Cluster semantics for modal logic

Abstract

One of the main branches of Yu.V. Ivlev’s scientific research was construction of natural semantics for modal logics. The approach to the analysis of alethic modal notions proposed by Yu.V. Ivlev was based on the idea of differentiation of factual and logical modalities and construction of logical systems, characterizing their features. The pivotal instrument of factual modalities modelling is a notion of so-called quasi-function (quasimatrix), which is non-trivial generalization of the well-known notion of function. Up to this time quasi-functional logics have been widely acknowledged in the world’s logical literature. Due to the nature of the represented modal notions the systems of quasi-functional logic are mainly non-normal. With a view to constructing natural theory of logical modalities Yu.V. Ivlev presented an idea of additional interpretation of propositional variables occurring in formulas with modal operators as denoting logically true, logically false or logically indeterminate propositions. The function of such interpretations in the theory of logical modalities is the same as the function of quasi-matrices in systems with factual modalities. The interpretations of this kind lead to the construction of finite sets of state-descriptions (or finite systems of such sets), substituting model structures of traditional semantics of possible worlds. The well-known Lewis’ system $S5$ proved to be a natural formalization of the proposed theory of logical modalities. As it has turned out recently by means of certain modifications of the initial strategy the semantics of the same kind can be constructed for the other normal modal systems $S4$, $Br$, $T$, $D$, $K$ as well as for the intuitionistic system Int. In contrast with quasi-functional logics this branch of Yu.V. Ivlev’s research is known to a much lesser extent. The proposed article aims to outline general strategy of semantics construction for the above-mentioned logical systems.