount to formalising rational common sense reasoning involving time. Moreover, Martha S. Palmer et al. [1993] have argued that the specific (i.e. context-dependent) interpretation of tense and other temporal expressions in na t ura l language often requires common-sense reasoning. Thus, studies in natural language understanding almost irresistibly call for techniques from temporal logic, which is in fact itself a theoretical field based on a study of valid commonsense reasoning. Time and again, we have exemplified this by scrutinising such deliberations, from Antiquity to the present day. From this kind of investigation into natural language understanding we are led to more general studies regarding the representation (and manipulation) of temporal knowledge. Such studies are immediately relevant for other issues within artificial intelligence, where temporal logic is important, for example w h e n prediction, explanation, planning or similar purposes are involved. In many expert systems, e.g. medical diagnosis systems, the representation of time is essential.

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L i k e w i s e , robotical s y s t e m s as a rule r e q u i r e t h a t some u n d e r s t a n d i n g of time m u s t be simulated. A m o s t i n t e r e s t i n g application of ideas f r o m t e m p o r a l r e a s o n i n g c a n be f o u n d in p l a n n i n g s y s t e m s . F o r t h e c o n s t r u c t i o n of linear as well as non-linear 'planners', the r e p r e s e n t a t i o n of time is fundamental [Shoham 1994, p. 199 ff.]. I n our opinion, all such applications of temporal logic should be viewed as u l t i m a t e l y inspired from the study of valid commonsense r e a s o n i n g as embodied in elements of n a t u r a l language. This view can be s u p p o r t e d by the careful work of R. S. Crouch a n d S. P u l m a n [1993], who have specifically d e m o n s t r a t e d how a n a t u r a l l a n g u a g e interface to a p l a n n i n g s y s t e m can be c o n s t r u c t e d . In this connection t h e y argue t h a t t h e t a s k of building a n a t u r a l language interface to an information system is one of modelling the domain in question as a reasoning s y s t e m . I n d e e d , t h i s w a y of r e l a t i n g n a t u r a l l a n g u a g e to r e a s o n i n g a n d logic m a y be seen as a m o d e r n version of the medieval conception of logic discussed in part 1. Time h a s shown, however, that the relevance of t e m p o r a l logic w i t h i n c o m p u t e r science extends far beyond n a t u r a l language u n d e r s t a n d i n g and artificial intelligence. It can also be applied in various phases of t h e system development process. Usually, this process is divided into four consecutive phases: analysis of t h e p r o b l e m at hand, design, i m p l e m e n t a t i o n , a n d validation. ( S o m e t i m e s v a l i d a t i o n is not c o u n t e d as a s e p a r a t e phase; moreover, it is generally recognised t h a t t h e p h a s e s are not clearly distinct. Rather, they overlap and are usually reiterated). Temporal logic has proved its w o r t h w i t h i n each of t h e s e p h a s e s . In d e s i g n and i m p l e m e n t a t i o n , it is u s e d for specifying properties t h a t the system in question should possess. In a n a l y s i s , it p l a y s t h e s a m e r61e as in c o m m o n - s e n s e r e a s o n i n g . In validation, its use is n o r m a l l y r e s t r i c t e d to verification, i.e. the t a s k of proving t h a t the p r o g r a m does have t h e r e q u i r e d p r o p e r t i e s (the o t h e r p a r t of v a l i d a t i o n being v a r i o u s ways of empirically testing the program). Obviously, t e m p o r a l logic can t h u s play a very g e n e r a l r61e in s y s t e m development; it even appears that it m a y be a n a t u r a l candidate

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for i n t e g r a t i n g the various phases, or at least relating t h e m to each other. The fact t h a t temporal logic can be u s e d for specifying c o m p u t e r programs, and for reasoning about t h e m , has been k n o w n for some years [Burstall 1974, Pnueli 1977, M a n n a & Wolper 1981]. In particular, temporal logic h a s become a n i m p o r t a n t tool for the analysis of c o n c u r r r e n t (parallelistic) programs. The main idea is t h a t the execution history can be described in t e r m s of t e m p o r a l logic, w i t h o u t n e c e s s a r i l y r e f e r r i n g to specific program states or times. This m e a n s t h a t g e n e r a l properties of programs such as freedom of deadlock, m u t u a l exclusion etc. can be expressed in a v e r y nice way in t e r m s of formulae of temporal logic. Anyone who wants to use temporal logic has to choose between its two major paradigms, namely A- and B-logic. Of course, t h a t choice m a y be seen as a choice between a s y n t a x of t e n s e s (operator approach) and a syntax involving reification of time (quantifier approach). With a B-logic it seems t h a t one has to formulate a theory of time as a structure. With an A-logic, on the other hand, it is not required that any s u c h structure be specified - not immediately, at least. On the o t h e r hand, E r i k S a n d e w a l l [1992, pp. 609-610] has argued t h a t the reified a p p r o a c h should be p r e f e r r e d for r e a s o n s of n o t a t i o n a l convenience. A similar answer has been p r e s e n t e d by Yoav S h o h a m [1994, p. 234 ff.], who has suggested a so-called 'time m a p m a n a g e m e n t ' in which temporal reasoning is based on a B-logic w i t h 'points in time'. We agree t h a t in m a n y cases one would like to refer to specific moments of time. However, as d e m o n s t r a t e d in previous chapters such references can also be obtained in an A-logic, to which ideas from Prior's third grade are added. Roger Hale [1987] has used a well k n o w n p r o g r a m m i n g example k n o w n as 'The Towers of Hanoi' to illustrate some of t h e ideas in so-called t e m p o r a l logic p r o g r a m m i n g . In t h e following we shall make use of the same example in order to clarify how t e m p o r a l logic m a y be applied for specification purposes.

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C H A P T E R 3.7

THE TOWERS OF HANOI L e t u s a s s u m e t h a t N different rings are given. T h e sizes of t h e r i n g s a r e 1, 2, ..., N. We shall u s e t h e size of e a c h ring as its n a m e , i.e. t h e s m a l l e s t ring is called '1', and so forth. The rings c a n b e placed on t h r e e pegs. At t h e b e g i n n i n g all t h e rings a r e p l a c e d on t h e first peg in an o r d e r e d w a y as indicated on t h e figure:

I f N = 4 as in t h e above figure, this s t a t e can be r e p r e s e n t e d b y t h e following kind of proposition:

state([1,2,& 4],[],[]) In a n y s t a t e t h e rings on each peg form a 'tower'. In general, a s t a t e c a n be r e p r e s e n t e d by a p r o p o s i t i o n o f t h e f o r m state(A,B,C), w h i c h should be read as 'in the p r e s e n t s t a t e t h e f i r s t t o w e r c o r r e s p o n d s to t h e l i s t A, t h e s e c o n d t o w e r c o r r e s p o n d s to t h e list B, and the t h i r d t o w e r corresponds to t h e list C'. As can be seen we use P R O L O G - l i k e lists to describe t h e states. W i t h this s e t t i n g a g a m e can be introduced. T h e r e is only one rule in the g a m e , called MOVE: MOVE: t h e player is allowed to pick a n y ring at t h e top of one of t h e t h r e e pegs, a n d move it to any o t h e r peg, provided t h a t this m o v e does n o t place t h e ring on t o p of s o m e o t h e r r i n g w h i c h is smaller t h a n the one b e i n g moved.

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Strictly speaking, this is hardly yet a game, since no purpose or goal h a s b e e n s t a t e d ; b u t for now, w e w i s h to discuss s o m e properties of t h e g a m e at this general level. F r o m t h e s t a t e at the beginning w e c a n in two steps r e a c h for instance this s t a t e state([3,4],[2],[1]), graphically:

I

,H,

1

I

1

I

r 7

T h e r e are N 3 possible states, which m a y be n u m b e r e d as Sl, s2, ..., SM, w h e r e M = N 3. To simplify m a t t e r s , we shall c u r r e n t l y a s s u m e t h a t t h e 'first' p e g (A) is t h e one w h e r e t h e rings a r e initially placed: s l = state([1 .... ,N],O,D

U s i n g t h e s e atomic s t a t e propositions as primitives, w e c a n form a logical l a n g u a g e in t h e u s u a l w a y . W e m a y also f o r m i n s t a n t p r o p o s i t i o n s as m a x i m a l c o n s i s t e n t sets. Given t h e u n d e r l y i n g 'model', it is s o u n d to a s s u m e t h a t t h e s t a t e s a r e m u t u a l l y exclusive: I- Si D - 8 j ,

where i cj

This m e a n s t h a t each m a x i m a l c o n s i s t e n t set contains e x a c t l y one 'atomic' (specifically, u n n e g a t e d ) s t a t e proposition. T h u s , given an i n s t a n t proposition ai there is a unique state proposition si s u c h t h a t k a i ~ si

We w a n t to k n o w w h e t h e r t h e r e is a series of possible m o v e s w h i c h can b r i n g u s from the initial s t a t e described b y sl to s o m e o t h e r s t a t e si. In o r d e r to d i s c u s s t h i s p r o b l e m , we n e e d a description of t h e possible moves in a n y conceivable state. S u c h a description c a n be obtained in t e r m s of metric t e n s e logic. T h e

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permissible moves can be specified by the following implications, which in fact form a complete inductive definition of the possible future states, 'PFS': PFS:

(state([X]A],B,C) (state(~lA],B,C) (state(A,[X IB],C) (state(A,[X]B],C) (state(A,B,[X I C]) (state(A,B,[X I C])

A A ~ A A A

legal(X,B)) legal(X,C)) legal(X,C)) legal(X,A)) legal(X,A)) legal(X,B))

~ ~ ~ ~ ~ ~

MF(1)state(A,[XIB],C) MF(1)state(A,B,[X I C]) MF(1)state(A,B,[X] C]) MF(1)state([X]A],B,C) MF(1)state([XIA],B,C)) MF(1)state(A,[X] B],C)

where legal is defined by

legal(X,[]) for any X, and legal(X,[Y] L]) iffX
T(ai, MF(1)p) -(3aj: T(ai, MF(1)aj) A T(aj, p)) Intuitively, it is clear t h a t the rings are 'ordered' in the initial state of the game, and t h a t the one rule of the game in all cases preserves this order. When this condition is compared with the rules 'defining' MF(1), it should be intuitively clear t h a t at any instant there is a possible next state. This means t h a t there are no deadlocks, t h a t is, no states from which we cannot reach a n o t h e r state following t h e rule. We m a y in fact state this property as the axiom:

p ~ MF(1)P(1)p Let us now discuss the notion of 'order' more formally. In every state each tower m u s t be ordered such t h a t t h e smallest ring is at the top of the tower - in general, the sequence of rings

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as s t a t e d in its list-form [X,Y,...,Z] must be increasing. This ordering can be defined recursively:

ordered([]) ordered([A]) ordered([A,B IL]) -=--def(A
order ~def (state(A,B,C) ~ (ordered(A) A ordered(B) A ordered(C)) By inspection of the implications defining MF(1), it is easily shown t h a t

order ~ NF(1)order (where NF(1)p --def - M - F ( 1 ) p ) Since we have the following theorem for metric tense logic

NF(1)(p ~ NF(1)q) ~ (NF(1)p ~ NF(1)NF(1)q) we can in fact prove that once the order has been established, it m u s t be preserved forever, i.e. the theorem ORDER ~ order ~ NG(order) (where Gp ~def - F ( X ) - p , for any n a t u r a l n u m b e r X) Since the implications on which this result is based (PFS) formalise the one rule of the game, we have also shown t h a t this rule is order-preserving. We now wish to add a 'purpose' or goal to the game, n a m e l y t h a t the game has been successfully finished when the following condition obtains:

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CHAPTER 3.7 FINISH: The tower of Hanoi which was originally on t h e first peg, corresponding to the proposition sl, h a s been moved to one of t h e other pegs. (That is, all the rings are now placed in order on one of t h e other pegs, and this h a s been achieved by following t h e MOVE-rule only.)

This final state of success corresponds to the proposition

finish -~lef (state(lY,~,[1,...,N]) v state(O,[1,...,N],[])) In a computer science terminology, we say t h a t t h e problem h a s been solved if t h e game has been successfully finished; in general, our p r o g r a m or plan is said to solve the problem if a n d only if it will always lead to the game being successfully finished. Given our d e f i n i t i o n s so far, t h e s t a t e m e n t t h a t t h e tower problem can be solved is equivalent to asserting the provability of the statement P-SOLVE e state([1,...,N],[],[]) ~ MF finish (where Fp -~-defF(X)p, for some n a t u r a l n u m b e r X) We have a l r e a d y in PFS specified MF(1)p in t e r m s of all possible moves in a n y type of situation. We now proceed to establish a definition of F(1) corresponding to a plan for future actions, that is, s u c h t h a t F(1) in effect specifies which move to m a k e in order to 'approach' a solution. This definition of F(1) should be made such that SOLVE

~ state([1,...,N],[],•) ~ F finish

is provable. Obviously, any demonstration of SOLVE will also be a d e m o n s t r a t i o n of P-SOLVE. P F S is, as p o i n t e d out before, in effect a specification of t h e f u t u r e operator. However, the specification of F corresponding to t h e plan m u s t exclude loops. We therefore propose a r a t h e r different definition of the future operator (in effect, a 'select'operator), availing ourselves of two derived concepts:

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• smallest -~ief( s l v P(n)sP, where n is even, a n d • another --defP(1)smallest. With a view to practical p r o g r a m m i n g it should be noted t h a t t h e h i s t o r y of moves, i.e. t h e s e q u e n c e of s t a t e s d u r i n g computation, m u s t be recorded, when these concepts are to be used in connection with a given program. The specification now r u n s like this:

(smallest A state(J1 [A],B,C)) ~ F(1)state(A,[1 [B],C) (smallest A state(A,[1 [B],C)) ~ Y(1)state(A,B,[1 [C])) (smallest A state(A,B,[1 [C])) ~ F(1)state([1 [A],B, C) (another A state([1 [A],[X[B],C) A legal(X,C)) F(1)state([1 IA],B, IXI el) (another A state([1 [A],B,[X[ C]) A legal(X,B)) F(1)state([1 [A],[X[ B],C) (another/~ state([X [A],B,[1 I C]) A legal(X,B)) F(1)state(A,[X [B],[1 [ C]) (another A state(A,[X[B],[l lC]) A legal(X,A)) F(1)state([X[A],B,[11C]) (another A state([X[A],[1 [B],C) A legal(X,C)) F(1)state(A,[1 [B],[X[ C]) (another A state(A,[1 IB],[X[ C]) A legal(X,A)) F(1)state([X [A],[1 ]B], C) Intuitively, t h e proposition 'smallest' m e a n s t h a t t h e ring labelled '1' is t h e next to be moved, w h e r e a s t h e proposition 'another' implies t h a t a ring different from the one labelled ' r is t h e next to be moved. The plan inherent in this F in effect selects a specific 'goal peg' a m o n g t h e two e m p t y pegs (this is directly reflected in t h e t h e o r e m below). T h e r e is r e a l l y n o t h i n g s u r p r i s i n g about that, b u t of course any of the two e m p t y pegs will do for a 'goal peg'. The only important point is t h a t one such m u s t be selected from the beginning.

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In o r d e r to d e m o n s t r a t e t h a t S O L V E holds w i t h t h e a b o v e definition of F we prove t h e following: Theorem:

(a) ( s m a l l e s t A state([1...k],[],[]) D F(2k-l, state(H,[1...k],[]) A another), w h e r e k is an odd integer (b) ( s m a l l e s t A state([],[1...k],[]) ~ F(2k-l, state(l],[],[1...k]) A another), w h e r e k is a n odd integer (c) ( s m a l l e s t A state([1. . .k],[],[]) D F(2k- l,state([],[],[1. . .k]) A another), w h e r e k is an even integer (d) ( s m a l l e s t A state([],[],[1...k]) ~ F(2k-l,state([],[1...k],[]) A another), w h e r e k is an even integer Proof: The t h e o r e m is p r o v e d b y m a t h e m a t i c a l induction. The p r o o f is i m m e d i a t e for k = l and k=2. Let us first a s s u m e t h a t n is e v e n , and let t h e i n d u c t i v e h y p o t h e s i s be t h e a s s u m p t i o n t h a t t h e t h e o r e m h a s b e e n p r o v e d for k = l . . . n . We m u s t now s h o w t h a t the t h e o r e m also holds for the n+l case. Since n is even we h a v e (smallest A state(J1.., n,n +l],[],[]) F(2n- l , s t a t e ( [ n +l],[],[1. ..n]) A another)

since t h e ring n + l does not influence t h e first 2n-1 steps. In t h e next step, however, this ring is moved. In consequence, we h a v e (smallest A state([1. . .n,n +l ],[],[]) F(2n)(state([],[n+ l],[1.., n]) A another)

B y i n s p e c t i o n into t h e rules for t h e F-plan one can a s s u r e o n e s e l f of the fact t h a t this ring n + l will not be moved a n y m o r e , and w e do not h a v e to p a y further attention to it in the following steps. For this reason it follows that

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(smallest A state([1.. .n,n + l],~,[]) F(2n- l +2n)(state([],[],[1...n,n +l]) A another)

i.e. (smallest A state([1. . .n,n +l],[],[]) F(2n+l)(state([],[],[1...n,n +l]) A another)

So, if n is even the induction step has been established. The case with odd n is similar. Q.E.D. The above theorem obviously implies SOLVE, and hence also Po SOLVE.

THEOREM PROVING AND DECISION PROCEDURES We have by now given an example of using temporal logic for reasoning about program properties. But it might be said t h a t the proof above gives nothing more t h a n one can have with a n 'ordinary' technique, in which mathematical induction is also used. There is, however, one important modification to s u c h objections: consider again the theorem SOLVE

~ state([1,...,N],[],[]) ~ F finish

Loosely speaking, this theorem states t h a t the F-plan can lead to the desired result, provided that the initial state is as specified, and moreover, t h a t in time it will indeed achieve this result. We m i g h t i m p l e m e n t t h e F - p l a n in PROLOG, or in s o m e algorithmic language, say Pascal. In fact, the Towers of Hanoi problem is a c o m p u t e r science classic; an example of a v e r y simple PROLOG solution m a y be found in [Clocksin a n d Mellish 1984, pp. 146], and a Pascal solution m a y be found in [Grogono 1978, pp. 102-103]. Now call the i m p l e m e n t e d program - in whichever language - P: if we can prove t h a t P satisfies SOLVE, we have proved t h a t P can a n d w i l l solve t h e problem. This is in contrast to non-temporal techniques. In general, n o n - t e m p o r a l techniques only describe w h e t h e r a

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p r o g r a m solves a problem correctly, if t h e program t e r m i n a t e s 'in t h e desired state'; this is called partial correctness. In such a f r a m e w o r k (for instance, so-called operational semantics), one h a s to resort to other techniques in order to prove also t h a t t h e p r o g r a m will indeed terminate. When both conditions: (i) P can lead (only) to the desired final state (ii) P will in time achieve this state are proved, we have proved w h a t is u s u a l l y called t h e total correctness of P. The point in connection with t e m p o r a l logic is t h a t SOLVE states both conditions in one single formula, a n d hence, t h a t a proof of this formula is immediately a proof of t h e total correctness. We shall illustrate this with one more example below. On t h e basis of t h a t example we shall conclude t h i s c h a p t e r w i t h some fairly general observations on t h e use of t e m p o r a l logic in computer science. A p a r t i c u l a r l y i m p o r t a n t issue w h e n computation is involved is t h e question of decidability. It is desirable to h a v e g e n e r a l p r o c e d u r e s of t h e o r e m p r o v i n g for t h e t e m p o - m o d a l logics w h i c h we would like to u s e for specification a n d r e a s o n i n g purposes. In computer science some results in this respects have been obtained. As an example we m e n t i o n the discrete t e m p o r a l logic s u g g e s t e d by M. Abadi and Z. M a n n a [1986] a n d f u r t h e r s t u d i e d by H. Bestougeff and G. Ligozat [1992, pp. 267 ff.]. We also m e n t i o n the b r a n c h i n g time logic of Ben-Ari et al. [1981], for w h i c h t h e a u t h o r s m a n a g e d to e s t a b l i s h s o m e nice decidability and complexity results. T h e t e m p o r a l logic w i t h which we described t h e F - p l a n is discrete, and it only treats one temporal 'direction', t h a t is, it does n o t i n c l u d e any operator for the past. M a n y of t h e t e m p o r a l logics s t u d i e d in c o m p u t e r science s h a r e these l i m i t a t i o n s , especially t h e latter limitation. However, in reasoning s y s t e m s as well as for m a n y practical purposes we would like to have c o n t i n u o u s logics with operators for t h e past as well as for the future. For most of these logics, however, we do n o t h a v e any g e n e r a l decision procedure, but only partial procedures valid for f r a g m e n t s of t h e logics in question - like the one s t u d i e d in

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[ 0 h r s t r C m & K l a r l u n d 1986]. It would be i n t e r e s t i n g to have such r e s u l t s for a t e n s e logic corresponding to Prior's t h i r d grade a n d t h e Leibnizian t e m p o - m o d a l logic w h i c h we have p r e s e n t e d in chapter 3.3 (some recent findings w i t h i n n o m i n a l tense logic [Blackburn 1993] seem promising in this respect). As a n a l t e r n a t i v e to general procedures for t h e o r e m proving, one m a y u s e s e m a n t i c a l models for the evaluation of t e m p o modal s t a t e m e n t s . Most of t h e temporal logics we h a v e been s t u d y i n g h a v e in fact been proved to be decidable, i.e. t h e y have the finite model property. This m e a n s t h a t a n y f o r m u l a A is provable in t h e logic if a n d only if it is valid in a n y f r a m e c o r r e s p o n d i n g to t h e logic. T h e computational complexities of decision p r o c e d u r e s of s u c h logic have been s t u d i e d a n d i m p o r t a n t results have been found (see for instance [Sistla a n d Clarke 1985]). One h i g h l y i n t e r e s t i n g result w a s f o u n d by H i r o a k i r a Ono a n d Akira N a k a m u r a [1980]. T h e y h a v e considered some of the most well known tense and m o d a l logics with t h e finite model property. Let L be one of t h e s e logics. We t h e n define a function rL such t h a t rL(m) is the smallest n u m b e r r which satisfies the following condition: For a n y formula A with m modal (or tense) operators, A is provable in L if and only i f A is valid in every L-model w i t h at m o s t r worlds. W i t h r e s p e c t to 'pure' tense logics Ono and N a k a m u r a have shown t h a t we have the inequality m+l_~r(m) ~_m+3 for Kb. Their r e s u l t s leave us with a higher degree of u n c e r t a i n t y w i t h respect to l i n e a r tense logic, since they have only been able to deduce t h a t t h e inequality m+l<_r(m) ~_2m+3 holds for Kl. At any rate, s u c h results are i m p o r t a n t for the i m p l e m e n t a t i o n of evaluation procedures for tempo-modal logics.

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ONE FURTHER EXAMPLE Consider the following program, which computes t h e greatest c o m m o n d e n o m i n a t o r (GCD) of two n a t u r a l n u m b e r s A a n d B; w h e n t h e p r o g r a m t e r m i n a t e s , A = GCD of t h e two original i n p u t values: LO: LI: L2: L3: L4: L5:

start read A; read B; i f A = B then goto LIO else goto L4; if A < B then goto L 5 else goto L8; C:= A,"

L6: A'= B; L7: B:= C; L8: A:= A-B; L9: goto L3; LIO: write A; L l 1 : end.

T h e p r o g r a m is w r i t t e n in a very s i m p l e p r o g r a m m i n g l a n g u a g e k n o w n as flowchart l a n g u a g e , w h e r e a label is a t t a c h e d to each instruction. An o r d e r i n g a m o n g labels is assumed, as indicated by their numbering. L0 is called t h e start label, and L l l is called a terminal label (in principle, there m a y be several 'termination points' in programs). Since the p r o g r a m is supposed to c o m p u t e the GCD of two a r b i t r a r y programs, we can specify this crucial property of the p r o g r a m by the following definitions and conditions: GCD(A,B) = g -~def (A m o d g = O) i (B m od g = O) A Vx • N. ((A m o d x = O f B m o d x = O) ~ x ~_g)

The input condition can be specified as follows: CI = {GCD(A,B) = g i A , B , g • IV]

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Strictly speaking, g ~ N is implied by the definition of GCD. It should also be noted that the program is w r i t t e n such that g=l is counted as a denominator of any pair of n a t u r a l numbers. The output condition is this one: Co = [A=gJ

Any program which takes some input and computes an output can be described as an input-output function. T h a t is, if P is the p r o g r a m and H is the corresponding i n p u t - o u t p u t function, t h e n /7 is the semantical m e a n i n g of P. T h e s e notions are customary in so-called operational semantics, which is the least abstract kind of formal semantics for p r o g r a m m i n g languages. O p e r a t i o n a l semantics for a given p r o g r a m m i n g l a n g u a g e c o n t a i n s g e n e r a l rules for s y s t e m a t i c a l l y c o n s t r u c t i n g t h e input-output function of any program w r i t t e n in t h a t language. We here ignore these rules (in [Andersen et al., forthcoming] the full set of such rules are given for the same example). Let X denote the set of possible inputs to P, and Y the set of possible outputs. In our current case, X = N x N and Y = N. We can now state quite precisely the crucial p r o p e r t y of P, n a m e l y t h a t it computes the greatest common d e n o m i n a t o r among two n a t u r a l numbers: (a) Vx ~ X, Vy e Y: (Ct(x) A II(x) = y) ~ Co(y). We can consider (a) to be the formal specification of P, where H is the input-output function corresponding to P. Moreover, (a) is a necessary condition of the correctness of P: if we can show t h a t C o obtains, whenever the terminal label L l l is reached provided t h a t the input condition CI is satisfied from the start t h e n we have shown t h a t the program is partially correct. (For t h e c u r r e n t example this can be shown by a fairly simple inductive proof.) In general, correctness is a relation between a specification and a program. The specification states w h a t the p r o g r a m should do, and a concrete p r o g r a m in s o m e p r o g r a m m i n g l a n g u a g e is m e a n t to i m p l e m e n t t h i s specification. If we can prove t h a t the p r o g r a m in question satisfies the specification, we have proved its correctness.

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However, (a) needs to be a u g m e n t e d in order to state t h a t L l l will indeed be reached. We therefore propose this specification instead, in which we avail ourselves of temporal notions: (b) Vx • X, Vy • Y: ((at LO) A Ct(x)) ~ F ( ( a t L l l ) A Co(y)). In this formula we still refer explicitly to P by referring to its labels; we have introduced a predicate 'at' to be able to do so. However, we need not refer to t h e input-output function of P. A proof t h a t P satisfies (b) is a proof of P's total correctness. - In the c u r r e n t case, we have to do with a deterministic program, and a linear t e n s e logic will suffice. But in m a n y cases, notably for indeterministic or concurrent programs, we can do better with a b r a n c h i n g time logic. Moreover, the linear cases can also be described within this framework. Now let P be any program w i t h input conditions ~, o u t p u t conditions ~, 'starting point' fl (where computation begins), and a set of terminal points E (where computation ends). A quite general criterion for the correctness of P can t h e n be stated as (c) Vx • X, Vy • Y: ((at fl) A ~o(x)) ~ ( N F ( ( a t e) A ~(y))), fore • E In fact, (c) does not refer to P at all. It can be used to refer to any program for which we have an identifiable 'starting point', and an identifiable set of 'termination points'. Loosely, we m a y read (c) as follows: if computation begins and the input conditions are satisfied, t h e n for all branches we 'reach' some terminal state e such t h a t the output conditions are satisfied (and computation ends). Note t h a t quantification over b r a n c h e s is implicit here. However, we omit the semantical details of this branching time

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logic, but these can be found in Ben-Ari et al. [1981]. General and t h o r o u g h overviews of how to use t e m p o r a l logic in verification can be found in [Emerson 1990, Stirling 1992]. Given any p r o g r a m P with ~0, ~, fl, a n d E as above, a proof t h a t P satisfies (c) is a proof of its total correctness. Such a proof m a y of course avail itself of all the techniques in any tense-logical axiomatisation of this branching time logic. W h e n i n v e s t i g a t i n g t h e properties of a p r o g r a m , we are interested not only in correctness, but also in being able to reason in general about its properties. For instance, in the loop from L3 to L9 t h e G C D - p r o g r a m m a y p e r f o r m a n u m b e r of subtractions. Hypothetically speaking, this could lead to A becoming negative. The 'swap operation' performed in L5-L7 is designed to p r e v e n t this. Again, we can use t e m p o r a l logic to describe this 'local' property of the p r o g r a m ((at L3) A (A > O)) ~ NG(A > O)

Here, we have also used the observation t h a t 'after' t h e L3-L9 loop, A is never changed. We could h a v e proceeded in smaller steps, first stating a weaker 'invariant' of the loop, a n d t h e n we could have d e d u c e d t h e above r e s u l t u s i n g o t h e r logical s t a t e m e n t s about t h e program. B u t at any rate, t h e above f o r m u l a holds a n d is a strong s t a t e m e n t of one i m p o r t a n t property of the GCD-program. Especially for reasoning about concurrent programs, t e m p o r a l logic h a s proved to be the suitable tool. Let P1 and P 2 be two concurrent processes, sharing some c o m m o n resource R - say, a printer, or t h e C P U for t h a t matter. In principle, P1 can be p r e v e n t e d from ever getting to use R by P 2 s n a t c h i n g it j u s t before P1, w h e n e v e r P1 requests it. If t h e concurrent p r o g r a m is to work with satisfactory results, t h e n the following condition m u s t be satisfied: requests(P,R) ~ NF(access(P,R)), where P is any of the processes involved.

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In g e n e r a l , t h e salient f e a t u r e s of concurrent p r o g r a m s can be described w i t h temporal logic, for instance • f r e e d o m of d e a d l o c k - e.g. if a n u m b e r of p r o c e s s e s simultaneously request the same resource, then the r e s o u r c e is allocated to one of t h e m (they do not all begin to w a i t for each other); • m u t u a l exclusion - it m u s t in some cases be prevented t h a t m o r e t h a n one process is allowed 'into' a certain 'region'; for instance, P1 and P2 cannot both w o r k on t h e s a m e p r i n t e r at t h e s a m e time (the results could be imagined); • f a i r n e s s - t h e p r o p e r t y briefly discussed above t h a t a n y p r o c e s s w h i c h r e g u l a r l y r e q u e s t s access to some r e s o u r c e does o b t a i n access sooner or later. This condition can be r e f i n e d in m a n y ways. A (conceptually) simple r e f i n e m e n t w o u l d be t h a t p r o c e s s e s a r e g r a n t e d access in t h e s a m e order as t h e y have r e q u e s t e d it; • l i v e n e s s - t h e p r o p e r t y t h a t a n y process w h i c h h a s b e e n t e m p o r a r i l y s u s p e n d e d is sooner or later resumed. This list could be prolonged significantly, and m a n y interesting q u e s t i o n s could be raised. The notion of concurrency is not only c o m p u t a t i o n a l l y i m p o r t a n t , b u t it also h a s c o n c e p t u a l ° information-theoretic - implications. J u s t for instance, who h a s t h e privilege of giving w h a t i n f o r m a t i o n w h e n ? W h a t can a n d should processes be capable of predicting a b o u t each other? B u t we shall leave these issues here.

PERSPECTIVES The u s e of t e m p o r a l logic in c o m p u t e r science is a large a n d r a p i d l y e x p a n d i n g field. We h a v e b u t s u g g e s t e d its more central uses a n d issues, and it m u s t be admitted t h a t even this h a s b e e n done only in a s k e t c h y m a n n e r . N e v e r t h e l e s s , it s h o u l d h a v e

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become evident that temporal logic is very versatile in computer science. It is useful, sometimes crucial, at all levels of computer science, which we might sum up as follows: • theoretical computer science: Temporal logic is useful for program verification, specification, and for reasoning about programs in general. In connection with concurrency it is crucial for such purposes. It is also the n a t u r a l formalism for expressing a generalised idea of total correctness. • programming languages and their theory: Here, the crucial question is t h e development of temporal p r o g r a m m i n g languages. This question is closely associated with the question of decision procedures for temporal logics (cf. F r a n k LeBke [1991]). Some t e m p o r a l logical p r o g r a m m i n g languages already exist. We mention Tokio [Fujita et al. 1986], T e m p u r a [Moszkowski 1986], a n d the work of Dov Gabbay [1987]. A useful overview is given by M. A. Orgun and W. Ma in [Gabbay and Ohlbach 1994, pp. 445-479]. A concomitant but more g e n e r a l question is how to characterise the temporal properties of existing programming languages, and perhaps to e s t a b l i s h t e m p o r a l l y m o t i v a t e d c r i t e r i a for e v a l u a t i n g languages. The use of temporal logic in program synthesis - the a u t o m a t i c g e n e r a t i o n of p r o g r a m s from more g e n e r a l specifications - is also under investigation [e.g. Emerson and Clarke 1982]. Such endeavours, if successful, would tie together specification, programming, and verification in a very fruitful way. • applications: The most obvious use of temporal logic in computer science is perhaps within the field of n a t u r a l language understanding. A large n u m b e r of fairly advanced 'information systems' also call for t e m p o r a l logic, for instance p l a n n i n g systems, decision support systems, and diagnostical systems. All these kinds of applications may be seen as cases of artificial intelligence. Our

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CHAPTER 3.7

discussions of the CIMP system (chapter 3.4) and of conceptual graphs (chapter 3.6) are also examples of this kind. • media systems: In some systems, w h e r e i n aesthetical and communicative properties are especially important, time assumes a crucial r61e. For instance, in hypertext systems or multimedia the conscious control of timing and montage is crucial. Such systems are best understood as well as designed with explicit reference to time (see e.g. [Andersen a n d OhrstrCm 1994]). (The g e n e r a l theoretical study of such 'sign production' has recently become known as 'Computer Semiotics'; see for instance [Andersen 1990], [Andersen et al., forthcoming], [Hasle 1993], [Hasle 1995].) • system development: As pointed out at the beginning of this chapter, temporal logic can be relevant in problem and domain analysis, as well as in design, implementation and validation. Its r61e in analysis and design is particularly related to its r61e in philosophical logic (analysis of concepts and language). Its possible r61e in implementation and validation is a direct consequence of its relation to theoretical computer science and 'programming languages'. At the beginning of this chapter we saw t h a t Prior himself in the 1960's anticipated the use of temporal logic in connection with computers. He also observed t h a t t h e r e might be some practical gains from the study of tenses "in the representation of time-delay in computer circuits" [TR, p. 4]. This r e m a r k also seems to anticipate its use in program verification and even in h a r d w a r e verification. At the other end of the spectrum, it is clear that the general linguistic and philosophical motivation for t e n s e logic explains its obvious r e l e v a n c e for artificial intelligence and a d v a n c e d information systems. T h u s in general, it would be in a good Priorian spirit to have logicians provide computer science with a collection of logical systems dealing with aspects of time, tense, and modality.

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W h a t does all this come down to? Needless to say, the present a u t h o r s s t a n d in e v e r y d a n g e r of o v e r e s t i m a t i n g t h e i m p o r t a n c e of t e m p o r a l logic, and a s w e e p i n g conclusion is indeed tempting: it would seem t h a t temporal logic, or perhaps we should merely say 'temporality', is pivotal w i t h i n computer science. It e x t e n d s in a vertical direction, r a n g i n g from f u n d a m e n t a l t h e o r y to applications, a n d in a h o r i z o n t a l direction, r a n g i n g f r o m analysis to v a l i d a t i o n in concrete s y s t e m development. B u t even if we, in deference to computer science proper, have to go for less, it is certainly no exaggeration t h a t t e m p o r a l logic h a s proved its practical w o r t h in m a n y areas within computer science.

4. CONCLUSION T h e logic of t i m e provides one of t h e most striking e x a m p l e s of a fruitful i n t e r a c t i o n b e t w e e n a v a r i e t y of disciplines, w h i c h a r e n o r m a l l y k e p t apart, m o r e or less strictly. Philosophy, logic a n d c o m p u t e r science h a v e played the k e y p a r t s in this i n t e r a c t i o n , b u t w e c a n well refine t h a t picture considerably: we h a v e (to v a r y i n g degrees) been d r a w i n g - as h a s t h e development of t h e logic of t i m e - on w h a t a p p e a r s to be v e r y diverse s o u r c e s , namely: • • • • • • • • •

g e n e r a l philosophy ethical and theological considerations conceptual analysis linguistic considerations l i t e r a r y fiction t h e history of ideas mathematics physics c o m p u t e r science.

T h e p i v o t a l discipline for l i n k i n g t o g e t h e r our v a r i o u s observations has been logic in a broad sense, t h a t is, logic in a ' p r e s y s t e m a t i c ' as well as a fully symbolical form. T h i s is in accordance w i t h the conviction stated in the introduction t h a t in order to s t u d y t i m e we n e e d to establish a common l a n g u a g e for t h e discussion, and t h a t such a l a n g u a g e should be d e v e l o p e d w i t h i n logic. H o w e v e r , logic is n o t ' m e r e l y ' a m e d i a t i n g language: t h e reason w h y the logic of t i m e can t a k e i n p u t from all t h o s e v a r i o u s fields, and also contribute to them, is in o u r conviction t h a t logic in its broad sense is really active in all s y s t e m a t i c h u m a n thinking. This is, however, not to be t a k e n in a s t r i c t psychological sense, but r a t h e r as a p h i l o s o p h i c a l s t a t e m e n t - a n d we add t h a t h u m a n r a t i o n a l i t y in our opinion s h o u l d be s e e n as c o m p r i s i n g m o r e t h a n logic, r e s p e c t i v e l y s y s t e m a t i c t h i n k i n g ( j u s t for i n s t a n c e , social i n t u i t i o n , aesthetical sense, rhetorical skill).

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In the case of t h e logic of time, we believe t h a t this subject c a n only artificially be s e p a r a t e d into one p a r t belonging to t h e h u m a n i t i e s a n d another part belonging to n a t u r a l science. To be true, for practical purposes t h e subject m a y be isolated into for instance one linguistic discipline and another computer science discipline; b u t t h e i r m u t u a l relevance should not be forgotten, a n d t h e e n t e r p r i s e of the logic of time should still be seen as a whole. This conviction m a y be provocative for traditional h u m a n i s t s as well as traditional natural scientists. We first take the case of opposing h u m a n i s t s : for quite some time and in a good m a n y places people b r o u g h t up within the h u m a n i t i e s have been told t h a t logic is completely i r r e l e v a n t in a field such as, say, l i t e r a t u r e , a n d o u t r i g h t m i s l e a d i n g w h e n applied w i t h i n linguistics. Now such assertions raise m a n y issues, w h i c h we shall not deal w i t h in any detail here. But we m a y r e m i n d t h e r e a d e r of j u s t one example. The analysis of Borges' short story 'The G a r d e n of F o r k i n g P a t h s ' d e m o n s t r a t e d how c e r t a i n logical ideas were anticipated in a piece of literary fiction - ideas, which are i n d e e d formalisable as well as technically applicable. On the other h a n d , it also showed how logical concepts can be applied in a literary analysis: we t h i n k it would be m u c h m o r e difficult to discern and present the ideas and t h e structure of t h e story w i t h o u t those concepts. - The Borges-example m a y s e e m biased to t h e e x t e n t t h a t the story in question lends itself to logical considerations in an u n u s u a l degree; and certainly, we did choose t h a t example because it is particularly striking. B u t we also t h i n k t h a t t h e kind of two-ways traffic exhibited in this connection is q u i t e general: logical analysis plays a r61e in systematic t h i n k i n g and is therefore, rightly, manifest also in t h e h u m a n i t i e s - m u c h more so t h a n is n o r m a l l y recognised. And conversely, symbolic logics to a very h i g h degree reflect or e m b o d y p r e s y s t e m a t i c a n a l y s e s of c o n c e p t u a l s t r u c t u r e s , philosophical problems and linguistic examples. The logic of time is a particularly evident example, but the same observation also applies to such p r o m i n e n t f o u n d a t i o n s of m a t h e m a t i c a l logic as Boolean Algebra and Frege's Predicate Logic - w h i c h

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F r e g e h i m s e l f called 'Begriffsschrift', m e a n i n g a p p r o x i m a t e l y 'conceptual w r i t i n g . W i t h these r e m a r k s we also anticipate an a n s w e r to sceptical n a t u r a l scientists, who m a y actually be using t e m p o r a l logic in t h e i r field, b u t who hold t h a t its philosophical a n d historical b a c k g r o u n d are really irrelevant to t h e p u r p o s e s w i t h i n t h e respective discipline. In fact, such a view was vividly expressed by [Ben-Ari et al. 1981], who were a m o n g t h e first c o m p u t e r scientists to systematically apply branching time systems within p r o g r a m verification. In discussing t h e linear t i m e a p p r o a c h v e r s u s the branching t i m e approach within the field, they stated that T h e difference in approaches has very little to do w i t h t h e philosophical q u e s t i o n of the s t r u c t u r e of physical t i m e w h i c h leads to t h e metaphysical problems of d e t e r m i n a c y [sic] versus free will. Instead, it is p r a g m a t i c a l l y based on t h e choice of t h e t y p e of p r o g r a m s a n d p r o p e r t i e s one wishes to formalise and study. [p. 164] In t h e end, the choice between linear and b r a n c h i n g models cannot be made on philosophical grounds b u t instead should be dictated by the type of programs, execution policies a n d properties which one wishes to study. [p. 165] T h e r e is a point here which is m u c h too common-sensical to be b r u s h e d aside easily. Clearly, the philosopher, t h e linguist, the physicist, or t h e c o m p u t e r scientist u s i n g t e m p o r a l logic m a y a n d m u s t a d a p t this logic to specific purposes. In doing so, t h e b a c k g r o u n d of t e m p o r a l logic can sometimes be ignored - a n d n o b o d y would call a scientist using t e m p o r a l logic in this w a y i n c o m p e t e n t , because he or she did not know Diodorus Cronus! N e v e r t h e l e s s , the conceptual b a c k g r o u n d as well as still n e w conceptual analyses have proved to be an i m p o r t a n t source - we say, a crucial source - for innovations a n d progress in t h e field of t e m p o r a l logic. Such n e w developments in t u r n will h a v e an i m p a c t on t h e sciences for which t e m p o r a l logic is useful. But t h e r e is more to it t h a n t h a t fairly utilitaristic a r g u m e n t . It

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seems to become ever more recognised in our age t h a t t h e classical division between 'hard' empirical science on one h a n d and more qualitative conceptual considerations on t h e o t h e r h a n d is highly mythical. In this connection we shall as our last example reflect on the notion of 'information systems'. Such s y s t e m s a r e clearly related to artificial intelligence - for instance, t h e CIMP system of c h a p t e r 3.4 is an i n f o r m a t i o n s y s t e m as well as a case of artificial intelligence. B u t 'information systems' m a y be u n d e r s t o o d as comprising a broader range of applications than 'artificial intelligence'. The t e r m 'information systems' refers to a certain class of computer applications, which is rapidly gaining in importance (indeed t h e notion is integral to the idea of an 'information society'). The very t e r m indicates t h a t attention is shii~ed a w a y from the u n d e r l y i n g computer architecture and towards t h e information content of the system. Of course, t h a t does not m e a n t h a t the underlying architecture has become u n i m p o r t a n t - t h a t part still ultimately defines the possibilities as well as limitations in the construction of such s y s t e m s . Moreover, t h e r e is no sharp boundary between the construction of a n i n f o r m a t i o n s y s t e m a n d a 'classical' p r o g r a m development: the latter also models some kind of information process. The difference is a m a t t e r of degree: in classical program development, the central activity is the construction of an algorithm. Such an algorithm specifies step-by-step how the computer is to carry out its computation. To that extent it is fair to call t h e process machine-oriented. In the construction of information systems, on the other hand, such considerations only e n t e r at a very late stage, if at all: the emphasis is clearly p u t on t h e m o d e l l i n g of i n f o r m a t i o n , and w i t h m o d e r n d e v e l o p m e n t tools t h e algorithmic aspect comes to be of secondary i m p o r t a n c e (for instance, w h e n fourth g e n e r a t i o n languages, expert system shells etc. are used). Therefore, an appropriate analysis of information in the domain in question is the crucial foundation of the entire construction process of an information system; to t h a t extent it is fair to call the process information-oriented.

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CONCLUSION

Logic in its broad sense provides a bridge b e t w e e n a p r e s y s t e m a t i c analysis of information and a formalisation w h i c h can be implemented. W h e n focusing on the relation between logic and the analysis of information in a domain, it is w o r t h recollecting Prior's view of logic: logic in his opinion "is not primarily about language, but about the real world" [TR, p. 1] (cf. c h a p t e r 2.5). In a domain we will fred real p h e n o m e n a and relations between them. Logical analysis is not 'just' a language game, but an attempt at singling out crucial p h e n o m e n a and to capture the relations between them. However, this endeavour at least w h e n it becomes systematical - presupposes t h a t we formulate our initial ideas about t h e domain in language, and therefore logical analysis is mediated by language. According to P e t e r Geach [1970, p.187] the young Russell as well as Prior held t h a t "ordinary language is not the logician's master, but it m u s t be his guide". Critics of logic have often contended that it is a study of highly artificial linguistic examples - indeed we have tried to show that such protestations were abroad already in the Renaissance and a cause of t h e downfall of Scholastic Logic. Such sentiments m a y even be s h a r e d by scientists and o t h e r professionals working in the field of information systems. Some of those m a y consider formal logic to be a useful language, but they m a y at t h e s a m e t i m e hold t h a t t h e a n a l y s e s c u r r e n t w i t h i n philosophical logic are somewhat esoteric and 'out of bounds' for their purposes. To meet such objections we first point out t h a t on Prior's - a n d our - conception of logic, it m u s t always be r e m e m b e r e d t h a t t h e linguistic examples m e d i a t e a study of real p h e n o m e n a and relations. This means t h a t logic does not h a v e to be empirically faithful to n a t u r a l l a n g u a g e in all respects, but on the other hand it does not render logic irrelevant to the s t u d y of language, for the latter also h a s to deal with reality in a logically reasonable way. Logical analysis of the kind we h a v e been s t u d y i n g in this book is r e q u i r e d for t h e construction of most information systems. We assert that logical analysis is information analysis. The latter, of course, comprises m o r e t h a n logic, for instance statistical m e t h o d s , but any i n f o r m a t i o n analysis which should lead to a computerised

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s y s t e m m u s t from some stage be logical (or t r a n s l a t a b l e into a logical analysis). And t h a t is also evident if we t u r n our attention a w a y from t h e d o m a i n a n d towards computer i m p l e m e n t a t i o n . S u c h obvious vehicles for t h e i m p l e m e n t a t i o n of i n f o r m a t i o n s y s t e m s as r e l a t i o n a l d a t a b a s e s and logic p r o g r a m m i n g are based on relational logic, also in a technical sense. In general, the d e v e l o p m e n t of ever h i g h e r level p r o g r a m m i n g l a n g u a g e s r e f l e c t s h o w a t t e n t i o n is i n c r e a s i n g l y d i r e c t e d t o w a r d s m o d e l l i n g i n f o r m a t i o n a n d away from r e f l e c t i n g i n t e r n a l c o m p u t e r architecture (cf. [Andersen et al. {forthcoming)]). R e c e n t d e v e l o p m e n t s such as object-oriented p r o g r a m m i n g a n d c o n s t r a i n t p r o g r a m m i n g e m p h a s i s e t h i s t r e n d a n d its c o n n e c t i o n w i t h logic. T h u s in object-oriented p r o g r a m m i n g one strives to identify objects (phenomena}, their properties and t h e relations between them. Furthermore, the crucial notions of g e n e r a l i s a t i o n a n d specialisation are e q u i v a l e n t to logical implication, or set inclusion; and the notions of i n t e n s i o n and e x t e n s i o n crucial in object-oriented analysis a r e i m p o r t e d directly from t h e logical tradition. - C o n s t r a i n t p r o g r a m m i n g also strives to identify logical properties within programs. T h e fact that logic is active both in the analysis of a domain and its i n f o r m a t i o n c o n t e n t , a n d in relational d a t a b a s e s , logic p r o g r a m m i n g , object-oriented p r o g r a m m i n g etc., explains w h y t h e l a t t e r are p a r t i c u l a r l y well suited for t h e c o n s t r u c t i o n of i n f o r m a t i o n systems. A n d the fact t h a t t h e s e p r o g r a m m i n g p a r a d i g m s are constantly gaining importance at t h e expense of classical algorithmic approaches reflects how t h e u s e of t h e c o m p u t e r is i n c r e a s i n g l y becoming a m a t t e r of i n f o r m a t i o n h a n d l i n g r a t h e r t h a n 'brute' data transformation, for which the a l g o r i t h m i c approach was ideally suited. To fully u n d e r s t a n d t h e s e d e v e l o p m e n t s one m u s t have knowledge of m o r e t h a n p r o g r a m m i n g l a n g u a g e s and their principles: one m u s t also u n d e r s t a n d t h e relation between logical l a n g u a g e s and their c o n c e p t u a l b a c k g r o u n d , as well as t h e i r h i s t o r y . T h e d e v e l o p m e n t of t e m p o r a l logic is a brilliant and e x e m p l a r y case in point, in its historical as well as systematical aspects. We consider temporal logic as a field worth s t u d y i n g in its own right, and this would be our conviction even it h a d no 'practical

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applications' at all. We hope that this interest in the subject for its own sake has been conveyed to the reader. Nevertheless, with the case of temporal logic in computer science we have shown how concepts and formalisms originally developed for entirely analytical purposes have proved their worth within applied science. Thus, the movement from the historical background (part one), via the formal development of tenselogical calculi (part two), and into computational applications (part three) may also serve as a demonstration of a more general point, namely that the philosophical analysis of concepts, language and logic is highly relevant to the field of information systems. It seems to us that this point is particularly well exemplified with reference to time. Therefore, exactly by emphasising the internal unity of the concept of time our study may have shown how 'time is ubiquitous'.

APPENDIX The following is a s u m m a r y of the systems discussed in the chapters 2.9, 2.10, and 3.3.

1.1 NON-METRICALA-LOGIC (TENSE LOGICS) Given a set of propositional variables (denoted p, q, r, ...) and a non-empty subset of this set i.e. the i n s t a n t variables (denoted a, b, c ...). T h e n t h e l a n g u a g e of n o n - m e t r i c a l A-logics can be p r e s e n t e d by t h e following f o r m a t i o n r u l e s for well-formed f o r m u l a s (wff): (1) Propositional variables, p, q, r . . . are wff (2) I f p and q are wff, then - p , p A q, Pp, Fp, Lp are also wffs. (3) Ifp is a wff, then Va:p is also a wff. (4) Nothing else is a A-wff. Abbreviations/definitions: Hp -~def ~P - p Gp -~def ~F-p (P ~ q) --def ~(P A~q) (p v q) ~-def ~(-P A~q)

SYSTEMKt Axioms: (A1) (A2) (A3) (A4) (A5)

p, where p is a tautology of the propositional calculus G(p ~ q) ~ (Gp ~ Gq) H(p ~ q) ~ (Hp ~ Hq) p ~ HFp p ~ GPp

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APPENDIX Rules:

(RMP) (RG)

(RH)

I f ~ p and ~p D q , t h e n ~ q . If ~p , t h e n ~ Gp. If ~p , t h e n ~ Hp.

SYSTEM Kb Add to Kt the axioms: (A6) FFp ~ Fp (A7) FPp ~ (Pp v p v Fp) Some theorems in Kb: (A6x) PPp ~ Pp (A7x) (Pp A Pq) ~ (P(p A q) v P(p A Pq) vP(Pp A q))

SYSTEM K~ Add to Kb the axioms: (A8) PFp ~ (Pp v p vFp) (A9) Gp ~ Fp (A10) Hp ~ P p (All) Fp ~ F F p Some theorems in K] : (A8x) (Fp A Fq) (A1 lx) Pp ~ PPp

(F(p A q) v F(p A Fq) v F(Fp A q))

SYSTEMPrt Definition: An instant proposition from Kt is any set of Kt-wffs, w h i c h is maximal and consistent with respect to Kt. Well-formed formulae (wff): (1) (2)

Any Kt-wffis a Prt-wff. Any i n s t a n t proposition from Kt is a Prt-wff.

APPENDIX (3)

I f a a n d fl are Prt-wiTs, and x is an i n s t a n t proposition, t h e n ~a, a A fl, Vx:a, Pa, and F a and all Prt-wffs. T h e r e are no o t h e r Prt-wffs.

(4) Abbreviation:

(._~a:p ) --~clef ~ ( )i/a : ~ P )

Axiom: (I1)

3a: a

Rule:

(RI) For a n y i n s t a n t proposition a and a n y w f f p : I f n o t ~ a ~ p , t h e n ~ a ~ ~p Prior's q u a n t i f i c a t i o n rules: (HI) I f ~ ¢(x)~fl t h e n ~ Vx:¢(x)~fl. (1-I2) I f ~ a~q~(x) t h e n ~ a~Vx:¢(x), f o r x not free in a. Deduced rules: (El) (Z2)

I f ~ ¢(x)~fl, t h e n ~ 3x:(p(x)~fl, for x not free in ft. I f ~ a~¢(x), t h e n ~ a~3x:¢(x).

THE SYSTEM~Vl~ort W e l l - f o r m e d f o r m u l a e (wff): (1) (2)

(3)

375

A n y Prt-wff is a Priort-wff. I f a a n d fl are Priort-wffs a n d x is an i n s t a n t proposition, t h e n -a, a A fl, La, and Vx:a are all Priort-wffs. T h e r e are no o t h e r Priort-wffs.

376

APPENDIX

Axioms: (L1) (I2) (I3) (BF) (LG) (LH)

L(p Dq) ~ (Lp DLq) -L~a L(a ~p) vL(a D-p) L(Va: ~(a)) ~- Va: L(~(a)) Lp ~Gp) Lp ~Hp)

Rule: (RL) If ~ p , t h e n ~ Lp In t h e same way, we can construct t h e s y s t e m s Priorb a n d Priorlfrom Kb a n d K1, respectively.

APPENDIX

377

1.2 NON-METRICAL TEMPO=MODAL LOGICS In t h e non-metrical tempo-modal logics the modal operator L is t a k e n into account. The s t a n d a r d modal logics are M, $4 and $5. T h e y can be presented as axiomatic systems.

SYSTEM M Axioms: (L1) (L2)

L(p ~ q) ~ (Lp ~Lq) Lp ~ p

Rule: (RL)

If ~p, t h e n ~ Lp.

SYSTEM S4 Add to M the axiom: (L3) Lp ~ LLp

SYSTEMS5 Add to $4 the axiom: (L4) - L - L p ~ Lp

THE McARTHUR SYSTEM Add to Kl the axioms: (L1) L(p ~ q) ~ (Lp ~ Lq) (L3) Lp ~ LLp (LG) Lp ~G p (LP) p ~ LPp, where p contains no occurrences o f F a n d t h e rule (RL)

If e p, t h e n ~ Lp.

378

APPENDIX

1.3. LEIBNIZIAN TENSE LOGIC L T Axioms of LT: (A1) (A2) (A3) (A4) (AS) (A6) (A7) (A8) (A9) (A10) (A11) (A12) (A13)

A, w h e r e A is a tautology of the propositional calculus G ( A ~ B) ~ (GA ~ GB) H ( A ~ B) ~ (HA ~ H B ) A ~ HFA A ~ GPA FFA ~ FA F P A ~ (PA v A v F A ) P F A ~ (PA v A v F A ) GA ~ FA HA ~ PA FA ~ FFA NGA ~ GNA PA ~ NPA, w h e r e A contains no occurrences o f F

In addition w e h a v e t h e $5 axioms for N: (N1) (N2) (N3) (N4)

N ( A ~ B) ~ ( N A ~ N B ) NAsA NA ~ NNA M N A ~ N A , w h e r e M -=clef - N ~

Rules in LT:

(RMP) (RG)

(RH) (RN) (nl) (n2)

I f ~A and ~ A ~ B, t h e n ~ B. I f ~A , then ~ GA. I f ~ A, then ~ HA. If ~ A, t h e n e N A . I f ~ ~(x)~fl, t h e n ~ Y'x:~(x)~fl. If~ o~O(x), t h e n ~ a~Vx:O(x), f o r x not free in a.

APPENDIX

379

1.4. NON-METRICAL B-LOGIC (INSTANT LOGICS) Let T I M E be a n o n - e m p t y set. The elements in T I M E are caUed instants, dates or j u s t times. Assume t h a t t h e r e is defined a relation, < , on T I M E . T h e expression tl < t2 is read 'tl is before t2'. The language i.e. t h e well-formed formula (wiT): (1) (2) (3) (4)

I f p is a propositional variable and t is an instant i.e. t e T I M E , t h e n T(t,p) is a wff. If T(t~) a n d T(t,q) are wffs and t~ and t2 are instants, then T(t,-p), T(t~o A q), tl < t2 are also wits. I f X a n d Y are w i t s and t e T I M E , then -X, X A Y, 3t:X, Vt: X , are also wffs. Nothing else is a wff.

We shall use the s a m e abbreviaitons and definitions as in t e n s e logics. B-logical tense-defintions: (DF) T(t, Fp) -----def-~l: (t
MINIMALB-LOGIC, Bm Axioms: (Wl) T(t,p A q) -- (P(t,p) A T(t,q)) (T2) T(t,~p) ----T(t,p) Some theorems in Bin: (DG) T(t, Gp) ~- VQ: (t
380

APPENDIX

BRANCHING TIME LOGIC Bb

Add the following axioms to Bra: (B1) (B2)

(tl < t2A t2 < t3) ~ t l < t3. (tl < t2A t3 < t2) ~ (tl < t3 v t l = t3 v t 3 < tl)

LINEAR TIME IA)GIC Bl:

Add t h e following axioms to Bb: (B3) (B4) (B5) (B6)

(t2 < tl A t2 < t3) ~ (tl < t3 v t l = t3 v t 3 < tl) V t i ~ 2 : tl < t2 Vtl~t2 : t2 < tl Vtl Vt23t3 : tl < t2 ~ (tl < t3 A t3 < t2)

EXTENDED B-LOGIC: A n y of t h e systems Bm,Bb, and B! can be e x t e n d e d by t h e axioms: (TXl) (TX2) (TX3)

(Vt: T(t,p)) ~ p (Vtl: T(tl,p)) ~ T(t2, Vt3: T(t3,p)) T(tl,p) ~ T ( t 2 , T ( t l , p ) )

and t h e rule: (RT)

I f e p , then e T(t,p) for a n y t

For t h e Leibnizian system we need the following B-logic: Definition:

A Leibnizian structure is a quadruple ( T I M E , < , _ T ) w h e r e T I M E is a non-empty set with two relations < and = s u c h that (B1)

(tl < t2 A t2 < t3) ~ tl < t3

APPENDIX (B2) (B3) (B4) (B5) (B6) (B7) (B8) (B9) (BIO)

(Q < t2 A t3 < t2) ~ (tl < t3 v t l = t3 v t 3 < tl) (t2 < tl A t2 < t3) ~ (tl < t3 v t l = t3 v t 3 < tl) V t ~ 2 : tl < t2 V t l ~ 2 : t2 < tl VtlVt23t3 : t l < t 2 ~ ( t l < t3A t3 < t2) t=t tl = t 2 ~ t 2 = t l (t~ -~t2 A t2 =t3) ~ tl -~t3 (t l =t2 A t3 < t2) D 3 t4: (t3 -~t4 A t4 < t l)

a n d a t r u t h operator T such t h a t (T1) (T2) (T3) (T4) (T5) (T6) (T7)

T(t~A A B ) - (T(t,A) A T ( t , B ) ) T(t, - A ) - - T ( t , A ) Vx:T(t~A) - T(t, Vx:A) where x is foreign to t. T(t, F A ) - 3 Q : (t < t~ A T(t~,A)) T(t, P A ) -~ 3t1: (tl < t A T ( t l , A ) ) T(t, N A ) - Vti: (ti -- t ~ T ( t l , A ) ) Vt~ Vt2: (ti ~ t2 A T(tl, P A ) ) ~ T(t2,PA)

where A contains no occurrences of F.

381

382

APPENDIX

2. METRICAL A-LOGIC(TENSELOGICS) The l a n g u a g e of MT is based on a set of propositional variables ( d e n o t e d p, q, r...) a n d a n o n - e m p t y set of i n s t a n t v a r i a b l e s ( d e n o t e d a, b, c ...). The set of w e l l - f o r m e d f o r m u l a e can be presented by the following definition: (1) (2)

(3) (4)

Propositional variables are wff. If a a n d fl are wff, and x is a positive number, t h e n

-a, a_~fl, aAfl, a v fl, Vx:a, La, P(x)a, and F(x)a are all wff. If a is a wff, t h e n Va:a is also a wff. T h e r e are no o t h e r wff.

We shall u s e t h e s a m e q u a n t i f i e r s y m b o l s for q u a n t i f i c a t i o n over n u m b e r s a n d i n s t a n t variables a n d w e shall use t h e s a m e a b b r e v i a t i o n s / d e f i n i t i o n s as in t h e s y s t e m s above w i t h t h e addition of t h e following definitions: (DGF) (DHF) (DUF) (DUG) (DUP) (DUH)

G(x)a --def -F(x)~a H(x)a --def -P(x)-a. Fp -~def3x:F(x)p Gp -~-defYx:G(x)p Pp --~def3x:P(x)p Hp -~defYx:H(x)p

The axioms of t h e s y s t e m MT are: (MT1) (MT2) (MT3)

G(x)(p ~ q) ~ (G(x)p ~ G(x)q) F(x)H(x)p ~ p F(y+x)p ~ F(y)F(x)p

APPENDIX

383

T h e rules of t h e s y s t e m MT are: (RM) The ' m i r r o r image' of a n y t h e o r e m (in w h i c h all occurrences of P a r e replaced by F a n d vice versa) is also a theorem.

(RMP) If ~A a n d ~ A ~ B, then ~ B. If ~ A, t h e n e G(x)A. (RF) (rI1) If ~ ¢(x)~fl t h e n ~ Vx:~(x)~fl. If~ a.~(x) t h e n e a~Vx:~(x), f o r x n o t free in a. (FI2) S o m e t h e o r e m s in MT: (MT4) (MT5) (MT6) (MT7) (MT8)

H(x)(p ~ q) ~ (P(x)p ~ P(x)q) p ~ G(x)P(x)p P(x)G(x)p ~ p Vx: G(y)G(x)p ~ G(y) Vx: G(x)p Vx: G(y)H(x)p ~ G(y) Vx:H(x)p

!JEIBNIZIAN METRIC TENSE LOGIC MLT T h e axioms of t h e s y s t e m MLT are: (LT1) (LT2) (LT3) (LT4) (LT5) (LT6) (LT7) (LT8) (LT9) (LT10)

G(x)(p ~ q) ~ (G(x)p ~ G(x)q) F(x)H(x)p ~ p F(y+x)p ~ F(y)F(x)p H(x)(p ~ q) ~ (H(x)p ~ H(x)q) P(x)G(x)p ~ p P(y+x)p ~ P(y)P(x)p F(x)-p - ~F(x)p P(x)~p - -P(x)p NG(x)p ~ G(x)Np P(x)p ~ NP(x)p, w h e r e p contains no occurrences o f F .

In addition w e a s s u m e t h e S5-axioms hold for N.

384

APPENDIX

3. THE SYSTEM OF PRIOR~S 3RD GRADE Given a set of propositional v a r i a b l e s (denoted p, q, r, ...) a n d a n o n - e m p t y subset of this set i.e. t h e i n s t a n t variables (denoted a, b, c ...). T h e n t h e l a n g u a g e of n o n - m e t r i c a l A-logics c a n be p r e s e n t e d b y t h e following f o r m a t i o n r u l e s for w e l l - f o r m e d f o r m u l a (wff):

(1) (2) (3) (4)

Propositional variables p, q, r ... are wff I f p and q are wff, t h e n -19, p A q, Pp, Fp, L p a r e also vcffs. I f p is a wff, then Va:p is also a wff. Nothing else is a wff.

Abbreviations]definitions as in 1.1 a n d in addition: ( D E ) (Ja:p) --=-def~(Va:~p) (DB) a < b -~defL(a ~ Fb ) (DT) T(a,p) --~-defL(a ~ p) Axioms for i n s t a n t variables: (I1) (I2) (I3)

Ja: a ~L ~a L ( a ~ p) v L(a ~ - p )

Some t h e o r e m s : ( D L ) 'Ca: T(a,p) - Lp (DG) T(a, Gp) - Vb:( a
APPENDIX

385

3 . 1 . INSTANT-LOGIC AND METRIC T E N S E - L O G I C SYSTEM MT* Add to MT an modal operator for which the following axioms hold: (BF1) (BF2) (LHX) (LGX)

Vx:L~(x) ~ L(Vx: ~(x)) 3x:L~(x) ~ L(3x: ~(x)) Lq ~ H ( x ) q Lq ~ G ( x ) q

Definition: (DB1)

before(a,b,x) -~-defL(a wF(x)b)

Some theorems in MT*: (L5) (L6) (T8) (T9) (T10) (Tll) (T12) (T13) (T14) (T15) (T16) (T17) (T18)

Lp Lp

Hp

before(a,b,x) - L ( b ~P(x)a) a
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INDEX A-theory 203 ft., 243-56 Abelard see: Peter Abelard Adams, Marilyn McCord 96 Adamson, Robert 112 Agricola, Rudolf 110 Albert of Saxony 42 Allen, J. F. 64, 303, 306, 311 ampliatio 9, 39-42, 112 Anselm of Canterbury 88, 90-95, 103, 107-108 Aquinas see: Thomas Aquinas Aristotle 3, 6, 10-14, 23, 24, 52, 57, 58, 69, 81, 88, 92, 117, 149-153, 170, 171, 193,243 Arnauld, Antoine 113 arrow of time 2 Ashworth, E. J. 85, 110, 112 Augustine 1, 4 Avicenna 72, 73 B-theory 203 ft., 243-256 backwards linearity 207 Bacon, Francis 114 Barcan Formula(e) 224, 229, 234 Becker, O. 23 Ben-Ari, M. 368 Berg, Harmen van den 320, 332 Bergson, Henri 119, 256,257 Bochenski, I. M. 70, 170 Boehner, P. 38, 89 Boethius 7, 72, 92, 95 Boethius de Dacia 35, 87 Boole, George 118, 119, 120, 124, 127, 171 Boolean Algebra 367 Borges, J. L. 120, 181-88, 367 branching time 31, 103, 120, 180-196, 207, 234, 281, 368 Broadie, Alexander 65, 68 Burgess, John P. 180, 202, 262 Buridan see: John Buridan 405

406

INDEX

Burleigh see: Walter Burleigh Capek, Milic 135 Carnap, R. 199 Carroll, Lewis 118 causalis 71 ceteris paribus clause 283 Chrysippus 19 Cicero 88 CIMP 285 communication 256, 258 completeness 214, 221, 236 computer science 241, 242 conceptual graphs 242, 387 connectives 72 consciousness 249 consequentia 38, 65, 68 contingent future 87, 141, 215, 266 contingent propositions 152 continuity 133 Copleston, F. 18 cosmic time 200 counterfactuals 282-302 Cresswell, M. J. 177 Crouch, R. 282, 328 cumulativity 307 databases 301, 320 De Interpretatione 10-14, 149, 153 De Morgan 130 deadlock 350 decision support 302 deontic logic 193 Descartes, R. 114 desinit 38, 52 determinism 28, 29, 120, 140, 141, 149, 241 diagnosis 302 dialectica 7 Diodorean implication 25 Diodorus Cronus 8, 15, 19, 23, 25, 33, 67, 103, 170, 171, 368

INDEX

407

Diodorus Cronus, the Master Argument of 33, 67, 103 dissectiveness 307 Dorp see: John Dorp Dowty, David 216, 303 Dumitriu, Anton 110, 114 durational logic 43-52, 74, 303-320 Edward, Jonathan 169 Einstein, Albert 199 empty time 132 Epictetus 15, 19 Epicureans 149 ethics 179 events 197 ft., 311 excluded middle 267 existential graphs 320 fatalism 28, 88 Faye, Jan 14 Findlay, John 167, 170, 171, 188 forward linearity 209, 235 freedom, human 9, 87-108, 116, 119, 139, 140, 241, 257-270, 368 Frege, Gottlob 118, 124, 367 future contingents 10-15, 87-108, 148, 264 future perfect 155, 159 Galton, Antony 64, 303, 319 Gardies, Jean-Louis 171 Geach, Peter 170, 172, 173, 370 God's omniscience 87-108, 116, 139-140, 265 Hale, Roger 303 Hamblin, C. L. 130, 176, 178, 305, 306, 307, 309 Hayes, Patrick J. 303, 306 Henry, D. 93 homogeneous 311 Hopkins, J. 90, 91, 95 hypotheticals 71 H¢ffding, H. 124 Ibn Sina, see Avicenna incipit 38, 52-64 indeterminism 107, 149, 241, 257-69

408

INDEX

indeterministic tense-logic 13, 257-69 instant propositions 221-230, 236-240, 272-281, 318, 373 interval semantics, see durational logic INUS-condition 287, 300 James, William 139, 140 Jespersen, Otto 158-166 John Buridan 36, 37, 43, 46, 49, 50, 68, 70, 72,74, 77, 303, 313 John Dorp 75, 76, 77 John of St. Thomas 42 Juan Luis Vives 110-112 Jungius, Joachim 114 J~rgensen, J~rgen 118, 124, 127 Kant, Immanuel 64, 117, 130 Kenny, A. J. P. 167, 170 King, Peter 66 Klocker, Harry R. 138 Kneale, W. and M. 14, 52, 70, 117 Knuuttila, Simo 53 Kraft, Jens 117 Kretzmann, Norman 52, 54, 56, 68 Kripke, Saul 120, 173, 178, 189, 194, 197, 261 Kroman, K. 122, 123 La Mettrie 257 Ladkin, Peter 303 Lavenham see: Richard of Lavenham law of contradiction 146 law of excluded middle 14, 145 Lawrence, N. 1, 2, 251 Leibniz, Gottfried Wilhelm 8, 89, 102-108, 113, 114, 115, 116, 185, 196, 269, 270 ft. Lemmon, John 173 Lewis, David 284, 285 Link, G. 83, 306 Lucas, J. R. 200 Lukasiewicz, Jan 12, 120, 124, 149-154, 171, 188, 193 MacKay, D. M. 259, 261, 265 Mackie, John 282, 285, 287 Malebranche 114

INDEX Mansel, H.L. 124 Massey, Gerald 201 materialism 140 Mates, Benson 15, 18, 23, 170 McCall, Storrs 149, 152, 192 McTaggart 172, 203, 216, 217, 243, 254-255 Melanchthon 112 Mendelssohn, Moses 64 metric tense logic 62, 231-40, 276 ft., 285, 382, 383, 385 Michael, E. 128 Michael, F. S. 16 Mill, J. S. 124 molecular propositions 71, 85 Montague, R. 124 Moody, E. A. 38 Moszkowski, Ben 303 Murdoch 53 natural language 1, 5, 7, 73, 110, 123-125, 145, 160-161, 165, 167, 173, 203, 216, 242, 256, 280, 293, 301, 303, 306, 319, 327-328, 345-346, 363-364, 370 natural scope 284 necessary future 15 ft., 137-138, 142, 143, 145, 146, 195, 247 Newtonian mechanics 259 Nicole, Pierre 113 Nishimura, Hirokazu 195, 268, 270 non-monotonicity 283 non-permanent existents 176 Normore, Calvin G. 39, 41 nowness 3, 43, 175, 192, 251, 255 Nuchelman, G. 33, 37, 113 object-oriented programming 371 Ockham see: William of Ockham Ockham-model 14, 108, 194 omniscience see: God's omniscience organon 6 passage of time 3, 192, 256 Paul of Venice 35, 54, 71, 84, 85

409

410

INDEX

Peirce,Charles Sanders 13-14, 119, 120, 124, 128-148, 171, 191, 262, 265, 320 Peirce-model 13-14, 147, 195 persistent 311 Peter Abelard 13 Peter Aureole 89 Peter Fonseca 85 Peter Lombardus 8 Peter of Spain 52 Peter Ramusll0, 112, 114 Pf~inder, Alexander 122 Philo of Megara 19, 67 physics 2, 141, 197-202, 251-254 planning 301 Plato 3, 6, 36, 78, 79 Poidevin, Robin Le 246 possible chronicles (futures) 145, 192, 195, 264, 269 Pnueli, A. 344, 347 predicate logic 367 Priest, Graham 41 principle of bivalence 13 principle of contradiction 144 principle of excluded middle 146 Prior, Arthur Normann 5, 13, 20, 32, 109, 119, 120, 127, 129, 139, 143, 152, 167, 194, 197, 198, 202, 203, 216, 217, 218, 231, 243, 304 program specification 242, 358 ft. program verification 358ff., 368 progressive time 134 prophecy 93, 97 quadrivium 7 quando 72 quantification 233, 236, 382 quantum physics 2, 141, 241 Read, S. 41 reality of time 255 Reichenbach, Hans 155-166, 172, 252 relativity 120, 198-202, 252 ff.

INDEX Renaissance 109, 110 Rescher, Nicholas 13, 18, 26, 29, 89, 191, 152, 218, 231 Richard Kilvington 54 Richard of Lavenham 12, 72, 53, 88, 98-102 RSper, Peter 310, 313 Rosenthal, Sandra B. 133 Rudolf Agricola 110 Salamucha, J. 153, 154 Scholasticism 8, 33 ft., 110 ft. Sedley, David 15 semiotics 128 Sextus Empiricus 19 Sherwood see: William of Sherwood Simons, Peter 303 simultaneity 197 ft. Smith, John E. 140 sociotemporality 3 sophism 36, 38, 50, 59-64 Sorabji, Richard 23 Sowa, John 320, 321 Spade, P.V. 72 tacit knowledge 4 Tarski, A. 153 temporal asymmetry 131, 256 temporal intervals 56, 303 ft. temporal logic programming 347 temporalis (temporal propositions) 9, 71, 72, 73, 79, 86 thermodynamics 2 Thomas Aquinas 34, 95, 95-96, 102, 265 Thomas Bradwardine 53 Thomason, Richmond H. 193 Tichy, P. 319 time and existence 91, 175 time of utterance 13, 155, 247 timeless knowledge 95-96, 251 Trentman, John A. 112 trivalent logic 153 trivium 7, 8, 128

411

412

INDEX

truth-value gap 247 Urmson, J. O. 111 Urquhart, A. 218 Valla, Laurentius 110 van Benthem, J. F. A. I~ 1 van Frassen, Bas C. 167 Vendler, Zeno 64, 303 Venn, John 127 Vives, Juan Luis 110, 111 Wagner, Gabriel 115 Walker, A. G. 304, 306 Walter Burleigh 38, 39, 41, 53, 67, 68, 71, 72, 81 Wegener, M. 200 Weizs~icker, C. F. von 253 Weyl, H. 252 Whately, R. 124 Whitrow, G. J. 200 William of Ockham 7, 12, 39, 40, 65, 66, 71, 72, 73, 77, 81, 83, 88, 97-102, 130, 141, 191, 262, 264, 265, 266 William of Sherwood 54, 55, 59, 62, 63 Zabrella, Jacobus 114

Some Important Logicians and Philosophers Aristotle (384-322 BC) Diodorus Cronus (340-280 BC) Boethius (480-524) Ibn Sina (Avicenna) (980-1037) Anselm of C a n t e r b u r y (1033-1109) Peter Abelard (c. 1079-1142) Peter Lombardus (c. 1095 - 1164) William of Sherwood (c. 1200-1270) Peter of Spain, later Pope John XXI (d. 1277) Thomas Aquinas (c. 1225-1274) Boethius of Dacia (ft. 1275) William of Ockham (c. 1285-1349) John Buridan (c. 1295-1358) Richard Kilvington (d. 1361) Richard of L a v e n h a m (d. 1399) Paul of Venice (c. 1369-1429) J u a n Luis Vives (1493-1540) Peter Ramus (1515-1572) Jacobus Zabrella (1533-1599) Francis Bacon (1561-1626) J o h n of St. Thomas (1589-1644) Joachim Jungius (1587-1657) R. Descartes (1596-1650) Antoine Arnauld (1612-1694) Gottfried Wilhelm Leibniz (1646-1716) George Boole (1815 - 64) Lewis Carroll (1832-1898) Charles Sanders Peirce (1839 - 1914) Gottlob Frege (1848-1925) Otto Jespersen (1860-1943) J a n [.ukasiewicz (1878-1956) Hans Reichenbach (1891-1953) A r t h u r N o r m a n n Prior(1914-1969)

413

Studies in Linguistics and Philosophy 40. C.TJ. Huang and R. May (eds.): Logical Structure and Linguistic Structure. Cross-Linguistic Perspectives. 1991 ISBN 0-7923-0914-6; Pb: 0-7923-1636-3 41. M J . Cresswell: Entities and Indices. 1990 ISBN 0-7923-0966-9; Pb: 0-7923-0967-7 42. H. Kamp and U. Reyle: From Discourse to Logic. Introduction to Modeltheoretic Semantics of Natural Language, Formal Logic and Discourse Representation Theory. 1993 ISBN 0-7923-2403-X; Student edition: 0-7923-1028-4 43. C.S. Smith: The Parameter of Aspect. 1991 ISBN 0-7923-1136-1; Pb 0-7923-2496-X 44. R.C. Berwick (ed.): Principle-Based Parsing. Computation and Psycholinguistics. 1991 ISBN 0-7923-1173-6; Pb: 0-7923-1637-1 45. F. Landman: Structures for Semantics. 1991 ISBN 0-7923-1239-2; Pb: 0-7923-1240-6 46. M. Sidedts: Indian Philosophy of Language. 1991 ISBN 0-7923-1262-7 47. C. Jones: Purpose Clauses. 1991 ISBN 0-7923-1400-X 48. R.K. Larson, S. Iatridou, U. Lahiri and J. Higginbotham (eds.): Control and Grammar. 1992 ISBN 0-7923-1692-4 49. J. Pustejovsky (ed.): Semantics and the Lexicon~ 1993 ISBN 0-7923-1963-X; Pb: 0-7923-2386-6 50. N. Asher. Reference to Abstract Objects in Discourse. 1993 ISBN 0-7923-2242-8 51. A. Zucchi: The Language of Propositions and Events. Issues in the Syntax and the Semantics of Nominalization. 1993 ISBN 0-7923-2437-4 52. C.L. Tenny: Aspectual Roles and the Syntax-Semantics Interface. 1994 ISBN 0-7923-2863-9; Pb: 0-7923-2907-4 53. W.G. Lycan: Modality and Meaning. 1994 ISBN 0-7923-3006-4; Pb: 0-7923-3007-2 54. E. Bach, E. Jelinek, A. Kratzer and B.H. P ~ (eds.): Quantification in Natural Languages. 1995 ISBN Vol. I: 0-7923-3128-1; Vol. II: 0-7923-3351-9; set: 0-7923-3352-7; Student edition: 0-7923-3129-X 55. P. Lasersohn: Plurality, Conjunction and Events. 1995 ISBN 0-7923-3238-5 56. M. Pinkal: Logic and Lexicon. The Semantics of the Indef'mite. 1995 ISBN 0-7923-3387-X 57. P. Ohrstrem and P.F.V. Hasle: Temporal Logic. From Ancient Ideas to Artificial Intelligence. 1995 ISBN 0-7923-3586-4

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Studies in Linguistics and Philosophy 20. F. Heny and B. Richards (eds.): Linguistic Categories: Auxiliaries and Related Puzzles. Volume II: The Scope, Order, and Distribution of English Auxiliary Verbs. 1983 ISBN 90-277-1479-7 21. R. Cooper. Quantification and Syntactic Theory. 1983 ISBN 90-277-1484-3 22. J. Hintikka (in collaboration with J. Kulas): The Game of Language. Studies in Game-Theoretical Semantics and Its Applications. 1983; 2nd printing 1985 ISBN 90-277-1687-0; Pb: 90-277-1950-0 23. E.L. Keenan and L. M. Faltz: Boolean Semantics for Natural Language. 1985 ISBN 90-277-1768-0; Pb: 90-277-1842-3 24. V. Raskin: Semantic Mechanisms of Humor. 1985 ISBN 90-277-1821-0; Pb: 90-277-1891-1 25. G.T. Stump: The Semantic Variability of Absolute Constructions. 1985 ISBN 90-277-1895-4; Pb: 90-277-1896-2 26. J. Hintikka and J. Kulas: Anaphora and Definite Descriptions. Two Applications of Game-Theoretical Semantics. 1985 ISBN 90-277-2055-X; Pb: 90-277-2056-8 27. E. Engdahl: Constituent Questions. The Syntax and Semantics of Questions with Special Reference to Swedish. 1986 ISBN 90-277-1954-3; Pb: 90-277-1955-1 28. M.J. Cresswell: Adverbial Modification. Interval Semantics and Its Rivals. 1985 ISBN 90-277-2059-2; Pb: 90-277-2060-6 29. J. van Benthem: Essays in Logical Semantics 1986 ISBN 90-277-2091-6; Pb: 90-277-2092-4 30. B. H. Partee, A. ter Meulen and R. E. Wall: Mathematical Methods in Linguistics. 1990; Corrected second printing of the first edition 1993 ISBN 90-277-2244-7; Pb: 90-277-2245-5 31. P. G~denfors (ed.): Generalized Quantifiers. Linguistic and Logical Approaches. 1987 ISBN 1-55608-017-4 32. R.T. Oehrle, E. Bach and D. Wheeler (eds.): Categorial Grammars and Natural Language Structures. 1988 ISBN 1-55608-030-1; Pb: 1-55608-031-X 33. W. J. Savitch, E. Bach, W. Marsh and G. Safran-Naveh (eds.): The Formal Complexity of Natural Language. 1987 ISBN 1-55608-046-8; Pb: 1-55608-047-6 34. J. E. Fenstad, P.-K. Halvorsen, T. Langholm and J. van Benthem: Situations, Language and Logic. 1987 ISBN 1-55608-048-4; Pb: 1-55608-049-2 35, U. Reyle and C. Rohrer (eds.): Natural Language Parsing and Linguistic Theories. 1988 ISBN 1-55608-055-7; Pb: 1-55608-056-5 36. M.J. CressweU: Semantical Essays. Possible Worlds and Their Rivals. 1988 ISBN 1-55608-061-1 37. T. Nishigauchi: Quantification in the Theory of Grammar. 1990 ISBN 0-7923-0643-0; Pb: 0-7923-0644-9 38. G. Chierchia, B.H. Partee and R. Tumer (eds.): Properties, Types and Meaning. Volume I: Foundational Issues. 1989 ISBN 1-55608-067-0; Pb: 1-55608-068-9 39. G. Chierchia, B.H. Partee and R. Turner (eds.): Properties, Types and Meaning. Volume II: Semantic Issues. 1989 ISBN 1-55608-069-7; Pb: 1-55608-070-0 Set ISBN (Vol. I + II) 1-55608-088-3; Pb: 1-55608-089-1

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