Very weak subintuitionistic logics

TL;DR

This paper introduces a new propositional logic called very weak subintuitionistic logic (VF), establishing its semantics, properties, and relation to other logics, along with modal extensions.

Contribution

It defines VF, proves its soundness, completeness, and key properties, and explores its modal companions, advancing the understanding of weak subintuitionistic logics.

Findings

VF is sound and complete with respect to its semantics.

VF has the disjunction and finite frame properties.

VF is strictly weaker than the weak subintuitionistic logic WF.

Abstract

We introduce a new propositional logic, called very weak subintuitionistic logic $\mathbf{VF}$, by adapting the relational semantics of Fitting, Marek, and Truszczy\'nski for the pure logic of necessitation $\mathbf{N}$ to the propositional setting. We prove that $\mathbf{VF}$ and its closed negative extensions are sound and complete with respect to this semantics, and that they have the disjunction property and the finite frame property. We also prove that $\mathbf{VF}$ is strictly weaker than the weak subintuitionistic logic $\mathbf{WF}$ of Shirmohammadzadeh Maleki and de Jongh. Finally, we study modal companions of $\mathbf{VF}$ and its closed negative extensions via Corsi's modified G\"odel translation.