ON THE ATOMIC FORMULA PROPERTY OF HÄRTIG S REFUTATION CALCULUS

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1 Takao Inoué ON THE ATOMIC FORMULA PROPERTY OF HÄRTIG S REFUTATION CALCULUS 1. Introduction It is well-known that Gentzen s sequent calculus LK enjoys the so-called subformula property: that is, a proof without cut in LK contains only subformulas of the formula occurring in the end-sequent (Gentzen [2]). In this paper, we shall propose a similar property, namely the atomic formula property, which is weaker than the subformula property, and we shall remark that for every formula provable in Härtig s refutation calculus 1 HC, there is a proof of it in HC with the proposed property. The proof of the fact was already (implicitely) in Härtig [3]. 2. Preliminaries Let us begin with the definition of the atomic formula property. Definition 2.1 Let X be a formal system. For any formula A of X, let < A > X be the set of all atomic formulas occurring in A (we delete X from < A > X, if no ambiguity arises). The system X has the atomic formula property (for short, a.f.p.), if for any formula A provable in X, there is a proof of A in X which contains only such formulas, say B 1, B 2,..., B n that < B i > X < A > X holds for all i (we say that such a proof has the atomic formula property). The system X has the subformula property, if for any formula A provable in X, there is a proof of A in X with the subformula property. 0 This paper is dedicated to the late Dr. Diana Raykova. 1 Refutation calculus is sometimes called the method of axiomatic rejection, for short, axiomatic rejection. 146

2 Proposition 2.1 For any formal system X, if X has the subformula property, then it has the atomic formula property. Let CP be classical propositional logic. Take an arbitrary formulation for CP with (disjunction) and (negation) as primitive and fix it throughout the paper. We may assume at least that the language of CP has countably many distinct propositional letters p 1, p 2,.... For any formal system X and any formula A of X, X A ( X A) means that A is (not) provable in X. Härtig s refutation calculus HC ([3]), 2 which axiomatizes a set {A : CP A}, consists of the following axioms and rules: Axioms: Rules: (HC1) HC r for any propositional letter r, (HC2) HC r for any propositional letter r. (HC3) HC A, HC B, < A > HC < B > HC = HC A B, (HC4) CP A B, HC B HC A. Theorem 2.1 (Härtig [3] (cf. [5])) For any formula A of CP, HC A CP A. Without the precise definition, we have already used a (un)provabilitypredicate HC ( HC ). However, in order to make a precise argument later, we shall define them with the notion of proof in HC as follows. Definition 2.2 Let A be a formula of HC. A proof of A in HC is an ordered finite sequence of formulas of HC, say B 1, B 2,..., B n (n 1) such that (1) for any 1 i n, B i is an axiom of HC or a theorem of CP or an immediate consequence from B j and B k for some 1 j < i and 1 k < i by applying (HC3) or (HC4) to them, (2) B n is A. For any formula A of HC, if there is a proof of A in HC, then we write HC A, otherwise HC A. 2 We shall here take the same language of it as that of CP. 147

3 So we can easily give some examples of proofs which does not have the atomic formula property. 3. HC has a.f.p. Our theorem to be shown is the following. Theorem 3.1 The Härtig s refutation calculus HC has the atomic formula property. Proof. Let A be a formula of HC. Suppose HC A. Take a conjunctive normal form of A, say B 1 B 2 B k (k 1) (cf. e.g. [4], [14] and so on). In view of Theorem 2.1 and the completeness theorem for CP, one of the conjuncts of it is of the following form: m p iµ µ=1 n p jν (m 0, n 0, m + n 1), ν=1 where every number i µ is different from every number j ν. Let B I be the conjunct. It is obvious that CP A B I, since A is provably equivalent to the conjunctive normal form of it. Also we immediately see that < B I > < A >. if we observe every procedure to make the conjunctive normal form of A. It is easy to construct a proof of B I in HC with a.f.p., say π. Now we can construct a proof of A in HC as follows: π, A B I, A. The above proof of A clearly has the atomic formula property. The proof of Theorem 3.1 was already (implicitely) in Härtig [3]. We remark that my refutation calculi with Hintikka formulas as axioms ([5, 6, 8] (cf. [9])) have the subformula property. 148

4 Before we close this paper, we shall give some open problems. Problem 1. We know some axiomatizations for the set of formulas unprovable in intuitionistic propositional logic IP ([1], [10], [11], and so on). Do we further have a Hilbert-style refutation calculus for IP which has a.f.p.? Problem 2. Study proofs with a.f.p. from the viewpoint of theory of consequence operations (cf. [13]). References [1] R. Dutkiewicz, The method of axiomatic rejection for the intuitionistic propositional logic, Studia Logica, vol. 48 (1989), pp [2] G. Gentzen, Untersuchungen über das logische Schließen, Mathematische Zeitschrift, vol. 39 (1935), pp ; pp [3] V. K. Härtig, Zur Axiomatisierung der Nicht-Identiäten des Aussagenkalküls, Zeitschrit für Mathematische Logik und Grundlagen der Mathematik, vol. 6 (1960), pp [4] D. Hilbert and W. Ackermann, Grundzüge der theoretischen Logik, 6. Auflage, Springer-Verlag, Berlin, [5] T. Inoué, On Ishimoto s theorem in axiomatic rejection the philosophy of unprovability, (in Japanese), Philosophy of Science, vol. 22 (1989), Waseda University Press, Tokyo, pp [6] T. Inoué, On rejected formulas Hintikka formula and Ishimoto formula, (abstract), The Journal of Symbolic Logic, vol. 56 (1991), p [7] T. Inoué, An invitation to consequence relation and consequence operation, (in Japanese), Philosophy of Science, vol. 25 (1992), Waseda University Press, Tokyo, pp [8] T. Inoué, Cut elimination theorem, tableau method, axiomatic rejection, Abstracts of Papers Presented to the American Mathematical Society, vol. 14 (1993), p [9] A. Ishimoto, On the method of axiomatic rejection in classical propositional logic, (in Japanese), Philosophy of Science, vol. 14 (1981), Waseda University Press, Tokyo, pp

5 [10] D. S. Scott, Completeness proofs for the intuitionistic sentential calculus, Summaries of Talks Presented at the Summer Institute for Symbolic Logic at Cornell University 1957, 2nd. ed., Communications Research Division, Institute for Defense Analyses, Princeton, 1960, pp [11] T. Skura, A complete syntactical characterization of the intuitionistic logic, Reports on Mathematical Logic, vol. 23 (1989), pp [12] G. Takeuti, Proof Theory, 2nd. ed., North-Holland, Amsterdam, [13] R. Wójcicki, Theory of Logical Calculi, Basic Theory of Consequence Operations, Kluwer Academic Publishers, Dordrecht, [14] D. van Dalen, Logic and Structure, 2nd. ed., Springer-Verlag, Berlin, Ina Boudier-Bakkerlaan 117 II 3582 XP Utrecht The Netherlands 150