Semantic characterization of certain sets of intuitionistic logical connectives

Abstract

The present article is a continuation of previous papers [i, 2] devoted to the development of a semantic approach to the concept of an intuitionistic logical connective. A detailed motivation of this approach may be found in [2]. In brief, the semantic approach consists in a definition of a logical connective as a method of specifying truth values on model structures, which, in the case of intuitionistic logic, are Kripke models. A connective must satisfy the following natural conditions. a) monotonicity; b) a connective must possess an explicitly described domain of definition; c) a connective must not distinguish similar models. The domain of definition of the standard intuitionistic connectives is the upper cone [2]. The domain of definition of the connectives --" and L (which are dual to ~ and -~ [4, 5]) is a connected component [3]. These models have been defined in terms of elementary equivalence and p-morphisms. The concept of a p-morphism first appeared in the theory of nonclassical logic as a convenient tool for the study of such logics. It was subsequently proved that, from the algebraic point of view, the concept of a p-morphism corresponds to the intuitionistic logical connectives. That is, it was proved [6] that the presence of a p-morphism with one scale into another is equivalent to the possibility of an inverse embedding in the corresponding pseudo-Boolean algebras. The results presented in [1-3] and the present article show that the concept of a pmorphism corresponds to the intuitionistic logical connectives from the point of view of classical model theory as well. The following alternative versions of conditions (b) and (c) will be investigated in the present article. bl) the truth of a connective at a particular vertex depends solely on the connected component at this vertex; b2) unbounded domain of definition, i.e., the truth of a connective is determined by the model as a whole; cl) stability under everywhere defined surjective mappings that are monotone and preserve the upper cones and the truth of the propositional variables; c2) formulation analogous to (cl) with the addition of a condition under which the lower cone is preserved. Below we will present the following results. Conditions (a, bl, c2) are characteristic of the connectives of Heyting-Brouwer