A GENERALISATION OF THE TARSKI-HERBRAND DEDUCTION THEOREM. Si. SURMA

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1 Logique & Analyse (1991), A GENERALISATION OF THE TARSKI-HERBRAND DEDUCTION THEOREM Si. SURMA The paper provides a condition which characterises the implicational fragment of classical sentential logic in the same way as the Tarski-Herbr deduction theorem together with its conversion characterises the implicational fragment o f intuitionistic logic. The deduction theorem, Tarski-Herbr-style, can be looked at as providing a general framework for the formalisation justification of a wide class of intuitively sound, direct proofs. Typical logical theorems provable in this way may be called ascending conditionals or ascending implications because they fall under the schema (t) C a where n is a positive integer number C O is a conditional or implication with the antecedent a the consequent 0. Clearly, not all provable conditionals are ascending. For instance, Peirce's law CCCaOcia is not. The widest class of directly provable conditionals must include what is called here descending conditionals, ie. conditionals of the form (1) The purpose of this note is to come to a Generalization of the above mentioned framework which would allow for the formalisation justification of intuitively sound, direct proofs of all logical theorems which fall under schema ( t) as well as under schema (4). Clearly, a direct proof of a (1 )- conditional statement involves exclusively properties of the connective of conditional C alone. In practice, however, a typical proof of a (4 )-conditional statement is indirect in that it depends also on properties of negation /or disjunction. Below we make use of some of stard set-theoretic notation. In particular, symbols g, n, U, 0 a a intersection, union, the empty set the set containing ce its only members, respectively.

2 320 S i. SURMA Let S be the least set containing a denumerable stock of sentential variables closed under the formation of conditionals using the connective of conditional C as the only sentence-forming connective. Thus, if a E S 0 E S, then also Ca/3 E S. Let Cn be a closure operator on the power set 2s of S. This means that, for any X, Y, Z g S, (I) X g Cn(X), (ii) t h e fact that X g Y implies that Cn(X) g Cn(Y), (iii) Cn( Cn( X) ) g Cn(X). For the sake of simplicity, expressions of the form are abbreviated hereafter as Cn(Xt..) ta Cn(X, a,, a We also assume that lower-case Greek characters, with or without subscripts, belong to the set S while Latin capitals X. Y, Z, a r e subsets of S. The condition E Cn(X) iff E Cn(X, a) is the classical deduction theorem is due to A. Tarski [3] J. Herbr [1]. This theorem can be generalised to the following one. (Ert) Ca,Ca Ca...Ca It may be remarked at the outset that by calling (D'a I ) the deduction theorem we depart slightly from the prevailing terminological convention according to which the term "deduction theorem" applies only to the "if" part of the biconditional condition (12 valent to) the rule of detachment. To be sure, X is said to be closed under (generalised) detachment if, for any a 0, the fact that a E Cn(X)

3 A GENERALISATION OF THE TARSKI-HERBRANDDEDUCTION THEOREM Coef3 E Cn(X) implies that (3 E Cn(X). We have the following well-known fact which settles the scope of application of the Tarski-Herbr deduction theorem. LEMMA I The following conditions are equivalent. (i) ( 1 3 (ii) ( D (iii) C n ( 0 ) includes the set of all theorems of the intuitionistic implicational logic is closed under detachment. It may be noticed that condition (iii) of the above Lemma can be improved by making a reference to an explicit axiom system for the logic involved. For instance, we could re-write (iii) as follows. (iii') C n ( 0 ) is the least set closed under detachment containing the formulae CaCOot CCaCO7CCafiCory. Clearly, condition (Er f) does not cover the case of ( 4 ) other words, we cannot simply place C C... C C o l Ca,Cce,Ca,...Cce t We may proceed now to the problem of formalisation of intuitive proofs of ( )-conditionals. Consider the following condition. (Drt 4) I f n is even, then CC...CCe e,ce Cn(X, co n Cn(X, a Clearly, the formula on the right of the biconditional formula ( /) " abbreviates the following condition.

4 322 S T SURMA Cn(X, a Cn(X, a cn(x, a Cn(X, cx _ -_, )( Cn(X, a The substitution of n in (D" I) gives Cast, E Cn(X) i f Cn(X, a which coincides with ( D't ) which, in turn, is equivalent to C a Thus, (D"1) generalises the Tarski-Herbr deduction theorem. The theorem below establishes the scope of application of (D" THEOREM 1 The following conditions are equivalent. (i) ( U t ), for any even n: (ii) ( D " I), for any even n ; (iii) C n ( 0 ) is the set of theorems of the C-fragment (ie. the implicational fragment) of classical sentential logic, closed under detachment. As it is well-known, condition (iii) of the above Theorem can be rewritten equivalently as follows. (iii') C n ( 0 ) is the least set closed under detachment containing the formulae CaCfia, CCaCOTCCafiCay, CCCafiaa. To get this theorem proved we must do some preparatory work. First, let us define an auxiliary sentential formula A(a, a,. a where n is an arbitrary positive integer number, by making use of the following recursive schema.

5 A GENERALISATION OF THE TARSKI-HERBRANDDEDUCTION THEOREM (i) A ( a, ) = oti, (ii) A ( a The following property of A can be established easily using induction on LEMMA 2 If Crt(0) contains all theorems of intuitionistic implicational logic is closed under detachment, then, for any i such that 0 < i n, LEMMA 3 I. C C c e, O C c t CciA(oist,...œ If 07(0) is the set of theorems of the C-fragment of classical sentential logic, closed under detachment, then CCa,3CCoe2j3...CCunOCA(ce Proof The proof is inductive. That line belongs to 01( 0) follows directly from the assumption about Cn(0). To show that the fact that Lemma 3 holds true of k implies that it also holds true of k + 1, we assume that II. C C o t we need to show that III. C C e t 3 O C C O E To do so suppose that I. C o t 2. C a 3. Motice2---œkolk,i) From 3 the definition of A 4. C C a From II, 1 4

6 324 S i. SURMA 5. CCak+tA(ala2 a1313 From 2, 5 the logical theorem CCCaOTCCayy 6. so line III is proved. From I, II, HI the principle o f mathematical induction, Lemma 3 is proved true of any n. Q.E.D. LEMMA 4 Let n be even. I f 0 1 (0 ) is the set of theorems of the C-fragment o f classical sentential logic, closed under detachment, then, for any i E 12, 4, 6, n), E 0(0). Proof In case n = 2 Lemma 4 reduces to line I. C C a which follows directly from our assumption about the set 0 7 (0 ). To prove the lemma for n > 2 we proceed by induction on n. Accordingly, we show first that Lemma 4 holds true for n = 4, ie. that II. C C C C a, a To do so we first show that 1. C C C C a, a Assuming 1.1. CCCa1a2a3a a, we need to derive A(a 2 a 1.3. C a, a, From 1.1, 1.3 the transitivity property of C 1.4. C C C a a a ic e By another use o f the transitivity o f C we may infer from 1.4 the logical theorem CaCi3a that 1.5. CCCa1a2a3Ca,a2

7 A GENERALISATION OF THE TARSKI-HERBRANDDEDUCTION THEOREM From 1.5 the logical theorem CCCaOaa 1.6. C a, a From a so line 1 is proved. Next we show that 2. C C C C a Assume that 2.1. CCCa,a2a3a a, From 2.2 the logical theorem CaC(3a 2.3. C C a, a From Ce4 which proves line 2. Lines 1 2 complete the proof of 11. As the next thing we need to show that if Lemma 4 holds true of an arbitrary even p, then it also holds true of p 2. To do so we assume, as an inductive hypothesis, that for any i E [2, 4, 6, p ) Cn(0), Now, to prove line IV. C C C C... C C a ' a t) C fp +2 ) 01(0), for any i E (2, 4, 6, p, p+2) we assume that 1. C C C... C C a, a, a 2. a i - i we must deduce A ( c e C C a 3. C a (cticei+2ai+ a p )

8 326 S i. SURMA From I, 3 the transitivity of C 4. C C C... C C o / a a 0 0 M n ' - - p On the other h, from HI 2 5. From 4, 5 the logical theorem CCCaO7CCa-y-y 6. A (c e io ii, 2 a t, 4 The case a Thus line IV is proved. From H, HI, IV the principle of mathematical induction, Lemma 4 holds true for n > 2. Q.E.D. LEMMA 5 Let n be an even positive integer number greater than o r equal to 2. I f Cn (0 ) is the set of theorems of the C-fragment of classical sentential logic, closed under detachment, then CCan_3A(04,-2an)CCan.,a,,CC...CCee Proof The case of n = 2 is obvious. Assume that the Lemma is true of an even p suppose that 2. C a 3. C C... C C a, a, a From I the definition of A

9 A GENERALISATION OF THE TARSKI-HERBRANDDEDUCTION THEOREM C a C a C a From 4 5. C C u C C a C C a On the other h, from C C... C C c e An application of the inductive assumption to steps 5 6 gives 7. C C a From 7 the logical theorem CCCOace 8. C C O E Repetition of an argument similar to that leading to step 8 gives, finally, 9 It follows from that the conclusion of our Lemma holds true of p 2. Q.E.D. Proof of Theorem 1 The proof that (i) implies (ii) is obvious so let us proceed to the proof that (ii) implies (iii). Substitution of n = 2 in ( D ) gives ( D't ). Hence by Lemma ( 0 ) includes the set of theorems of intuitionistic implicational logic

10 328 S i. SURMA 2. C n ( 0 ) is closed under detachment Clearly, 3. Cn( 0) r ) Cn( c e) E Cn(a), Cn(a) E Cn(a) From 3 by (Di' 4), in which we put n = 4, 4. C C C oto ota E ) Formula CCCaf3cici extends the intuitionistic implicational logic to the C- fragment of classical logic. Hence, from I 4 5. C n ( 0 ) is the set of theorems of the C-fragment of classical sentential logic This proves that (ii) implies (iii). Proof that (ill) implies (1 have to show that, on the assumption of condition (iii) of Theorem I, conditions (*) (**) are equivalent where (*) (**) are as follows. (*) C C...CCa,a (**) Ctz ( X, c t for any i E (2, 4, 6, n ) n n cn(x. ce First, we will show that (*) implies (**) To do so suppose that, for any E (2, 4, 6, n ), 1.1. f i E Cn(X, C n ( x, a,) n ca(x, ce, )n n cn(x, a By Lemma 1, condition (iii) of Theorem 1 implies that (D't ) holds true. Hence from Cce,O, C o e, 1 3, C a

11 A GENERALISATION OF THE TARSKI HERBRANDDEDUCTION THEOREM From 1.2 Lemma C A ( a, a " n-1 From 1.3 the transitivity of C 1.4. C C u +2 From 1.4 Lemma CCCC...CCa,a2a,...an.2an-larga0 E Ca(X) F 1.6. Co t, 4 0 E Cn(X) From 1.6 ( D E Cn(X, Lines prove that 1. C n ( X, n cn(x, ai,2)ncn(x, a 0/(X, ai.,) so we may conclude that condition (*) implies condition (**). To see that (**) implies (*) assume that 2.1. C n ( X, a for any i E {2, 4, 6, n ) From condition (iii) of Theorem 1 Lemma A ( a A ( a. A(an2an) E Cn(X, f o r any j E {n-2, From , for any i E 12, 4, 6, n ), 2.3. A ( c e i c i ce Cn is a closure operator so a 2.4. a From (iii), , for any i E {2, 4, 6, n ), 2.5. C o l From 2.5, (iii) Lemma C C... C C a 1oi 2 -et 3 a doe E Cn(X) This proves that (**) implies (*) We may conclude, then, that conditions (*) (**) are equivalent so that (iii) implies (i). This proves Theorem I. Q.E.D.

12 330 S i. SURMA As we have seen, condition ( " ), which appears in the above proof, is both sufficient necessary for the provability of a conditional referred to in condition (*). It may be noted, in this context, that a related condition, presented in the Polish paper [2], although sufficient was not necessary, besides, it was built into a less simple metalogical framework. It follows from Lemma 1 Theorem 1 that condition (DPI) sts to the C-fragment of classical sentential logic in the same relation as condition (Et ) sts to the C-fragment of intuitionistic implicational logic. It is for this reason that (TY' ) may be called the deduction theorem for classical sentential logic. Clearly, our classification of conditional statements into ( t )-conditionals (4 )-conditionals is not disjoint. For instance, condition Cu p, is both ascending descending. To get the classification improved in this respect we may simply treat Cce (TY' I') (EY' ) does not consist in that the first condition generates as logical theorems exclusively ascending conditionals because one o f the consequences of ( U t ) is that, for any n, CCCC...CCCce,a,u,a,...ceace a where, clearly, C C C C... C C C a, c t However, the point here is that, unlike ( U as logical theorems also all those( t )-conditionals which are intuitionistically unprovable. Let us distinguish between even ( 4 )-conditionals odd ( 4 )-conditionals according to whether n is even or odd. Clearly, Theorem I provides characterization of even ( 4 )-conditionals only. However, it may be remarked that no genuine odd ( 4 ) of the classical sentential logic. Indeed, n o conditional o f the f o rm CC... C C a logic where n is odd ct,, ce,, ce Modifications o r specifications o f (D" ) can be used to provide a full characterization o f the C-fragments o f some non-classical logical systems. One such modification of ( U t ) provides a full characterization o f the C- fragment of system S5 of C. I. Lewis. Let S' denote the set of all conditionals of S, ie. S' = [Cafi: oe, 13 E S}

13 A GENERALISATION OF THE TARSKI-HERBRANDDEDUCTION THEOREM let us consider the condition ( D C We have the following theorem. THEOREM 2 The following conditions are equivalent. (i) ( D C ' ) for any even positive integer number n: (1i) ( D C ' ) for any even positive integer number n 4 : (iii) 0 1 ( 0 ) is the set of theorems of the C-fragment of Lewis's system S5, closed under detachment. The proof of this theorem is similar to that of Theorem 1. It follows from Theorem 2 that the logic, which condition (DC' d e t e r - mines, is precisely the C-fragment of Lewis's sentential logic S5 based on the rule of detachment as the only inference rule. University o f Auckl ACKNOWLEDGMENT I would like to thank the anonymous referee for the insightful comments. REFERENCES 111 H E R B R A N D, J., Recherches sur la théorie de la démonstration. Prace Tow. Nauk. Warsz., Wydz. III, Nr 33, 1930, [2] S U R M A, S. J., Twierdzenia o dedukcji dia implikacji zstepujacych. Studia Logica, vol. 22, 1968, [3] T A R S K I, A., Ober einige fundamentale Begriffe der Metamathematik. Sprawozol. Tow. Nauk. Wars.z., Wydz. ill