ON PURE REFUTATION FORMULATIONS OF SENTENTIAL LOGICS

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1 Bulletin of the Section of Logic Volume 19/3 (1990), pp reedition 2005 [original edition, pp ] Tomasz Skura ON PURE REFUTATION FORMULATIONS OF SENTENTIAL LOGICS In [1] J. Lukasiewicz introduced the following refutation rule: If α β is not provable and β is refutable then α is refutable, which employs the notion of provability as well as that of refutability. In this paper we give refutation formulations for a large class of logics using only pure refutation rules, i.e. rules of the form: If α is refutable then β is refutable. This syntactical way of presenting a logic is similar to the semantic one in that both methods are negative: A formula is a theorem if it cannot be refuted by a syntactical (semantical) device. Because of the generality of the result we are using sequents rather than formulas. Let F be a fixed algebra of formulas, i.e. F = (F or, f 1,..., f k ), where F or is the set of all formulas generated from the set V ar of sentential variables by the connectives f 1,..., f k. For every α F or, the symbol S(α) will denote the set of all subformulas of α, and for any X F or, S(X) = {S(α) : α X}. With every set Y f F or ( X f Y stands for X is a finite subset of Y ) we associate a one-to-one mapping g Y from S(Y ) into V ar in such a way that g Y (p) = p for each p S(Y ) V ar. For every β S(Y ), g Y (β) will be denoted by p β. If γ = f(β 1,..., β n ), where f is a connective, then we write F (γ) instead of f(p β1,..., p βn ). A sequent is a pair (X, α), where X {α} f F or. A rule is a subset of the set {(Σ, S) : Σ {S} is a finite set of sequents}. Let Q be a set of sequents and let R be a set of rules. Then a sequent (X, α) is said to be derivable from Q by R iff there is a finite sequence of sequents S 1,..., S m such that S m = (X, α) and for each 1 i m, either S i Q or S i is obtained from some preceding sequents by a rule of R.

2 On Pure Refutation Formulations of Sentential Logics 103 By a sentential logic we mean a structural consequence operation on F, i.e. a function C : {X : X F or} F or such that for all X, Y F or the following conditions are satisfied: X C(X); C(C(X)) C(X); C(X) C(Y ) whenever X Y ; and e(c(x)) C(e(X)) for every substitution e : F or F or. A sequent (X, α) is said to be provable in a logic C iff α C(X). A matrix is a pair M = (A, D), where A is an algebra similar to F and D A. We say that a matrix is countable iff its algebra is countable. A set K of matrices (called a matrix semantics) is countable iff K is a countable set of countable matrices. Every matrix induces a consequence operation Cn M defined thus: α Cn M (X) iff for every valuation v : F or A in M, if v(x) D then v(α) D. If K is a matrix semantics then Cn K is defined as follows: α Cn K (X) iff α Cn M (X) for every M K. With every countable matrix semantics K we associate a one-to-one mapping g K from Z = {A : (A, D) K} into V ar. For every x Z, g K (x) will be denoted by p x. A logic C is said to be equivalential (cf. [2]) iff there is a finite set E(p, q) of formulas in two distinct variables such that the following conditions are satisfied: (i) E(α, α) C( ), (ii) E(β, α) C(E(α, β)), (iii) E(α, γ) C(E(α, β), E(β, γ)), (iv) β C(E(α, β) {α}), (v) E(f(α 1,..., α n ), f(β 1,..., β n )) C(E(α 1, β 1 )... E(α n, β n )), where f is a connective and n is the arity of f. For every countable algebra A similar to F and for every finite set E(p, q) of wffs in two variables we define the description of A in terms of E (cf. [5], [4]) as follows: D E (A) = {E(f(p x1,..., p xn ), p f(x1,...,x n)) : f is a connective, n in the arity of f, x 1,..., x n A}. Moreover for every countable matrix M = (A, D), we define the following sets of sequents: Q E (M) = {( X, p a ) : X f D E (A), a A D}, Q E (M) = {( X Z, p a ) : X f D E (A), Z f {p d : d D}, a A D}. Further if K is a countable matrix semantics then we put Q E (K) = {Q E (M) : M K},

3 104 Tomasz Skura Q E (K) = {Q E (M) : M K}. Observe that if K is a recursive matrix semantics then the sets Q E (K), Q E (K) are recursive. We will consider the following rules: r sb (e(x), e(α)) (X, α) r ex (E(β, γ) X, α) (X(β/γ), α(β/γ)) (e is a substitution) where X {α, β, γ} f F or. As usual α(β/γ) is the formula obtained from α by replacing some occurrences of β with γ. Of course X(β/γ) = {α(β/γ) : α X}. For every sequent (X, α) we define the normal form of (X, α) (cf. [3]) thus: N((X, α)) = ( (X {α}) {p γ : γ X}, p α ), where (Y ) = {E(F (β), p β ) : β S(Y ) V ar} for every Y f F or. Lemma. If (X, α) is a sequent then the rule is derivable from the rule r ex. Proof. that N((X, α)) (X, α) If S(X {α}) V ar = the N((X, α)) = (X, α), so we assume S(X {α}) V ar = {γ 1,..., γ m } for some 1 m and γ i γ j for all 1 i j m. Consider the sequence of sequents: S 0, S 1,..., S m, where S 0 = N((X, α)), S i = t i (/S i 1 / i ) (1 i m) and (i) For 0 i m, t i is the substitution defined thus: t 0 (p) = p (p V ar), and for 1 i m, { ti 1... t t i (p) = 0 (f(γ i )) if p = p γi p otherwise. (ii) For every sequent (Y, β) and 1 i m, the symbol /(Y, β)/ i denotes the sequent (Y E(t i (p γi ), p γi ), β).

4 On Pure Refutation Formulations of Sentential Logics 105 By an easy induction on i it can be shown that for each i 0, S i has E(t i... t 0 (F (γ k )), p γk ) as its part for all i < k m. Therefore it is clear that for every 1 i m, S i is obtained from S i 1 by deleting E(t i (p γi ), p γi ) and replacing all occurrences of p γi with t i (p γi ), which is clearly an application of the rule r ex. We should show that S m = (X, α) it is easily to see that S m = t m... t 1 (({p γ : γ X}, p α )). Now we note the following Proposition. For any 0 k m, if 1 i m and p φ S(t k... t 0 (F (γ i ))) then φ S(γ i ) {γ i }. Proof. Indeed, Proposition is true for k = 0. Assume that k > 0. Case 1. p γk S(t k 1... t 0 (F (γ i ))). So if p φ S(t k... t 0 (F (γ i ))) = S(t k 1... t 0 (F (γ i ))), then by ind. hyp. φ S(γ i ) {γ i }. Case 2. p γk S(t k 1... t 0 (F (γ i ))). Then by ind. hyp. γ k S(γ i ) {γ i }. Assume that p φ S(t k... t 0 (F (γ i ))). If p φ S(t k (p γk )) then p φ S(t k 1... t 0 (F (γ i ))) and φ S(γ i ) {γ i } by ind. hyp. If p φ S(t k (p γk )) = S(t k 1... t 0 (F (γ i ))) then by ind. hyp. φ S(γ k ) S(γ i ) {γ i }. Since, by Proposition, p γi S(t i (p γi )), it is easy to show by induction on the complexity of β that for every β S(X {α}) we have t m... t 0 (p β ) = β. Hence S m = (X, α). For any logics C, C we write C = f C instead of C(X) = C (X) for every X f F or. Theorem 1. Let C be an equivalential logic such that C = f Cn K for some countable matrix semantics K. Then for every sequent (X, α) we have (X, α) is not provable in C iff (X, α) is derivable from Q E (K) by the rules r sb, r ex.

5 106 Tomasz Skura Proof. ( ) It is easy to see that Q E (K) {(Y, β) is a sequent: β C(Y )}. (E is the set existing by the definition of an equivalential logic.) Moreover the set of unprovable sequents of C is closed under rules r sb, r ex. ( ) Assume that α C(X). Then v(x) D, v(α) D for some valuation v in some (A, D) K. Let χ = (T {p v(γ) : γ X}, p v(α) ), where T = {E(f(p v(β1),..., p v(βn)), p f(v(β1),...,v(β n))) : f is a connective, n is the arity of f, f(β 1,..., β n ) S(X {α})}. Obviously χ Q E (K). Now let e be a substitution such that e(p β ) = p v(β) (β S(X {α})). Then e(n((x, α))) = e(( (X {α}) {p γ : γ X}, p α )) = χ. Hence N((X, α)) is derivable from Q E (K) by r sb from χ. Lemma (X, α) is derivable from Q E (K) by r sb, r ex. Therefore by If we identify a logic C with the set C( ) of its theorems then we have the following Theorem 2. Let C be an equivalential logic such that C( ) = Cn K ( ) for some countable matrix semantics K. Then α C( ) iff the sequent (, α) is derivable from Q E (K) by the rules r sb, r ex. Proof. Similar to that of Theorem 1. References [1] J. Lukasiewicz, Aristotle s syllogistic from the standpoint of modern formal logic, Clarendon Press, Oxford [2] T. Prucnal and A. Wroński, An algebraic characterization of the notion of structural completeness, Bulletin of the Section of Logic vol. 3 (1974), pp [3] M. Wajsberg, Untersuchen über den Aussagenkalkül von A. Heyting, Wiadomości Matematyczne, vol. 46 (1936), pp (English translation in: Logical Works by M. Wajsberg ed. by S. Surma, Ossolineum, Wroc law 1977.)

6 On Pure Refutation Formulations of Sentential Logics 107 [4] A. Wroński, On cardinalities of matrices strongly adequate for the intuitionistic propositional logic, Reports on Mathematical Logic vol. 3 (1974), pp [5] V. A. Yankov, On the relationship between deducibility in the intuitionistic propositional calculus and finite implicational structures, Doklady AN SSSR, vol 151 (1963), pp Department of Logic Uniwersytet Wroc lawski ul. Szewska 36 Wroc law, Poland