Knowledge and ignorance in Belnap--Dunn logic

TL;DR

This paper introduces new modalities for modeling knowledge and ignorance within Belnap--Dunn logic, addressing limitations of the standard necessity operator and providing a formal semantics and proof system for these concepts.

Contribution

It proposes non-standard modalities for knowledge and ignorance in Belnap--Dunn logic, with semantics, proof systems, and analysis of their interrelations and definability.

Findings

Introduced modalities $lacksquare$, $ullet$, $lacktriangledown$, and $ extbf{I}$ for knowledge and ignorance.

Established soundness and completeness of the proof systems for the extended logics.

Showed non-definability of standard necessity $ox$ by the new modalities and vice versa.

Abstract

In this paper, we argue that the usual approach to modelling knowledge and belief with the necessity modality $\Box$ does not produce intuitive outcomes in the framework of the Belnap--Dunn logic ($\mathsf{BD}$, alias $\mathsf{FDE}$ -- first-degree entailment). We then motivate and introduce a non\-standard modality $\blacksquare$ that formalises knowledge and belief in $\mathsf{BD}$ and use $\blacksquare$ to define $\bullet$ and $\blacktriangledown$ that formalise the \emph{unknown truth} and ignorance as \emph{not knowing whether}, respectively. Moreover, we introduce another modality $\mathbf{I}$ that stands for \emph{factive ignorance} and show its connection with $\blacksquare$. We equip these modalities with Kripke-frame-based semantics and construct a sound and complete analytic cut system for $\mathsf{BD}^\blacksquare$ and $\mathsf{BD}^\mathbf{I}$ -- the expansions of $\mathsf{BD}$ with $\blacksquare$ and $\mathbf{I}$. In addition, we show that $\Box$ as it is customarily defined in $\mathsf{BD}$ cannot define any of the introduced modalities, nor, conversely, neither $\blacksquare$ nor $\mathbf{I}$ can define $\Box$. We also demonstrate that $\blacksquare$ and $\mathbf{I}$ are not interdefinable and establish the definability of several important classes of frames using $\blacksquare$.