Relational and Neighborhood Semantics for Intuitionistic Modal Logic
Relational and Neighborhood Semantics for Intuitionistic Modal Logic

Summary/Abstract: We investigate semantics for an intuitionistic modal logic in which the “possibility” modality does not distribute over disjunction. In particular, the main aim of this paper is to study such intuitionistic modal logic as a variant of classical non-normal modal logic. We first give a neighborhood semantics together with a sound and complete axiomatization. Next, we study relationships between our approach and the relational (Kripke-style) semantics considered in the literature. It is shown that a relational model can be represented as a neighborhood model, and the converse direction holds under a slight restriction. Also, by considering degenerate cases of neighborhood and relational semantics, we demonstrate that a certain classical monotone modal logic has relational semantics, and can be embedded into a classical normal bimodal logic.

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