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Distribution-Free Normal Modal Logics

Chrysafis Hartonas

TL;DR

This paper develops a semantic framework for distribution-free normal modal logics using implicative modal lattices and a uniform, two-sort relational semantics. It establishes canonicity and completeness for the minimal distribution-free logic with the $K$-axiom and extends these results to standard modal axioms ($D$, $T$, $B$, $S4$, $S5$) within refined frame classes and canonical extensions. The construction of frames for distribution-free logics—via intermediate structures, MacNeille extensions, and K-frames—enables a robust treatment of the interaction between implication and modal operators in a lattice-theoretic setting. It also analyzes distributive and Heyting base cases, showing how canonical/ frame results adapt when the propositional logic is distributive or intuitionistic, and outlines future directions for sorted correspondences and first-order characterizations in this distribution-free landscape.

Abstract

This article initiates the semantic study of distribution-free normal modal logic systems, laying the semantic foundations and anticipating further research in the area. The article explores roughly the same area, though taking a different approach, with a recent article by Bezhanishvili, de Groot, Dmitrieva and Morachini, who studied a distribution-free version of Dunn's Positive Modal Logic (PML). Unlike PML, we consider logics that may drop distribution and which are equipped with both an implication connective and modal operators. We adopt a uniform relational semantics approach, relying on recent results on representation and duality for normal lattice expansions. We prove canonicity and completeness in the relational semantics of the minimal distribution-free normal modal logic, assuming just the K-axiom, as well as of its axiomatic extensions obtained by adding any of the D, T, B, S4 or S5 axioms. Adding distribution can be easily accommodated and, as a side result, we also obtain a new semantic treatment of Intuitionistic Modal Logic.

Distribution-Free Normal Modal Logics

TL;DR

This paper develops a semantic framework for distribution-free normal modal logics using implicative modal lattices and a uniform, two-sort relational semantics. It establishes canonicity and completeness for the minimal distribution-free logic with the -axiom and extends these results to standard modal axioms (, , , , ) within refined frame classes and canonical extensions. The construction of frames for distribution-free logics—via intermediate structures, MacNeille extensions, and K-frames—enables a robust treatment of the interaction between implication and modal operators in a lattice-theoretic setting. It also analyzes distributive and Heyting base cases, showing how canonical/ frame results adapt when the propositional logic is distributive or intuitionistic, and outlines future directions for sorted correspondences and first-order characterizations in this distribution-free landscape.

Abstract

This article initiates the semantic study of distribution-free normal modal logic systems, laying the semantic foundations and anticipating further research in the area. The article explores roughly the same area, though taking a different approach, with a recent article by Bezhanishvili, de Groot, Dmitrieva and Morachini, who studied a distribution-free version of Dunn's Positive Modal Logic (PML). Unlike PML, we consider logics that may drop distribution and which are equipped with both an implication connective and modal operators. We adopt a uniform relational semantics approach, relying on recent results on representation and duality for normal lattice expansions. We prove canonicity and completeness in the relational semantics of the minimal distribution-free normal modal logic, assuming just the K-axiom, as well as of its axiomatic extensions obtained by adding any of the D, T, B, S4 or S5 axioms. Adding distribution can be easily accommodated and, as a side result, we also obtain a new semantic treatment of Intuitionistic Modal Logic.

Paper Structure

This paper contains 25 sections, 34 theorems, 28 equations, 4 tables.

Table of Contents

  1. Introduction
  2. Implicative Modal Lattices and Logics
  3. Implicative Modal Lattices
  4. Languages, Logics and Algebraic Semantics
  5. Frames and Relational Semantics
  6. Frames and Models
  7. Full Complex Algebras
  8. Soundness of the Fusion Logic $\mathbf{\Lambda}_{\Box\Diamond}^\rightarrow$ in the Relational Semantics
  9. Implication
  10. Necessity
  11. Possibility
  12. Frames for Distribution-Free $\mathbf{K}^\rightarrow_{\Box\Diamond}$
  13. Intermediate Structure and Lower and Upper MacNeille Extensions for $\hbox{$\largesquare\space\Box\;$}$
  14. Refined Frames
  15. K-Frames
  16. ...and 10 more sections

Key Result

Theorem 2.3

For any similarity type $\tau$ the minimal logic $\mathbf{\Lambda}_\tau$ is sound and complete in the variety $\mathbb{NLE}_\tau$ of normal lattice expansions of similarity type $\tau$.$\Box$

Theorems & Definitions (79)

  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3
  • Definition 2.4
  • Proposition 2.5: Definability
  • Proposition 2.6: Completeness
  • Proposition 2.7: Duality
  • Lemma 3.1
  • proof
  • Definition 3.2
  • ...and 69 more