On the relation between modal and multi-modal logics over Łukasiewicz logic

Abstract

In a previous paper, it was shown that the (minimal) modal logic MŁc n with fuzzy accessibility relations over the finite-valued Łukasiewicz logic Ł n and a corresponding multi-modal logic mMŁc n (with a modality □ a for each value a in the n-valued Ł n -chain) had the same expressive power when the language is extended with truth-constants. In this paper we partially extend these results when replacing the underlying logic Ł n by the infinite-valued Łukasiewicz logic (with rational truth constants in the language). We prove that the (standard) tautologies of the modal logic MŁc n (resp. mMŁc) are in fact the common tautologies of all the logics M Łc n (resp. all the logics mMŁn) when letting n vary over N. This fact opens the door to show an alternative proof of the finite model property for these logics and hence their decidability.