Relational semantics for flat Heyting-Lewis Logic

2026-03-30

AI Summary

This study addresses flat Heyting-Lewis logic (HLC^flat)—a modal system extending intuitionistic logic with a Lewis-style strict implication whose first argument does not satisfy the disjunction property—and presents the first relational semantics for this logic. By integrating relational semantics, modal logic techniques, and algebraic methods, the authors establish strong completeness and the finite model property for HLC^flat and several of its axiomatic extensions. This work not only fills a gap in the semantic characterization of HLC^flat but also uncovers a fundamental semantic divergence between this flat variant and its sharp counterpart, thereby providing new tools for the refined investigation of intuitionistic modal logics.

Abstract

We introduce relational semantics for "flat Heyting-Lewis logic" $\mathsf{HLC}^{\flat}$. This logic arises as the extension of intuitionistic logic with a Lewis-style strict implication modality that, contrary to its "sharp" counterpart $\mathsf{HLC}^{\sharp}$, does not turn meets into joins in its first argument. We prove completeness and the finite model property for $\mathsf{HLC}^{\flat}$ and for several extensions with additional axioms.

Problem

• relational semantics
• Heyting-Lewis logic
• flat implication
• intuitionistic logic
• strict implication

Innovation

• relational semantics
• Heyting-Lewis logic
• strict implication
• finite model property
• intuitionistic logic