Bi-modal G\"odel logic over [0,1]-valued Kripke frames

TL;DR

This paper explores a bi-modal G"odel logic over fuzzy Kripke frames with [0,1] valuations, establishing strong completeness results and axiomatizations for various modal systems, and providing a representation theorem for bi-modal G"odel algebras.

Contribution

It introduces a new fuzzy Kripke semantics for bi-modal G"odel logic and proves completeness and axiomatizations for several modal variants, including T, S4, and S5.

Findings

Proves strong completeness of Fischer Servi intuitionistic modal logic IK with fuzzy semantics.

Provides axiomatizations for bi-modal G"odel logic variants T, S4, and S5.

Establishes a representation theorem for bi-modal G"odel algebras.

Abstract

We consider the G\"odel bi-modal logic determined by fuzzy Kripke models where both the propositions and the accessibility relation are infinitely valued over the standard G\"odel algebra [0,1] and prove strong completeness of Fischer Servi intuitionistic modal logic IK plus the prelinearity axiom with respect to this semantics. We axiomatize also the bi-modal analogues of $T,$ $S4,$ and $S5$ obtained by restricting to models over frames satisfying the [0,1]-valued versions of the structural properties which characterize these logics. As application of the completeness theorems we obtain a representation theorem for bi-modal G\"odel algebras.