A Gödel Modal Logic Over Witnessed Models

2026-06-30

AI Summary

This work addresses the lack of the finite model property in traditional Gödel modal logics under standard Kripke semantics, which stems from their reliance on limit behaviors that yield non-constructive interpretations. To overcome this limitation, the paper introduces GW logic, equipped with a novel witnessed Kripke semantics that requires the truth of every modal formula to be explicitly witnessed by some accessible world, thereby eliminating non-constructive limit cases. Building on this semantics, the authors establish the first Gödel modal logic framework enjoying the finite model property and develop a corresponding refutation calculus together with a terminating backward proof-search algorithm. The calculus is proven sound and complete, enabling automated reasoning and countermodel generation, and substantially enhancing the constructivity and computability of the logical system.

Abstract

We introduce GW, a Gödel modal logic based on Kripke models in which the value of each modal formula is witnessed by an accessible world. This witnessed semantics eliminates the limit-based phenomena that preclude the finite model property in the usual Kripke semantics for Gödel modal logics, thereby yielding a more constructive semantic framework. We present a sound and complete refutation calculus for GW and design a terminating backward proof-search procedure with countermodel generation. As a direct consequence of this procedure, GW enjoys the finite model property.

Problem

• Gödel modal logic
• finite model property
• witnessed semantics
• Kripke models
• limit-based phenomena

Innovation

• Gödel modal logic
• witnessed semantics
• finite model property
• proof-search procedure
• countermodel generation