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46. on the Completeness O F the Leibnizian Modal System with a Restriction by Setsuo SAITO Shibaura Institute of Technology,Tokyo (Comm

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46. on the Completeness O F the Leibnizian Modal System with a Restriction by Setsuo SAITO Shibaura Institute of Technology,Tokyo (Comm 198 [Vol. 42, 46. On the Completeness o f the Leibnizian Modal System with a Restriction By Setsuo SAITO Shibaura Institute of Technology,Tokyo (Comm. by Zyoiti SUETUNA,M.J.A., March 12, 1966) § 1. Introduction. The purpose of this paper is to show the completeness of a modal system which will be called Lo in the following. In my previous paper [1], in order to show an example of defence of _.circular definition, the following definition was given: A statement is analytic if and only if it is consistent with every statement that expresses what is possible. This definition, roughly speaking, is materially equivalent to Carnap's definition of L-truth which is suggested by Leibniz' conception that a necessary truth must hold in all possible worlds (cf. Carnap [2], p.10). If "analytic" is replaced by "necessary" in the above definition, this definition becomes as follows: A statement is necessary if and only if it is consistent with every statement that expresses what is possible. This reformed definition is symbolized by modal signs as follows: Opm(q)[Qq~O(p•q)1, where p and q are propositional variables. Let us replace it by the following axiom-schema and rule: Axiom-schema. D a [Q,9 Q(a •,S)], where a, ,9 are arbitrary formulas. Rule of inference. If H Q p Q(a •p), then i- a, where a is an arbitrary formula and p is a propositional variable not contained in a. We call L (the Leibnizian modal system) the system obtained from the usual propositional calculus by adding the above axiom schema and rule and the rule of replacement of material equivalents. (Qa is regarded as the abbreviation of a). This system is easily proved to be equipollent to the system obtained from the usual propositional calculus by adding the following axiom-schema and rule and the rule of replacement: Axiom-schema. (a)(E1a 0,G). Rule. If a is a tautology, then H a. We call Lo the latter system with the restriction that if a is a formula of Lo then a does not contain i. We shall discuss the completeness of Lo in the following sections. § 2. Main results. We write a, R, 7, ... for the formulas of Lo No. 3] Completeness of Leibnizian Modal System with Restriction 199 which do not contain 0. We write a', ~3', 7', • • • for general formulas of Lo. (a', ,9', • • • are composed of propositional variables and 0 a, ORS ... with ', •, V, i.) Let 7 be an arbitrary formula not containing . We call a 7-valuation a manner of value assignments which satisfies the following condition: t true, if 7 a is a tautology; ruth value of Li a = f alse otherwise where a is an arbitrary formula not containing L , and 7 is called the axiom. For a general formula, the following definition is given: a' is a 7-tautology, if and only if, for a fixed axiom 7, a' is true for all 7-valuation. (a, which does not contain L , is a 7-tautology if and only if a is a tautology.) We now state the following theorems: Theorem 1. I f a' is provable in Lo, then a' is a 7-tautology for all 7. Theorem 2. I f a' is a 7-tautology for all 7, then a' is provable in Lo. § 3. Proof of Theorem 1. If 7 (a ~ 9) is a tautology and 7 3 a is a tautology, then 7 1),5) is a tautology. Therefore 0 (a ~ 9) ( L a L /3) is a 7-tautology. If a is a tautology, then 7 i a is true for an arbitrary 7-valuation. Therefore, if a is a tautology, then Li a is a 7-tautology. § 4. Proof of Theorem 2. Let us mention the following lemmata for the sake of the proof of Theorem 2: Lemma 1. I f a'•9' is 7-tautology, then a', ,9' are 7-tautologies. The proof is evident. Lemma 2. I f o does not contain 0 and (I) ..Da1V,., La2V ... V,.LamV L/9iV OR2V ... V ORnVo is a 7-tautology, then (II) ~' La1V' Da2V ... V ^' DamV L/9iV 0/92V ... V ONn is a 7-tautology or is a tautology. Proof. If o is not a tautology, then there exists a 7-valuation by which 3 is false. For such a 7-valuation, (II) is true. Therefore, (II) is a 7-tautology, because truth value of (II) depends only on a1, a2, , am, /91,N2, , /n, 7 and is independent of 7-valuation, q.e.d. An arbitrary fomula a' is reduced to a conjunction of the formulas of the form (I). If in (I) is a tautology then (I) is provable in Lo. From Lemma 1 and Lemma 2, therefore, for the proof of Theorem 2, it is sufficient to consider (II) as a'. Now take a1 •a2 • • • • • am as axiom 7 (if m=0 then it means a tautology). Then, since 0 ai, L a2 t ..., Dam are true, 200 S. SAITO [Vol. 42, ~R1VOR2V ... VLJi is true. Therefore for some i (1 <_i <_n) Li ~2 is true, that is (a'1. a2. ... am) h'i is a tautology, accordingly so is a1 (a2 (... (am i9i) ... )), Therefore, DaiD(Da2D(. " (Damp C7R2) ...)) is provable in Lo, and ' Da1V' Da2V ... V rv L1amV ORiV ER2V ... V Ei9 is provable in Lo. I express my hearty thanks to Prof. S. Maehara who gave me significant advices which enabled me to improve my original manuscript. This work was supported by the Grant-in-Aid from the Ministry of Education. References [1] S. Saito: Circular definitions and analyticity. Inquiry (Oslo), 5,158-162 (1962). [2] R. Carnap: Meaning and Necessity. University of Chicago Press, Chicago (1947)..
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