Unified inverse correspondence for LE-logics

TL;DR

This paper extends Kracht's internal describability theory from classical modal logic to all LE-logics, using algebraic and duality techniques to connect formulas with frame conditions.

Contribution

It introduces a generalized framework for internal describability in LE-logics, broadening the scope from classical modal logic to lattice expansion-based logics.

Findings

Generalizes Kracht's theory to LE-logics

Establishes algebraic and duality-based correspondence methods

Provides a unified approach for frame correspondence in LE-logics

Abstract

We generalize Kracht's theory of internal describability from classical modal logic to the family of all logics canonically associated with varieties of normal lattice expansions (LE algebras). We work in the purely algebraic setting of perfect LEs; the formulas playing the role of Kracht's formulas in this generalized setting pertain to a first order language whose atoms are special inequalities between terms of perfect algebras. Via duality, formulas in this language can be equivalently translated into first order conditions in the frame correspondence languages of several types of relational semantics for LE-logics.