Theory of Language Syntax: Categorical Approach

THEORY OF LANGUAGE SYNTAX

Nijhof International Philosophy Series

VOLUME 42

General Editor: JANT .J .SRZEDNICK Editor for volumes on Logic and Applying Logic : STANILWJ Editor for volumes on Contributions to Philosophy : JANT Assistant to the General Editor : DAVI WOD

. SURMA .J .SRZEDNICK

Editorial Advisory Board : R .M . Chisholm (Brown University, Rhode Island) ; Mats Furberg (Goteborg University) ; D .A.T. Gasking (University of Melbourne) ; H .L.A. Hart (University College, Oxford) ; S . Korner (University of Bristol and Yale University) ; H .J . McCloskey (La Trobe University, Bundoora, Melbourne); J . Passmore (Australian National University, Canberra) ; A. Quinton (Trinity College, Oxford) ; Nathan Rotenstreich (The Hebrew University, Jerusalem) ; Franco Spisani (Centro Superiore di Logica e Scienze Comparate, Bologna); R. Ziedins (Waikato University, New Zealand)

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

Urszula Wybraniec-Skardowska

Theory of Language Syntax Categorial Approach

WKAP-ARCHIEF

I NM KLUWERACDEMICPUBLISHER DORDRECHT / BOSTON / LONDON



Library of Congress Cataloging-in-Publication Data Wybraniec-Skardowska, Urszula . [Teorie ,igzykow syntaktycznie kategorialnych . English] Theory of language syntax : categorial approach / by Urszula Wybraniec-Skardowska . p. cm . -- (Nijhoff international philosophy series

: v . 42) Translation of : Teorie ,igzykow syntaktycznie kategorialnych . Includes bibliographical references and index .

ISBN 0-7923-1 42-6 1 . Categorial grammar . 2 . Formal languages . and Mathematical . I . Title . I . Series . P161 .W913 1991

3 . Logic, Symbolic

415--dc2O

91-64

ISBN0-7923-142-6

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.JunkadMTP res

.

Sold and distributed in the U.S .A . and Canada by Kluwer Academic Publishers, 101 Philip Drive, Norwell, MA 02061, U.S.A . In al other countries, sold and istributed by Kluwer Academic Publishers Group, P.O. Box 322, 3300 AH Dordrecht, The Netherlands .

Printed on acidfree paper

This book has originally been published in Polish, entitled Theorie Jgzyk6w Syntaktycznie Kategorialnyck with PWN, Warsaw, 1985. Translated from the Polish by Olgierd Adrian Wojtasiewicz

All Rights Reserved © 1991 Kluwer Academic Publishers No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner . Printed in the Netherlands

To the m ory of Jerzy Sfupecki



CONTENTS

Introduction

ix 1 . Main assumptions, objectives, conditionings ix 2 . Intuit ve foundations



xv

ChapterI The Axiomatic Theory TLTk of Label Tokens I Sec .I

.i . Primitive concepts

1

Sec .I

.2 . Label tokens and their equiformity 3

Sec .I

.3 . Concatenation

Sec .I

.4 . Vocabulary

10

Sec .I

.5 . Word tokens

12

Sec .I

.6 . n-componential words ; length of word 17

Sec .I

.7 . Generalized concatenation

23

Sec .I

.8 . Methodological remarks

29

5

ChapterI The Axiomatic System TSCL of Simple Categorial Languages Sec .I

. 32

.1 . The principal objectives of construction the theory of syntactically categorial languages . 32

Sec .I

.2 . Connections between TLTk and TSCL 37

Sec .I

.3 . Categorial indices and their indication 42

Sec .I

.4 . Expressions

46

Sec .I

.5 . Well-formed expressions

51

Sec .I

.6 . Syntactic categories

61

Sec .I

.7 . The fundamental theorems of the theory of syntactic categories

66



viii

CONTENTS Sec .I

.6 . The algorithm of checking the syntactic correctness of expressions 78

ChapterI The Theory TSCe-L of Categorial w-languages 90 Sec .I

.1 . Introductory remarks



90

Sec .I

. 2 . The foundations of TSCe-L 95

Sec .I1

.3 . Operator expressions

Sec .I

.4 . Well-formed expressions

103

Sec .I

.5 . Fundamental theorems

109

Sec .I

.6 . The algorithm of checking the syntactic

97

correctness of expressions 120 ChapterIV Dual Theories Sec .IV

132

.1 . The double ontological character of linguistic objects and the biaspectual approach to language

132

Sec .IV

.2 . The theory TL Tp of label types 139

Sec .IV

.3 .

Interpetaioni

TLTp of Tarski's axioms of

metascience

148

Sec .IV

.4 . The theory TETp of expression types 151

Sec .IV

.S . The dual theory DTSCL

164

Sec .IV

.6 . The dual theory DTSCe-L

171

Final Remarks

181

Annex

186

Notes

233

. .

.

References

237

INTRODUC

1 . Main assumptions, objectives and conditionings

1 .1.

The present book is concerned with certain problems

in the logical philosophy spirit

the

of

tradition, categorial

and

Polish shows

of language .It logical,

two

languages :

the

is written in the

philosophical,

conceptions theory

of

of

and the

simple

semiotic syntax

languages,

of i .e .,

languages which do not include variables nor the operators that bind them (for instance, large fragments languages Aristotle's definable

of

well-known

traditional algebras),

and

languages which include latter
calculi,

sentential syllogistic, the

of natural languages, the

languages

theory

of

language

of

of

equationally

w-languages, i .e .,

operators and variables

bound by the

the language of the functional

calculus

and languages of the theories based on that calculus) . The considerations on languages to be found in this book are very general and, e .g ., do not depend on the notation used in the

recording

of

expressions

in

the

languages

under

consideration. The influence of Polish logic

is reflected in the

strictly

formal manner of presentation, and in particular in the use of the axiomatic method . It is com only known that those important factors dominated the enquiries of eminent

representatives of

the Warsaw school of logic, such as Stanislaw ix

Lesniwski,Jan



x

INTRODUC

Lukasiewicz, Alfred Tarski, Jerzy Slupecki, and others

.

Reference categorial

to

Polish

semiotics

approach to language,

manifests i .e.,

itself

in

the

the approach which in

the syntactic description of language takes into account the general

principles

of

Leigniewski's

theory

of

syntactic

categories [1929,1930] in the version improved by Kazimierz A jdukiewicz [1935] . It is worth noting that Le6'niewski's theory of syntactic categories was constructed under the influence of the ideas of pure grammar, advanced by Edmund Husserl [1900,1901], and also under the influence of the concept of logical type, for the language of protothetics and ontology [1929,1930] . That formulates

general

principles

of

the

theory

classification

of

expressions into syntactic categories, its main objective being the

formulation of criteria of the meaningfulness,that is

well-formedness of expressions . In that spirit it was developed or used by Bochei'iski

Tarski [1933], [1947],

A jdukiewicz [1935],

Tadeusz

Kubii ski

InocetyM

.

[1958],

Witold

Marciszewski (1977, 1978a], and others . For similar purposes, although independently of Les'°niewski, a theory of syntactic categories was presented and developed for the needs of

the

so-called combinatory logics by Haskell B . Curry [1961,1963] . The categorial approach to language, used in the present book, refers to A jdukiewicz's conception [1935] and also to his ideas to be found in [1960], and is based on the method of indexation of expressions, that is the method of assigning them categorial

indices

and

the

resulting

defining

of

their

INTRODUCTION xi syntactic categories . the i .e .,

algorithm

method

makes

which allows one to

syntactic

terminology,

That

correctness, their

of

syntactic

it

examine

possible

the

expressions

to

apply

well-formedness, Cin

connectedness>.

A jdukiewicz's

The

categorial

approach makes it possible to treat language as a language generated by

the appropriate

categorial

whose idea

grammar,

goes back to A jdukiewicz [19357 . The unquestionable achievements of Yehoshua Bar-Hillel [1950, 1953,1964], who shaped the concept of categorial grammar and popularized Ajdukiewicz's conceptions, and thus made a great contribution

to

the

development

of

the

foundations

of

categorial grammars, and the equally unquestionable attainments in

that

respect

of

Joachim

Lambek

[1958,19617

and

the

continuators of these two prominent researchers, bear a certain relation to the approach of the foundations of categorial languages suggested in this book, even though they do no play any essential role . The philosophical trend of the theoretical reflections to be found in the present book manifests itself in the biaspectual treatment of language : as language

language of expression-tokens

of expression-types,

and as

the latter being interpreted in

an abstract way . This is accompanied by the claim that it is the expression-tokens, accessible to sensory cognition, which are the basis of considerations on language, while the abstract expressions

(expression-types),

of

which

the

former

are

representations, are derivative and definable constructs . This approach

is

linked

to

the

nominalistic

Cconcretistic)

xi INTRODUCTION standpoint

in

the

philosophy

particular

by

such

Polish

S .Ledniewski,

T .KotarbirSski,

language, represented in

of

logicians and

and

also

philosophers

J .Slupecki

as

(during

the

last years of his life) . These aspects which determine the main assumptions this

book,

namely

philosophical both

aspects,

theories

categorial

the

logical,

result

of

in

a

categorial

grammars .

the

linguistic,

two-level

languages,

Formalization at

the

made in and

the

formalization or

theories

first

of of

level pertains

to languages of expression-tokens and yields the theory TSCL of simple

syntactically

theory TSCw-L I)

.

of

the

Formalization

expression-types DTSCL

categorial

w-languages at

and

languages

the

yields,

and DTSCca-L (Chap .IV)

.

of


such

second

and

the

expressions
level

respectively,

pertains

the

dual

Those theories are

to

theories

expansions of

the relevant theories formalized at the first level. One

of the

relationships

basic objectives between

is

theories

to demonstrate formalized

at

the

mutual

these two

different levels .

1 .2.

Comprehension of the formalisms which yield the

said

theories requires only an elementary knowledge of mathematical logic in the broad sense of the term, but it assumes a certain skill

in

the

consistent

apparatus .

This

is

why,

acquainted

with

the

formulate

below


application in order

to make

subject matter Sec .2)

the

of

of

intuitions

the the the

formal

logical

Reader

better

book

we

pertaining to

shall the

INTRODUC philosophical to

the

formalisms

explaining theories book

and linguistic aspects

the of

referring

main

the

should

themselves .

also

ideas

syntax be

They

underlying

of

by

latter

are,

in

the

used

and

prove

useful

in

two

The

suggested

formal

understanding

numerous

to intuition, in most cases definitions,

framework

may

the

language .

facilitated

recordings of axioms, the

of the

verbal

of

the

comments,

accompanying the

symbolic

and

theorems


proofs

of

overwhelming

majority,

transferred

to

the Annex to make the perception of the main text easier) .

1.3.

The thus outlined two-level

system of

treating both

theories of language has, as it were, its precedent in the theory of algorithms formulated by A .A .Markov [1954], which, however, is not an axiomatic theory . An endeavour to axiomatize it was undertaken by Jerzy Slupecki

. It was on his in tiative

and under his guidance, and with reference to Markov's ideas, that the axiomatic theory of concrete and abstract words was formulated Csee C3 .Bryll and S .Miklos 11977]) .Itwasfol wed

-

for other purposes, connected with the theory of syntactic categories

-

by the axiomatic theory TLTk of label tokens,

constructed by the present author Csee 11981a,1983a]) . These goals coincided in the 1970's with the revived interests in categoriel grammars, which had their semantic and philosophical grounds. The fact was stressed by Witold Marciszewski [1978b] and found manifestations in publications of such authors as P .Geach R .Montague

11970],

M.J .Cresswell

(1970,19731,

and

[1972,1976],

A .Nowaczyk

11978] .

D .Lewis That

Ci9721, trend

in



xiv

INTRODUC

research had been anticipated by R.Suszko 11958,19601 . The theory TLTk has become the basis for considerations on syntactically categorial languages . It also forms the core of all

the

theories

of

language

presented in this book and

mentioned previously . It is hown in Chap

.I .Itisalsoan

outline of the intuitions connected with its formalization that opens the preliminary presentation of the contents of the book (Subsection 2 .2) .

1.4 .

But

first

a

brief

explanation is needed of the

relation which the English-language version bears to the Polish original . The essential content of the book has remained unchanged . Only the Introduction, the References and the Final Remarks (which now form a separate item that follows Chap .IV) have been expanded. This has been done for three essential reasons : to make the book accessible to a much broader category of Readers, to bring it closer to the world literature of the subject, and to outline new solutions thematically connected with the original version of the book . This, it seems, places the book in a better perspective, especially in the light of the recent achievements in the field of categorial grammars, whose

turbulent

development

originated

in

the

1950's

and,

revived in the 1970's, has continued up till now .Itmustbe added here that this development has been influenced by the still vital research on formal grammars in the spirit of Noam Chomsky, going on incessantly since the publication of his book 119571 that marked a breakthrough in linguistics .



INTRODUCTION xv It

would

referring the

not

sphere

I

possible

here

to

indicate,

with

a

view to

the present book to historical and present studies in of

coincidences why

be

categorial

between

take the

this

grammars, book

the the

and

liberty to refer

the

differences said

Readers

studies . above

and This

all to

the is the

entry "Categorial grammar" in the Dictionary of Logic edited by Marciszewski [1981], the anthology bearing the same title and edited by Buszkowski, van Benthem, and Marciszewski [1988], and the selected publications by the leading contemporary students of

categorial

andJ

grammars,

W .Buszkowski

[1982a-c,1984,1985,1986]

.van Benthem [1984,1986], which appeared when the Polish

version was

being

prepared

for

the

press

or after its

publication . The essential part of the original version was ready in 1981, and the main ideas of the underlying theoretical conceptions were presented at the Conference on the History of Logic <see [1981b]>,

held

production

of

in

Cracow Kazimierz

and

dedicated

Ajdukiewicz .

to The

the Polish

scientific version

appeared in 1985 . Profes or Jerzy Slupecki, who sug ested to me the problems analysed in it, had died before the publication of the English-language version . Let me, -therefore, dedicate it to his memory .

2. Intuit ve foundations

2.1.

The controversy over the problem of the double

ontological nature of linguistic objects is linked to the

well

xvi INTRODUCTION known

controversy

over

universals .

The

preceding sentence are concrete physical sight .

The

first

sentence

within

the

second

and

in

the

this

quotation

treated as concrete

twentieth paragraph,

mark

in

egvi form.

the

words

written

objects

word like

term

the

perceivable by

which the

in

occur

word

in

the

occurring

"controversy"

may

be

representations of that name . That

name is often treated as the name of a certain abstraction,a word

type,

that is the class of all the words which are

equiform with the word in the quotation marks . The class of such concrete words is then an abstract word, or a certain ideal entity . The controversy over the ontological nature of the objects designated by the name "controversy" arises when we want to decide which of the kinds of words denoted by that name has original existence, and which has derivative existence . Without making any decision in that matter we shall point to the fact that the demonstrated duality in the interpretation of words or expressions of language, which has become common especially owing to the important study of A .Tarski [1933], takes place in logic and semiotics . Note also that following the distinction between [1931-1935]

(see

also

token

and

Y .Bar-Hillel

made

by C .S .Peirce

[1953,1954])

inscriptions,

type

words, expressions understood as concrete objects are included in the class of

tokens,

and when understood as classes of

equiform tokens -

are included in the class of

types

CR .Carnap

[1942] used, respectively, the terms events and designs . In ac ordance with the wel known division made by Ch

.Morris

[1938] logical semantic analysis can be carried out in three

INTRODUCTION xvi aspects : the syntactic, the

semantic,

and

the

pragmatic .

Such

an analysis is sometimes applied to concrete and sometimes to abstract

linguistic

objects.

For

instance,in

syntactic

studies,

when formulating rules of constructing coherent expressions we make use of expression types, but if we want to examine the grammatical expressions

correctness, and

use

i .e.,

for

syntactic

that

connectedness,

purpose,

e .g .,

of

A jdukiewicz's

algorithm [1935] we make use of tokens . At the pragmatic level linguistic analysis

pertaining

to the functioning

is linked to the use of expressions situations numerous

(see

J .Pelc

functions

emotiveness,

of

linguistic

suggestiveness,

in definite contexts and

[197113 .

[1967],

Reference

expressions

decisiveness>,

tokens .

On

the contrary,

same

when

it

comes

to

(descriptivity,

functioning

basic

of

in

is based on the

(potential inscription) . such

the

requires the use

the

pragmatics of the concept of communication concept of expression type

to

substantiation

statements, and in particular theorem proving of expression

of language

semantic

Itishe

concepts

as

designation, denotation, truth, and meaning . The said ambiguity in the use of linguistic

inscriptions in

the analysis of language has also its broader dimension. This is so because in the literature of the subject it is not always clear whether expression

type

is understood as

equiform tokens, sometimes there is even no concept of equiformity even though the expression

types are

meant .

a class of

reference to the

context indicates that

W.Marciszewski

claims

concept of type has several different interpretations

that

the

and need



INTRODUC

Xviii

not be understood in an abstract way . Note in this connection that when speaking about the dual ontological nature of linguistic tokens

and

types

in

objects

we

shall refer

to

accordance with the distinction made by

Peirce . The dual ontological nature of linguistic inscriptions and the possibility of using them in two ways in semiotic analysis results in the necessity of the biaspectual description of language

in

the theoretical, logical conception of language, as

a language of expression tokens and a language of expression types . Yet in formalizing a given conception of language we must choose either tokens or types as the point of departure . In the two ap roaches to the theory of language sug ested here we refer to the nominalistic Cconcretistic) interpretation and construct those theories first as the theories TSCL and TSCarL of languages of expression tokens (the first level> ; in doing so we start from token

(label

as a primitive,

token>

undefined, concept and several other concepts connected with the former . When constructing those theories as theories of expression types (the second level) we obtain the dual theories DTSCL and DTSCCrL,

respectively,

in

which

the

fundamental

concepts include such abstract concepts as label type, word type,

expression

type,

which

are

defined

and

derived

constructs .

2.2.

The theory TLTk of label tokens (Chap .I)isthepoint

of departure for the formalization of both conceptions of the

INTRODUCTION xix syntax

of

language .

primitive,

undefined,

appropriate the

In

that

theory

whose

meaning

axioms . They are :

relation

::~: of

four

the

set

the

equiformity,

is Lb

concepts

appear

explained of all

vocabulary

by

label

as the

tokens,

distinguish

V,

from the set Lb, and the relation C of concatenation . The set Lb is the universe of an arbitrary but fixed simple language

W

or

w-language

w- .x',

expressions . The elements of

which

Lb,

called

has

tokens

label

as

its

tokens,

are

concretes and may (but need not), be inscriptions . Label tokens are any objects perceivable by sight, extensive in time and space, from which by

egvi formi ty

one can obtain elements of

expressions of graphic languages . Out of label tokens we choose the simplest words of the language .

or w-X, namely the words

which form the vocabulary V of .LP or G)-X, and from them in turn we form compound words of that language, the

expressions,

including the well-formed expressions of which consist X or w-X_ Label tokens also include the categorial indices which make it possible to determine the syntactic categories of expressions, to demonstrate their syntactic roles, and to examine grammatical coherence . The existence of label tokens and their varietes listed above follows from the axioms adopted in the theories TSCL and TSCw-L . The tokens .

relation of equiformity

It

is

holds between pairs of label

interpreted very broadly by imparting it a

pragmatic sense : it depends on the objective which we have in view in order to obtain definite results . The equiformity of label tokens may, but need not, be determined by physical

xx

INTRODUC

similarity .

For

instance, for

a

visual

artist,

a

graphologist,

or a printer the concrete words : ontology, ontology, ONTOLOGY Glower

case,

lower

case

italics,

capitals)

are

diversiform,

whereas from the point of view of a person who is learning the meanings of words they may be treated as equiform . On the other hand,

in

categorial analysis

correctness

of

expressions,

intended to

examine the

for •instance

of

the

syntactic

tautological

sentence The creative man is creative the adjectives which occur in it may be treated as diversiform in view of their enviroment : the first modifies a noun and is a name-forming

functor,

while

the

second

is

independent

and,

being a name, belongs to another syntactic category and has a different

categorial

index

(see

H .Hi :C

11960,1967,19681) .

Should

we resort to a metaphor, we might say that it is like with one and the same deciduous tree in summer which differs from itself in winter belongs

to

formalized with

when it has no leaves . Note also that a word which

at

the

language least

communication categorial

vocabulary

index

a

analysed

at

word

used

one

and

of

has of

a

the

natural the

former,

token level

earlier

categorial

language

in

index

and

the

or

also

a

equiform

is

process

equiform

hence

even

with the

of the same

grammatical category. It

is further assumed that the relation :-- of equiformity is

reflexive, problem

symmetric, whether

that

and

transitive,

postulate

could

without be

considering

replaced

by

a

the less



INTRODUC

xxi

rigorous one . Itis

also postulated that a label token which is

equiform with a word from the vocabulary of a language .L° or w-Z is also a word from the vocabulary of that language . The vocabulary V is used to generate the set words of 2 or c,-X . That language is determined by all

its

meaningful,

i .e .,

well-formed

W of

all

the set

expressions,

the S

of

which is a

subset of W. The set W is generated from the vocabulary V by the

relation

C

of

concatenation,

which holds between any two

label tokens and the label token which is their

concatenation .

W is the least set, which includes V and every label token which is the concatenation of simple word tokens,

i .e .,

the ordinary

concatenation of two simple words from the vocabulary V, or the generalized concatenation of more than two simple words from V.

At the

token

level

the relation of concatenation need be a

function because two given label tokens may yield many equiform concatenations . This is so because intuitively concatenation may be understood as the relation of writing, on the right side <e .g ., in European ethnic languages) or on the left side in Semitic languages), and at the same level, of a label token equiform with the first label token in a given pair, of a label token equif orm with the second label token in that pair . But it may also be a relation of non-linear juxtaposition of two label tokens which thus form a single label token . This may take place,

e .g .,

in hieroglyphic script and in many exact sciences .

Upon the relation of concatenation we impose axiomatically certain intuitive conditions related to A .Tarski's axioms of metascience [1933] <see also e .g . H .Hi± [1957]), but in doing

x i INTRODUCTION so

we

have

relation

to

is

bear

defined

for

of the theory TLTk the

in

mind

the

tokens,

guarantee,

vocabulary V of 2 or w- .L°,

fact

and

that

not

of course, unlike

for

in

our

types .

that

case The

that

axioms

the words

from

the compound words

from

the set W'.V of that language, are not concatenations of any two label tokens . The generalized concatenation of label tokens is an ordinary concatenation or the concatenation of an ordinary concatenation and a given label token. A compound word of .L° or always

a

c,- .e is thus

generalized concatenation of simple word tokens,

because it can be obtained by a single or multiple application of ordinary concatenation to simple words from V . A compound word is thus formed of a finite number of vocabulary words . The number of such words which are parts of a given words is defined as its length.

2.3.

The second stage in the formalization of the theories

TSCL and TSCw-L of categorial languages

',

and correspondingly

w- .i°, consists in the construction of the theory TETk of expression tokens . Its presentation is to be found in Chap

.I

where the theory TSCL is shown . Now the theory TETk is obtained from TLTk by the introduction of two new essential concepts, that of expression token and that of categorial

index. Here are

the principal intuitions connected with the formalization of that theory . Categorial

indices

are

used

to

carry

out linguistic

categorial analysis in accordance with the ideas advanced by



INTRODUC

xxiii A jdukiewicz [1935,1960] . Such an analysis pertains to any compound expression

tokens

exclusively

that form the subset

Ec

of

the set W of word tokens . In ac ordance with the principle originating from G .Frege [18791 every expression from the set E q-

has the functor-argument structure

-

one can distinguish in

it its constituent called the main functor and its constituents called the arguments of that functor . The proper categorial analysis of a compound expression token


examination

preceded by its functorial

of its analysis :

syntactic

correctness)

is

after having distinguished

in a given compound expression the main functor and its arguments we continue the process of distinguishing the main functor and its arguments with reference to all

constituents

of

a given expression which are compound expressions . By way of example, the functorial analysis of the sentence "John ardently loves Helen" is illustrated by two diagrams of a tree .Inthe first, the bold lines indicate the main functors ; in the second the method of parsing used by Ajdukiewicz is taken into consideration (see Fig .1 and Fig .1a below) .



xxiv

INTRODUC John

John

ardently e loves Helen

John

/\ ardently loves Helen

a .~ d+es +. . i y

ardently loves Helen

ardently loves

loves

ardently

John

Helen

laves

Fi g .1

Fig .1a

The categorial indices (types) by means of which we examine the syntactical correctness of compound expression tokens are label tokens of Lb . But they are not in the set W of the words of Z or w-X. They are words from metalanguages of those languages and, to use Buszkowski's terminology, are used for the typisation, or determination of the syntactic categories, of expressions in those languages and for the examination of their grammatical correctness . For the formal syntactic description of the categorial language

.R .,

or w- .L°, we introduce

a new primitive concept, namely that of auxiliary vocabulary I that is the

o set

of

all

one technical symbol, ),

basic

e .g .,

indices, which includes at least

slant /, colon

:,

comma

(

or

etc . It is, of course, postulated, that the sets 1

V

have no common elements . From the vocabulary 1

by the relation C of concatenation, the

„

bracket 0

set

I

0

and

we generate,

o f all

ca t egor i a l



INTRODUC

xxv indices,

as

the

W is generated from V. The set I

set

contains all basic indices from I

thus

and all indices of functors.

0

The latter are generalized concatenations

of basic indices. The

principles of the construction of the indices of functors of a given categorial language

or w- .e are

.I°

fixed by special rules .

For instance, of the basic indices s and n, used, respectively, for the typization of sentences and names in such a language, and

the

auxiliary

concatenation

of

symbol label

juxtaposition, we quasi-fractional

tokens

can

form,

/,

on

assumption

that

provides

for

right-side

indices

of

functors

form e .g .,

the

s/nn

and

s/nn//s/nn,

respectively the index of the sentence-forming

the linear

of

which

a are,

functor of two

name arguments, and the functor-forming functor, which has the above functor as its only argument and forms such a functor . The typization of definite words of X or w-Z is obtained by the relation

.

of indication of indices of words, in other

words, the relation

of

typization .

That relation is

a new

primitive concept in the theory TETk . It is postulated that it is a function which assigns to definite

word tokens one index

token. It is also axiomatical y as umed

that words which are

equiform with a word that

has an index have

their indices

equiform with that of the said word . From the set of those words which have indices, that is from the domain D(e )o of the relation

. , we distinguish the set

E of

all expression tokens of . or w-X, and hence both the set E of s all simple expressions of such a language, which are words of V and have an index, and the set

E

c

(obtained from E ) of all s



xxvi

INTRODUCN

compound expression

tokens

the

of

formation,

expressions rules .

In

by

single

a

formation

of

out such

theoretical one,

of that language . The principles of

words

a

that

language

are

considerations

namely

the

have

indices,

fixed

those

one-to-one

of compound expression

is defined as the counter-domain

of

compound

by its rules

func

syntactic

are

tion

replaced

p

o f the

of .L° or w- .I° . The set Ec

tokens

D1 Cp) of that relation .

The function p is the third and the last primitive of

the

theory

TETk .

We

associate

intuitions . It al ows us to as ign to Cp , p o

p ),

i n

with

it

the

concept following

any finite configuration where

categorial indices,

n>i,

of words of

such a generalized

.

or w- X

that have

concatenation of

them

which in the language of TETk is recorded as C6)

p<po , P1 , . . . , Pn >

and termed a compound expression token of

.L° or w-.L° consisting

of the words po,p1,

This is so because (6) is treated as . . .,pn . a substitute of any compound expression of such a language which is constructed of the main functor p o and its successive arguments P1 , . . .,p n .

In doing so we do not consider the formal

structure of such an expression and disregard the symbolism which it is recorded and how it is formed as the generalized concatenation

by rules valid

inscription <6) thus may denote

in

.L'

in

appropriate

or

w-X.

The

various expressions of d° or

w- .k', built according to various rules, on the condition

that in

each such expression the number of words is the same and that among those words we can distinguish the main functor

and its

INTRODUCTION x vi arguments .

Moreover,

such

expressions

need

not

be

of one

and

the same syntactic category ; they may be sentences, names, and functors . Further, synonymous Sec .IL4)

may

also

replace

different

but

in

different

notations

(see

expressions recorded

.

The TSCL

(6)

theory

and

TETk

is

enriched

by

new

concepts,

specific

of

TSCu -L, the most important of which is that of a

meaningful

expression, i.e., welt-formed expression token . That

concept is defined differently for X and for w-Z .

2.4 . language

The

theory

TSCL

describes

.L° of expression tokens .Its

any

simple

categorial

welt-formed expressions

are elements of the set S, which in that theory is defined as containing

all

simple

expressions

from

Es

and

all

those

compound expression tokens of .LP, which have the property that both they and each constituent of theirs which is a compound expression of that language satisfy the rule

m

which in a free

formulation states :

The categoriat index of the main functor of a compound expression is obtained as a generalized concatenation from the index of the expression, i .e., the index of the expression which that functor forms together with its arguments, and the indices

of all successive arguments of that functor .

This definition of well-formed expression makes it possible to describe an algorithm of examining the syntactic correctness of linguistic expressions which comes close to A jdukiewicz's algorithm

[19351 .

This

is so because

categorial

analysis



INTRODUC consists

in

finding

whether

the

rule

m

holds

for

every

constituent (which is an expression of . ') of a given expression of

.L° that consists of functors and arguments .Ifthis so,

then a given compound expression of X is in S and hence - is a well-formed expression . By referring to the above example of an expression whose functorial analysis was illustrated by Fig .i and

Fig .ia,

we

can

easily

check

its

well-formedness

by

assigning categorial indices to its constituents . For that purpose we can use the tree of categorial indices of that expression shown in Fig .2 and Fig .2a below, where s and n, and the quasi-fractional inscriptions are interpreted as explained previously .

s/nn//s/nn

s/nn

s/nn//s/nn

s%nn

Fig .2

Fig .2a

expressions of a categorial language

.L° can be generated in this

It can also easily be se n that the set S of al wel -formed

way by the system

INTRODUC GX . < Es ,I

o,

e, m >,

which may be treated as a reconstruction of classical

grammar,

whose idea goes back to A jdukiewicz (1935] . Such a grammar is rigid (see W .Buszkowski (1986]), which is to say that a given

expression of .L° has only one index (type), and hence only one syntactic category, assigned to it . It wil be as umed about a syntactically

described

unambi guous ly

assumption

categorial

language

typizable

(see

W .Buszkowski

quite

natural

because

is

X

that

(1986]) .

it Such

finitely

an

typizable

languages, which we usually consider to be natural languages, can always be disambiguated by the "separation" of words or expressions to which a finite number of indices is assigned. That

can

be

achieved

if

we

provide

them,

for

instance, with the appropriate numerical indicators, letters, or diacritical signs . Moreover, as has been said earlier, we can very broadly interpret the concept of equiformity, which substantiates the assumption made above . The categorial nature of the language with

the

logical

partition

well-formed expressions of disjoint

syntactic

of

CtCS)

.

is also connected

the

set

S

of

all

.Lp into non-empty and pairwise

categories .

Syntactic

categories

are

determined by categorial indices and are sets of expressions which have equiform categorial indices .Inthefamily

CtCS)

all syntactic categories we can distinguish and

functoral

expressions, i .e .,

while

the

The

categories .

latter

expressions include

former

with basic

func tors,

i .e .,

basic

include indices

of

categories

all from

basic

Zo,

expressions with



INTRODUC

XXX

functoral

indices

from

the

setI'

0

.

The

set of

all

basic

expressions of .L° and the set of all its functors have no common elements

and the

sum of

those two

sets

cover

all

well-formed

expressions of d°. The concept of syntactic category bears a certain relation to the relations of

repiaceabiiity

of expressions, which is a

defined concept in TSCL . In the traditional interpretations a syntactic category is the set of all expressions mutually replaceable in generally,

in

arbitrary all

sentential contexts,

well-formed

contexts .

The

or,

more

traditional

definitions of that concept (see S .Led'niewski 11929], A .Tarski 11933]

,

K.Ajdukiewicz

,

11935]

I .M.Bocheiiski

11949])

imply

problems because they do not preclude a vicious circle, which is due to the fact that the concepts of sentence or meaningful (well-formed>

expression

and

that

of

replaceability

of

expressions function in them as undefined . Moreover, as was stated by R .Carnap [1937] the principle of replaceability of expressions in arbitrary well-formed contexts, assumed in the traditional

definitions

questioned (cf . also

of H .Hiz

syntactic [1961]) .

categories,

The

may

be

correctness of such

definitions can be refuted by examples . In ac demic handbo ks of logic,

e .g .,

the definition of names guarantees to proper

names and personal pronouns belonging to the category of names, whereas it can easily be seen that well-formed expressions, respectively, a nominal and a sentential one (se J

.Lambek

11958] >, BigJohn,

BigJohnisher,



RMODUCTION xxxi on the replacement in them of the proper name by the personal pronoun "he" yield meaningless <not well-formed) expressions Big he,

Big he is here .

On the other hand, for instance, both expressions "2+2"

are

well-formed,

but

the

latter

is

"2=2" and

obtained

from

the

former by the replacement in it of the sentence-forming functor _"

by

a

functor

belonging

to

another

syntactic

category,

namely the name-forming functor "+" . Even though we deviate here from the traditional definitions of the concept under consideration it remains in an important connection with replaceabiixty . That connection is described in Sec .IL7

by the fundamental

categories,

at first

theorems o f the theory o f syntactic

intuitively

discussed

in

Sec .ILi,

where

the objectives of the construction of the theory of categorial languages are outlined .

2.5. The

theory TSCa-L

describes

an

arbitrary

categorial w-language t,--w of expression tokens . Its expressions

belong

to

the

set

defined

S,

but

fixed

well-formed in

a

slightly

different way than in the theory TSCL, because its expressions may

include

variables

definition of S

should

and

operators

imply all

that

that as

bind

has

them.

been shown

2 .4, makes it possible to carry out the categorial

analysis

The in of

.LP that could be correct from the point of the expressions of (AYview should

of

the

theory

accordingly

of

enable

syntactic us,

in

algorithm of the examination of

category .

Such

particular,

to

the

syntactic

a

definition

formulate

an

correctness of

x xi INTRODUCTION expressions containing The

endeavours

operators

to formulate

and

variables

such

an

bound by

algorithm,

them .

undertaken

already by A jdukiewicz, as far as I know deviate more or from

the

less

general

principles

of

categories . An algorithm which described in Sec .I Inorde

the

theory

of

syntactic

refers to those principles

is

.6 .

to

give

a proper

definition

of a well-formed

expression in w- .e we add to the theory TETk new concepts, which do not occur in TSCL. They are those

of the

set

0

of

all

operators, the set Vr of all variables, and the relation Cfv) of being a free variable in a given expression token . The first

two of them are ptimitive

concepts in TSCa-L. The intuitive

0) and

sense which we impart to operators (elements of the set

to variables (elements of the set Vr) does not deviate from the interpretation

of these concepts in the exact

sciences . To

simplify the matters we assume that one operator binds only one variable while token .

its scope

Operators

expressions in w-X.

and

is limited variables

to only one are

treated

Operators as such are

vocabulary of w- .L° and cannot

as

simple

elements of the

be concatenations of any

tokens, and moreover they can be assigned only index each . Thus, for instance,

expression

label

one categorial

we treat as an operator the

existential quantifier V or 3, and not the graphic symbol X or 3x, and we can assign it one categorial index (the same applies to any inscription

equiform with it) .

This seems to be

contradiction with the so-called typical ambiguity <see, e .g.,

R .Suszko

11958],

11964]),

with their

in

of operators being

treated



INTRODUC

xxxiii as

categorial

as

open

functors

context-dependent

[1989]) .

For

quantifiers

such

<see

D .R.Vanderveken

functors an

occurring,

(see,

e .g .,

interpretation, for

[1976]),

W .Marciszewski

graphically

instance, in

the

arithmetical expressions and an expression

and

similar

following

of

two

the sentential

calculus with quantifiers : X x < Y,

(i)

have

different

categorial [1989]

X y x < Y,

syntactic

indices .

The

P

categories first

two

and

Cp

hence

because

-, -, P) also

in

different

Marciszewski's

interpretation the propositional functions x < y and

n

x 5 y have different syntactic categories, and the third y because, unlike the first two, it binds the variable p which has the category of a sentence, and not of a name . This problem can be eliminated as previously

<see

2 .2),

those quantifiers as diversiform because

either by treating

of their environment,

or by providing them with some diacritical

signs which would

make it possible to treat them as diversiform . Problems with assigning to the operators that bind variables their syntactic categories have been known since the Ledniewski

[1929],

the interpretation

Tarski

[1933],

and

suggested here

A jdukiewicz

they will be

times of [1935] .

treated

In as

functors of a special kind, which according to the context have a definite categorial index each .

That binding role of theirs

co- called operator manifests itself in definite expressions of .LP expressions ; they are ordinary compound expression tokens consisting of three

constituents

each: an operator,

called the

xxxiv

INTRODUC

main

operator

variable,

and

of

that

expression,

an

expression

a

token

variable

called

called

the

indexical

scope

of

that

operator . Thus the main operator is the main functor of such an expression, while the indexical variable and the scope of the operator are its successive arguments . The operator in an operator expression may be empty-binding if in its scope it has no any

free

free

variable or if it

does not

bind

variable in its scope (which is to say that no free

variable in its scope is equiform with its indexical variable . The concept of free variable in a given expression is, of course, defined by means of the relation variable which bears the relation

C fv)

Cfv>,

namely as a

to that expression, if

it is a constituent of that expression and does not occur at the same time in the scope of any operator in which it would be equiform with its indexical variable . The definition of the set

S of all

well-formed

expression

tokens of w-X takes into consideration the degree of the complexity of the expressions which are its elements . The simplest well-formed expressions expressions

in that set are all its

(including operators and variables> . The simplest

compound well-formed expression in that set is either -operator expression w- .LP

simple

i0

a non-

which is a compound expression token of

consisting of simple expressions and such that its main

functor is not an operator and also such that it satisfies the rule m, or 2' an operator expression such that the scope of its main operator is the simplest non-operator expression containing the variable which that operator binds, and such



INTRODUC

xxxv that this expression

also satisfies

the rule m

(the index of

the main operator of that expression is a concatenation of the index

of

that

expression and

the

indices

of

the

iridexical

variable of the main operator and that of its scope> . More complex

welt-formed

non-operator

from the simpler well-formed non-operator

expressions

expressions only by

being constructed of at least one compound

(operator or non-

-operator) well-formed expression, and more complex operator expressions

well-formed

differ from simpler ones by the fact that

their scope is not a simplest well-formed compound Note that the set S thus includes

expression.

all simple expressions of

w-X, all its non-operator expressions,

and all its

expressions .

compound

Note

differ

also

that

every

operator

well-formed

expression of w-X and every constituent of such an expression satisfy the rule

m, which makes it possible to generate the set

S of w- X by the categorial grammar G w_ X _ ( ES ,I

o , e, m >,

and also to describe the algorithm of the examination syntactic

connectedness

of

expression

A jdukiewicz's algorithm for expressions

that

of the

resembles

without operators that

bind variables . When applying that algorithm we start from the parsing of a given compound expression token of functorial analysis of that If unctorial an lysi

w-.'

by carrying out the

expression as described under

brings us to simple expressions

2 .3 .

(which

is to say, if the ends of the tree are simple expressions)

of

x xvi INTRODUCTION w- .L', then we find whether there is among them an empty-binding operator . Ifit

is not so, then we apply categorial analysis in

the proper sense of the term by assigning to every expression obtained

by

parsing

its

categorial

index

and

check

every compound expression thus obtained satisfies If

this

is so, then the

well-formed.

A

given compound

precise

description

whether rule

m.

expression of w-l°

is

of

that

the

algorithm

examples of its application are to be found in Sec .IL6

and

.

Note also that if we apply the same designation with the index s of sentences

and propositional

functions and if

assign the same index n to names and name functions the same principles of forming 2 .4>,

then

the

quantifiers

we


index functors as in 2 .3 and

which

occur

in

the

expressions in Ci> have the index s/ns, whereas

first

two

the quantifier

which occurs in the third expression in has the index s/ss . This is not

at variance

with the fact

category with the index s/ns

that the syntactic

covers such cases in indirect

speech as "thinks that" and "knows that",

while

the category

s/ss covers truth functors of two arguments each . This is so because we do not adopt

here the traditional definition

syntactic category . By the

way, in order to distinguish

syntactic

category

closed

propositional functions in which occur, we can accordingly

sentences

s/ns i,

s/nsi

and

that

the of

one, two, etc ., free variable

use the typization:

such a case the said existential indices

from

of

s,s,,s etc . In 2P quantifiers would have the

s/s i s i ,

respectively,

and

the

universal quantifier in the second expression in Ci> would have



INTRODUC

xxxvii the

index

s/ns

(see

2

W .Marciszewski

1198913 .

The

above outlined

algorithm obviously does work in such a case . It must be emphasized at this point that al the properties which make it possible to describe, in terms of LP

TSCL,

language

as a categorial language are applicable to language

well .Inthe ory

TSCw-L

63,- X

as

all the analogues of the theorems

for well-formed expressions of £ hold, too .

2 .6 . DTSCw-L

the dual theories

The formalization of

DTSCL

and

takes place at the second level, at which we are

interested

in

types

and

relations

among

them.

Those

formalization pertain, respectively, to an arbitrary but fixed simple language fixed (L-language

of expression types and an arbitrary but

Y

w-Y

of such expressions .

All the linguistic elements of expression tokens of w- .e, e .g .,

label

token,

word

token,

.L° or

expression token, well-

-formed expression token, variable token, operator token, and object token which make it possible to carry out the categorial analysis

of

expression

tokens,

and

hence

categorial

index

tokens, have analogues in induced concepts which are elements of expression types in expression

type,

etc .

.Lp

or w-.L, and,

for

namely label type, word type, categorial

analysis,

index

types . All of them are defined as classes of abstraction of the corresponding equiform tokens . Likewise, all the relations holding among tokens have their analogues in induced relations holding among types . The latter are defined so that they hold among types if and only if

Xx vi i INTRODUCTION relations

with

the

same

intuitive

meaning

hold

among

representations at the token level Crepresentatives

their

of classes

of abstraction) . All the

properties

of

concepts

definitions, and theorems of TSCL

grasped

by

the

axioms,

which describe

categorially the syntax of .' of expression tokens pass upon the induced concepts . This makes categorially the syntax of

a

Cor

w-Y>

it possible to describe of expression types . We

find the full analogy between

the syntactic concepts of the

languages in question analysed

at two different levels .

in the syntactic description of language the existence

of

object

tokens

as

the

Since

assumption of the

fundamental

linguistic

elements served as the point of departure, we can conclude from the above that a theoretical categorial syntactic description of language does not require language is a

the preliminary assumption that

language of abstract

language of certain ideal entities .

expressions,

that

is

a

Hence in syntactic analyses

of categorial language we may abstain from making

assumptions

about the existence of ideal linguistic objects . This conclusion can be substantiated

more strongly in the

light of recent studies carried out by the present author

<see

Final Remarks> .

Acknowledgements

Professor

Grzegorz

Bryll

typescript of the present

was

the

first

reader

of

the

book in its original version

and

INTRODUCTION x xix checked the

formal

and

factual

correctness

of my

work .

I

owe

for

his

him my cordial thanks for that . My thanks are initiative particularly efforts also

to

also due to

have

grateful

accompanying

thank all

its appearance .

those

the to the

Professor

book Professor

Srzednicki

published

in

Stanislaw

publication

nameless

Jan

friends

English . J .

Surma

of the, English who

have

Iam for

his

version .

contributed

I to

CHAPTER

I

THE AXIOMATIC THEORY T L T k OF LABEL

S e c . I.l. Let

Primi-tlve us

Imagine t h e

concepts

situation

who w a n t s t o w r i t e a l e t t e r who

wants

to

formulate

formalized

language.

labels

ol'

a

labels

that

follow

Linguistic

physical,

Cin

events;

Latin

see

precise

the

we t o

usually

begin

with

labels.

Thus

our

universal simple

in

fixing

set

statfidard

Lt> of

of

want

words

and

those

to

label

[1942]>.

we

A

objects

face

an

are label

also

objects.

formulate

them.

namely

material

which

abstract

complex

user

English

a

simple

reproduce

objects,

those

In

out

more

R.Carnap

of

ideal,

proofs

language

in

language

mathematician

write into

given

than

We

when face

intuitive

language. syntactically

out of

or

V,

label those 1

a n y l a n g u a g e *f ?

indicating

departure

Mocabulary all

a

properties

writing

the

ahd

concreto,

characterize

point

a

produce

[1931-19351;

we

ethnic

of

them

perceivable

difficulties

c h a r a c t e r i s t i c of a given

consists

by

properties

an

and • f r e q u e n t l y

in

the

when

to

combine

Gh.S.Peirce

difficulties

How a r e

select

them

sensorily

greater

we e s t a b l i s h

of

used

of

situation or

another,

of

a user

theorems

tlabels>

explanation

encounters

such

one

of

or the

shape,

expressions

concrete, tokens

Both

definite

TOKENS

in

which

simple

standard

characterizing is

tokens

and

labels

which

a

subset

consists are

We

of of

e
£ the such with

2

CHAPTER I

t-hem. T h e

set

defined

as

of

set

the

tokens

a ¥

of

yacXl—foTme
all

vord

the

the

form

of

< Lb;

by

the

W; %,

for

the

by t h e Like

S

relation

it set

theory

laibel t o k e n s

will S

called

the

the

earlier

first

works

be

symbolized

as

of

meanln(;ful,

i.e.,

simple

categorial

the

syntaix,

set

of

and

label

of

labels

Thus,

characteristic

of

equiformity

of

in

Jf

a

takes

five to

concepts

be

the

present

TLTk.

The

sixth

well-formed

called

of

presented

language

of

labels,

and

labels.

by

catsgorial language

W is

of

be

>,

of

in

rules

now

slxtuple

properties of

the

Jf c a n

concatsnation.

c, f o r t h e r e l a t i o n of c o n c a t e n a t i o n of The

of

vocabulary,

their

syntactic

C;

the

by

where

from

ordered

V.

a; s t a n d s

tohens,

former

the

expressions

specified

words

formulation,

<*>

where

ail

of

from

simplified

of

cert.adn

formed

obtained

on t h e

S

CJf> a r e

in

this

author

that

expressions for

oi-1 angxiage,

for

the

is

chapter.

C1981a,1983a]

concept,

and

fixed

of

the

the

so-

syntactically

described

by

the

a p p r o p r i a t e t h e o r i e s p r e s e n t e d In Chap.II a n d Chap.III. The tokens

problem was

Qrzegorz the

Bryll

theory

and

of

thematlcally

It

differs

It

by

close the

with

its

is

thus

Slupecki

application a

different

in

theory first

with

in the

by those

reference

to

theory and

respects

of

label by

Bryll

theory

solution

of solved

119541. T h e

formulated

primarily

and

11977]

A.A.Markov

that

former

axiomatic

Miklos

of

to

an

Jerzy

Stanistaw

algorithms

from

connected

categories.

constructing

formulated

is

are

of

of the

TLTk Miklos. which

syntactic problem

THE AXIOMATIC THEORY TLTk OF LABEL TOKENS forrnxjlat-ed presented The

in this

theory

narrower It

is

the

by Slupecki

TLTk

based

C.

those

of sets

in

shall

specify

S.Mlklos

detail

a n dover

C197r]

of

in

on

s e t theory.

those those

four

primitive

concepts

Caxloms>

by

of t h erelations

t h e successive

t h e conditions

classical

a n d [1981a.l983a]

Sec.1.85

Lb a n d K a n d

which

*

shall

be

sections;

we

those

concepts

Label tokens and their equlformity

tokens

are

material

products

of

indicated

by them. They are

a

given

given

place

chalk on

only

also

t h e

satisfy.

Label

pencil

of t h e analyses

over

identity

meaning:

greater

S e c . 1.2.

of

with

s e e

The intuitive

also

superstructured

calculus

author;

discussed

should

is

Ccf. G.Bryll,

present

and

I s t h e foundat-ion

book.

functional

concepts:

aind

3

in

ethnic, in

a

drawings,

The

that set

Lb of

not

a

a

on

and

all

within which we single out

of

language, or

further

of

etc.,

of

is

a

labels

a

with

pen

or not

but

also

states,

and

is

token

down

in

include

label

certadn s e t s

nature

ballpoint

signals,

tokens

of

printed

pi-operty

label

either

expressions

essential a

are

written

pen,

they

light

definite

frontiers

The

part

only

book,

paper;

They

products

fountain

ranges,

leaves.

every

of

smoke

or

formalized

with

mountain of

thus

copy

place

configurations

label.

given

products.

beings

or

definite

of

is

formal,

of

a

tokens

human

blackboard

definite

contours

a

empirical

of

also

label such

universal

a

set,

which make

it

4

CHAPTER I

p o s s i b l e f o r u s t^o d e f i n e a g i v e n languaic^e. The

relation

TLTk the

is

of

a binary r e l a t i o n

counter-domain.

holds a r e

catil&d

different

places.

sentence

and

paragraph is

»
not

an

p.ll2>,

when

needed

for

wrote

in

e
For

seventh

word

same

identity

relation. the

letter

of

with

those

y has

inscriptions

is

word

in

"...An

when

form

situated

of in

the a

this

C11981], concepts

relationships,

inscription

speaking

in

X and y a r e t w o t o k e n s X

first

equiformlty

the

syntactic

has

the

letter

X

of

one

and

of

the

form

place

is

ordinary

"A", b u t

different

two

in

Grzegorczyk of

and

in

the

of

of

relation

sentence

sense

that if,

that

may o c c u r

relation

various

y

the

which

second

the

instance,

the

domain

AndrzeJ

this

For

the

Thus

aun i n s c r i p t i o n

"A" and

both

labels

intuitive

description

inscription.

concept

thirteenth

in

labels.

connection

primitive

between

instsuice,the

l a n g u a g e , o n e would s a y t h a t the

labels

equlform

the

another

Two e q u l f o r m

explaining

with

-

i n which Lh i s

two

»(fui/orin.

the

are

Any

ty

one

from

of that

i n which t h e o t h e r i s ..." The

equlformity

geometrical but

recorded

of

similarity.

labels Two

with equlform

is

words letters

not

always

printed of

in

different

alphabet

occurring

s a m e o r d e r may be e q u l f o r m . The e q u l f o r m i t y g i v e n languaige i s r e l a t e d t o t h e s y s t e m o f The say

the

fact

that

relation

ais t h e e x p r e s s i o n q

stre

equlform,

the of

labels

p

equlformity

and

or:

the

label

of

read: t h e p

types in

expressions

<j au^e e q u l f o r m

token

by

the in

a

caligraphy.

holds b e t w e e n

p % Q, which i s

determined

is

them>

Cthat is

to

recorded

label t o k e n s equlform

is

p amd

with

the

THE AXIOMATIC THEORY TLTk OF LABEL TOKENS label t.oken An

q.

import-ant.

stated

by t h e

AXIOM

5

property

of

the

relation

of

equiformlty

is

axioms:

I.l

a. p % p , b. p % 9 •* 4 % p , c.

p % « j ^ 9 S ; r ^ p % r .

In

the

recording

of t h e Convention

of

axioms

I.l.

The

we

have

p , Q, T, s,

axioms

I.l.a-c

an equivalence in

S e c . 1.3.

t, u, V,...,

Telatiorx

of

its

three

of

concatenation

a

European

label

level,

state

that

the

Lb.

relation

of

equiformlty

is

Lb.

Concatenation

The

the

ourselves

letters

with or without subscripts, range over the s e t The

availed

I.l

C o n v e n t i o n Cv^y

these

of domains

equal

to

ethnic

equiform

the

concatenation

two

p

by

label

is

A label

laibels

languages,

with

successive

Lb.

c

is

any

writing equiform

which

on

its

with

Eiffel,

beatrs

a

concrete

token la Tour

ternary

called

o f t h e l a b e l s p a n d <j. For i n s t a n c e , t h e label

a

relation. the

relation

concatenation. label

right,

qf i s

a

obtained on

All

the

In from same

concatenation

6

CHAPTER I

which

is

a

French

language

expression,

and

any

equiform

name

obt.ained f r o m t-he l a b e l t.okens 1

1 a

Ct-he

lower

Semitic

ethnic

obtained the in

in

left. a

preceding languages

a

similar

Not

linear

the

in

but

languages

Numerous

TLTk, which we s h a l l

Csee,

e.g..

concatenation

has

concatenations relation the

need

of

cCp,cj,r>,

be

of

TLTk. which

same

recording

a

label

w i t h q, o r ,

concatenation.

two

second

are

that

not

them

thus: token more

language in

the

shows

we

by

shall

the

equiform briefly,

the

that

form of

may

that

further the

p

label

be

that of

specific expression

token

with

the

properties

use

label

down

of

labels

and o t h e r

fixed

on

relation

equiform

which

are

added

written

recorded

two

are

the

In

tokens is

are

that

These

label

element

of

labels,

a function.

read

of

also

property

is

of

equiform

use,

concatenation In

r

concatenations

Note

the

not

concatenation token

the

of

relation

axioms

I.2>.

the

u

their

expressions

theory

Axiom

is

concatenations

way,

all

form.

upper>

T o

r

and token

is a r

a

label is

a

c o n c a t e n a t i o n o f l a b e l t o k e n s p and q. And

here

come

c o n c a t e n a t i o n c;

the

axioms

which

describe

the

relation

of

THE AXIOMATIC THEORY TLTk OF LABEL TOKENS AXIOM

7

1.2

V c.

For

any

two

label

tokens

there

is

a

label

mhich.

is

their

concatenation. AXIOM

1.3

C ^ -1 Cr 5: p )

A label is

not

token

ecfuiform

AXIOM

i^ich

is

with, any

/s -1 Cr S; q>.

a concatenation of

of

tvo

tokens

them.

1.4

c ^ c C r , s , w > / \ p ^ r ^ < f ^ s * t

Concatenations second

label

of

elements

two

pairs

which

are

of

label

tokens

pairwise

^ xi.

with

eQuiform

first

are

and eQuiform

labels. AXIOM

1.5

CCp,9,s) ^ c<.s,r,t> // a

label

token

concatenation label of

of

token

labels AXIOM

is

lajbel

u is tf and

t

^ CCv> ^ CCp,v,u> =* f % t». a concatenation

tokens

p and

a concatenation r,

then

the

// one

and

concatenation

the of

concatenation only

label

label t and

a label

p and

which

token a

u cere

r.

is

a

and

a

concatenation eifuiform.

1.6

cCp,q,0 ^ cCr,s,f>

and


of lcU>els

of

of if

q and

•* Cp % 2- «* q % s).

same

label

labels

p

and

q

and

s,

then

labels s are

r

token

equiform.

is,

on

and, p and

the

on r

one the

are

hand,

a

other,

a

equiform

if

8

CHAPTER I AXIOM

1.7

cCp,Q,r) <^ s % r 1* cCp,Q,s>. A lotbel

idhicA

concatenation IabeIs

cCp,
it

two

labels

with is

a

lab^l

also

token

tii/iich.

a concatenation

is

of

a

those

1.8

^ cCr,s,u'>

^ t %w-»pS;rv-^

two ecfxiiform

label

of

etfaiform

•

AXIOM

and

of

is

tokens r are

with

e(fui/orm

other

The

explanation

s e e m s , be l e f t

are

elements

or

and of

several

tokens

first

the

example

label

at

of

one

label

the

without

CCp,v,i->>.

concatenations

p and r,

least

some

CcCr,v,p> v

of

of

respectively,

them

is

of

then

a

p

concatenation

token.

ii^t-uitlve

sense

of

these

axioms

a commentao^y. We s h a l l g i v e

theorems

pairs

that

follow

from

the

can,

by way o f

axioms

adopted

above. Axioms 1.3 aind I.la y i e l d i m m e d i a t e l y CONCLUSION

I.l

-> ^ C<:p.q»/>> ^ "* q cCq(,p,p>. No token

label

is

a concatenation

nor a concatenation

This labels

conclusion is

theories which the

token

not of

lets

assumed

formal

Conclusion

analyses

of

I.l

carried

some

us

note

in

TLTk,

languages. follows, out

of label

itself token

that

the

unlike

in

The a d o p t i o n has

in t h i s

no

and and

the of

but

case Axiom

of.

empty

of

most

1.3,

from

importance is

label

itself.

existence

essential

chapter,

some

in

for

agreement

THE AXIOMATIC THEORY TLTk OF LABEL TOKENS

with

int.uit.lon

and

is

useful

syntactically categopial THEOREM

9

for

reflections

token

concatenation

u

labels

this

labels under

% r ,

WlicAi is

of

To p r o v e

hence,

theory

of

I.l

A label

the

the

languages.

C
for

on

and

r

and s

the

a concatenation

mttich

theorem

are

it

there

by

is of

of

equiform

suffices

assumption

further,

CCr,s,'u>.

to

a label

two

with note r

labels

a

by Axiom

1.2

them^ that

such

that

this

theorem

and

1.7,

cCr,s,-u),

which

Axiom

is

by

c C r , s , r '>, Axiom

was

1.4,

to

be

demonstrated. Axioms 1.6 and 1.7 y i e l d CONCLUSION

1.2

cCp,<j, t> /s CCr,s,«> /N t % « ^ C p % r * * q S ; s>. If

eQuiform

label are

tokens equiform

label

toAtens

p and

Q and

if

only

and

r if

are

and

concatenations s,

of

respectively,

q and s are

then

pairs

of

p and

T

equiform.

We shall now record one more theorem, to be used further in the text. THEOREM

1.2

cCp,<j, O

yN C
*u*»piferA<j5:sv

v V V CCp,-u,r> />, c<-u,s,q>)). Two label tokens only

if

tokeTts

which

are

concatenations

p and

q and

r and

s,

either

the

first

elements

respectively, of

of are

those

pairs

of

equiform

pairs,

i.e.

label if

,

p

and and

10

r,

CHAPTER I

are

and

eqrtiform

s,

are

i.e.,

eifuiform,

p or

label,

and. the

r.

is

xithile

respec

remaining

Annex.

In

a

one of

element

of

of

those

of

the

the

pairs, first

other

the

pair,

of

that

i.e.

.

q

elements,

and

a

certain

i.e.,

s

label

or

Q,

and

the

s.

difficult,

that,

least

concatenation


rather

at

second

is

label

The

or

elements

a concatenation

the

t ively,

second

proof

proof use

of

is

t.hls

made

theorem

of

Is

Axioms

given

in

the

1.3,

1.5,

1.6,

I.l,

a n d 1.8.

S e c . 1.4.

Vocabulary

The vocabulary the It

point is

of

the

departure

postulated

t h e following AXIOM

V -

about

last

in

primitive

the

that

concept

description

set

that

it

of

a

satisfies

of given

TLTk

-

is

language.

in

particular

of

all

axioms:

1.9

0 ^ K S L6. The vocabulary

is

a non-empty

subset

of

the

set

label

tokens. AXIOM

1.10

p^V/^q^p^q^V. A

label

vocabulary The alphabet

uift.ic/i is

is

also

included

vocabulao^y or

equiform

of set

with in

formal of

letters

the

a

label

inc luded

in

the

vocabulary.

languages Csee,

is e.g.,

also

often

called

A.A.Markov

11954],

THE AXIOMATIC THEORY TLTk OF LABEL TOKENS

H

M.Gross, A.Lentin [19673, J.E.Hopcroft., J.Ullman [1969J>. Elements

of

the

vocabiilary

are

Simple w o r d s cannot, h a v e a s p a r t s mereology) words t h a t used

to

form,

expressions as

such

as

of

are

a

are

not

simple

Cln t h e

sense

different

their

given

called

On t h e

concatenations

of

tokens.

Le^nlewskl's

f r o m t h e m . Simple w o r d s

concatenations,

language.

vord

of

compound contrary,

any

label

are

words simple

tokens.

and words

This

is

not

be

p r o v i d e d f o r by t h e f o l l o w i n g axiom: AXIOM

1.11

cCp,q,r> ^ r « K. Concatenation The

of

properties

properties

of

two

labels

which

any

is

simple

not

a simple

word

tokens

have

Axiom

1.11

labels. Note

that

«iK>rd.

simple word tokens, but is not true for just any Theorem

1.3 describes

some

properties

of

may is

true

for

label tokens. simple

words

and

confirms the comment made above. THEOREM

1.3

^ c -^ p ^ r ^ Q ^

c
a that

tokens, are

label

token their

then

the

is

first first

a. concatena.tion or and

of

second the

two

elements

second

paiz-s are

elements

of

S.

of

labels

simple

\i)ord

those

pairs

eqxiiform^ Theorem

and 1.11.

1.3 follows

easily from

Theorem

1.2 and

Axioms

I.la

12

CHAPTER I

S e c . 1.5.

Word t . o k e n s

A word given


ianc:ua£;e

vocabulajr-y.

also

need

Those

not.

uord

au^e i n t h e s u p e r s e t of

all

word

definition

be

of

-

Wort; In

course

(oltens

¥ of

tokens

will

s t - r i n g ; i n Ger.

in

be

which

an

we

Fr.

element,

are

a

preceded

secondary

by

a

of

a

of

the

interested

here

t h e v o c a b u l a r y V. The s e t is

mot.>

concept

W in

convention

and

letter

with

the

set

TLTk.

Its

an

auxiliary

definition. C o n v e n t i o n

I.lb.

The

X,

or

without

s u b s c r i p t , i s a v a r i a b l e which r a n g e s o v e r t h e f a m i l y 2 DEFINITION

I.la

XCX> • • K S X ^ A set if

X satisfies

X contains

of

^ <.p,Q e X A C. p , q , r '^'^ '^'^'

the

the

sentential

vocabulary

expression

V and

is

closed

\CX> if under

and

the

only relation

concatenation. The d e f i n i t i o n o f V i s a s DEFINITION

follows:

I.l

¥ - n < X 1 \CX> >. The which

fiet

of

all

satisfies

Note

word the

that

-

under Lb

I.la

-

accordance

with

both

of

ail

label

is

expression

Definition

set

tokens

Axiom

definitions which

least

set

of

label

tokens

\CX>.

satisfies

tokens

the

1.9,

Conventions

I.la,b,

the

condition

X.<X>.

¥

the

subset

is

contains

leaist

V and

is

and

Thus

closed

of

in the

under

THE AXIOMATIC THEORY TLTk OF LABEL TOKENS

the

relation

theorems closed

of

on t h e

under

a

13

concatenation least

sets

given

C.

which

relation

This

and

contain Csee,

a

the

well-known

given

e.g.,

set

and

L.Borkowski

are

[19701,

p.222-4> y i e l d C0NCI.US10N

1.3a

y ^ w s Th& set subset

of

of

<3.XX \ooTdL tokens

the

set

CONCLUSION

of

all

Lb.

contains

label

the

a itord

yihich

of

ttto

tooTd. tokens

is

also

contaiTied

in any

set

closed

under

and Lb

Hence

element

I.3C

A

W and

1.11 r

a concatenation

V ^ W^

vocabulary

r

is

¥.

token.

¥ is

an

is

I.3b

CONCLUSION

I.3a.

and

tokens.

p,<j e V A C
vocabulary

of

is

are

cp,q

non-empty

the

set

the

non-empty

such

that

€ X -^ cCp,q,r> ^ r e X) ^ It- £ X.

WW

c(.p,p,r

is

of

label the

sets

tokens

relation under

non-empty, set

W, t h e n

> and

under

irfiic/i contains of

Axiom 1.9 too, under

concatenation. aund

because Axiom

Conclusion

if

an

1.2 t h e r e

is

1.3b

1.4

a.

W ^ 0 ^^ Lb J' 0,

b.

W^V ^ 0.

E l e m e n t s o f It'SK will b e t e r m e d compound

veord

Conclusion is

€ Wsy. T h i s y i e l d s

CONCLUSION

the

tokens.

p

and

Axiom

a

14

CHAPTER I

Two m o r e a x i o m s hold In TLTk. They a r e a x i o m s which

describe

t h e p r o p e r t i e s o f W. AXIOM

1.12

^ ^ ^^ Every word

compo-und

->' p , ^ h , CCp,<,.r>.

\M>rk tohsn

is

a concatenation

We axioms

pair

of

1.13

r e K ^ cCp,q,r> ^ p,q

those

a

tohens.

AXIOM

If

of

a tiK>rd token

is

label

are

tokens

shadl

now

adopted

a concatenat vtord

ion

e W.

of

t\oo label

tokens,

then

tokens.

give

several

t-heorems

above.

Axioms

1.12

and

that, 1.11

follow

and

from

the

Conclusion

I.3b

yield THEOREM

1.4a

A compound tvx> uiord THEOREM

word

token

is

a label

Wiic/i is

simple

word

Theorem

of

tokens. 1.4

r e l t ' + ^ r e K v A label

a concatenation

token or 1.4

is

a laord

a concatenat follows

^ . ^ cCp,Q,r).

token

if

ion

some

of

directly

and

only two

from

if

word

it

is

C o n c l u s i o n 1.3a, Axiom I.ll and T h e o r e m 1.4a y i e l d

a

tokens.

Theorems

C o n c l u s i o n 1.3a.

either

1.4a

auid

THE AXIOMATIC THEORY TLTk OF LABEL TOKENS THEOREM

15

I.4b r e V ^ r ^ y ^ - i

A labol toKen

token

uthich

is

is not

a simple

V

^tord

a concateruxtion

cCp,q,r).

if

oaui only

if

of

any

of

pair

it

is

a word

viord

tokens.

T h e o r e m 1.4a and Axioms 1.10, 1.7 and C o n c l u s i o n 1.3a y i e l d THEOREM

1.5

A label

token

word

wtticfi

is

equiform

with

a word

token

is

also

a

token.

From Axiom 1.13 and C o n c l u s i o n I.3b we obt.aln immediat.ely THEOREM

1.6

C -•
to is,

token

which

is

a concatenation

a word

token

if

and only

In t h e

proof

of

-theorems of

avail o n e s e l f

of

a certain

as

it

adopt t h e

were,

an

auialogon

if

those

of

labels

TLTk i t

is

metatheorem of

the

two

are

lcU>el

word

somefclmes of

principle

that of

tokens

tokens. convenient

theory,

which

induction.

We

convention

C o n v e n t i o n numbers

e l»0.

are

marked

1.2. with

Theses an

m e t a t h e o r y TLTk, and r e c o r d otir m e t a t h e o r e m

thus:

Caccepted s e n t e n c e s )

asterisk

are

theses

of

whose the

16

CHAPTER I THEOREM

IcLTXguage

of

such

that

TCr)

be

in

it

TCp)

any

free

q

and

and

if

for

respec

^„

r for any

r

Then,

xaord

expression

of

not

a free occur

TCp> by if

any

p and

TCp).

in

P,QGW

P r o o f .

simple

word

q hihich

satisfy

TCcfy.

p

satisfies TCpy TCr),

and then

symbols: •*

A

r-r^r

Assume

Let

and

substitution

satisfies

In

r

it.

the p

a proper

concatenation

p satisfies

of

variable

^ ^ ^ CT

/s TC<j> /s CCp,q,r> ^ T C r »

peK

T
p^W

t-hat peV

and

use do

tokens

their

ssntsntial

the

from

p.

tively,

^

a

<j and

obtained

token

TCp)

be

with,

variables

expressions

\eord

TCp)

-recOTdued.

TLTk

of

TC
Lot

I.l .

'^

that

C2>

^ W ^ ^ rC<j) ^ CCp,q,r> •* TJ. Let

that

W

be

the

set

of

ail

those

words

which

satisfy

TCpy,

which

contains

is,

C3>

W We

shall

-

< p e

demonstrate

that

W \ T
is

>.

a

set

the

1

vocabulary In

V

fact.,

formula

and

it (3)

is

closed

follows

from

under

the

relation

assumption

CI),

of

concatenation.

Conclusion

1.3a

and

then

the

that

C4>

K c If . 1

On

the

definition

other C3>

of

hand, W

and

if

p,q

formula

e

W

<2>

^

yield

cXp,Q,r>, T<.ry,

where

reW

in

i

accordance r

e

y .

with

This

Conclusion

shows

that

I.3b. V

is

Hence, closed

in

view under

of the

formula relation

C3>, of

THE AXIOMATIC THEORY TLTk OF LABEL TOKENS

'^

c o n c a ' t e n a t . l o n , i.e..

From f o r m u l a s <4> and C5>, u n d e r C o n c l u s i o n I.3c, we o b t a i n W ^ W, which. In a c c o r d a n c e

wit-h <3>, y i e l d s

the

thesis

be

principle

of

the

theorem

being proved. Theorem the

least

the

I.l set

for

This

is

with

the

is

thus

the

set

of

induction

v o c a b u l a r y and i s and

hence

the

closed

for under

principle

of

W.

n-conq>onentlal words; l e n g t h o f word

simple so

of

of

given

by

of

follows

compound initial

their

mereology>, concept

language

which

any

those

determined

the

a

words,

because

use

Lei^niewski's define

the

concatenation,

Compound w o r d s of

the

of

S e c . 1.6.

set

termed

which c o n t a i n s

relation

induction

may

are

from

word

words.

are

finitely

Definition

is

simple

componential

Cin t h e words. word

the

of

W.

exclusively

"structure"

parts

from I.l

recorded

The

component

which

generated

of

words

sense Before

we

refer

of we to

t h e usually adopted convention: C o n v e n t i o n or

without

subscripts,

1.3. L e t t e r s are

variables

Ars<0> o f naturail n u m b e r s w i t h o u t We s h a l l

now

introduce

the

i,

J,

which

k,

I,

m, n,...,

range

over

the

concept

with

the

set

of

the

zero.

definition

of

18

CHAPTER I

set

V of

will

be

they

all

called

are

n—compoTiential

xaord

n-componential

xiiorcl

usually termed

"n-letter

tokens.

Elements of

tokens.

In

that,

formal

words". The s e t

set

languages

V is

defined

by i n d u c t i o n a s f o l l o w s < s e e G.Bryll, S.Miklos [19773): DEFINITION

1.2

a.

*K -

b.

r e

A

"""V «•

ial

We s h a i l

now

definition

\Dord token

c.

token is

word

is

a label

token

formulate adopted

n-componentlal

V

word

an n-componential

the

V

one-componential

n-H-component of

V,

and

word t o k e n s .

theorems

and

Their

tohich

a simple

several

aibove

a

describe inductive

simple

word;

is

concatenation

a

word

token.

which

result

the

an

from

properties

proofs

are

of

given

in

t h e Annex. N o t e f i r s t t h a t t h e f o l l o w i n g t h e o r e m holds: THEOREM

Every

1.7

n-componential

word

token

is

a word

token.

Theorem 1.5 has its analogon in THEOREM

1.8

pe A word token

is

token also

which

l^^q%p^<je is

equiform

an n-componential

Two other theorems follow:

with word

K. an

token.

n-componential

word

THE AXIOMATIC THEORY TLTlc OF LABEL TOKENS THEOREM

19

1.9

/N CCp,Q,T> If

a

la&el

m—componential

token one

and

an m*n-componential THEOREM

is

label

is

is

I.IO

then

that

can

It

essential as

replace

shown,

a

word

of

a simple

token

shall

that-

Definition

definition

whether

an

concatenation

of

now

I.2b

from

example,

only

and

an

because,

as

this

n+f-componential an

word

n-componential

formulate

the

theorem

which

theorem. token

word

and

is a

word. refers

to

the

Section.

1.11a

p e V =• V p e n word

In t h e

is

and

token

above,

follows

c o m m e n t made i n t h e i n i t i a l p a r t o f t h i s THEOREM

if

word

s i m p l e word o r a s i m p l e word and a n n - c o m p o n e n t i a l

Every

label

token.

be

We

words.

\ccp,<,,r>.

a cOTtcateruztion

can easily not

**A

an n*1-componential

word

Theorem

obtained

one,

two

vord.

token

n-componential

is

of

1.10

A label that

V.

concatenation

n-componential

'-"''^

if

a

^ r e

token

proof of

of

is

a fini

this

Theorem

tely

componential

theorem

I.l ,

that

V.

we is

avail the

word.

ourselves, principle

by

of

way

of

induction

f o r W. Note fact

that

follows

the from

present

theorem

Definition

I.2a.

is

true Let

us

for

any

now

peK, assume

which "by

20

CHAPTER I

Induction" t h a t

it

is

true

for

word

tokens

p

ami

p

1

that

it

is

true

for

the

word r

which i s

their n

follows

from

the

assumption

that

p

1

e

hence by Theorem 1.9 we have that V j. e I.l

1

and

show

2

concatenation. n

2

V and

p

It

e

2

V,

and

K. By applying; Theorem

we find that the theorem being proved is true for any word

token. Theorems 1.11a and 1.7 yield THEOKEM

1.11

W-

U

V.

k.~1

The

set

component

of

ial

utord

Theorems in

a

I.ll

set

of

a

set

such

affirmative

all

xDord

tokens

is

the

set

of

all

finitely

tokens. arul

I.lla

finitely

have

shown

componential

determined

that

words.

every Is

unambiguously?

word

the

The

token

is

membership

in

answer

in

the

i s p r o v i d e d by

THEOREM I . l i b

p € »•' ^ Vi p e n "^ Every one

set

word of

token

finitely

is

a nnrd

xnhich

componential

V. is

a meitJber

of

precisely

words.

The p r o o f i s g i v e n i n t h e Annex. By

this

theorem

"componentiaiity"

an

other

n-componential

than

n,

and

hence

word

cannot

catfinot

be,

have

e.g.,

an

replaced

by

m - c o m p o n e n t i a l word i f m ^ n. The

rather

clumsy

word

"componentiaiity"

will

be

THE AXIOMATIC THEORY TLTk OF LABEL TOKENS •the

word

commonly uKtrd that

"lengt.h". used

token

p

concept

in

the

will

be

takes

DEFINITION

In

on

21

accordance

literature recorded the

of as

the

symbolism

subject

\p\.

In

the

TLTk

the

most

length

of

definition

a. of

form:

1.3

word

n-componential The

the

following

p € V •» <.\p\ A given

wit-h

token

has

the

^ n *¥ p ^

V).

length

and

n if

only

if

it

is

an

word.

correctness

of

this

definition

is

gU3u:
by

Theorem

I.lib. N o t e t h a t C o n c l u s i o n I.3a, D e f i n i t i o n s 1.3 a n d I.2a y i e l d CONCLUSION

I.5a

p e V ^ p e W ^ A simple Note

u«ord token

further

that

is

a word

Theorem

\p\ token

I.lla

and

«1. of

length

i.

Definitions

I.2a

and

1.3

imply CONCLUSION

I.Sb

p € yr i>. A compotind natural

number

Further, yield

word

token

greater Definition

is than 1.3

a word

token

of

a length

which

is

a

1. and

Theorems

1.5,

I.lla

and

1.8

22

CHAPTER I CONCLUSION

1.6

p ^ W ^ q ^ p ^ A inox-d tuftich. same

length

as

is

)Q| «

a e t j u i / o r n t with

the

\p\.

a given

word

token

has

the

lat(er.

D e f i n i t i o n 1.3 and T h e o r e m s 1.11a, 1.9 and 1.7 y i e l d CONCLUSION

1.7

p,Q e W ^ CCp,Q,r:> ^ \r\ The concatenation the

s'um of

the

lengths

The l e n g t h o f than

zero,

of

two word of

which

follows

\p\

tokens

those

a word t o k e n

-

+

has

\
the

length

etfual

to

words. Is always a natural

from

Definition

1.3,

number

Theorem

I.lib

other and

C o n v e n t i o n 1.3. T h i s amd C o n c l u s i o n 1.7 yield a c c o r d i n g l y CONCLUSION

1.8

p,Q € V ^ C •• | r | The than

concatenation

the

length

of

of

two

words

either

of

those

C o n c l u s i o n 1.7 a l s o y i e l d s CONCLUSION

>

has

u/vi'c/i

length

\q\. is

greater

words.

immediately

1.9

p,q Concatenations same

> I P I ..-, \r\

€ W ^ CCp,<|,r> /s CCq,p,t) •» \r\ of

word

tokens

p and

length.

Theorem I.4a and C o n c l u s i o n 1.7 yield

Q and

-

|t|. q and

p have

the

THE AXIOMATIC THEORY TLTk OF LABEL TOKENS CONCLUSION

23

1.10

r e W\V «» ^ V

cc ^ | r | -

| p | + |.

p , ^fcPr

i4 compo-und

word, token

fino }))or
of

such,

that

is

that

a word xafiich the

sxun of

is

a concatenation

their

lengths

of

ecfuals

the

word.

We s h a l l a l s o g i v e o n e m o r e t h e o r e m which d e s c r i b e s

words of

a d e f i n i t . e lengt-h > 1. THEOREM

1.12

reW .^ | r | - 7. > 1 • •

^A^

p,^W

^ ' ^ ' " k ^ \
I ^ C
k+lmn A

word

token

of

concatenation

of

each

their

e q u a l s n.

lengths

Sec.

1.7.

Sec.1.6

we

which

states

that

have

But t h e d e f i n i t i o n s to state

word,

pair

word

shall

generalized The

1

word

is

a

tokens

word

such

formulated

a

tokens

finitely

the intuitively

from

make

the

lise

relation expression

are

which

that

obvious

concatenation

of

theorem

formulated fact

i s a word o f

order formally t o w r i t e we

of

>

and t h e o r e m s

f o r n > 1, t h a t

obtained

n

the

is

a

sum of

Generalized concatenation

In

us

length

of

the

term

of

concatenation

c

<-PM>PJ>--,P

C

simple Just ,

words.

s o f a r do n o t allow

that

down t h e t h e o r e m

I.lla>

componential

a length n

CTheorem

an

n-componential

n > 1, i s word

a

label

tokens.

formulated

which

In

above

denotes

the

o f n+2 a r g u m e n t s . +.,»P^

is

read:

p

is

a

24

CHAPTER I

concat.enat.ion tokens

of

n-K

P.>Pj>—>P ^^-

t o be t h e

successive

Oeneraiized

element.s,

The

labels

coinponents

relation

namely

of

of

successive

P^»P_»-..>P ^^

of

that

are

label

then

said

concatenation.

concatenation

c

l>

is

defined

by i n d u c t i o n . DEFINITION

1.4

a.

n " 1 •• Cc

(-P ,p ./>> «• CCp ,p ,p>>,

b.

n > 1 =• Cc"

«*

In

accordance

concatenation "ordinary"

of

with two

this

label

concatenation

n+1-element

l>

which

n-element

Pl'P2'

'^n ^"^ ^ label

only

those

if

p

means

it

of is

Cn>l!)

is

the

two

concatenation

2und

an

tokens of

P^>P^>->P >P ^A i f 1 2 n n+1 is

definition

a

a

two-element

same

as

labels;

p

a

p

is is

label

an

tokens

concatenation

concatenation

an

of

label

of

labels

p^_^^.

And here is the announced THEOREM

p ^ W^

I.13a

\p\

A \jtoTd. token concatenation In

m n+1 ^

of

length

of

certain

accordance

with

possible

to

token

a

is

formulate word

V

which

n*1

is

simple

C

a

label

• itord

Conclusion the is

following a

Cp ,p ,...,p

which

is

^ ,p).

an n+1

-element

tokens.

I.5b,

Theorem

conclusion:

concatenation

of

a

1.13a a

makes

compound

finite

it

word

number

of

THE AXIOMATIC THEORY TLTk OF LABEL TOKENS

25

component.iaI

simple

word

t.okens,

component.ial

simple

words

equals

and t.he

t-he

number

len(;Ch

of

of

the

t>hose

word

in

quest-ion. The p r o o f o f T h e o r e m 1.13a i s c i v e n i n t.he Annex. The f o l l o w i n g t-heorem a l s o h o l d s : THEOKEM

A

1.13b

compound,

coTicatenation The

xaord. of

a finito

simple

Conclusion

tohsn

is n-vmber

implication

I.5b,

a.

Theorem

of

in

by

lemma

induction,

I.13a,

Pi

and

a

certain

tokens. 1.13b

follows

Conclusion

to refer

I.3a.

To

from prove

to

is

a

^ ^ ^ """"^^^Pi'Pz'- 'f>n^'P^ generalization

based

on

that

D e f i n i t i o n I.4a,b, i s l e f t t o t h e Conclusions theorem

word

is

I.l

l
vhich

Theorem

t h e c o n v e r s e implication i t s u f f i c e s LEMMA

lahol

1.3a,

Cnext t o

V o f all w o r d s :

1.5b

Theorems

and

of

-* p

Concliision

conclusion.

'S>f-y. 1.3b.

Its

Axiom

I.ll

proof and

Reader. Theorem

1.4 and I.ll>

1.13a

yield

one

which d e s c r i b e s

the

more set

26

CHAPTER I THEOREM

1.13

A iDord

tok&n

is

either

IS a concatenation

of

a simple

a finite

inord

n'umber

token

of

simple

Simple word t.okens which a r e c o m p o n e n t s word

are,

up

to

equiformity,

or

determined

of

^ c < r , s , t > ^ p,r,
word

which

tokens.

a given

compound

unambiguously.

s o b e c a u s e t h e r e holds a g e n e r a l i z a t i o n o f t h e cXp,q,t>

a label

This

is

theorem

e V ^ p ^ r ^ q S : s ,

which i s a w e a k e n e d f o r m o f T h e o r e m 1.3: THEOREM

1.14

//

one

and

generalized

the

sajne

word

concatenation

tokens, in

the

l < v < n + l fi'^fi

then same

the

of

components

places

are

^ ^ -» l
token

two of

can

be

"r

presented

configurations those

'',• *

of

as

simple

configurations

a word

occurring

ecfuiform.

The p r o o f o f T h e o r e m 1.14 i s g i v e n i n t h e Annex. The

concept

practical later is

to

in be

of

generalized

importance, this

work.

seen

generalizations

in of

being For

our

concatenation useful

in

purposes

has

analyses its

particular

in

certain

theorems

formulated

a

considerable

to

practical theorems for

the

be

found

usefulness which

are

"ordinary"

concatenation. The

following

theorem

is

a

generalization

of

Axiom

1.7;

it

THE AXIOMATIC THEORY TLTk OF LABEL TOKENS

27

ToUows Trom t-hat. axiom and D e f i n i t . i o n I.4a,b. THEOREM

1.15

J4 lab»l

tok&n

concatenation

ufiich

is

n+1

label

of

n-H those

equi/orm

with

tokens

is

a label also

a

token

t/tat

is

concatenation

a of

labels.

Theorem

1.16

is

a

£eneraiizat,ion

of

Theorem

I.l;

it

of

n*1

follows

f r o m t h e l a t t e r and D e f i n i t i o n I.4a,b. THEOREM

1.16

^""'^^^I'^'Z' '''n+l'^' " l < i < n + l

If

a

tokens, labels components

given then which of

The s i m p l e

label it

is are

the

token

is

also

a

pairvnise initial

proofs

'*i ^ Pi

a

^

concatenation

concatenation etfuiform

of with

label

n+f

successive

the

successive

concatenation.

of

Theorems

I.IS

and

1.16

are

left

to

the

Reader. The t w o a b o v e t h e o r e m s y i e l d i m m e d i a t e l y CONCLUSION

n+1 C ^Pt'Pz'

in

accordance

1.11

'Pn*i'Pn^2^

with

» ^ l < i < n + 2 ^i ^ " i

which

if

the

relation

-

of

concatenation

of

28

CHAPTER I

n+2

arguments

holds

holds

between

bet, w e e n

successive

given

labels

with t h e elements of t h e f i r s t

label

which

tokens, are

padrwise

tokens

such

pairwise

that

which any

eqvtiform,

are

of

are

concatenations

thjeir

i-th

To c o n c l u d e

this

g e n e r a l i z a t i o n of

C

section

we

of

pairwise

place,

of

n+1 labels

each

Cl

are

omitted. a

Cq^,^^, ,<J„+i,q>

-

theorem

which

is

a

1.18

etfuiform

two configvirations

same

•* P ^ <*•

C o n c l u s i o n 1.2:

- C

two

^ Qi

components

formulate

^ p * q ^ l < j < n + l ^

If

Pi

etfxiiform.

The e a s y p r o o f o f t h i s t h e o r e m i s

THEOREM

equiform

1.17

Two label are

also

in

-^ l < i < n + l

and

it

concatenation.

Axiom 1.4 h a s i t s i n d u c t i v e g e n e r a l i z a t i o n THEOREM

then

except equiform,

label such for then

tokens that the those

l =• Pj

are all J-th j~th

n*i-element components

fj

*

concatenations occurring

components components

too. The proof of Theorem 1.18 is given in the Annex.

in

Cl<_/, are

equi

the are form,

THE AXIOMATIC THEORY TLTk OF LABEL TOKENS

S e c . 1.8.

The

29

Methodolosical remarks

t.heory

TLTk,

presented

In

based on t h e s y s t e m o f primitive
preceding;

sections

and

concepts

Lb, %, c , V equivalent

tl981a,1983ail of

t.he

to

tokens

author

and

2dt>stract

words

[19773;

C/9>

like

Lb,

which d i f f e r e d of

label

present

system of primitive

instead

of

by t h e

concrete

S.Nflklos

theory

the

latter

in t h e

papers

amd modelled

on t h e

theory

formulated

theory.

It

by

was

O.Bryll

based

on

and the

concepts %, c , W

f r o m Coi> by h a v i n g

the

outlined

vocabulary

V;

t h e s e t of

the

latter

a l l word t o k e n s W

is

defined

by

the

set

K,

formula CD

r^V^TGWys-i The

length theories.

definitions I p I,

and

Axioms

of the I.1-I.8

V the

following

relation are

c
C

are

identical,

the

identical

too,

and

-1.13 of TLTk a r e replaced by t h e axioms C2>

C4> C5>

0 ;^ K £ If c

y S X ^

^ ^, ^
Lb,

^ r e X^ ^ W S X,

cCp,<j,r) ^ Cr e V ^ p,tt e l»0.

in

Axioms

both 1.9-

30

CHAPTER 1 Thus

t.he

following

t.heorem

THEOREM 1.2 . system.

Cft>

Tho

of

consisting

of

Definition

Ciy

is

P r o o f . Cl>

and

expression

Cl>

theory of

is

from are

based

Axioms

adopt

the

on

set

and

definition I.la>

set

y which

the

other

any

set

-

on

h a n d , If Y which

XCX>,

which

I.9-I.13

of

of

the

are

reX,

from also

directly

from

Definition

CI),

and

finally

expression

C2)

I.3c,

the

Axioms

Axiom

I.ll

the

if in

we TLTk

for

every

<X|\CX».

say

that

and

theory

because

-

C4) ^

in

consisting

s a m e as

because C2>

holds.

then

W S

re¥,

Axiom

system

Condition

the

Theorem

theorem

relK,

to

and

adso

a

and the

Conclusion

hence

In

on

because

I.3a,

if

then

1.1-1.8

Conclusion

X<.Xy t h e

and

concepts, C2>,

the

axiom

of

and

Definition

Axioms

i n TLTk i s

theorems

on

TLTk,

Conditions

Condition

based

Inclusion

the

the system

C2y—CSy,

of

G H <X|XCX», w h i c h i s

primitive

CS>,

I.S,

strength

from

Condition

and

because

directly

from

1.9

\CX>,

axiom

TLTk,

expression

\<X),

the

on

TLTk.

in

In

CD a n d

satisfies

follows

TLTk

system

r

theory

W adopted

the

satisfies

the

converse

C2>-CS>,

of

theory

Theorem

Definition

1.1-1.8

tDefinition

C^>

the

the

I.4b

The

based

expressions

conslstln^^

Axiom

tokens on

contained

Theorem

C3>-C5)

I.l o f

to

that

Is

respectively.

Definition

anot

systems

immediately

expressions 1.6,

first

C2>-C5>

label

concepts,

equivalent

axiom

expressions

follows

tive

of

I. 1-1.8

Not-e

the

theory

primi Axioms

holds:

W

reX

for

satisfies

C5>. baised

On

Axioms on

the

Axiom

1.9

follows

and

1.12

follow

1.13

follows

1.10

follows

Immediately easily

from

THE AXIOMATIC THEORY TLTk OF LABEL TOKENS Condit-ions In

C2> a n d

Definition Theorem

C3>, a n d A x i o m s 1.12, 1.7, a n d

t-he paper

demonstrated

[1983a3

that Cl>

interpreted

the

set

number

1,

the

of

author

label

I.1-I.8,

of

TLTk is

also

in the arithmetic

are

as

Axioms

present,

tokens

<2>-CS>

The theory TLTk is

The consistence

y,

t h e

I.ll.

is

i t was based

on

consistent.

yields

T H E O R E M 1.3 .

interpretation

by

t h e theory

and

1.2 t h u s

31

as the the

set

relation

the function of addition.

set

N of

<1>, the ^5:, as

of

consistent.

immediately natural

natural only

numbers if

numbers

element

identity,

obtained

and

of

the

by

its

Lb and W

without which relation

zero,

Is

the c

as

CHAPTER 11 THE AXIOMATIC S Y S T E M TSCL OF S I M P L E CATEOORIAL

Sec. II.l.

LANGUAGES

The principal o b j e c l l v e s o f t h e c o n s t r u d i o n I h e -theory o f s y n t a c t i c a l l y c a t e g o r i a l

The

discussions

languages,

which

concerned tokens

with

which

languaige expression on

its

is an

do

£,

of to

not

be

in

accordance

grammar

tl935]

Sec.I.l

we

a

which the

consists, vocabulary,

equiformity, meaningful, elements theory

V, V; ^,

the

word

tokens

set

of

all

relation

of

concatenation,

well-formed

presented is

a

imposed

of

characteristic

of

TLTk,

of

K.AJdukiewicz's

conceptions

set

C;e>, e x c e p t

language

those

the in

of

any

C; S >

the

i.e.,

The

sixtuple

of

of

expression

the

successively,

the

operators.

of

be

expression

conditions

Idea

simplified

language in t h e form of t h e < Lb;

of

will

t o his ideas.

gave

CJ?>

the

and

£

by

certain

categorial

chapter,

cat&gorial

by

with

this

bound

simple

languages

simple

language

variables

a

of in

fixed

determined

r e s e c u ^ c h e r s who r e f e r r e d In

but

Include

is

TSCL

constructed

here

tokens,

categorial

theory

arbitrary

called

syntax

the

of

all

tokens,

the

and

expression

label

the

tokens.

tokens,

relation set

of

All

of all the

last

one,

are

described

to

Chap.I.

The

set

S

well-formed

concept

which 32

is

of

all

defined

in

the

be

the

theory

THE AXIOMATIC SYSTEM TSCL OF SIMPLE CATEGORIAL LANGUAGES TSCL to

and

the

tokens The

adso

<see

categoTial which

theories

may include

of

of

the

because

categorial

S

set

of

used

the

in

expression of

syntactic

connexion

of

expressions

They

are

categories to

carry

of

categories The set

I

o/ -^

o elements

of

functoral,

of

all

I

categorial

are

The

are

are

and

is

auxiliary they

are

in

the

the

syntactic

Secs.II.8

the

S

all

well-formed

Csee

of

of

useful

language

C t CS>

indices

indices.

used

to

The

indices

is

and

syntactic

and enable into

us

syntactic

generated

basic

indicate

remaining

univocally

termed

symbolized language

given

Sees.II.2

a

indicating

cisissif i c a t i o n

basic

syntactically

function

a

both

because

language

in

into

an

of

them.

the

indices

basic

determine

from

the

which

are

categories the

of

derivative,

categories.

Indices

words

logical

of

only

that

given all

expressions

all

expressions.

given

a

correctness,

above

perform

and

by

foundation

concept

III.4>

a

Csee

the

not

the

expression

bound

indices

they

and

of

of

C s e e S e c s . I I . 6 a n d 111.53.

set

I

used

the

out

of

pertaining

introduction

are

essentiaJ

Secs.II.5

checking

III.6>.

the

indices

definition

(see

languages

categorial

is

TSC«»-L,

a n d variables

Although

role

theory

i.e.,

require

languages. their

the

operators

categorial

function,

in

<ji-langvages,

definitions

II.3>,

Ghap,III>

33

by and

that

categorial

the

L. T h e its have

assigned

definite

language

/-unction set

E

of

subset

S

will

indices,

to

that

of

by

word

means

indication

all

is,

selected from

of

of

expression

be

tokens

token from

the

a

of

certain

indices of the

domain

a

and a

given

set of

of the

34

CHAPTER II

function A

t.

cert-ain

put-

synt.act.ic

t-oget.hei>,

concatenation, that and

have is

useful

vocabulary ainalogon (p

4>

the

obvioijs

that

well-formed

not

any

every

greatly

TSCL

language

constructed

in

show

and £, the

that

< Lh;

V, V; •^, C; I

satisfy

is

has

any

its

two

word

which

have

have

this

and

an

<^ ) ^ C<^ >.

to

index, It

way

sentential

definite

the

language

present

Cje >

of

whose

,

in

be

is is

a

calculus.

conditions

must

words.

requires

which

complex s y s t e m ,

elements

assured, the

to

which

formed

language

categorial

much m o r e

whose

be

indices of

made a b o v e

<(> ,

index

will

p

calculus,

concatenation

the

to

simplified

categorial

and

an

contemporary

operation

that

tokens

calculus,

concatenation

expression

of

is

syntactically

of

has

the

particular

four-element

expression

remarks

is

word

language,

in

t.o

appropriate

that

sentential the

possible

of

the

instance,

brackets,

seven-element

it.

the

token of

of

ip ,4>

by

word

language

tokens

makes

configuration

For

which

the

which

token

language

of

be imposed upon t h e

of

E.

round

well-formedness

The

set

the

word

ordered compound

operation

assigns

assigns

If

the the

an and

three

a

p,

combine

expression

includes

index,

any

into

of

in

hence

finite,

defining

descriptions

an

any

compound

in

tokens

and

indices

a

operat.ion

the

syntactic in

the

characteristic

form

completion. foundation

chapter,

is

of

(JfJ

The of

Is

simple

the

theory

characterized

by

a

namely , 1;

certain

L, p;

E, S; C t C S )

aussumptions

>,

postulated

in

that

THE AXIOMATIC SYSTEM TSCL OF SIMPLE CATEGORIAL LANGUAGES

35

t-heory. It, any

is

wort/h

noting

ci>-languaee,

form

of

an

to

that

which

ordered

the

the

system

syntactic

theory much

characterization

TSCw-L

more

pertains,

complex

than

of

has

the

CJP >

(see

Sec.III.l). The the

analyses

theory

III.5>

in

cairrled

of

syntactic

two

theorems

of

out

TSCL

which

theory

and

categories

theorems

t/ie

In

of

TSC«»~L c l e a r l y

and

might

syntac

tic

result

by

(see

termed

categories,

refer

to

Secs.n.7, /'undamiental

aibbreviated

as

rttsc. Their

formulation

is based

Cay

on t h e

expression

rCpyq^s,

whose

corresponding

Important The from

definitions

expression

the

by t h e Now

CcO

is

s

by

expression

p.

Co) i s

common

the

has



r,

the s

p, the

r

the

are

its

expressions

schema

of

the

replacement

of

The second

as

II.

has

the

TSC«-L

its

r

its

is

are

fttsc.

One

second

same

syntactic

in

the

form

of

of

the

same

syntactic

theorem

under

y.

implication a y^ y

^

ft

of

q

those

assumption:

the

the

obtained

constituent

of

a ^ ft ^ ftlsc

of

both

implication I.

and

expression

expression

thesis

TSCL

the

assumption

expressions

has

in

theories.

read:

following

are

theorem

Cj'J Thus

in t h o s e

expression

theorems

That

definitions

category.

expression: category.

consideration

has

the

36

CHAPTER II

as

it,s

schema.

Under

the

assumpt.ion

Now Cft^ h a s Cft'i

The

t - h e t w o

its

concretlzatlon

r,s

are

expressions

of

sentences.

fttsc

In t h e

of

the

aa-e

converse

theorems.

expression:

syntactic

category

implications

CIO

a y^ ft' •* r

or dll}

are

a ^

schemata

theory

of

categories requires

of the definitions

syntactic as

of

both

belongs

the

ft'y

C;'>, often

The

expressions

introduction

categories to

of

^

of

categories.

sets

the

Cf

into

Cot) and

syntactic

of

the

of

syntactic

replaceable

In

sentences

of

syntactic

theory

concept

category

in

treatment

the

the

adopted

of

expression

sentences, that

that

is, the

concept of sentence. Including the concept of

the

theory

of

of sentence

syntactic

in the

categories

possibility of defining it. It seems

that

primitive

would

concepts

preclude

It is the

the

defining

of

the concept of sentence that should be one of the fundamental tasks

of

natural definition

the and

theory symbolic

of

sentence

of

syntactic

categories

languages.

It

seems

should

satisfy

the

as also

applied that

condition

to the

which

enables us to formulate the algorithm of checking the syntactic connectedness

of

.^Jdukiewicz. But condition, when

a

expressions, definition

CI'> and

anadogous

of

CIII) are

sentence treated

to which

that

given

satisfies

ais a schema

of

by this the

THE AXIOMATIC SYSTEM TSCL OF SIMPLE CATEGORIAL LANGUAGES

definition

of

belonging

syntactic

categor-y,

circle.

constructing

In

languages is

we

treated

certain chapter.

try

as

constructed

expression

in

involve

essential

IV.6

we

the

with

the

purpose

DTSCL

and

ideas

in

more

vicious

endeavour and the

next

They

underlie

describing

its

theories

DTSCw-L.

which

same

categorial

TSCL

outlined two

of

a

Our

theory

discuss

the

of

difficulties.

of

the

syntactically

TSCw-L,

languages:

accordance

for

goal

to

danger

of

those

theory

and

the

theory

overcome

the

expressions

of are

TSCL

languages

of

types.

S e c . II.2.

theory.

the

two

the

categorial

TSC«»-L

Now

to

Secs.IV.S

syntactically

and

would

modification, In

of

37

C o n n e c t i o n s b e t w e e n TLTk a n d TSCL

TSCL Hence

substitution,

like

TLTK

all are

is

logical valid

based rules,

on in

classical

logic

and

set

particular

the

rule

of

in TSCL b e c a u s e

TSCL i s

superstructured

o v e r TLTk. Hence a g a i n REMARK II.1.

The c o n s t a n t s y m b o l s o f t h e v o c a b u l a r y o f TLTk Lb,

y,

W, %, C

a r e c o n s t a n t s y m b o l s o f t h e v o c a b u l a r y of

TSCL.

The v a r i a b l e s y m b o l s o f t h e v o c a b u l a r y o f TSCL i n c l u d e C-u i

p, i^ T, s,

t, XI, x>, w i t h o r w i t h o u t

subscripts,

and Cu )

X, w i t h o r w i t h o u t and

Cv > r a n g e ,

subscript;

respectively

Csee

Convention

I.la,b>,

over

38

CHAPTER II

Lb a n d t,he f a m i l y 2 The

variables

of

all it.s subset^s.

list-ed

under

a n d

do

1

list,

of

t-he

variable

symbols

of

not

exhaust,

t-he

Z

t-he

vocabulary

of

t-he

language

o f TSCL < s e e Sec.II.S, C o n v e n t i o n I1.3b>. We o b s e r v e t h e p r i n c i p l e Csee C o n v e n t i o n I.2> C o n v e n t i o n whose

numbers

are

II.1.

marked

Those t h e s e s with

an

Caccepted

asterisk

are

m e t a t h e o r y o f TSCL, a n d n o t e c o r r e s p o n d i n g l y THEOREM II.l . a thesis

of

Both tokens. being the

In

shall

both

now

under

theories

it

account.

From

into

ff*, b u t

see

thosis

sentences)

theses

in

the

that

Cacc&pted

sentence)

of

TLTk

is

TSCL.

theories

taken

set

Every

that

also t h e

CSec.II.3,

set

consideration is

Lb that

I of

Theorem

which set

all

II.l),

/

pertain

is we

the

has

out

set

not

indices;

no

word

universal

single

categorial

to

common

only

as

we

elements

w i t h W. Now recorded

/

is with

TSCL, w h i c h

a the

defined use

denotes

of

the

concept a set

new of

in

TSCL.

primitive all

Its

term

basic

definition

is

I , specific

of

indices

CI

includes o

a t leaist one t e c h n i c a l symbol). The

set

/

will

be

defined

with

reference

to

the

set

/

in o

t h e f o l l o w i n g m a n n e r Ccf. D e f i n i t i o n

I.la):

THE AXIOMATIC SYSTEM TSCL OF SIMPLE CATEGORIAL LANGUAGES

DEFINITION

II.la

XCXy'^ ** I The set and

only

of if

relation

39

Z X ^

label it

of

^

Cp.QS

tokens

incltides

X ^ c ^ r e X).

satisfies the

set

the I

expression

and

is

\<Xy

closed

under

if the

concatenation.

Hence we have
II.l

/ - n < X I \ao^ >. The set

of

all

categorial

tokens

tthich

satisfies

By

these

definitions

which

includes

indices

the

/

is

expression

I

is

and

is

/

are

the

\CX>

the

set

of

label

.

least

closed

least

set

of

under

the

label

tokens

relation

of

o concatenation. The three

properties of

the

of

following

described

five

above

axioms

of

all TSCL

by

the

Ccf.

first Axioms

I.9-I.13>: AXIOM

II.l

o The set AXIOM

of

indices

is

non-empty

iTidex.

of

label

tokens.

II.2

p e l A label

set

tnhich

is

etfuiform

y \ q S i p ^ q e I . with

a basic

index

is

also

a

basic

40

CHAFFER II AXIOM

II.3

cCp,q,r> A basic

ind»x

is

not

a

" • r e / .

concatenation

of

any

two

label

a concatenation

of

tokens. AXIOM

II.4

p,
o A categorial a pair

of

AXIOM

index

lahich

categorial

is

not

basic

is

indices.

II.5

r e / /s cCp,<j,r> ^ p,qi € // those

a categorial labels

are

We s h a l l I

of

index also

now

all

is

a concatenation

categorial

introduce

/.

a

n—componential

of

two

labels,

then

indices. definition

by

categorial

induction

indices

of

the

Ccf.

set

Definition

I.2a,b>. DEFINITION a.

*/

II.2 -

/

, o

b.

r e

"**/ • •

V

V

pe I ^ A one-componential n-t-1-component concatenation

ial of

cCp,q,r>. o

categorial categorial

index

index

an n-componential

is

a

is

a

label

categorial

basic token index

index; xahich and

a

an is

a

basic

i ndex. The II.2a,b

syntactic above

respectively,

with. which

analogies

of

Definitions are,

Definitions I.la,

I.l,

successively,

and

II.la,

II.1,

I.2a,b

of

definitions

of

atfid TLTk, the

THE AXIOMATIC SYSTEM TSCL OF SIMPLE CATEGORIAL LANGUAGES expression

\CX>,

The

same

the

correspondine

those of

a

set,

W

to

the

analogies

appiies

Axioms

observations dv.al

analogon

DEFINITION TLTk

the

which

II.l . next

t-he

set

V,

between

are

self-evident.

Axioms

11.1-11.5

I.9-I.13

of

TLTk.

We

shall

more

precisely

by

making

use

of

of

sxpression

an

/ /

to

and

41

(p

is

of

any

variables

formulate

the

concept

TLTk.

expression

may

and

of

include

the

the

language

symbol

Lb

of

and

the

only

if

symbols <1>

K,

then

the


is

each

expression

4>

obtained place

from

of

¥. is

a

tp f>y

symbols

dual tf*-^

analogon

of

replacement,

Cl> respectively

I . I,

<1>'^

"v.

by

and

the

latter,

in

are

dual

symbols

"l.

o It

is self-evident t h a t

REMARK

II.2.

analoga

of

Axioms

II.1-II.5

Definitions

Definitions are

Il.la,

I.la, dual

II.l,

I.l,

and

and

analoga

II.2a,b

I.2a,b, of

respectively.

Axioms

1.9-1.13,

respectively. N o t e t h a t , b y D e f i n i t i o n II.l*, REMARK

II.3.

Axioms

I.la-c,

I.2-I.8

have

themselves

as

t h e i r dual anadoga. T h e o r e m II.l THEOREM

defini

t ions

a n d R e m a r k s II.3 a n d II.2 y i e l d

II.2 .

The

accepted

in TLTk are

Hence we h a v e

dual

analoga

theses

of

in

all

TSCL. ' '

axioms

and

42

CHAPTER II THEOREM II.3 .

a thesis The

Every dual

analogon

of

a thesis

in TLTk

is

in TSCL. pract-lcal

list., by way of

values example,

of

Theorem

t^hose

II.3

theorems

can of

be

shown

if

we

TSCL which we shall

use in the proof of the theorem given in Sec.II.3 below. We adopt the following convention: C o n v e n t i o n TLTk

has

the

same

II.2.

A duad analogon

number,

but

that

number

of is

a

theorem

marked

of

by

the

semiotics

by

superscript d. Here are the theorems mentioned above: CONCLUSION 1.3

a

I CONCLUSION

o

S I S Lb.

I .4^>

I\I

7t 0 . o

T H E O R E M I.IO

'^ THEOREM

o
1.11

U Ifc-1

Sec.

II.3.

Categorial indices and t h e i r

Categorlal Kazimlerz the

indices

AJduklewicz

hierarchy

of

were

introduced

11934,1935].

semantic

He

into

used

Csyntactlc)

indication

them

categories

in

describing and

in

his

THE AXIOMATIC SYSTEM TSCL OF SIMPLE CATEGORIAL LANGUAGES investigations

of

the

syntactic

43

connectedness

of

linguistic

expressions. Categorial called

Just

often

been

the

in

that

they

that

do

students

be

TSCL

it

not

describe

axiom

is

kinds.

It

is

to

indices

assume,

are

as

has

label

tokens

are

exhaust,

the

dual

of

analoga

course,

properties

of

the

of

Axioms

I.9-I.13

list

those

aodoms

of

categorial

indices.

The

next

II.6

disjoint

with

Now I

V the

set

of

a.

of

V

of

a

from

(Definition

I.l>

well-formed

expressions.

the

set

the we

/

the

auxiliary

the

well-formed

of

which

is

part

vocaittilary

vocabulary

K)

single a

/

>,

a of

of

X.

language a

of

way

(Definition the

set

relative

Jf.

out

similar

indices

lang-aage

X

is

to

the

indices.

language

expressions of

categorial

object

In

vocabulary

expressions

0.

i t s basic

shall

all

-

a n auxiliary

given


o

given

all

may b e t e r m e d

voc2d}ulary

language

various

convenient

categorial

which

vocabtilary

course,

of

languages

numbers.

is

V n I

from

categorial

II.l, Sec.II.2>.

which

The

of

objects

are natural

earlier,

II.l-II.S,

AXIOM

may

of

Definition

TLTk,

some

purposes

done

Axioms

by

categories,

assumed

For

(cf.

indices,

of

Indexing other

metailanguage

From all

the we II.1>

the

set

word

set can

S

tokens of

all

single

out,

(generated

from

expressions expressions than

It*

£,

relative

indexing are,

namely to

Jf

of

of

a

(see

44

CHAPTER II

Sec.n.4>. The

theorem

announced

In

Sec.II.2

sheds

much

light,

upon

t.he

vith

th»

s»t

of

all

Jf

by

the

a b o v e pemark: THEOREM

II.1

¥ n I m 0. The

sot

of

all

catsgorial

word

tokens

is

disjoint

indices.

The p r o o f I s g i v e n i n t h e Annex. Gat.egoriaI relation

indices

t

of

are

assigned

indicating

relation

Is

satisfies

t h e following axioms:

AXIOM

the

the

second

a

ftmc tion

a subset AXIOM

of

A label

with

the The

we

of

of

Miord

primitive

tokens.

concept

In

That

TSCL.

It

of

one

L is

a fxinc

arg'ument

tion.

which

maps

a subset

of W

I.

II.8 p e

index

words

II.7

L is

an

the

indices

specific

L S Wxl and

into

to

DCL'>

token is

xnhich

also

index

^ < J % p ^ q f € D C i , >

is

ecfuiform

a word

of

that

interpretation

note

that

the

are

words

to

that

word of

/s

L *; <. Cp>.

with

has

an

a word index

token

which

that

is

has

equiform

token. Axiom

elements

of

II.8

the

becomes

domain

comprehensible

DC i. > o f

the

if

function

<7>

L

elements

of

the

which

an

index

counter-domain

has

been

D <(.> o f

assigned,

that

function

while

the

are

the

THE AXIOMATIC SYSTEM TSCL OF SIMPLE CATEGORIAL LANGUAGES

indices

of

-those

words.

This

foUows

45

iTnmediat.eiy

from

Ajdom

word

tokens

II.7 i n t h e v i e w o f CONCLUSION

Il.l

a.

DCt? = ¥ ,

b.

D'*C0

The domain laftile

its

of

that

arbitrarily assigned

of

is

11.7

whether

to

decide

words o f

a

the

s»t

sxibset

of

and

¥ of of

indices

technical,

a given

whether

Conclusion

categorisd

structural,

all

the

set

I

of

all

do

are usually

Csuch

as

indices

not

or

signs

language

categorial

Il.l

decide

not

are

Included

brackets);

are assigned

in

n o r do

only t o

the

practice

we

£.

Conclusion not

II.la

seems

Inclined

concatenation that,

a stibs&t

Axiom

vocabulary

are

L is

indices.

Note

they

1.

counter-domain

categorial

the

£

of

however,

expressions

to

natural assign

t h e simple

with

also

a

categorial

words

reference

(Ger. AusdrUcke>

of

to those of

because

that

a

given words

language.

in index

to

every

language.

We do

which a r e

termed

Expressions

will

b e d i s c u s s e d i n Sec.II.4. Axioms II.8 and I.lb i m m e d i a t e l y y i e l d CONCLUSION

II.2

a.

p e Ddy

b.

p,q

We

shall

occasions.

e

^ q ^ p •* tf e

DCL^ /\

avail

(f ^ p

^

ourselves

DCty,

i.

% t < q > .

of

Conclusion

II.2b

on

many

46

CHAPTER 11

S e c . II.4.

The

Expressions

one-to-one

operation and of

the

forming

last,

all f i n i t e

Oc L > o f to

of

all

them. I t s

funct-ion

p

compotind concept

and g r e a t e r

than

tokens,

variable, tokens

called is

the

t-he third,

of

TSCL. I t s

domain

is

the

sum

one

Cartesian

powers

of

the

set

which

counter-domain

one

expression

primitive

word

of

have

a

categoriaJ

is a subset

of

DCi.^\y.

index

assigned

Hence

we

add

t h e f o l l o w i n g axiom t o TSCL: AXIOM

II.9

p:

U DCL>

»

DCL^W.

*-2

The c o m p o u n d w o r d t o k e n d e n o t e d

which

is

the

value

of

n>\

and

(~p^,p.,.-,p > for O 1 n -domain

of

compoMnd tokens

p,

and

misunderstanding, compoxind To

,

in

an

of

£.

the

compound

expression

will

be

not

precise,

thus

which

set

consisting

order.

If

is

Their g e n e r a l i t y makes precision

the

of

will

be

is

connected

certain

Justified

by

impossible.

no

termed

make

counter-

is

n+1

there

argument

the

DCtJSK, of

intuitions

shall

for

element

the

expression

we

p

Jf,

appropriate <6)

an

of

of

that

such

operation

also

token

expression

activate

is

hence

expression P,^,P.,--,P

the

by

called

<7»>1>

of

briefly

remarks.

their

word

risk

with

a

a

the They

character.

THE AXIOMATIC SYSTEM TSCL OF SIMPLE CATEGORIAL LANGUAGES

We b e g i n w i l h s e v e r a l

terminological

The s e q u e n c e

tokens

C*> of

o f word

''o'^l' which

the

compound

word

functor <xTgvm»nts that

p

of

of

C6>, of

conventions.

'''n expression

be called t h e e x p a n s i o n The

47

of

expansion

the

p^, t h e

£

Is

constructed

will

<6).

the

and

<<5> o f

(.e^

words

word

p.

will

be

P^iP^r-tP

Cl£i

called

the

be

called

will

-

the

i-tK

n>ai.-n the

argument

of

functor.

By

analogy

functor"

au^d

functor Index

to

of

of

the

commonly

"name-forming

The

functor,

existence

terms

functor"

<6> t h e a-forming

C<5>.

used

of

we

shaill

where

that

"sentence-forming

index

a

call

is

the

the

follows

main

categorial

from

Axioms

II.9 a n d I I . 7 . The

label

treated

as

compound functor

from a

the

translation

expression pu

and

"translation" intuitive

language

Its

is

sense

into of

X

of the

which

successive

left

TSCU

language is

of

arguments we

with

given

that

by

to

by

such

a

the

main

•

The

term

n

assume It

of

be

of

P^rP'^f-tP \ Z

merely

C6> m a y

TSCL

constructed

undefined;

agrees

denoted

that

Alfred

its

Tarskl

i n h i s w e l l - k n o w n m o n o g r a p h [19331. In

the

translation TSCL

does

expression whether

present of

not is

a

a

book

compound depend

recorded;

bracket

the

essential

expression upon In

notation

the

of

X

into

symbolism

particular, is

fact

used

it in

is X

is

that

the

the

language

in

which

independent or

not.

of

that of That

48

CHAPTER II

t
depends

compound

sense

of

merely

expression

Le^nlewski's

on or

which

the

mereology>

words

of

the

expansion

expansion

of

its

part

which

is

a compound

of

(in

the

expression

are main functors and which aire their arguments. For of

instance,

each of

the

the

same

following

metallnj^uistic

label

is

a

translation

synonymous compound expressions

of

the

language of the sentential calculus: p .«
letters

and the

p,q,r

are

expressions

notation

introduced

in

here,

given

which Is now

o-Cp,oCq,r»; of

above

most

[19291,

CpEqr.

course, are

sentential

written

commonly used,

aurad

variables,

successively

Le^niewski's

Lukasiewicz's

in

the

notation

parenthesis-free

notation, called Polish, tl929]. Axiom formulate

II.8

Csee

an

important

same applies t o the AXIOM

If

p

is

a. word

it

if

of

n+1 word

same

II.2b>

property

of

makes compound

it

possible

to

expressions.

The

following

11.10

sv.ccessi'oe

and only

equiform

Conclusion

with

place.

compo-und expression

consisting

tokens,

token

if

it

then is

also

tokens

and

every

the

word of

the

a

label

q

of is

n+1

ecfuiform

a compound expression word

of

expansion

the of

Cn>l^

expansion p occxirring

with

consisting of

cf in

is the

THE AXIOMATIC SYSTEM TSCL OF SIMPLE CATEGORIAL LANGUAGES

A x i o m 11.10 l e a d s CONCLUSION

49

\,o

II.3

^ Cp %
Tiiio compoxindL expressions \aoTdi tokens their

each,

are

expansions

consisting

equiform

t^ich

if

occ-ur

in

of

and the

the

only

same

p^

same

if

any

place

are

% Q^>.

nvanber

of

terms

of

t\fo

equiform.

The p r o o f i s g i v e n i n t-he Annex. We f o r m u l a ' t e t h r e e f u r t h e r DEFINITION

a.

definitions:

II.3

E

m V n

DCiy,

s b.

£

-

D'*<:p>,

G

c.

£•

E -

U E .

s The tokens

c

elements of

£.

counter-domain

of

the

The of

set

elements the

E

are

called

aa>e of

function

t e r m e d compound e x p r e s s i o n set

E

the p,

tokens

expression

simple

In

accordance

expressions

expressions,

of

of

with £

is

sum

of

of

Jf.

the

earlier

been of

the

Frequently,

when

and E we s h a l l J u s t

refer

II .3c of

i.e.,

The e l e m e n t s

expressions,

Definition the

E , c

expression

already

X.

tofcens

compound

simple

set

have

speaking about t h e elements of E , E to

called

the

the set

and set of

all

expressions. E its

e x p r e s s i o n s and t h e s e t o f a l l i t s compound e x p r e s s i o n s . D e f i n i t i o n 11.3a, Axiom I.IO, and C o n c l u s i o n II.2a yield

of

all

simple

50

CHAPTER n

CONCLUSION II.4a

p G E A label also

token

vihich

a simple

y^
is

ecfuiform

iiiith

a simple

expression

is

Conclusion

II.2a

expression.

Definition

II.3b,

Axioms

11.9

and

11.10

and

yield CONCLUSION II.4b

p G E A label is

also

token

mhich

a compo-und.

is

/\
with

a compo-und

expression

expression.

D e f i n i t i o n II.3c and C o n c l u s i o n s II.4a,b y i e l d CONCLUSION

A label also

II.4C

token

\uh.ich

an expression

of

is

etfui/orm

with,

an expression

of

£

is

X.

D e f i n i t i o n s II.3a,b,c and Axiom II.9 y i e l d CONCLUSION a.

£• c

b.

E

The of

set

all

II.5 S OCL>N^',

S

of

D(.L:>.

all

compound

expressions

is

compound words

included

expressiojxs

which in

have the

set

is an of

included

index. all

The

words

in

the

set

set

of

all

have

an

which

index. In

accordance

with

Conclusion

categorlal index assigned to it.

Il.Sb

every

expression

has

a

THE AXIOMATIC SYSTEM TSCL OF SIMPLE CATEGORIAL LANGUAGES To want,

conclude t o

Lhe discussion

point,

construction,

t.o a

carried

fragment

t h e fragment

of

which

51

out^ i n

t.his

t h e theory describes

sect.lon

TSCL

y/e

now under

t h e system

of t h e

form: <£ >

< Lb; V, y; 5:, C> Z , I,

c In

L

, p; E >.

o doing

and

so

we

Axioms

the

also The

theory TETk

TSCtr-L,

to

II.l-II.lO

fragment 11.13.

refer

and

is

under

tokens

t h e foundation

analysed

(jm languages,

in

that

definitions

II.la

and

of

TLTk

I1.1-II.3.

That

11.12

and

II.8-II.10,

consideration

will

be

called

t h e

theory

and denoted by TETk.

of

t h e

is

and

Definition

now

expression

axioms

Definitions

includes

fragment of

the

TSCL

next

and also

chapter

languages

with

of

and

pertaining

operators

which

t o bind

variables. TETk the of

most

i s enriched important

by new concepts, of them

o f TSCL.

o f a vtell-formed.

One

of

expression

£.

S e c . II.5.

We

Well-formed

begin

with

many o c c a s i o n s

a.

further

We r e s t r i c t

C-u !) t h e s e t E.

expressions

formulating

C o n v e n t i o n

to

is that

specific

p,<j,r,s,

several

conventions,

t o be used on

in t h e text. II.3 t h erange

of t h e variables

with

o r without

subscripts,

52

CHAPTER II

b.

The

letters

Cv >

a,b,c,
,0,

with or without

subscripts,

3

I.

range over the set c. and

i.

" a,

for

any

i

e

i. " c ,

ma.,

(.Co.) "ft".,

We e n l a r g e

Convention

with

zero.

Convention concerning

•

d,

the

range

t"c.,

of

the

i»J»fc>i »"»>'>»•••> (see

(.<s>

//u<0>

tCp.) d.

(.C<j> " 6 ,

I.3>

II . 3 a

the

to

the

is

set

not

in

variables

tCs.)«d..

variables

with

or

NLKO>

of

without all

naturail

contradiction

of

the

type

subscripts,

with ^'*'^^

numbers

Remarlc Csee

II.1

Sec.II.2>

because CONCLUSION

II.5C

C S V .

The alI

set

word

of

all

expression

tokens

is

included

in

tfie

set

of

tokens.

T h i s f o l l o w s f r o m C o n c l u s i o n s Il.Sb and II.la. The

correctness

of

Convention

11.3c

follows

from

Conventions

II.3a,b and C o n c l u s i o n II.5b. We now -formed

formulate

expression

£.

In

doing

of

the

of

order

set n

so

the tokens

we

avail

S consisting CTI>0>.

definition of

the

a given simple

ourselves of

of

all

of

the

well-formed

set

S of

all

categorial

welllanguage

inductive

definition

expression

tokens

THE AXIOMATIC SYSTEM TSCL OF SIMPLE CATEGORIAL LANGUAGES DEFINITION

53

II .4

s b.

p e

$

^

p e

S ^

V Po^Pj,

,P„€*5

00

c.

S -

By

Oefinlt,ion

well-formed

U "S. n-0

II.4a

a

expression

well-formed

simple

of

expression

order

token

of

is

£

are

has

with

already an

p satisfies the

c accordance

functor

of

of

index

the

that

or

II.4b

set

is

a

every

of

S, a b o u t which we a

Is

order assume

compound

expression

that

all

its

order

h,

S. F u r t h e r ,

the

such

tokens the

Definition

is

Pr^yPtyP'.'-fP

to

^

of

words

that

is

expression

condition

m in

If

the set

expression

which b e l o n g

By

of

o r d e r fc+1 e i t h e r

defined,

expansion

well-formed

expressions

been

0.

p of

k, and t h u s i s an e l e m e n t o f that

expression

Ca,a^,...,a^,ajj>,

with

p, t h a t a

functor;

which is

the

and t h e in

m

the

categorial

a-forming

indices the

of

index

functor, all

a.

index is

of a

be

main

concatenation

successive

should

the

arguments followed

of

by

a

P definite

technical

AJdukiewicz's

index.

symbolism

f u n c t o r of p, t h a t i s f r a c t i o n a/at.,a.,...,a . \ Z n By D e f i n i t i o n

II.4c,

Note the

the S is

in

categorial

a-forming the

this

sum

connection index

of

Index, h a s t h e of

all

sets

of

that the

form

in main

of

the

well-formed

54

CHAPTER II

expression

tokens

of

a

finit-e

order

Cgreat-er

about-

well-formed

t^han o r

equal

to

tokens

we

zero). Instead

of

speaking

s h a l l h e n c e f o r t h r e f e r Jvist t o w e l l - f o r m e d The

set

following

S

of

we

well-formed

AJdukiewlcz

syntactically Now

all

in

S

can

adopt

be

a

expressions.

expressions

that

connected

expression

respect

-

of

J? we

call

the

shall se t

of

expressions.

defined

otherwise.

successive

To

substantiate

definition,

which

is

this

claim

auxiliary

In

character. DEFINITION

II.S

s

p n>l

p^,p^,

. . . ,p^eX

A, C The set

X satisfies

a superset

of

expansion condi

uhose

tion

We

the

m

is

P

also

give

tfte set

all also the

E words in

Ca,a.,...,a

condi

and

'^

tion

'^ ^O '^l ,a > •* p e

SCXy

if

e.very compo-and are

that

in

*^n

X and

Xi.

and

only

if

expression which

it

is

with

an

satisfies

the

set.

theorem

which

could

replace

the

adopted

definition of S. THEOREM

II.2

S - n < X I 6<X> >. S is The

the

least

proof

ourselves of

is

set given

that in

satisfies the

the

Annex. In

condi that

tion

6CX>.

proof

we

avail

THE AXIOMATIC SYSTEM TSCL OF SIMPLE CATEGORIAL LANGUAGES LEMMA

II.1

S S

m <: n ^

Jl\e-

set

greater

of

than

expressions The

55

all n

of

lemma

II.4b, '^s S

well-formed

is

expressions

included

order

S.

of

in

the

set

of

from

the

fact.

ca\

order

all

not

-well-fornted

n.

follows

easily

'that.

, by

Definit.ion

S f o r any At.

Definitions

II.4a,

II.3c

and

II.4b,

Concliislon

Il.Sb,

Axiom

II.9 and D e f i n i t i o n s II.3b,c y i e l d by i n d u c t i o n LEMMA

II.2

* S £ £ = D,

f r o m which, by D e f i n i t i o n II.4c, we a r r i v e

at

CoNcuusioN II.6a S £ £ /N S S D C O .

The the

set

set

that

of

of

have Let

us

all

all an

veil-formed its

expressions

expressions

and

of

the

set

£ of

is

included

all

its

in words

index. call

the

of

Jf.

expressions

elements

S\ S

of

Definitions

compound

II.4c,a,b,

Lemma

well—formed II.2,

Axiom

II.9, Definition II.3b, and Conclusion 11.5a yield C O N C L U S I O N II.6b

c The subset of

set of

of

all

compound

the

set

of

at I compound

word

all tokens

well-formed its

compound that

have

expressions expressions a categorial

of and

£ the

index.

is

a set

56

CHAPTER II

A t h e o r e m a n a l o g o u s t o C o n c l u s i o n II.4c holds: THEOREM

II.3

An expression expression

is

Theorem

of also

X vittich

is

etfui/orm

a xiiel I-formed

II.3 f o l l o w s

with

its

xaell-formBd

expression.

immediately

from

the

lemma

whose

proof

expression

of

by i n d u c t i o n i s i n t h e Annex: LEMMA

II.3

An expression order

h is

Prom arrive

of

also

X vifiich

is

a vtel I—formed

Conclusions

II .6a

etfuiform

with

expression

and

its

of

II.2b

order

2u^d

Convention

II .3c

we

at

CONCLUSION

II .7

p,q Ecfuiform

eS'Np%<)^a%&.

well-formed

By C o n c l u s i o n

the

set

of

S is

£ Kave of

h a v e a c a t e ^ f o r i a i index. N o t e t h a t we a r r i v e

at

CONCLUSION

II.6a,

expressions

set

Ttie set

of

all

in common with Conclusion and

Axiom II.7.

a

e
indices.

expressions

that

II.8

S n (.CS> -

II.1

h.

Dtel l-formed

the II.8

follows

set is,

from

of as the

0.

expressions

of

£ has

all

the

indices

of

these

it

were,

a

latter.

specification

Conclusions

no

elements expressions.

of

II.6a,

Theorem 11.5c,

and

THE AXIOMATIC SYSTEM TSCL OF SIMPLE CATEGORIAL LANGUAGES

The s e t , i.<S> m i g h t b e t e r m e d t h e set the

well-formed

Axiom

II.6>.

expressions

57

of

expressions

o f Jf Csee t h e c o m m e n t s

By C o n c l u s i o n

II.8,

n o t included i n t h e e x p r e s s i o n s

the

indexing

that

follow

expressions

o f X. They b e l o n g

of a metalanguage o f t h e o b j e c t language

indexing

to a

are

fragment

£.

In TSCL. we a d o p t o n e m o r e axiom: AXIOM

11.11

o There basic

is

a compotrnd

categorial

Let

us

of

that

the

set

indices

which

are

expressions

to

equiform

sentences

that

is

o r sentential

consideration

of

X which

has

I

of

basic

indices

consists

Axiom

equiform

o f Jf which h a v e

with

that

expressions 11.11

with

letter,

the

letter

a n index are

"s".

assigned

usually

called

o f X. In t h e c a s e now under

postulates

the

existence

of

s e n t e n c e which i s a compound e x p r e s s i o n o f X. Axiom 11.11 l e a d s u s s e l f - e v l d e n t l y

to

CONCLUSION II.9a

S\^S * 0. There

is

a compound

C o n c l u s i o n II.9a y i e l d s

well-formed

expression

of

X.

immediately

CONCLUSION II.9b

S i< 0. The set

a

o

Those well-formed them

expression

index.

say '

exclusively

veil-formed

of all

vtell-formed

expressions

of

X is

non-empty.

a

58

CHAPTER II

Further, sets

S,

where

11.4c,

S

II.9b.

Thus

arrive

should

would

S««0, t h e n ,

Ai > 0, would also

in

be

be

empty,

accordance

Definition

empty,

which

with

too.

would

II.4b,

each

H e n c e , by

II.4a

the

Definition

contradict

Definitions

of

Conclusion

and

II.3a

we

at

CONCLUSION

II.9C

° S - £• m s The s e t that

by

have

of

vords

of

an index,

is

£,

r) y !H 0.

DCL)

that

is

the

set

simple

iiiord

tokens

non-empty.

C o n c l u s i o n s II.9a,b a n d II.6a,b y i e l d t h e CONCLUSION

of

II.9d.

The

sets

DCiy,

successive f)Ci.J>\K,

E,

and

E

are c

non-empty. We shall now adopt definitions of two sets: the set B of basic

expressions

of £

saxd the s*f

F of

all

functors

all

of that

language. DEFINITION

II.6

'^ The

set

of

xoell-formed

that It II/4

set

have

'

o

expressions of

is

£ vthich

have

the

set

a basic

of

all

those

index.

II.7

of an

follows and

basic

expressions

DEFINITION

The

all

II.3

'^ is the

functors index

which

from

the

that

is

not

' set

a basic

definitions

the

index

o well-formed

of

of of

expressions

index. B and a

F and

functor

from is

Axioms

always

a

THE AXIOMATIC SYSTEM TSCL OF SIMPLE CATEGORIAL LANGUAGES

concatenation whereas

of

the

They

of

are

immediately

least

index

concatenation sets.

Cat

of

indices. also

from

two

-

cf.

a

basic

This

shows

non-empty.

Definition

Theorem

I.

expression that

The

II.6

59

never

B and

Axiom

is

F are

non-emptiness

and

indices,

of

11.11

a

disjoint B

follows

because

we

have CONCLUSION

11,10 CSN^'sJ n

There

is

basic

a compound

S ^

well-formed

0.

expression

of

X which

is

a

expression.

The

non-emptlness

II.9a,

Definitions

of

F

follows

II.4c,a,b,

immediately

Convention

from

Conclusion

II.3b,

Theorem

1.13'^b,

Definitions

II.6

and

II.7,

that

the

sum

and D e f i n i t i o n II.7, It

also

Conclusion

follows I1.6a,

immediately

Axiom

II.7,

from

and

Conclusion

1.3

o f B and F e q u a l s t h e whole s e t S. Hence we h a v e THEOREM

II.4

S - B u f / ^ B i « 0 / N F ^ 0 ^ B n F « i 0 . The

set

of

two

non-empty

and

that Since,

sets

k

of by

0 S \ S,

demonstrated

all and

all

well-formed disjoint

func

on t h e

setst

that

of

of

all

£

is

the

basic

sum

of

expressions

tors.

Conclusion where

expressions

fe

II.9a, >

0,

strength

II.1. T h i s i s why t h e s e t

S\ S is of

Is

non-empty,

non-empty, Definitions

too,

each which

II.4c,b,a

SN S, which i s equal t o t h e

and

set

of can

the be

Lemma

60

CHAPTER II

^ c

Ca,a.,...,a

,a^» >

1

71

O

c a n n o t be e m p t y . Hence t h e r e a

finite

Theorem

Is a p

number

e

of

1.13 b, t h e r e

S such t h a t

indices. Is

a

Hence,

is

by

p e S such

s o t h a t , by D e f i n i t i o n II.7, p

a

that

a concatenation

Definition

II.4c

P^^eS aind a

e

of and

I\I

,

find

a

X;

it

Is a f u n c t o r . This yields 0

CONCLUSION

11.11

^S n F J' 0. There It

in £ a simple

follows

basic

from

expression

follows that

is

Conclusion aunong

uhich

11.10

that

the

compound

Conclusion

II.lt

and

always

a functor

fi'om

we s h a l l

expression

find

is

we

a /unc

shall

always

expressions

Definitions among

the

tor.

of

II.4a

and

elements

of

11.3a the

v o c a b u l a r y o f Jf. The

following

theorems

have

close

connections

with

Theorem

II.3: THEOREM II.5a

peB^Cf^p^qcB. An expression

and

expression is

of also

£

a basic

wttich

is

equi/orm

expression.

with

its

basic

THE AXIOMATIC SYSTEM TSCL OF SIMPLE CATEGORIAL LANGUAGES THEOREM

Il.Sb

An expression also

61

a f tine

of

£

vttich

is

etfuiform

with

its

functor

is

Theorem

II.3,

tor.

Theorem

II.5a

follows

Conclusion

II.7

and

Deflnlt-loti

n.7.

Theorem

from

Axiom

Definition

II.6,

Theorem

II.5b

II.2;

II.3,

Conclusion

II.7,

follows

from

Theorem

1.4 a,

and Axiom 1.7.

S e c . II.6.

Synlact.ic c a t e g o r i e s

The

concept.

E.Husserl's 11900/013.

syntactic

category

Bedoufungskategorien Husserl

the point of sentence,

of

and

used

interchangeable

CSemantic

distinguished

view o f

his

semantic

s y n t a x , namely t h e

them

originated

with

role

reference

Creplaceable>

to

from

categories)

categories

from

t h e y c a n play i n classes

expressions

of

in

a

mutually sentential

contexts. A similar semantic

intuitive

category

syntactic In

that

of

the

was associated

by S.Le:^niewskl,

categories

C193S1.

sense

course

"syntactic

119291, of

the

A.Tarski

time

the

category"

the

author

of

11933),

and

more

was

with

the

of

theory

of

K.AJdukiewicz

adequate

adopted

concept

term,

by

namely

I.M.Bochetiskl

11949], y.Bar-Hlllel 11930,1964], and c o n t e m p o r a r y l o g i c i a n s . The l a t t e r agrees

with

do n o t its

always u s e

origin.

In

that

term

in t h e

the

present

manner

book,

too,

which when

62

CHAPTER II

formulating usual

t-he

definition

intuitive

sense,

of

which

that is

term,

due

we

to

deviate

certain

from

its

difficulties

in

symbolized

by

d e f i n i n g t h a t t e r m , o u t l i n e d i n Sec.II.1. A syntou:tic Ct

.

We

category

observe

an

Conventions

definition of t h a t DEFINITION

with,

index

^

II.3a-c

will

and

be

adopt

the

following

concept:

II.8

Ct^ - < p I a % ? >. A syntac

tic

category

of

£ whose

expressions The

intuitions

category

defined

with, index

is

connected in this

an

index etfuiform

with

way

f

the

are

is

the

with

f.

concept

explained

by

set

of

of

syntactic

the

those

two

fttac

given in Sec.II.7. Now Ct

will be used to denote the family

categories

of

expressions

of

Jf,

in

of

all

syntac

tic

accordance

with

the

following DEFINITION

II.9

Ct m i ct The is

the

indices By

family

of

family of

of

syntactic

all

expressions

Conclusion

demonstrate

Ct that

agreement.

categories

syntactic of

Il.Pd

that

which d e t e r m i n e s categorial

all

I ? e t<Ey >. of

categories

expressions whose

of

X

indices

are

We

shall

£. the

is

set its

partition The

E

is

logical is

the

definition

of

non-empty. partition.

relation the

The « of

relation

relation syntactic

»

hais

the

THE AXIOMATIC SYSTEM TSCL OF SIMPLE CATEGORIAL LANGUAGES

following

form:

DEFINITION

11.10

P ^ 9 «• The

63

expression

p >• q i s

y P>9 e Ct .

r e a d : p and

q belong

to

the

saune

c

syntactic In to

category.

accordance

the

same

with

Definition

syntactic

category

11.10

if

two

and only

expressions if

they

belong

belong

to

a

s y n t a c i c c a t e g o r y w i t h a c e r t a i n index. Definitions I.la-c

11.10

aind

II.8,

Conventions

II.3a-c,

and

Axioms

yield

CONCLUSION

11.12

p -<)«<> a % b. T«i» e x p r e s s i o n s belong only

if

they

have

to

t/te same

e
tic

category

if

and

indices.

C o n c l u s i o n 11.12 and Axioms I . l a - c CONCLUSION

syntac

yield

11.13

a.

P ' P'

b.

P '

C

t '^ Q = P. c

c

c.

p » < j ^ c j « r ^ p " r . c

c

The relation symmetric In

of

and

c

syntactic

transilive

accordance

categorial in

with

the

set

Conclusion

agreement

is

reflexive,

E. 11.13

the

relation

»

is

an

subsets.

It

c

equivalence

relation

in

E

and

In

each

determines t h e partition of E into c l a s s e s of It

can

easily

be

seen

that

Conclusion

of

its

abstraction. 11.12

and

Definition

64

CHAPTER II

II.8 yield CONCLUSION

11.14

M - - c'„c

A class

of

abstraction

vtith

respac

t

to

tKe

rotation

> is

a

c

syntactic

category

a representative Thus

vith of

that

Definition

previously

tho

ind»x

of

the

expression

that

is

class.

II.9

and

Conclusion

11.14

yield

the

announced

THEOREM II.6. which

has

is

The family

etfual

to

the

Ct

of

(fuotient

all

syntactic

family

E/

categories,

,

is

a

logical

c

classification the

(logical

relation

',

parti

ithich

is

tion) to

of

the

say

that

set Ct

E determined is

a

family

by of

c

syntactic

categories

Cl>

^

it/tich

satisfies

^ 0

- i t

Ct

the

following

consists

of

condi

tionst

non-empty

categories Cli>

_. ^. ^ . CCt, p' Ct ^ Ct^ n Ct Ct„ ,Ct ^:t F a t a -



Let the

U Ct eCt

the

basic

of

- and whose

Ct<.S'>, Ct<.B>, of

well-formed

expressions

follows:

and

the

Ct:

CtCF>

subfamilies

categories,

defined a s

m E

are

pairwise

disjoint

svm equals

E.

'

symbols

following

categories

Ct

which

0>

family

the

of

denote,

family of all

Jf,

respectively,

of the

all

syntactic

family

functor

of

all

categories,

THE AXIOMATIC SYSTEM TSCL OF SIMPLE CATEGORIAL LANGUAGES DEFINITION

11.11

a.

Ct<S>

m < ct

I ^ € tCS) >,

b.

CKfi) - < Ct

I i; € tCB> >,

c.

CtCF) - < Ct

I ? e

It

can

65

easily

be

seen

(,CF> >.

that.

Definitions

11.11,

11.6

and

11.7,

T h e o r e m 11.4, Axiom I.la and D e f i n i t i o n 11.8 y i e l d CONCLUSION l l . t S

CtCS) - CtCBy U CfCF) ^ C*CB>i*0 ^ CtCFys^B /v C( r?ie

family

expressions of

syntactic

of

all

of is

all

the

syntactic

swm of

categories!

functor

The s e t s

two that

categories non-empty

of

all

of

and

basic

0.

\oell-formed,

disjoint

families

categories

and

that

categories. S, B, F a r e n o n - e m p t y s u b s e t s

II.9b

and

II.6a

and

Theorem

11.13

that

the

relation

•

11.4).

is

an

It

of

E Csee

Conclusions

from

Conclusions

follows

equivalence

in

each

of

those

c

sets.

By

easily

obtain

under

consideration,

ll.l-Il.ll, langxiage.

Definition

has

the

11.11

theorem

been

and

Conclusions

which

described

by

called

a

explains the

11.14 why

system

and

the <Jf )

syntactically

ll.lS

language and

we Jf

Axioms categorial

66

CHAPTER II THEOREM

Il.da.

The

relation

«

determines

a.

logical

c

classi

fication

pairwise is

Clogical

disjoint

to say that

parti

tioTt>

CtCS>

syntactic categories the following

whose

conditions

«*^

Ct^.d^'^^t^S^



U Ct • S. Ct e C t < S > ^

of S into

This

STim equals S.

are

*^'? " ^ ' a ^ ^ ' ?

non-empty,

satisfied!

^ ^ ^ - «^'

F u r t , h e r , S c a n b e p r e s e n t . e d a s t.he s u m c o n s l s t . i n g o f t,he s u m of

all basic

functor

cat.egorles

categories

which

which

yield

yield

B a n d t,he s u m o f a l l

F,

so

that

the

following

condition i s satisfied: Civ)

S-

U Ct^Ct

Ct U U Ct ^ Ct eCtCF) ^

^

U Ct - S /v U Ct - F. eCt<.B> ^ Ct eCtCF> ^

Sec.

II.7.

The fundamental theorems of the theory of syntactic categories

As results several

has in

been two

mentioned

theorems. Their

definitions,

formulation

of

in

fttsc

which but

Sec.II.l

the

discussion

formulation are

not

will

be

TSCL

preceded

by

for

the

essential

conclusions drawn from

of

them

enable

us

to b e t t e r comprehend the nature of both theorems. We

begin

consti tuent

with

of n-th

the

definition

of

the

order of an expression

set

C of all P p and the definition

THE AXIOMATIC SYSTEM TSCL OF SIMPLE CATEGORIAL LANGUAGES of -the set

C

of

all

The expresislons

consti

tuents

q e C

of

and <j e

67

the expression C

are

read,

p.

respectively:

(j

i s an element, of n-t-h order of p, «j Is an element of p. The definition of C i s inductive. 'P DEFINITION

a.

f

11.12

e

C

« » / > % t .

i4 const i/went equiform

with

o/

that

order

zero

of

a

given

expression

expression.

< e C* •» V "^ . „, X fP " pCt„,t^,...,t > ^ p n>l tj^,t.,...,t eD<(.) '^ '^ 0 1 n

b.

0<j
if

tuent

p is

eQuiform with

of

first

order

of

a compound expression

some \tord of

a given and t is

the expansion

h+i

c.

A consti d.

const i txient tvent

of

of

first

V t reC*

expression

p.

e C ).

order

of a consti

of

p,

tvent

when of

k-th

k>0,

is

a

order

of

p.

t e C ** '^ t e c'^. p n p

order

CONCLUSION

b.

if

a word xehich is

of

of

that

a given

expression

is

a constituent

expression.

The definitions adopted and Axiom 11.10 easily lead us t o

a.

p

P

k+i-th order

of

J

1

fc>0=».CfeC«» ^

A constituent finite

is

11.16

t e C P t ^ C P

^ Q ^ a p ^ t e C , «J x v t i i % ; t * v e C . P

of

a

68

CHAPTER II Pi>om

11.5b

Oefinlt,ion

and

II.2a

11.12,

we

arrive,

Convent.lon by

II.3a

applying

and

Conclusions

mat.heinat.ical

induction,

at. CONCLUSION II.17a

P

P

in accordance wit.h which auny const.it.uent.s of

an expression

of

£

are words of £ which have a cate^^orlal Index. If

p

belong

a

compound

t.o the

belong II.9>

is

t.o

it.s

are

applying

set. E

well-formed

equal

induction

det.ermined

expressions

and

by

Theorem II.3 we easily arrive

t,hen

p does

S, and t.he const.it.uent.s

unambiguously

well-formed

expression,

by

of

expansion

referring

to

p which

Csee

Definilions

not.

Axiom

II.4c,a,b.

Definition

By

11.12

and

at

CONCLUSION II .17b

p « S ^ c " £ C P P in

accordance

with

which

any

=S,

constituents

of

well-formed

expressions are well-formed expressions. Every some

constituent

syntactic

of

a

position.

compound Further,

expression to

every

occupies

in

constituent

it

of

a

<17>

given

compound

sequence ot makes

it

expression

we

oaxx assign

a

finite

natural numbers greater than or equal t o

possible

to

determine

its

syntactic

ordered

zero

position

which

In

that

of

th.6

expression. Let consti

the tuents

symbol of

the

C expression

Cn>0>

denote

the

p Cof order other

set

than 0) u^^lc/l

THE AXIOMATIC SYSTEM TSCL OF SIMPLE CATEGORIAL LANGUAGES

occupy

in

It

the

adopt, t h e f o l l o w i n g

syntactic

definition

69

position of this

y-Jt,J^,->J^>-

We

term:

D E F I N I T I O N 11.13

** p e p word of the expansion of p,

the

a.

f < s C

b.

Ac > 0

^ Ct e

Juj.^""*- word of

C ^ P

'^

E c

^

t

is

eaxiiform

C P

4* t

the expansion

of

II.13a

and

with

is

the

J.-th 1

»quifoT«t

with

of

set

a constituent

the

" >.

From

Definitions

and

b

II.12b

and

c

we

easily

arrive by induction at the following conclusions: CONCLUSION

11.18

n>0=*C

* ^

" S C " P

P The

constituent

determined

by

of

the

the constituent

p

n-term

of n-th

which

occupies

setfuence

order

of

of

in

it

natural

the

posi

tion

numbers Cn>Oy is

p.

C O N C L U S I O N 11.19 N.

rt

^^

^n

*

P Every the

consti tuent

syntactic

natural

We further

.


n~th

position

^

J„>

order

determined

Cn>0) of by

1

^

n ^

P

an

p occupies

n-term

in

seifuence

it of

numbers.

now proceed analyses,

replacement

of

a

to formulate of

the definition, important for

the four-argument

consti

tuent

of

an

relation

expression

of token.

Its

70

CHAPTER II

formulat.lon reltxtion 0)

of

The

requires

adopt.ion,

Cy"> of

ttxs Teplacsmeixt

a givon

expression

expression

obt.alned

t.he

from

of

lnduct.ion,

a constituent

Is

read:

expression

s

by

t.he

expression

t.he

n-th

order

the

expression

rCp/tfys

is

read

r

replacemen-t

c o n s t i t u e n t Q o f n - t h o r d e r by t h e e x p r e s s i o n The

of

of

tohen.

rCp/q> s

the

by

Is

of

its

s,

but

p.

analogically

as

rK.p/'tfi

w i t h t h e o m i s s i o n o f t h e p h r a s e "of n—th o r d e r " . Here

are

the

definitions

of

the

relations

under

consideration: DEFINITION

a.

rCp/<j> s 4* s,r

The

expression

replacement expression such

11.14

that

of

its

p if

and

tf is

is

obtained,

from,

constituent only

eQuiform

if

the


s and

with

r

s ** s,r

b.

7»>1

r

€S^
of

r are

s and

expression xero

s

order

by

laell-formed

p is

by

eQuifortn

the the

expressions with

r.

e S y->

s ^ , s . , . . . , s e S 0 1 n ^

'^

0

1

'^

n

0

i

n

ro,r^.....r^eS

0
expression

replacement expression expressions

of p

if with

r its and

is

obtained

from

constituent only

expansior\s

if

tf s

and

consisting

the of

r

expression first

are of

order

compound the

same

s

by by

the the

well-formed number

of

THE AXIOMATIC SYSTEM TSCL OF SIMPLE CATEGORIAL LANGUAGES

veil-formed

expressions,

while

an expression

etfuiform

the

of

expansion

r

al I expressions same

in

replacement and

than

expression

only

r

of if

constitvent

j

is

obtaiT\ed

t

of

k-th the

t by

of

order

by

d. The

finite

is

J which

in

with

p;

equiform

and

r

which

have

the

V

Cr

from q of

from

order

the

expression

k+i-th

s by

s /v vCp/jjJ

by an

replacement

order

the

s

Ck>0?

it

by by

replacement

expression

in

t».

of

v

its

the p

if

of

its

which

is

consti

tuent

q

p.

expression

r

of

above

is

its

from order

The

nvMber

there

rC/>/q)s • • V rCp/q> *.

replacement obtained

s

s

equifornu

obtained

r

the

of

expression

s «•

constitvent

from

first

is

expansion

of

are

its

obtained

the

expansions

k > 0 ^ CrCpyqy

The

lt.s

the

the

q arid having

determines

numbers other

c.

with

in

71

s

by

consti the

Cgreater readings

Just.ificat.lon

obtained

in

tvent

from

the

q

p

by

replacement than

or

of

t^he

the

equal labels

conclusions

of

expression if

and

its

only

by if

constituent

to 0^ by

the r

is

q of

a

p.

rCp/(}> s which

s

and

follow

r<.p/'q'>s directly

has from

D e f i n i t i o n s 11.14 and 11.12 amd C o n c l u s i o n II.6a: CONCLUSION

11.20

a.

rCp^q)

s -^ q e C

b.

rCpyq^s

^ q G C

In a c c o r d a n c e replaced

in

a

with given

^ p e C , ^ p G C .

these

conclusions

expression

is

the its

expression

which

constituent

is Cits

72

CHAPTER II

c:onst.it.uent. a

of

definit,e

const.it.uent.

expression

Cits

that

same

order.

obtained

replacing

Moreover,

expressions

and

const.lt.uent.

is

replaced and the

ordei>>,

of by

the

are

replacing

a

definite

are

order)

of

Further,

constituents

constituents

same

expression

replacement.

constituents

they

occupy

t,he

which

syntactic

of

is the the the

in

compound

position

because

Definitions II.14b and c, 11.13a and b yield by induction CONCLUSION

11.21

r

0 " » , .

.

. ^ <. a e C

^^

'h'Jz V ,).

/^ p € C '^ r When

r
interc/ian^ea6le justification

p

and

in

in

s

the

qi

are

often

r.

Such

and

following

called

replacBoble

terminology

conclusion

that

or

has

is

its

obtained

easily from Definition 11.14 and Axiom I.lb: CONCLUSION

11.22

r

r. //

r

is

obtained

constitvient

Q

replacement Those

of

by its

p.

from

s

then

e

constituent

constituents

which

by is

the

replacement

obtained.

from

r

of

its

by

the

p by tf. are

replaceable

in

expressions

are

likewise well-formed expressions because we have CONCLUSION

11.23

rCp/<j)s •¥ s,r,q,p which follows

from

the

fact

that

if

G S, rCp/<j)s,

then

s

and

r

are

THE AXIOMATIC SYSTEM TSCL OF SIMPLE CATEGORIAL LANGUAGES

well-formed Ccf.

by

Definition

Conclusion

II.20>

11.14, .and are

73

-their

constituents

well-formed,

too,

p

in

and

view

q of

Conclvision II.17b. Once

we

agreement of

have

with

described

our

expressions

certain

intuitions,

we

formulate

properties,

connected

and

prove

with the

which the

are

in

replacement

first

of

the

two

fttsc. THEOREM 11.7a

CI f t t s c >

rCp/g>s

^ r « s ^ p « q c

//

an

category

expression as

replacement
wttich

an expression of

to

r,

its

the

P r o o f .

belongs is

constituent

same It

s,

c

to

obtained

the

from

scone the

latter

Q by an expression

syntac

tic

suffices

to

syntactic

p,

then

by

the

p

and

category. prove,

by

applying

the

principle

o f i n d u c t i o n , t h a t t h e f o l l o w i n g lemma holds: LEMMA

11.4

c

I

fttsc

follows

immediately

from

c

that

lemma

by

Definition

II.14d. Now when n"0 II.14a,

Theorem

Lemma 11.4 i s II.3,

obtained

Conclusions

C o n v e n t i o n s II.3a,c. Assume t h a t Cla)

rCpy s

and
r

-

St.

directly

II.7,

11.12,

from

Definition

II.13b,c,

and

74

CHAPTER II

We s h a l l

prove

t.hat>

p

m q

and

hence

t.hat.

Lemma

II.4

holds

Definition

II.14b

c

for

n"l. We

from

conclude

the

assumption

Cla>

and

that C2)

r,s e S

and

that

there

are

s^,s

expressions

,...,s

e

S

Cn >1>

and

e S such that

r«
and r - pCrQ,r^,...,r^ >

C4)

and t h a t f o r a c e r t a i n n a t u r a l number j '

such

that

•'I

<5> we

0 < jf

< n

have

<6>

if ^

s

A

and C7>

p % r

and

also

Since s formulas II.9

-

and r a r e C2>-C4!>,

they

than

zero

well-formed

Conclusion

belong

II.4a> and a r e a t

an

A

to

the

unambiguously,

Hence and

II.6a, set

Definition

SN S

<see

II.4c>.

the in

By

Axiom

expansions

view

of

of

11.3b,

and

Definitions

t h e same time e x p r e s s i o n s

CDefinition

injection.

and compound e x p r e s s i o n s

of

II.9 s

and

Definition

II.4b

r

and

the

atnd

greater

function are

by

Axiom

II.3a

an o r d e r

the

-

p

is

determined formulas

THE AXIOMATIC SYSTEM TSCL OF SIMPLE CATEGORIAL LANGUAGES

C3>

and

C4>

s

respectively.

and

r

Under

sat.isfv

t.he

Convention

75

condlt,ions

II.3c

we

tn s

atnd

record

tn , r

these

conditions, respectively. In the form COS

c

cd,d^,d^,...,d^

C


,d^y

and <10) Since

the

expansions holds,

constituents

are

in

of

well-formed

view

of

s

and

,Cjj>.

r

which

expressions

Convention

II.3c

belong

and

we

the

can

to

their

formula

state,

C8>

on

the

s t r e n g t h of Conclusion II.7, t h a t

*"'

0
follows

referred under

from

to

so

the

far,

Convention

assumption

from

II.3, t h e

Conclusion indices

d

and r, r e s p e c t i v e l y , are equlform s o C12) accordance

ISJ Sn .

In

the

generalized

CIO),

C12>

and

with first

11.12

and c

and of

has

not

been

Axiom

I.lb

that,

the

expressions

s

that

C6>

-

i<J
C7) This

generalized

p

that

II.7, •

time

we

<j. It it

Cll),

d .

d

be

find,

% c .

so

in

refer

order % c

by

H.13b,c

to

,

two

cases:

ourselves

likewise

suffices

that

avail

and

Is

in

catft

and

11.12

concatenation

C12) and

there

case,

Cll>,

11.23,

and

C5>

concatenation

Conclusions

CIO),

which

d m c.

In

on


to from

of

the

Theorem formulas

that

-

and the

in

find, which

view

the

second

to

J^"0

or 1.17 C9), of

formulas case,

for

Theorem

1.18

in

of

C9>,

follows

that

it

view

on

76 p

CHAPTER II -

<,.

We h a v e t.hus demonst.rat.ed t-hat. Lemma II .4 h o l d s f o r Assume

now

that,

a l s o t h a t rCp/Q>

this

lemma

s and r ^

holds

Tor

k>0

and

n«l.

n—h. A s s u m e

s.

c

We s h a l l d e m o n s t r a t e hence t h a t

it

follows

that

from

under t h e s e

the

a s s u m p t i o n s p ^ Q, amd c

assumption

that

f o r n-M 0> t h a t i t a l s o h o l d s f o r n-ft+1. By Definition II.4c it follows from r

s

that

s

and

k rCv-yty

there

1 w Cp/«j> t .

the assumption stating V Cp/<}> (

are

expressions From

that

r

and Lemma II .4 h a s

the

m s c

Lemma II.4

the

v

and

inductive

we i n f e r

assumption t

such

that

assumption

that

been proved for

holds

v

n^l,

and

• t 1 c* 1

Since

we h a v e

that

P - <»• c

This c o n c l u d e s t h e i n d u c t i v e p r o o f o f Lemma II.4. We now p r o c e e d t o f o r m u l a t e and t o p r o v e t h e s e c o n d

fttsc.

THEOREM II.7b CII f t t s c > rCp/9)s If

an

expression

r

of

const

replacement syntac

tic

its

category

syntactic

as

is

^pmif'^rms.

obtained i txtent

if,

from

an

expression

ef by p tiihich

then

r

and.

s

s

belongs belong

to to

by the

the same

the

same

Definition

II.14d

category.

P r o o f .

The

second

fttsc

follows

from

r

S/%p"<}i»r"is,

and LEMMA

II.5

whose proof,

as

in t h e

casie

of

Lemma II.4, i s

by I n d u c t i o n

with.

THE AXIOMATIC SYSTEM TSCL OF SIMPLE CATEGORIAL LANGUAGES

77

r e s p e c t fco t h e o r d e r n o f t h e c o n s t i t u e n t <j o f For

n-0

II.14a,

Lemma

Theorem

II .5

follows

II.3,

s.

self-evidently

Conclusions

II.7,

from

11.12,

Definition

II.13b,c

and

C o n v e n t i o n s II.3a,c. N o t e a l s o t h a t f o r n"l t h e Cla)

assumption

rCp/'q)

yields

the

and t h e

formulas

C2>-<11)

s

used

in

the

proof

of

Lemma

11.4,

assumption



P •

9

c

and t h e f o r m u l a s C6> and C7> y i e l d t h e C13)

d.

Cln

view

of

formula

%c .

Conclusions

11.23,

II.7,

11.12

and

II.13b,c

and

in

for

n"!

suffices

a c c o r d a n c e w i t h C o n v e n t i o n II.3c). In to

order

to

find

that

demonstrate

that

r

Lemma

•

s.

11.5 h o l d s

In f a c t ,

since,

by

it

C5>, 0<j'
and

1 1

c

t h e f o r m u l a s Cll> and C13> hold, we h a v e A

d

* c

0<>i
the and

formulais

C9> and

conclude

that

*

k'

<10> hold

d

%

c,

we

may

so

refer

that,

by

C o n c l u s i o n 11.12 and C o n v e n t i o n II.3c we h a v e t h a t T ^ c It

remains

to

prove

that

if

Lemma

to

II .5 h o l d s

Theorem

Axiom

I.lb.

s.

for

a

natural

number n"fc 0>, i t a l s o h o l d s f o r a n a t u r a l number n"*+l. Assume

accordingly

that

t h e same applies

shall

demonstrate

that to

the

the

inductive

formulas

assumption

r<.py

s

holds

and p > <). We c

that

in

such

a

case

r

m s.

and

Since

rCp/Q>

k+1

s

c

we

can

infer,

by

Definition

II.14c,

that

there

are

expressions

78

CHAFTCRII

V, and 1

t^ s u c h 1

t.hat.

It

rCu^/t^) s 1 1

1

and

-U-Cp/q) t^. 1 1

1

Since

v^Cp/Q> t , 1 1

h o l d s we i m m e d i a t e l y h a v e t h a t , v^ • t^ b e c a u s e we h a v e a s s u m e d 1 c 1 t h a t P " <J and Lemma II.5 h a s b e e n p r o v e d f o r nmi. Since c

r ( u / t ) s,

r

m

1 1

s

follows

immediately

from

the

inductive

c

assumption. The t w o f t t s c THEOREM

t a k e n J o i n t l y make i t p o s s i b l e t o

prove

II.7

r

s ^ Cp — <j • • r » s ) . c

Two expressions if

and only

-formed which

if

of

on replacing

expression belongs

belonging confirms

of

to

Theorem

II.7

of

X heloixg

the

one

to

the

same

of

them

by

£ u>e ohtain same

cannot,

syntac

of

tic

correctness

to

the

be

other

as

used

same

the

tic

category in a

xnell-

expression

category

the

of

syntac

a well-formed

course,

two expressions

the

c

the

as

a

X

former. definition

syntactic

definition

of

category. of

of It

syntactic

c a t e g o r y a d o p t e d i n S e c t i o n 11.6.

S e c . II.8.

The a l g o r i t h m o f c h e c k i n g t h e

syntactic

correctness of expressions

The l a n g u a g e theory

TSCL,

should

be

expression of

X.

The

procedure

X,

which

ahould

an

be

is

to

algorithm defined

in

object

decidable,

algorithm

belongs

the

which

enabling

the

set

to

be

us

S of

investigation is

to

say

to

check

the

well-formed

described

AJduklewicz

of

below

C1935]

that

whether

is and

by

a

the

there given

expressions

a

generalized

intended

to

THE AXIOMATIC SYSTEM TSCL OF SIMPLE CATEGORIAL LANGUAGES

investigate or

the

syntactic

certain

ideas

pertains

syntactic

connectedness,

correctness

of

to

in

be

found

about

precision,

but

it

we

there

shall

AJdukiewicz

not

seems

i.e.

expressions.

t o compoiind e x p r e s s i o n s o f

comments

79

It

also

refers

[19603. T h a t

be

observe

no

point

the

in

to

algorithm

Jf. When f o r m u l a t i n g

always

to

well-formedness

further

rigour

making

of

those

comments more p r e c i s e . We

begin

algorithm, which

with

several

which,

are

it

usually

general

seems, made

are

on

remarks

somewhat

such

an

on

the

concept

different

occasion.

from

We

do

of

those

not,

of

operations

or

c o u r s e , claim them t o be original. An algorithm operations directives.

is

(IP)

fini

and

the

of

should

Cthe

determined

should

have

number

indicate

earlier one

the

them the

directive of

should a

given

should

define

successive

the

essentiad

directives

should

are

appropriate

several

performance

of

operations about

by

Ceach

effective

Cone o f

should

inform

psychophysical

machine,

an

teness

the

of

per/ormaibility of

teness

others

results

directive

as

defini

a

directives

possibility

operation),

operation,

by

Those such

the

finite>,

sequence

performed

properties secure

a

the

first

operation

known,

and

characteristic

be

once

the

last

feature

of

t h e final operation of a given algorithm). When

formulating

algorithm of a given about

language

the

lad>el o f

checking

the

Jf we

symbolism

Jf i s

the

of

directives

which

well-formedness

avail £.

ourselves First

of

recorded with a finite

of

from all

we

determine

the

the

expressions

of

several

assumptions

assume

that

number o f

every

s y m b o l s . Next

we

80

CHAPTER II

assume they

t.hat.

are

them i s that the

about,

any

two

equiform,

and

about

the

concatenation

about

any

vocabulary

and w h e t h e r for

any

parts to

it

it

and

the

additional and

second is

labels,

remaining

decide

it

that

of

£.

Further

sense

its

of

X

we

Le^niewski's

which

of of

parts

that

of

Jf we c a n

amy

word

has

an

indicate

that which

correspond

corresponds

we

the

index,

aussume

correspond

Finally

from

indicate

of .it

of

assume

a

we

can

it

functor.

is

mereology>

e x p a n s i o n , which p a r t and

whether

whether

of

whether

t w o . We aJso

an e x p r e s s i o n

expression

decide

to

to

the

assume

that

categorial

index

it.

in

with

by the

the

way

label,

treated

belonging t o

the

a

is

made not

not

functors which

structurally

following

to

checking:

calculus, we

force

compound

the

subject

because

binding

a word

above

sentential

x+y • z ,

the

main

are

the

label

are

however, as

of

about

labels

them

example,

notation

information

in

assumptions

of

arithmetical

+. T h e s e

indicate

can

word

expression

laJ>els,. c h o s e n label

three

the

can

we

a

compound

In a c c o r d a n c e

the

of

£

is

Cin t h e

sissigned t o

any

whether

arguments

every

of

it

functor,

successive

we

£,

is

words of

main

for

of

given

of

the

Its

label

labels

of

their

has

ambiguous

an

index name

The s e q u e n c e o f c a t e g o r i a l

I s c a l l e d t h e intermBctiats

conventions. indices

sxpansion

of

tfte

exprossion

the

functors we

cannot

arguments.

arithmetic.

We a d o p t t w o t e r m i n o l o g i c a l

lack

the

expressions;

and

^-i,

because

The it

expression

THE AXIOMATIC SYSTEM TSCL OF SIMPLE CATEGORIAL LANGUAGES

c6y

pcp^.p^

81

p^j)

which has i t s expauislon in t.he sequence of e x p r e s s i o n s

If and only i f f o r any i CO
i-th

term of t h e sequence

Note t h a t

in accordance

with Conventions

II.3b,c e v e r y

i-th

<0 term of t h e sequence CC> i s t h e c a t e g o r i a l index of

the

i - t h t e r m of t h e sequence <«>. It

is

said

condition of

about

the

sequence

C^> t h a t

it

satisfies

tKe

tnp i f and only' i f t h e index f^ ^0 i s t h e concatenation index a of t h e e x p r e s s i o n p and of t h e indices

the

?-,?_,...,? 1 ^ n, s o t h a t t h e following holds: It

is

assumed

indices t h a t

it

about

every

i s possible t o

finite

sequence

of

decide whether i t

categoriad

does, or

does

not, s a t i s f y t h e condition Itt . Note t h a t i f of

the

t h e sequence

expression

condition

*«,r-. i f

C^> i s

C£> t h e n

the

and

if

only

an intermediate

latter that

sequence

condition

expansion

satisfies

is

the

satisfied

by

CC? Csee Condition I.ll>. Before

we

algorithm

for

shall the

describe t r e e of

compound

formulate checking an

the the

auxiliary

derivation

expression

directives

well-formedness al{;orithm,

Such

a

termed t r e e I of a given expression.

of

namely

Calso termed t h e token.

which

determine

the

expressions,

we

that

of

dendrite)

of

diagram

will

drawing a

given

henceforth

be

82

CHAPTER II

By r e w r i t i n g we

the

want, t-o c h e c k

writing to

we o b t a i n

immediately

write,

the

the

expansion

the

second

compound

below,

expression

the

on one

words equiform of

the

line

of



and t h e

with t h e

expression tree

first

I.

If

whose

well-rormedness

line

same

of

level,

successive

under

tree from

such

By left

components

consideration

every

I.

word

of

we

obtain

a

simple

is

e x p r e s s i o n o r i f s o m e o f t h e m i s a compound w o r d which i s n o t compound

expressions

decomposed tree the if

I of

into the

its

Cand main

not

every

compound w o r d s

of

words

which

expansion, way

we

until

are

in t h e

obtain we

perform

of

the

tree

I

that

such

an

follows

from



one

a

simple

expansion

of

the

a

line

are

whose

expression

case

I.

We

all

on

of

we

words

line

proceed

of

its

In

this

this

way

two. in

elements

write

are

simple

t h e m i s a compound w o r d s which i s n o t By w r i t i n g

the

is

fact is

tree

If,

given expression

successive

of

the

and

the

three

then

expression,

with

element

there

is

in t h e

be

completed.

equiform

compound

the

been

such

line

we

has

word

way a s

cannot

arguments),

each

compound e x p r e s s i o n s . line

such

its

that

under

e x p r e s s i o n s or some of

a

and

word

then

same

obtain

a

in q u e s t i o n

the

compound e x p r e s s i o n s ,

is

functor

expression

contrairy,

hence

last

down t h e

operation

expression obtained that

in

an element

in

in

a

last

of

element

the

finite line

which i s

of

The

a

such

construction

question.

each

a

of

conclusion

number

of

except

the

steps last

r e c o r d e d by m e a n s

a g r e a t e r number o f s i m p l e w o r d s t h a n e v e r y e l e m e n t o f t h e

of

next

line. If

in

tree

I

of

a

given

expression

some

of

its

final

nodes

THE AXIOMATIC SYSTEM TSCL OF SIMPLE CATEGORIAL LANGUAGES

is

not

a simple

expression), of

that

then

Ajduklewicz final

simple

the

is

nodes

is

not

tl93S]

it of

thoroughly

of

operations


a

given

the

set

then

of

checking

the

arrangedy.

of

If

expression

Cin

under

all are

AJdukiewicz's

consideration

proceed

an

under

terminology

S

the

not

well-formedness

compound

we

is

expression

\a»ll

expression

arranged^,

the

the

thoroughly

from the

u>el I

successive

I

word t h a t

checking

completed:

not

tree

if

of

well-formed

is

expressions

terminology,

Cis a compound

algorithm

expression

consideration

the

expression

83

to

is

perform

well-formedness

the

of

that

expression. The

elements

expression are

in

higher

are

tree

I

ordered

so

that

lines

that

different

of

l i n e , and o u t

o n e i s t h e earlier Once t r e e might

be

compound

of

I has

any t w o

been

the

expression.

out is

thoroughly of

any

the

Tree

above,

of

of II

we

elements which i s

in t h e

saune

in

it,

all

its

obtained in

accordance

elements

by

in

the

line

tree

indices from

which

that

left.

construct

ca.tegoria.1 is

well-arranged

two

e a r I i er

elements

completed,

tree

replacement

established

one

a

which i s s i t u a t e d m o r e t o t h e

termed

successive

of

II,

which

a

given

of tree

I

with

by

the

their

the order

categorial

indices. Before the

formulating

algorithm

of

well - a r r a n g e d intermediate

the

checking

the

compound

expansions

formed of p a r t s of t r e e

directives

the

well-formedness

expression of

of

compound

II. Of any

of

£

princlpad of we

expressions

a

part

of

thoroughly order

of

tree

two such expansions t h a t

the I, one

84

CHAPTER II

is

t,he »ar I i er

which is

an

expansion

of

an earlier

element,

of

expansions

of

t-ree I. Those parts of all

compound

given

tree

II which form intermediate

expressions

expression

are

s ,s

checked

» so

that

with the order fixed for

intermediate

do,

the

or

do

encounter

not, the

satisfy first

the expressions

the

m

condition

correctness that

the

of

of

of

tree

for

find

-

expansions m

tree in -

I

whether

CO<J.

expansion

out

of

the

of

checking

been performed,

consideration

is

of

the

a

accordance they

Once

we

the

C0<^<*> which does not

has

under

of

such

satisfy

syntactic

and we conclude

not

well-formed.

If

any compound expression s . CO<J
tree

II

s

algorithm

expressions

I its

satisfies

checking

the

and

conclude

we

the

expression

we find that element

,

we

conditions

intermediate

expansions of

Cfc>l)

Intermediate the

condition

well-formedness that

the

of

expansion m

,

then

expressions

expression

formed the

has

under

of

a

part

adgorlthm

been

of

performed

consideration

is

well-formed. By

making

use

of

the

algorithm

described

above

we

shall

check now the syntactic correctness of two expressions from the arithmetic sentential use

of

real

calculus.

brackets

numbers In

the

as technical

and

one

recording

of

expression those

from

the

expressions

we

symbols which have no indices and are

used in the ordinary way. E x a m p l e

II.1.

range over the s e t

Assume

that

the

variables

x

and

y

of read numbers, and the symbols +, <, and 0

THE AXIOMATIC SYSTEM TSCL OF SIMMX CATEGORIAL LANGUAGES

denot,e, "less

respecMvely,

than",

and

t-he

the

operat.lon

number

syntactic correctness of the Cni>

zero.

85

of

addiMon,

We

proceed

t,he t-o

relat.ion

check

t-he

expression

X + (y

< 0>.

We c o n s t r u c t t r e e I o f t h e e x p r e s s i o n CTT1> Csee Flg.II.l). .

Fig.Il.l. As

can

be

well-arranged.

T r e e I o f t h e e x p r e s s i o n

seen, We

x+Cy<0)

then

expression

accordingly

Crrl>

construct

is

tree

thoroughly II

of

the

c a t e g o r l a l I n d i c e s o f t h e e x p r e s s i o n Cnl> Csee Pig.II.2>.

•> n

s/'nn

Fig.II.2.

of

T r e e II o f t h e e x p r e s s i o n (rTl>

Trees

I and II o f

three

l i n e s e a c h . The l i n e s o f

expansions: t h a t expression equiform

being

of

the

the

and

letters

under

tree

expressions

checked

w i t h y<0. The

expression

that

II f o r m

which of

consideration

the

is

two

Intermediate

equiform

expression

which atre e q u i f o r m

consist

with

with

the

which n and

is s

86

-

CHAPTER II

and

of

which

are

part-s

expressions

which

expressions,

and

category

sentential

the

of

Indices

of

recorded symbols index the

which

indices carrying

out

checking case

which

the

the

of

determined

the

root

of

tree

I

II main

is

Cthe

not

root

I s n o t

a well-formed

expression

II.2. from

not

/

forms,

of

satisfy

that

I)

check

sentential

reasons

indices)

the

and

that

are using The

index

the

indices

we

the

functor.

the

When

algorithm

find

of

that

in

of the

Intermediate

expansion

the

condition.

required

expression of

and

fuunctor.

arithmetical

the

successively,

of

the

to

symbols.

indicates

first

of

name

notation

technical

expressions

tree

We

the

Cn2> Its

of

as

concatenation

of

arguments

E x a m p l e

a

certain

indicate,

the

of

typographical

operations

Crtl)

indices

belong

linear"

functor

functor

successive

which

of

arguments

does

category

For

slant

of

respectively,

the

/

symbol

essentiad

tree

of

the

the

given

expression

index

expression

a

with

successive

The

t-o

Cconcatenations

well-formedness

by

are,

expressions

that

the

the

-

expressions.

before

after

of

of

equiform

occurs

occur

II

"quasi-fractional

are

expression

which

indices

the

tree

belong

functors

in

which

of

which

the

the Hence

is

the

index

of

that

indices

of

the

the

expression

expression.

the

syntactic

correctness

of

calculus:

p •• C-i <-i) V p>. tree

expression shown in

I

Fig.II.4.

is

shown IS

in

Fig.II.3.

thoroughly

It

can

well-arranged.

be

seen Its

that

tree

the II

is

87

THE AXIOMATIC SYSTEM TSCL OF SIMPLE CATEGORIAL LANGUAGES p-»C-iC-i>vp>

Pie.II.3.

T r e e I o f t h e e x p r e s s i o n

sySS

•. S

OS

s/s

Flg.II.4. Trees now

under

that

accordance first

index

of

index

of

expression that

and

II

T r e e II o f t h e e x p r e s s i o n Cn2> of

the

consideration

expression

carrying

the

I

out with

the

of

expression

The

three

of

satisfies

functor

second

of

lines

intermediate

those the

of

the

desired is

that

Indices o f

expression each.

the

intermediate

a

Tree

II

adgorithm -

we

condition

so

forms

successive expso^sion

-

In that

that

the

of

the

Cl.e.,

the

arguments also

of

When

find

concatenation

functor

Cn2)

expansions.

expansions

implication which

calculus

four

operations

order

Crr2>> and t h e

functor.

consist

basic

the

expansion the

sentential

determines the

s/s

of

satisfies

CHAPTER [I

88

the

desired

the

contrary,

condition

condition the

tn

-

In t h i s

third

.

Hence

expansion we

under c o n s i d e r a t i o n i s n o t E

x

a

m

p

l

e

case

can

the

does

c o n d i t i o n TO , -

not

conclude

satisfy that

the

the

On

desired

expression

well-formed. II.3.

We

shall

nov

examine

the

w e l l - f o r m e d n e s s o f a n o t h e r a r i t h m e t i c a l e x p r e s s i o n , namely

X • Cy + < - « » -

In r e c o r d i n g pertaining the

that

to

the

brackets.

expression

determines eight

v> + <x • C-e)>.

expression

we

observe

constants

•,

+,

-,

tree

I

We

aind

Cx

construct

note

that

each

the

the

tree

usual

conventions

variables

and has

tree

x,y,s,

II

five

of

this

lines

and

expansions.

X- < y - K - « ) ) - C x y ) + < x - C-«>)

Cxy)+Cx

Flg.II.5.

T r e e I o f t h e e x p r e s s i o n C7T3)

and

<-«))

TOE AXIOMATIC SYSTEM TSCL OF SIMPLE CATEGORIAL LANGUAGES

89

n

« rkynn

n/'n

• Fig.II.6.

In

carrying

out

T r e e II o f t h e e x p r e s s i o n the

checking t h e s y n t a c t i c the

eight

I

of

which

basic

operations

correctness

intermediate

n^n


algorithm

<7T3> we f i n d t h a t

of

main

o f e a c h compound e x p r e s s i o n which i s a n e l e m e n t o f

tree

Cf73> that

Is

a

condition

concatenation

functor

forms

and

so

of

that

the

the

the

index

of

index

of

the

expression

indices

of

the

successive

a r g u m e n t s of t h a t f u n c t o r . This s h o w s t h a t c o n s i d e r a t i o n Is well-formed.

by t r e e

each

the

desired

determined

of

(nS)

the

expansions

of

of

." ^

II o f

satisfies functor

on

the expression

under

CHAPTER

III

THE THEORY TSCto-L OP CATEOORIAL o-LANGUAOES

S^c. III.l.

Inlroduct-ory remarks

The

t-heory

TSCL

described

certain

sense.

This

is

mathematical new

theories

expressions,

namely e x p r e s s i o n s variables.

It

retaining observed the

formulated in

therefore,

correct

natural

the

that

modify

so

The

point

view

in

include

languages, those

TSCL -

while

Sec.II.l

and

in

that

a

bind

it

should

syntactic

of

in

which

categorial

to

-

poor

formulated

languages

described

above.

from

are

and o p e r a t o r s

constructed

mentioned

in

simple

objectives

when TSCL w a s

expressions,

theorems

occurring

main

lau-iguages

most

with variables

seems,

the

which

because

are

not

languages

cover

amalysis theory

of

of

the

of

use

of

categorial

of

splitting

s y n t a c t i c c a t e g o r y , should be r e s p e c t e d , t o o . In

such

indices

an

in

analysis

we

accordance

usually

with

make

the

principle

e x p r e s s i o n s i n t o "main f u n c t o r s and t h e i r The

method

structure and

of

operators

mentioned trouble

of

using

lioguistlc that

already

connected

by with

of

other

"typical

the

the

in

the

expressions

bind

connectedness is

indices

them

the

ambiguity" 90

of

can

the

syntactic

include

variables

many

problems,

encountered

in

119331. One

algorithm

expressions

study

which

has

AJdukiewicz

arguments".

now of

of

of

checking under

the

them the

is

syntactic

consideration;

operators

the

which

the bind

THE THEORY OF TSC(0-L OF CATEGORIAL (D-LANGUAGES

variables for

belongins

instance,

the

if

we

arithmetic

syntactic variables sets,

to

then

that the

expressions

the

natural

categories and

different

admit

of

of

of

e.g.,

syntactic

distinction,

numbers,

name

<x | x>5>

and

the

between

which

operators

categories. in

variables:

variables

equiform

as,

91

language

of

<X | XSN>

of

two

different

of

individual

that

stsind

Thus,

for

the

names

of

abstraction

in

would

different

have

such

indices. I t i s l i k e w i s e when we c o n s i d e r e x p r e s s i o n s s u c h '^ Cp V - i p > p

in

which

eqxjlforni the

the with

category

operators, the of

a n d '^ X = X

namely

symbol

^,

sentences

the

bind,

and

a

as

X,

universad

respectively, variable

of

quantifiers,

a

the

variaJile

of

category

of

names. The

elimination

ambiguity" o f of into

a

of

operators

syntactically diversiform

conventions


^

in

in t h e

operator

accordance

R.Suszko

and which

quantifier

which

conception

of

typical

variables

may c o n s i s t

related

C19583,

the

"typical

appropriate

matrking

and with

to

thus

splitting

definite

tl964]>.

For

it

caligraphic

instance

the

1

quantifier

the

difficulties

ambiguous

signs

0 symbols

the

^ binds binds

could a

belonging

to

used

sentential a

L.Borkowskl ambiguity

be

of

name [1958]

denote

variable

variable, the

Cproper>

different

to

au-id

universal

the

universal

respectively.

obstacle

which

categories

r e m o v e d by t h e a d o p t i o n o f t h e c o n v e n t i o n t h a t

In

connected

quantifiers

syntactic

the

the

could

index o f

the with bind be the

92

CHAPTER HI

synt-act-ic of

t.he

cat.egory

of

quant.ifler

quant.ifiers

variable

C-the

is

independents

variable

which

of

t,he

index

accompanies

t,he

q u a n t i f i e r >. The

problems

connected

syntactic

categories,

could

eliminated

be

defined

by

in

C1935!l

occur.

the

result

full

system

by

obtained

by

the

the

not

quauitiflers

of

infinity

This

from

the

system

there,

language

occur.

Cwithout

which

do

not

simple

system do

the

simple

suffices

to

however,

a

terms

this

it

of

which

bound

practice, which,

while

syntactic problem include

and

system

of

the

infinity).

As

it

form

the

possibility

have

become

it

therefore

is

retaining

categories, of

the

operators

the

would

syntactic and

every

eliminating

the

make

rooted

ideas it

type

theory

the

the

full

axiom

of

mathematics.

Is

quantifiers

in

and

bound

mathematical

seeking of

possible

connectedness variables

eliminating

mathematics?

worth-while basic

of

logical

operators

known,

of

the into

a

and

with

foundations

deeply

of

simple is

of

brought

thesis

enriched

of

be

do

mentioning

thesis

possibility

be

the

operators

variables

full

theory

could

defining

which

can

to

type

the

variables

worth-while

theory

o t h e r o p e r a t o r s from t h e language of Operators

is

the

quantifiers.

of

in

[19351:

type

bind

Include

possibility

of

with

operators

with a synonymous

of

the

and h e n c e

which

points

the

axiom

the

J.Stupecki

correspondence

indices,

if

with

one-to-one in

operators

means

In c o n n e c t i o n

of

the

mentioned

quantifier

not

the

general

expressions

AJduklewicz universal

of

with

of by

the to

a

solution

theory approach

expressions them

by

of the

which applying

THE THEORY OF TSCoyL OF CATEGORIAL m-LANGUAGES

93

AJduklewicz's inet.hod o f making u s e o f It.

is

subject

worth

endeavours

method

so

that

operators more

or

C19581,

it

that less

syntactic

could

cover

them. the

categories.

For

C1964]

Polish

made the

Those

modify of

in

objects

and

the

and

the

deviate

theory

works as

the

AJduklewicz's

however,

of

the

used

of

variables

principles

the

[19781

literature

endeavours,

are

and

to role

instance,

indices

A.Nowaczyk

the

general

expressions

also

in

been

bind

[1960],

that

have

from

co-ordinating <see

no-ting

indices.

of

of

R.Suszko

Instruments

which

B.Stanosz

they

and

of

denote

A.Nowaczyk

[1976]>. In

the

modifying ideas

present TSCL

chapter

in

such

and complying

syntactically be

with

categoriad

able

as

to

of

expression tokens

by

them

To

simplify

may

set -

while

goals

of

languages, the

ourselves

Such

analysis

one

in

we those

variable

the

each

and

languages

variables

termed

may

that

that

include

the

bound

oy-languages.

convention

languages

of

Sec.II.l

categorial

be

of

theory

in

and t h e

will

adopt

a

discussed

syntactically

task

AJduklewicz's

constructing

as

languages

the

retaining

i n which o p e r a t o r s

occurring

bind o n l y

way

the

discuss

occur. our

expressions which

to

a

we

the

operators

ratnge

of

such

(23)

operators however, could include of

is that

not

be

confined the

and

one

expression.

considerations

generalized

expressions

vau>iables

to

with the

so

as

carried to

quantifiers

languages

cover that

which

o p e r a t o r s t h a t h a v e a f i n i t e number o f

It out

does in

the bind

Include

not

this

seem, chapter

languages a

finite

which number

expressions

arguments.

with

94

CHAPTER III

It.

Is

t-he

theory

TSCe»-L

which

is

the

conception

pertaining

to

the

oi-languaiges.

any

u-language

Jf.

That

language

fixed

ordered

It is

is

theoretical

concerned

characterized

with by

an

system

Cu-J? )


whose

formal

V, Vr,

elements

axioms

and

O, W; is, c;

satisfy

I

,1;

L, p,

definite

definitions

of

C / v > ; E, S,

conditions

TSC«»-L.

In

CtCSy>,

formulated

, c

next

by

the

to

the

concepts occurring in the ordered system < Lb; V, V; %, C; 7 , i; (,, p; E >,

CC ) c

o

described

by

the

theory

TETk

of

expression

tokens

Csee

Sec.II.4> and t o t h e c o n c e p t s o f t h e s e t S o f a l l w e l l - f o r m e d e x p r e s s i o n s and t h e f a m i l y C t CS> o f a l l s y n t a c t i c c a t e g o r i e s o f S,

which

occur

described are

In

new

in

CJf ) c

the

expression Inclusion

set

the

are,

for

differently

Vr

of

all

Cfv>

token.

in

but

somewhat

O, and t h e relation

given

Their

TSC«»-L

concepts:

operators a

also

They

of

conceptual

than

in

be

the

a fi-ee

of

there set

variable

discussed

apparatus

reasons,

TSCL,

variablss, being

will

obvious

in

of in

Sec.III.3.

TSC»»-L

shows

t h a t i t s s t o r e o f c o n c e p t s i s r i c h e r t h a n t h a t o f TSCL. Before of

TSC«*-L

that of

we p r e s e n t we

shall

theory to

the

axioms, basic definitions

point

TLTk and i t s

to

certain

and

relationships

expansion, t h a t

e x p r e s s i o n t o k e n s , which i s a f r a g m e n t

is the

and a t

theorems which

theory

the

same

link TETk time

t h e c o r e o f TSCL CSec.III.2). The adopted

definition in

of

TSCwL

the

concept

CSec.III.4)

Is

of a

well-formed modification

expression of

the

THE THEORY OF TSCo)-L OF CATEGORIAL (O-LANGUAGES

definition It

of

t-hat,

possible

describe and

to

the

which

II.7a,b

CI

describe

the

syntactically

II

in

to

fttsc>. of

correctness)

of

Theorems

checking those

it the

expressions

The Toundations or TSC««-L

tokens

TSC«»^L i s batsed on t h e

while t h e

latter

is

in

II.2,

turn

it

theory

based

II.6a

and

possible

to

connectedness

which

may

TETk of

on t h e

which

CSec.III.S)

II.4,

syntactic

Sec.

makes

theorems

makes

bind variables CSec.III.6).

The theory

It.

u-lamsuages

operators that

III.2.

CSec.II.5>.

fundamental

Further

algorithm

TSCL

categorial

analogous

and

Csyntactic

adopted

formulate

are

the

concept,

95

include

expression

theory

TLTk of

label tokens. Note having

<see its

concepts

Sec.II.4>

apparatus and

by

Definitions II.12a-d,

that

enriched

axioms

Il.la,

II.13a,b>

TETk

and

by

new

obtained

the

TLTk

and

CAxioms

II.2a,b,

establish

from

primitive

definitions

II.l,

which

is

by

secondary

II.l-II.lO

II.3a-c,

and

11.8-11.10,

properties

of

those

concepts. In

connection

with

this

and

in

accordance

with

Conventions

I.la.b and II.3b t h e following remark is t o be made: REMARK III.l. TETk,

namely

The Lb,

W, %, c,

symbols

of

, / , L, p, o c o n s t a n t symbols of t h e vocabulary of TSC»»-L. The variable

V,

constant

symbols

of

the

/

vocabulary

of

the E,

vocabulairy Ct,

•, c

of

C p

are

TSCwL include

the

96

CHAPTER III

let.t.ers Cw

>

p,tj,r,s,t,vi,v,...,

w h i c h rani^e o v e r



The

over

remarks

t w o metatheorems

a n d Remark pertaining

III.l

enable

us

to

t o TSC«»-L.

Every tKesis

Caccopted

sentenced of TETk

is

Every thosis

Caccsptsd. sentenc&> of TLTk

is

of TSC«»-1^

T H E O R E M III.2 .

Note

subscript,

t h e family 2

T H E O R E M III.l .

a thssis

subscripts,

/; t h e letter

preliminary

formulate

a tKesis

fC, w i t h o r w i t h o u t

X, w i t h o r w i t h o u t

which r a n g e s

subscripts,

Z.6; t h e l e t t e r s

a.,b,c,d.,^

which ran^e o v e r

with o r without

of TSC«»-L. that

we

have

retained

a

convention

su-ialoguous

to

Conventions 1.2 and II.1, namely C o n v e

n

t

1

o

n

III.l.

The

theses

(accepted

sentences) whose numbers are marked with an asterisk are

theses

of the metatheory of TSCt^L. Theorem

III.l

enables

us

to

arrive

at

the

conclusion

to

which we shall refer on numerous occasions. CoNci^usioN III.l . TSC*r-L li.ea.b.

inctudei

Il.iOa.b.

Axioms

II.3a-c.

Conclusions

Il.la.b. II.17a,

The

Caccepted

II.1-II.10;

11.8-11.10. Il.Sa.b. 11.18.

theses

Definitions

II.iaa.-d. II.3,

11.19.

II.13a.b;

II.4a-c.

sentences) II.la,

of II.i.

TKeorem II. 1;

11.5a-c,

II.1B-14.

THE THEORY OF TSC
By

ret-aining

analogon in.2 , III.l

of

the

an

expression

Remarks -

the

deflnit-lon

II.3

and

conclusion

of

97

Csee TLTk

II.2

made

which

is

Sec.II.2>

we

of

obtain

in

-

Sec.II.2,

analogous

the by

and

to

dual

Theorem Conclusion

Theorem

II.3 ,

namely CONCLUSION (accepted, In

Every

sentence)

in

connection

with

convenient, of

III.2 .

this

when

TLTk is the

to

analogon

a thesis

in

TSCt*-L

observe

is

the

of

a.

thesis

TSCt»-L.

relationships

presenting

chapter,

dual

shown

the

above

successive

conventions

it

is

sections

introduced

in

Chap.II. H e n c e we h a v e C o n v e n t i o n

III.2.

We

retain

Convention

II.2

CSec.II.2) a n d C o n v e n t i o n s I I . 3 a - d CSec.II.5).

Sec.III.3.

Operator expressions

We now concepts

add

to

the

conceptual

which

do

not

occur

in

TSCL

c o n c e p t s i n TSC«r-L. T h e y a r e : t h e CO-language £,

a n d t h e set

Kr of

The

sense

we

intuitive

sets,

that

deviate of will

from

formal be

variaible.

is,

which

respectively,

the

Thus,

set all

to for

Note,

denote

impart

to

however,

instance,

label by

of

TETk

which

are

all

and to

that which

the

of

the

two

that

elements variables,

them

new

primitive

operators

variables

operators

a

and

O of

o n e w h i c h we i m p a r t

disciplines. used

apparatus

of

the

language. of

those

does

not

In t h e

languages

term

"operator"

the does

universal

not

include

quantifier

a we

98

CHAPTER III

mean t.he graphic sign ^ and not. such graphic symbols a s ^, ^, and by t.he algebraic sign o f summation we mean ^ and n o t , say, 00

z fc«l This i s s o because we adopt AXIOM III.l

O u Vr ^ E . s

The set

of a l l o p e r a t o r s ccnd. th.e set

s u b s e t s of the set In

accordance

of simple with

expression

Axiom

III.l

of

a l l •uaria<>les ore

tokens. and

Definition

II.3a

we

arrive a t CONCLUSION III.l a.

O S K /N O £ DCi.>,

b.

Vr S V ^ Vr S DCL:>.

Thus o p e r a t o r s and variables a r e simple word tokens of t h e CO-language Jf under consideration which have Indices. In

the

operators

vocabulary that

Conclusions

V of

would

II.2b

and

the

be Ill.la

(<>-language

"typically and

JP t h e r e

are

ambiguous", Convention

no

because

11.3c

Csee

Convention III.2> yield CONCLUSION in.2a p,<}

The indices Likewise, yield

^ 0 / \ p ^ Q ^ a S i b .

of eqvi/ornt operators Conclusions

II.2b

and

are Ill.lb

etfuiform. and

Convention

I1.3c

THE THEORY OF TSC(0-L OF CATEGORIAL CO-LANGUAGES

99

CONCLUSION III.2b

p,q The

indices

We a l s o

of

e Vr

eqxiiform

assume

about

^ p % q ^ a % 6 . variaAlss

the sets

ars

eQui/orm.

O and Vr t h a t

they

satisfy,

r e s p e c t i v e l y , a x i o m s which a r e anaiog^uous t o Axiom 1.10: AXIOM III .2

p e O / v t % p > » ( € 0 . A Itxbel

token

latiich

is

etfuiform

with

an operator

is

also

an

operator. AXIOM III.3 p e Vr

A label

token

lahich

/\ ( % p ti^ t €

is

equiform

with

Vr.

a variable

is

also

a

variable. It is often emphasized of

the operators

that

is that

binding role of an operator which

include

cj-language

operators.

operators

can

is revealed

To

Jf we distinguish

one of the essential

describe

bind

variables. The

in definite

such

functions

structures

structures

in the

the set E of all expression

from

tokens of Jf its subset pCOxKrxO, whose elements will be termed operator

expression

operator

expression

tokens

of

the to-language

of that language. Operator

J? or, briefly, expressions are

thus the values of the function p restricted to the set

OxVrxE.

They are thus compound expressions consisting of an operator, a variable, and an expression token, in that order. In

our

further

analyses

we

shall

make

use

of

the

100

CHAPTER III

a b b r e v i a t . i o n i n t . r o d u c e d by t h e m e t . a l i n ( ; u i s ' t i c DEFINITION p which

p^Cp^,p^,p^> is

thesis

In

K

other with

the

of

a

an

side

of

that

said

expression The

sign

standing

elements

of

sign,

by

an the

to

with

of

In

one

side

standing

expression

every

on

an

of

the

on

the

equiform

the

one

substitution that

the

P-^^^'

construction,

standing

of

^

replace.

the

an

appropriate

expansion

p^eVr

that

replace

side

t-o

on

expression

other

^

under

standing;

and the

us

now

equiform

for

on

allows

TSCw-L that

A p^eO

side

for

the

sign.

operator

expression

P *^^0'^1''*2 '

i.e.,

the

elements

successively,

- the

xohich

p

-

indexicgil

The

token"

the

the main

the

sequence operator

of

the

operator

Pn>P^>P^> of

the

operator p ,

p

are

called,

expression

<<5 ) ,

p_ o r t h e -

the

variable

scope

of

the

p . expression

consisting its

of

variable

accompanies

operator

and

vhlch

with

the

t<5 >

p

rule

expression

substitution

the

p m pCp^,p^,p^->

theory

equiform

by

a

^

fact,

of

expression sign

III.l

of

the

scope we

p .

shall

C<5 > i s main

the

operator

Instead often

read:

of use

operator

p ,

the

its

term

the

expression

indexical "operator

shorter

token

variable

p.

expression

form

"operator

expression'". The

adopted

terminological conventions

will

be

used

later

in

t h e v e r b a l f o r m u l a t i o n o f f u r t h e r t h e s e s o f TSC«»-L. The

following

conclusion

is

a

specified

version

of

THE THEORY OF TSC(0-L OF CATEGORIAL 0>-LANGUAGES

Conclusion

II.3;

it

follows

101

from

the

latter

and

Definition

III.l*. CONCLUSION III.3

p - p^
-^ <J - p'^C<jg,^^,<j2> •» •»

Two operator main

expressions

operators,

ths

scopes

the

of

We h a v e

those

otre e
indexical

variables

operators

are

t h e following

if of

and

only

those

if

their

operators

and

equiform.

theorem

which

is

analoguous

to

Axiom

token

Q is

11.10: THEOREM III.l

p » p^
If

p is

etfuiform

an operator vtith

expression

the

that

it

ajui

operator, the of

if

operator

-

and

then

only

expressions

indexical

operator,

III.l ,

expression

the

indexical

The

•»<<}% p «^

are

variable

-uariable equiform of

if

an expression q

of

its

of

that

wt th

that

is

also

an

expansion

-

operator the

operator

main and

operator the

main

and the

scope

operator scope

of of

p. that

respectively. proof Axioms

of

this

11.10,

theorem III.2

and

is

obtained

III.3,

from

Convention

Definition 11.3a

and

Conclusion II.4c. The concept of the relation

C/v.> of

being

a free

variable

in

102

a

CHAPTER III

given

expi^ession

TSC«-L.

That,

t.oken

relat,ion

is

is

a

secondary

defined

by

Cderived>

the

concept,

expression

in

p
which i s r e a d : p i s a f r e e varlatble i n t-he e x p r e s s i o n t o k e n Q. DEFINITION III.l P
•»

p ^

^'^ ^

^''•o'^l''2 p is p is

is

^

variattls

a variable

there of

a fT«s

the

the

scope

of

and

at

with

the

^

0

1

2

the

q

same

p is

main

indexical

C

expression

variable the

e

in an expression

no operator

tf and

p

time

the

operator

variable

of

that

r

it

time

and

that

q, if

"

is

i

and only

a constituent

same

of

r^

token

r such at

^

of a

if

q and constituent

a constituent

of

a variable

eefuiform

operator.

D e f i n i t i o n III.l and C o n c l u s i o n II.Ida i m m e d i a t e l y yield CONCLUSION III.4

pCfv'yef y. s S: tf ^ If

p is

a free

a free

variable

The

variable

in an expression

in an expression

concept

of

a

bound

associated

with

A variable

which i n a g i v e n o p e r a t o r

main

operator

is

of

a

operator

and e q u l f o r m

operator

expression

variable

a free

free

p

such

with

by

variable

variable

with

token

etfuiform

variable

that

pCfv^s.

its

a

within

bound by t h e main o p e r a t o r

• p

it

is

cf. given

expression

p

thjen

operator

in a given

indexical

that

q.

is

expression.

i s bound by i t s

the

scope

variable.

of

Thus

p ^Pf.fP^,Py'>, if

and only

then

said

if

that In

Po

an

^^^ ^

P^^fW>p

(24)

and

p_

%

p .

The operator

p

Is

to

bind

the

THE THEORY OF TSC(i>-L OF CATEGORIAL(i>-LANGUAGES

variable

p_.

The o p e r a t o r to

103

bind any

voidly

in

free

binding

a

given

variable

in t h a t

operator

in

its

operator

scope.

is

voidly

binding

in

a

there

is

no

free

variable

in

its

which

include

binding

We t h e n

expression.

operator

voidly

expression

given

The

scope.

operators

of

the

definition

of

of

ordieT

n

we

shall

refer

the

the

of

the

set

<<>-language set

to

the

be

replaced

of

all

will

be

all

it

is

We a l s o s a y t h a t

an

expression

Operator

are

not

expressions

included

in

based

recording

II.3b,c

the by

\aell-/ormed

DEFINITION

Csee

term the

on

the

inductive

expression

tokens

of

this

definition

Convention

"well-formed

shorter

term

III.2>.

"well-formed

III .2

p e

S 4 i » p € S v

c"**
.

p^,p^,p^,p^

At

^ P2 ^

S ^ p^C/v^p^

^P '

P"
3

'^ P3 =K P j -^ C

<.a,a^,a^,a^>y,

in

expression

s

.

the

expression

expression".

b.

if

X.

itell-formed

symbolic

Convention

formulations

will

S

Jf

S o/

C7i>0>. In

verbal

token"

that

not

Weli-formed e x p r e s s i o n s

definition

tokens

say

operator

well-formed e x p r e s s i o n s of t h e oj-language

S e c . III.4.

may h a p p e n

^

104

CHAPTER III 00

c.

S -

In

U "S. nmO

accordance

with

Definition

III.2a

a

e x p r e s s i o n of order 0 Is a simple e x p r e s s i o n of

well-formed the

oi^-language

X. In

accordance

with

Definition

III.2b

a

well-formed

e x p r e s s i o n of order k+i i s 3tn e x p r e s s i o n which s a t i s f i e d

one of

t h e following t h r e e conditions: 1

p is

a well-formed

element of t h e s e t 2 all

expression

of

order

k,

and hence

S,

p i s a compound e x p r e s s i o n of t h e oj-lanjfuage £ such t h a t elements

of

its

expansion

PntP^r—rP

are

well-formed

e x p r e s s i o n s of order Ai, which i s t o say t h a t they are of

an

the

set

S; f u r t h e r ,

the

main functor

p^ of

elements

p Is

not

an

o p e r a t o r ; finally p s a t i s f i e s t h e condition m

c

In accordance functor

of

indices

a ,...,a

with


which

the

l>.

cate^^orial

p i s a concatenation of of

all

index a

of

t h e index a of

successive

arguments

expression,

the

of

the

main

p and the

the main

functor, 3

p is

an

operator

matin o p e r a t o r

which binds a c e r t a i n variable p„ which i s f r e e which i s a well-formed

e x p r e s s i o n of order

p

of

in i t s scope p^

k.; t h e

index a„

of

t h a t o p e r a t o r i s a concatenation of t h e index a of p, t h e index a

of t h e Indexical variable p

of

that

operator

a^ of i t s scope p ; p t h u s s a t i s f i e s t h e condition

and t h e

index

THE THEORY OF TSCco-L OF CATEGORIAL oo-LANGUAGES

In a c c o r d a n c e all

sets

of

with

Definition

well-formed

105

III .2c t h e s e t - S i s t h e s u m o f

expressions

of

a

finite

order

C^reater

t h a n o r equal t o zero>. We

now a d o p t

two

terminological

e x p r e s s i o n p f o r which t h e r e satisfies compound

the

condition

non-operator

conventions.

i s a naturaJ. number

2°

will

be

expression.

A

3

We s h a l l

u s e t h e symbols

well-formed all

will b e t e r m e d

a

well-formed

compound

well-formed

a well-formod

S

and S

non-operator

operator

well-formed

At s u c h t h a t

termed

f o r which t h e r e i s a n a t u r a l number k s u c h t h a t condition

A

^aell-formed expression

p satisfies

operator

t o denote,

expressions

expressions,

p

p the

expression. t h e s e t of all

and t h e

set

respectively.

of The

d e f i n i t i o n s of t h e s e c o n c e p t s a r e recorded a s follows: DEFINITION

III.3

p ^ S"'^ ^)^

n>l

^ p^,p^,...,p^^

y. p DEFINITION

^ O ^ C

fc.

^^ " PtPo'Pi' 'P„> S

Ca,a ,...,a

.<XQ>5-

III.4

s " •• y k

V pQ'Pi'p2'P3

cp • p^(.p^,p^,p^^

^ ^2 ^

^ •^

3 '^ P^^f'*>^p2 -^ P3 * Pj -^ *= *-°'°l'*'2''*0''"*' Note compound the

that

the

concept

non-operator

concept

of

the

of

the

expressions set

SS S

set is of

S

of

all

well-formed

i n TSC«»-L am a n a l o g o n all

well-formed

of

compound

106

CHAPTER III

expressions. be

not-ed

Introduced t.hat.

operator-free

a

In TSCL, ( s e e

non-operat.or

expression there

main f x i n c t o r

of

expression

expression

o n e : i n t-he e x p a n s i o n

non-operator

that

which

of

a

It. must, need

furt-her

not,

well-formed

be

an

an o p e r a t o r

operator

an

compound

may o c c u r , a s a n a r g u m e n t

expression,

has

Sec.II.S>.

of the

expression

expressions

o r an

as

its

constituent. In f o r m u l a t i n g have

availed

compound

this

ourselves

non-operator

remark, of

and e a r l i e r

the

fact

expression

is

t h e Condition

that a

2 , we

every

well-formed

compound

expression.

D e f i n i t i o n 111.3, LEMMA III.l

"^S £ E c o c o . D e f i n i t i o n II.3b and Axiom II.9 y i e l d CONCLUSION 111.5a

c Lemma III.l

is

an analogon

of

Lemma

II.2. I t s p r o o f

is

given

i n t h e Annex. Definitions

III.4,

III.l*,

Conclusions

III.la,b

and

11.5b,

D e f i n i t i o n II.3b and Axiom II.9 y i e l d a l s o CONCLUSION 111.5b

c E v e r y e l e m e n t of Note t h a t

th.e s e t S

t h e main f u n c t o r

S , unlike t h e main f u n c t o r

is

a. compounti

expression

of each expression

o f any e x p r e s s i o n

from

from

token. the

set

the s e t S

,

THE THEORY OF TSC<0-L OF CATEGORIAL (0-LANGUAGES

i s a n o p e r a t o r . The s e t s is

a

one-to-one

also

disjoint

III.5a,b

CAxiom

the

fact

a r e thus disjoint

II.9).

the s e t E ,

from

C o n c l u s i o n II .Sa.

and S

function

with

and

S

107

Each

which

of

follows

that,

by

because

p

sets

is

these

from

Conclusions

Definition

II.3a

and

E n E =0. c s

We a d o p t a n axiom a n s d o g u o u s t o Axiom 11.11, namely AXIOM

III.4

.cs""> n I In

the

(x>—I angtiags

non-operator

expression

We a l s o AXIOM

X

there

which

the

and

otherwise,

oi-langvage which S

£

The

compotind index.

and S

Moreover,

follows

earlier analyses

yield

0.

is

applies III.2a,b,

t o o , and t h u s .

t h e sum o f

^

a

uiel l-/ormed

categorial

same

III.4, t h e s e t s S

that

categorial

In

to each

view

operator

index.

a r e non-empty in accordance

by D e f i n i t i o n s

it

o

there

has a basic and S

III.5.

empty,

III.4

\tell~ formed.

III.5

The s e t s

be

tx

adopt

expression

III.4

is

has a basic

tCS"^? n I

In

^ 0.

o

the of

of

set

with E

the sets

Axioms

because In S would

Definitions

III.3

and

III.2c,a,b,

III.3

and

would b e e m p t y .

E ,

from

Definitions

s " " and s " equals

S. T h i s

and t h e

108

CHAPTER III THEOREM

111.2

s - °s u s"" u s" ^ °s »« 0 ^ s"^ .< 0 ^ s" ^ 0 ^ ^ °S n s " " - 0 .V °S n s " - 0 ^ s"^ o s " - 0.

is

The

set

the

SUM of

set

of

S of

S

of

that

all

veil-formed

three

all

non-empty

single titel l-formed

language

and of

This

and

expressions

all

expressions

expressions

that

pairv»ise of

compound

the

set

S

of

the

the disjoint

all

J?

setsi

oi-langttage

non-operator of

ui-language

£,

the

the

set

expressions

of

well—formed

operator

language.

t-heorem.

Definition

III .2c

and

Lemma

III.l

yield

the

u>-language

Immediately CONCLUSION

111.6

0 ^ S ^ E ^ S ^ DCi.y. The is the

set

of

all

a non-empty set

of

Since

set

laord S

III.l

III.l

that

tokens

have

an

it.

follows

O

and

Vr

III.2a,c

of

that

language

X and

index. from

Definition

are

non-empty,

these

sets

are

III.4

and

too.

By

subsets

of

III.7

0

b.

0 i< Kr = S.

i< O =

accordance

operators

of

at

a.

In

which

Definitions

S. Thus we a r r i v e

expressions

expression

non-empt-y

au-id

CONCLUSION

of

tokens

is

Metadeflnition Axiom

viel I-formed

S,

with

and t h e s e t

of

Conclusions

III.7a,b

the

set

of

all v a r i a b l e s a r e non-empty s u b s e t s

all of

THE THEORY OF TSCo>-L OF CATEGORIAL CO-LANGUAGES

109

t-he s e t - o f a l l v e i l - f o r m e d e x p r e s s i o n s o f X.

S e c . III.5.

In the

this

Fundainent.al t^heorems

section

well-formed

to

the theorems

we sh&U

expressions

formulate of

oj-laneuages.

o f TSCL. p e r t a i n i n g

of simple c a t e g o r i a l

theorems

pertaining

They

are

t o well-formed

to

analoguous expressions

languages.

T h e o r e m II.3 h a s i t s a n a l o g o n i n THEOREM III.3

An expression veil-formed

of

the oi—language

expression

This

theorem

is

follows

which i s analoguous

also

£ uiftich

is

equiform

a ttel I-formed

from

Definition

iitith.

its

expression. III.2c

and

the

lemma

expression

of

t o L e m m a II .3:

LCMMA III .2

An order The

expression k is

equiform

also

with

a well—formed

inductive

proof

of

a

well-formed

expression this

of order

lemma

is

given

k. in

the

Annex.

T h i s lemma i s u s e d i n t h e p r o o f s o f t h e f o l l o w i n g t w o t h e o r e m s : THEOREM III.3a

p ^ S An

expression

non-operator non-operator

^qKp'^qcS

equiform expression expression.

is

with

a

well-formed

compound

also

a

well-formed

compound

110

CHAPTER in THEOREM

111.3b p e

An

expr0ssion

expression The

is

of

of

Theorems

omitted.

In

of

well-formsd

operator

of that

Convention

a

III.3a,b,

t h e inductive

therefore

view

with

a well—formed

quite similar t o t h e proof on t h e b a s i s

^ « j % p ^ < j € S .

equi/orm

also

proofs

S

operator

expression.

baised

on

Lemma

III.2,

lemma i n t h e c a s e s

assumption

II.3c,

2

ajid 3

( s e e Annex>. They

Conclusions

III.6

ar-e

and

are

II.2b

yield CONCLUSION

III.8

/>,qt Tfie

indices

of

e S ^ p % « j ^ a f e 6 .

ecfui form

well-formed

expressions

are

eqvt I form. We

shall

now

Definitions

formulate

II.6

and

c o n c e p t s : t h e set

B of

Jf and t h e set DEFINITION

F of

all

definitions

II.7.

They

al I basic fuTic tors

are

of that

set

well-formed index.

are

analo^a

definitions

expressions

of t h e

of

of two

co^language

language.

III.S

» - < p e S | a e / The

which

of

all

basic

expressions

expressions of

the

o is

oi—lang-aage

>. the

set

X which

of

all

have

those a

basic

THE THEORY OF TSC(D-L OF CATEGORIAL CD-LANGUAGES

111

DEFINITION III.6

The

sst

of

expressions

alt

of

a basic

functors

is

the oi-language

the

set

£ \eh.ich

of

have

all

veil-formed

an iTidex

that

is

not

index.

Definition

III.5,

Theorem

III.3,

Conclusion

III.8

and

Axiom

II.2 y i e l d THEOREM

111.3C

An expression basic

OQUiform

with

a basic

expression

is

also

a

expression.

Definition

III.6,

Theorem

III.3,

Conclusion

III.8,

Theorem

1.4 a and Axiom 1.7 y i e l d THEOREM Ill.Sd

An expression

equiform

The d e f i n i t i o n s

of

with

a functor

B and F make

it

is

also

a

functor.

possible

to

formulate

t h e o r e m which i s a n analog^on o f T h e o r e m II .4 i n TSCL. B e f o r e record

it

we

shall

first

draw

several

a we

conclusions

from

these

Definition

III.5

yield

definitions. Axiom immediately

III.4,

Theorem

the

following

C o n c l u s i o n 11.10:

III.2

and

conclusion

which

is

atnaloguous

to

112

CHAPTER III

CONCLUSION III.9 S"^ n B ^ 0.

7"ftere

is

a itel l-formed

t/ie ui-langriage

compo-und

£ u/tic/t. is

non-opsrator

a basic

expression

of

expression.

Axiom III.5, T h e o r e m IH.2 and Definit-ion IIl.S y i e l d CONCLUSION III.10

s'^ n B >' 0. There

is

a well-formed

operator

expression

which

is

a

basic

expression. Since,

by C o n c l u s i o n

expression,

auid,

expression

consists

three-element Conventions operator

by

III.10,

Definlt-ions of

an

concatenation III.2

whose

and index

there

11.2

of and

is

a

III.4

well-formed

aaid

operator

whose

indices,

by

Conclusion

belongs

to

III.l ,

each

Theorem

set

such

index

III.2 >

the

operator

is

I.13'S>

there

/\/

.

a <see

is

an

Thus,

by

C o n c l u s i o n III.7a and D e f i n i t i o n III.6, we h a v e CONCLUSION III.ll

O n F 7' 0. At

least

This

conclusion

assumption however,

one operator

that

is an

a cosequence

disjointness

of

of at

the o^-lang-uage variance

operator of

O £>nd F could,

adoption of a different

with

cannot

the here of

X is a func the

be

adopted course,

a

tor.

often

adopted

functor.

definition

of

be guaranteed

d e f i n i t i o n o f F, f o r i n s t a n c e

F • < p € S'SO I a € I\l

>. o

It

is,

F. The by t h e

THE THEORY OF TSCio-L OF CATEGORIAL o)-LANGUAGES

Since

S

is

Definitions

a

III.3

non-empty and

113

subset

III.2c,

of

S

Theorem

(Theorem

LIS^S)

and

III.2>,

by

Definition

III.6 w e a r r i v e a t CONCLUSION III.12

f S O ^ 0. There

is

a functor

la/tich.

is

not

a.n

operator.

We s h a l l r e c o r d o n e m o r e c o n c l u s i o n p e r t a i l n l n g t o f u n c t o r s . CONCLUSION III.13

° S n F i* 0. There

is

a simple

expression

of

the

U)-language

£ \ahich

is

a

functor. Conclusion be

proved

III.13

by

is

an

analo^on

reference

to

III.11-III.13

do

of

Conclusion

Conclusion

III.ll,

11.11.

Axiom

It

can

III.l

and

D e f i n i t i o n III.2a. Conclusions possibility follows hand,

from the

Conclusion III.6

of

and

including each

of

operators

them

that

non-emptiness III.9. Axiom

This, II.7

a n a l o g o n o f T h e o r e m II.4:

not,

of

of in

Definitions yield

the

F is B

course, set

preclude of

non-empty.

follows

immediately

functors. On t h e

Immediately

III.5, the

the

III.6,

It

other from

Conclusion

afore-mentioned

114

CHAPTER IH THEOREM

III.4

The

of

is

set

the

all

svan of

basic

well-formadi

two

expressions

ftinc

expressions

non-empty

and

of

language

that

of

disjoint

the

(ji-language

sets, and

that

that

of

of

all

all

its

other

than

tors. Note

t>^lal

Definition

If

we

adopt.ed

111.6 ^ i n s e r t e d

a

derinit.lon

earlier,

and

of

an

F

additional

(.CO>£7N/ , we could r e p l a c e T h e o r e m I1I.4 by t h e f o l l o w i n g The

set

is

the

sum of

of

all

basic

of

func tors, The

all

vtell—formed

three

and

that

<•)-language

of

all

£

that

syntactic

the

axiom: one:

i^-language

disjoint

language.

sets:

that

of

that

all

consideration

can

well-formed

be

split

in

this

The

set

into

syntactic

expression

S

belongs

we f o r m u l a t e

the

to

of

one

theorem

which

the

its

exactly

appropriate

that

a

categories

includes

states

is

all

T h e o r e m II.6a i n TSCL we n o t e t h a t

theorem

its

chapter

which i s a n a n a l o g o n o f the

£

operators.

language.

category. Before

of

pairwise

that

its

under

expressions

every

and

of

categorial

well-formed

expressions

non-empty

expressions

syntactically

so

X

TETk

relation

^

of

c

categorial

agreement

Conclusions

II.13a-c)

identify

syntactic

abstraction

of

and

an also

categories

that

Conclusion

III.6,

proved

reference

by

is

the

relation set to

E is the

equivalence the

theorem

with Csee

the

which

in

II.14>.

(eatrller

available

in

E

enables

appropriate

Conclusion

non-empty means

relation

that TSCL:

Csee us

to

classes

of

Since,

by

fact

was

Conclusion

THE THEORY OF TSCa>-L OF CATEGORIAL ti>-LANGUAGES

II.9d II.6

follows is

from

also

syntactic

a

by

determines

a

By

theorem

the

in

-. c

partition

of

categories

of

the

CtCB) of and of

family

those the

sets:

well-formed

the

all

basic

family

that

III.4,

family

of

of

course,

all

subset S,

B,

E. The

CtCS> of

of

of

of

E.

auid

F,

logical

oij-language J?,

functor of

also

syntactic

expressions

The definitions

E

following

all

the

aU

of

respectively,

expressions categories

of

partition

sets

are,

CtCFy of

language.

the

Theorem

Ct

non-empty

subsets

Ct

the

family

relation,

every

non-empty

II.9a,b>,

logical

That

are

the

The

a

TSC»*-L,

of

expressions

thus

Theorem

of

and

TSCw-L.

and

partitions

language,

is

II.6a,b

III.6

subfamilies

family

in

relation

logical

Conclusion

defined

Conclusions

categories

determined

115

of

caregories those

that of

concepts

are analoguous to Definitions Il.lla-c: DEFINITION III.7

a.

CtCS) = < C(

I f e t CS) >,

b.

CtCBy

m < Ct

I ^ e t,

c.

CtCF> = < Ct

I ? e UF? >.

The comments made above allow us t o formulate THEOREM

III.S.

c lassi fication well-formed categories their

The (logical

of

the

iiihich are non-empty,

consisting

of

B and

sum of

S.

Further,

the sum of all

"

parti tioJt)

expressions

sum ecfuals

the

relation

all

functor

determines of

the

set

(x>-language £ pairwise

5 can basic

be

of

all

syntac

tic

and such

presented equal

equal

logical

S

into

disjoint

categories

categories

a

to

as

that

the

sum

to

the

set

the

set

F.

116

CHAPTER III

Thus,

the following

"^*

conditions

ct,,ct''^tcs>

Clll)

U

av>

sm

m S,

^

u

c t u

Ct eCKBy

/undouiten t a { Those

Ct m B ^

and

II

of

the

ai-e

latter as

by in

by

of

the

is

we on

of

adopt

the

the

the

of

p.

In

TSCL, of

expression of the 00-language £.

of

of

Csee

is

It

the

order

fttsc and

obtained

constituent

Inductive

here

I

Sec.II.l

r

the

two

categories.

defining

the

n-th

of

abbreviations

expression

replacement

constituent

formulation syntactic

rCp/q> s , which denotes a

- f.

^

expression

expression

case

the

theory

read:

by the

the

the

expression

replacement

concluded

based

s

ct

Ct eCKPy

which

expression

ourselves, of

for

rCp^cfys, which

the

the

be

U

^

theor&ms of

fttsc,

Sec.II.7> from

will

theorems,

ct

Ct^tCPy

U

section

U

'

ct eCtCB> This

satisfiads

^^'j; ^ ^*a •* ^'? ^ ^ ^ - ^^'

Ct

ct sctcsy

are

we

avail

definition

relation

of

4

a

of

given

THE THEORY OF TSCtO-L OF CATEGORIAL (O-LANGUAGES DEFINITION

117

I1I.8

a.

r

* • • s , r

eS/\<j%S/^p%r,

b.

rCpy^y

e S

- n>l

s ** Cs,r

^

s „ , » , , ' ' . . ,B € S <^»-P<«o'*l- ' V 0' 1 ' " --

" r-p
^

0
-^ «Q

%»

ys p * r^ /«. s

a: r

/N s

% r_> v C«j % s

/^ p % r

-^

^ 0 V. Cq % s ^ - P * r^ - SQ % r^j ^ Sj % r ^ ^ ^ ,

c.

At > 0 ^ CrCp/q>'''*'*s 4» j V ^ € r < v / 0 * s /^ v<.pyy,

d.

r < p / q > s 4* V rCp/<j> s .

Definit.ions

III.8a,c,d

formulat.ions accordance from

the

as with

have

t.he

Definitions Definition

expression

s

ror-ms

II.14a,c,d,

111.8b

by t h e

saane

the

of

its

r

with

s

and r

a r e w e l l - f o r m e d compound n o n - o p e r a t o r

expansions

expressions

of

consisting the

of

(>>-language

with t h e same numbers o t h e r the

J-th

expression

the

of

the

X

that

same and

number

such

that

is

of

s

is

4

either expressions

of

well-formed

all

components

a c e r t a i n J a r e equiform,

exp2insion

In

obtatined

constituent

o f t h e f i r s t o r d e r by t h e e x p r e s s i o n p i f and only i f 1

verbal

respectively.

expression

replacement

and

equiform

with

and «j

118

CHAPTER III

and t-he J-i-h expression of t h e expansion of r Is equiform wlt-h p,

or 2° s and r a r e wel-formed o p e r a t o r e x p r e s s i o n s such t h a t one

of t h e t h r e e foUowlns conditions Is s a t i s f i e d : Ca) qf i s equiform with t h e main o p e r a t o r of s, p is with

the

main o p e r a t o r

of

r,

and

the

indexical

equiform

variaJsles

of

t h o s e o p e r a t o r s and t h e i r s c o p e s a r e equiform, Cb>
equiform

o p e r a t o r of s,

with

the

indexical

variable

of

the

main

p i s equiform with t h e indexical variable of

the

main o p e r a t o r of r, and t h e main o p e r a t o r s of s an r and t h e indexical

variables

and

the

scopes

of

those

operators

are

equiform, Cc>
s,

p i s equiform with t h e scope of t h e main o p e r a t o r of r, and t h e main

operators

of

s

and r

and

their

indexical

variables

are

equiform. Note t h a t

in t h e

aT& equiform those

with

case the

2 Cb) i f

the

corresponding

replaceable

indexical

variables a r e equiform. In such a c a s e

expressions

variables, s

and r

then

are,

by

III.8a-d

we

does

not

Conclusion III.3, equiform, t o o . Note can

also

arrive

that at

on

the

conclusions

strength whose

of

Definitions

symbolic

recording

differ from t h a t of Conclusions I1.20a,b, II.21-II.23. We now p a s s t o t h e two f t t s c which are e s s e n t i a l f o r TSC«*-L. Their graphic form does not differ from t h e two f t t s c TSCL CTheorems II.7a and I1.7b).

valid in

THE THEORY OF TSCo)-L OF CATEGORIAL avLANGUAGES

119

T H E O R E M I I I . 6 a CI f t t . s c > c

If

an expression

expression syntac

s

tic

of

to

The fttsc

p,

then

of

the

o^-lang^iage

language,

by

the same

proof

of

that

category,

an expression belong

r

the the

£

uhic/t

replacement

is

obtained

belongs

of

reploLceable

syntactic t.hls

c

its

to

from the

same

constituent

expressions

on

q by

p and q

also

category.

theorem,

modelled

after

t h e proof

of

I

I n S e c . I I . 7 , i s g i v e n i n t h e Annex.

THEOREM

III.6b r<.pyq')s

// an expression expression

s

constituent

q

syntac

tic

syntac

tic

The

r of

of

that

by

an

category

y - p i ^ q ^ r ^ s . c c

the

(ji-language

language

by

expression

as q,

£ the

obtained

replacement

p v>hich

then

is

belongs

r and s also

to

belong

to

from

an

of

its

the

same

the

same

category.

proof

of Theorem

III.6b can be easily

reconstructed if

we follow the proof of II fttsc given in Sec.II.7, and also the proof of Theorem III.6a. It is therefore omitted here. The two fttsc taken jointly enable us to prove T H E O R E M III.6

rCp/qJs -> Cp » Q •» r « s>. Two syntactic in

expressions category

a rtiell-formed

well-formed

of if

the

and only

expression

expression


of

if of

that

by replac the

£

belong

the

same

ing one by the

other

ij>-language

language

which

to

£

we obtain

belongs

to

a the

120

CHAPTER III

saume syntac

Sec

tic

category

III.6.

as

ths

T h e SLlgorithm o f c h e c k i n e t h e cor-recloess of

The

algor-it.hm

expressions given

in

of

t.o

the

Sec.II.8

be

and

preliminary

CO-language

label c o n s i s t s

able,

referring

t w o . We a l s o is

index,

in

whether

expression. position of

its

to

one of

It

If

is

it

arguments

of

categorial

index

that to

similar

the

to

compound

them

an is

chapter,

Sec.II.8.

We

number

are is

t.o

compound

the

algorithm

expressions

of

which p a r t s

functor. every

expansion

of

the

assume

of

that

to

it

symbols, decide and

of it We

about is

or

compound

of

the

a

a

about about

a concatenation

operator,

parts

of

an

eqvilform

whether

part

symbolism

accordingly

we c a n d e c i d e

a

the

are

criteria,

they

which

which

pert.alns

concerning

finite

that

establish

and

a

vocabulary,

expansion,

functor,

in

not,

assume

the

It. i s to

this

empirical

or

labels, whether

it

in

of

to

here

language.

formulated

whether

£.

assumptions

every

laibels

described

pertaining

Jf, a n a l y s e d

assumptions

synt.act.tc

expressions

OJ-language

any s i m p l e c a t e g o r i a l The

latter.

of

it

it

expression

a l s o a s s u m e t h a t a b o u t any s e q u e n c e o f

two three

remaining whether

word

that

has

variable,

or we

correspond

to

corresponds

to

also of

any

label

correspond are

are

any

expression,

of

we

any

the

that

the

Indices

to able

the to

a

an

compound

are the

in

a

words

the

main

successive assign

o>-language

X.

a We

THE THEORY OF TSCci>-L OF CATEGORIAL w-LANGUAGES

12'

we can decide whether it. sat.isfles the condition

where a i s the index of the expression p. We

ailso

retain

the

term

compound expression". Let

"intermediate

vis recall

that

expansion

the

sequence

and

only

of

a

i s

an

intermediate expansion of the expression

such

that

Pf.>P^t—rp are

such

that

0
f .

expressions

s: a.

and

a. .

if is

the

if

index

for

of

any

the

i

i-th

element of the expansion of C6>. Thds

if

a

compound

expression

of

the

co-languaige

£

is

an

operator expression of the form f*

>

P

then,

in

accordance

III.l ,

the

,

with

intermediate

the

convention

expansion

of

adopted

(.6 >

is

and

any

Definition

sequence

of

indices

such t h a t for any i, The

ailgorithm

compound auxiliary

0
of

expression

checking of

algorithm, namely

expression operations

Csee of

that

Sec.II.8>.

the

where a. syntactic

the

ci>-language

that

of

In

» (.Cp.5. correctness

£

begins

cons tr-uc ting

carrying

algorithm we find

out

whether

trse the

tree

of

with I of

a an

that

successive I of

a given

122

CHAPTER III

compound e x p r e s s i o n i s , o r i s not,, a t-ree o f arranged

expression.

•we o r d e r

the

elements

Sec.II.8,

and

that

checking

of

If

next

the

order

that

we

pass

to

the

existence

with

that

whether

at

operator.

If

is

question,

which

this

included

in

the

is

languages

we

Sec.II.8.

If,

of

I of

tree

transform

not

of

the

In

underline the

then

described

algorithm,

the

of

namely in

upon

it

we

all

check,

in

the expression

in

first

tree of

I of

its

final

then

the

and

nodes

is

an

expression

in

does

not,

since

oo-language

simple

to

of

well-arranged,

the

contrary,

I of

that

order

it

J? which

operations

at

one

least

of

in q u e s t i o n ;

be

categorial

described

the

tree

can

final

I' by

in

nodes

then

we

proceeding

directives;

for

which

into

of

satisfies

syntactically

further

expression

fixed

variables

expression

in

operators

t h e e x p r e s s i o n i n q u e s t i o n i s an o p e r a t o r ,

tree

all

afflrmat-ive,

binding

operators,

in accordance with t h e following 1

well-

manner

•uoidly

case,

binding

imposed

the

one

thoroughly

submit

on

t-he

successive

algorithm

fragment

conditions

in

in t h e

the

leaist

voldly

that

tree

the elements of

question,

include

is

thoroxigh.ly

consideration.

fixed for

course,

answer

of

t h e e x p r e s s i o n under In a c c o r d a n c e

t-he

a

the

aure t h e those

elements final

of

tree

nodes

of

are

their

variables

tree

I

we

I,

of

own

free

variables. 2

Once we know all

underlined,

in

the

binding o p e r a t o r s transformation,

the

free

successive -

we

in t h e find,

variables

expression

*-th while

line of

-

which o c c u r , which

tree

observing

have

I being

the

order

and no

voidly

subject from

are

to the

THE THEORY OF TSCtO-L OF CATEGORIAL(i>-LANGUAGES

123

left, i^o t.he rights, whet,her in any compound expression p in t-he A-l-t-h

Une

binding

situated

operator,

above

and

in

we

that,

tree

establish

there

what

is

a

possible

voidly further

o p e r a t i o n s should be made in trainsformins t r e e I. Ca> If t h e main functor of p i s not an o p e r a t o r , t h e n p does not

include

a

voidly

binding

operator.

Then in

every

part

of

t h a t e x p r e s s i o n which corresponds t o t h e component p.

Ci>0) of

the

which

expansion

of

equiform

with

syntactic

position

expression

is

p we underline

the

free in t h a t

equiform

every

variadsle

that

expression

with

p..

such

of

The

variable

occupies the

A-th

variables

is

the

same

line,

which

underlined

In

t h a t manner are all f r e e varistbles in p. If t h e main functor of p i s an operator expansion in t h e sequence p_,p^,p_, then in t h e occur

expressions


equiform,

aind p has fc-th

line

its

there

respectively,

with

p„,p ,/>^ and such t h a t e i t h e r Cb > «j

is

not

eqviiform

with

any

free

variable

in

q

or

t h e r e i s no f r e e variable in tf , or Cb > qf, i s equiform with a variable q In t h e tree

I'

caseCb > t h e of

the

operator

expression

which i s f r e e in q .

p„ in p i s

under

voidly binding

consideration

has

and been

constructed. In t h e c a s e Cb„) t h e o p e r a t o r p^ in p binds a variable which is

eqxiiform

with

the

variable

Then p has no voidly binding part

of

variable

p

which

which i s

corresponds equiform

q

and i s

operators. to

with

p a

we free

within In t h a t

its case

underline variable

scope

in

p^.

in

that

every

such

q ,

other

124

CHAPTER HI

t.Kan cf^, which has

t,he

same

synt.act
posit.ion.

The

vai^iables

underlined in p are all f r e e variables in p. 3°

The

construcMon

carrying

operations

operator

or

expression

of

1

tree

and 2 ,

we reach t h e

1'

we

first

under consideration,

is

terminated

encounter

line of

a

tree

when,

voidly

in

binding

I, which i s

and find t h a t

it

has no

the

voidly

binding operator. If in t r e e I' of t h e e x p r e s s i o n under consideration t h e r e an

element

algorithm has

which

of

been

well-formed. which

successive

checking

the

completed:

the

If

has

includes

no

a

expression

I' proves

voidly

binding of

binding

well-formedness

tree

operations

voidly

to

in

be a

operator,

checking

of

the

operator that

the

expression

question

is

t^ee

of

an

then

we

pass

syntactic

is

not

expression to

correctness

the of

t h a t expression. For

that

elements their

purpose,

of

tree

categorlal

catsgorial

I

in we

indices.

indices

accordance successively We obtsdn

of

a

given

tree

will

be

termed

Sec.II.8,

we

proceed

Sec.II.8>. That

with

the

fixed

replace in

this

its way

compound tree

II

order

of

elements

by

the

tree

expression of

the

of <see

expression

being checked. As

in

expansions p a r t s of

of

the

to

order

compound e x p r e s s i o n s

t r e e II. Out of

of

two intermediate

the tree

intermediate I,

formed

expansions t h a t

of one

i s t h e eourlier which i s an intermediate expansion of an e a r l i e r element of t r e e I. The

description

of

the

last

part

of

the

algorithm

of

THE THEORY OF TSC(D-L OF CATEGORIALW-LANGUAGES

checking

the

compound

well-formedness

expression does

Sec.II.8.

accordance

In

expansions which

is

we an

not

with

check

which

Rt . P

of

does

checking

When

not

the

we

I,

find

satisfy

being

intermediate

expansion

the

tree

syntactic

the

for

well-formedness

the

the

indices

examples

not

which

(including elements form,

a

main of

as

before

tree

equiform

is

forms}

that

equlfoi-m

To

of

the

indices the

index

of

a

s/ns,

and

to

has

the

the

functors which

are

occurs

expression occur

after

airguments o f

which that that

an i n d e x w h i c h

quantifier

the

I

been

of

which

which

h a s an i n d e x n we a c c o r d i n g l y with

tree

quasi-fractional

index

assign

the

checking

main

linear

successive

the

well-formed.

the

which

we

of

expressions

The

index o f

expression

the

the

Sec.II.8.

the

indices

in

p of

question

of

of

If

concatenations

compound

in

indicates

the

s.

indices

written

To a s e n t e n t i a l with

the of

case

/

functor

name v a r i a b l e which

are

the

slant

symbol i n d i c a t e functor.

I

the

to

operators)

was

the

given

correspond

below

the

completed:

algorithm in

which

algorithm

expression

the

p

expansion

well-formed.

expression

given

II

p satisfy

the

completed, too: t h e e x p r e s s i o n being checked i s In

tree

been

in

expression

of

m

has

then

given

intermediate

compound

condition, of

parts

binding

intermediate

compound

first

is

any

for

the expression

correctness

of

desired

any

condition

checked

voidly

description

adopted

those

the

the

well-arranged

include

the

order

expansion of

expression

satisfies

the

thoroughly

not

from

whether,

element

a

does

differ

f o r m thfc i n t e r m e d i a t e condition

of

which

operators

125

index

binds

assign of

a

a

is a

label

quantifier

126

CHAPTER III

which

binds

equiform the

of

of

in

a

contains

it

is

a

a

is

sentential

name

variable

variable

sentential

it

variable

of

and

variable

functor

the

and

a

a

a

free

of

that-

second

a

that

is

a is

functor

which

expression

which

it

about

the

second

it

is

the is

in

forms

together

expression

it.

arguments,

free

which

in

which

variable

sentential

functor

is

say

the

name

say

sentential

which

and

is

label

two

index

that we

a we

of

a

is

Likewise,

index

label

contains

nsane

the

assign

variable,

words,

expression.

we

former

name which

other

that

sentential

the

sentence-forming

In

with

variable

About,

expression

it.

together

that

s/ss.

which

sentential free

sent.ent.lal

wit-h

index

first

a

that

forms

a

label

with

a

contains

a

a

sentential

expression. The

above

possible which

to

are

description

bring

out

operators

of

the and

the

indices

specific to

of

operators

character

distinguish

of

them

makes

those

it

functors

from

"ordinary"

of

checking

functors. We

shall

now

well-formedness the

recording

their

are

apply

to of

three

those

ordinarily

E x a m p l e correctness

of

algorithm

compound

arithmetical

expressions

we

use

expressions.

brackets

In

the

In way

used.

III.l. the

this

We

shall

first

check

the

syntactic

is

thoroughly

expression

Cnl>

/^ V X > y .

y y In

order

well-arranged

to

find we

whether construct

the

expression

tree

I

of

Cnl> that

expression

THE THEORY OF TSCa>-L OF CATEGORIAL Ci>-LANGUAGES

CFlg.III.lJ.

All

expressions,

terminal and

accordingly

nodes

hence

transform



127

or Is

tree

I

tree

thoroughly

into

i n c l u d e s v o i d l y binding o p e r a t o r s

that,

tree

(see

are

simple

well-arranged.

I'

and

see

We

if

Cnl>

in

the

Fig.III.1>,

.'^A

> Fig.III.l. Note

that

expression equlform quantifier because second

the

in

variable

in

the

first

place

variable is

which

expression

the

not

line y.

line

which

in

binds

variable

contrary,

of

the

tree

occurs

in

with

that

Let

a

occurs

the

us

the

any

line

being checked i s n o t 111.2.

which

On

equlform

occurs

in

the

is

the

is

universal

voidly

second

free

which

binding

line

in

the

variable

in

the

third

place.

Hence

well-formed.

check

now

the

well-formedness

expression

Cfi2>

A V X + Cy • « ) X y

We can

quantifier

second

the

E x a m p l e of

existential

the

" y

I' o f t h e e x p r e s s i o n Cnl)

with

the

expression

the

Tree

o x

be

construct seen

tree

that

I and

the

then

expression

tree

I'

being

of

Cn2>

checked

<:Fig.III.2>. It is

thoroughly

128

CHAPTER in

well-arranged must,

and

accordingly

does construct,

compound

expression

form

int^ermediat.e

t.he

not.

include tree

p of

tree

voldly

II I

of

of

operat^ors.

C7i2> and s e e

t.hose

expansion

binding

part.s

of

p sat.isfy

if

for

t-ree

t-he

We any

II

which

condit.ion

m .

T r e e II o f C7T2> i s s h o w n i n Fig.III.3. Those form

part.s

the

the

tree

intermediate

s a t i s f y in

of

third

the line

of

that

'"vv-i-r expression

expression

and,

vairiable

that

of

which

expansion

condition which

compound e x p r e s s i o n condition

II

m

•>*

^^e

is

not

with

index

Pig.III.2.

intermediate

of

the

the

the the

index index

the of

of

and with

indices

expansion

the

well-formed.

y

T r e e I' o f t h e e x p r e s s i o n

a the

quantifier

index o f

its

of

satisfy

existential of

line

equiform

contrary,

a concatenation

and

second

^ x+ do n o t

t h e e x p r e s s i o n being checked i s not

jc

its

an e x p r e s s i o n

the

successively, quantifier

of

in

. On t h e

form

equiform

occur

that

indexical

scope.

Hence

129

THE THEORY OF TSCoJ-L OF CATEGORIAL CD-LANGUAGES

T»XnT»

Flg.III.S. E x a m p l e correctness

of

Tree

II o f

III .3. the

We

the

first

construct

thoroughly

expression

shall

X

has • the

n

check



now

the

syntactic

expression

<;TI3>

which

-

functor tree

of

Implication

I of

well-arranged.

y as

that We

•'

as

expression accordingly

'

Its

main

functor.

and f i n d t h a t transform

We

CrrS) I s

tree

I

of

t h a t e x p r e s s i o n I n t o I' Csee Flg.III.4). The Is

operator

equiform

operator.

expression

with

The

^

x+v<2

two

line

of

tree

^

and

^

^

operators.

Hence

does

Include

not

therefore,

I'

x+y
perform

does

operator

second x>0

which

is

and

not

are, do only

voidly

binding

the

successive

In

the

include

expressions

also not

occurs

any

which

third

line

voidly occur

binding In

respectively,

equiform

not

voidly

include

thoroughly operators, operations

of

the with

binding

well-arrau-iged either.

and

We

checking

but must, the

130

CHAPTER III

well-formedness

of

fche

expression

under

consideration.

For

that

p u r p o s e we c o n s t r u c t t r e e II o f Cn3> . In

accordance

with

expansions

we

expression

p

of

determined

by

parts

the

check,

order

established

successively,

tree of

I

the

tree

II

whether

for for

expressions

Cn3> i s a w e l l - f o r m e d

of

tree

I

any

expansion

of

p

satisfies

condition

tn .

It

the

satisfies

that

expamsions of condition.

expression.

o ^ x>0 ^ /N V x + y < 2 ' S^ y z - ^

Fig.III.4.

compound

Intermediate

turns out t h a t each of the seven Intermediate compound

Intermediate

T r e e I' o f t h e e x p r e s s i o n Cn3>

the

Hence

131

THE THEORY OF TSCOJ-L OF CATEGORIAL lo-LANGUAGES

Fig.III.S.

Tree

II o f

s/n/i

", 71

Tiynn

on

t-he e x p r e s s i o n

CrrSJ

on

*> n

CHAPTER

IV

DUAL THEORIES

S e c . IV.l.

The d o u b l e ont.ologicad. ctvaract.er o f

lincui8t.lc

o b j e c t s and I h e b i a a p e c t u a l apppoacti t o l a n g u a g e

Thei<e t-he

atre

t.erm

discrepancies

"lan(;ua|;e".

understanding;

of

expression.

and

They

such

from

expressions

are

well

G.S.Pelrce as

t.he

semiotlc related

linguistic

Particularly

originating

in

to

the

components known

and

as

is

C1931-353,

to/tens

lnt.erppet.at.lon

as

ambiguous label,

the

between typos

of

word,

distinction, laheis,

Calso:

words,

events

and

d e s i g n s , L a t . i n c o n c r e t o a n d In s p e c i e ) . Tok»ns

are

entitles

concrete

which

are

material

classes

of

tokens.

linguistic

objects,

belonging

categories,

are

Independently

semlotlc and

and

logic

objects

we

logical refer,

and t o

analyses Csee

used

[19671),

expressions

in in

the

hand,

meaning,

truth,

with

the

as

the

is

expressions,

functioning

of

denotation, 132

and

abstract kinds

different

to

of

ontological

instruments

becaiise

of

in

semiotics

concrete

linguistic

Instance,

connected

contexts

reference,

so

functional is

are two

convenient

footing, For

typos These

two

This

equal

which

concrete

to

objects.

definite

interested other

an

aJ^stract

concerned

J.Pelc

analysis. on

objects,

In t h e

approach with

pragmatic

to the

situations,

i.e.,

expression

such

semlotlc

communication

language use

of

we

are

tokens.

On

concepts

as

process

is

DUAL THEORIES

133 (ZS)

based

on

t.he

linguistic

concept, of

product.

equlphonlc

expression

which

expression

is

the

tokens.

t.ype

class Such

of

expressions

clearly, in the contemporary analysis. The of

graphic

can

always

Many

labels,

words,

context

that

authors

do

expressions, they

refer

to

interpretation

of

more

whether

explain though

them

as

or

with

as

an

less

language

expression

understood

not

even

or

concerned

abstraction of expression tokens determined equiformity.

equiform

seen,

literature

are

all

abs'tract

natural of all. Such

be

doubt, however, arises languages

of

an

expression types seems to be the most interpretation

, i.e., an

types

classes

of

by the relation of how

it

they

interpret

follows

types

from

the

without,

however,

"label",

"word",

making use of such concepts as equiformity. This

duality

In

"expression" does not language.

Theoretical

literature

of

take

account

into

the

using impose

the any

terms

definite

constructions

subject that

and

logical conception

encountered

pertaining

aspect

of

to

language

in

language which

of the

usually

makes

It

possible to speak about its expressions as types. In the preceding three chapters we have been interested in a different point of view. We have been concerned with the formal determination and

of

expression

components concrete

of

the

properties

tokens

of

languages,

objects,

that

of

label tokens, word

graphic

languages.

being

perceivable

is, tokens.

The

These by

theories

tokens,

syntactic sight

TLTk,

are TETk,

TSCL, TSC««r-L thus describe language as a product of a different kind, as a language of expression tokens.

134

CHAPTER IV

The t.he

double

double

analysis

ont.ologlcal

way

of

speak,

chaj>act.ei>

making

use

however,

of

If

not. want, t-o b r e a k

one

does

should of

language

not. c o n f i n e

expression

tokens

only.

single

theoretical

have

Both

precedents

also

in

the

Jerzy

as

of

the

whose the

object

indirectly,

as of

be

can

t.he

be

and

words,

by

G.Bryll

chapter

we

inquiry

they

said

outline

is

that

take

into

biaspectual

S.Miklos

in

several the

of

namely

the

TLTp,

four TETp,

extension

of

preceding

chapters.

definitional theory the of the

the

account,

approach

DTSCL,

Thus,

directly

to

of

and

TSCLj

and

of or

syntactically dual.

in

DTSC**-L,

theory

this is

discussed

in

chapter,

definitional one

of

the

theory

TLTp of

label

types

the

theory

TLTk

label

tokens;

the

definitional

extension

of

; the

theory

types

expression

categorial

analysed

theories

the

expression

t h e o r y TETk o f

theory

theories

corresponding

extension

TETp of

simple

dual

by

119771;

description

c a t e g o r i a l l a n g u a g e s . T h e s e t h e o r i e s will b e t e r m e d Each

inspired

axiomatic.

shall

feature

a

algorithms

abstract

and

in

We

and

presented

expression

attained.

of

concrete

one

language

reflected

of

specific

the

should

aspect.s.

pract-lce

119541 t h e o r y

and

of

semiot.ic

language

goal

t-heoretlcal

double

language

language That

it.s

and

llnsuist.ic

such

A.A.Markov's

present

principal

from

obJect,s

log^ical

of

the

t h e l a t t e r , unlike Markov's t h e o r y , i s In

In

explain

away

conception.

theory

Slupecki

could

or

auspects

in

Ihem

t.o t . r e a t . l n g

only

Un(;uist.ic

favour

which

oneself

types

of

in

concept.lons

of

is

a

tokens

languages

is

the

theory

dual

a

of

dxial

definitional DTSC«»-L

is

DTSCL

extension of

a

of

categorial

DUAL THEORIES

135

ui-lang'u.ages The

i s a d e f i n i t . i o n a l e x t ^ e n s i o n o f t h e ' t h e o r y TSC«»-L.

definitions

which

are

added

TSCL a n d TSC«»-L t o p r o d u c e t h e It

possible

to

characterized

look by

at

to

theories

corresponding

language

the

the

not

ordered

TLTk,

TETk,

duatl t h e o r i e s

as

a

sixtuple

product

Csee

make

earlier

Sec.I.l

and

Sec.II.l> < Lb;

<£->

V, W; %, C; S >,

but as a product whose syntactic characteristic is described by the ordered sixtuple < IE;

C£->

consisting

of

vocabulary

V of

the

the

relation

The

sets

equiformity of

equiformity is

holds

the

etfuali

Lb,

of

V,

Lh

types,

of

a.X.X label

tfie

ty

and

W,

^/-^> the

in

the

S

are

'•'/-o.'

set

W of

Lb, set

the S

abstraction

any

Lb,

of

label

three-au>gument

V,

W, of

Cequlvalence by

relation

label

t.hat

element

%, d e t e r m i n e d ordinary

defined

^Z—*

sets

%. H e n c e

between

abstraction a

of

Lb,

Lb^^^

quotients

•

iiiord

set

types,

all

word

types,

relation

of

the

c

all

of

ttell-formed

types.

families

class

unlversad

in

expression

W; -, C; S >,

simple "

concatenation

V,

a

types, tokens.

Cternaur-y?

respective

is

as

S

by

each

class>

given

of

as

The

is,

relation

relation

the

relation

of

the

of

sets

The

equality

of

is

a of

relation sets.

classes

concatenation holds

of

relation

token.

between

which

respective

those

extensional that

the

of

label

quotient

between

It of

C

is

label

136

CHAPTER IV

types. The

concepts

respectively, defined

to

Lb, the

V,

W,

concepts

and

Lb,

c,

V,

in TLTp CSec.IV.2>. We s h a l l

which

W, and

correspond,

C in

demonstrate

TLTk.,

that

In

one can r e p r o d u c e - If a c e r t a i n a s s u m p t i o n i s made axioms a

of

Tarski's

concept

which

C1933] m e t a s c i e n c e

is

defined

<Sec.IV.3>.

separately

in DTSC«»-L CSec.IV.6). The l a s t

in

The

are TLTp

all

set

the S

DTSCL <Sec.IV.5>

two dual t h e o r i e s

are

the latter primitive the

by t h e adding t o i t and s e c o n d a r y

definition

that

set.

followinif

of

We

here

concepts: t h e

s e t I of a l l abatTCLct indices

of

expression

vord

/ , the It

is

set

above

all

Z

btisxc

of

indices, the

function

that

the

quotients

%, and t h a t

determined

by

the

t,

sets of

basic

the

definitions

relation E of

L of of

function

T , o

/, I

E ,E

t/ie

forming

compound

types. of

by

classes

These the

p , and t h e

are

the

indicating

the

of

set

set

defined

as

€. the

relation

t h e a r g u m e n t s of t h e r e l a t i o n appropriate

of

of

the

concepts

the

introduce

indices,

expression the

from

properties

aJbstTtxct

p all

/ ,

to

the

t h e r»la.tio7%

on

of all analoga of

TETk, needed

determine

and t h e set

/, the

corresponding

are

of

analoga, r e s p e c t i v e l y , of

natural

equlformity

set

types,

types,

concepts a r e

concepts

S and t o

meaun

of d e f i n i t i o n s

and

based

TETp CSec.IV.4), which i s t u r n based on TETk and o b t a i n e d

is

t and

abstraction

of p of

label t o k e n s . Generally

speaking,

te, each p r i m i t i v e definitional

or

extension

except defined yields

for

the

concept

concept

of

one

the

of

of

a given dual

the

relation

theory,

whose

theories

under

DUAL THEORIES

137

consideration, concept the

in

that,

d e n o t e d by t h e

term

line

has

denoting

placed

given

the

aJ>ove

dual

corresponding

theory

Its

t e r m which d i f f e r s concept

that

theory

dual

that

Each

such

be

said

to

Cterm)

of

to

theory

term.

will

concept

in

as

analogon its

be

the

the

shape

by t h e

concept

In

from

horizontal

tterm)

indvcod

of

by

corresponding

a the

original

theory. In next

the to

dual the

described consider

simple

by

the

the

described

theory

categorial

system

simple

by

DTSCL, w h i c h

the

is

superstructured

language

CJf > C s e e

categorial

X

Sec.II.1),

language

following

of

system

on

expression we

J? o f

will

TSCL., tokens,

be

able

expression

consisting

to

types,

of

concepts

an

extension

i n d u c e d b y t h e c o n c e p t s o f Cjf ): CJf >

< Lb;

V, V; - ; c ; Z , 7 ; I , p ; E, S , C t C S ) >.

c

o

Likewise,

in

the

dual

theory

DTSC«*-L,

o f TSC«»-L, n e x t t o t h e (oi-Jf c ) <see Sec.III.l>,

w-languaige we shall

categorial

£,

following of

o>-language system

consisting

d e s c r i b e d by be concerned

described of

is

the system with the

syntactically

concepts

Induced

by

by t h e

the

concepts

Cco-Jf >: c

Cco-^ > < i 6 ; V, VT,

In

every

properties

O, W; - ; C ; 7 , 7 ; 7 , p , C / v ) ; £ , S,

dual

of

the

theory

serving

as

its

the

concepts

initial theory pertaining and

£

which

theorems

induced

by

which the

establish

concepts

to the languages of expression

foundation, will hold. Those

be syntactical analoga

CTCS:».

of

the

corresponding

of

the the

tokens

theorems

will

axioms, definitions

138

CHAPTER IV

and

theorems

by

the

Initial

of

the

initial

appropriate theory.

concept

of

an

What

definitions,

does

mean?

it

expression

expression

called

aiicioms,

defined in a r i g o r o u s given

theory,

induced

a

mainner. B u t t o

and

the

and

For

by

expressions theorems

every given

TSCL, TSC«»-L, and

Let let

T be

DT be

etny

induced

of

the

by

it

dual

T.

tf>^n (p is

theory

the

can

be

theory

it

a

suffices

rigorous:

theories

the

the

analogy b e t w e e n

t o a d o p t a g e n e r a l d e f i n i t i o n , which i s n o t v e r y DEFINITION IV.
dual

of

expression

show t h e

expression

induced

TLTk,

obtained

TETk,

from

the

latter. 1/

induced

an expression

in

by 4> ^f caxd- only

replacement

in

Ca) of

the

Cb) of

other

the

latter

term

% by

if

(p is

obtained

in every the

constant

DT a n

from

expression the

correct

place

symbol terms

in

", of

T by

the

terms

induced

by

them, Cc> of by

the

the

variables

Cd> of variables An dual

letter

the

ranging,

theory

a

over

ranging the

expression version

ranging

respectively,

variables

ranging induced

variables

family has

which

t h e e x p r e s s i o n by w h i c h i t i s Before

we

proceed

to

DTSCL, DTSCur-L we a d o p t

in is

the

sets

Lb,

over

the

sets

Lb,

the

family

2

J, I,

E E,

by

the

in

the

2 its quite

verbal

formulation

similar

to

the

version

of

induced.

outline the

over

over

the

dual

convention,

as

theories in c a s e

TLTp, of

TETp,

previous

DUAL THEORIES

139

c h a p t ^ e r s , s t - a t - i n ^ t.ttat. C o n v e n t i o n whose

numbers

are

IV.l.

marked

m e t a t h e o r y of a given In

constructing

theory

we

dvial

The

concepts

Caccepted

asterisk,

are

sentences)

theses

of

the

as

the

an

extension

conventions

of

a

applied

given

at

the

theory.

T h e t h e o r y TLTp o f l a b e l t y p e s

theory

TLTk o f

an

theory

observe

p r e s e n t a t i o n of t h a t given

S e c . IV.2.

with

theses

theory.

a

shall

The

TLTp o f

label t o k e n s induced

label

types

by a d d i n g

by

the

to

is

obtained

the

primitive

former

and

from

the

theory

definitions

of

the

concepts

of

the

derived

latter. THEOREM IV.l . a th»sis

of

This this It

Every

thesis

Caccepted

sentenced

of

TLTk

is

occasions

later

in

of

TLTp.

theorem

will b e

referred

to

on

many

of

TLTk

chapter. follows

from

equiformity

is

accordingly

produce

relation.

The

determined will b e

an

by

Axioms

equivalence classes

class the

denoted

by

/L6/_ C t h e q u o t i e n t

I.la-c

of

relation of

that

the

relation

the

set

Lb.

in

abstraction

abstraction

of

relative the

We to

token

p (representative

of

F f

1 • A^

^^

^^

quotient

i6

by t h e

of

the

set

known,

the

relation

that

relation

label

that

*) is

can

%,

claiss), family the

set

140

of

CHAPTER IV

all

classes

represent.at.ives

of

abst.ract.lon

are

definite

label t y p e s Is defined DEFINITION

of

label

t-he

t.okens.

relat.lon The

%

set

whose

Lb

of

all

thus:

IV.l

,

Lb m Lb/^

w h i c h c a n b e r e p l a c e d by CONCLUSION

IV.l

p e I F «• V p _ [ p ] . A label determined,

type is

the

class

by a certain

of

label

abstrac

tion

of

the

relation

%.

token.

We adopt the following convention Csee Convention I.la>: C o n v e n t i o n p, <j, T, s,

IV.2a.

The symbols

t, %i, V, with or without subscripts,

range over the set Lb. The

expressions

respectively,

that

induced

the

relation

by

Axioms

of

equality

I.la-c "

is

state, reflexive,

symmetric, and transitive in Lb. W e accordingly have THEOREM TLTk are The

IV.2 . theorems

The in

expressions

by

Axioms

I.la-c

of

TLTp.

three-argument

defined thus:

induced

relation

of

concatenation

in

Lb

is

DUAL THEORIES

141

DEFINITION

IV.2

c<:p,5,r> • • p ^ ^ J. *^P • J4 l<xb«l if

and

tho

tyf>& r

only i /

relatioTX

is

[p]

-^ 5 -

[Q] ^ r -

t/ie c o n c a t e n a t i o n of

t/ie t a & e l s p . g .

of

l<xbel

types

r a r e c l a s s e s of

% sucA. t/iat a ropresontative

concatsnation

[ r ] /v cCp,Q,r)).

rspr»ssntatives

of

of

abstraction

the

tA.e classes

p and q

class

o/

r is

p and

th.&

q.

D e f i n i t i o n IV.2 c a n b e r e p l a c e d by CONCLUSION

IV.2

p •« [ p ] A Q which

follows

[ q ] /s r •

from

t-his

[ r ] "• tcCp,^,F> *^ c
definition

aund

Theorem

I.l

and

Axiom

1.7. It can be d e m o n s t r a t e d t h a t c i s a two-argument

operation

in

Lb. In f a c t , t h e e x p r e s s i o n i n d u c e d by Axiom 1.2, namely THEOREM

IV.l

V

c(p,Q,r>

r i s a t h e o r e m i n TLTp. If

p

and

Conclusion p " [Pil there is

and

Convention

tokens

it

any

have

if

we

that

there

label

label

C Cp,qi, fr T >,

IV.2a, assume

follows

are

types,

then

it

follows

from

p^ and a. s u c h t h a t 1 1 ^^'i 9 " E'^il' ^^ a l s o f o l l o w s f r o m Axiom 1.2 t h a t a laJ^el t o k e n r^ s u c h t h a t CCp. ,q, , r . ) . By C o n c l u s i o n 1 1 1 1

we

then

are

IV.l t h a t

IV.2

Further,

<j

from

Pi''*!'*!'*"!'*!'^!

we that

tokens

from

which,

by

Conclusion

immediately

arrive

at

Theorem

CCp,q,t>,

Definition ="*=*^

cCr,s,Ti5

IV.2

*'^^*-

that ^[^i]'

and

p"r

there

IV.l IV.l.

and <j™s, are

^ [^^i]'

label ^"['i]'

142

CHAPTER IV

r»[rj, q s» .

^-[sj], This

^"["j]'

and

Axiom

cCp^,<j^,r^>, 1.4

yield

c,

that.

t Jteu ,

Pj^^^i

which

in

^^'^ turn

y i e l d s t h e e q u a l i t y t"H. Hence we h a v e THEOREM

IV.2

C
is

induced

by

Axiom

1.4.

From

Theorems

"^i, IV.1

and

IV.2

we

at

CONCLUSION IV.2a

c: IbxLb —+ TS. The relation

c is

a two-argwnent

operation

in t6.

The v a l u e o f t h e f u n c t i o n c f o r t h e a r g u m e n t s p and q c a n be s y m b o l i z e d by p <j by a d o p t i n g <•>

that

r « p qi «•

c.

W e shall however abandon here this notation for the value of the function C because our intention is to demonstrate that the expressions

induced

by

the

axioms

and

theorems

of

TLTk

are

theorems in TLTp. In view of Definition IV.2 and

the properties of the classes

of abstraction

[p] - [9] •• P == 9 we easily arrive at the following THEOREM of

TUTk are The

Expressions

IV.2 b. theorems

theorem

in

which

induced

by

Axioms

1.3,

I.

6-1.8

can,

with

TLTp. is

induced

by

Axiom

1.5

DUAL THEORIES

143

r e f e r e n c e t o t h e f o r m u l a <*>, b e r e c o r d e d i n t h e s i m p l e r

form

Cp qf> r • p <<j r ) . This

theorem

states

that

the

relation

of

concatenation

a n a s s o c i a t i v e o p e r a t i o n i n t h e s e t Lb o f a l l l a b e l The vocabulary DEFINITION

V of

simple

word

types

c

is

types.

is defined

thus:

IV.3

This d e f i n i t i o n c a n be r e p l a c e d by CONCLUSION

IV.3

P ^^ p

is

abstrac

a

simple

tion

cer tain

uwrd

of

simple

^ p^ir^'

type

the

relation

word

token.

if

W

and

of

only

if

etfui fortuity

p

is

a

class

determined

of by

a

C o n c l u s i o n s IV.l a n d IV.3 a n d Axiom 1.9 y i e l d THEOREM

IV.3

0 * V S which

is

an

easily

obtain

expression

THEOREM I V . 2 C. are

theorems The s e t

in

induced

Expressions

by

Lb, Axiom

induced

IV.4

of

Axioms

TLTp.

W of all word t y p e s i s defined

DEFINITION

by

1.9

thus:

TLTk.

I.10

We

and

also

1.11

144

CHAPTER IV

which c a n b e r e p l a c e d by CONCLOSION I V . 4

of

p is

a word

the

relation

t y p e if of

and only

if

eQvifoTmity

"p is

a class

d&tsrmined

of

by

a

aJbstTax:tion certain

laord

t oken. We

adopt,

a

convention

which

Is

analogical

to

Convention

lib. C o n v e n t i o n

IV.2b.

The l e t t e r

X, w i t h o r w i t h o u t

subscript,

r a n g e s o v e r t h e family 2 The

expression

Induced

by

Definition

I.la

Is

the

definition

o f t h e e x p r e s s i o n XCX>: DEFINITION

IV.5

XCX> **^-^'^_ClBy

applying

Conclusion

we c a n d e m o n s t r a t e THEOKEM

*^^'^ € X ^ CCp,q,r) ^ F e X>. IV.4,

Definitions

I.la,

I.t

and

IV.S

that

IV.4

i? - n < X I xcxy >. This

theorem

somewhat difficult It then are

follows there then

is

is

induced

by D e f i n i t i o n

I.l

of

the

set

W. The

p r o o f o f t h i s t h e o r e m I s g i v e n i n t h e Annex.

from

Conclusions

a n r eWW

word t o k e n s

p

such and tj

IV.4 that

and

IV.3

that

r " fr T. By Axiom

such

that

If 1.12

rePsK, there

cCp ,q ,r >. Hence,

In

DUAL THEORIES

view

145

of

Conclusions

C^C fp 1 , [qf 1 , r ) , have t h u s

so

IV.2

that

and

we

can

that

[.**ll

^

^

^'^'^

^

C.

We

a. pair

of

TLTk

also

IV.5

? e W\V ••

xaord

state

[p/]'

proved

THEOREM

Ev»ry

IV.4,

compound

word

typ&

_ ^ is

C.

the

concatsTUition

of

types.

T h e o r e m IV.S I s induced by Axiom 1.12. The f o l l o w i n g

t h e o r e m i n d u c e d by t h e

last

axiom o f

holds: THEOREM

IV.6

r e IK ^ CCp,«j,r> ^ p,q « K. / / a word those To view

labels

are

prove

this

of

Since

type

is word

by

and

aissumption

IV.2,

that

1.13,

that

are

p

cCp ,<j ,r >. p ,q e y .

two

label

we

note

there

is

an

r eV

IV.2a,

it

follows

Convention

there

of

types,

then

types.

IV.4

IV.l t h a t the

concatenation

theorem

Conclusion

p,tfsLb

the

In

amd Q

that

such t h a t

ret** by

that

c"Cp,5>?^>

Since

r elf'

we

accordance

with

such

In

^"F^JI-

Conclusion

and q«« [ q . ] . T h i s

yield,

find,

that

from

^ [p.]

stating

assumption.

in

Conclusion

by

Conclusion

view

of

IV.4

we

Axiom have

p,5eS'. The c o n c e p t is

induced

by

of the

the

set

concept

"V of of

word t o k e n s . I t i s d e f i n e d t h u s :

the

all set

n-co»ponential of

all

word

types

n-component ial

146

CHAPTER IV DEFIKITION

IV.6

which y i e l d s CONCLUSION

IV.5

p ^""V ^

P - [p].

\ pe V

The

following

expressions

D e f i n i t i o n I.2a,b o f t h e s e t THEOREM

^V m V,

b.

F e ""^^F «•

IV.3,

easy

IV.l

Induced

by

the

inductive

"v;

IV.7

a.

The

are

proof

and

IV.2

V

of

V

Theorem

and

on

cCp,g,F>. IV.7,

based

Definitions

on

Conclusions

I.2a,b,

is

left

to

p will n o t

be

symbolized

IV.S, the

Reader. The l e n g t h \p\,

as

concept by

I p I.

of

could

a given

be

induced

by

Definition

word

excepted a of

given

in

type view

concept,

that

concept

of

the

definition

formulated is

in

induced

by

of

the

Sec.IV.l,

but

by

Definition

1.3: DEFINITION

IV.7

p e

W •* i.\p\

m n ^

p G

V:>.

The g e n e r a l i z e d d e f i n i t i o n o f t h e r e l a t i o n o f concatenation —h c o f fc+1 a r g u m e n t s Ck>2>, which h o l d s b e t w e e n l a b e l t y p e s , i s defined thus:

DUAL THEORIES

147

DEFINITION

The

following

Definition holding

IV.8

two

I.4a,b

between

THEOREM

of

-theorems the

label

are

Induced

generalized

relation

b.

n > 1 =*
Theorem

IV.8a

<^Pi'P2'' ' ^ n + l ' ^ " * **

Convention

Note

follows

Theorem

Axiom

IV.Sb

IV.2a,

directly follows

Conclusion

also

that

Definition

Definitions

Definitions

Definition

IV.8,

IV.8

IV.2,

IV.8 a n d C o n c l u s i o n

[p^]

^P2

_jl[ _

_

-

[P2]

-

_

1.4a

and

I.4b,

Theorem

1.1

conclusion have

1.11

yield

can

replace

demonstrated induced

Csee

Theorems

IV.6).

Moreover,

by

an

^ Pk"

^ P '

[P]

^

<:Pj,P2>-.Pj|j.P»

Definition

IV.8.

that

in

the

axiom

of

TLTk

IV.2*a,IV.l,

[^ft]

j^

=* Pj|j.P> •> c

expression

from

from

IV.l,

cc5.P„+j.P>»

IV.6

ft > 1 ^ p^ -

We

'^

I.la.

CONCLUSION

This

concatenation

tokens:

** % <^"P2'-'Pn'^^

and

of

Inductive

—n+1 — — — _ — CO Cp^,P2>P> ** <^P2'P^^>

n « 1 ^

IV.2.

the

IV.8

a.

and

by

IV.2,

dual is

IV.2*b,

theory a

TLTp

theorem

IV.3,

in

IV.2*c,

every TLTp IV.5,

148

CHAPTER IV

analogon

In

expression Definition

the

form

being

a

IV.5,

of

an

expression

definition

Theorems

or

IV.4

a

and

induced

by

theorem

in

IV.ra.b,

it.,

such

TLTp

Definition

an Csee

IV.7

and

T h e o r e m IV.8J. T h u s w© o b t a i n THEOREM de/init

is

IV.2 .

ion

in

fTvery

TLTk i s a thesis

This t h e o r e m

implies

THEOREM I V . 3 .

Every

a thesis

Sec.

expression

in

induced

in

by

an

axiom

or

a

TLTp.

expression

induced

by

a

thesis

in

TLTk

TLTp.

IV.3.

Interpretation

i n TLTp o f T a r s k i ' s

axioms

of m e t a s c i e n c e

A.Tarski suffices

to

found

expressions primitive "sign

of

of

for

the

of

in,

of

consisting

system

of

expressions

Cand

concrete

following

the

"sign

of

logical

inclusion", of

two used,

sum", of

axioms

x,y,z,t

sequences

of

which

of

are

^UfW

the those

of

which

signs

and

are

the

those

axioms: "universal

fc-th

expressions

abbreviations the

of

classes of

respectively,

variables

labels

of

"variable

successive

axioms

names

algebra notation

o n e . In r e c o r d i n g the

specific

the

Tarski

that

The

of

in

v , a n d x'-y, a s

last

of

language

"sign

assumed

classes

a

occurring

"expression". qu,

also

the

negation",

"expression

sm,

gave

metascience.

terms

quantifier",

and

C1933]

shape", x

and

symbols:

y", ng,

terms

except

metascience

Tarski

range

expressions) equiform

over

the

set

treated

as

with

a

given

DUAL THEORIES

label,

149

and

that

the

vax-iable

X ranges

over

the

class

of

such

expressions;

no

two

expressions. We q u o t e h e r e T a r s k i ' s a x i o m s o f AXIOM

1.

ng,

sm,

(fu

amd

in

expressions out of t h e s e four are AXIOM 2.

v

is

differs

if

x^y

is

an

from

A is

a

natural

ng,

sm,
and

y

i f At x 1.

expression

e x p r e s s i o n s ; x'^y d i f f e r s

and only i f

from t h e expressions

and i n and f r o m e v e r y e x p r e s s i o n v AXIOM 3.

aire

identical.

an e x p r e s s i o n

number o t h e r t h a n 0; v

metascience:

the

if

and

only

expressions

if

ng,

x sm,

are

QU and

in

x^yas'^t

if

and f r o m e v e r y e x p r e s s i o n w. . AXIOM and

4.

only

CcO x"« and

If if

x,y,2

one

and y " t ;

t"u^y;

and

of

(

the

t h e r e

are

expressions,

following is

conditions

an e x p r e s s i o n

Cj') t h e r e

Is

an

S Cprinciple

of

mathematical

then

expression

u

is

u such

such

satisfied: that

that

x"«'-u

z"x^u

and

yM-U*^ t.

AXIOM satisfies in^X-f

the

If

following Ac i s

a

conditions:

natural

induction). Ca>

number

ngeX,

other

If

a

smeX,

than

O,

class

<}ueX then

X and

v.eX; AE

Cj') i f

xeX and y^X, t h e n x ' ^ e X , t h e n e v e r y

expression

is

in

the

c l a s s X. Tarski's G.Bryll

and

latter

by

that the

axioms

of

S.Miklos the

thesis

terms

metascience

119771) ng,

sm,

by

can

enriching


in,

-u

be

found

the CfceW)

in

language and

by

TLTp

Ccf.

of

the

assuming

150

CHAPTER IV

o

which

states

infinite this

t-hat,

the

enumerable

thesis

if

we

vocabulary

set,

is

postulate

valid in

of

simple

in

TUTp.

TLTk

that

word We

types

can

there

is

an

arrive

at

are

infinitely

e n u m e r a b l y many d i v e r s i f o r m s i m p l e word t o k e n s . In

such

Tarski's

a

we

axiom

ing,sm,(fVi,in,v x,y,z,t

case

,u,

variable

obtain

system

,v ,...y^y

and

respectively X

expression

by

the

r"p tj,

i.e.,

by

interpretation

given

above

by

by

variable

an

X,

the

expression

vai^iables

p,q,r,t

expression c

of

setting

the

variables

the

TLTp

by

replacing

the

in

,u,

z«x^y Csee

the

by

the

formula

>, and t h e t e r m " e x p r e s s i o n " by t h e t e r m "word t y p e " . Since,

by

expressions

Theorem

Induced

by

IV.3 ,

Conclusion

I.4a a r e v a l i d i n TLTp, we a r r i v e CONCLUSION

IV.7

CONCLUSION

IV.8

c<.p,Q,r> CONCLUSION

can

I.3a

and

which

Theorems

1.6

are and

at

IV.9

eeusily

find

that

_ ^ CCp,q,F). p,<j«el? for

the

T a r s k i ' s Axioms 1-3 b e c o m e t h e s e s i n TLTp. The

theorems

=» p,qi e >»0,

F € !?v7 •• We

those

expression

above

interpretation

DUAL THEORIES

151

CCp,qi,s> yN c C r , t , v > V

Is

also

a

i * C s " v * » p « r A < j "

t

V V c^Cp,t/,r> ^

theorem

in

TLTp,

Induced

by

v cC«,t,q)>>

Theorem

1,2.

From

this

e x p r e s s i o n , by C o n c l u s i o n IV.8, we aarrlve at. CONCLUSION IV.IO

p,tf,T,t

e h' -^ CCp,qfS)

N/

The <*>, i s

^

/N CCr,t,v'>

^ C s « i i * » p " r y s q »

CcCr,u,p> ^ c'Cu,5>'t'» V

above

conclusion,

an e x p r e s s i o n

after

into

V

the

t v

Cc .«> c'.

application

which T a r s k l ' s

of

Axiom 4

the

formvila

turns

for

the

i n t e r p r e t a t i o n given above. We h a v e above, fact

to

Tarski's because,

demonstrate

Axiom 5 b e c o m e s by

C o n c l u s i o n 1.3c, t h a t CONCLUSION

that,

Theorem

a

IV.3 ,

for

the

theorem the

interpretation in

TLTp. I t

conclusion

is

given so

induced

In by

is

IV.11

K S X .-N _ A ^ ^ F e )?> =• 5? £ X, i s a t h e s i s i n TLTp.

S e c . IV.4.

The the

T h e t h e o r y TETp o f e x p r e s s i o n t y p e s

theory

theory

TETp

TETk o f

of

expression

TLTk o f l a b e l t o k e n s . We n o t e

expression

accordingly

types

tokens,

is

and

superstructured hence

on t h e

on

theory

1S2

CHAPTER rv

THEOREM and

in

Every

IV.4 .

part

icular

of

thesis

(accepted

TLTk, is

sentence)

a thesis

in

of

TETk,

TETp.

Now TETp is obtained from TETk by the joining to the latter of

all concepts

of

TETk.

concepts

The

induced

by

concepts

primitive

induced

of TLTk, which

been defined in TLTp

the

is a

by

the

and

derived aand

derived

TETk, have

already

primitive

fragment

of

concepts

CSec.IV.2>. It can thus be seen that

TETp

is based on TLTp and the following theorem holds: THEOREM a thesis

Every

IV.5 .

in

task

those

(accepted

sentence)

of

TLTp

is

induced

by

TETp.

Theorems IV.4 Our

thesis

and IV.S

now

primitive

reduces and

concepts in TLTk

will hereafter be used tacitly. to

derived

and

to

defining

the

concepts

of

formulating

concepts TETk

the

which

are

basic theorems

not which

describe the properties of the concepts so introduced. The concept

of

the

set

I

of

all

basic

abstrac

t

indices

-

o induced

by

the

defined

thus:

DEFINITION

concept

of

the

set

/

of

all

basic

indices

-

is

IV.9

o

o' %

w h i c h c a n b e r e p l a c e d by CONCLUSION

IV.12

^ ^ ^o •* '^(^I o ^ ' A

basic

consisting

abstract of

all

indices

index

is

equiform

W

the

class

with

a certain

of

abstraction basic

index.

DUAL THEORIES

The

153

set

T of

all

abstract

indices

-

the

induced

by t.he

concept, o f t h e s e t / o f a l l i n d i c e s - i s d e f i n e d t h u s : DEriNlTlON IV.IO

^ - v^' which c a n b e r e p l a c e d by CONCLUSION

IV.13

An abstract all

indices

index eqtiiform

is

the

with

class

of

abstraction

a certain

consisting

of

index.

By making u s e o f DEFINITION

IV.11

K'^ • • 7

S X ^

A

tp^q e X ^ cCp,q,F> ^ r e X),

which i s induced by D e f i n i t i o n II.la, we c a n p r o v e THEOREM I V . 9

7 - n < X I x:cx>'* >, induced

by

Definition

II.1

of

the

theory

TETk.

Its

proof,

a n a l o g i c a l t o t h e p r o o f o f T h e o r e m IV.4, i s o m i t t e d . The b a s i c p r o p e r t i e s

of the s e t s

7

and 7 a r e e s t a b l i s h e d by

t h e o r e m s induced by Axioms II.1 and II.3-II.5; THEOREM IV.IO

0 ?< 7

s o

Tb, '

THEOREM IV.ll

CCp,9,r> -» r ^ 7 ; o

154

CHAPTER IV THEOREM

IV.12

r e 7sT THEOREM

•*

V

cCp,q,F>j

IV.13

r e / ^ C •» p,<j s Theorem Axiom

II.l

Deflnit-ion

and

IV.12

and

follows

lmmecUat.eIy

Conclusion

IV.2,

Theorem II.4

IV.IO

Axioms

follows

II.3

from

Definition

IV.2.

IV.13,

Definition

Conclusion

IV.l.

7.

from

Theorem

and

Conclusion IV.ll

follows

from IV.12.

11.2

and

Conclusion

Conclusions

IV.13

and

Theorem

IV.13

IV.2,

follows

Axioms

1.7

IV.12,

IV.12,

Axiom

directly

from

and

II.5,

and

C o n c l u s i o n IV.13 a g a i n . We s e i f - e v i d e n t l y h a v e t h e '^

formula

o

o

which I s i n d u c e d by Axiom 11.2 o f TETk. The indices

concept CT\>1)

of -

the

set

induced

I by

of

the

all

n—component

concept

of

the

ictl set

defined thus: DEFINXTION

IV.12

"7 . "//^ This d e f i n i t i o n i s e q u i v a l e n t CONCLUSION

to

IV.14

p e '^r •»

V pel

p "

[p].

The f o l l o w i n g t h e o r e m s a r e induced by D e f i n i t i o n s II.2a,b:

abstract /

-

is

DUAL THEORIES

155

THEOREM

IV.14

a.

*r « 7 ,

b.

o F € "**7

«•

V pe T

V tf^T

c(p,i,F>. o

The

easy

and

proofs

IV.12,

of t>hese theorems, based

Definit-ions

II.2a,b,

Conclusions

on Conclusions IV.14 IV.2

and

IV.1,

and

Convention IV.2a, are omlt-t-ed. Note

now

Theorems

that

Remarks

IV.2*a,b,

IV.l

II.2

and

and

IV.2,

II.3,

Definition

Definition

IV.ll,

IV.Ca>,

Theorems

IV.8-IV.14, and formula <»••> yield THEOREM of

all

The

IV.6 .

axioms

and

expressions

de/ini

tions

induced adopted

by in

the

TLTk

dual

analoga

are

theses

by

the

in

TETp.'^*" Theorem IV.6 THEOREM analogon

of

implies

IV.7 .

Every

a thesis

in

expression

TLTk is

In view of Theorem IV.7 expression

induced

a thesis

in TETp.

we can see, for instance, that

I £/Si6, induced

by

dual

Conclusion

1.3 a

Cdual

the

analogon

of Conclusion I.3a>, is a theorem in TETp. Note further that the theorem induced by Axiom II.6 in TETk holds in TETp: THEOREM

IV.15

V n Jo • 0. The common

vocabulary with

the

of set

a of

simple all

basic

word

types

abstrac

t

has

no

indices.

elements

in

156

CHAPTER rv

Theorem

IV.15 follows

easily

from

Conclusions

IV.3 and IV.12

and Axioms 1.10 and 11,6. The relat.ion primit-lve

i

of lndicat.ing

concept

indicating

in TETk.

ttte indices

of

t,he indices

The concept, \eord types,

of word

of

t.okens

t-he r e l a t i o n

is a L of

induced by t h e former, i s

defined thus: DEFINITION 1V.13

'^^'5^ *• p€^ct> ' ^ " M -^ ^ - b^p^V Definition IV.13 and Axiom II.7 yield immediately CONCLUSION IV.15

p e Dcr> •• p ^ c o p - \p\' b.

5 € o'^co 4»

The

relation

y

i. has a s i t s domain

abstraction

of t h e r e l a t i o n

which

an index.

all

have

classes

of

?"['»]•

It

t h e s e t of all c l a s s e s

% determined

by t h o s e

word

has a s i t s counter-domain

abstraction

of

t h e relation

of

tokens

t h e s e t of

S: determined

by t h e

indices of word tokens. It II.2b IV.15a, from

follows that II.la

(.

from is

a

the

definition

function.

and IV.4 t h a t

Conclusions

IV.lSb,

Since

its

of it

domain Il.lb

L and from follows is

a

and

counter-domain i s a s u b s e t of I, we obtain THEOREM IV.16

I S W-X.I A 1. i s a function.

from subset

IV.13,

Conclusion Conclusions of that

W,

and its

DUAL THEORIES

157

T h i s t h e o r e m i s i n d u c e d b y Axiom II.7 o f TETk. On d e n o t i n g by i.C/>> t h e v a l u e o f domain

we

obtain,

a n d II.2b, t h e

by

Definition

L for

the argument

IV.13

and

p of

Conclusions

Its

IV.15a

following

THEOREM IV.16a

p "

[ p ] •^ P ^ DCty •* TCp) -

It is self-evident THEOREM TETk is

expression

p

of

forming

compound

expression

of

relation

are

and t h e elements

types.

In

p

the holds

word

p,

recording expression between

or:

p

is

a

definition

sequence

t h e binary

sequences

of

the

rP^> ^° of

II. 8

expression

, in t h a t

order.

tokens

of

of

of

the

p

of

domain

definite

word

a r e definite

word

read:

p

we

the

type

has

relation

relation

^^

words

compound

shall

relation

>

and

the

of

the

consisting

IV.14

n > 1 =» < p
- Pi - [',] from

^

t^,t^,."',t 0 1'

^D<.y

follows

this

definition

a

one-to-one

function.

IV.14,

Conclusion

IV.lSa,

Further

Axiom

II.9,

^h

-

['Q] -

' n

- P„ - C'n] - P - [F^'o'^i'

It is

Axiom

expression

i t s counter-domain

P^Pr.'P.t>--'P

the

w o r d t y p e s Pf.,p^,—,p DEFINITION

by

The e l e m e n t s

ordered

of

the

of

types.

finite

types,

use

compound

in t h e induced c o n c e p t

forming that

induced

in TETp.

operation

analogon

that The

a. ttteorem

The its

I V . 8 a.

[tCp>].

and it

''n^P

Conclusion follows

Conclusion

from IV.3

*

II.3

that

p

Definition and

Axiom

158

CHAPTER rv 00

1.10 t h a t a

that,

subset

of

function

has

DCL)W

as

the its

— k. U DCi.> as At-2

set

counter-domain.

its

This

domain

and

yields

the

f o l l o w i n g t h e o r e m , i n d u c e d b y Axiom II.9 o f TETk: THEOREM

IV.17

-

On

denoting

areuinent

by

-

*"

p:

>

of

its

-

—-.» D\V. * *

P^Pf.,P^,--,P

^Pf.,p^,--,P

-

fc

U DCO At-2

>

the

value

domain

we

of

p

obtain,

for

by

the

Definition

IV.14 a n d C o n c l u s i o n s IV.IS a n d I I . 3 , THEOREM IV.t7a

n > 1 ^ ' o - ' l - ' ' n « Ot^> - Po - Pn A word of

compound

relation

tokens' („,*.,,. ..,t 0 1 n %, represented words

t^.t^,...,t 0 1

=» P ^ P o ' ^ 1 ' • ' ^ n ^ "

expression

Pf.rP^>-tP

types

the

[*J

C'o] '^ ^1 -

type

-

consisting,

> determined

*, represented,

^<^'o'S'

by

classes

respectively,

, in n

that

expression order, that

token

^

'*n^] successively, of by

- is a class of abstraction

by compound

C'l] '^

abstraction indexed

of the

consisting

is of

of

the

word

relation of the

functor

t. 0

and its successive arguments t ,. ..,t 1 yx S e l f - e v i d e n t l y we h a v e THEOREM

IV.8 b .

TETk i s a theorem We

pass

concepts expression

of

now the types

r/ie

expression

induced

by

Axiom

II. 10

of

by

the

in TETp. to sets

definitions

of

E , E

E. T h e set

is defined

and thus;

concepts

induced E

of

all

simple

DUAL THEORIES DEFINITION

159 IV.15 ^s

•

^ s /' -% ^

This definit-ion c a n b e r e p l a c e d by CONCLUSION

IV.16

"^ For t-he s e t

E

of

all

s

compoMnd

expression

types

we h a v e

DEFINITION I V . 1 6

c

c'a

which c a n be r e p l a c e d by CONCLUSION

IV.17

^ ^ ^c •• p ^ "^

For t h e set DEFINITION

E of

all

expression

c

? types

W we h a v e

naturally

IV.17

f - E/^ , which cau^ be r e p l a c e d by CONCLUSION I V . 1 8

Conclusion

IV.16,

Definition

11.3a,

Conclusions

IV.3

and

IV.lSa, and Axiom 1.10 y i e l d THEOREM IV.lSa

E

m y n DCO.

s Conclusion

IV.17,

Definition

II.3b,

Axiom

IV.17a, C o n c l u s i o n IV.lSa and T h e o r e m IV.17 y i e l d

II.9,

Theorem

160

CHAPTCRIV THEOREM

IV.18b c

Finally,

by

Conclusion

IV.18,

Definition

II.3c

and

respectively,

by

C o n c l u s i o n s IV.16 and IV.17, we h a v e THEOREM IV.lSc E m E

Theorems Definitions We We

IV.18a,b,c

U E . c

induced,

II.3a,b,c.

now

shall

are

s

adopt

conventions

henceforth

II.3a-c, t h e

have

analogical to

to

observe,

Conventions

next

to

II.3a-G.

Conventions

following

C o n v e n t i o n

IV.3a.

p,(f,r,s,

The v a r i a b l e s

with or without

subscripts,

r a n g e o v e r t h e s e t E. b.

The v a r i a b l e s a.,h,c,d,^ ,a, w i t h o r w i t h o u t

range over the s e t c. .

The the

/.

LCp> m a

^

iCq'i

^ tCp.i

•

a.

,/^r^^

syntactic

category

formula DEFINITION

IV.18

subscripts,

•

6

^

tCr)

•

y. ( . C Q . >

«« 6 .

^

Ct=- with,

c

^

i.Cs> •

i-Cr.) «

the

index

d.

c.

^

f

LCS.>

is

>•

defined

d..

by

DUAL THEORIES

161

An expression abstract the

type

irxdex

relation

tohen

such

xehich

is

relation

that

is

and

% which it

in

has

is

% iithich the and

the

only

in

if

as

of

determines

syntactic

p is

its

the

a representative

From II.3a-c

Jf if

p

category

a class

of

tic

that

class

the

index

an

category of

the

abstraction

representative

syntac

vith

of expression

with

the

abstraction

index of

the

^.

above

definition.

Definition

IV.3a-c,

Conclusion

II.5b,

II.8,

Conventions

Theorem

IV.16a,

and

set

all

C o n c l u s i o n s IV.18 and IV.13 v/e o b t a i n THEOREM

IV.19

CT- The

syntactic

expression

category

types

whose

< p I 5 - f >. with

index

is

the

index

eQual

to

? the

is

the

index

of

f.

N o t e t h a t T h e o r e m IV.19 i s i n d u c e d by D e f i n i t i o n II.8. The

concept

Induced

by

the

family

c a t e g o r i e s of expression tokens i s t h a t syntactic

categories

DEFINITION

of

expression

of

types,

Ct

of

all

syntactic

t h e family defined a s

Ct

of

all

follows.

IV.19

CT - < C T - I f e T<E> >. K The

relation

•

of

syntactic

categorial

agreement

of

c

expression

tokens

has

its

analogon

in

the

relation

=

of

c

syntactic

categorial

agreement

of

expression

types

t h e f o l l o w i n g d e f i n i t i o n Induced by D e f i n i t i o n 11.10;

described

by

162

CHAPTER IV DEFINITION

IV.20

p = q «• V p , 5 e CT-.

r types

Two expression category

if

abstrac

t

they

?

p and q belong

belong

to

to a syntactic

the same

category

syntac

tic

the

same

with

index.

When formulating the theory TETk w e have adopted three other concepts: that of the set C

of

all constituents

of the

n-th

P

order- of an expression

token

constituents of an expression token p, <J^J2' C which

Jn^

of those

occupy

in

constituents

it

the

of the set C

p, that

and that

of the set

of aun expression

syntactic

of atU

positions

token

<7.,J,.,--,J > 1 2

Cn>0).

The c o n c e p t s

respectively,

of

those

by

induced

C—, C— and P P

concepts

DEFINITION t

by t h e s e

are

IV.21a e C- *^ ^"^ p t ,p

concepts

will

C— P

p

be

The

n

denoted,

definitions

a s follows.

Ct

"

[tl L J

^ p ^

-

[p'] L^J

^

t

G C ). p

ft"! ^ p \. i f

fpl \ri

^

t

e

C >. p

G

C

D E F I N I T I O N IV.21 7

DEFINITION

€

C- •• V p t,p

cT •

IV.22

*» ,rp ^' - [0 - p = w -

'^ ^ ' p

^

t

>.

P It

can be

II.12a-c,

proved,

Conclusions

by r e f e r e n c e IV.18

and

to IV.1,

Definitions Theorem

IV.21a a n d IV.17a

and

DUAL THEORIES

163

Conclusions

IV.lSa

and

II.16a,

that.

t.he

expression

induced

by

D e f i n l t - i o n s I H 2 a - c a r e t h e o r e m s i n TETp. H e n c e we h a v e THEOREM

IV.20

a.

_ _0 t e C— ^

p B

b.

^ ^ ^ " • * * _ _

t, ^ _

_ < p « p < 7 , 7 ,...,7 ) /v

- 0<)
A > 0 =»
«*

V

'^ The

following

"^ = ' ' / '

f e C - ).

reC—

expression

induced

by

Definition

II.12d

f o l l o w s i m m e d i a t e l y f r o m D e f i n i t i o n s IV.21, II.12d a n d IV.21a: THEOREM

IV.20d

7 e C - «* V T g c p n p By

making

use,

in

particular,

of

Definitions

IV.22

and

II.13a,b a n d T h e o r e m IV.17a we c a n p r o v e THEOREM IV.21

_ _0-,> _ _ « e C— ** p G £ and p c of p,

a. expansion

t

is

the

j^-th -'i

xaord

of ^

the

taord

of

the

-^A'-'2'•'-'fc+1^ b.

t

expansion The

€ C—

of last

^

an element

of

theorem

gives

the

t

is

set

C

the

expressions

J.

.-th for

Induced

k>0. by

Definition

II,13a,b. It TETk

is self-evident has

its

that

analogon

in

every an

axiom Induced

and e v e r y expression

definition which

is

of a

164

CHAPTER IV

t ^ h e s i s i n TETp. T h e f a c t II.l-II.lO the

of

basis

IV.15,

TETk of

IV.20,

and

IV.14a,b,

Il.la,

in

IV.17,

Theorems

Definitions

expressions TETp

IV.10-IV.13,

IV.8*a,

IV.9,

the

theorems

Theorems

IV.16,

Theorems

are

that,

II.l,

and

definition

IV.8 . of

Every

TETk is

T h e o r e m IV.8

It

in

follows a

the

Ct

is

family a

Every

logical

IV.ll, and

induced

II.8,

by

11.9,

11.10,

axiom

or

substantiates by

an

a

TETp.

expression

induced

by

a

thesis

of

TETk

TETp. from

particular,

Theorems

implies

THEOREM I V . 9 . i s a thesis

on

IV.19

are

induced

in

and

Definitions

II.3a-c,

expression a thesis

Axioms

Definition

IV.21a,b

II.2a,b,

by

established

C***>,

IV.19,

II.12a-d, a n d 11.13a,b, r e s p e c t i v e l y . T h i s THEOREM

be

IV.8*b.

IV.18a-c,

IV.20a-d

can

formula

and

induced

Theorem

theorem of

1V.9

induced

all

syntactic

classification

of

that

by

if

£i«!0 t h e n

Theorem

II.6.

categ^ories the

set

of

£,

we

It

have,

states

that

expression

determined

in

types by

the

e q u i v a l e n c e r e l a t i o n =. C

Sec.

IV.5.

T h e d u a l t l i e o r y DTSCL

The t h e o r y tokens

Is

latter

of

including expression

'

TSCL o f

obtadned one the

simple c a t e g o r i a l

from

the

theory

axiom

CAxiom

definition

of

tokens

of

a

given

TETk

11.11) the simple

languages by

and

set

S

the

of

expression

joining

to

the

several

definitions,

of

well-formed

categorial

aJ.1

language

£.

The

DUAL THEORIES

dual

-theory

language obtained all

165

DTSCL,

It

described

from

concepts

TSCL.

by

by

concepts

of

makes the

TSCL by j o i n i n g Induced

The

concepts

which

TETk

the

possible

system to

the

primitive

induced

have

It,

adready

by

CJf > latter and

the

been

t.o Csee the

on TSCL b u t o n TETp a s well. We a c c o r d i n g l y

and

IV.IO .

Evory

in partic-ular

thesis

of TETk,

in

definitions

of

concepts

of

and

derived

TETp. I t

follows

superstructured

have

(accepted

i s a thesis

is

primitive

defined

t.he

Sec.IV.l),

derived

t h e r e f o r e f r o m t h e a b o v e t h a t DTSCL i s n o t only

THEOREM

analyse

sentence)

of

TSCL.

in DTSCL.

and THEOREM IV.11 . a thesis

Every

thesis

Caccepted

sentence)

of

TETp

is

in DTSCL.

Ve shall h e n c e f o r t h

make t a c i t

references

to

Theorems

IV.IO

and IV.ll*. The definition

theory of

DTSCL

the

set

includes,

S of

all

above

^aell—formed

all,

the

expression

following types

of

t h e l a n g u a g e X: DEFINITION

IV.23

5 - S/^ , which c a n b e r e p l a c e d by CONCLUSION

IV.19

^ ^ ^ *• p ^ ^ A well-formed of

abstraction

expression consisting

type of

all

of

W the

language

expressions

Jf i s

equiform

a

class

with

a

166

CHAPTER IV

c e r t a i n w e l l - f o r m e d e x p r e s s i o n t-oken o f The Cn>0>

see -

a

Introduced quotient

S of

conceptin

of

all

well-formed, Induced

TSCL

-

the set

expression

by

is

£.

t-he

defined

types

concept-

as

a

S by t h e r e l a t i o n

of

of

order

n

t-he

set-

S

quotient

family:

of equiformlty.

the

H e n c e we

have D E F I N I T I O N IV.23a

T h i s d e f i n i t i o n c a n b e r e p l a c e d by t h e f o l l o w i n g C O N C L U S I O N IV.20

pe S By m a k i n g u s e o f t h e d e f i n i t i o n s of

all

to

inductive induced

demonstrate Definition

by t h e

that

II.4a,b

Definition

the

adopted above expressions

of

the

set

S

IV.4c

of

the

set

induced

and S

we h a v e

the are

first

by

the

expression theorems

in

DTSCL. THEOREM

rv.22

a.

S m E s

b.

p .

*^^S -

P - *S .

^V^ _ _ ^ _ P^,P^,...P^
^ S

Ca,aj,...,a^,aQ»,

oo

c.

S .

Theorem and

IV.16

IV.22a and

U "S.

follows

Definition

immediately II,4a.

The

from somewhat

Conclusions more

IV.20

difficult

DUAL THEORIES

proof

of

167

Theorem

follows

easily

IV.22b

from

is

given

in

Conclusion

t-he

Annex.

IV.19,

Theorem

Definition

IV.22c

II.4c

and

C o n c l u s i o n IV.20. In of

the

the

theory

TSCL we

condition

states

that

condition.

(5CX)

the

In

set

we

have

is

the

least

S

expression

in t h e

T h e o r e m II.2 i s valid. N o t e REMARK

IV.l.

Definition

II.5

In of

the

adopted

and

DTSCL t h e

c a n be s h o w n t h a t

have

latter

an

auxiliary

proved

Theorem

set

6<:X)

theory

which is

definition II.2

which

satisfies

that

induced

by

6CX).

It

a theorem analogical

to

that

DTSCL

an

condition

expression <5CX> i s

induced

the

by

the

of

the

definition

c o n d i t i o n <5tX>. The

following

theorem

induced

by

Axiom

11.11

is

also

a

t h e o r e m in DTSCL. THEOREM IV.23

o Theorem II.4a,

IV.23

follows

Conclusions

IV.19,

easily

from

Axiom

IV.20,

II.4a,

IV.12

11.11, and

Definition II.6b

and

T h e o r e m IV.lfia. The s e t of

all

of

the

functors defined

B of

functor set of thus:

all

basic

types B of £,

all

expression

types

of £ are concepts expressions

respectively.

These

of

of

J? and t h e

induced by t h e

J? and concepts

the

set

set

F

concepts F

of

self-evidently

all are

168

CHAPTER IV DEFINITION

IV.24

DEFINITION

IV.25

and

F- f4 . We u s e t h e r e s u l t . i n g CONCLUSION

IV.21

CONCLUSION

IV.22

conclusions

and

t-o

formulate

theorems

which

are,

respectively,

induced

by

D e f i n i t i o n s II.6 and II.7, a d o p t e d i n TSCL: THEOREM IV.24

a " < p < = S | a e 7

>

o

and THEOREM IV.25

F » < : p e S

I a e 7 N 7 >. o

By the

Theorem

set

index types

of

IV.24

those

i s a basic is

the

the

set

of

well-formed index.

set

of

all

expression

By T h e o r e m all

basic

those

expression types

whose

IV.25 t h e s e t well-formed

types

of

is

abstract

all

expression

functor types

w h o s e a b s t r a c t index i s n o t b a s i c . Theorem

IV.24

follows

from

Conclusion

IV.21,

Definition

DUAL THEORIES

169

II.6,

Conclusion

II.2,

Conventions

II.3c

IV.25

foUows

from

Conclusion 1.5 ,

IV.19,

IV.19,

Axiom

Theorem and

IV.3c

amd

IV.16a,

Conventions

Conclusion

IV.22,

IV.12.

and

Axiom Theorem

Definition

Conclusion

II.3c

II.6a,

Conclusion

Conclusion

Theorem

II.2,

IV.16a,

II.7,

II.6a,

IV.3c,

and

Theorem Conclusions

IV.13 a n d IV.12. The of

concepts

of

well-formed

basic

CtCS), those

of

CtCB),

CtCS>,

£

are

and

II.lla,b,c,

In

and

of

the

induced,

of X,

CtCf>

writing

the

are

syntactic

the

family

C t CF>

respectively,

Without

DTSCL

all

family

we merely n o t e

by

out

categories

CT
of

all

the

the

of

all

functor

concepts

of

definitions

of

that

definitions

expressions

of

induced

the by

concepts Definitions

respectively.

now

Join

to

the

four-argument

relations:

cortsti

n-th

the

CTCS)

types and

CtCf).

IV.2.

CKBy

We

X

of

induced c o n c e p t s

REMARK

family

expression

categories

categories

the

tuent

of

relation

Cy)

DEFINITION

IV.26a

DEFINITION

IV.26

FCp/cjfJs **•

V

of

order

theory the

DTSCL

r&lation

Cfor n>0> of

replacing

a consti

two



definitions of

replacing

an expression t-uent

of

of

an

type,

a

and

expression

type.

( ^ ^p'J ^ q » |"^j ^ r » ^J^2 ^ J » ^s"j ^

r(.pyq'}s:>.

170

CHAPTER IV

By r e f e r r i n g t-o t,he d e f i n l t - i o n s a d o p t e d a b o v e we c a n p r o v e THEOREM

IV.26

a.

rCpyefy

s ** s,i-

b.

r<J>?^y^s 4» s,r

e S ^ t f ^ s ^ p m r , e S ^

V

V

<s m pCSjj,s^,...,s^> >N r • pcFp,r^,...,r^> y^

- 0 < y < n f ? - ^^ - P - r^ -

fc^j

^j, -

?fc»-

0<*
c.

* > 0 ^ Cr***s 4* _v_ c F*s ^ vcp7^>*7 >>,

d.

rCpytf^s

Theorems

«• » rCp/qi) s .

IV.26a-d

are

expressions

induced,

respectively,

by

D e f i n i t i o n s II.14a-c and II.14d. Theorem

IV.26a

follows

Immediately

from

Definitions

IV.26a,

II.14a and C o n c l u s i o n s IV.19 and IV.I8. Theorem

IV.26d

follows

immediately

from

Definitions

IV.26,

II.14d and IV.26. The p r o o f s o f T h e o r e m s IV.26b,c a r e g i v e n i n t h e Annex. Prom t h e a n a l y s e s c a r r i e d o u t aibove we o b t a i n THEOREM definition

IV.12 . of

£very

expression

TSCL i s a thesis

induced

by

an

axiom

and

IV.ll

IV.12 and

a

in DTSCL.

0

Theorem

or

41

can be proved

IV.22a-c, Remark

IV.2 and Theorems IV.26a-d.

by reference IV.l, Theorems

to Theorems IV.8 IV.23-25,

Remark

DUAL THEORIES

l''!

Theorem IV.12 THEOREM

yields

IV.13 .

TSCL IS a thesis By availing

Every

expression

indxtced

ourselves

categories

of

Theorem

of

IV.13

well-formed

logical

partition

of

types

of

syntactically

the

partition

a.

thesis

of

in DTSCL. we can s t a t e

DTSCL we have a theorem which says that syntactic

by

being

the

set

determined

S

the

all

the

types

well-formed

categorial

by

family C t <S5 of

expression

of

that-

=

all a

expression

language

relation

is

in

X,

Ccf.

that

Theorem

c

II.6a>.

The

II.7a,b>,

analoga

that

syntac tic

is

of

the

both

fttsc

/-undaanental

categories

for

Csee

Sec.II.l

theorems

expression

types

and

of of

the

Theorems theory

X, are

also

of valid

in DTSCL.

Sec. IV.6.

The

The duail t h e o r y DTSC«»-L

dual

co-language

theory X

of

DTSCtar-L

expression

C<jy-X > given

in

Sec.IV.l.

by the

joining

to

the

induced

by

primitive

based

on

derived

its

TETk,

concepts

and of

the

and in particular

possible described

DTSC«*-L

of

the

latter

is

to

analyse

by

the

obtained of

concepts.

Since

induced have

by

already

the

all

the

system

from

definitions

derived

concepts

the

Every of

and

it

types,

Now

latter

TETp, i t i s self-evident THEOREM IV.14 .

makes

TSC«*-L

concepts

TSCcj^L

primitive

been

defined

is and in

that the following metatheorems hold: thesis

TETk,

is

(accepted a thesis

sentence)

in DTSC«*-L.

of

TSC«*-L,

172

CHAPTER rv

THEOREM IV.15 . a

thesis

in

€very

thesis

(accepted

sentencey

of

TETp

is

DTSCMP-L.

T h e s e t w o met-at-heor-ems will h e r e a f t - e r b e u s e d taclt-ly. It- c a n a l s o to

be s e e n

TETp t h e d e f i n i t i o n s

that

DTSCiar-L cam b e o b t a i n e d

o f all c o n c e p t s

induced

by j o i n i n g

by t h e

concepts

i n t r o d u c e d i n C h a p t e r III. The of

first

two definitions

the concepts

to

be

given

induced by t h e c o n c e p t s

of

here

are

definitions

the sets

O and Vr,

respectively. DEFINITION

IV.27

ODEFINITION

O/^

IV.28

The concepts thus defined are termed the set types

and the set

of

all

variable

types,

of

alt

operatoT

respectively.

Definition IV.27 is equivalent to C O N C L U S I O N IV.23

^ ^ ^ •* p ^ ^ D e f i n i t i o n IV.28 I s e q u i v a l e n t CONCLUSION

M

to

IV.24

p e P 7 *• ^ ^ ^ p - [ p ] . From

Conclusions

IV.16 we o b t a i n

IV.23

immediately

and

IV.24,

Axiom

III.l

and

Conclusion

DUAL THEORIES

173

THEOREM

IV.27

O U Vr ^ E . s The types

set

of

are

all

operator

subsets

of

the

types set

of

and all

the

set

simple

of

all

variable

expression

types.

Theorem IV.27 i s induced by Axiom III.l o f TSCta^L. It- i s s e l f - e v i d e n t

that

THEOREM

The

and

III.

IV.16a .

3 of

TSC«*-L are

The c o n c e p t its

of

element type

record

the

type,

a

type

set

of

of

induced

expression an

the

operator

is

from type.

metalinguistic

definition

which

of

the

has

III.

in

S

£

Is

defined

in

makes

to-language

as

an

type,

a

III,

we

possible

to

Chapter it

An

expression

operator

As

DTSCu-L type-

a compound

an

expression

expressions

Axioms

expression

as

an

operator

token

a>-language

successively,

and

by

in DTSCw^L.

pCOxVrxO, t h a t

consisting,

variable adopt

concept

expression of

theorems

an o p e r a t o r

anaiogon in t h e

operator

expressions

J? i n

a

simpler

form. DEFINITION

The p

is

IV.1

expression an

operator

on

operator p ,

the scope p Definition

the

the

left

side

expression indexlcal

of t h a t IV.l*,

of

type

variable

this

definition

consisting p

of

that

of

is

read:

the

main

operator,

and

operator. Conclusions

IV.23,

IV.24,

II.5b, T h e o r e m IV.17a and D e f i n i t i o n III.l* y i e l d

IV.18,

Ill.ta.b,

174

CHAPTER IV THEOREM

IV.28

p - •^<.p^,p^,p^:>

^ p ^ , p ^ y p ^ , p cp^ -

^ ^2 "

An operator p , its

expression

iTidexical

abstraction

of

expression

token

the

main

are,

tion

We being

the

adopt a free

type

consisting and

of

its

the

followlne

a

scope

%,

determined

the

following

is by

the

in an expression

a class an

p

types of

operator of

operator

expression

tokens:

and

of

expression

-

main

p

variable

definition

[pj

P^^

of

representatives

determine

variable

its

indexical

respectively,

the

^ P ' I P ^ ^ P -

relation

p ,

that

[^2]

p

consisting

operator

p , which abstrac

variable

[Po] - Pi -

its

scope

the

classes

p^,p.

and

relation

of p .



of

induced

by

type:

DEFINITION IV.29

pCjvyq This

4^

definition

V

Cp - [p] ^ q - M

yields

the

^ p
following

theorem

D e f i n i t i o n III.l: THEOREM IV.29 pQ

** p

e

Vr

y. p

e

C— -^ ~i _

_

Y_

_


/N r

e

C— A p e

^ p

C—

Cr , r , r

A p «

> ^

r^>.

The proof of this theorem is given in the Annex. The S ,

B,

concepts F,

corresponding

induced

defined

in

quotients

by

the

concepts

TSCu-L,

are

of

those

sets

of now

by

sets: defined

the

S,

S,

S

as

relation

,

the %.

We

DUAL THEORIES

accordingly

'75

have

DEFINITION IV.30

and CONCLUSION

IV.25

pe S DEFINITION

IV.31

S -

S /

and CONCLUSION

IV.26

^ ^ ^ *• p^S ^ " DEFINITION

IV.32

and CONCLUSION

IV.27

pes DEFINITION

IV.33

and CONCLUSION

IV.28

pes DEFINITION

IV.34

W'

176

CHAPTER FV

and CONCLUSION

IV.29

P ^ ^ ** ^ DEFINITION

P -

\P\'

IV.35

and CONCLUSION IV.30

P ^^ ** P^FP ' By

referring

following

to

Conclusions

theorems

which

IV.2S

are

M and

Induced

IV.26

by

we

prove

Definitions

the

III.2a,b

and III.2c: THEOREM IV.30

a.

°S - £ , s

b.

p 6 ***S • • p e * S s/ V ^ '^ n>l — —

V — h.-^

Cp - pCpp,p^,...,p^> ,v p^ e O A c"" Ca,a^,...,a^,aQ)> v

dp m p <-PQ,P^,P^'>

^ P^ e

S ^ P^f'^^P^. •^

pQ'Pi'Pi'P^

CD

s -

U '*s. n«0

Theorem Definition simply

IV.30a III.2a

from

follows and

immediately

Conclusion

Conclusion

IV.26,

IV.16.

Definition

from

Conclusion

Theorem III.2c

IV.30c and

IV.25, follows

Conclusion

DUAL THEORIES

177

IV.25. The p r o o f o f T h e o r e m IV.30b i s g i v e n i n t-he Annex. By

availing

111.3

and

find

that

by

ourselves

following

the

of

the

following

Conclusion

proof

of

theorem,

IV.27

Theorem

induced

and

Definition

IV.30b

by

we

Definition

easily III.3,

i s valid: THEOREM

IV.31

P ^S""^

**X

n>i

PQ'P^'

^

k^^P 'Pjx^ 5

- P^pQ'Pi'

^ Pjj « O ^ c" Likewise,

by

applying

111.4 and by f o l l o w i n g

to

the

Conclusion

proof

of

'Pr?

^

Ca,a^,...,a^,aQ». IV.28

Theorem

aund

IV.30b

Definition

we

find

that

t h e f o l l o w i n g t h e o r e m , i n d u c e d by D e f i n i t i o n III.4 i s vzdid: THEOREM

IV.32

p e S " « » V _ _V_ _ P0'P\'P2'P3

< p - ^*^^0'^1'^2'' ^ ^2 ^

^ '^

•^ P^'-f'^^P^ ^ P3 " Pi -^ '^ *^"'*'l''*2'*'0'*'* From and

Conclusion

IV.12,

IV.29,

Theorem

Definition

IV.lfia,

III.S,

Conclusion

Conclusions III.6

and

IV.26 II.2b,

C o n v e n t i o n s II.3c and IV.3c and Axiom II.2 we o b t a i n THEOREM IV.33

B « < p e S | a e 7 From IV.13

and

Conclusion IV.12,

IV.30,

Theorem

Definition IV.16a,

o

>.

III.6,

Conclusions

II.3c and IV.3c and Axiom II.2 we o b t a i n

Conclusions III.6,

IV.26,

Conventions

178

CHAPTER rv THEOREM

IV.34

F « < p e S

I a e 7 \ 7 >. o

The

last,

t-wo t h e o r e m s

a r e induced

by D e f i n l t - i o n s

III .5 and

III.6, r e s p e c t - i v e l y . The

following

two

expressions,

induced

by

the

last

two

a x i o m s o f T S C M ^ L , r e s p e c t i v e l y , a r e t h e o r e m s i n DTSCw-L: THEOREM

IV.3S o

and THEOREM

IV.36

rcs"> n To ^ 0 . Theorem IV.12,

111.5a

follows

IV.35

follows

and

Il.Sa

immediately

from and

from

Axiom

III.4,

Theorem

Axiom

Conclusions

IV.16a.

III.S,

1V.27,

Theorem

Conclusions

IV.36

IV.28,

IV.12,

111.5b and II.5a and Theorem IV.16a. In TSC«*-L w e h a v e CftBJ,

adopted

Ct and d e f i n i t i o n s

definitions

o f t h e families

of t h e r e l a t i o n s

CtCS>,

f / i " and CV).

We

now a s s u m e t h a t REMARK IV.3. are

defined

In DTSCw-L t h e c o n c e p t s

by

expressions

induced

by

cTcS), c7(S>

and cTcFi

Definitions

III.7a,b,c,

respectively. The e x p r e s s i o n s r < p / g > s and TCp/cf>s DEFINITION

r^pyq^i^s

4«

ax-e d e f i n e d t h u s :

IV.36a

V^ ^ <^P^\t>\ y- «»-[<»] ^ '•"[''3 ^ * • [ * ]

^ r
>.

DUAL THEORIES

179

DEFINITION

1V.36

r

* 4«

^

Cpm [p]

O y ul j T j S

Definlt-ions immedlat-ely 111.8a.

The

on

IV.28, and

Definitions

Theorems

Definition The

Definitions

state

that

In

is

by

IV.16 b.

view

Theorems

The

IV.36,

a n d Ill.Sd of

proof

are

been

definition

IV.16 . of

Every

TSCianL is

the

induced

in

IV.26b,

is and

and

IV.18

induced

Theorem

follows We

holds

IV.17a

IV.26

of

by

IV.26c.

immediately

can

by

I.la

Definition

Theorem

theorem

proof

IV.36a.

accordingly

De/ini

tions

in DTSC»*-L. said

IV.27,

expression a thesis

above

IV.16*a,

IV.3 and T h e o r e m IV.16 b we c a n s t a t e THEOREM

also

Ill.Sd

and

theorems

IV.IS*,

III.8b

Conclusions

Definition

hois

by

Theorems

the

Axiom

induced

111.8b,

of

on

and

of

IV.3a,

III.8d

IV.26

proof

Expressions

what

IV.8*,

the and

and

is

fc»j

Definition

to

modelled

induced

from

THEOREM

11.3a

which

by

IV.36a

III.3a,b.

m.8c

theorem

Ill.8a.-c

theorem

similai^

" ~

Conclusion

induced

proof,

Conventions

" "

111.8a,

the

theorem

DTSCiw-L. I t s based

IV.36a,

yield

^ qm [jj] ^ F" [ r ] ^ s » [ s ] ^ r

s >.

" "

and

referring

IV.29-IV.36,

to

Remark

that induced

by an

axiom

or

a

in DTSCw-L.

This leads u s t o THEOREM

IV,17 .

TSC«»-L i s a thesis

Every

expression

induced

by

a

thesis

of

we c a n

state

in DTSCw^L.

By a v a i l i n g o u r s e l v e s

of

t h e above

metatheorem

180

in

CHAPTER IV

part-lcular

theorem

stating

categories is

a

t.hat.

of

t-he

that

dual

the

well-formed

logicad

expression

in

of

that

family

expression

classification

types

theory

of

DTSC»»-L

C t CS> types

the

set

we

of

of S

all

the of

languaige, d e t e r m i n e d

have

t-he

syntactic

Ci>-language J?

all

well-formed

by t h e

relation

= c

of

categorlal

agreement

metatheorem

also

Induced

both

III.6a,b>, syntac

tic

by that

is


authorizes fttsc

us

hold in DTSC«»-L,. The

for

to

given

/xindamental

categories

Theorem

in

types £

that

Chap.III

theorems

expression

oj-language

state

III.5>.

Csee

of of

the

the the

The

said

theorems Theorems theory

of

oi—language

£

can, therefore, correctly

be

referred to as a syntactically categorlal language.

FINAL

We

have

languag^es

presented pertaining

respectively, variables theory In

i.e.,

working

out

to

trend

nominalism.

question

accord

that

both

to

the

fact

linguistic

on

the

theories

in

have

that

logical

a

u-languages,

operators

been

we

based

have

ontology

done

in

and

of

and

on

the

Le^niewski-Ajdukiewicz-Gurry.

We h a v e

objects

one

of

the

conceptions

contain

conceptions

t o say, recognizing the f a c t 1

languages

which

categories

of

axiomatic

simple

languages

syntactic

consideration

in

to

formal

bound by t h e m . B o t h

of

refers

two

REMARKS

so

dual

of

into

language

because,

and

taken

which

without

semiotic

ontological

caulling

practice

status,

that

hand,

inscriptions,

including have

words

a

s e n s o r y c o g n i t i o n , i n a word, a r e c o r i c r e t e on t h e o t h e r , linguistic e n t i t l e s , expressions, are classes of and a r e t h u s a b s t r a c t , formalizing

that of

toh»ns, language

entitles. ontological

both

theories

we

and h e n c e c o n c r e t e and

Typos,

taking

we

postulated

that

nature,

by r e f e r e n c e t o Thus,

ideal,

were

is

words

material

and

objects,

entities, started objects, their

abstract

assigned

to

entities,

including

equiform

and

definite

s h a p e , a r e e x t e n s i v e i n s p a c e and t i m e and a c c e s s i b l e

in

is

that

e x p r e s s i o n s , are physical o b j e c t s t h a t

2

we

from ax-e

the

the

basic

existence objects

derived

aissumption

of

existence

stratum

as

primitive

a

different

and

defined

tokens. into

consideration 181

the

nominal

is tic

trend,

182

FINAL REMARKS

which

is

otherwise

in

agreement,

LeSniewski

and

ontological

formalized

both

theories

tokens

and

that

accordingly tokens

of

Ctofcen

extension)

at

level

languages of expression

types.

demonstrated

languages theories

of of

important set

of

types,

level,

among

them.

because

property

sets

every

of

all

or

an

t ype

I eve J,

which

the

theories

also

based

on

light

of

corresponding

the

is

the

axioms

a

the

describes

of

natural.

The

level,

i.e.,

them,

that

concepts

latter. hence

at

token

which

dual

the

This

also

is

every tokens,

level,

when

describes

induced

is

relations

expression

the

dual

the

conceptCs>

theory,

hence

categorlally,

but

self-evident

that

as

a a

types. the

syntax and

of

type

and

the

language

theories

the

corresponding of

the

and

language of

theorem

of

tokens, of

expression

theories

among

object

at

dual

the

the

definition,

expression definitional

quite

relations

induced of

is

at

properties

We

corresponding

and

every

properties

the

of

level.

of

their

types

property or properties

into

theories

we

that

tofcen

from

concepts

tokens,

the

axiom,

language of e x p r e s s i o n In

to

a given theory of

translated

theory

types,

Inherit

which d e s c r i b e s

the

the

the

Cas

to

expression

that

object

i.e.,

theorem of

at

is

of

from

passing

Stanistaw

levels,

formulated

tokens

induced r e l a t i v e

token

so

languages

we

of

Kotarbiriski,

different

next

the

expression

point

concepts

that

views

Tadeusz

axiomatic

and

type

We

the

two

first

the

of

starting

level),

at

relsm

types,

constructed

with

of

above

it

languages

definitions

is of

which

expression are

types

expressions

both can

be

induced

FINAL REMARKS

183

by t-he axioms and definitions of the corresponding languages

of

expression

considerations

tokens.

pertaining

to

Hence

the

in

theories of

purely

syntactic

theoretical

and

categorial

description of language we cain disregard the ontological nature of its objects. In such a case the postulatlon at the outset of the

existence

necessary.

of

But

linguistic

from

concrete

the

point

objects

of

view

Ctokens>

of

is

philosophy

not the

adoption of such an assumption a«nd the basing of the theory of the syntax of language on it is of essential importance because It points to

the

existence

abstract

of

The

above

more

linguistic

philosophical

considerations much

possibility o/ avoiding

presented

si.rongly,

objects

in

thesis,

in

which

the

this is

postvilation syntactic

which

book,

shown

is

can by

of

the

analyses.

a

be

the

result

of

substantiated latest

studies

carried out by the present author.

it

We

shall brieffy

is

known,

the

substantiate logical

the

conception

opinion

voiced

of

ontology

the

above. As of

the

syntax of language should provide formal foundations that would enable one to formalize the theory of language at two levels by taking

into account

the

dual nature

of

linguistic

objects. The

theory of language can thus be constructed if we start from the assumption that the study of language is based on object types, that

is

ideal

representations

entities in

intuitively

tokens,

that

interpreted

is

concretes

entities derived from

the former. In such

refer

philosophical trend, namely

we

to

begin

a

different

the

formalization

of

the

as

theory

treated

a case

of

having

we

as

obviously

Platonism, and the

syntax

of

184

FINAL REMARKS

languag^e

object

typ«

at. t.he

t-ypes and

types. Later, by level C token

level

by

constructing making

level')

we

post.ulat.lns

first

use

of

the

Its

construct

t.he

theory

of

extension,

the

exlst.ence

theory

expression

at

of

of

the

the

second

syntax

language of expression tokens by defining the concepts at

of

that

level by reference to types. Then

the

problem

arises: would

the

descriptive

potential

of

such a theory of the syntax of language be an argument

to the

detriment

in

of

that

theory

of

the

syntax

formalization of which the noihinalistic Let the nature of that The

relation

of

langusige

the

approach was used?

problem

concatenation

of

be explained by an

between

types

is

example.

a

function,

whereas when defined on tokens it is not a function. Should not that

fact be of an essential importance

in the

description

of

the syntax of language based on the Platonixing

approach? Does

it

new,

not

allow

one

to

describe

some

essentiatlly

different, properties of objects at the

type

different properties of objects at the token

level,

and

even

and new

or

level"?

The answer to this essential question is in the negative if we

take

into

discussed carried 1990],

in out

and

account

this by

the

categorlal

book. This the

developed

is

present in

11988

approach

demonstrated

author, and

by

indicated

19891. The

to the in

example

formal theories of simple categorlal lau-iguages, which the outlined opposing nominal

is tic

and Platonisiing

language, studies

[1986

and

of

two

represent

approach to

the syntax of language, respectively, demonstrates that the set of theses

Caxioms, definitions, au^d theorems>

of

both

theories

FINAL REMARKS

is

the

185

same: t h e

shown

that

every

respectively has

its

the

type

of

Cor

level,

other

in

first

property

at

is

theories

allow

a

for

languages though

ontological linguistic The

two-level

in

an

the

theories

and

conclusion

of

reference

level, can by

property

theory

described be

a

framework

of

manner,

the

fofcen) has

at

the of

such

a

theory.

interpretable

and

of

simple

categorially,

start

matter

the

first

syntax

is

consideration

of

thesis

into

the

the

that

thesis

concepts by t h e

of

at

Ci> e i t h e r

transposed

thesis

a

respectively

unvtvally

in

to

both

of

from the

formulated

other

that

in

categorially

existence

described

of

the

even

opposing

nature

syntax

interpretability

stating

approached the

theories

mutual

thesis

c a n be

to-languages

of

their

languages

assumption

other

by

theory,

of

objects.

philosophical

simple

type

in a s e n s e ,

under

assumptions

concepts

t h e corresponding

or

analogical

equivalence

languages

are

of

Cor

typei

Cii>

property

described

is Cor

one

level,

the

token

of

the of

the

it

thesis

every

described

Thus t h o s e

by a

Moreover,

at

conversely,

by a t h e s i s

which

concepts

to/ten>

concepts

theory,

ecfui-ualent.

corresponding

And

Cor r e s p e c t i v e l y

the

of

of

analogue in a p r o p e r t y of

token

the

the

theory.

described

are

described

respectively

corresponding

its

theories

property

type)

analogue

the

level

said

of

ideal

also

with

categorially,

theories

of

thus

syntactic one

can

linguistic reference and

language,

t h e o r i e s o f f o r m a l l a n g u a g e s in Chomsky's s p i r i t .

simple

consolidate research

on

eliminate

the

objects.

This

to

probably for

of

the

theory

also

instance

with the

ANNEX

We g i v e

here

t-he

proofs

of

certain

t h e main t e x t . The numbering o f the text are THEOREM

the

t-heorems

formulated

t h e o r e m s and t h e i r

in

order

in

observed.

1.2

c
% r i * » p % r ^ Q % s v

•V V <:c ^ cCv,q,s> v cCp,v,r) ^ c. Now T h e o r e m 1.2 f o l l o w s f r o m t w o THEOREM

theorems:

I.2a

c C p , Q , 0 ^ CCr,s,xi> ^ t

*v=*pa;r/N<j5!;sv

V ^ ^ c<:v,q|,s> vx c < p , v , r ) ^

CCv,s,qi'>

and THEOREM

1.2b

CCp,cj, O yN c C r , s , u ) ^ Cp % r -~. <j % s V V V
V cCp,v,r> /v c> =• ( ~ «.

o f T h e o r e m 1.2a.

Let

From Axiom 1.8 and t h e f o r m u l a Cl> we o b t a i n p % r V V CcCr,v,p) V c>. If

p

is

Conclusion theorem

equlform

1.2 is

qf

true.

is

with

r,

equiform

Let

us,

then with

s

therefore, IS6

by and

assumption the

assume

thesis

Cl>

and

of

the

additionally

that

ANNEX

187

cCr,i> , p ) V C a n d c o n s i d e r Cl.l>

case

when

CCr,t»^,p>.

It follows

from

<1> a n d Cl.l>

C1.2)

cCr,v

Further,

have

cCv^,q,v^:>

Hence, that,

by

Axiom

1.5,

that

tp") /v c C p , < j , t ) .

b y A x i o m 1.2, w e

C1.3>

^

c.

formulas

b y C1.3> a n d A x i o m 1.7, i t

C1.4> and

the

C1.2>

follows

and

C1.3>

yield

tftt

so

that

CCr,v^,ty, since

cCr,s,ii)

^

t%u

by

assumption

Cl>,

by

Axiom

1.7

we

obtain C1.5)

cCr,s,ty.

In t u r n

<1.4> amd <1.5> a n d A x i o m s I . l a a n d 1.6

<1.6)

s

By the

applying

Axiom

relationship

thesis

of

from Since

analogical as

the

<1.7>. in

in t h e

We n o w p r o c e e d <1)

once

more

to

C1.3> with

and

C1.6)

Cl.l>

we

obtain

yields

c C r , u ,p> ^ CCv , < j , s ) .

immediately cCp,u , r ) .

%V

cCi> , q , s > , w h i c h t o g e t h e r

C1.7> The

1.7

yield

theorem Ve

that case to

under

have

still

to

case

the

proof

when cCr,v , p ) , i t

prove cCp,cj, O

consideration analyse

is

T h e o r e m I.2b. L e t ^

C

of

the

us

case

the

omitted

follows

thesis here.

assume

that

when is

18S

ANNEK

and C2>

p f t r ^ Q ^ s v ^

CC ^ C < v , q , s )

v

v C ^ C
the first

element

of t h e disjunction

t h e theorem

follows

Immediately

us, therefore,

assume

from

C2> h o l d s

the

CD a n d Axiom

thesis

1.4. L e t

that

CCr,u , p ) y- CCu , q , s ) v c C p , v , r > ^ CCu , s , q > and c o n s i d e r

t h e case

Cl.l>

c C r , v , p ) /^ CCv

By a s s u m p t i o n s <1.2)

Cl> a n d Cl.l> w e o b t a i n

availing

ourselves

from



Cl.l>.

In t h e o t h e r

that

the thesis THEOREM

The the

,tf,sy.

CCr,^ , p > ys c C p , q , t ) ^ cCu ,«j,s> ^ C C r , s , u > .

By

of

when

of

Axiom

t5:u, w h i c h case,

I.S

concludes

when

cXp,v

we

obtain

t h e proof ,ry

^

i s analogical. I t i s t h e r e f o r e

immediately

for the

cCv ,s,q>,

omitted

the

case proof

here.

1.7

proof

of

this

theorem

is

by induction

with

respect

to

v a r i a b l e n. P

r

o

o

Definition

f

.

For n«l

the

1.2a a n d C o n c l u s i o n

theorem

1.3a. L e t

is

true

us, therefore,

Ac inductively Definition assumption

that I.2b, p.et*',

in

view

of

assume

k+i VGW and,

p e V, and

additionally,

<J-<sf', by

and

that

CCp ,<j ,r>.

Conclusion

I.3a,

re By «7-eV.

V. the By

Then,

by

inductive availing

ANNEX

189

ourselves

of

Conclusion

1.3b

k the assumption

we

have

reW.

Thus

it,

follows

from

h+i VSW t h a t

I'Sh', and s i n c e

the

theorem

is

true

immediately

from

f o r n"l i t i s t r u e f o r any n a t u r a l number n. THEOREM

1.8

p e P r o o f . Axiom It

I.IO

follows

p eV.

For

and

n"!

Definition

from

Since

V/\qi^p^qe

tf^p,

the

theorem

I.2a.

Assume

Definition by

v.

I.2b

Axiom

1.7

follows that

that

we

n>l,

p€ V

cCp ,p , p ) ,

obtain

and

(}%p.

V

and

p e

cCp ,p ,Q>. H e n c e ,

by

D e f i n i t i o n I.2b, «je V, which w a s t o b e d e m o n s t r a t e d . THEOREM

1.9

p e The

inductive

V ,^ q> e proof

I' .^ cCp,<j,r> ^ r e

of

this

theorem

K.

is

carried

out

with

r e s p e c t t o t h e v a r i a b l e n. P r o o f , view

of

inductively

Note f i r s t

Definition that

it

that

for

I.2a,b. is

n"l

Let

true

for

the

us, n^k

theorem

therefore, and

P e "V,

C2>

t

C3>

cCp,Q,r).

e

k+i

V,

By D e f i n i t i o n I.2b we o b t a i n f r o m C2>

C5)

Q2 ^ "'

true

that

in

assume

demonstrate

v a l i d i t y f o r nmk+t. We a c c o r d i n g l y a s s u m e i n a d d i t i o n CI)

is

its

190

ANNEX

C6>

CC«J^,By a v a i l i n g

ourselves

of

Axiom

<8>

1.2 w e

obt.3iin

ccr^,q2''2* It.

Tollows

C4)

and

is

true

<7>

that,

for

Definition

By

from r e

I.2a

and

Inductive V.

n"l,

referring

<7> a n d

t-he

Since

the

C6>, i n t h a t

C8)

Theorem order,

CIO),

yield r e

Axiom

I.lb

and

find

CD,

consideration

the

formula

C5),

^

to

formulas

CSi

and

<:3>,

of

Theorem

and

that

r.

and

C9>,

V, which c o n c l u d e s t h e

THEOREM

formulas

under

conclusion,

1.2

r

theorem

and

yield

we

CIO)

Hence

the

above

formula

to

assumption

in

view

I;8,

proof.

I.IO

P r o o f .

We

shall

first

prove

a b o v e e q u i v a l e n c e i s r e p l a c e d by t h e T e ^^V The p r o o f

of

this

^

V p^ '^

lemma i s

the

lemma

in

which

the

impllcat.ion

V cCp,q,r). n,, '^'^' qe V

by i n d u c t i o n

with r e s p e c t

to

the

v a r i a b l e n. For

n^l

the

assume

the

truth

lemma of

the

is

true

by

lemma f o r

Definition rv^k.. We s h a l l

I.2a,b. prove

Let.

us

that

it

ANNEX

191

holds

for

n»k+l.

Assume

Derinltions

I.2b,a w e

accordingly

P, €

***V,

C2>

P2 e

*K,

of

the

inductive

assumption

C4)

<jj e

K,

C5>

q>2 «

*'''

We the

now

apply

labels

<j

Axiom

and t h e

1.2

to

the

a n d CI) w e

labels

concatenation

C7>

V.

By

of

<j^ a n d

again

to

have

p

and

<j_ a n d p . We

obtain

'='^92''*2''l*'

Now

r*r

and

C8>.

t e

K

of

re

have

<1>

a n d In v i e w

that

follows

Hence follows

from

Axiom

cC<j , t , r ) from

C4> w e c a n s t a t e

in

I.S

view

Theorem

and

formulas

C6>,

<8>

and

and

(5),

C2>

and

C7>,

in

view

t h e lemma i s t r u e

for

n«Aj we

have

1.9

Axiom

V

s h o w n i t s t r u t h f o r n"«At+l, which c o n c l u d e s i t s i n d u c t i v e

converse

easily of

the

Since

that

Thus on t h e a s s u m p t i o n t h a t

can

1.7.

C7)

of


It

C3>,

be

seen

lemma

that

proved

the above

implication is

also

proof.

which true,

T h e o r e m 1.10 i s t r u e by T h e o r e m 1.9 and D e f i n i t i o n I.2a.

is so

the that

192

ANNEX THEOREM

I.lib

P r o o f . suffices

t,o

By

p e If ^ Vi p e

V.

Theorem

pelf

deinonst.rat.e

t-hat,

I.lla, peV

•»

V

t^ p e V ^

p^v.

amn.

Hence

it.

proof

of

The

t h i s lemma will b e by i n d u c t i o n w i t h r e s p e c t t o t h e vau^iable n. The lemma i s pe y

and

ns^l,

true

for

nm\,

then

by

Definitions

I.2a,b,

the

one

hand,

be

Axiom 1.11 p, on word t o k e n s


concatenation results

in

lemma i s

of

a

any

for

If

it

would

could

pair

of Let

labels us

7i"At. We s h a l l

so

that

Convention

other,

would

(because

assume

prove

be

a concatenation

and, on t h e

contradiction.

true

for

its

1.3

and

of

two

not

be

p e V>,

inductively truth

pev,

for

a

which

that

the

n^k+i.

Let

accordingly

CD

P « "V,

C2>

p e By

Definition

I.2b,

we

**V. obtain

-

in

view

of

C2>

-

the

expressions <:3>

Pj e

'^y,

C4>

p^ e

K,

CS>

cCp^,P2,p>. Now

view

C55 of

therefore, the

yields

p«V

Definition apply

relationships

by I.2a.

Definition

Axiom Since L2b

to

I.ll, Cl>

and

hence

holds,

formula

Cl>.

also nOl.

Then

p« V We we

in can,

obtain

ANNEX

193

We n o w a p p l y T h e o r e m

1.3 fco C5>, <8>, < 4 ) a n d C7) a n d

C9>

p^ % t j ,

while

b y T h e o r e m 1.8 f r o m

C6) a n d < 9 ) w e

C10> now

we have

apply

the

THEOREM

T^W -^ | r |

-

n > 1 «*

in

the

*^'^'

- * - ^

carry

out

only

shadl

view

T^¥

of of

implication

demonstrate

because

from

that

consequent

PX^W

implication,

directly

accordingly and

|r \ -n-2.

Definitions the

implication

under

that

it

I.2b,a is

consideration

is

true

for

and

C3>.

demonstrated.

iQl

-

follows

from

Definition

Definition I.2b -

1.3

yields T^ e

I.3b

Then and

true. is

"V,

|r| and

J ^

Then

cCp,q,r».

inductive

converse and

by

1.3

and

true

proof

for

of

implication 1.7.

Definition

Assume

7»"/B+1. L e t

T ^ W and

in view o f

CIO)

be

an

the

Conclusions

and

tl> It

to

m^fc+1, w h i c h w a s t o

^A^

We

simple

follows

assumption

1.12

P r o o f . the

inductive

B»-l»/t, a n d h e n c e

obtain

"^V.

Pj e

We

C2)

obtain

Theorem

Assume 1.3 1.7

inductively n"!m>l.

2 re V

We

the that shall

accordingly

= m+l. Cl>

that

re

V,

which

-

194

ANNEX

Further,

in view

of

T h e o r e m I.IO, w e

C5)

t

€

y,

C6>

t^

e

"V,

C7y

have

c<.t^,t^,ry. The

inductive

and D e f i n i t i o n

and in view

Assume

assumption

yields,

in

view

of

<2>,

Theorem

1.7

^

or

1.3,

of

C6)

additionally

that

i+J"m+l.

Then

Ci-l!>+j«m

i-l>l

i + C J - l > " m /v j - t > l . Let above

us

consider

disjunction

Cl.l> In t h i s

first

the

holds s o Ci-1)+J

case,

a.2>

in view Pj,q^ e

of If ^

case

when

By a p p l y i n g

m m ^ i-1 <9>, w e IPjl

-

obtain

> 1.

have i-1

c
first

that

and <1.3>

the

^

iQjl

-

J

part

of

the

ANNEX

195

Cl-4>

CCtj,Pj,Uj5,

C1.5>

CCvi^,q^,u^>,

which

yield

r*u

in

view

a n d <1.5) y i e l d , b y A x i o m

of

C1.3>

and

C7)

and

view

of

Axiom

1.5.

This

1.7,

C1.6)

CC« ,<j^,r>.

Since

N . |"i

Conclusions

I.3a,b

|«j !•_/

g.elt'

C1.6>

and

and

ueV

and

1.7

in

in and

view

of

<5),

Definitions

C1.2>,

we

I.2a

can

<:i.2), and

st-at-e,

1.3,

<1.4>, and

considering

that

<1.7)

V

<|p|

-

i

^

1,1

-

J ^ cCp,<|,r)>.

p,
now t h e

C2.1>

second

i+<j-l>

By above

availing

ourselves

assumption

<:2.2>

case,

-

that

is

m />. jf-1 > 1.

of

formula

C8)

~ ^ ^

\
we

obtain

from

the

that

/>2,q2 ^ ^ ^

\P2\

'

-'"^

and C2.3>

CCp^,q2,r^>.

By a p p l y i n g

A x i o m 1.2 w e

have

C2.45

C<(t^,r2,-u^y,

C2.5)

CCp^jVjjV^).

By of

availing

C2.3),

C4>,

A x i o m 1.7, w e

ourselves C2.4)

have

and

of

Axiom C2.5>.

1.5,

Hence,

we in

obtain view

m^^u of

in <:2.5)

view and

196

ANNEX

C2.6>

C
It, follows Iv. |>J and

from

follows

C3:>,

C2.2> t.hat. p^eW and

from

Conclusions

Definlt.ions

I.2a

and

lp->l"ij

I.3a,b

1.3

and

and

whereas

1.7,

v.^W and

formulas

formula

C2.4>.

C2.2)

Hence

in

view of C2.6> we can st-at-e t,hat< <2.7>

^ ^ ^ <.\p\

Thus

for

the

-

i ^ 1«,| ~ j

assumption

that.

r. c
i+j«m+l

in

both

cases

under

consideration, i.e., Cl.lJ and <2,1>, we have p,^H' C|p| -

i" .N | q | - J

A cCp,<j,r)>,

and hence i^j i+j-m+l The l a s t of

the

p , ^ N ' *^l^l " ' '^ I*"! " J ^ CCp,<,,r».

expression

implication

proves

for

that

n"m>l

the

assumption

results

in

its

being

n"m+l. This concludes t h e inductive proof of t h a t THEOREM

» n+1 «*

P r o o f .

2

pe V

by

the

truth

true

for

implication.

1.13a

p G W ^ \p\

induction.

of

For

n"l

Definition

V

We

refer

the

expression

1.3

and

equivalent, by Definitions

to

c

the

principle

peh' ^

Theorem

I.2b,a, t o t h e

<.p ,p ,...,p _^,,p>.

1.7,

of

mathematical

\p\"2

is

equivalent

and

that

in

turn

to is

expression

p^,^3.K^<^l'''2'^^ Thus,

in

view

of

Definition

I.4a,

the

theorem

is

true

for

n»«l.

ANNEX

197

We sissume truth

for

that,

it

is

n«fc+l. For

equivalences

which

true that

for

n«At>l and s h a l l

purpose

follow

we

from

write

demonstrate

out

a

Definitions

its

sequence

1.3

and

of I.2b,

T h e o r e m 1.7, t h e i n d u c t i v e a i s s u m p t i o n and D e f i n i t i o n I.4b: k+Z «je*^K

^ft+2^'^

* ^

h+i. ** )( ^

It

^

follows

^

^

^v

from

^^

these

^Pi'P'>>- >Pb*t''*'^

equivalences

^ CCg,p

that

,p»

the

4^

Inductive

assumption Implies that the theorem under consideration is true for n™*+l. Hence the theorem is true for any natural number n. THEOREM

1.14

n+1,

^

n+1.

l
it

is

1.4a true

It that for

follows the

immediately

theorem

is

true

nmh. To d e m o n s t r a t e

f o r n»fc+l we a s s u m e a d d i t i o n a l l y Cl>

^i'*»i ^ '^ =• l < . < n + l from for that

Theorem n"l. it

is

that

C**2cp^,P2,. .,Pj^^j,Pj^^2'P' ^ '=**^'^'»1''»2' ''»*H-l'''it+2'^'

and

We s h a l l d e m o n s t r a t e

that

f'i

We

^ '^i 1.3

and

assume

then

true

198

ANNEX

l < i < k + 2 Pi. * " i In v i e w o f <1> it- f o l l o w s f r o m D e f i n i t i o n I.4b t h a t ,

and

By T h e o r e m 1.3, f o r m u l a s C4> and C2> yield

and ^*^

^fc^2 * In

view

of

t,he

theorem

''M*2

which

is

a

generalization

of

Axiom

1.7 Csee T h e o r e m I.15> f o r m u l a s C3), C5) and <2> y i e l d

-

I < t < * + 1 ''i-'^i ^ '''

w h e n c e by t h e i n d u c t i v e a s s u m p t i o n we o b t a i n

Since,, m o r e o v e r , <6!> h o l d s , we n o t e

that

l
concludes

consideration.

the

inductive

proof

^i' of

the

theorem

under

ANNEX

199

THEOREM

1.18

The p r o o f I s by lnduct.ion w i t h r e s p e c t , t o t h e v a r i a b l e n. P r o o f .

Note

immediately

from

therefore,

assume

first

that

for

n«l

the

Conclusion

1.2

and

Definition

its

for

n"fc

and

truth

theorem 1.4a.

demonstrate

h o l d s f o r n*iJt-t-l. F o r t h a t p t i r p o s e we a s s u m e a d d l t o n a l l y

follows Let

us,

that

it

that

and C2> and

p % «j, also

C3>

\ < J < Jit+2

and

It follows

from

Definition

I.4b a n d f o r m u l a

<1>

that

and «S>

•=<^'l'<'At+2'''^ '^ *=*^*2'*»*+2''»* Let

us

consider

now

two

cases,

we s e e ,

in view

namely

when

of

that

1<J<*+1. In t h e

first

case

C4>

Jmh-*-2

and

when

200

ANNEX

By T h e o r e m holds

from

1.17,

in

view

Conclusion

of

1.2

<5),

and

we

have

formula

'^*'o>

<6)

we

^nd

since

obt-ain

C2>

Pi,4.->^j,+o-

Thus <7>

J " >t+2 ^ p . % Q .. In t h e s e c o n d c a s e Ct<j, we h a v e by <4> ''*+2 * '»fc+2 ^ ^

Hence, also,

by by

Conclusion the

1.2

inductive

p .%o . in v i e w

and

I.+1 ^ i * " t

fofnailas

C2> and

assumption

and

C6>,

formula

f %t C5>,

Hence

we

have

of ^

D

Si a

.

i^J

Therefore C8) From t h e

implications

self-evldently for

n"k+l

is

that

1 < J < fc+l ^ p

:s q .

<7> and

formula

p . ^ . and

true.

This

C8> and

that

the

concludes

thesis

the

C3) we

of

obtain

Theorem

inductive

1.18

proof

of

p^W ^

P^I •

T h e o r e m 1.18. THEOREM

II.l

V n / - 0. P r o o f . Then,

in

view

Let of

us

aissume

Axioms

1.9

that

pe¥nl

and

II.l

so and

1.3 a, I.4b and 1.4 t> we h a v e o n e o f t h e f o l l o w i n g (1)

p ^ V ^ p e I , '^ o

that

Conclusions cases:

I.3a,

ANNEX

201

Cll>

p e

r\K

Clii)

p e

K /s p e

Civ>

p e

l«^K -N p e

Now

case



are

excluded,

II.4

and

follows

is

excluded

respectively,

11.11.

We

from

have,

Theorems

Civ),

Definition

1.2a

that

pe

pe

K

Theorems

/N p e

and

by

/\/

,

/\/

the

where

I.IO a n d I.IO , t h a t

-

Axiom Axioms

and

and

p^eK<^q>^e"^K-N

,

therefore,

I.ll

I,

by

/

11.6.

Cases

1.12

and

to

I.ll

-

and

there

are

in

view

p.,ct

^ q^ By we

applying

find

-

in

of Axioms

Axiom

view

of

I.ll and

1.8

to

the

the

fact

II.3 - that

that

by

contradiction proof

of

=

P<-PQ,P^,-

P r o o f .

I.IO

with

Theorem

CONCLUSION P

Axiom

that

Axiom

1 ^

p^^P^ 1

follows

p.el'

p^'^V

II.6.

This

^

Assume

q

"

P
••>Qj^> •*

that

<1>

P =

C2>

<j =.

P C P Q . P J ^ - .P„>>

pCq^,q^,...,q^)

latter

by

that

cCp ,<j ,p>

^

and

and

p^Gl

^

o P^^I 2

Po^^ >

concludes

II.l.

condition

cCp^,q^,pi.

1

II.3 .P^

I ^

P^^

and

It

follows,

such

2

2

Axioms

^

e

formula

it



Clv>.

the

Hence

€: I ^

and

the

to

,p^,
and

case

of

dual

n>l.

c ^ p^

II.3

analyse

definition m>l



cCp

,q

in

,p) view

• Hence

it

o which the

is

in

indirect

202

ANNEX

and a d d i t i o n a l l y thatCl.l)

P % q-

Then by Axiom 11.10 f r o m f o r m u l a s <1) and Cl.l> we o b t a i n CI 2 )

«j -

pCr^,r^,...,r^)

and 0<>i
by

Axiom

II.9 t h e

k

"^k

function

p Is

a

one-to-one

function

f r o m C2) and t l . 2 ) we o b t a i n

" -*'

0
which by <1.3> y i e l d s ^'^'

0<^
^2*^

0
Then by Axiom 11.10 and f o r m u l a s <1>, <2> and C2.1) we h a v e C2.2>

q « P

w h e n c e by Axiom I.lb we o b t a i n C2.3> Since

P * 9 the

assumption

Cl.l>

yields

C1.5>

and

the

assumption

C2.1> y i e l d s C2.3> t h e p r o o f o f t h e c o n c l u s i o n i s concluded. THEOREM

II.2

S - n < X I 6CX) >. The proof of Theorem II.2 Is based on the two inclusions:

ANNEX

203

THEOREM

II.2a S

THEOREM

S: (^ < X

is

<5<:X>

>,

II .2b

n -t X o

I

P

r

o

f

of

to

be d e m o n s t r a t e d

Ca>

II.2a.

I 6CX) > S S.

In

accordance

with

Definition

II.4c

it

that

r e

U

S ^ A C6<X) ^ r e

X>.

fcaO

It

suffices

to

demonstrate

that

the

following

implication

holds: Cc<'>

r €

The

proof

Definition

of

II.4a,

true

for

by

be

is

reC ,

self-evidently implication

Ca'>

* S =* ^ <:<5CX> =» r € X>.

true

kml+i.

induction.

whence

the

Definition for

Assume

II.4b a n d C o n v e n t i o n

by

fe">f.

Cl>

shall

accordingly

II.3c w e r

fe»0,

consequent

II.5. We

For

re S

of

and,

Ca')

follows

Let,

therefore,

prove

that

that

re

S.

it By

by

is

the also

Definition

liave G

's

or C2)

7»^ > 1 ... r^,r^,..,r^

e

S ^ r

^ c

In

case

<1>

c a s e C2> we h a v e

Ca'J

is

true

by

the

•* p C r ^ . r ^ , . . . , ? ^ )

^

C c , C j , . . . , c ^ ,Cjj).

inductive

assumption.

In

204

ANNEX

C3)

n^ > 1 ^ r^,r^,...,r^

e

S

and C4)

r

and,

-

p(r^,r^,...,r^

> ^ C

Cc,c^,...,c^

b y C3> a n d t - h e i n d u c t - l v e a s s u m p t i o n ,

C5J

0 <6CX) •» r „ , r ^ , . . . , r X

Thus obtain

when from

O

X Is

any

C5!>

that

o b t a i n by D e f i n i t i o n In

both

cases,

Eissumption concludes P

r

that

the

o

o

proof f

set.

and

Cot'>

is

of

1

eX,

the

and

C2>,

it

true

for

This

(oi), t h a t

theorem

A <6CX> ^ r 6 X> ^ r

satisfies

the

<4)

we

the

inductive statement

is

to

the

equivalent

to

the

implication

6

U

S.

*-0 antecedent

Definition

following

we

of

This

is

^

with

from

Ifl+l.

<.ft>

accordance

view

£CX),

. II.2b.

the

in

follows

to

that

condition

reX.

implication converse

Assume

have

X> ^ n^ > 1.

satisfies

r _ , r . ,...,r

Cl>

we a l s o

7*.

1

that,

11.5 t h a t

of

e

,c^)

II.5,

of r

C^>

is

true

so

is

in

any

set

that, X

in that

conditions:

r e x ,

CD

s

p n>l

In

order

to

PQ,P^,•••,P„«^ ^ C

<.a,a^,...,a 1

demonstrate

that

O 1 ,a_J ^ p e n o "^ At r« U S

fc-0

it

n X>.

suffices

to

prove

ANMEX

205

t-hat

t-he

it

"* k U S

set

satisfies

satisfies

condition

conditions

Cl>,

Cl>

" k E £ U S,

i.e.,

^

Definition

S^E . To demonstrate s condition C2> let us assume that

n > 1 ./v ^o'^i'-'^n

In

fact,

because

that

"" k

^

<2>.

by

Jlt=0

II.4a

<:3>

and

U

It

satisfies

S

and

By

C3>

number

I

we out

expressions

can of

state

the

Definition °° k p e U S.

>

1

II.4b

The

^

the

"* At U S

set

C/3> i s

e

P r o o f . Definition



C5>

and

satisfies

accordingly

the

pe

S,

have

C4!)

the

yield

required

condition

true.

II.4a us

The and

assume

true

for

k S ^ Q ^ p = * q e

proof

of

this

Conclusion its

k^l+l.

being

II . 4 a

true

the

for

accordingly

p €

'*^S

Q ^ P-

S.

lemma

Let

and <2)

of

S.

k

also being

order

II.3

p e

Let

the

natural

>i=0

<2). The I m p l i c a t i o n

k^O.

greatest

determine

PQ,P^,->P^

formulas

At-O

LEMMA

which

is

C n > l ) . T h u s b y L e m m a II.l w e n

whence

there

numbers

Pf.,p^,--,p

C5>

By

that

is

by

lemma

AE»{ a n d

Induction. is

true

demonstrate

By for Its

206

ANNEX

By D e f i n i t i o n

II.4b it. f o i i o w s

C3>

P €

from

Cl> t.hat.

^S

or

Po'^1'

Now

for

<3> t - h e l e m m a

assumpt^ion view

'^n^ S

in

view

of

follows C2)

o f C o n v e n t i o n II.3c, w e

and

immediat-ely

from

DeflnK^ion

II.4b.

t-he

induct.lve

For

C4),

in

have

<5)

n^ > 1 A r^,r^,...,r^

<6)

p -

e

S,

p
By A x i o m 11.10, <6> a n d <2> y i e l d <:8>

<j -

p<s^,s^,...,s^

)

and 0
C5> and C9> y i e l d ,

<10)

k.

by t h e I n d u c t i v e

n^ > 1 A s^,s^,...,s 1

By

k

Convention

Definition

II.3c,

0 it

1

follows

e

Since a t

the

same

time

*

S.

n from

II.4c and C o n c l u s i o n s II.2b and II.6a 0
assumption,

A

CIO), that

C5>

and

C9),

ANNEX

207

<12>

6

by

CD,

Lemma

follows

II.2,

from

C7>,

% a

C2>, Cll>,

Axiom <12),

II.8

and

Convention

Axiom

I.lb

and

find,

by

II.3c,

It

Conclusion

I.ll

that

In

view

that

Q€

of S.

inductive

C10>,

Hence

<8>

the

assumption,

and

C13>

lemma true

we

under

for

Definition

consideration

fc"l+l

also

in

is,

II.4b,

under

the

C4>.

This

case

c o n c l u d e s t h e i n d u c t i v e p r o o f o f t h e lemma. LEMMA III.l "s

P r o o f .

In v i e w

of

£• £ that

it

suffices

to

DCO.

C o n c l u s i o n 11.5b w e

<1> so

S £• S

DCO,

demonstrate "S

C2>

have

£

that

£•

holds. The

proof

immediately inductively it

is

pe cases

C2i

from that

true S.

of

Then,

CI ,2 ,3 >,

definition.

In

accordance

with

is

induction.

ntk.

Let

us

assume

listed

the

1°,

III.2b,

in

the

when

inductive

For

and

for

Definition

case

III.2a

true

n""k+l.

by

by

Definitions it

for

is

II.3c.

We s h a l l

we

C25

follows

us

assume

Let

demonstrate

for

that

have

one

verbal p e S,

n"0

purpose of

the

formulation formula

assumption.

In

that

C2)

case

2 ,

that three

of

that

holds p

in is

a

208

ANNEX

compound

expression

in

accordance

by

CD

is

in

they

Finally,

its

p^D~'^Cp:> from

II.3b>

is

an

scope,

inductive

to

be

belong

t,o

S;

aussumptlon

they

are

C,

and

Axiom

and

in

of

by

and an

DCL^.

and

operator,

its

in

case

II.3b,c, that

II.3c).

hence, it

is

by a

indexlcal

3 ,

by

pe£.

<2)

p

Axiom

Thijs

is

true

it for

III.2

P r o o f . lemma

being

thus

(Definition

Cl>

Definitions assumption

Cp>;

formula

Thus

At

assume

E

pcD

in

demonstrated.

p e

the

II.9,

expression,

III.la,b

in

the

that,

operator

all

hence,

element.s

by

consisting

and

n»fe+l, w h i c h w a s LEMMA

Hence,

Conclusions

expression and

follows

p

of

inductive

DCt).

3

III.l ,

compound

II.9,

in

case

Definition

the

(Definition

In

variable

with are

£• c

consisting

that true

The

is

proof

true the

for

k

S ^ q ^ p ^ q e

by

of

this

Definition

lemma

is

true

h^l+i.

For

for

that

S.

lemma

is

ni.2a

and

k^l

by

and

purpose

induction.

For

*i"iO

II.4a.

We

demonstrate

its

Conclusion shall

we

assume

additionally

that Cl>

p ^

'*'S

and C2>

g = p. By

three

Definition cases

holds:

III.2b

it

follows

from

<1)

that

one

of

the

ANNEX

209

p^.p^,...,p^^

S c

o In

case

Definition

9€

S

in

the

Is

S

III.2b.

by

proof of

in.2c.

III.2b a n d t o

expression

q,

on

the

of

III.3>, i s

also in

we

to

of

that

that

C4>.

Conclusion

the

the

consider

It

«je

S

suffices

II.6a,

Lemma

respectively, III.l

and

inductive

expression which

case

Pf^,p.,p~,p^ P

<4)

p^

<:5>

is

-

3 . In

such

In

to

Definition

assumption

p

that

the

e

*S,

p^C/t»p^, P3 * P j .

c

Ca,aj,a2>«o^-

that

that

P^P2^'

and <7>

fact

set

the

satisfies S

Csee

s"".

<3>

<^6^

C2>,

the fact

case

Lemma

of

expression

expressions

the

refer,

III.6,

with

Definition

sequence

in

II.4c,

strength

2 ,

have

of

of

formula

A x i o m III.2.

condition

Finally

an

II.3

to

assumption, proof

proof

Lemma

II.4b,

equiform i.e.,

of

the

the

Conclusion

make u s e

that

to

2 ,

Definition

Definition

Definition

inductive

case

inductive instead

the

In

analogical

and

Note

qf€

quite

merely, II.2

I+l

1 ,

and

^.

case

there

is

a

210

ANNEX

It there

follows exist

from

Theorem

expressions

III.l

and

qf_,<J^,<J_ s u c h

formulas

C3>

and

C2)

that

and

formulas

that

and

It

follows

a n d <9>

from

the

Inductive


that

<10>

^^2 ^

It

assumption

follows

formulas

in

turn

'^

from

C5) a n d C9> t h a t

Axiom

there

Cll>

is

I.la.

Conclusion

a cj„ s u c h

III.4,

and

and

<9>

the

pei)Ct>.

This

and

that

Q^C/vyt}^

and C12>

,3

By

Axioms

I.lc,b

we

* P3.

obtain

from

<12>,

<6>

formula <13>

Qg a: qf^.

It

follows

formula

from

Cl>

and

III.l

that

C2) y i e l d , b y A x i o m II.8 a n d C o n v e n t i o n

<14>

II.3c,

6 % a.

Since, is

Lemma

also

III.l)

by

Convention

guaranteed

they

belong

making

use

again

on t h e

strength

of

of

11.3a,

Pn'P4>P-y

by

Definitions

to

the

Axiom

C9>, a t

set II.8

the

are

III.l D

and

formula

expressions and

CConcliision

Convention of

II.3c

the

II.3c

form

aoid

Cwhich Axiom

II.5b). we

By

arrive,

ANNEX

211

"^^

0
formulas

<7>,

(14)

and

C15)

we

obt-ailn,

by

Axiom

I.lb

a n d C o n c l u s i o n 1.11, C16>

c^Cb,b^,b2,b^>.

Thus in view

of

C17>

<8>, CIO), <11>, C13> a n d C16> w e V

c, -

p^Cq

^0'^1'^2'^3

have

,
u 1 z

S ^

z 3

and

in

accordance

with

Definit.ion

i+1 obtain *i«l+l

<j€

III.2b,

in

view

of

C17), we

o S. Thus a l s o

follows

from

in case

3

t-he t r - u t h o f t-he lemma f o r

t h e assumption

of

its

being

true

for

At" I.

The lemma i s a c c o r d i n g l y p r o v e d . Note in

also

case

which

3 ,

is

that when

equiform

i t follows peS

from

by

with

p

is

t h e inductive

Definition also

in

III.4, S

assumption an

because

that

expression

<j

f o r m u l a C17)

holds. THEOREM III.6a CI

fttsc) c

P r o o f .

The t h e o r e m

c

follows

immediately

from

Definition

III.Bd and LEMMA III.3

rCp/qi) s ^ r — s - ^ p ^ q , c

c

which will b e p r o v e d by i n d u c t i o n . For III.8a,

n">0 Lemma Theorem

III.3

III.3,

Is

true

Convention

on t h e s t r e n g t h II.3c

and

of

Definition

Conclusions

III.8,

212

ANNEX

11.12, and II.13b,c. Assume t>hat. for

n"l


jrCp/ij) s

and
r • s. c

We

have

t.o

demonstrate

that

P"<J-

In

accordance

with

c Definition they

111.8b

are

Definition

and

the

assumption

cases

III.8.

Note

1

a>nd

2 ,

that

in

case

expressions from the s e t consideration which

in

quite

substantiates

successive and

is

steps

II.4a,b,c TSCL,

and

to

in

Conclusions

to

Definitions

and

Conclusions

III.6

case

2 . We have

then

and

C2>

r,s

and there are s ,s ,s

verbal

comment

analyse

each

of

to

and r ,r ,r

s

are

of

suffices of

in

Lemma only,

III.2a,

the II.14b

are

TSC«*-L,

us,

in

Definitions which

II.4

valid

that

III.3,

therefore,

is, ni.2c,

consider

such that

p^<:r^,r^,rQ>

conditions Definition those

and

e s"

r the

proof

III.8bCl°), Let

after

the lemma now under

II.7,

analoga

<4> of

of

and

s ^ p Cs^.s^yS^y,

one

when

instead

<3>

and

r

It

III.8.

possible;

1 ,

TSCL.

II.6a

are

verbally

the

their

cases

described

to

proof,

to

two

, the proof

fttsc the

refer

respectively,

S

analogical

I

of

Cla>

, III.8b

conditions

Cb),

Cc),

holds. we

formulated

Before

have

to

we

in

proceed

note

that

the to It

ANNEX

213

follows

from

III.7a,b Since

that. s

formula -

the

r,

given

expressions

also

determined Definition

^A'^A

by

s

C3) -

Conclusion

C2> a n d D e f i n i t i o n s

Definition

and

unaanbiguously

C2>,

Note

<4>,

= ,r

and

determined

C3),

and

and

C4>,

have

their

Defintion

Definition

11.3b

and

III.4 a n d I I I . 2 c . V e a c c o r d i n g l y

have

that

the

accordance

the

form,

with

the

expansions

it

follows

conditions

Convention

expansions

from

tjl

of

and

II.3c

III.l ,

Axiom

s

from

in

view

and

r

<2>-C4>

m

s In

expressions.

follows

are

unambiguously,

Conclusions

well-formed

too,

since

r

and

well-formed,

that

III.4

^'^^

which

111.5b,

III.l*

are

II.9 of

are and

satisfied.

r

these

conditions

take

on

respectively,

and C7)

C Note

also

that

<:8>

rf

follows

Cc,c^,C2,c^:>.

from

% c

the

assumption

Clb>

and

Conclusions

to

conditions

11.12

and

11.13b. We

now

assume

proceed

first

Ca)

that

the

analyse following

condition

is




n.3c

that

it

follows dj=scj

'from

and

^^^^

C5>,

Conclusions

This

yields,


We

satisfied:

, j j % Sjj ^ p % r^j ^ S j * r^ . . s ^ * Then

and

III.8 by

r^. and

C8:>,

<;6>

Convention and

C7>

214

ANNEX

and

Theorem

1.17,

d^^ix: , U

in

view

III.8,

of

C5>,

11.12

and

and s i n c e

p*r-,,

U

Theorem II.13b,c.

III.3, Thus

c o n d i t i o n (a> i s s a t i s f i e d . Let us consider t h e next Cb>

«J^5»-, and

O

Convention the

II.3c

lemma

is

can

holds,

by

IlI.e

we

easily <5),

have

be

seen

b!>xi.,

in

view

of

Cb)

III.3,

aSix.

1

and

1

have

still

to

implies

Convention dtsx..

Conclusions for

n-1

Thus,

if

if

This

*

'i

s 5a",. II.3c

and

yields

Cb)

Conclusion

immediately

p^
11.12

and

t r u e f o r n"l i f c o n d i t i o n
and

1

Conclusions

pmq

condition

that

Theorem

have

c

true

g s: S j >s p 5: r^ ^ 0
we

U

II.13b,c.

Thus

the

lemma

is

satisfied.

demonstrate

that

p^
if

the

following

c

c o n d i t i o n holds:

ct ^ s^ ^ p ^ r^ ^ s^ ^ r^ ^ s^ ^ T^. Now

let

Convention d.ysc

in

psa-^,

we



hold.

II.3c

we

view

of

have

Then have

<8>,

p*(}

in

view

d^ix

and

C6>, C7) and

in

view

of

of

C5),

d.^ac , Theorem

CS!),

Conclusion so

that

III.8 we

1.18. S i n c e

Theorem

III.3,

assumption

of

and

obtain

tfiis

and

Convention

II.3c, and C o n c l u s i o n s III.8, 11.12 and II.13b. Thus Lemma III.3 i s t r u e f o r The

proof

being

true

quite

similar

assumption

for

of

of

the

fact

n"l.

that

n»At Cfc>0> r e s u l t s to

the that

proof lemma

D e f i n i t i o n II.14c i t s u f f i c e s

of

the

in i t s

being

Lemma

II.4

being

true

for

true for

that for

T»"fc+1,

n«At.

n««fe+l on

Instead

t o r e f e r t o D e f i n i t i o n 111.8c.

T h i s c o n c l u d e s t h e i n d u c t i v e p r o o f o f Lemma III.3.

lemma is the of

ANNEX

215

THEOREM I V . 4

i? •

n < X I x"<5?> >.

To p r o v e t-hls t h e o r e m it. s u f f i c e s LEMMA

t-wo l e m m a t a :

IV.l

y and i t s

t-o p r o v e

p,H,T

converse

LEMMA I V . 2

/ V < F s X / v _ / V _

P r o o f

Cp,Q e

X ^ CCp,q,F>

o f L e m m a IV.l. A s s u m e

Cl>

s

and additionally

= » r € X ) ^ s e X > ^ s e i ? .

that

e i?

that

C2)

7 S X

and C3>

_ l!^ _ ^ ^ ' ^ e X x^ c ' C p , q , r ) ^ F e X ) .

We h a v e It label

t o demonstrate

follows token

s

from such

<1>

C6>

Assume

that

e X.

and

Conclusion

s^ e

y

C45 w e o b t a i n ,

^ CK S X /«.

s

IV.4

that

there

is

that

C4>

From formula

that

/^

Cp,q e

by D e f i n i t i o n s

I.l a n d I . l a ,

X ^ cCp,q,jr> ^ r e X> ^ s

€ X>.

a

216

ANNEX

C7>

XQ

It.

can

eaislly

Csee D e f i n i t i o n In then

fact, it

< p

seen

I

that

p^y,

follows

then

e

X >.

X

satisfies

from

i"^

view

<2> a n d C7> t h a t

peX

the

condition

\<.X ')

[p]^*'^

of

Conclusion

IV.3

and

have

C8)

V S X If,

then have IV.2a

on by

the C7>

other [p]

and

^^ [ p ] ' [ ' * ] ' r ^ l •*• and

Conclusion

C7>, r e X

hand, [q] ^y IV.l)

we

^

belong

to

applying we

that

X

and

p,<}€X by

formula

obtain


frleX

and

c,

Definition

<3>

Csee

so

IV.2

we

Convention

that,

X„ /s c C p , q , r ) ^ r €

p,«j,r From

assume

in

view

of

Hence

C9>

C6>,

C8> a n d

a n d <S> w e h a v e

CD

[p]

I.la>.

if

We t h u s

be

-

0 C9)

we

0

obtain

s e X . L e m m a IV.l h a s

s eX-, thus

satisfies In f a c t

to

the it

demonstrate

that

seW

it

from

C2>

Conclusions

in

view

of

<7>

proved.

^ r

e

suffices

c o n d i t i o n XCX> C s e e D e f i n i t i o n follows

whence

been

P r o o f o f L e m m a IV.2. A s s u m e t h a t ^ < V s X ^ _ ^ _ C p , 5 € X -N c'Cp,^,?)

In o r d e r

X„>.

X) * s

to

e

show

X).

that

W

IV.5>.

IV.3, IV.4 a n d I . 3 a

that

K c iP. I t cam a l s o

C3>

be proved

that

_ '^ _ Cp,q e P»9»r To

do

so

let

us

»? /s c"Cp,q,r> ^ F e JtO.

assume

that

p,
and

cCp,<j,r5.

Then

it

ANNEX

217

follows q

from

such

that

Convention label are

and

r

such

tokens,

1.3b,

and

been

proved.

Now

IV.4

P" [PJI

IV.2a

token word

Conclusion

hence

r.

^"'^

there

^[^i]"

Conclusions that is

rei?

formulas

that,

C I ) , <2>

and

there

is

<^tp ,<} , r ) .

Since

P^rQ^

IV.4.

imply

and

that

IV.2

token

<3>

p

and

a

Conclusion

tokens

it

^^'^

word

word

Further

^"[^4!

also

by

IV.l

are

in

view

Formula

that

follows

seW,

of

from a

Conclusion

C3>

has

what

yraes

thus

to

be

demonstrated. THEOREM

IV.22b

Cp P r o o f .

p ^ c "

TO p r o v e

the

CI)

direct

p e We s h a l l

Cv>

demonstrate

P ^^"^

-

Ca,a^,...,a^,a^:>y.

implication

we a s s u m e

-

P
that

S.

that

r,>l

-

-

""

Po'Pi

K,"^

'

>P„>

-

P„« ^ _n+i^ - x-> .-. c Ca,a^,...,a^,aQ».

By such

Conclusion

IV.20

it

follows

from

Cl>

that

there

is

a

<j

that

C2)

p - [«,J

and C3>

<j It

two

follows cases:

from

C3>

and

e

S.

Definition

II.4b

that

there

may

be

218

ANNEX

Cl>

q

« *S

or

^ c

In case Conclusion

Cl> pm [q 1 in IV.20

and

Cv)

accordance is

true.

C6j,a^,...,a^,aQ».

with

C2) and hence

Consider

t-hen

case

pe S by Cii5. Now

C2>, Cii) and Convention II.3c yield

1

1 ^ c

C6^,c^,...,c^ ,Cjj) ^ n^ > 1,

Assume that.

It

follows

from

formulas

C6>,

<2>,

<4>,

Lemma

II.2,

Theorem

IV.16a, and Conventions II.3c and IV.3c t h a t

In

view

of

Conclusion

IV.20 and

formulas

C4) and

C6) we

see

that

Since r, ,r,.,...,j~

in

accordance

eDC'. >,

IV.17a t h a t

it

with

follows

Lemma II.2 and formula from

(6>

and

(S>

C4i and

",-1

and

Theorem

ANNEX

219

<9>

p " p<.7^,7^,...,T^ >. Further,

in

view

of

C7)

and

C4>

and

Conclusion

IV.6

we

obtain —n+1 C Ca,c

C10>

— ,...,c

— »c ) /\ n

> 1.

1 The follows Now the

t-rut-h o f from formulas

i n t-he s e c o n d <8)-C10!).

let

It

first

follows


hold.

constituent

immediately

therefore

the

Then t h e r e

is

is

of

the

from

second

to

be

of

two

possible

demonstrated

disjunction

Conclusion

constituent

a n n >1 a n d t h e

the


IV.20

of

and

the

following

s^,s^,...,s^

e

C2>

p = pCs^,s^,...,7^

then

Lemma

disjunction

formulais



— fc+1— pe S.

that

holds,

are

cases

p€

II.1. Cv>

If

k+i— S Let hold.

valid:

S,

> 1

and C3>

c^'*'^Ca,d~,...,d

In that

view

of

there

are

Conclusion s^,s.,...,s U

1

IV.20 such

n

,d„>.

1

71^

O

and

formula

<1)

we

can

state

formula

C4),

that

1 and

Since s

,s

,...,s

n >1 eOCt>,

T h e o r e m IV.17a,

and, we that

by find,

Lemma by

II.2

referring

and to

<2>

and

C5>

and

220

ANNEX

The from

expression Axiom

Theorem

II.9. By

applying

IV.16a w e o b t a i n

From

C5)

Conventions

By

p < s , s ,...,s

we

)

is

in

Convention

OCi.>, lV.3c

which

and

follows

referring

t.o

f r o m C6)

obtain

immediately,

by

Theorem

IV.16a,

IV.3c a n d I I . 3 c ,

referring

and t h e f a c t

to

that

Conclusion

n +1>1 w e

IV.6,

formulas

<7>,

C8)

and

<3)

obtain

n+1 C < t C p < : s ^ , s ^ , . . , s ^ » , d ^ , . . . , d ^ ,dQ>.

C9>

In that

view

of

pCs-.,s.,...,s •^ 0 1

Conclusion

Definition n

>e

IV.20,

S, we

II.4b

and

formulas

whence, obtain

in

view

— Ac+1— pe S,

<4> a n d of

C9> w e

formula

which

was

C6> to

find and be

demonstrated. THEOREM

rCpyay '^ ^

IV.26b

s ** s.r

e S ^

<« •

^. n>l

^ — — ^0'*!'

— ^r '*n^

pC?^,?^,...,*^) .^ F •

pC?Q,F^,...,r^) A

^ 0
^j

*jk - rj^».

0
The

following

list

of

equivalences

results

from

ANNEX

221

Definitions

IV.26a

and

II.14b,

Conclusion

II.6a,

Theorem

IV.17a

and C o n c l u s i o n s IV.19, IV.18 and II.7: FCp/Q) s

**

p,qyr,s

/N Tdp^c^^ s ) <•

*^ •

[P] '^ i

-

[<j] '^ r «

V^ s ^^ " [P] -^ ^ " [<»] ^ r -

''O'^l

k

k

p,tt,r,s

n>l

«J •

rVj]

0<j
'^

L^J

TQ , r J , . - . , r^eS

A T "

Frl

^

s

•

fsT

yv s , r

e

S ^

s

^ r - pCrp,r^,.,.,r^> ^ q « s^. ^ p % r^ .. «»

V

V 0

1

p C s ^ , s ,...,s

^Qj

s^ % r^)

0
y -^

71 ^

•

0
[s]

Ci}a:s.-Np%r.^ J J

S Q , S ^ , . . . ,s^eS

0
^

[ r ] /s s -

[s]

'n^^

^ r • pCr„,r,,...,r ) ^ r.^^^ 0 1 n 0<j<7i

k^J

[r] ^ s -

n>l

-

-

D

^

-

i

n

^

'^^''^

e

S ^

s

0<_/

.00<*
«». - i'^»

-

pCSp,s^,....s^)

J

J

> -N

*•

^

^

222

ANNEX According

to

the

above

list

of

equivalence

both

sides

of

the

IV.26a

and

equivalence under consideration are equivalent. THEOREM IV.26C r-

h+i

fe > 0 •»
Conclusion

r Cpytf'> s

and

Let

Axioms

^ 11.10

IV.l

Jb

s ** _y_ fc>0. and

We

r Cp/q> s and

refer

the

to

i

to

r, %r_ 1 2

follows

prove

».

Definitions

relationship

(which

I.lc>

I

(. r s ^ v

t

from

the

-^ s^5i»_ 1 2

Definition

following

^

II.14b

list

of

equivalences and implications: Tc'p?qy

s

^ rCp/q>

-V

V

««

p.afr.s

s 4> p^q^r^s

c rCu/0*s

*"^ "

[p] •-- 5 " [ Q ] '^ r •

[r"] ^ s " [ s ]

^P "

[p] -^ 5 "

[r] .^ s -

/s u

*(

qi = [cj] /v r «" fr] ^ s «

>>

[<»] -^ ^ "

«•

V

[ s ] xv rCvyti

cp «

[ s ] /^

fpl .^

s ^ v

t ,v

^ \

%,t^r,s

A rCu/t' ^ * '

^ 7 -

[f J LIJ

f" -

M

-

' -

^ " P,<J,v^,«/^ -

^ w (p/jjj^t » 1"^^ 1

^

[0

-

«• -

[p] - g -

V p,
[r] -

'

[9] - V -

'1

-

M

-

[x,J ^

^

^

ANNEX

223

.^ Cf "

[cfl

A T "

fr")

^ s

•" r « l

- ^^CP/<J>^1 > -» p , q , r y s , t , v

A rCv/O

S / ^ v % V / v t % t ^

[P] - 9 -

[9] - ? -

[r]

-

— k i /^ s " rs"] /s r C v / O s j^ u

t ). It.

follows

from

implications

that

Cr(p/<j>

implies

the

equivalence

right the

s

side

the

for

in

the

the the

above

list.

of

assumption

expression theorem

At>0

which

under

equivalences

is

the

the

and

expression

right

consideration,

side and

of that

in t u r n

i m p l i e s a n e x p r e s s i o n which i s e q u i v a l e n t t o jt+i— CrCp/cj> s. This concludes t h e p r o o f o f Theorem

expression

IV.26C. THEORFM

pC/vyq

IV.29

^ p e Vr ^ p m C— .-> -1



V

.^ r e C— .^ p e C—

In

the

following

proof

of

this

theorem

we

avail

^ p = r^).

ourselves

lemmata:

LEMMA I V . 3

t e C— ^ f = f f l .^ p « rpT =» f e C , which f o l l o w s f r o m Theorem IV.20d, D e f i n i t i o n II.12d and LEMMA I V . 3 '

7 e C^ ^ 7 -

[ f ] /^ p -

[p] -» t e C"

of

the

224

ANNEX LEMMA

IV.4

P which follows LEMMA

«- J

f r o m T h e o r e m IV.20d, D e f i n i t i o n

IV.3'

follows

P r o o f

of

the

and

direct

[P]

Definition

directly

from

IV.21a

and

Definition

implication.

Let

the

Conclusions

IV.21a. antecedent

"pCjvyq true.

IV.29 t h a t

In

view

there

<2>

of are

this

assumption

expressions p "^3]

^^^

p^,p^ 9 -

it such

follows

from

Definition

that

[P2]

stnd

In v i e w o f <4>

<3> w e o b t a i n P3 e

from Definition

Kr ^ P 3 e

III.l

C^^

and '''0''l''2

^

O' 1

2

P2

^ P 3 e C^^ ^ P 3 *

F r o m C 2 ) - C 4 ) w e o b t a i n , b y C o n c l u s i o n IV.24 a n d L e m m a C6>

of

implication

<1> be

'- -'

from

I I . 1 6 a , b . L e m m a IV.4' f o l l o w s

that

II.12d

IV.4'

P Lemma

[P]

p iB Vr

^ "p e C—. 9

r^>.

IV.4,

ANNEX

225

A s s u m e t.hat, Cl.l)

V

cF -

p ^ c F , r , r > y. r

T,r^,r^,r^

0

s

C--

1 2

9

^ p e C—

It. f o U o w s

from

Cl.l>

« -

<1.3>

s

C1.4>

p e C- , *2

C1.5)

p »• s .

are

view

of

^Q'^-I'^2'^3

C1.6>

C1.2) such

Jg - [*J

it

-^ p = r . >.

that

<:i.2>

In

^

'p^Cs^,'s^,s^y,

e

C-, 9

follows

from

Theorem

IV.28

that

there

that

- =2 • [*2] -" ^3 "

- ?^ - [ s j

[=3]

and

By L e m m a IV.3 i t

foUows

<1.8)

from s

e C ^

a n d b y L e m m a IV.3 i t

foUows

in view o f

from

C1.4>, C2> a n d C1.6)

that

*2

( l . S ) , C2> a n d C1.6> w e

a.lO>

obtain

Pg % S j .

But C5).

, ^2

"* Further,

C1.4>, C2> a n d C1.6> t h a t

formulas

Hence

C1.7>-C1.10>,

assumption

Cl.l)

taken results

Jointly, in

a

contradict

formula

contradiction,

and

226

ANNEX

t h u s t-he r o r m u i a -I

is

V Cr B p ^ r e C— ^ p e C— ^ p « r^> —— — — 0 1 2 Q ^ r^ ^ 1

true.

That, f o r m u l a ,

t h e implication under P r o o f

in

viev

of

<6>, y i e l d s

the

consequent

of

consideration.

of t h e c o n v e r s e implication. Let

CI)

p e

^ p 6 C-

FT

and ,— —00,— — — Cr « p Cr„,r^,r_) /s r e C— ^ p e C— /N p • 0 1 2 Q r_ '^

C2)

r^) 1

'^'''0'''l'''2 Assume t h a t C3>

-. . By D e f i n i t i o n IV.29, a s s u m p t i o n C3) r e s u l t s

C4)

^ In

view

of

CD,

[ p ] /N q •> [<j] •* -1 Cp<:/v)q>).

Conclusion

t h a t t h e r e a r e p.rP^rP^

such

IV.24

p = [p^] ^ p^ €

C6>

p -

C8>

and

Definition

[Pg] - 5 -

C9>

and

[P2]'

C6> a n d C5> w e

C9>

Vr.

Pg e to

see

Vr,

a n d b y A x i o m III.3 a n d f o r m u l a s

referring

we

have -I

By

IV.21

that

C5>

By C4> and C6) we

in

Definition

III.l

and

obtain

formulas

C8>,

ANNEX

227

<7> w e c a n s t a t e

that

there

are

CIO)

S3 -

Cll>

s ^

<:i2>

,s

such

that

e

C

, *2

Pg a: s^.

Now

let

<14>

s

-

[53]

-

^Q -

[ s j

By T h e o r e m IV.28, T o r m u l a s <:iS>

s

It

follows

<:i3> a n d C14>

from

<1S>

Cp -

-

[ s j

-

=2 "

[*2]

p^C?Q,s^,S2^ IV.4

and

and

C16>

G C— ^ p e

yield

C2). T h i s c o n c l u d e s

THEOREM

Sj -

C14> a n d C10> y i e l d

Lemma

s

Now

-

formulas

Cll>,

C12>,

the

a

C—

formula

,s p =

s,.

which

is

in

contradiction

proof.

IV.30b

p<:pjj,p^,...,p^> .N pjj « O ^ c "

Cp » 'p^<.p^,p^,p^:>

< a , a ^ , . . . , a ^ , a j j ) > •-•

/>. P 2 «

s

^ PgC/vSp^

Po'Pi'P2'P3 -^ Pg P r o o f .

C6),

that

<16>

with

,s

e , P2

^ <:i3>

,s

p'^<s^,s^,if^>, e

p

s

TO prove

the

-3^ - ,., p^ ^ c C a , a ^ , a 2 , a Q ) ) .

direct

implication we assume

that

228

ANNEX

CO

p . It

follows

such

from

Cl> and

***S. Conclusion

XV.2S

that

there

is

a

<j

that

C2>

p - [<»^]

and J^+1

Thus,

in

view

t h e following

of

Definition

III.2b

and

formula

C3),

one

of

t h r e e c a s e s can hold:

Ci>

q^ e * S ,

Cii>

^^,

V


Po-Pj,

P ^ p ^ O ^

,P„e S ^ c

Ca,a^,...,a , a „ ) ) , 1 n o

-^ P3 * P^ -«• c Ca,a^,a2'"o In

case

,

Conclusion

— fe—

/>G S

IV.2S,

in

and

view

of

hence

C2)

the

and

in

accordance

disjunction

which

c o n s e q u e n t o f t h e i m p l i c a t i o n under c o n s i d e r a t i o n i s In c a s e the

Cii),

direct

the

proof

implication

being

proof

of

formula



IV.22b.

It

suffices

merely

of

Conclusion

IV.20

in

and

and a l s o t o make u s e o f Let

us

of

accordingly

the

fact

proved

is

case to

to

Cli>

refer Lemma

that true

of to III.l

the is

the

the

true. consequent

modelled

proof

Conclusion Instead

is

with

on

of the

of

Theorem

IV.25

instead

of

Lemma

II.2,

view

of

Axiom III.2 and Conclusion IV.23. consider

case

Ciii>.

In

ANNEX

229

C o n v e n t i o n II.3c a n d Cill) w e C4)

q^ -

It

p -

follows

Definition

from

III.l

we

r

^0 "

and

follows C3)

formulas IV.Sc w e

C4>

5 ^ ^^'^f^^^2

that

[^o] from

and C6>

9 j " P <:r_,r , r >.

e

O /s r

e

^ 'l

"

•" ^ 2 "

Conclusions

Lemma and

[^ll

,

III.l

Kr A r

e

Hence

by

£.

Theorem

['2]

III.la,b

that

r

IV.16a

'^ ^ 3 "

aind

['3]

II.5b,

formulas

C5>

Hence

by

II.3c

and

, r , r , 4 eD. and

^ ^2~

•=0 " t o ] '^ ^1 • t J In

view

state

that

of

Theorem

IV.28

and

Conventions

view

of

Definition

IV.29

[''2] ^ " " [**!]

formulas

F r o m C o n c l u s i o n IV.2S a n d f o r m u l a s

in

'l

have

<7>

and

'^ ""s *

that

<*' It

r^ e

have

C5) Note

have

and

Formulais

formulas

TgCTuSr . C4> a n d C6> y i e l d

C2>

C6) a n d C4> w e

have CIO)

(65,

immediately

C6)

and

C4>

we

C4>

we

obtain

and

230

ANNEX

Further a n d C4>

Now the

conclude

formulas

Ciii>.

Definition

IV.8

and

formulas

C7>

of

the

taken

Jointly

implication

concludes

the

allow

being

proof

us

proved

of

the

to is

state also

direct

that

true

in

Implication

IV.30b.

prove

disjunction

the

converse

which

consideration must

C8)-C12>

This

Theorem To

from

that

consequent

case of

we

is

is

implication

the

true.

right

Hence

side

one

of

we of

the

aissume the

that

the

theorem

following

under

three

cases

hold:

XJ^+I^

(.***->

_

_ V^

_ *^? "

-

p"*^^o'^l'^2"* "^ ^ 2

—

"^

^

^3*^^-*^2

P j '^ C


S in each of these

It must be demonstrated that pe

^

'^ ^ ^ ^ ^

—^

'^ P 3 •

^

^^

three

cases. In and

case

Cv>.

— h-*A—

pe

Definition

analogical we

C*>

to

assume It

III .2b.

the

the

S

proof

truth

suffices

to

follows The of of

immediately

proof the the

refer

of

left

to

— >t-H— p« S

side

second

from

of

in

case

Theorem

constituent

Conclusion

Conclusion

of

IV.25

<*•>

IV.22b

IV.25 is when

disjunction instead

of

ANNEX

231

Conclusion

IV.20

and

to

Definition

III .2b

Instead

of

Definition

II.4b, and a l s o t o make u s e o f C o n c l u s i o n IV.23. C o n s i d e r now c a s e (••*>. Then we h a v e

<1>

P - i°"<^*o'*l'*2'*'

C2>

s^

e

<3>

*S,

s^CjG>s^,

C4>

.3

By

Theorem

IV.28

it

-

.^,

follows

from

Cl>

that

there

are

Conclusion

IV.25

*0'*1'*2''*1 ^""^^ t h a t C6)

=^0 " C*o] ^ *1 " [ ' J ^ ^2'

1^2^

and

By

formulas

<3>

and

<2>,

Definition

and Lemma 1II.2 t h e r e a r e s^.s^ s u c h 4 4

IV.29,

that

^2 - [*43 " ^3 " [^5]

^8^ and

5

It

follows

from

C6)

and

4

C8>

4

that

'o**^?

and

hence,

if

we

assume t h a t (10) then

p^ -p"CSjj,s^,s^>, by

C7>,

Axiom

I.la

and

Then i t f o l l o w s f r o m C7) t h a t

Conclusion

III.3

we

have

p Sip

232

ANNEX

CIO

p -

[pj.

I t a l s o f o l l o w s from <4>, C8> and C6> t h a t <12>

* g S. *^.

It

can

belong

to

easily DC L >.

be

seen

Hence

II.3c and in view o f

In

By

availing

the

accordance

expressions

with

Conventions

^ri'^i'^a.'^-? IV.3c

and

Cll), C6), <8> and Theorem IV.16a we have

- M

CIS)

that

- '^o - C''o] - ''i - K ] - '^2 - M

ourselves

of

Conclusion

IV.6

we

obtain

III.2b

and

formulas

S,

whence

from

C5)

and C13>

By C12> and

referring

and

C14>

formula

The p r o o f

we

Cll>

to

Definition

find we

that

have

h a s t h u s been

p e

— k+l— pe S, completed.

which

was

by to

C10>,

Conclusion be

C9), IV.25

demonstrated.

NOTES

only. theory

When A

speaking

formalized of

C2>

<:3>

The

C1981b]

read

dedicated

here

at

the

the

27th

works

word

of

graphic

languages

languages

Can

axiomatic

[19671.

"part"

in

the

sense

drawn

in

my

paper

of

Logic,

11927-311.

TSCL w a s

Conference

of

mean

by T.Bat6g

mereology

outline

we

phonic

the

Le^niewski's

essential

to

of

was presented

interpret

from Stanisiaw

languaiges

t-heory

phonology)

We

about

presented

on

Kazimierz

the

History

AJdukiewicz,

on

October

16,

1981. C4>

See

M.J.Cresswell <;5>

The

for

instance

£19611,

Y.Bar-Hillel

[1964],

[1973,19771. definition p e

is

J.Lambek

dual t o

Definition

<6>

See

for

/

=• <\p\

« n «» p e

7)

1.3. instance

R.Montague

[19701,

M.J.Cresswell

1 1 9 7 3 , 1 9 7 7 1 , A . N o w a c z y k [19781. <7> OCt> i s

We

shall

later

a non-empty

C8>

See

for

demonstrate

Csee

Conclusion

II.9d>

that

set. instance

T.Kubirtskl

[1959],

B.Diankov

[1974,

19791. <9>

See

for

instance

K.AJdukiewicz

[19351,

Y.Bar-Hillel

[19531. <10> guarantee

Note

in

that

connection

with

compound

words 233

this in

that JP

Axiom are

II.9 its

does

not

compound

234

NOTES

expressions.

Hence

there

an

which

we

are

not.

and

its

arguments.

index

the

In

main

functor

w h i c h ax-e

structurally

Cll)

These

C12>

See

C13)

may

are

K.AJdukiewicz for

in

able

£

such

words

unambiguously We

mean

which

to

have

distinguish

here

expressions

ambiguous.

definitions

See

occur

given later

on in t h i s

chapter.

tl93S].

instance

M.J.Gressweil

[1973],

p.71

and

C1977]

p.258. C14)

Cf. R . S u s z k o C19S8], p . 2 3 6 .

C15> next

An

essentlatl

chapter.

(not

very

The

use

adequate

a

differs

LeSniewski's

mereology. Cf.

these

intuitive

AJduklewicz>

C16>

of

sense

definitions of

translation

from

the

K.AJdukiewicz

the of

meaning

119601

is

word the

of

and

the

made

in

the

"constituent"

term

used

by

word

"part"

in

W.Marclszewskl

l\977,

1978a]. C17>

Not n e c e s s a r i l y

<18> the

We

zero

can

order

<19>

adopt

of

Cf.

unambiguous.

the

the

convention

expression

J.Stupecki,

that

p occupies

K.Halkowska,

the the

constituent

of

p o s i t i o n O.

K.Pir«5g-Rzepecka

[1976]

p.278. C20)

Cf. P . D e m b l r t s k l , J . M a i u s z y r t s k i

C21> section. and

Examples Those

hence

compound derivation

of

trees

concrete

trees have

as

labels

expression tree

are

Csee

p

is for

to

their

<see a

[1981], pp.11-2. be

found

elements

Figs. graphic

instance

further

in

expression

II.1-II.6).

Tree

illustration J.E.Hopcroft,

this tokens

I of

of

a the

J.O.Ullman

NOTES

235

C1969])

treated

as

a

Is

set.

all

those

t-he

Definition

of

II.12>,

and

subordination formula

q—•r

the

linear

CC ,—•>, P

relation

p

of

can

of

of

graph

p

which

relatvon

easily

p which i s

the

the

all

are

might

of

of

(.C ,—•>, are

of

Its

where parts

direct

[19601>,

be

elements

set

defined

C Csee

syntac

Its

the

that the

part

tree

tical

by

I

nodes

ordering

relation

formal,

formulated

its

of

constituents

parts;

termed

be

of

those

its

position.

definition element

ordering

which

the

ordering

which

tic

is

a

the

^ QGC .

consisting

expression

syntac

an

as

K.Ajdukiewicz

4* q , r e C

Such

namely

elements

—y

Csee

C22> to

graph,

is

of

the

tree

of

a

given

is of

done

we

occupies

in

a

earlier

complicated,

consider it

by

an

somewhat if

related

that

every

a

fixed

syntactic

of

one

argument.

position. (23) They

Such

operators

include,

operator,

the

e.g.,

are

ordinary

algebraic

such

also

tfor

p.22>.

For

discussed instance

o p e r a t o r s of two C24>

The

Some

quantifiers, symbol

operators

those

the J] of

,

description the

integral

n att>guments

are

Tt>l>

<see

L.Borkowski

C1958,

19703

quantifiers

with

restricted

range

are

arguments.

symbolic

formulation

of l i t t l e p r a c t i c a l u s e , i s o m i t t e d <25)

operators

summation

s y m b o l S, e t c . Next t o being

called

comments

on

that

of

this

definition,

which

is

here. matter

are

to

be

found

both

the

in

my

p a p e r [1983b], p p . 3 0 8 - 9 . C26) TSCL

of

As we h a v e s e e n ,

TETk i s

simple

languages

categorial

the

core

and

the

of

theory

theory

TSCc»-L

of

236

NOTES

a>- I a n j ^ u s f ^ e s . (27) the

Conventions

sets

Tb,

I,

IV.2a,b,

IV.3a-c>.

<28)

This

theorems

of

(29)

For

definition

E,

is

dual

of

length

am a b s t r a c t

(305 sequence

why

The

verbal

will o f t e n of

later

be

range

<see

Conventions

of

over

induced

omitted. we

/>,

is

It

which

rormulations

precision

pe

1.3.

variables

given

the

«^

Definition

the

are

sake

•* C\p\mn

analogon of

2

theories the

p^I

concerning

which defines

add

to

TETp

the

by

the

dual

concept

of

the

induced the

index.

expamsion

of

P/^yp^f-,P

of

word

The

expressions

the

expression

types

of

P^Pf.,p^,--,p

which

that

5

is

the

expression

consists. <31> analogically

rCpycfys.

to

the

r

s

corresponding

and

T

s

expressions

are

rCp/q)

read s

and

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Nijhoff International Philosophy Series 1. N. Rotenstreich: Philosophy, History and Politics. Studies in Contemporary English Philosophy of History. 1976 ISBN Pb 90-247-1743-4 2. J.TJ.

STUSAIUCVX:

Elements of Social and Political Philosophy. 1976 ISBN Pb 90-247-1744-2

3. W. Tatariciewicz: Analysis of Happiness. Translated from Polish by E. Rothert and D. Zielirfski. 1976 ISBN 90-247-1807-4 4. K. Twardowski: On the Content and Object of Presentations. A Psychological Investigation. Translated from Polish with an Introduction by R. Grossmann. 1977 ISBN Pb 90-247-1926-7 5. W. Tatarkiewicz: A History of Six Ideas. An Essay in Aesthetics. Translated from Polish a.o. by C. Kasparek. 1980 ISBN 90-247-2233-0 6. H.W. Noonan: Objects and Identity. An Examination of the Relative Identity Thesis and Its Consequences. 1980 ISBN 90-247-2292-6 7. L. Crocker: Positive Liberty. An Essay in Normative Political Philosophy. 1980 ISBN 90-247-2291-8 8. F. Brentano: The Theory of Categories. Translated from German by R.M. Chisholm and N. Guterman. 1981 ISBN 90-247-2302-7 9. W. Marciszewski (ed.): Dictionary of Logic as Applied in the Study of Language. Concepts - Methods - Theories. 1981 ISBN 90-247-2123-7 10. I. Ruzsa: Modal Logic with Descriptions. 1981

ISBN 90-247-2473-2

11. P. Hoffhian: The Anatomy of Idealism. Passivity and Activity in Kant, Hegel and Marx. 1982 ISBN 90-247-2708-1 12. M. S. Gram: Direct Realism. A Study of Perception. 1983 ISBN 90-247-2870-3 13. J.T.J. Srzednicki, V.F. Rickey and J. Czelakowski (eds.): Le^niewski's Systems. Ontology and Mereology. 1984 ISBN 90-247-2879-7 For 'The Le^niewski Collection' see also Volume 24. 14. J. W. Smith: Reductionism and Cultural Being. A Philosophical Critique of Sociobiological Reductionism and Physicalist Scientific Unificationism. 1984 ISBN 90-247-2884-3 15. C. Zumbach: The Transcendent Science. Kant's Conception of Biological Methodology. 1984 ISBN 90-247-2904-1 16. M. A. Nottumo: Objectivity, Rationality and the Third Realm: Justification and the Grounds of Psychologism. A Study of Frege and Popper. 1985 ISBN 90-247-2956-4 Volumes 1-8 previously published under the Series Title: Melbourne International Philosophy Series.

Nijhoff International Philosophy Series 17. i. Dilman (ed.): Philosophy and Life. Essays on John Wisdom. 1984 ISBN 90-247-2996-3 18. J. J. Russell: Analysis and Dialectic. Studies in the Logic of Foundation Problems. 1984 ISBN 90-247-2990-4 19. G. Carrie and A. Musgrave (eds.): Popper and the Human Sciences. 1985 ISBN Hb 90-247-2998-X; Pb 90-247-3141-0 20. C. D. Broad: Ethics. Lectures given at Cambridge during the Period 1933-34 to 1952-53. Edited by C. Lewy. 1985 ISBN 90-247-3088-0 21. D.A.J. Seargent: Plurality and Continuity. An Essay in G.F. Stout's Theory of Universals. 1985 ISBN 90-247-3185-2 22. J.E. Atwell: Ends and Principles in Kant's Moral Thought. 1986 ISBN 90-247-3167-4 23. J. Agassi, and I.C. Jarvie (eds.): Rationality: The Critical View. 1987 ISBN Hb 90-247-3275-1; Pb 90-247-3455-X 24. J.T.J. Srzednicki and Z. Stachniak: S. Le^niewski's Lecture Notes in Logic. 1988 ISBN 90-247-3416-9 For 'The Le^niewski's Collection' see also Volume 13. 25. B.M. Taylor (ed.): Michael Dummett. Contributions to Philosophy. 1987 ISBN 90-247-3463-0 26. A. Z. Bar-On: The Categories and the Principle of Coherence. Whitehead's Theory of Categories in Historical Perspective. 1987 ISBN 90-247-3478-9 27. B. Dziemidok and P. McCormick (eds.): On the Aesthetics of Roman Ingarden. Interpretations and Assessments. 1989 ISBN 0-7923-0071 -8 28. J.T.J. Srzednicki (ed.): Stephan Korner - Philosophical Analysis and Reconstruction. Contributions to Philosophy. 1987 ISBN 90-247-3543-2 29. F. Brentano: On the Existence of God. Lectures given at the Universities of Wurzburg and Vienna (1868-1891). Edited and translated from German by Susan F. Krantz. 1987 ISBN 90-247-3538-6 30. Z. Augustynek: Time. Past, Present and Future. Essays dedicated to Henryk Mehlberg. (in preparation) ISBN 0-7923-0270-2 31. T.Paw\owskv. AesthetiQ Values. 19^9

ISBN 0-7923-0418-7

32. M. Ruse (ed.): What the Philosophy of Biology Is. Essays dedicated to David Hull. 1989 ISBN 90-247-3778-8 33. J. Young: Willing and Unwilling. A Study in the Philosophy of Arthur Schopenhauer. 1987 ISBN 90-247-3556-4 34. T. Z. Lavine and V. Tejera (eds.): History and Anti-History in Philosophy. 1989 ISBN 0-7923-0455-1

Nijhoff International Philosophy Series 35. R.L. Epstein: The Semantic Foundations of Logic. Volume 1: Prepositional Logics. With the assistance and collaboration of W.A. Camielli, I.M.L. D'Ottaviano, S. Krajewski and R.D. Maddux. 1990 ISBN 0-7923-0622-8 36. A. Pavkovic (ed.): Contemporary Yugoslav Philosophy. The Analytic Approach. 1988 ISBN 90-247-3776-1 37. A. Winterboume: The Ideal and the Real. An Outline of Kant's Theory of Space, Time and Mathematical Construction. 1988 ISBN 90-247-3774-5 38. K. Szaniawski (ed.): 77i^ Vienna Circle and the Lvov-Warsaw School. 1989 ISBN 90-247-3798-2 39. G. Priest: In Contradiction. A Study of the Transconsistent. 1987 ISBN 90-247-3630-7 40. J. Wolenski (ed.): Kotarhinski: Logic, Semantics and Ontology. 1990 ISBN 0-7923-0865-4 41. P. Geach (ed.): Logic and Ethics. 1991

ISBN 0-7923-1044-6

42. U. Wybraniec-Skardowska: Theory of Language Syntax. Categorial Approach. 1991. ISBN 0-7923-1142-6

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