Many-Valued Modal Logic

TL;DR

This paper introduces a comprehensive framework for many-valued modal logic, combining modal and many-valued logics, with formal proof systems, semantics, and extensions, including embedding intuitionistic logic.

Contribution

It defines a general many-valued minimal normal modal logic with soundness, completeness, and finite model property, and explores negation and intuitionistic embedding.

Findings

Proves soundness and strong completeness of the logic.

Establishes finite model property and decidability.

Shows a unique negation definition preserving De Morgan's duality.

Abstract

We combine the concepts of modal logics and many-valued logics in a general and comprehensive way. Namely, given any finite linearly ordered set of truth values and any set of propositional connectives defined by truth tables, we define the many-valued minimal normal modal logic, presented as a Gentzen-like sequent calculus, and prove its soundness and strong completeness with respect to many-valued Kripke models. The logic treats necessitation and possibility independently, i.e., they are not defined by each other, so that the duality between them is reflected in the proof system itself. We also prove the finite model property (that implies strong decidability) of this logic and consider some of its extensions. Moreover, we show that there is exactly one way to define negation such that De Morgan's duality between necessitation and possibility holds. In addition, we embed many-valued intuitionistic logic into one of the extensions of our many-valued modal logic.