Many-Valued Modal Logic

2024-12-28

Electronic Proceedings in Theoretical Computer Science

AI Summary

This paper addresses the long-standing separation between many-valued and modal logics by constructing the first minimal normal many-valued modal logic for finite linearly ordered semantics. Methodologically, it introduces a Gentzen-style sequent calculus that independently characterizes the necessity and possibility operators under Kripke-style many-valued semantics, rigorously establishing their De Morgan duality. Strong soundness and strong completeness are proven, along with the finite model property and strong decidability. The main contributions are threefold: (i) the first unified axiomatic framework integrating many-valued and modal logics; (ii) a proof that negation is uniquely determined by duality within this framework; and (iii) a faithful embedding of many-valued intuitionistic logic into the modal system, thereby providing a more robust modal foundation for many-valued reasoning.

Technology Category

• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Knowledge Representation and Reasoning: Nonmonotonic Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Web Mining and Content Analysis: Mining multimedia, multimodal, multilingual, cross-lingual Web data

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.

Problem

• Multi-valued Modal Logic
• Necessity and Possibility Independence
• De Morgan Duality

Innovation

• Modal Logic
• Many-Valued Logic
• Gentzen-style Proof