From Kripke to Neighborhood Semantics for Modal Fuzzy Logics

The majority of works on modal fuzzy logics consider Kripkestyle possible worlds semantics as the principal semantics despite its well known axiomatizability issues when considering fuzzy accessibility relations. The present work offers the first (two) steps towards exploring a more general semantical picture, namely a fuzzified version of the classical neighborhood semantics. First we prove the fuzzy version of the classical relationship between Kripke and neighborhood semantics. Second, for any axiomatic extension of MTL (one of the main fuzzy logics), we define its modal expansion by a square-like modality, and, in the presence of some additional conditions, we prove that the resulting logic can be axiomatized by adding the (E)-rule to the corresponding Hilbert-style calculus of the starting logic.