A S I M P L I F I E D FORMALIZATION OF P R E D I C A T E

L O G I C W I T H I D E N T I T Y *

By ALFRED TAI~SKI in Berkeley, Calif.

I n t r o d u c t i o n

Two of the not ions commonly used in describing the formal i sm of (first- order) p red ica te logic exhib i t less s imple in tu i t ive con ten t and require def- init ions more careful and involved t han the remain ing ones. These are the no- t,ion of a var iable occurring free a t a given place in a fo rmula and the re la ted not ion of the proper subs t i tu t ion (or rep lacement) of one va r i ab le for ano the r in a given formula. The re la t ive ly compl ica ted charac te r of these two no- t ions is a source of cer ta in inconveniences of bo th prac t ica l and theore t ica l na- tu re ; th is is d e a r l y exper ienced bo th in teaching an e l e me n ta ry course of m a t h e m a t i c a l logic and in formalizing the s y n t a x of p red ica te logic for some theore t ica l purposes.

The not ions discussed seem to be essent ia l ly involved in the formal- izat ion of pred ica te logic. Nevertheless , we shall show in this paper tha t , b y in- eluding i den t i t y in p red ica te logic and mak ing essential use of i ts p roper t ies in the der iva t ions of logical theorems, even of those in which i d e n t i t y is no t involved, we can s impl i fy the formal iza t ion in such a way t h a t the use of the not ions discussed proves to be considerably reduced or even ent i re ly e lba ina t - ed . 1

More specifically, we shall set up the foundat ions for two formal sys tems of pred ica te logic with iden t i ty , referred to as ~ and ~2. The sys t em ~ pro- vides a complete ax iomat iza t ion for all un iversa l ly va l id sentences (i. e., for- mulas wi thou t free var iables) ; in t i le sys tem ~ such an ax ioma t i za t ion is p rov ided for all un iversa l ly val id formulas. As is well known, an ax iom sys- t em for p red ica te logic wi th iden t i t y can be ob ta ined f rom a comple te set o f ax ioms for o rd ina ry p red ica te logic b y adding the usual i den t i t y axioms. Our sys tems ~1 and | have been ob ta ined b y s impl i fying such a sys tem,

* Eingegangen am 30.10.62. The main results of this paper were s tated for the first Lime (without proof) in

[14] ; see also [6]. (The numbers in square brackets refer throughout to the biblio- graphy a t the end of the paper.) The paper was prepared for publicat ion when the author as well as Donald Monk and Wil l iam Pit t , who were assisting in the work on the paper, were engaged in a research project in the foundations of mathe- matics supported by the National Science Foundat ion (Grants G6693 and G14006).

62 Alfred Tareki

namely the one derived from Quine's set of axioms in [ 12]. Following Quine, we use detachment (modus ponens) as the only operation in deriving theorems from axioms in the system ~1 ; in ~ we apply in addition the operation of gen- eralization (universal quantification). The notion of a free occurrence of a var- iable and tha t of proper substitution are not involved in describing the axioms or in defining derivability for either of the two systems. In formalizing ~1 we have, however, to use the notion of a variable being free in a given formula; this notion (which is essentially involved if only in the definition of a sentence) has a simpler content than the notion of a variable occurring free at a giv- en place in a formula, and can be defined independently of the latter notion. In the formalization of ~2 the notion of a free variable is not involved at all. Instead of the generM notion of proper substitution we use, in describing ~ and ~ , a much more elementary and special notion: tha t of replacement, of one variable by another, in an atomic formula.

The main results of this paper are the completeness theorems for ~j and ~ . In order to establish these results i t will clearly suifiee to show that our formalization is equivalent to one for which a completeness proof is avail- able in the literature; specifically we shall refer to the completeness proof in [3].

Several variants of the system ~1 will be briefly mentioned, and one of them, the system ~] (structurally close to ~2), will be discussed in some de- tail. We restrict ourselves in this paper to systems of predicate logic in which predicates (relation symbols) occur as the only non-logical constants. Howev- er, the extension of our results to predicate logic with operation symbols and individual constants is almost immediate ; ef. [7], [8]. At the end of the paper we shall briefly discuss the extension of our results to predicate logic without identity.

Intuit ively underlying our whole discussion is the following simple obser- vation. Let ~ (a/E) be the formula obtained from the formula ~ by proper substitution of the variable fl for the variable a. (The precise meaning of the last sentence will be explained below.) Then, assuming tha t a and fl are two distinct variables, ~ ~a/fl) proves to be equivalent to the formula

(where A is the universal quantifier, ------ is the identity symbol, and --> is the implication symbol). In other words, both implications

q~ (a/E) --> A a ( (a ~ t ) --+ cp) and A a((a ~- t ) -+ ~) -~ ~ (a/t)

are universally valid. Thus, in a sense, the operation of substitution reduces to basic operations used in constructing formulas. I t may be noticed that the formula

A ~ ((<x ~- #) ~ ~)

can serve as an adequate substitute for ~ (af t) also in those cases when proper substitution of/7 for a in ~ cannot be performed {for the reason that,

A Simplified Formalization of Predicate Logic with Identity 63

as a result of the substitution, fl would occur bound at some places at which a occurs free).

This simple observation, which was a heuristic source o f the results in this paper, has also some further implications, with no direct bearing on the pres- ent discussion. I t shows, e. g., that , wi thout any loss in the power of ex- pression and proof, we could restrict the formalism of predicate logic with ident i ty by considering only formulas in which every predicate (with the ex- ception of the ident i ty symbol) is followed by a fixed sequence of variables. I n consequence, it leads to a simple characterization of cylindric algebras, i.e., those algebraic structures which are constructed with the purpose of providing an adequate apparatus for an algebraic analysis of predicate logic with identity. Cf. here [4], pp. 86, 94.

This paper will be followed by two articles with a closely related content. I n Kalish-Montague [8] it will be shown tha t in our axiom system for ~2 one schema can be derived from the others and tha t the system obtained by omitt ing this superfluous schema is independent ; the article will also give precise indications concerning the extension of our results to predicate log- ic with operation symbols and individual constants. In Monk [10] our axiom system for ~1 and some of its variants will be shown to be independent; more- over the article will contain an axiomatizat ion of predicate logic in which every predicate is followed by a fixed sequence of variables.

The author is great ly indebted to Donald Monk for his help in preparing this paper for publication and to William Pi t t for his work on the final draft .

T e r m i n o l o g y

We use throughout various familiar set-theoretical notions and symbols. By (o we denote the set of all natural numbers (non-negative integers). Each natural number n is identified with the set of all smaller numbers :

n : {m : m < n};

the formula m ~ n is thus equivalent to the condition : m e n and m, n e w. I n particular, 0 is the number zero and at the same time the empty set, 1 ~-- {0}, 2 = (0, 1}, etc. BA denotes the set of all functions on B to A (i. e., with do- main B and range included in A). The members of 'OA are what are called simple infinite sequences, with all terms in A. I n case ne e0, the members of nA are referred to as finite n-termed sequences, again with terms in A. The consec- utive terms (function values) o f a finite or infinite sequence / are denoted by /0 , / 1 . . . . . /n,.... Every finite sequence ] e (in e ~ nA uniquely determines the number n such tha t ] e nA ; n is called the length of / and is denoted by l / [" ( a ) is the sequence / with I / I = 1 and /0 = a; (a, b) is the sequence / with ] ~ I ~ 2, ]0 ~ a, ]1 = b; etc. Given two finite sequences / and g, we denote by /~ 'g their concatenation, i.e., the finite sequence h determined by the conditions :

64 Alfred Tarski

I h l = i l l + l g l ; h . = 1. for n < Ill; h i . , ' I + n = gnforn< Ig I"

W h e n descr ibing the fo rmal i sm of p red ica te logic, we shall use va r ious meta logica l symbols denot ing symbols of this logic, sets of symbols , sequen- ces of symbols , re la t ions be tween symbols and sets of symbols , etc. We shal l never use the symbols of p red ica te logic itself, and we shall no t need to speci- fy the shapes of these symbols .

S M will denote the set of all symbols of p red ica te logic. We assume t h a t this set is d iv ided into three d is jo in t subsets : the set VR of variables, the set LC of logical constants, and the set P R of predicates (relation symbols). VR is assumed to consist of inf ini te ly m a n y symbols which are a r ranged in a s imple infini te sequence v 0, v 1 . . . . w i thou t repea t ing te rms. ~ ' e shall use a,/~,

. . . . to represen t a r b i t r a r y e lements of VR; thus a, fl, y . . . . are meta logica l var iables used to represent a r b i t r a r y logical var iables . The set LC consists of four e lements : the implication symbol i, the negation symbol n, the univer- sal quantifier symbol q, and the identity {equality) symbol r Iqo s t ipula t ions re- garding the number of e lements of PR are m a d e : the set PR m a y be finite, infinite, or even non-denumerable . We assume t h a t wi th every e lement ~ PR a na tu r a l number r (~) is cor re la ted which is called the rank of ~. P R m a y be referred to as the set of non-logical constants. (The elements of P R funct ion as cons tants in our formal iza t ion of predica te logic since, when deriv- ing consequences f rom a g iven formula , we are no t pe rmi t t ed to subs t i tu t e any th ing for predica tes occurr ing in th is formula.) The iden t i t y symbol e, which is no t inc luded in the set PR, m a y nevertheless be aiso regarded as a predicate , and in fact as a p red ica te wi th r ank 2.

F in i t e sequences all t e rms of which are symbols are called expressions. E X is the set of all expressions; t hus

E X = U . e , ~ nSM.~

The sets S M and E X are assumed to be dis joint . The conca tena t ion ( e ) - ( a ) - ( f i ) where a and fi are a r b i t r a r y members of

VR is called the equation with terms a and fi and is deno ted b y a ~ ft. (The sym- bol -~ should not be confused wi th the symbol ~ ; the l a t t e r denotes the log- ical re la t ion of iden t i ty , a n d the former a meta logica l opera t ion defined on cer ta in couples of symbols and y ie ld ing an expression.) I f ~ is a p red ica te wi th r ( ~ ) ----- n and ~0 e n VR, t hen the expression ( u ) ~ 0 ~ l l be deno ted b y

For some reasons of both intui t ive and formal nature i t m a y be useful to t rea t the sets SM and E X and the operation ~ as undefined metalogical notions and to characterize them by means of a suitable axiom system. In this case the necessity of distinguishing between a symbol a and the one-termed sequence (a ) vanishes, and the set SM is t reated simply as a subset of EX. See here, e.g., [13] pp. 173, 282.

A Simplified Formalization of Predicate Logic with Identity 65

~r o ~. T h e express ions a ~ fl a n d :~ o ~v j u s t de sc r ibed a re ca l led atomic jot- mulcts; t h e se t of a l l a t o m i c fo rmu la s is d e n o t e d b y A T .

I f ~v a n d F are a n y express ions a n d a is a n y va r i ab l e , t h e n the exp res s ions ( i ) - ~ - y : , ( n ) - ~ , a n d ( q ) - ( a > - c p are r e s p e c t i v e l y d e n o t e d b y 99 --> ~v, -7 ~, a n d A a 99. B y a formula we u n d e r s t a n d an exp re s s ion ~c w h i c h be- longs to e v e r y se t F sa t i s fy ing the fo l lowing cond i t i ons : (i) A T C F, (ii) ~9 -2 y~ e I ' w h e n e v e r 99, ~f z F ; (iii) -~ 99 e F w h e n e v e r 99 e F ; a n d (iv) i ~x 99 e F w h e n e v e r 99 e F a n d a ~ VR. The se t of al l f o r m u l a s is d e n o t e d b y F M ; t h u s the se t F M can be c h a r a c t e r i z e d as the sma l l e s t class F s a t i s f y i n g cond i t ions (i)-(iv). F r o m now on t h e v a r i a b l e s 99, F . . . wi l l be u sed a l m o s t ex- c lus ive ly to r e p r e s e n t fo rmulas .

OC @) is t he se t of a l l v a r i a b l e s oecu~rk~g in ~, i . e . , t he set, o f , i : ~ z VR such t h a t a = ~n for some n < I 99 1" We s a y t h a t a occurs ]ree at the nth place in 99, in s y m b o l s OF (a, n, ~v), i f a e VR, q~ e F M , n < I 9 9 [, qJn = a, a n d t h e r e a re no ~v a n d m sa t i s fy ing the fo l lowing cond i t i ons : (i) ~fl e F M a n d

~,, < n < m + I ~ I < I ~ 1; (ii) ~0 = q a n d ~ = a ; (iii) ~k = 99~+k for e v e r y k < [ ~v ]. We s a y t h a t a is a / r e e variable o/99, in s y m b o l s a e F V (99), i f OF (a, n, 99) for some n. The fo l lowing l~ropert ies of F V (99) eas i ly fo l low f r o m these de f in i t ions : (1) i f ~0 e A T t h e n F V @) = OC (99); (2) i f 99, ~v e F M , t h e n F V (q: --> ~v) =- F V (q:) w F V (~) ; (3) ff ~ z F M , t h e n F V ( ~ ~v) -: F V (~); a n d (4) i f 99 e F M a n d a e VR, t h e n F V (A a V) is t h e s e t - t h e o r e t - ica l d i f ference of F V (99) a n d {a}. F r o m (1)-(4) i t is seen t h a t F V (99) can be de f ined d i r ec t l y b y an e a s y recurs ion , w i t h o u t r e fe r r ing to t he c o m p l i c a t - ed de f in i t i on of OF (a, n, 99).

A f o r m u l a ~0 w i t h F V (99) = 0 is ca l led a sentence, s y m b o l i c a l l y 99 z ST . A sen t ence ~o is s a id to be a closure of a f o r m u l a 99, in s y m b o l s ~v z CL (~v), i f t h e r e is a n n e ~o a n d an i e n(o such t h a t ~v = A vi0. �9 �9 A Vtn_ ~ 99. (A prec i se re- curs ive de f in i t ion of A Vio... AV~n_~ can eas i ly be g iven. ) I n case {v~j : j < n} = F V (~v) a n d i~ < i]+~ for e v e r y ] < n - - 1, y~ is ca l led the Quine closure of ~p a n d is d e n o t e d b y [(,v] ; cf. [12].

We s a y t h a t t he a t o m i c f o r m u l a F has been obtained/rom the atomic ]ormula by replacing some occurrence o] the variable a by the variable fi, in s y m b o l s P

( ~ # , ~ # , a , D ) , i f ~ v , ~ v e A T a n d ] cp l = t ~ v ], a, fl e VR, and t h e r e is a n n < [ T i such t h a t Vn = a, ~pn = fl, a n d 99k = Vk for e v e r y k < [ ~v [, k ~: n. I n case 99 a n d ~ a re a r b i t r a r y f o r m u l a s we i n t r o d u c e a m o r e gene ra l n o t i o n : ~v is obtained ]rom ~ by replacing a ]ree occurrence o] the variable a by a ]ree oc- currence o] the variable [~, in s y m b o l s R @, y', a,/~). T h e de f in i t i on runs as fo l lows: R (~, ~v, a, fl) i f a n d on ly i f qv, ,y a F M a n d [ cfl [ = I F ], a, fi a VR, a n d t h e r e is an n < [ ~ [ such t h a t OF (a, n, 99), OF ~ , n, ~), a n d 99~ = W~ for e v e r y k < [ ~v ], k ~: n. A r e l a t e d n o t i o n is t h a t of p r o p e r s u b s t i t u t i o n . The f o r m u l a ~v is s a id to be obtained ]rom the/ormula q~ by proper substitution o / the variable f l /or the variable a, s y m b o l i c a l l y S (~0, ~v, a, fi), i f ~v, ~v e F M a n d ] 99 ] = ]~v[, a, f l e V R , and , for e v e r y k < I q0 [, e i t he r O f ( a , k , ~ ) a n d OF (fl, k, ~v), or else OF (a, k, ~) does n o t ho ld a n d ~0~ = y~k. O b v i o u s l y

5 Mathematische Logik 7/1-2

66 Alfred T arslei

S (q, ~o, a, fl) can be defined in terms of R (~v, yJ, a, fl) by a simple recursion. As is easily seen, for any ~ e F M and any a, fl~ VR there is a t most one ~o such tha t S (T, ~o, a,/7). On the other hand such a yJ does no t always exist (i. e., we cannot always perform a proper subst i tut ion of/3 for a in q with- out first " renaming" some bound occurrences of/3 in ~0).

F r o m the last paragraphs it is seen tha t the definitions of the notions OF (a, n, ~v), R (~0, ~, a,/3),.and S (~0, ~o, a,/3) are indeed ra ther complicated. As was s ta ted in the introduction, the use of these notions will be avoided in our formalization of predicate logic. Nevertheless these notions will be essentially involved in the proofs of our main results.

i f / " is any set of formulas, we denote b y QF (/ ') the smallest set A ~_ F such tha t ~0 --> ~ e A and ~ ~ e A whenever ~0, yJ e zJ ; members of QF (F) are said to be quantifier-/ree combinations of formulas of / ' . By a valuation on1" we unders tand any function ] e r2 satisfying the conditions: if q, ~p, ~v --> ~o e F, then ] (~0 -~ ~o) = 0 if and only ff ] (~0) = 1 and ] (y~) = 0; ff ~0, -7 ~ e F, then ] (-7 ~v) = 0 if and only ff / (~v) = 1. We say thu t a formula ~0 is a tautology on F, symbolically ~0 e TA(F), ff q~ e QF (F), and ] (~) -~ 1 for every funct ion f which is a valuation on QF {F). I f ~v e TA (FM), or ~v e TA (ST), we say simply t h a t ~ is a tautological formula, or a tautological sentence, respectively.

~Ve shall denote by UF the set of all universally valid formulas. The usual definition of this notion, which is ra ther involved, need no t be given here, for reasons s ta ted in the remarks following Theorem I I below. The set of all uni- versally valid sentences will be denoted by US; thus US = UF n ST.

Given any set /" of formulas, we denote by F the set of all formulas deriv-

able f rom those of /" by using only the operat ion of detachment, and b y / ~ t h e set of all formulas derivable f rom those o f f by using the operations of detach- ment and generalization. T h u s / i i s tbe smallest set zJ ~_ F such t h a t y~ e A when-

ever ~0, r --> ~ e A ; ~ is the smallest set A ~_/' such tha t yJ e A whenever ~, ~ --> v 2 e 4 , and A a ~ e zJ whenever r e A and a e VR. Ins tead of ~ e

we usually write F ~-- ~, and instead of r e F we write F ~ ~. A se t / " of sentences is called complete f f P : US. Similarly a set/" of formnlas is called

complete ff F : US'. Since every sentence is a formula, the term "complete" jus t in t roduced is clearly ambiguous when applied to a set of sentences. When applying it, however, to a set o f sentences, we shall always use it in the first of the two senses defined above; obviously a set of sentences is never complete in the second sense. I n informal passages we shall apply the term "complete" no t only to a set /" of sentences or formulas, bu t also to the sys- t em of predicate logic for which/" serves as the set of axioms.

To establish the main results of our paper we shall use the following two completeness theorems:

THEOREM I (COMPLETENESS THEOREM FOR SENTENTIAL LOGIC). Given any set F C FM, let A be the set consisting o/all formulas of the following three kinds:

A Simplified Formalization of Predicate Logic with Identity 67

(i) @ -+ y~) --> ((~ --> Z) -+ @ --> Z) ), where 9, y3, )~ e Q F (F) ;

f f i) ( -~ 9 --~ 9) --> 9, where 9 ~ Q F (F) ;

(iii) 9 -+ ( -7 9 --> ~) , where q), y~ e Q F (I ') .

Then ~ : T A (F) .

This is the main result of •ukasiewicz [9] ; a proof may also be found, in outline form but more accessibly, on p. 167 of [2].

THEOREM I I (Co~meLETE~ESS THEOREM FO~ PR~DICATV. LOGIC). Le t /1 be the set o / a l l / o r m u l a s o / the ]ollowing seven k inds :

(i) ( 9 - - ) - y J ) - - > ( ( 9 - - > ( y ~ - - > ) ~ ) ) - - > ( 9 - - > g ) ) , w h e r e 9 , y s , ) : e F M ;

(ii) 9 --> (~ -~ 9) , where 9, YJ e F M ;

(ii i) (--1 9 --> -7 yJ) -+ (y~ --> 9), where 9, Y~ ~ F i l l ;

(iv) A a (9 --> Y~) -+ (9 -> A a yJ), where 9, Y; ~ F M , a e VR , and a 6 F V (9) ;

(v) /~ a 9 -~ Y~, where S (9, YJ, a, fl) ;

(vi) a ~ a, where a e V R ;

(vii) a ~-- fl --> (9 --> ~) , where S (9, Y~, a, fl).

Then LJ is complete, i. e., ~ ~ UF.

This result is essentially due to GSdel, but in the form just given it is for- mulated and proved in [3]. For the purposes of our further discussion the

formula ~ ~ U F stated in the conclusion of Theorem I I may be treated as the definition of UF. ~;e shall take as granted certain simple consequences of this conclusion; more specifically we shall assume without proof tha t various simple formulas belong to UF.

I t may be observed that the conclusion of Theorem I remains valid upon replacing formulas ( i ) - ( i i i ) of Theorem I by formulas ( i ) - ( i i i ) of Theorem I I ; see [2].

w 1. T h e s y s t e m ~1

As mentioned in the introduction, the sys tem ~ is intended to provide a complete axiomatization for all universally valid sentences of predicate logic with identity. The set ~'~ of axioms of ~1 consists of all sentences fall- ing under one of the following nine schemata, where a, fl z V R and 60, Y~,Z e F M are arbi trary unless otherwise stated:

(A1) [(9 ~ W ) ~ ((~ -+Z) ~ (9 ~ Z ) ) ] ;

(A2) [(-7 9 --+ 9) ---> T];

(A3) [9 -> (-7 9 --> ~v)];

(A4) [A a / \ f 1 9 -->/~fl A a g ] ;

(As) [A a (9 -~ ~) -~ (h a 9 -+ h a ~0];

(A6) [A a 9 -+ 9];

(AT) [9 --> A a 9], where a 6 F V (9);

5*

68 Alfred Tarski

(A8) [-1 A a ~ a---- fl], where a 4= fl;

(A9) [a ~- fl -+ (~0 --> ~p)], where P (% ~p, a, fl).

The sentences falling under (A 1)-(A 3) are the sentential calculus axioms, those falling under (A4)-(A 7) are the quantification axioms, while those com- ing f rom schemata (AS)-(A9) are the equali ty axioms. As will be seen, however, these three classes are used interdependently in the essential deri- vat ions of the system ~1. The system ~1 has ~71 as its set of axioms and de- t achment as its only rule of inference 8. The main task of this section is to show t h a t ~1 is complete. To prove this (in Theorem 1 below), we shall give first a number of necessary lemmas. I n these lemmas and th roughout the remainder of the paper, ~0, % Z, and 0 are arbi t rary formulas, and a, fl, and y are arb i t rary variables, unless otherwise stated.

We begin with three lemmas the proofs of which are essentially the same as the proofs of *111, "112, and "113, respectively, in [12].

L r ~ M A 1. I [ I 1 ~ - [qJ ~ ~] and X 1 ~-- [~0], then • ~-- [F].

LV.M_MA 2. I [ 2:1 ~- [~] and a e VR , then X 1 ~-- [A a ~0].

LE~aMA 3. Jr/ '~'1 ~" [~0 -+ ~] and ~x ~-- [Y) --> g], then Z 3 ~-- [qJ --> g].

From Lemma 1, schemata (A 1)-(A3), and Theorem I we obtain:

L v . M ~ 4. I [ q) i s a tautological lormula, then Z 1 b-- [~0].

The following lemma can now be easily proved:

L E m ~ 5. I [ a r F V (~), then 11 ~ [A a (~0 -+ ~v) --> (~0 -> A a ~v)].

PROOF. We have:

2:~ ~-- [ h a (~0 -> ~) --~ (A a ~ -> A a ~)] by (A5);

2:1 ~-- [~0 -+ A a ~] by (A7);

(A o~ (~p -+ ~p) -+ (~ -+ A a ~p) ) )] by Lemma 4;

271 ~- [A a (~ -> Ip) -> (~0 --> A a ~)] by Lemma 3.

L~.~m~ 6. X1 ~- [a------ a]-

PROOF. Let fl be ~ variable distinct from a. Then

"~'1 ~ [fl ~ {% -O- ( ~ ~ 0~ --)" 0~ ~ 0~)] by (A9);

2:1 ~- [-7 a ~- a --> ~ fl ------ a] by Lemmas 1 and 4; )

2:1 ~- [A fl ( 9 a-= a -+ -7 fl ~ a)] by Lemma 2;

s The axioms of ~x differ somewhat from t~hose found in our abstract [14]. The only important difference is tha t in the abstra~:t the schema (A 7 ) was provided with the condition a ~ OC (~) instead of a ~ F V (q~). This was, however, evident- ly a mistake; the stronger condition a ~ F V (~0) seems to be needed to insure com- pleteness; cf. [10].

A Simpli f ied Formal izat ion of Predicate Zogic with Iden t i t y 69

Z 1 ~-- [-7 a ~ a --> A fl -7 fl ~ a] b y L e m m a s 1 ~nd 5;

Z l ~ - - [ ~ A f t - T f i ~ - a - > a - a ] b y L e m m a s 1 a n d 4;

Z 1 ~ - [a ~ a] b y (AS) a n d L e m m a 1.

The n e x t few l e m m a s l ead up to L e m m a 12, in which a s t r o n g e r v e r s i o n o f t he e q u a l i t y l aw (A9) is g iven .

LEMMA 7. Z 1 ~ - [a--~ fl --> fl ~ a].

PROOF. W e h a v e

271 ~ - [a ~ fl --> (a ~ a --> fl ~ a] b y (A9) ;

271 ~ - [a ~ a --> (a ~ fl - * fl ~ a)] b y L e m m a s 1 a n d 4;

271 ~ - [a ~ fl --> fl ~ a] b y L e m m a s 1 a n d 6.

U s i n g L e m m a s 1, 4, a n d 7, we can eas i ly p r o v e the fo l lowing t w o l e m m a s :

LEMMA 8. I ] Z 1 ~-- [fl ~-- a --> (V --> ~)], then Z 1 ~ - [a ~ fl - * ( ~ T --> ~ V)]"

LEMMA 9. I ] Z 1 ~ - [fl ~ a ---> (~v --> ~)], then 272 ~ - [a ~ fl --> ((of --> Z) --> (9 -~ z))].

Aga in , a p p l y i n g L e m m a s 1 a n d 4, we o b t a i n :

LEMMA 10. I / Q b-- [5 ~ ~ ---> (~) - ~ ~)] , then Z a ~ - [5 =:/~ --> ( (g --> of) - - >

(z -> ~)].

LEMMX 11. I / 271 ~ - [a ~ fl - * (~ - * ~v)] and 5, fl * Y, then Z, 1 ~-- [5 ~ fl -->

(A y ~ -> A y ,?)]. PROOF.

Z~ ~--- [A 7 (5 ~ / ) - > ((p -+ ~p) )] b y L e m m a 2 ;

Z 1 ~ - [5 ~ fi --> A y (~ --> ~f)] by L e m m a s 1 a n d 5 ;

271 ~ - [a ~- fl --> (A y ~ - * A r ~) ] b y (Ab) a n d L e m m a 3.

LEMMA 12. I / R (~, ~, 5, fl), then Z~ ~ - [5 ~ fl --> (q~ --> ~v)].

P ~ o o F . L e t F be t he class of al l ~ e F M such t h a t , for a l l ~ e F M a n d al l 5, fl e V R , i f R (~, % 5, fl), t h e n 27~ ~-- [ a ~ fl --> (~ --> ~o)]. F o r each n e co l e t On be t h e se t of al l ~ e F M such t h a t for e x a c t l y n n u m b e r s p < ] ~ I we h a v e q% e {~, ~, q}. Us ing (A9) a n d L e m m a s 8-10 , i t is eas i ly seen b y i n d u c t i o n t h a t On C F for each n e co ; s ince F M : Une~ On, t h e t h e o r e m fol lows.

L~M~A 13. I / S (q), 9 , 5, fl), then 271 ~ - [5 ~ fl --> (of ---> ~v)].

PROOf. I / ~ = ~v, t he conc lus ion fol lows b y an a p p l i c a t i o n of L e m m a 4. On t h e a s s u m p t i o n t h a t ~ ~ % t h e r e is a n n e co w i t h n 4= 0, 1 a n d t h e r e is a

e n F M such t h a t g0 = q0, gn-~ = % a n d R (Z~, Z~+I, a, fl) for a l l i < n - - 1 . One can t hen p r o v e t h a t

271 ~ - [5 ------ fl --> (~ --> Z~)] for each i < n

b y i n d u c t i o n on i, m a k i n g use of L e m m a s 1, 4, a n d 12. Our l e m m a fol lows fo r t h e p a r t i c u l a r case i ~ n - - 1 .

5~

70 Alfred Tarsld

LEMMA 14. I ] a ~= 8 and S (~0, ~, a, 8), then 2:1 ~- [A a ~0 -+ ~v].

PROOF. Note t h a t a r F V (~). Hence

2:~ ~- [a =-- fl --> (~o -+ ~o)] b y Leroma 13 ;

271 ~- [~0 -+ ( 7 ~ -+ ~ a ~ /~) ] b y L e m m a s 1 and 4;

2:1 ~- [A �9 ~o -+ A a (-~ v/--> -7 a------ B)] b y L e m m a s 1 and 2, and (A5);

2:1 ~- [A a ~0 -+ ( 7 ~ - + / ~ a ~ a -~ fl)] b y Lernmas 3 and 5.

2:1 ~ - [-7 A a -7 a- - - 8 - ~ (A a ~ - ~ ~)] b y L e m m a s 1 and 4;

2:1 ~- [A a ~ -~ v/] b y (A8) and L e m m a 1.

F r o m L e m m a 14 and (A6) we obta in :

LrMMA 15. I 1 S (~, W, a, fl), then 2:1 ~ [A a ~ --+ ~].

The following two l emmas are no t needed for our fur ther discussion, but , as ment ioned in the introduct ion, t hey fo rm a heuristic basis for the construc- t ion of the sys tems ~1 and Ca.

LE~rM, 16. I [ a * 8 and S (q~, ~, a, fl), then 271 ~ [~ --+ A a (a ==- fl --+ ~0)].

PROOF. Clearly S (-7 ~0, -7 ~, a, fl) and a r F V (ta). Thus

2:1 ~- [a ~ 8 -+ ( 7 ~0 -+ -7 W] by L e m m a 13;

2:1 ~- [ v / -+ (a-~ fl --> ~)] b y L e m m a s 1 and 4;

2:1 ~- [A ~ (W -+ (a------ fl -+ ~o))] b y I~.mma 2;

2:1 ~-- [~ -+ A a (a ~ 8 -+ ~0)] by Lem,nas 1 and 5.

LEMMA 17. I ] a =~ 8 and S (~, ~, a, fl), then 271 ~- [A a (a --= 8 -+ ~0) -+ V].

PROOF. Clearly S (a ------ 8 -+ ~, 8 ~ 8 -+ V, a, 8).

Hence

2:1 ~- [A �9 (~ ~ 8 -~ ~) --> (8 ~ 8 -~ ~)] b y L e m m a 15;

271 ~- [A �9 (~ ~ 8 --> ~) -~ ~] b y L e m m a s 1, 4, and 6.

THEOREM 1. The set 27~/8 con~/e~, i. e., ~ i = US.

PROOF. L e t / t be the set of formulas described in Theorem I I . I t can easily be

seen t h a t ~ C z~. Since ~ h a s the p rope r ty t h a t ~ e z~whenever A a ~ e A,

i t follows t h a t {~ : ~ e ~ ' M and [~] e ~1} _C zJ. F r o m L e m m a s 4, 5, 15, 6,

13, 1 and 2 we see t h a t the opposite inclusion z~ C {~0: ~ e F M and [~] e ~1}

also holds. Thus z~ = { ~ : ~ e ~ M and [~] e ~ } , so ~ S T = ~ r The- orem I I now implies t h a t ~ = US, i. e., 2:1 is complete.

The sys t em ~1 can be subjected to various modifications. The mos t impor-

A Simplified Formalization of Predicate Logic with Identity 71

taut of these concerns the axiom schema (A 6) ; it turns out that this schema can be replaced by much weaker schemata, which we shall now describe. 4

First let Z* be the set of all sentences falling under schemata (A 1)-(A5) and (A 7)-(A9) together with all instances of the following weakened version of (A6):

(A6*) [A a ~ --> ~], where a r F V (q~).

Now in the proof of Theorem 1 the schema (A6) was used only in the proofs of Lemma 1 and Lemma 15. The proof of Lemma 1, however, ~etually in- volves only the weaker schema fA 6*). Hence Lemmas 1 through 14 hold upon replacing 2:1 by 27" ; these modified lemmas may be referred to as Lemma 1", Lemma 2", etc.

L~m~A 15". Z~I ~ [A a ~ - ~ ~].

I>ROOF. Let fl be a variable such that f lr OC (A a q~). Then there is a yJ e F M such that S (~, ~, a, 8) ; it follows that also S (~, ~,/~, a). Hence

271" ~- [A a ~ -+ ~] by Lemma 14" ;

~- [A a (A a ~ --> yJ)] by Lemma 2* ;

~- [A a ~0 ~>/~ a ~] by Lemmas 1" and 5* ;

~-- [A a ~ -~ ~] by Lemma 1 4 " ;

~- [A a ~ -+ ~] by Lemma 3*.

Thus every instance of (A6), and hence every member of 2: 2, is a member of ~*. The following theorem is, therefore, an immediate consequence of Theorem 1 :

T~EORE~ 2. The set ~* is complete, i. e., ~"l -~ US.

A somewhat deeper modification of the system ~1 is as follows. We shall consider subsets O of US satisfying the following condition :

(A6') With each a e VR there is associated a /ormula ~ (~) such that F V (~ (a)) C {a} and -7 A a ~ (~) e O.

(Notice that (Aft) does not contain any stipulation concerning the relation- ship between ~ (a) and ~ (~) for two different variables a and/~.) We shall show that every such set O containii~g in addition all the sentences (A 1)-(A 5) and (A 7)-(A 9) is complete.

In the following lcmmas we assume tacitly that 0 is a fixed subset of US containing the sentences (A 1)-(A5), (A 7)-(A 9) and satisfying the condition

4 The modifications concerning (A 6) were found only recently. The author, know- ing from [6] and [7] that the schema (A6) is redundant in the related system ~2 to be discussed in w 2, tried to eliminate this schema in system ~1 as well, and succeeded at least in weakening it. As is shown in [10], the complete elimination of (A6) in system ~1 is impossible.

72 Alfred Tarski

(Aft). Moreover, whenever we use the symbol ~ (a), we assume that it de- notes a formula with the properties stated in (Aft). Note that it is not pos- sible to prove the analogue of Lemma 1 at once, since the proof of that lemma involves (A6).

From Theorem I and (A 1)-(A3) we deduce first:

LEMMA 18. I f q~ iS a tautological sentence, then 0 v-- q~.

Using (A 1) and detachment, we have:

L~M~Lr 19. I / 0 v-- q~ --> ys and 0 v-- ~ --> ~, then 0 v-- qn ---> •.

LEMMA 20. I/ A %... A Ol~-I (~ --~ ~Y) = [~ ~ ~/;], then 0 v-- A ao... A a n - i

(~ --> ~Y) --> (A 0~0.-. A 0~n-1 99 --)- A a0--- A an-1 ~/)).

PROOF. For n : 0 the statement is a particular case of Lerama 18. Now as- sume the statement of the lemma to be true for n (inductive premise) and the hypothesis to hold for n q- 1. Then

A %--- h an-1 (h an (~ - ~ ) -~ (h an ~ -~ h an~)) = [A an (~ - ~ ) -~ (A an ~ -~ A an ~)];

O ~- A %--. A an-1 (A an (~ -~ ~) -~ (A an ~ -~ A an ~)) -~ (A %.. . A an (~ -+ ~) -+ A %.. . A an-1 (A an ~ -+ A an ~)) by the inductive premise;

o ~- A %... A an-1 (A an (~ -~ ~) -~ (A an ~ ~ A an ~) ) by (A5);

0 b---- A (XO''" A an ((P --~ ~/)) ~ A ao.-- A o~n-1 (A o~t (p --~ A o[~ ~/)) by detachment;

O ~- A %... A an-1 (A an ~ -+ A an ~) -+ (A %..- A an ~ -+ A %. . . /~ an V) by the inductive premise;

O ~ - A %.. . A an (~ - ~ ) -~ (A s0... A an~ -+ A %.. . A an ~) by Lemma 19.

This completes the inductive proof.

L~.MMA 21. I f a is a biunique /unction with domain n e w and range F V (~), and i / lc is a permutation on n, then

0 ~ A ao..~ h an-~ ~ --> A a~o.., h a~,~_~ ~.

PROOF. The lemma holds for n = 0, 1 by Lemma 18. Now assume: n > 2, the theorem holds for every m < n, and the hypothesis holds.

First consider the case in which ]r is a transposition of a number and its successor, say, ]c = (0 .. . . , m -~ 1, m ... . . n - I ) . Now there is a permutation I on m such that

h ~/0' '" A ~ltli-- 1 h 0lin... h ~li-1 ~19 = [ h ~lt . . , h an-1 ~19]. Thus

O I--- A aZO''" A O~lin_ 1 (A ~ ' ' " A an--1 ~O --~ A 0ltlt'l'i A ~ ' ' ~ A an-~ ~) b~ (A4);

A Simplified Formalization of Predicate Logic with Identity 73

O ~- A %.- - A a~_~ A a~ . . . A a~- , ~ -+ A % . - . A a~m_ ~ A am+~ A am.. . A an-1 q~ by Lemma 20 and de tachment ;

O ~- A ~0--. A a,__ ~ ~ A ~0"" A % _ _ A am... /~ an-~ cf by the inductive premise;

O ~- A a~o... A %~_, A a,,+~ A a~ . . . A 0~n-i 99 --~ A 0~0''" A a~+~ A am... A an-~ ~ also by the inductive premise;

O ~- A ao... A ~ - ~ ~ -+ A ao... A ~,~+~ A a~. . . A ~ - ~ b y Lemma 19;

O b--- A g 0 ' ' " A a n - i ~ --> A a/r O .... A gk?~_ 1 ~9.

Thus the conclusion of the lemma holds for every transposit ion of two ad- jacent elements. Hence, by induction using Lemma 19, it holds for every per- mutat ion.

L~MMA 22. I] cf ~ ST , 0 ~-- -7 ~o, ~ e QF ({~0}) and O ~-- A %. . . A an-1 ~o, then 0 ~-- v 2.

P~OOF. Let F be the set of all Z ~ QF ({~}) such tha t either O ~- 5/or O ~-- ~ g" By hypothesis, ~ e F . Obviously ~ g e F whenever 2 e F. I t is easily verified using Lemma 18 t h a t x --> 0 ~ F whenever Z, 0 ~ F. Hence /" = QF ({~}) and in particular ~ e F . I f O ~- ~ ~p, then f rom the assumption O _C US and the hypothesis of the theorem we would have 9, ~ ~ e UF, which is impossible. Therefore O ~- ~o.

L~MMA 23. I f a 6 F V (q~) and yJ ~ TA ({A a ~, ~0, A a ~ (a)}), then 0 ~-- [~o].

PROOF. Let F = {A a ~0, ~, A a ~ (a)}, and let z] consist of all formulas in the schemas (i)-(iii) of Theorem I. Then, by Theorem I, ~ = T A (F). Let E = {Z e A : O F- [Z]}. To prove our lcmma it suffices to show t h a t ~ C E. F rom (A1)-(A3) we infer t h a t A _C E.

Now suppose Z, Z -+ 0 e E. Thus Z, Z -~ 0 e A, 0 ~-- [Z], and O ~- [Z --> 0]. Therefore 0 e ZI and )~, Z --> O, 0 ~ QF (F). Let [~] = A rio... A fin-1 gO. N o w i f [Z -+ 0] = / , --> 0, then [X] = Z, [0] = 0, and by de tachment we have O ~-- [0], so t h a t 0 ~E. In the other case we have [Z ~+ 0] = Aft0--- A fin-1 {Z -+ 0). Then, by Lemma 20 and detachment ,

(1) O ~- A / ~ o - . - A # , - - X -+ ADo.- . A # , - - 0.

Now ff Ix] = A rio-.- A fin-1 Z, then from (1) we obtain by de tachment

(2) O ~ A t~0.-- A #n--, 0.

I f on the other hand [Z] = Z, then by assumption O ~-- 25, and several appli- cations of (A7) and Lemma 19 give 0 ~- A f lo. . . /~ fln-~ Z; then de tachment applied to (1) yields (2) again. Thus (2) holds in either of the two possible cases. I f [0] = A rio.-- Afln-~ O, then (2) means thu t O ~- [0], so t h a t 0 e E. I f [0] ~ 0 and n > 0, then ~ r S T and consequently 0 e QF ({A a $(a)}), and O ~- [0] follows from (2) by Lemma 22 ~nd (Aft) ; thus again 0 e E.

74 Alfred Tarsk~

Thus E is closed under the operation of detachment, so ~ C E, and the proof is complete.

L~MMA 24. I / a ~ F V (q), then 0 ~-- [A a q~ --* q~].

PROOF. Let [~o] = A/70... A fin-1 q~. I f F V (~r = {a}, then, by Lemma 21,

(1) O , - [~ ~ (~ ~ -->- ~(<<))] ~ A ~o... A t~,,-~ Aa (~ ~ ( ~ ~ ~ ~ (~,))).

If, on the other hand, ~(a) e ST, then

O ~- [(~0 -+ (-7 q --~ ~(~))) -~ /~ a (q -* (-7 q --~ ~(a)))] by (A7);

then {I) follows by Lemma 20 and detachment. Thus (1) holds in either case. Hence

0~-- Aflo. . . Afln-1/~ a (~ --~ (-~ ~ --~ ~ (a))) by (1), (A3), and detachment;

O ~-/~/70.. . A/~n-1 (A a q -~ A a ( ~ ~ --~ ~(a))) by (A5), Lemma 20, and detachment;

0 ~- A Bo... A ~ - ~ (A a ~ - ~ ( A ~ ~ ~ - ~ A ~ (~))) by (A5), (A1), Lemma 20, and detachment;

O ~ - A ~ o . . . A ~ - I ( - ~ - ~ A a ~ ) b y ( A 7 ) ;

O ~ A ~o... A/7,,-1 ( ( A a -~ ~ ~ A a~ (~<)) ~ (-7 ~ - + A a ~ (<,))) by (A1), Lemma 20, and detachment;

O ~- A ~ o . . . / ~ / ~ - ~ ((A a ~--~ ( 7 ~v --~/~ a t ( a ) ) ) by (A1), Lemma 20, and detachment;

O,-- A flo... A fl,~-~ (CA a ~ ~ ( - ~ ~ ~ A ~ (,,,))) (--, A a ~ (~) --+ (A a ~ --+ ~) ) ) by Lem~a 23 ;

O,-- A~o. . . A ~,,- , -~ A a~ (~,) ~ [ A am --->- ~] by (A1), Lemm~ 20, and detachment;

O ~- A rio... A fln-~ -~/~ a ~ (a) by (Aft), (A7), and detachment;

O ~- [A a ~ -~ q] by detachment.

XVe have now shown tha t every sentence falling under (A6*), and hence every member of ~ , is a member of D. Thus Theorem 2 implies the follow- ing theorem:

THEOREM 3. I f 0 is a subset o~ US such that all the sentences o] schemata (A 1) -(A5) and (A7)-(Ag) are members o/O, and q 0 satisfies the condition (Aft), then 0 is complete, i. e., 0 : US.

Several natural choices are available for the formulas ~ (a) satisfying con- dition (A 6'). For example, we can let ~ (a) = 7 a ~- a ; in other words; the axiom system obtained ]rom ~ by omitting (A6) and deleting the provision "where a 4= fl" in (AS) is complete. We can also let ~ (a) ---- -7 (~ --> ~) where

is a fixed sentence. In addition to the Quine closure [~] some other related notions have been

discussed in the literature, in particular that of the Berry closure (see [1]). As

A Simplified Formalization of Predicate Logic with Identity 75

is easily seen, all the proofs above remain valid if [99] is understood to be the Berry closure, rather than the Quine closure, of 99.

In the article [10] it is shown that both in the system ~1 and in allits modifi- cations mentioned above the underlying sets of axiom schemata are independ- ent. Thus in particular the replacement of Quine closure by Berry closure does not make the commutat ivi ty schema (A4) redundant; the situation is different here than in the case of the original system of Quine 5.

To conclude we may mention that in all the systems discussed so far the schemata (A 1)-(A3) can be replaced by the much stronger stipulation accord- ing to which [99] is an axiom for every tautological formula 99. Applying this modification together with the replacement of (A6) by (Aft), the proof of Theorem 3 becomes simpler since Lemmas 22 and 23 become superfluous. On the other hand, however, by using the term "tautological" in the formal description of a system of predicate logic, we introduce a notion which is of at least as complicated a nature as those notions whose elimination is the main purpose of this article.

w 2 T h e s y s t e m s ~ a n d ~ .

Like ~1, the system ~ is intended to provide a complete axiomatiz- ation for all universally valid sentences of predicate logic with identity. The axioms of @1 are obtained by taking Quine closures of all formulas in a cer- tain set. Using an idea of Fitch [5], we shall obtain the axioms of ~ by tak- ing all closures of members of a related set of formulas.

The set 2:~ consists, namely, of all closures of formulas of the following eight kinds :

(B1) (99 --> y~) --> ((~p -+Z) --> (99 -~ Z));

(B2) ( 7 q -+ q) --> 99; (B3) q -+ (-7 q --> ~v);

(B4) A ~ (99 -+ ~,) -~ (A ~ 99 - § A a w);

(B5) A a99 ~99; (B 6) 99 --> A a 99, where a r OC (99) ;

(B7) ~ A a ~ a ~ fl, where a # /~ ;

(B8) a ~ ~ --> (99 --~ ~fl), where P(99, ~v, a, fl).

~ is the system of predicate logic with ~:~ as its set of axioms and detach- ment as its only rule of inference.

The system ~ is closely related to ~1- In fact, since [99] ~ CL (99) for any 99 e F M , each instance of the schemata (A1)-(A3), (A5), (A6), (A8), and

5 CL [ 1 ]. I t may be mentioned that the presence of the commutativity schema (A 4) seems Ix) be essential even under Berry closure if another of Quine's original axioms (the one corresponding to (v) of Theorem II) is weakened in the way indi- cated in [11].

76 Alfred Tarski

(A9) is a member of Z~. Thus it is convenient to prove the completeness of ~ by using the fact already proved that ~1 is complete.

LEMMA 25. I / n e w, a e "VR , A % . . . A an-1 (qJ --> 9) c CL (q~ -+ 9), Z~ ~-- Aao... Aan-1 (~-->9) and Z ~ - A a 0 . . . A a n - 1 % then ~'1~-- A ao ' " A an-19. PROOF. For n = 0 the conclusion follows by applying the operation of de- tachment. Now assume the lemma to be true for n and suppose the hypo- thesis holds for n -t- I. Then

X~ ~-- A a0.-- A an-1 [A an (~ ---> ~) -+ (A an ~ -+ A an 9)] by (B4); 2:~ ~- A a0... A an-1 (A an ~ -~ A an 9) by the inductive premise;

Z~ ~- A %.-. A an yJ by the inductive premise.

This completes the inductive proof.

From Lcmma 25 we obtain at once

L ~ . ~ A 26. I] CL (q~ ---> 9) -C 2,-~ and CL (q~) _C Z~, ~' then CL (9) C_ Z~.-'

This lemma is the analogue for the system ~ of Lemma 1. The analogue of Lemma 2 is as follows:

L ~ . ~ 27. I ] CL (T) _C Z i and a e VR, then CL (A a el) C Z~. ~'

Since CL (A a q~) C CL (~), this lemma is trivially true. On the basis of these three lemmas the following lemma can be established

with essentially the same proof as Lemma 14:

LEMMA 28. I ] a =~ fl, S (T, % a, fl), and a r OC (9), then CL (A a ~ -> 9) _C ~ .

LEmu~ 29. I] a 4 fl, S (q), 9, a, fl), f l r OC (q~), and a r OC (9), then CL (A a - ' Z~. --> A f ig ) C _ Z ~ a n d C L ( A f l g ~ A a ~ ) C_

I~ooF. From Lemma 28 we have that CL (A a ~ -> 9) C 2~. Hence, by Lemmas 27 and 25, and by (B4), CL (A fi A a ~ -~ A fl 9) _C ~ . But fl ~ OC (A a ~), so by (B6) we deduce CL (A a ~ -+ A fl ~) _C Z~. By symmetry, C L ( A f l g ~ A a~) C ~'

The following lemma represents the main step toward proving that Z~ is complete.

L E ~ 30. I [ a e VR and q~ e F M , then there is a 9 e F M satis/ying the [ollow- ing /our conditions:

(i) CL (q~ --> 9) C_ Z~;

(ii) CL (9 -+ q~) C_ - "

(iii) i] n < [ 9 I and ~fn = a, then OF (a, n, 9);

~iv) ~ V (q~) = F v (9).

PROOF. Le tT be the set of all r e F M for which there is a 9 satisfying (i)-(iv). I t is easily established that A T C 1" and that -7 % ~ --> g e ]~ whenever ~, g e T .

A Simplified Formalization of Predicate Logic with Identity 77

Suppose ~ e F and f le VR. Let ~p be a formula satisfying (i)-(iv). Then by use of (B4) it is easily proved tha t

(1) e L (h tff ~ -> h fl ~v) _c z-~

and

(2) CL (A fi~v -+ A fi~) _C Z~.-'

If/~ :~ a, then A fi ~ and A/~ ~v also satisfy conditions (iii) and (iv) ; hence /~ fi ~ e F. Assume fi = a. Let 7 be a variable occurring neither in A a ~v nor in ~. Then there is a formula 0 such tha t S (~, 0, a, y). Therefore a 4= and a ~ OC (0). Hence by Lemma 29 the following two s ta tements hold :

(3) e L (h a ~f -+ h y 0) C_ - "

(4) CL (A r 0 -+ A c, w) c_ ~;.

Combining (1)-(4) with the fact t ha t a ~ OC (A y 0), we see tha t (i)-(iv) hold with % ~v replaced by A a q, /~ y 0, respectively. Hence /~ a r e F. Thus F = F M , and the proof is complete.

LEMMA 31. I1 a ~ F V (~), then CL (q~ --> A a q~) C_ Z~.

PROOF. Let ~ satisfy conditions (i)-(iv) of Lemma 30. Then a ~ OC {F). Hence

CL @ --> ~) C Z~ by Lemma 30 (i) ;

CL (y~ --~ A aye) C Z~ by (B6);

CL (yJ --> q~) CC Z'~ by Lemma 30 (ii);

CL (A a~f -~ /~ a~) _C ~,~ by Lemma 27 and (B4);

$;, thus CL @ -+ A a q~) C as desired.

~1. LEMMA32. C L ( A a A f l ~ - ~ A f i A a ~ ) C - '

X~. Then by P~OOF. By use of (BS) we deduce tha t CL (A a A fl q: -> ~) _C - ' Lemma 27 and (B4) we infer tha t CL (~ a ~ a/~ fi q~ --->/~ a 99) C - ' . X~, ap- plying Lemma 31, it follows tha t CL (A a A fl ~ -~ A a ?) c X~. Going through this procedure once more leads to the desired conclusion.

F rom Lemma 31 and 32 we see t h a t X* __C Y,~. The completeness of the sys- t em +~ follows:

THEOREM 4. X~ i8 complete, i. e., ~'~ = US.

Let 2:~ consist of all formulas (B 1)-(B 8). The system ~ of predicate log- ic has 2:2 as its set of axioms and de tachment and generalization as its rules of inference.~ Thus, as mentioned in the introduction, ~ is described wi thout

s The system ~2 differs mainly from the system described in the abstract [14] in that the commutativity schema *101 has been dropped. The possibility of deriving *101 from the remaining axioms of ~2 was known to the author at the time of publication of [14], but was discovered also independently by Kalish and Monf~ue in [6] and [7].

78 Alfred Tarski

the use of the involved notions of free variable and proper substitution. The completeness of ~ is immediate from Theorem 4:

TH~OI~EM 5. Z~ is complete, i. e., ~ = UF.

I t turns out tha t in the system 6~ the schema (B 5) can be derived from the remaining schemata; omitting (B 5), the resulting schemata are independ- ent. These results are proved in [8].

The results of this paper can be extended to predicate logic without identity. Assume first for simplicity tha t we are interested in predicate logic in which a binary predicate g occurs as the only non-logical constant. Given any two formulas r and ~v, we shall denote, as usual, the formula -~ (~ --> -~ ~v) by

A ~v and the formula (T --> ~v) A (~v --> ~) by ~ ~-~ ~v. Let a ~ fl denote the f o m u l a

A ~ ( ( ~ o ( a , ~ ) ~ o (fl, ~)) A (~ o (~, a ) ~ o (~, 8))) ,

where a and 8 are two arbi t rary variables while ~ is the first variable in the sequence v0, vj . . . . which is different from a and 8. Consider the following schemata (in which a, 8, ~ e VR and ~, ~v e FM):

(A8') [ ~ A a -7 a ~ 8 ] , w h e r e a ~ : 8;

(A9') [a ~ 8 --> (~ -~ ~)], where P (~, ~o, a, 8);

(A9") [a ~ f l - - > ( a ~ r - > 8 ~ ) ] ;

(A9'") [a ~ 8 --> (r ~ a --> ~ ~ 8)]"

The relation of these schemata to (A8) and (A9) of the system ~1 is clear. Let Z + be the set of all sentences of our predicate logic without identity which are instances of the old schemata (A1)-(A7) and the new schemata (AS') - (A9 '") , and let 6 + be the system which has 2: + as its set of axioms and the rule of detachment as its only rule of inference. Obviously, all the sentences in 27 + are universally valid, and from Theorem 1 we easily conclude tha t the set 2: + (and hence the system 6 +) is complete. Various possibilities of simpli- fying and modifying the system 61 present themselves in a natural way but will not be discussed here.

In an entirely analogous manner we can transform all the other systems discussed in this paper into corresponding systems of predicate logic without identity. With obvious changes the same procedure can be applied to predi- cate logic with any finite number of non-logical constants. Finally, we can construct a complete set of axioms for a system 6 of predicate logic with in- finitely many constants by forming the union of the complete axiom sets for all "subsystems" of 6 containing only a finite number of constants.

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A Simplified Formalization of Predicate Logic with Identity 79

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UNIVERSITY OF CALIFORNIA, BERKELEY