Kripke Semantics for Fuzzy Logics

TL;DR

This paper explores whether Kripke semantics can be applied to fuzzy logics, finding that only G"odel Logic and its extensions are sound and complete with respect to certain Kripke frames, thus providing a semantic characterization.

Contribution

It establishes a Kripke semantics framework for fuzzy logics, identifying G"odel Logic as uniquely sound and complete within this setting.

Findings

G"odel Logic is sound and strongly complete with respect to reflexive, transitive, and connected Kripke frames.

Only extensions of G"odel Logic admit Kripke semantics among fuzzy logics.

Provides a semantic characterization of G"odel Logic in the context of fuzzy logics.

Abstract

Kripke frames (and models) provide a suitable semantics for sub-classical logics, for example Intuitionistic Logic (of Brouwer and Heyting) axiomatizes the reflexive and transitive Kripke frames (with persistent satisfaction relations), and the Basic Logic (of Visser) axiomatizes transitive Kripke frames (with persistent satisfaction relations). Here, we investigate whether Kripke frames/models could provide a semantics for fuzzy logics. For each axiom of the Basic Fuzzy Logic, necessary and sufficient conditions are sought for Kripke frames/models which satisfy them. It turns out that the only fuzzy logics (logics containing the Basic Fuzzy Logic) which are sound and complete with respect to a class of Kripke frames/models are the extensions of the G\"odel Logic (or the super-intuitionistic logic of Dummett), indeed this logic is sound and strongly complete with respect to reflexive, transitive and connected (linear) Kripke frames (with persistent satisfaction relations). This provides a semantic characterization for the G\"odel Logic among (propositional) fuzzy logics.