On the relation between modal and multi-modal logics over Łukasiewicz logic
Francese Esteva, Lluı́s Godo, Ricardo Óscar Rodríguez · 2017
In a previous paper, it was shown that the (minimal) modal logic MŁncwith fuzzy accessibility relations over the finite-valued Łukasiewicz logic Łnand a corresponding multi-modal logic mMŁnc(with a modality □afor each value a in the n-valued Łn-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 Łnby the infinite-valued Łukasiewicz logic (with rational truth constants in the language). We prove that the (standard) tautologies of the modal logic MŁnc(resp. mMŁc) are in fact the common tautologies of all the logics MŁnc(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.