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Failure of Esakia's theorem in the monadic setting

Guram Bezhanishvili, Luca Carai

TL;DR

This paper shows that Esakia's theorem fails for the monadic fragment of predicate logic by constructing two Kuroda-inspired modal logics, GKur and LKur, as modal companions of the monadic intuitionistic logics. It proves that while both logics are modal companions in their respective contexts, their join cannot yield a greatest modal companion for the monadic IPC, thereby showing that no greatest modal companion exists for MIPC. The authors develop semantic characterizations via descriptive MS4-frames and skeletons, revealing a fundamental imbalance between the MIPC and MS4 descriptive-frame settings that drives the failure. They also address Naumov's claim in the monadic setting and discuss potential remedies and open problems in the predicate case, suggesting directions for restoring a maximal companion with additional axioms.

Abstract

Esakia's theorem states that Grzegorczyk's logic is the greatest modal companion of intuitionistic propositional calculus. We prove that already the one-variable fragment of intuitionistic predicate calculus does not have a greatest modal companion, yielding that Esakia's theorem fails in the monadic setting.

Failure of Esakia's theorem in the monadic setting

TL;DR

This paper shows that Esakia's theorem fails for the monadic fragment of predicate logic by constructing two Kuroda-inspired modal logics, GKur and LKur, as modal companions of the monadic intuitionistic logics. It proves that while both logics are modal companions in their respective contexts, their join cannot yield a greatest modal companion for the monadic IPC, thereby showing that no greatest modal companion exists for MIPC. The authors develop semantic characterizations via descriptive MS4-frames and skeletons, revealing a fundamental imbalance between the MIPC and MS4 descriptive-frame settings that drives the failure. They also address Naumov's claim in the monadic setting and discuss potential remedies and open problems in the predicate case, suggesting directions for restoring a maximal companion with additional axioms.

Abstract

Esakia's theorem states that Grzegorczyk's logic is the greatest modal companion of intuitionistic propositional calculus. We prove that already the one-variable fragment of intuitionistic predicate calculus does not have a greatest modal companion, yielding that Esakia's theorem fails in the monadic setting.
Paper Structure (6 sections, 34 theorems, 37 equations, 4 figures)

This paper contains 6 sections, 34 theorems, 37 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Global Kuroda logic
  4. Local Kuroda logic
  5. Failure of Esakia's theorem for $\mathsf{MIPC}$
  6. Conclusions

Key Result

Theorem 2.4

$\mathsf{MIPC}$ is Kripke complete.

Figures (4)

  • Figure 1: The frame $\mathfrak{H}$
  • Figure 2: The frame $\mathfrak{K}$
  • Figure :
  • Figure :

Theorems & Definitions (103)

  • Definition 2.1
  • Definition 2.3
  • Theorem 2.4
  • Definition 2.5
  • Definition 2.6
  • Remark 2.7
  • Definition 2.8
  • Remark 2.9
  • Theorem 2.10
  • Remark 2.11
  • ...and 93 more