Completeness results for many-valued \Lukasiewicz modal systems and relational semantics

TL;DR

This paper explores adding modal operators to ukasiewicz many-valued logics, establishing completeness results for Kripke semantics using canonical models, including finitely-valued and infinitary systems.

Contribution

It introduces a class of modal many-valued logics, their Kripke models, and proves completeness results, including for systems with infinitary deduction rules.

Findings

Completeness achieved for modal finitely-valued logics.

Two distinct classes of Kripke-complete modal many-valued logics.

Canonical model construction used for completeness proofs.

Abstract

The paper is dedicated to the problem of adding a modality to the \Lukasiewicz many-valued logics in the purpose of obtaining completeness results for Kripke semantics. We define a class of modal many-valued logics and their corresponding Kripke models and modal many-valued algebras. Completeness results are considered through the construction of a canonical model. Completeness is obtained for modal finitely-valued logics but also for a modal many-valued system with an infinitary deduction rule. We introduce two classes of frames for the finitely-valued logics and show that they define two distinct classes of Kripke-complete logics.