Characterizing Fuzzy Modal Semantics by Fuzzy Multimodal Systems with Crisp Accessibility Relations

In [1] the authors considered finitely-valued modal logics with Kripke style semantics where both propositions and the accessibility relation are valued over a finite residuated lattice. Unfortunately, the necessity operator does not satisfy in general the normality axiom (K). In this paper we focus on the case of finite chains, and we consider a different approach based on introducing a multimodal logic where the previous necessity operator is replaced with a family, parametrized by truth values different from zero, of necessity operators each one semantically defined using the crisp accessibility relation given by the corresponding cut of the finitely-valued original accessibility relation. This multimodal logic is somehow more appealing than the original modal one because axiom (K) holds for each necessity operator. In this paper we axiomatize this multimodal logic and we prove that, in the case the starting residuated lattice is a finite BL chain, the modal and the multimodal languages have the same expressive power iff this algebra is an MV chain. Keywords— many-valued modal logic, fuzzy modal logic, Łukasiewicz modal logic, fuzzy logic.