Regular non-normal modal classicalities

TL;DR

This paper explores the use of a consistency operator to extend non-normal modal classicalities within a Kripke framework, introducing a four-valued logic and its modal extension with novel semantic tools.

Contribution

It introduces the logic $ ext{B}_4^ ext{o}$, a four-valued Boolean logic with a consistency operator, and extends it to modal logic using an eight-valued algebra, revealing new frame properties.

Findings

Modal logic exhibits both normal local and non-normal global consequence relations.

New modal formulas capture accessibility notions unattainable in traditional frameworks.

Frame properties like transitivity and reflexivity require novel semantic characterizations.

Abstract

We present a novel investigation into the consistency operator ($\circ$), traditionally associated with paraconsistent logics, as a means of capturing non-normal modal classicalities within the Kripke framework. By semantically reinterpreting $\circ$ as an operator that distinguishes top and bottom values from other values in the algebra, we extend its applicability beyond paraconsistency into classical and modal logics. We introduce the logic $\mathcal{B}_4^\circ$, a four-valued Boolean logic augmented with the consistency operator, and provide a sound and complete axiomatization. Building on this foundation, we extend the semantics to the modal domain using the many-logics modal logic (MLML) framework. Specifically, we construct Kripke frames based on an eight-valued Boolean algebra that contains three distinct four-valued subalgebras, each representing a different world type. Our analysis reveals that the resulting modal logic exhibits a normal local consequence relation alongside a non-normal global consequence relation. Consequently, the characterization of frame properties --such as transitivity, reflexivity, and Euclideanness --deviates from modal logic $K$, requiring novel semantic tools. We further identify new modal formulas in an extended language that capture previously unavailable kinds of accessibility, leading to frame characterizations unattainable in traditional modal frameworks.