Positive Modal Logic Beyond Distributivity

TL;DR

This paper introduces a duality theory for non-distributive modal lattices and develops weak positive modal logic with new semantics and correspondence results, expanding the understanding of modal logic beyond traditional distributive frameworks.

Contribution

It develops a duality for non-distributive modal lattices and introduces weak positive modal logic with a new relational semantics and Sahlqvist correspondence.

Findings

Established a duality for non-distributive modal lattices.

Proved that weak positive modal logic is $$-persistent.

Developed a new relational semantics with Sahlqvist correspondence.

Abstract

We develop a duality for (modal) lattices that need not be distributive, and use it to study positive (modal) logic beyond distributivity, which we call weak positive (modal) logic. This duality builds on the Hofmann, Mislove and Stralka duality for meet-semilattices. We introduce the notion of $\Pi_1$-persistence and show that every weak positive modal logic is $\Pi_1$-persistent. This approach leads to a new relational semantics for weak positive modal logic, for which we prove an analogue of Sahlqvist correspondence result.