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Many-Valued Modal Logic

Amir Karniel, Michael Kaminski

TL;DR

This work develops a comprehensive framework for many-valued modal logics by unifying modal and many-valued reasoning in a Gentzen-style calculus. It defines mv-K over a finite linearly ordered set of truth values and interprets $\\Box\\varphi$ as an infimum and $\\Diamond\\varphi$ as a supremum of successor values, proving soundness and strong completeness w.r.t. many-valued Kripke models and establishing a finite model property. The authors extend mv-K with standard modal axioms (mv-D, mv-T, mv-K4, mv-S4, mv-B, mv-S5), discuss decidability via filtration, and identify a unique negation that enforces De Morgan duality between $\\Box$ and $\\Diamond$, making certain rules redundant. They also show how many-valued intuitionistic logic embeds into mv-S4, linking modal and intuitionistic reasoning in the many-valued setting. Overall, the paper offers a unified, decidable, and extensible theory of many-valued modal logics with explicit treatment of dual modalities and a constructive canonical-model approach.

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.

Many-Valued Modal Logic

TL;DR

This work develops a comprehensive framework for many-valued modal logics by unifying modal and many-valued reasoning in a Gentzen-style calculus. It defines mv-K over a finite linearly ordered set of truth values and interprets as an infimum and as a supremum of successor values, proving soundness and strong completeness w.r.t. many-valued Kripke models and establishing a finite model property. The authors extend mv-K with standard modal axioms (mv-D, mv-T, mv-K4, mv-S4, mv-B, mv-S5), discuss decidability via filtration, and identify a unique negation that enforces De Morgan duality between and , making certain rules redundant. They also show how many-valued intuitionistic logic embeds into mv-S4, linking modal and intuitionistic reasoning in the many-valued setting. Overall, the paper offers a unified, decidable, and extensible theory of many-valued modal logics with explicit treatment of dual modalities and a constructive canonical-model approach.

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.
Paper Structure (7 sections, 17 theorems, 72 equations)

This paper contains 7 sections, 17 theorems, 72 equations.

Table of Contents

  1. Introduction
  2. Many-valued modal logic
  3. The canonical model theorem and the proof of the "if" part of Theorem \ref{['t: main']}
  4. Extensions of mv-K
  5. Decidability of mv-K and its extensions
  6. Duality of $\Box$ and $\Diamond$ via negation
  7. Embedding many-valued intuitionistic logic into mv-S4

Key Result

Theorem 1

Let $\hbox{\boldmath$\Sigma$}$ and $\Gamma \rightarrow \Delta$ be a set of sequents and a sequent, respectively. Then $\hbox{\boldmath$\Sigma$} \vdash \Gamma \rightarrow \Delta$ if and only if $\hbox{\boldmath$\Sigma$} \models \Gamma \rightarrow \Delta$.

Theorems & Definitions (17)

  • Theorem 1
  • Lemma 2
  • Lemma 3
  • Theorem 4
  • Lemma 5
  • Corollary 6
  • Lemma 7
  • Lemma 8
  • Corollary 9
  • Theorem 10
  • ...and 7 more