Mitio Takano

Strong completeness of lattice-valued logic

Abstract. Strong completeness of S. Titanis system for lattice valued logic is shown by means of Dedekind cuts.

Intermediate Predicate Logics Determined by Ordinals

For each ordinal α>0, L(α) is the intermediate predicate logic characterized by the class of all Kripke frames with the poset α and with constant domain. This paper is devoted to a study of logics of the form L(α). It will be shown that for each uncountable ordinal of the form α+η with a finite or a countable η(>0), there exists a countable ordinal of the form β+η such that L(α+η)=L(β+η). On the other hand, such a reduction of ordinals to countable ones is impossible for a logic L(α) if α is an uncountable regular ordinal. It will be proved that the mapping L is injective if it is restricted to ordinals less than ω ω

Ordered sets R and Q as bases of Kripke models

Those formulas which are valid in every Kripke model having constant domain whose base is the ordered set R of real numbers (or, the ordered set Q of rational numbers) are characterized syntactically.

Proof theory for minimal quantum logic: A remark

It is remarked that the inference rule (‘ → ’) is superfluous for the sequential system GMQL introduced by H. Nishimura for the minimal quantum logic.

An interpolation theorem in many-valued logic

?1. A many-valued logic version of the Craig-Lyndon interpolation theorem has been given by Gill [1] and Miyama [2]. The former dealt with three-valued logic and the latter generalized it to M-valued logic with M ? 3. The purpose of this paper is to improve the form of Miyamas version of the interpolation theorem. The system used in this paper is a many-valued analogue (Takahashi [4], Rousseau [3]) of Gentzens logical calculus LK. Let T = { 1,..., M} be the set of truth values. An M-tuple (F1,..., FM) of sets of formulas is called a sequent, which is regarded as valid if for any valuation (in canonical sense) there is a truth value P e T such that the set F, contains a formula of the value ,u with respect to the valuation. (In the next section, and thereafter, we change the format of the sequent for typographical reasons.) Miyamas result is as follows (in representative form): (I) If a sequent ({A}, 0,. .., 0, {B}) is valid, then there is a formula D such that (i) every predicate or propositional variable occurring in D occurs in A and B, and (ii) the sequents ({A}, 0, . ., 0, {D}) and ({D}, 0,.. ., 0, {B}) are both valid. What shall be proved in this paper is the following (in representative form): (II) If a sequent ({A, }, {A2}, ..., {AM}) is valid, then there is a formula D such that (i) every predicate or propositional variable occurring in D occurs in at least two of the formulas Al,-.. , AM, and (ii) the following M sequents are valid:

A semantical investigation into Leśniewski's axiom of his ontology

A structure A for the language L, which is the first-order language (without equality) whose only nonlogical symbol is the binary predicate symbol ɛ, is called a quasi ɛ-struoture iff (a) the universe A of A consists of sets and (b) aɛb is true in A ↔ (∃[p) a = {p } & p ε b] for every a and b in A, where a(b) is the name of a (b). A quasi ɛ-structure A is called an ɛ-structure iff (c) {p } ε A whenever p ε a ε A. Then a closed formula σ in L is derivable from Leśniewskis axiom ∀x, y[x ɛy ↔∃u (u ɛ x)∧∀u; v(u, v ɛ x→ uɛv)∧∀u(u ɛ x→ u ɛ y)] (from the axiom ∀x, y(xɛ y → x ɛ x)∧∀x, y, z(x ɛ y ɛ z → y ɛx ɛ z)) iff σ is true in every ɛ-structure (in every quasi ɛ-structure).

Subformula property in many-valued modal logics

?0. Introduction. Fitting, in [1] and [2], investigated two families of many-valued modal logics. The first, which is somewhat familiar in the literature, is that of the logics characterized using a many-valued version of the Kripke model (binary modal model in his terminology) with a two-valued accessibility relation. On the other hand, those logics which are characterized using another many-valued version of the Kripke model (implicational modal model), with a many-valued accessibility relation, form the second family. Although he gave a sequent calculus for each of these logics, it is far from having the cut-elimination property (CEP) or the subformula property. So we will give a substitute for his system enjoying the subformula property, though it is not of ordinary sequent calculus but of the many-valued version of sequent calculus initiated by Takahashi [7] and Rousseau

Gentzenization of Trilattice Logics

Sequent calculi for trilattice logics, including those that are determined by the truth entailment, the falsity entailment and their intersection, are given. This partly answers the problems in Shramko-Wansing (J Philos Logic 34:121–153, 2005).

Embeddings between the elementary ontology with an atom and the monadic second-order predicate logic

Let EOA be the elementary ontology augmented by an additional axiom ∃S (S ɛ S), and let LS be the monadic second-order predicate logic. We show that the mapping ϕ which was introduced by V. A. Smirnov is an embedding of EOA into LS. We also give an embedding of LS into EOA.

Axiomatization of a Basic Logic of Logical Bilattices

A sequential axiomatization is given for the 16-valued logic that has been proposed by Shramko-Wansing (J Philos Logic 34:121–153, 2005) as a candidate for the basic logic of logical bilattices.