Simplified Kripke Semantics for K45-Like Gödel Modal Logics and Its Axiomatic Extensions

Abstract

In this paper we provide a simplified, possibilistic semantics for the logics K45(G), i.e. a many-valued counterpart of the classical modal logic K45 over the [0, 1]-valued Gödel fuzzy logic mathbf{G}. More precisely, we characterize K45(G) as the set of valid formulae of the class of possibilistic Gödel frames langle W, pi rangle , where W is a non-empty set of worlds and pi : W mathop {rightarrow }[0,1] is a possibility distribution on W. We provide decidability results as well. Moreover, we show that all the results also apply to the extension of K45(G) with the axiom (D), provided that we restrict ourselves to normalised Gödel Kripke frames, i.e. frames langle W, pi rangle where pi satisfies the normalisation condition sup _{w in W} pi (w) = 1.