G\"odel-McKinsey-Tarski and (not quite) Blok-Esakia for Heyting-Lewis Implication

TL;DR

This paper explores Heyting-Lewis Logic, extending intuitionistic logic with strict implication, and establishes duality and decidability results by translating it into classical modal logic.

Contribution

It introduces a semantic framework for Heyting-Lewis Logic, establishes a duality with algebraic structures, and adapts a translation to classical modal logic to prove key properties.

Findings

Categorical duality between algebraic and frame semantics

Finite model property for a large family of Heyting-Lewis logics

Decidability results obtained via translation to classical modal logic

Abstract

Heyting-Lewis Logic is the extension of intuitionistic propositional logic with a strict implication connective that satisfies the constructive counterparts of axioms for strict implication provable in classical modal logics. Variants of this logic are surprisingly widespread: they appear as Curry-Howard correspondents of (simple type theory extended with) Haskell-style arrows, in preservativity logic of Heyting arithmetic, in the proof theory of guarded (co)recursion, and in the generalization of intuitionistic epistemic logic. Heyting-Lewis Logic can be interpreted in intuitionistic Kripke frames extended with a binary relation to account for strict implication. We use this semantics to define descriptive frames (generalisations of Esakia spaces), and establish a categorical duality between the algebraic interpretation and the frame semantics. We then adapt a transformation by Wolter and Zakharyaschev to translate Heyting-Lewis Logic to classical modal logic with two unary operators. This allows us to classical results to obtain the finite model property and decidability for a large family of Heyting-Lewis logics.