Set-theoretic Semantics for Heyting’s System Int

Abstract

The article aims at analysis of the new method of construction of set-theoretic semantics for Lewis‘s systems $S4$, $S5$, which doesn‘t use such notions as ‘possible world’ and ‘model structure’. The initial idea is to interpret each elementary proposition occurring in some formula in the terms $\{N, C, I\}$, i.e. as logically true, logically indeterminate, logically impossible. Such restrictions of possible truth values of the variables of some formula lead to certain restrictions of the original set of state descriptions (s.d.) for the formula, namely on the basis of metavaluations $\{N, C, I\}$ restricted, additionally and relatively restricted sets of state descriptions ($RSSD, ARSSD, RRSSD$ respectively) are constructed. These sets substitute model structures of the semantics of possible worlds. The possible world is interpreted as classical s.d. The proposed semantics involves only traditional logical notions such as truth, false, (in)compatibility of the truth values of elementary propositions etc. Besides that the number of $RSSD, ARSSD, RRSSD$ for the formula is always finite. The algorithms of characterization and enumeration of such constructions for the formula are proposed in the article. On the basis of the translation of formulae $Int$ into $S4$ implemented by McKinsey, Tarski set-theoretic semantics of the same sort for $Int$ is also proposed. The possible world in this semantics is interpreted as classical s.d, and model structures are substituted by finite ordered sets of s.d. Sense of intuitionistic logical connectives is modeled by classical metalanguage with quantifires over s.d. and their sets.DOI: 10.21146/2074-1472-2016-22-2-9-26