On Three-Valued Modal Logics: from a Four-Valued Perspective

TL;DR

This paper introduces a systematic four-valued perspective to define and interpret three-valued modal logic, providing a versatile methodology with practical applications in deontic and epistemic contexts.

Contribution

It proposes a novel four-valued approach to three-valued modal logic, including natural deduction systems for deontic and epistemic cases, enhancing understanding and application.

Findings

Developed a four-valued interpretation method for three-valued modal logic.

Presented sound and complete natural deduction systems for deontic and epistemic logics.

Demonstrated the approach's potential to integrate deontic and epistemic notions into temporal logic.

Abstract

This paper aims at providing a comprehensive solution to the archaic open problem: how to define semantics of three-valued modal logic with vivid intuitive picture, convincing philosophical justification as well as versatile practical usage. Based on an existing line of work concerned with investigating three-valued logic out of innovative angles of view, we adopt a detour approach to interpret three-valued logic from a four-valued perspective, which results in the invention of an universal and systematic methodology for developing, explaining as well as utilizing three-valued modal logic. We illustrate our method through two concrete cases, one deontic and another epistemic, for both of which a sound and strongly complete natural deduction proof system is also presented in detail. We perceive our three-valued modal logic as a lightweight candidate to merge deontic or epistemic notion into temporal logic, without heavier burden of multiple modalities.