Positive modal logic beyond distributivity

Nick Bezhanishvili; Anna Dmitrieva; Jim de Groot; Tommaso Moraschini

doi:10.1016/j.apal.2023.103374

Positive modal logic beyond distributivity

Nick Bezhanishvili, Anna Dmitrieva, Jim de Groot, Tommaso Moraschini

Faculty of Science

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1 Citation (Scopus)

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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 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.

Original language	English
Article number	103374
Journal	Annals of Pure and Applied Logic
Volume	175
Issue number	2
Early online date	26 Sep 2023
DOIs	https://doi.org/10.1016/j.apal.2023.103374
Publication status	Published - Feb 2024

Keywords

Duality
Modal logic
Non-distributive positive logic
Sahlqvist correspondence
Weak positive logic

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10.1016/j.apal.2023.103374Licence: CC BY

Bezhanishvili_etal_2024_APALFinal published version, 746 KBLicence: CC BY

Cite this

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title = "Positive modal logic beyond distributivity",

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 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{\textquoteright}s correspondence result.",

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AU - Moraschini, Tommaso

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N2 - 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.

AB - 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.

KW - Duality

KW - Modal logic

KW - Non-distributive positive logic

KW - Sahlqvist correspondence

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JF - Annals of Pure and Applied Logic

SN - 0168-0072

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Positive modal logic beyond distributivity

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Keywords

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