Duality via Truth for Some Fuzzy Modal Logic

Abstract

Duality via truth is a kind of correspondence between a class of algebras and a class of relational systems (frames, following terminology well-known in non-classical logics). The first class is viewed as an algebraic semantics of some logic, whereas the other class constitutes Kripke-style semantics of this logic. The duality principle underlying the duality via truth states that algebras and their corresponding frames provide equivalent semantics for this logic in the sense that a formula is true with respect to one semantics if and only if it is true with respect to the other semantics. Consequently, the algebras and the frames express the equivalent notions of truth and in this sense they are viewed as dual structures. In this paper we develop duality via truth for a fuzzy modal logic. The MTL logic, introduced by Esteva and Godo, is taken as a basis. Several axiomatic extensions, motivated by well-known schemas of modal logic, are also considered.