The Notion of Analytic Truth [Reprint 2016 ed.] 9781512817829

Citation preview

The Notion of Analytic Truth

THE NOTION OF ANALYTIC TRUTH by

R. M. Martin

PHILADELPHIA UNIVERSITY OF PENNSYLVANIA PRESS

© 1959 by t h e Trustees of t h e University of Pennsylvania Published in Great Britain, India, and Pakistan by t h e Oxford University Press London, Bombay, and Karachi Library of Congress Catalogue Card N u m b e r : 58-9516

Printed in Great Britain by W . & J. Mackay & Co Ltd, C h a t h a m

For RUDOLF CARNAP

with gratitude and affection

That sir which serves and seeks for gain, And follows but for form, Will pack when it begins to rain. And leave thee in the storm. But I will tarry; the fool will stay, And let the wise man fly: T h e knave turns fool that runs away; T h e fool no knave, perdy. —King Lear,

II, iv

Contents Introduction

xiii

I.

Semantical Substructure A. Object-Languages B. Syntax C. Multiple Denotation D. Non-Translational Semantics

1 2 5 11 15

II.

Analytic Truth A. Semantical Rules and Analytic Truth B. The Formal Definition C. Identity D. Logical Theorems and Analytic Truth E. An Alternative Definition F. Logical Truth and Logical Constants

22 22 25 31 56 39 41

III. Adequacy A. Non-Logical Axioms and Analytic Truth B. Set Theory C. Other Object-Languages D. Adequacy of 'Anlytc' E. Godel's Completeness Theorem

46 46 47 56 58 64

IV. L-Semantics A. Carnap's Postulates B. Some Derivative Notions C. Quasi-Intensions D. Carnap on State-Descriptions E. Meaning Postulates F. On Abstract Entities in Semantics

73 73 78 80 82 87 91

V.

Some Alternative Characterizations A. Frege on 'Analytisch' B. Validity C. Tarski on the Notion of Logical Consequence

95 96 99 102

C O NT E N T S

D. Quine on Vacuous Variants E. Analytic Sentences and the A Priori F. Some Concluding Remarks

107 113 116

Short Bibliography

119

Index

121

Special Symbols

123

The Notion of Analytic Truth

Introduction

I

n one way or another notions of logical or analytic truth have played an important role in logic, and hence in philosophy, since the time of Aristotle. An analytic truth is usually described as a statement true in virtue of logic, or true in virtue of the meanings of the terms occurring in it. A synthetic truth is then described as one which depends for its truth fundamentally upon matters of fact. Another way of stating this, that of Leibniz, is that analytic truths (vérités de raison) are true in "all possible worlds", whereas non-logical or synthetic truths (vérités de fait) are true only of the actual world. All of these characterizations are somewhat vague and in need of a precise formulation. But howsoever formulated, the distinction between analytic and synthetic truth has in one form or another been very important historically and is essential for the contemporary philosophy of science. Several recent attempts have been made to develop a theory of analytic truth in a precise way and in accord with modern standards of rigor. For this an extensive use of the new logic is essential. All of these methods, however, make use of logicomathematical devices of great complexity. The so-called functional calculi of higher order are employed fundamentally. In fact, the methods usually used are so powerful as to be roughly equivalent with those employed in constructing the mathematical theory of real numbers from a foundation in integers. That the theory of analytic truth, which is and should be essentially simple in structure, should require such powerful methods seems somewhat anomalous. In this little book a theory of analytic truth is developed using methods much simpler and more economical than those usually xiii

xiv

INTRODUCTION

employed. T h e underlying logic is of first order, and the semantical assumptions are of an especially simple and straightforward kind. Under investigation is a wide class of object-languages, namely, the simple, applied, functional calculi of first order in essentially the sense of Church. This class of languages is thought to be of especial interest for applications to science and philosophical analysis. All t h e meta-languages to be formulated here are also first-order languages of this kind. Further, they are extensional in an appropriate sense, so that neither intensional meta-languages nor meta-languages of higher order are needed for t h e theory of analytic t r u t h for first-order object-languages. T h e semantical basis needed has been formulated in detail elsewhere. In order to make this book self-contained, however, a brief sketch of the underlying semantics is given in Chapter I. In Chapter II the definition of 'analytic in L', for first-order L, is given. T h e adequacy of this definition will be discussed in Chapter III and in fact shown to hold in an appropriate sense. In Chapter IV, various notions depending upon that of 'analytic in L ' are defined. T h e theory of these notions constitutes essentially what Carnap calls L-semantics, the semantics of logical notions or concepts. Finally, in Chapter V, the definition of 'analytic in L' put forward is compared and contrasted with several alternative definitions. T h e advantages of the present definition will t h e n appear. Also we shall see how it incorporates certain desirable features to be found in one or the other of these alternatives. T h e author is beginning with rigorous terms Hilbert-Bernays, Church, Quine,

very m u c h indebted to the various writers who, Frege, have been attempting to clarify in the notion of analytic t r u t h , in particular, Hilbert-Ackermann, Gtidel, Tarski, Carnap, Beth, Scholz, Henkin, and Remeny. More

INTRODUCTION

XV

specifically, t h e definition to be put forward here m a y p e r h a p s be regarded as a " r a t i o n a l reconstruction" of t h e kind of definition given by Hilbert-Bernays. But the semantical m e t a l a n g u a g e s here, in which t h e definition of b e i n g analytic m a y be g i v e n , are not only fully formalized, they are of a very restricted kind. A very considerable reduction in the axiomatic s u b s t r u c t u r e as well as in the n u m b e r and kinds of entities needed as values for variables is achieved. T h e interest in such restricted s e m a n tical m e t a - l a n g u a g e s is perhaps mainly philosophical in character, but there are important conceptual links with interest in constructivistic and intuitionistic methods in m a t h e m a t i c s . T h i s book is written primarily for philosophers and methodologists of science who are interested in theoretical semantics. It is self-contained, no previous familiarity with the subject being presupposed. Proofs of m a n y of the most i m p o r t a n t t h e o r e m s will be given in outline, although s o m e will be omitted altogether. But there will be no difficulty in s u p p l y i n g these w h e r e needed, only the most elementary logic being presupposed. T h e author wishes to thank Professor E. W . Beth for s o m e helpful criticisms and suggestions, and M r . G. F. R i e m a n , J r . and M r . L a r s Svenonius for a Gegenbeispiel to a purported theorem. Also t h e author is indebted to the Research C o m m i t t e e of the University of Pennsylvania for a g r a n t covering clerical aid, and to the University of Chicago Press, H a r v a r d U n i v e r s i t y P r e s s , Oxford University Press, the A m e r i c a n A c a d e m y of Arts and Sciences, Blackwell's, L o n g m a n s , G r e e n & Co. L t d . , and t h e Chelsea P u b l i s h i n g Co., for kind permission to quote f r o m works published by t h e m .

I Semantical Substructure

T

o give an exact and careful definition of the notion of being an analytic truth, we must characterize first the languagesystems under consideration. To be analytically true is to be analytically true within a given language-system. The languagesystem is the object-language, and presumably no definition of being an analytically true sentence can be given unless it is relativized to some object-language or other. The language in which the definition of being analytically true is given, on the other hand, is a meta-language, more particularly, a semantical meta-language containing a syntactical meta-language as a part. To give a definition of analytic truth relativized to a specified object-language presupposes that the syntax and semantics of that object-language have been fully formulated. In the syntax of a language definitions of 'term', 'formula', 'sentence', 'axiom', 'theorem', 'sentence', and similar notions are given. And then in semantics the expressions of the language are interrelated with the objects about which it speaks. Certain of the expressions are said to denote certain objects, and others, the variables, may be said to range over a certain domain of objects. Also within the semantics we may introduce the semantical truth-concept by means of an exact definition. Analytic truth is a special kind of truth. One might reasonably hope, therefore, to characterize a sentence as analytically true in N.A.T.-B

1

2

THE

NOTION

OF A N A L Y T I C

TRUTH

a given object-language L if and only if it is true in L (in the sense of the semantical truth-concept) and, moreover, such and such other conditions hold of it. Such a definition we attempt to formulate here. Any such definition can be given only within a semantical meta-language of L already containing a truthconcept for L. In this Chapter the semantical meta-languages employed throughout are sketched very briefly.1 In §A the object-languages studied throughout are outlined. The syntax of these object-languages is discussed in §B. One method of handling the semantics of these languages is based on the relation of multiple denotation. This method is sketched in §C. Finally, in §D, another method of providing the semantics of L is considered, of a non-translational kind. A. Object-Languages. The object-languages L we consider are simple, applied, functional calculi of first order.2 For the moment we may suppose them to contain no primitive (nonlogical) functional or individual constants. As non-logical primitive predicates (or predicate constants) of L, we may use ip> Y) . (Y => X))'. As rules governing the truth-functions we have the following. TFR1. TFR2. TFRJ. TFR4.

v Y).* l - ( l v l ) 3 X. h (X v Y) => (YY X). M X => Y) => ((Z v X ) => (Z y Y)). (X

1 The use of bold-face letters as informal syntactical variables is due to Church. Single quotes enclosing a symbolic context containing a boldface letter are used in the sense of Quine's quasi-quotation. See his Mathematical Logic, Revised edition (Harvard University Press, Cambridge: 1951), pp. 55-37. • The ' h ' is Frege's sign of truth-assertion and is to be read 'is a theorem'. It is to absorb all single quotes around the symbolic context to which it applies.

4

T H E NOTION OF A N A L Y T I C

TRUTH

Each of these rules stipulates an infinity of f o r m u l a e of L as axioms. An occurrence of a variable x in a f o r m u l a X is said to be a bound occurrence of that variable in X if and only if it occurs in X in a part of t h e f o r m '(x)Y'. Otherwise an occurrence of x in X is said to be free. Concerning the quantifiers w e need the following two rules, each stipulating an infinity of axioms of L . QRl. I- (x)X => Y, if Y differs f r o m X only in containing free occurrences of a variable wherever there are free occurrences of x in X. QR2.

I- (x)(.Y v F ) .=>. X v (-r)F, if there are no free

occurrences of x in X. As Rules of Inference Rule of Generalization. MP. Gen.

we need only Modus

Ponens

and the

If h X and h X ^ Y then h Y. If I- X t h e n h {x)X.

An operation of abstraction is to be taken as a primitive operation also, and Axioms of Abstraction as additional logical axioms. L e t ' y v X ' be a one-place abstract, '3' being an additional logical primitive. 1 T h e Axioms of Abstraction needed are then as follows. Abst. h ysX x . = . Y, where y is a variable not free in Y, and X differs from Y only in containing free occurrences of y in 0 or more places where there are free occurrences of the variable x in Y. In a similar way, two-place, three-place, etc., abstracts m a y be introduced. Concerning higher-place abstracts, we a s s u m e corresponding Axioms of Abstraction. T h u s 1

The '3' is used here approximately in the sense of Peano.

SEMANTICAL

5

SUBSTRUCTURE

h xysX

zw . = . Y, if (etc., as needed),

V xyzzX

wux'

. = . Y, if (etc., as needed),

and so on. T h e s e rules provide what we may call Extended Axioms of Abstraction. W e let lFn', ' G n ' , etc., hereafter designate arbitrarily any primitive «-place predicate constant or any «-place abstract containing no free variables. One-place abstracts stand for what we may call informally virtual classes, two-place abstracts for virtual dyadic relations, and so on. Virtual classes and relations are not values for variables and hence cannot be regarded as genuine classes or relations within the systems here. But they behave very much like genuine classes and relations, and it will be convenient therefore to have a notation for them. If desired, the identity sign ' = ' may be taken as an additional logical primitive. As rules of logic we have then also the following, stipulating Axioms of Identity. IdRl.

I- x =

x.

IdR2.

b x = y . z3Xx

: => : z3Xy,

if X is an atomic formula.

W e m a y let 'F1 =* G 1 ' abbreviate

x . = . G1

x)'.

This definition introduces a notation for the identity of virtual classes. In addition to the rules providing for the logical axioms of L, we have also some non-logical rules characterizing specifically the primitive predicates. But for the moment these need not be given. B . S y n t a x . T h e syntax of L may be provided within a formalized syntactical meta-language by an adaptation of the methods

6

T H E N O T I O N OF A N A L Y T I C

TRUTH

of Chwistek, Tarski, and Quine. 1 As primitives of this metalanguage we need structural-descriptive names of each of the primitive signs of L: 'lp' for '(', 'rp' for ')', 'pee' for 'P', 'ex' for 'ac' for 'vee for V , 'tilde' for W , and 'invep' for V . (If ' = ' is a primitive of L, we should need a structural-descriptive name, say 'id', for it here.) Also an operation of concatenation, symbolized by T V , is taken as undefined. L e t 'a', 'b', 'c', 'd', and V , with or without numerical subscripts or accents, be the syntactical variables ranging over the expressions of L. Then (aC\b), read 'a concatenated with b', is the result of writing the expression a followed immediately by b. This syntactical metalanguage we may call M . T h e underlying logic of M is that of the preceding chapter with identity. In addition to the rules of the underlying logic including identity, we have Syntactical Rules as follows.* SynRl and SynR2 are Rules of Distinctness. SynR) gives the circumstances under which two concatenates are identical, and SynR4 is a Rule of Infinite Induction. SynRl. h ~ lp = rp . : F 1 c .v. (Ed)(Ei)(d Pr bc.e Pr b c . A)) . {c){F* c . =>. G 1 c) . (cXdXeXG1 d . G1 e . A : : G 1 c) Gl a, where A is a formula of M containing 'c', 'd\ and as its only free variables. Also TB2. h T h m a . T h m b . c MP a,b :=> : T h m c, TB). f-Thm a . b Gen a :'=>: T h m a, TB4. h Fmla a . Vbl b : =>: T h m (a hrsh b exisqu a), TBS. I- Fmla a . Fmla b . Vbl c . ~ c FV b : = : T h m (c qu (ia vee b) hrsh (c qu a vee b)). TB6. h Fmla a . Fmla b . Fmla c . Vbl d . ~ d FV a . ~ d FV c . T h m (a hrsh (b vee c)) : 3 : T h m (a hrsh {d qu b vee c)). These few theorems are cited merely as examples. Some of them will be useful below. C. Multiple Denotation. We may now form a semantical meta-language for L by augmenting the syntax language M to include L itself (or a translation of L) and a theory of multiple denotation. In addition to the primitives of L, the semantical meta-language includes then those of M and 'Den', to symbolize

12

T H E N O T I O N OF A N A L Y T I C

TRUTH

multiple denotation. The semantical meta-language, to be called S M p , thus contains roughly speaking four parts, (1) a basic logical part, (2) a syntactical part, (3) a translational part (i.e., L itself), and (4) a specifically semantical part. 1 We have two styles of variables in S M t h e translational variables 'x', 'x", etc. (as in L), ranging over the objects of L, and the expressional variables 'a', etc. (as in M), ranging over the expressions of L. 'a Den x' is to be significant and is to express that a denotes the object x. 'a Den x' is of course significant for all a, but the important uses of phrases of this kind will be where a is a one-place primitive predicate constant or a oneplace abstract containing no free variables. To say that a Den x is to say in effect that the predicate a extends over or applies to the object x. Note that x is not a class here, or a property or relation, because only individuals are denoted in the sense considered. The Rules of SMfe are of four kinds, corresponding to the four parts of the meta-language distinguished above. There are logical rules providing for a simple, applied, functional calculus of first order with two styles of variables, 2 the Syntactical Rules SynRl—SynR4, and Translational Rules stipulating as axioms translations of the descriptive axioms of L or other non-logical axioms from which these descriptive axioms may be proved. Finally, there are specifically Semantical Rules governing multiple denotation. These are the following. 1 This meta-language is, in essential respects, an adaptation of one studied by Tarski. See Der fVahrheitsbegriff, §5, where a predicate for truth rather than ' D e n ' is taken as a primitive. See also Martin, op.cit., pp. 9 9 - 1 4 2 . * See A. Schmidt, " U b e r Deduktive Theorien mit Mehreren Sorten von G r u n d d i n g e n , " Mathematische Annalen 115 (1958): 4 8 5 - 5 0 6 ; and H. W a n g , " L o g i c of Many-Sorted T h e o r i e s , " The

Journal

of Symbolic

Logic. 17 (1952): 105-116.

SEMANTICAL

13

SUBSTRUCTURE

SemRl. h (x)(a Den x . = . X), if X is a sentence or formula from the translational part of SMLD the only free variable of which (if any) is x, and 'a' is taken as the structural description of the abstract 'xsX'. SemR2.

I- a Den x .

PredConOne a.

From these two Rules, presupposing those of the underlying first-order logic together with the Syntactical and Translational Rules, the requisite semantical properties of L may be proved. The consistency of SemR\—SemRl relative to that of M and (the translation of J L may easily be shown. 1 In terms of 'Den', the semantical truth-concept may immediately be defined. 'Tr a ' abbreviates Den x))'.

'(Sent a . (EZ>)(Vbl b . (x){b C^invep C\a)

This truth-predicate may be shown adequate in the sense that b Tr a . = . X, where X is any sentence from the translational part of SM^ and 'a' is taken as the structural description of the sentence of L of which X is the translation. The proof of adequacy is by SemRl— SemR2. Thus we have TCI. (-Tr a . ^ . X, if (etc., as required). Some further important consequences of SemRl may be listed as follows. TC2. TC). TC4. TC5. TC6. 1

and SemR2

(- Sent a : =>: ~ Tr a . = . Tr (tilde a). (- Sent a . Sent b : =>: Tr (a vee b) . = . (Tr a .v. Tr b). h (Sent a . Ax a) .v. (Eb)(Ax b , a Clsr b) Tr a. h Tr a . b Gen a Tr b. V Tr a . Tr b . c MP a,b : = : Tr c.

See A. Tarski, Der IVahrheitsbegriff, §5.

14

T H E NOTION OF A N A L Y T I C

TC7. TC8. TC9. TC10.

TRUTH

h Sent a . Thm a P : T r a . I- Thm a . b Clsr a : =>: Tr b. Y Sent a .=>. ~ (Tr a . Tr {tilde a)). h ~ (E«)(Sent a . Thm a . Thm (tilde a)).

Within SM£ we may introduce abstracts, just as we did in L and M. Translational abstracts will be translations of those of L, and expressional abstracts those of M. Here also we may have mixed abstracts if needed. E.g., 'byiX ax' may be regarded as significant and as equivalent to an appropriate Y. (In writing translational expressions, we shall frequently use y , 'z', etc., as variables in addition to 'x", 'x'", etc.) A sentence is false in L if and only if it is not true in L. 'Fls a' abbreviates '(Sent a . ~ Tr a)'. If one or more individual constants together with ' = ' are available as primitives of L, we may define within SMfc the relation of designation for individual constants as follows. Let 'InCon a' abbreviate that a is a primitive individual constant of L. Then 'a Des x' may abbreviate '(InCon a . (EA)(Vbl b . (b(~\invepC\ br\idf\a) Den x))\ Concerning designation we have the following principles. TCll. I- a Des x .=>. InCon a. TC12. h a Des x, where 'x' is taken as a primitive individual constant in the translational part of the meta-language and 'a' is taken as its structural description. TCI). I- a Des x . a Des y : : x = y.

SEMANTICAL

SUBSTRUCTURE

15

This brief sketch of SM£ is merely by way of example. Any alternative semantical meta-language of the same general type may be presupposed equally well in its stead. D. Non-Translational Semantics. The semantical metalanguage of L outlined in the preceding section contains as a part a translation of the whole of the object-language L, as we have already noted. We now consider another kind of semantical metalanguage, a so-called non-translational one, which lacks this crucial feature. Non-translational semantics consists essentially of the theory of concatenation of §B augmented by a Boolean algebra of one-place predicate constants (including abstracts containing no free variables).1 First let us formulate very briefly a non-translational metalanguage, and then note that it contains a genuine semantics in the sense of the preceding section. The meta-language here we shall call SM£, the semantical meta-language for L based on comprehension, which will figure as the sole semantical primitive. Within SMfc we could have defined 'a Cmprh b' as '(P re( lConOne a . PredConOne b . (x){b Den x .=>. a Den x))'. 'Cmprh' in this sense will now be taken as the sole semantical primitive of SMand the significance given to it is precisely that provided by this definition. ('a Cmprh b' may be read 'a comprehends f ) . Because contains no translation of L, has only three parts, a basic logical part, a syntactical part, and a Boolean algebra of comprehension. As variables we thus need only the expressional variables 'a', etc. 1

See M a r t i n , op.cit.,

pp. 1 7 9 - 2 1 2

16

T H E NOTION OF A N A L Y T I C T R U T H

In addition to logical rules providing for a simple, applied, functional calculus of first order with identity, we need Syntactical Rules and some Rules of Comprehension. As Syntactical Rules we take Syn Rl—Syn RJ but we may drop SynR4, the Rule of Infinite Induction, and take in its place the Rule of Framed Ingredients, TBI above, and the following. SynR4'. h (a){Kb)(a F r l n g b . (c)(c F r l n g b :=>: P S c .v. (Ed)(Ee)(d Prfc c . e Pr& c . c = (rfn«)))). These five rules suffice to provide the syntax of L. And although SynR4 is needed for semantical purposes in SMfc, it is not as needed in SMfc. Here we may accomplish what we wish with SynR4' and TBI instead. As Semantical Rules we need the following General Rides of Compreh ension. GRC1. GRC2. GRC). c',d . b =

h ( E b ) ( a Cmprh b .v. b Cmprh a) . e= . PredConOne a. I- a Cmprh b . c Cmprh a c Cmprh b. V (E£')(Ec')(Ed)(SentFuncOne b',d . SentFuncOne (d(~\invepC\b') . c =

( d C \ i n v e p ( ~ \ c ' ) . a = (d(~\invepC\

(b' vee c' ))).=>. a S u m b,c, where 'SentFuncOne a,b' c =

abbreviates '(Fmla a . Vbl b . (c)(c FV a

b)y,

and 'a S u m b,c' abbreviates '(a Cmprh b . a Cmprh c . {d)(d Cmprh b . d Cmprh c d Cmprh a))\ GRC4. I- ( E A ' ) ( E c ' ) ( E d ) (SentFuncOne b',d . SentFuncOne c',d . b = (dC\invepC\b') . c = (dr\invepC\c') .a = (d(~\invepr\ {b' dote'))) . a Prodb,c,

SEMANTICAL SUBSTRUCTURE

17

where a Prod b,c' abbreviates '(6 Cmprh a . c Cmprh a . (d)(b Cmprh d . c Cmprh d : a Cmprh d))\ l

GRCS.

I- (Ec)(Ed)(SentFuncOne c,d . a =

(dninvepHc)

.

b = (dC\invepC\tilde c)) . =>. b Neg a, where 'Null c' abbreviates '(¿XPredConOne b . =>. b Cmprh c)', 'Univ c' abbreviates '(¿>)(PredConOne b . ~=>. c Cmprh b)', and 'b Neg a' abbreviates ' ( ( c ) ( a Cmprh c . b Cmprh c := : Null c) . (c) (c Cmprh a . c Cmprh b : = : Univ c))\ GRC6. h d Neg b . c Prod a,d . ~ b Cmprh a :=>: ~ Null c. GRC7. h d Sum a,b . d Cmprh c . (e)(c Cmprh e . (a Cmprh e .v. b Cmprh e Null e) Null c. GRC8. h(Ec)((Sent c . a = c) .v. Sent c . a Clsr c) : LogAx c) . V b l i Univ (bC\irwepC\a). GRC9. I- Univ a . Null b : => : ~ b Cmprh a. GRC10. I- SentFuncOne b,a . Vbl c . Univ ( a r \ i n v e p C \ b ) : ^ : Univ ( a C i i n v e p C s c qu b). GRC11. h Sent a . Vbl b . Vbl c . ~b = c:=>: (b C\invep C\a) Cmprh (c O invep C\a). GRC12. I-Sent a . Vbl b :=>: Univ ( b r \ i n v e p C \ a ) .v. Null (ib(~\invepC\a). GRC1). I-SentFuncOne a,b . b ¥V a . d SFJ a . — b = c :(br\invep(~\a) Cmprh (c C\invep(~\d). GRC14. I- pee Cmprh (ex n i n v e p Opee O e x ) . (ex C\invep C\ pee O ex) Cmprh pee. N.A.T.-C

18

T H E NOTION* OF A N A L Y T I C

TRUTH

Some of these Rules are akin to some of Tarski's postulates for Boolean algebra based on inclusion. 1 T h e semantical truth-concept for L may be defined w i t h i n SM¿ as follows. T r a' abbreviates '(Sent a . (E¿)(Vbl b . Univ

(br\invepC\a)))'.

Note that the adequacy of this truth-predicate cannot be stated or proved, because SM£ contains only structural descriptions, not translations, of the expressions of L. But this predicate is clearly in accord with the criterion of adequacy because of the meaning given to 'Cmprh'. W e note that TC2, TC), TO, TC6, and TC9 (of §C above) may be proved in S M ¿ as consequences of GRCl-GRC14. But TC4, TC7, TC8, and TCÍ0 may be proved only with 'LogAx' replacing 'Ax' and 'LogThm' replacing ' T h m ' . T h e more general form of these theorems may be proved only on certain hypotheses, as we shall see in a moment. TC4 and TC7 of §C are of course extremely important semantical theorems. To prove them in SM¿ certain additional assumptions would be required. Corresponding to each non-logical axiom of L we might, e.g., assume that some given abstract of L is universal. If the axiom of L is a sentence a, the corresponding rule of S M c might state that the abstract (bC\invepC\a), where b is a variable, is universal. Some such rules would seem essential to allow us to prove, first, that all the non-logical axioms of L which are sentences are true, and t h e n that all the non-logical theorems of L which are sentences are true. T h e General Rules of Comprehension do not depend upon the non-logical axioms of 1 See A. Tarski, "Grundzüge der Boole'schen Algebra I," Fundamenta Mathematicae 24 (1935): 177-198, esp. pp. 181-182. Also pp. 320-341 in Logic, Semantics, and Metamathematics.

SEMANTICAL

SUBSTRUCTURE

19

the object-language and thus in no way characterize its nonlogical part. The purpose of non-translational semantics here is merely to provide a general framework for a semantical definition of truth. The rules of the non-translational meta-language should contain nothing "dubious" and should not depend upon any non-logical assumptions. Note that the general theory of comprehension does provide such a framework, but is not of sufficient power to determine which sentences of the object-language are in fact true, other of course than those sentences which are logical theorems. Further, the assumptions GRC1—GRC14 are consistent with the underlying syntax. (The proof of this statement is omitted.) Thus these assumptions presumably contain nothing dubious, and none of them is in any way determined by the non-logical axioms of L. Hence we shall not add any further axioms to S M c and thus within S M c w e shall not be able to prove that all the theorems of L which are sentences are true. But sentences stating that certain one-place abstracts of L are universal may be formulated and taken as hypotheses if needed. We shall thus be able to prove on certain hypotheses what can be proved otherwise by assuming rules to the effect that those axioms of L which are sentences are true. 1 (This device of using such hypotheses where needed is essentially that of Principia Mathematica with regard to the Axiom of Infinity and the Multiplicative Axiom.) The hypotheses required are called Specific Hypotheses of Comprehension. To take an example, suppose •(*)(P* .=»• (Er')P'xx')" is a non-logical axiom of L. Corresponding to this we could define 1

Cf. Truth and Denotation, pp. 203-204.

20

THE

NOTION

OF A N A L Y T I C

TRUTH

'SHCl' as l(a)(b)(a — (ex qu (pee Hex hrsh ex(~\ac exisqu peeC\ acr\exr\exriac)) . Vbl b : Univ (bCiinvepria))''. And similarly for the other non-logical axioms of L. Suppose there are just nine such axioms. Then we may suppose iSHC2', lSHC)', etc., to be defined appropriately. Let SHC'

l

abbreviate '(SHCl

. SHC2

SHC9)'.

It is then easy to prove the following. TD1.

h SHC . DesAx a . Sent a :=>: T r a.

TD2. TD). TD4.

V SHC . Sent a . T h m a T r a. h SHC . T h m a . b Clsr a : =3: T r b. h SHC . Sent a :=> : ~ (Thm a . T h m (tilde

a)).

These last three theorems correspond with TC7, TC8, and TC10 respectively. A theorem corresponding with TC4 may now easily be proved using TDl. W e have taken 'Cmprh' here as the sole primitive merely by way of an example. There are several alternative primitives to consider also. But appropriate axioms governing any such primitive would presumably yield the same theorems which GRC1GRC14 yield. Also it may be shown that non-translational semantics provides a minimal semantics in an appropriate sense. Roughly speaking, a minimal semantics is one presupposing as little as possible with respect to the number and complexity of primitives, and with respect to the number and kinds of objects taken as values for variables. Also, as little as possible should be assumed in the axioms and rules. Non-translational semantics provides a minimal semantics in this sense. No essentially weaker type of first-order meta-language providing for the semantics of firstorder L may be formulated. The theory of analytic truth to be

SEMANTICAL

21

SUBSTRUCTURE

put forward is therefore also minimal in the sense that it cannot be formulated within a weaker meta-language. 1 If L is taken to contain one or more individual or functional constants as primitive, some slight changes must be made in the material of this Chapter. But these present no essential difficulty and may readily be supplied. 1

For a more careful statement of this, see Truth

pp. 276-278.

and

Denotation,

II Analytic Truth

W

e now turn to our main task, the formulation of an exact definition of 'analytic in L\ for all first-order L, within the narrow confines of S M c or S M ^ . In §A the connection between semantical rules and analytic truth is brought out. In §B the formal definition is put forward. Identity is then introduced in appropriate ways in §C. In § D it is shown that all logical theorems of L which are sentences are analytic in L in the sense defined. An alternative definition is given in §E, and finally, in §F, the distinction between logical and non-logical constants is discussed more fully than above. T h e next chapter will be devoted to establishing the adequacy of the definition of being analytic put forward here. A. Semantical Rules a n d Analytic T r u t h . We have seen in the brief sketch of specific semantical meta-languages above that the semantical rules ( S e m R l - S e m R 2 or GRC1-GRC14) are an essential part of the deductive apparatus of semantics. They play a role within formalized semantics similar to that of arithmetical axioms or rules within a formalized arithmetic. Within a translational semantics based on multiple denotation the semantical rules stipulate properties of that relation. One of them states the conditions under which certain expressions denote certain objects. T h e semantical rules of a non-translational semantics stipulate 22

ANALYTIC

23

TRUTH

g e n e r a l properties of comprehension. B u t t h e semantical rules of either kind of a semantics do not provide a basis for d e t e r m i n i n g which statements of the object-language L are in fact t r u e . For this other assumptions or hypotheses are needed. B u t t h e s e m a n tical rules do determine which s t a t e m e n t s of L are t r u e wholly in virtue of first-order logic. First let us characterize analytic trutli within the f r a m e w o r k of a translational semantics. T h i s m a y be done s o m e w h a t provisionally as follows. (A) A sentence a of the object-language L can be said to be analytic or analytically true in L if and only if the f o r m u l a ' T r a' (of t h e semantical m e t a - l a n g u a g e ) is provable in t h e semantical m e t a - l a n g u a g e wholly by means of the logical, syntactical, and semantical axioms. T h i s statement (A) is not intended as a definition of 'analytic', being itself a statement within t h e appropriate m e t a - m e t a l a n g u a g e , b u t rather as a partial condition or criterion u n d e r which a definition of 'analytic* m i g h t be regarded as adequate. ( S e e Chapter I I I , § D . ) C a r n a p has proposed definitions of 'analytic' (or ' L - t r u e ' ) which are no doubt in accord with this condition. 1 T h e meta1

S e e R . C a r n a p , Logical

Foundations

of

Probability

( U n i v e r s i t y of

Chicago Press, Chicago: 1950), pp. 82-89. Also see his Introduction to Semantics (Harvard University Press, Cambridge: 1946), pp. 60—70, 79-80, and 134-140; and Meaning and Necessity (University of Chicago Press, Chicago: 1947), pp. 8-10. Cf. also his Logical Syntax of Language (Kegan Paul, London: 1937); A. Tarski, "Uber den Begriff der Logischen

Folgerung,"

Actes

du

Congrès

International

de

Philo

sophie Scientifique (Hermann et Cie., Paris: 1936), Vol. VII, pp. 1-11 (also

pp.

409-420

in

Logic,

Semantics,

J. Kemeny's review of Carnap's Logical The Journal

of Symbolic

Logic

and

Metamathematics)\

Foundations

16 ( 1 9 5 1 ) : 2 0 5 - 2 0 7 .

of

and

Probability,

24

T H E N O T I O N OF A N A L Y T I C

TRUTH

languages which Carnap employs, however, are of higher order and thus contain very powerful logical (and hence semantical) modes of expression. If L is a finite language-system, containing only a finite number of primitive individual constants designating just a finite number of distinct individuals, Carnap's type of definition can be reformulated within a restricted semantical meta-language such as SMfc. But if L is an infinite language, as many important languages in fact are, it is doubtful whether Carnap's type of definition can be reformulated here. The reason is that his method makes fundamental use of non-denumerable totalities (classes) of expressions, if L is infinite, and such totalities cannot be handled here. Hence it would seem that the full effect of Carnap's method of definition cannot be mirrored within so restricted a meta-language as SMfe. (But see Chapter IV, §§A-D.) Let us now consider analytic truth within a non-translational semantics. What does it mean to say within such a semantics that a sentence a of L is analytic? Again, we shall not (for the moment) give a definition, but rather will lay down a general condition under which any definition of 'analytic' might be said to be partially adequate. (B) A sentence a of L can be said to be analytically true in L if and only if the formula 'Tr a' of the non-translational semantical meta-language is provable wholly by means of the logical and syntactical axioms together with the General Rules of Comprehension. Any suitable definition of 'analytic in L' within a nontranslational semantics would presumably be in accord with this condition. The Specific Hypotheses of Comprehension are not taken as axioms, and hence (B) can be simplified to read:

ANALYTIC

TRUTH

25

(C) A sentence a of L can be said to be analytically true in L if and only if the formula 'Tr a' of the non-translational semantical meta-language is provable. (Note again, of course, that (A), (B), and (C) do not provide definitions of 'analytic', being themselves statements within the relevant meta-meta-languages.) We accept provisionally criteria (A), (B), and (C), but later we shall incorporate other conditions also in defining what it means for a definition of being analytic to be regarded as adequate. The use of 'semantical rule' here has been ambiguous. We have spoken of SemRl and SemR2, e.g., as semantical rules of SMp. Strictly these are statements of the meta-meta-languages stipulating the semantical axioms of 5 M ¿ , and these semantical axioms in turn are the semantical rules of L, laying down the semantical properties of the expressions of L. No harm can come, however, of speaking of SemRl and SemR2 either as semantical rules of SMfe or as semantical rules of L. In the case of nontranslational semantics, as formulated, e.g., in SMthe General Rules GRC1-GRC14 can be regarded as semantical rules of either SM£ or of L likewise. But within non-translational semantics the Specific Hypotheses of Comprehension also play the role of semantical rules of L, although, being formulae within SM£ not about SMthey cannot properly be regarded as semantical rules of SM£. Hence in non-translational semantics it is important to distinguish the two uses of 'semantical rule'. B. The Formal Definition. Analytic truth is a special kind of truth and it is reasonable and natural therefore to define 'a is analytically true in L' as meaning that a is true in L and that such and such other conditions hold of a. Such a definition we

26

T H E NOTION OF A N A L Y T I C T R U T H

shall now put forward. This may be given in any of the (translational or non-translational) meta-languages which have been sketched above. The leading idea of the definition to be given is very roughly that a sentence a of L is analytic in L if and only if it is true and every sentence b which results from a by simultaneously replacing the primitive predicate constants of a by any primitive or definable predicate constants of the appropriate degree, is also true. In other words, we shall require that a be true and that every sentence formed from a in a certain way be true also. The first step in introducing a notion of analytic truth is then to define what it means to say that one sentence results from another by the kind of replacement required. Let us assume for the present that L is sufficiently rich in modes of expression and axioms so that some individual constants are available either primitively or by definition. More particularly, if the fundamental domain of L consists of just n objects, we assume that there are distinct primitive or defined individual constants of L designating uniquely each entity. If the fundamental domain is (denumerably or non-denumerably) infinite, we assume that there is a denumerable number of distinct primitive or defined individual constants designating a denumerable number of entities of L. Thus L may contain zero or more individual constants as primitive, or suitable primitive predicates in terms of which individual constants are definable. Clearly there is nothing objectionable in making this assumption. Many languages of sufficient power for mathematical, scientific, or philosophical purposes in fact satisfy it. First-order languages of this kind constitute an especially important sub-class of first-order languages, a class not too weak in axioms or modes of expression for most purposes and yet not so strong as to be

ANALYTIC

27

TRUTH

logically or philosophically objectionable. Further this requirement facilitates the definition of being analytic to be put forward. In fact it assures the availability of a sufficient supply of definable predicate constants in L. For a finite L, the requirement assures that all predicates (in extension) applicable to the individuals of L are definable. Let us assume for the moment that L contains ' = ' in one way or another. (Cf.§C below.) If the fundamental domain of L contains just three objects, e.g., and 'a,', 'a,', and 'a,' are available primitively as distinct individual constants designating respectively the three objects, then ' x 3 ~ x = x', kX-3X

=

i j ' , lx3(x

=

at .v. x =

a, .v. x = a,)' (i.e., 'xzx predicate constants of L, of L is co-extensive with two-place predicates, and

a , ) ' , a n d 'x?{x

=

5 j .v. x

=

= x') are in effect the only one-place i.e., every one-place predicate constant one or the other of these. Similarly for so on.

If 'a/, 'a g ', and 'a,' are not primitives of L, but defined say by means of Russell's theory of descriptions, we then assume that formulae containing occurrences of these constants are suitably defined. Further, the axioms of L are to be such as to assure the existence and uniqueness of the entity involved. Any formula of L containing a defined individual constant is to be understood hereafter in accord with these requirements. (Note that if an individual constant 'Sj' is defined by means of a description, to say that 'a,' designates the object Ij is a mere manner of speaking. Such phrases are informal and there will be no need here of an exact definition for them within the meta-language, descriptions being in no way a part of the primitive notation of L.) In §E below, a tentative definition of being analytic will be given for which the requirement that individual constants be available either primitively or by definition is not needed. In order to define the notion of the simultaneous replacement

28

T H E N O T I O N OF A N A L Y T I C

TRUTH

of predicate constants of any degree in terms of successive replacements of predicate constants of fixed degree, it will be convenient to use what we may call predicate parameters. These are neither variables nor constants, but certain special sequences of symbols introduced ad hoc for certain technical purposes. T h e predicate parameters will be of degree one, of degree two, and so on, but there will be no need for predicate parameters of degree greater than five, because we have assumed L to contain no primitive predicate constants of degree greater than five. W e may define 'PredParOne a ' as '(a = (ac(~\pee) .v. (EA)(AcString b . a = (iacC\peeC\b))y, ' P r e d P a r T w o a' as '(a = (ac(~\acC\pee) .v. (Eò)(AcString b . a = {acO>acC\peeC\b)))', and so on, so that, e.g., " P " " is a predicate parameter of one place, ... , and " " " P " " > ¿ s a predicate parameter of five places. L e t 'a R 1 P C 1 * d' express that a results from d by replacing one occurrence of c in d, where c is pee or a one-place predicate parameter, by the one-place predicate constant or one-place parameter b. 'a R l P C l * d' may abbreviate '((PredConOne b .v. PredParOne b) . (c = pee .v. PredParOne c) . (Ec')(E£')(c' Occ^ c . b' Occ a b . (e)(c' = (enc) = (eC\b) : d = (c' ne) . = . a =

(•b'ne))))'. One can then define 'a R l P C ? ' d ' by framed ingredients to express that a results from d by replacing 0 or more occurrences of c (where c is pee or a one-place predicate parameter) in d by the one-place predicate constant or parameter b. ' a R1PCPJ d' abbreviates ' ( E e ) { a F r l n g e . (a')(a :=>:a' = d.v. (Eb')(b'Prea' .a' R I P C l J b ' ) ) ) ' .

Frlng e

ANALYTIC

29

TRUTH

Finally, 'a R1PC* d' may be defined to express that a results from d by replacing all occurrences of c in d by b, where b and c are the appropriate kinds of expressions. 'a R1PC* d' abbreviates '(a R1PC?£ d : (Ed)d Occa c .=>. c = b)'. W e can suppose analogous definitions given for two-place predicate constants, where as above l P " is the only primitive two-place predicate constant of L. Thus, 'a R2PC1* d', 'a R2PC?£ d', 'a R2PC* d', may be defined appropriately. And similarly for three-, four-, and five-place predicate constants, where ' P ' " , ' P " " , and ' P ' " " are as above the only primitive such of L respectively. W e now introduce the notion of a predicate variant. W e let 'a PredVar b' abbreviate '(Sent b . Sent a . (Ec){a Fr Ing c. {d)(d Frlng c : d = b .v. {Ee){Ea'){Eb'){e Pr c d : (d R1PC«: c .v. d R2PC-: c .v v. d R5PC«: e)))))\ This notion, in effect that a results from b by the simultaneous replacement of predicate constants of any degree of L, is defined as a succession of replacements of predicate constants of fixed degree of L. Because a and b are required to be sentences here, no predicate parameter can occur in either of t h e m and every primitive predicate constant of b will be supplanted if at all by primitive or definable predicate constants of the proper degree. Note that because predicate parameters are available we can always accomplish the simultaneous replacements we wish. E.g., suppose that ' P ' occurs in 'xyzX' and that ' P " occurs in 'xs Y', where these abstracts contain no free variables. T h e n (1)

'(x)(z)(iv)(xysX

zw .v. xsY x)'

30

T H E N O T I O N OF A N A L Y T I C

TRUTH

is a predicate variant of (2)

'(x)(z)(w)(P'zw

.v. P * ) \

If we replace the occurrence of 'P' in (2) by 'x3 Y' directly, we cannot then gain (1) by replacing the occurrence of l P " by 'xysX'

because ' P " occurs in 'x3 Y'. But w e can first replace ' P ' of

(2) by a one-place predicate parameter, say " P " " ^ gaining '(x)(z)(^)(P'zu; .v. ' P " ' x ) \ Similarly we can next replace ' P " here by a two-place predicate parameter, say ' " P " " ^ gaining '(x)(z)(w)("P""zw

.v. ' P ' " x ) \

Then we can replace ' " P " ' " by 'xy3X', and in this result we can replace " P " " by 'x3 Y', thereby gaining (1). And similarly for more complicated replacements. In terms of 'PredVar' we may define directly the notion of analytic t r u t h for L as follows. 'Anlytc a'

abbreviates '(Tr a . (b)(b PredVar a .=>. T r b))'.

A sentence is analytically t r u e in L provided it is true and every predicate variant of it is t r u e also. 1 A s e n t e n c e is synthetic

o r synthetically

true

i n L, c l e a r l y , if

and only if it is true but not analytic. 1 In view of this definition, a sentence may be regarded as analytic in view of its form. Of course the reference to form here is somewhat vague. But if a sentence a is true and every predicate variant of it is true likewise, then clearly the truth of a in no way depends upon the non-logical predicate constants occurring in it. It is true rather because of the arrangement of its logical constants, in short, in virtue of its form. This definition seems therefore akin to definitions of the scholastics' bona da forma type. For what appear to be the historical roots of this kind of definition, see William of Ockham, Summa Logic ac, partis 2, pars 3, c.l ff., and Buridan, Consequentiae, I, ch. 4. Cf. also Albert of Saxony, Logica, IV, ch. 1.

ANALYTIC

31

TRUTH

'Synthc a' abbreviates '(Tr

a

, ^ Anlytc a)'.

A few lemmas which will be useful below may be listed as follows. TBI. TB2. TB).

h LogThm a . b PredVar a :=>: LogThm b. Y Vbl a . Sent c . b PredVar {a qu c) : =: b PredVar c. I- Anlytc a . Anlytc b . c PredVar (a dot b) : =>: Tr c.

C. Identity. In (I,A), it will be recalled, it was stated that ' = ' may perhaps occur as a logical primitive of L. If not, it may be taken as a non-logical primitive. Or, under suitable circumstances, it may be introduced by definition. We have, then, at least these three methods of handling identity within L. Let us consider these methods more closely now that the notion of 'Anlytc' is available. First we might introduce identity within L by definition as follows. We let 'x =

f

abbreviate

( z ) ( P ' z x .EEE. V'zy). = . Vzyw)

'((P* .= . P r )

(z){iu){Y'xzw

. (z)(P'xz . = . P'yz) .

.= . P"yzw) .

{z){w)(x'){y'){V"zwx'y'x

.= .

(z)(w)(Vzxiv P""zivx'y'y))'

To say that x = y in this sense is in effect to say that 'x' and iyy may be substituted for each other in all atomic contexts. However, for this definition we assume that not all of the properties or relations designated by the primitive predicate constants of L are null or universal. For suppose P is a null or universal property, P' a null or universal dyadic relation, P " a null or universal triadic relation, and so on. Then clearly I- Px . = .

Py,

h (Z)(P'*Z . = . P ' y z ) , 1 Cf. D. Hilbert and P. Bernays, Grundlagen (Springer, Berlin: 1934), pp. 3 8 1 - 3 8 2 .

der Mathematik,

vol. 1

32

T H E NOTION OF ANALYTIC T R U T H

and so on, so that every x and y would be identified according to the definition. But ordinarily the properties or relations designated by the primitive predicate constants are neither universal nor null, so that in practice the requirement laid down seems gratuitous. Using this definition of identity, we can easily prove the two basic laws IdRl and IdR2 of (I,A). If we introduce identity in this way within L, statements t r u e in the theory of identity are already included among the analytic truths of L as defined above. But we now assume that L contains an at least denumerable infinity of objects in its fundamental domain. T h a t this assumption is needed may be seen as follows. Suppose (1)

'WOOx =

y',

which states that there is at most one object in L , were true in L . It would t h e n also be analytic, using the definition of identity just given. Because we know that hx = y

F1x . = . F^y,

for any one-place predicate constant F1. And hence, because (1) is true, is also true. Similarly \x){y){z){G*xz

. = . G*yz)'

is true. And so on, so that (2) '(x){y)((F^x . = . . ( z ) ( G » « . = . G*yz) (y')(Hlzwx'y'x . = . H'zwx'y'y))'

(z)(w)(x')

is also true, for any F1, G1, etc. But any instance of (2) is a predicate variant of (1), expanded according to t h e definition of ' = ' . Hence if (1) is true in L it is analytically t r u e in L.

ANALYTIC

53

TRUTH

Similarly if ' 0 ) ( r ) 0 ) 0 = r .v. x = z .v. y = z)', which states that there are at most two objects in the fundamental domain of L, is true in L it is then analytically true in L. And so on. So that if L contains, just a finite number, say k, of objects in its fundamental domain, the statement (3) '(x l )...(x J [ + 1 )(x 1 = x 2 .v. xx = x s .V =

-V

v. X, =

x*+1

.V

v. x1 = xk+1 .V. X,

V. Xk =

Xk+1)',

which states that L contains at most k objects, would be analytic. On the other hand, the statement which says that there are at least k objects in L, if true in L, is not analytically true. For k = 2, e.g., such a statement is (4)

' ( E x ) ( E 7 ) ~ x = y\

If we replace 'x = y' here by its definiens we gain (5)

. Tr d) . a Gen b . e PredVar a :=>: Tr e. TD2b.

t- c MP a,b . Anlytc a . Anlytc b :

: Anlytc c.

For this we use TC6 of Chapter I and the law that h {b){b PredVar a' .=>. Tr b) . (b)(b PredVar (a' hrsh b') Tr b) :=>: (b)(b PredVar b' Tr b). Hence we have that TD).

I- c IC a,b . Anlytc a . Anlytc b

Anlytc c.

Using TBI above and 7X7(1), we have that TD4.

1- Sent a . LogThm a : ^ : Anlytc a.

This may also be established by TBl(l), TDl, and TD). TD4 establishes what we wish, namely, that all the logical theorems of L which are sentences are included among the analytic truths of L. A few further theorems which may be useful later are as follows.

38

T H E N O T I O N OF A N A L Y T I C

TRUTH

TDS. I- Sent a . Sent b :=>: (Anlytc a .v. Anlytc b) . = . Anlytc (a vee b). But clearly it does not hold that if (a vee b) is analytic where a and b are sentences, then a is analytic or b is. TD6.

h Anlytc a

~

Anlytc {tilde a).

H e r e again the converse does not hold. It is interesting to compare TDi and TD6 with the corresponding theorems concerning theoremhood and truth. TDS holds if we replace 'Anlytc' wherever it occurs by either ' T h m ' or ' L o g T h m ' . But if we replace 'Anlytc' by ' T r ' , we can gain a stronger theorem with ' = ' in place of the right-most '=>' of TD5. If L is consistent and we replace 'Anlytc' throughout TD6 by ' T h m ' , the resulting statement is clearly a semantical theorem. If we replace 'Anlytc' by ' L o g T h m ' the result holds because we know the set of logical theorems of L to be consistent. Finally, if we replace 'Anlytc' by ' T r ' throughout TD6, change the to an ' = ', and add as hypothesis 'Sent a ' , the result is also a semantical theorem. W e also have among others the following principles. TD7. I" Anlytc a . L o g T h m (a tripbar b) : => : Anlytc b. TD8. h Anlytc a . b PredVar a :=>: Anlytc b. TD9. Y Sent a . ~ Anlytc a Synthc a .v. ~ T r a. TD10. I- Anlytc a . Synthc b :=>: Synthc (a dot b). TDll. I- Sent a . Sent b :=> : Anlytc (a dot b) . = . (Anlytc a . Anlytc b). TD12. h Sent a . Sent b : =>: Synthc (a dot b). = . (Synthc a . T r b :v: T r a . Synthc b). TDl)a. I- Fmla a . Fmla b . c Clsr (a hrsh b) . d Clsr (tilde a) • e Clsr {tilde b) . ~ Anlytc d . Anlytc e : ^ : ~ Anlytc c.

ANALYTIC

TRUTH

39

TDlib. h Fmla a . Fmla b . c Clsr (a tripbar b) . d Clsr a . e Clsr b . Anlytc c . ~ Anlytc d : ^ : ~ Anlytc e. TD14. I- Fmla a . Fmla b . d Clsr b . Anlytc d . c Clsr (a hrsh b) :

: Anlytc c.

TD1S. h Fmla a . Fmla b . c Clsr a . d Clsr b . e Clsr (a hrsh b) . Anlytc c . Anlytc e : =>: Anlytc d. The proofs given within this section have been within SMp. Certain changes must be made, as we know from (I,D), if they are to be given within SM£. And some of them may be proved within SM£ only with the hypothesis lSHC'. E. An Alternative Definition. For the purposes of the definition of 'Anlytc' in §B above, we assumed that individual constants are available in L either primitively or by definition. But perhaps this assumption is not needed. Instead, we can let free variables play the role of individual constants. But in the replacements allowed in forming predicate variants, we must now use abstracts (containing perhaps free variables) rather than just primitive or defined predicate constants (which contain none). To see this we may proceed somewhat tentatively as follows. In the definiens of the definition of 'a R1PC1* d' we replace 'PredConOne b' by 'AbstOne b'. Let the result of this be written la R I P C l ' J d\ And similarly for 'a R2PCl£ d\ which now becomes 'a R2PC1'* d\ And so on. Also 'a R l P C ? * d' is to be defined similar to the way in which 'a RIPC?^ d' is defined above, but with 'a' R I P C l ' J d " replacing V R I P C l J d " in the definiens. 'a R2PC?'J d', etc., are to be defined similarly, 'a RIPC'J d' likewise is now to be defined similar to the way in which •a R1PC* tf is above, but with la RIPC?^ d' replacing 'a R1PC?£ d' in the definiens. 'a R2PC'J d', etc., are to be defined similarly. We let now

40

T H E NOTION OF A N A L Y T I C

TRUTH

'a AbstVar b' abbreviate '(Fmla a . Fmla b . (Ec)(a Frlng c . (d)(d Frlng c :=>: d = b .v. (Ee)(Ea'){Eb')(e Pr c d . (d R1PC-' e .v. d R2PCJ; e.v v. d R5PC'£ e) . ~ (Ec')(Erf')(c' FV a' . d' BOcc.•)))))'• The relation of being an abstract variant of differs only slightly from that of being a predicate variant of. Note that a and b here are formulae, not necessarily sentences. Note also that no free variable of a' can have a bound occurrence in e. This is to assure that no free variable of a' becomes bound by replacing an appropriate sign by a'. In terms of 'AbstVar' a closely related notion, that of being a closed abstract variant of, may be defined as follows. 'a ClAbstVar b' abbreviates '(Ec)(E^)(E«)(LogThm (b tripbar c) . c Clsr d . e AbstVar d . a Clsr e . ~ a = b)'. Note that 'AbstVar' holds between formulae. If a ClAbstVar b, on the other hand, a is a sentence formed from an expression b as follows. One picks out first a sentence c such that (b tripbar c) is a logical theorem and c is a closure of some formula d. a is then to be a closure of some formula e which is an abstract variant of d. We may perhaps define the notion of being an analytically true sentence of L as follows. 'Anlytc a' abbreviates '(Tr a . {b){b ClAbstVar a .=>. Tr b))'. A sentence a is analytically true in L provided it is true and provided every closed abstract variant of it is true also. The intuitive content of this definition may be seen more clearly as follows. If (a tripbar b) is a logical theorem, where a and b are formulae, we may for the moment say they are equivalent. (This is a syntactical notion of equivalence, not the

ANALYTIC

TRUTH

41

more usual semantical one.) To say that a sentence a of L is analytic in L is then to say that a is true in L and that every b is true, where b (is distinct from a and) is a closure of a formula which is an abstract variant of d, a being equivalent to a closure of d. Note that, although 'LogThm' occurs in the definiens of this definition of 'Anlytc' in a fundamental way, the definition of 'LogThm' does not depend upon the specific logical axioms of L chosen. It depends merely upon the character of L as a firstorder language, because the class of logical theorems as defined on the basis of any adequate set of logical axioms will coincide with that as defined on the basis of any other. This definition is put forward tentatively and merely as a basic for further study. Perhaps it can be simplified or improved upon in various respects. But it appears to be satisfactory, and thus to provide a basis for the theory of analytic sentences above. Also no doubt other alternatives along similar lines can be given. For subsequent purposes, however, we return to the perhaps simpler and more intuitive definition of (II, B). F. Logical Truth and Logical Constants. Let us pause for a moment to consider the distinction if any between logical and analytic truth as well as the related distinction between logical and non-logical constants. Some logicians, notably Quine, have distinguished between analytic truths and logical truths somewhat roughly as follows. The logical truths are first identified with the general truths of logic and thus with the sentences we have above called 'Anlytc'. Then a wider class of truths is introduced as the analytic truths in a new sense of this word. This class includes the logical truths as well as all sentences which arise from these logical truths by

42

T H E N O T I O N OF A N A L Y T I C

TRUTH

replacing one or more occurrences of an individual, predicate, or functional constant by any synonym thereto, respectively. T h e former type of truth is then regarded as respectable, but the latter is somewhat suspect because of the difficulties involved in giving a suitable definition of what it means for one individual, predicate, or functional constant to be a synonym of another. Logicians who make this distinction between logical and analytic truth are usually interested primarily in natural languages rather than in formalized language-systems. Perhaps the distinction between these two kinds of truth can be drawn satisfactorily for natural language. This is a matter we need not concern ourselves with here. But if one confines attention exclusively to formalized language-systems, the distinction between analytic and logical truth seems to be ill-founded. W e m a y see this as follows. Within formalized languages definitions m a y be regarded merely as conventions of notational abbreviation. Certain sequences of symbols are introduced as abbreviations for other sequences containing only primitive signs or signs already defined. T h u s , if a new predicate constant is introduced, by either explicit or contextual definition, we are free to use this constant or context wherever we wish in place of the expression which defines it. And we mean by this constant or by the context in which it occurs precisely what is meant by the definiens of the definition by which it is introduced. T h u s in a restricted way, definitions m a y be regarded as introducing synonymous expressions. But nothing is introduced into the system by means of a definition that is not already contained in it. T o speak of synonyms in this restricted sense is merely to speak of quite harmless notational abbreviations. Thus, when we confine ourselves exclusively to formalized language-systems as here, and further carry out all our meta-theoretic parlance within suitably re-

43

ANALYTIC T R U T H

stricted, extensional meta-languages as above, the distinction between logical and analytic truth vanishes. Hence we have spoken throughout merely of "analytic" truths, and have not introduced this gratuitous distinction. To take an example. Consider a system L , of the kind discussed throughout, containing 'Fe' and 'Fo' as primitive predicate constants. Let 'Fe' denote all females, and 'Fo' all foxes. We might at some point wish to introduce a definition to the effect that 'Vi x' abbreviates '(Fe x . Fo x)'. For x to be called a 'Vi' it is then necessary and sufficient for x to be a female fox, and we are free to write or speak ' Vi x' in place of '(Fe x . Fo x)' as notational or verbal shorthand (putting any other variable or individual constant throughout in place of 'x'). But, and this is the important point, nothing new has been introduced by this definition. We can, if we wish, read 'Vi' as the English word 'vixen'. But we have not defined that English word. To define the English word 'vixen' in an acceptable way without running into the circularity of the ordinary worddictionaries would be a task of immense difficulty. For one thing it would presuppose an at least partial formalization of ordinary English, with a clear enunciation of some primitives. Thus, within the L's under consideration, (1)

'(x)(Vix : = : F e x . Fox)'

is an analytic sentence, and clearly so, because we can gain it from the analytic sentence '(x)(Fe x . Fo x : = : Fe x . Fo x)', using the definition of 'Vi' above. But we do not therewith say that the English sentence 'Every vixen is a female fox, and conversely' is an analytic sentence. Nor do we claim even that

44

T H E N O T I O N OF A N A L Y T I C

TRUTH

(1) is a "correct" symbolic translation within L of this English sentence. To claim this would in effect presuppose an elaborate analysis of the structure of ordinary English within a comparative meta-language of a very complicated kind. It is doubtful whether any such meta-language has ever been satisfactorily formulated. Let us now consider a little more closely the distinction between logical and non-logical constants. For the definition of 'Anlytc' given above, and in the discussion throughout, we have presupposed a distinction, reasonably clear-cut, between these two kinds of constants. Certainly the distinction between analytic and synthetic sentences seems to rest upon a kind of prior distinction between logical and non-logical signs. There has been considerable discussion in recent years as to the legitimacy of this kind of a distinction between signs, and therefore we shall do well to look at it a little more closely. Under consideration throughout has been a vast infinity of language-systems, namely, first-order systems of a certain kind. These systems are distinguished from other systems primarily by the kinds of signs, constants and variables, which they are allowed to contain. We have not put forward here an exact semantical definition of first- and higher-order systems. But because such definitions have not been put forward, one would not therewith wish to contend that the widely used distinction between them is arbitrary or subjective or not "clear" in some epistemological sense. If we understand the distinction between first-order and other types of systems, it would seem that we then have a legitimate distinction also between logical and non-logical constants. Suppose we are given the vast infinity of first-order systems L discussed above. Further, given any formalized system at all, suppose we can tell whether it is in fact one of these systems or not. (For this,

ANALYTIC

TRUTH

45

of course, minor notational adjustments may have to be made, and slight differences of formulation taken into account. For what follows, we may assume that a suitable notational uniformity is achieved.) By examining this vast array of systems, one will note that certain constants are common to all. Other constants will be found to occur in at least one system but not in all. The former constants we call logical constants, the latter non-logical. In this way a very clear and simple way of making the distinction is provided. But nothing more ultimate, either of a metaphysical or epistemological kind, is involved in making the distinction in this way. In place of "logical constants" we could equally well speak here of "common constants". The distinction between logical and non-logical constants used throughout is thus seen to be harmless, being based merely on the classification of languagesystems as of such and such a kind.

Ill Adequacy

I

n this Chapter we consider the adequacy of the definition of 'Anlytc' given in (II,B) above. For this, of course, we must be clear as to what we mean by 'adequacy'. And any definition of it we might wish to give must be justified to some extent on intuitive grounds. A definition will be put forward here in some detail, and it may be that complete agreement concerning it will not be forthcoming. But several reasons will be given in favor of it. More specifically, in §A we consider the relation between the non-logical axioms of L and analytic truths, and in §B the particular case where L formalizes an axiomatic set theory. Still other object-languages are considered in §C. Finally, in §D and §E the adequacy of the definition of 'Anlytc' is established in the sense defined. A. Non-Logical Axioms and Analytic Truth. Consider now the non-logical or descriptive axioms of L . These in effect characterize in whole or part the properties or relations which the primitive predicate constants of L stand for, so to speak. Different properties and relations are characterized in different ways. Suppose that some descriptive axiom of L were analytic in the sense defined. It could then be regarded as characterizing not only the properties and relations designated by the primitive 46

ADEQUACY

47

predicate constants of L but all other properties or relations designated by primitive or defined predicate constants (abstracts) as well. If a descriptive axiom were analytic it would not succeed in its role of characterizing in whole or part the properties or relations designated by the primitive predicate constants occurring in it. Thus on intuitive grounds, we should wish the following to hold for any L under consideration. h DesAx a . (Sent a . b = a :v: b Clsr a)

~ Anlytc b.

To prove this, of course, for any specific L presupposes that the non-logical axioms of L have been explicitly enumerated or stipulated. And it may be that, for specific L, we should wish in place of this theorem one of a more complicated form, as we shall see. B. Set Theory. By way of an example, let us consider for a moment the non-logical axioms of the famous Zermelo-Skolem set theory (in the form due to Church). 1 Let this system be called 5 and formulated on the basis of the logic of (I,A), but with ' = ' regarded as a non-logical primitive. Let 'e' be its only other nonlogical primitive. Let us write the atomic formulae as 'x e y' and 'x =

y'

rather than ' e x / ' or ' = x y ' , as we would if we were to conform to the notation of Chapter I. 1 See E. Zermelo, "Untersuchungen über die Grundlagen der Mengenlehre I , " Mathematische Annalen 65 (1908): 261-281, and Th. Skolem, "Einige Bemerkungen zur Axiomatische Begründung der Mengenlehre," Wissenschaftliche Vorträge gehalten auf den Fünften Kongress der Skandinavischen Mathematiker in Helsingfors vom 4. bis 7. Juli 1922 (Helsingfors 1923): 217-232.

THE

48

NOTION

OF A N A L Y T I C

TRUTH

W e can then presume that two-place abstracts occur only in contexts such as 'x yzs w' rather than 'yzs

xiv .

The axioms of 5 are provided by the following rules. 51. H (x)(y)((z)(z Extensionality) 52. Set)

e x . = . z e y)

h ( E x ) ( j ) ~ y e x.

(Axiom

of

of the Existence of the Null w = y).

(Axiom

54. h ( x ) ( E y ) ( z ) ( z E y . = . (Eu;)(z e w . w e X)). of Sums)

(Axiom

of

53. h (x)(y)(Kz)(u>)(u> Pairing)

{Axiom

x = y).

Si. Axiom) 56.

z z := : w = x

h (x)(E^)(z)(z e y . = . (w)(u> e z

(Power

Y ( x ) ( ( E y ) y t x . (jy)(y e x.=>. (EZ)Z e y) . ~

( E z ) ( y £x,ZEx.~y

=

z . (Ew){w e y . w e z))

(z e x .=5. (Eu;)(u)((u Ez.uey). Choice) 57.

. w e x)).

h ( E x ) ( ( j ) ( ~

(Ez)

z

e y

= .u = .=>.

y

e x)

w))). . (y)(y

(Er)

( E y ) (z)

( A x i o m of e x

.=>.

(z)

((w)(w £ z . — . w = y) .=>. z E x))). (Axiom of Infinity) 58. h (x)ÇEy)(z)(z z y :=: z EX. A), where A is a formula of S not containing 'x' or ly' as free variables. (Axioms of Subset Formation) 59. h (x)((y)(y e x . = . ( E z ) ( w ) ( A . = .«*= z)) .=>. (Ez)(w) (u> E z . = . ( E y ) ( y e x . A))), where A is a formula of 5 not containing 'x' or 'z' as free variables. (Axioms of Replacement) S10. h (EX)A . =>. (Ex)(y* . ~ ( E y ) { y E X . B)), where A is a formula of 5 not containing ' y ' as a free variable, and B differs

49

ADEQUACY

from A only in containing free occurrences of 'y' wherever and only where there are free occurrences of 'x' in A. {Axioms of Consolidation.)1

The syntax and semantics of 5 we can formulate in a precise way as in (I,B) and (I,C-D). In particular we may suppose that 'ExtAx a ' is defined to express that a is the Axiom of Extensionality. I.e., ' E x t A x a'

abbreviates

qu (ex C\ac C\ac C\ep C\ex ex n id C\exC\

a

=

l

tripbar

{ex

qu exC\ac

qu

(exC\acC\ac

ex C\ac C\ac C\ep C\ex C\ac)

hrsh

ac))'.

Similarly ' N u l l S e t A x a' tilde

may abbreviate

'a

=

(ex

exisqu

exC\ac

a n d s o o n f o r S3-S7.

Also

'SubsetFormAx a' may abbreviate '(EA)(Fmla b . ~ FV b . ~

qu

exC\acC\epC\ex)\

ex F V b . a =

(exC\acC\acC\epC\exC\ac

(ex tripbar

qu exDac (ex Oac

exisqu

(exC\ac)

exC\acC\ac

C\ac C\ep Oex

dot

qu b))))\

and similarly for S9 and SiO. We can now prove that TBI.

I- E x t A x a . = . ~

Anlytc

a.

If, in the Axiom of Extensionality we replace simultaneously the primitive predicate constant 'e* by 'u>u3(w = w . u = u)' and ' = ' b y 'wu3(u>

E u .

u> e u)'

and then use the extended

Abst

concerning two-place abstracts, we gain the sentence T h e s e rules m a y be simplified if desired. See, e . g . , W . A c k e r m a n n , " M e n g e n t h e o r e t i s c h e B e g r ü n d u n g der L o g i k , " Mathematische Annalen 115 ( 1 9 3 7 - 8 ) : 1 - 2 2 ; and H. W a n g , " O n Z e r m e l o ' s a n d non N e u m a n n ' s A x i o m s for Set T h e o r y , " Proceedings of the National Academy of Sciences 35 ( 1 9 4 9 ) : 1 5 0 - 1 5 5 . 1

N.A.T.—E

50

THE

'(*)(r)((zX(* =

N O T I O N ' OF A N A L Y T I C

Z .x = x) . = . (z =

TRUTH

z . y = y)) .=>. (x e y .

~xEy))',

which is clearly false in 5 (by 7X7(1) and TC2(l) in translational semantics or by the analogous laws in non-translational semantics.) Similarly we have that TB2.

h NullSetAx a .=>. ~

Anlytc a.

We see this if we replace 'E' in the Axiom of the Null Set by WU3{W =

1

W. U =

u)'.

In the Axiom of Pairing ( S J ) we can simultaneously replace ' E ' by lvu3(v

e u . ~

v t u)' and ' = ' by ' v u z i v =

v . u =

u)'.

T h e n we see that TB).

h Pair Ax a .=>. ~

Anlytc a.

Also we have that TB4.

hSumAx a

~

Anlytc a.

W e see this if we replace 'e' in the S u m Axiom by ' u v u — v' and then derive from this in 51 the falsehood '(Ex)(.z)(Eu/)(~ z = w • ~ w = x) •=>• ( E j r ) ( z ) ~ z =

y\

T o show that the Power Axiom is not analytic we replace 'e' throughout by 'WU3(U> Z u . ~ w E U ) ' . This result is clearly false. Thus TB5.

I- Power Ax a

~

Anlytc a.

We let 'x c: y'

abbreviate '(z)(z e x .'=>. z E y)'.

T h e n in the Axiom of Choice, stipulated by S6, we replace ' E ' by 'xy3x

c y

and ' = = ' by 'xy3(x

From this result we can easily derive

=

x .y = /)'.

ADEQUACY

51

(1) < ( E r ) ( ( E r ) r . (Ez)z c y) . ~ ( E r ) ( E z ) (y . (Er)(Ejr)(z)(z c x .•=>. (Ew)(u)((u . (z e x \z)A

=

TRUTH

z e x . A)',

.=>. (x)(Ey){z){z ey

:^:zcx.

A)',

if ' x ' is not free in A, and hence (6) '(>,)...(ivn)(z)A .=>. K ) . . . ( ^ „ ) ( x ) ( E j ) ( z ) (z e y : =: z t x . A)\ if 'x' and 'y' are not free in A. If (3) is a logical theorem, then (7)

.. (wn){x){Ry){z){z

e y : = : z £ x . A)'

is likewise. But (7), for a specific A, is merely the closure of an instance of the Subset Formation Rule. Hence if (3), for that A, is a logical theorem, any closure of the appropriate instance of the Rule is also and thus in no way helps to characterize 'e'. Hence the Subset Formation Rule may now be stated with the proviso that (3) is not a logical theorem. But in view of TD4(ll), now that the notion 'Anlvtc' is available, we may also restate the rule as follows. If (3) is not analytic, then (7) is a theorem. We now wish to show that if (3) is not analytic, then neither is (7). Let 'E' throughout any instance of this Rule be replaced now by luv3(u E v .v.