Axiomatizing logics of finite G\"odel-Kripke models

TL;DR

This paper resolves a 15-year open problem by establishing completeness of certain modal G"odel logics with respect to finite G"odel-Kripke models through new axiomatizations.

Contribution

It introduces novel axiomatizations that achieve completeness for finite G"odel-Kripke models, addressing longstanding open questions in the field.

Findings

Natural candidate axioms do not restore completeness.

New axiomatizations are complete for finite models.

Characterizes intermediate witnessing conditions for basic logics.

Abstract

We investigate completeness for modal G\"odel logics with respect to finite G\"odel-Kripke models, along with related aspects. It is well known that the logics studied in [4, 11] fail to be complete with respect to finite G\"odel-Kripke models. We show that the natural candidate axiomatic extensions do not restore completeness, thereby resolving a 15 year open problem posed in the aforementioned works. We then provide new axiomatizations that are complete for finite models and characterize intermediate witnessing conditions that hold for the basic logics.