Positive modal logic beyond distributivity

• Published in: Annals of pure and applied logic Vol. 175; no. 2; p. 103374
• Main Authors: Bezhanishvili, Nick, Dmitrieva, Anna, de Groot, Jim, Moraschini, Tommaso
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01-02-2024
• Subjects: Duality; Modal logic; Non-distributive positive logic; Sahlqvist correspondence; Weak positive logic; Sahlqvist correspondence; 06D50; Non-distributive positive logic; Duality; Modal logic; Weak positive logic; 03B45; 06B15; 03G10
• ISSN: 0168-0072
• DOI: 10.1016/j.apal.2023.103374

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 Π1-persistence and show that every weak positive modal logic is Π1-persistent. This approach leads to a new relational semantics for weak positive modal logic, for which we prove an analogue of Sahlqvist's correspondence result.1