Knowledge and ignorance in Belnap-Dunn logic

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 ($\textsf{BD}$, alias $\textbf{FDE}$-first-degree entailment). We then motivate and introduce a nonstandard modality $\blacksquare $ that formalizes knowledge and belief in $\textsf{BD}$ and use $\blacksquare $ to define $\bullet $ and $\blacktriangledown $ that formalize the unknown truth and ignorance as not knowing whether, respectively. Moreover, we introduce another modality $\textbf{I}$ that stands for 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 $\textsf{BD}<^>{\blacksquare }$ and $\textsf{BD}<^>{\textbf{I}}$-the expansions of $\textsf{BD}$ with $\blacksquare $ and $\textbf{I}$. In addition, we show that $\Box $ as it is customarily defined in $\textsf{BD}$ cannot define any of the introduced modalities, nor, conversely, neither $\blacksquare $ nor $\textbf{I}$ can define $\Box $. We also demonstrate that $\blacksquare $ and $\textbf{I}$ are not interdefinable and establish the definability of several important classes of frames using $\blacksquare $.

Automated summary

This paper discusses the limitations of the traditional approach of using the necessity modality $\Box$ to model knowledge and belief in the Belnap-Dunn logic framework. It introduces a nonstandard modality $\blacksquare$ to address these limitations and formalizes knowledge, belief, unknown truth, and ignorance. The paper provides a Kripke-frame-based semantics and a sound and complete analytic cut system for the introduced modalities, as well as demonstrates the non-definability and definability of certain classes of frames using $\blacksquare$.