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 $\mathrm{\square }$ does not produce intuitive outcomes in the framework of the Belnap–Dunn logic ($\mathsf{B}\mathsf{D}$, alias $\mathsf{F}\mathsf{D}\mathsf{E}$ – first-degree entailment). We then motivate and introduce a non-standard modality $\mathrm{■}$ that formalises knowledge and belief in $\mathsf{B}\mathsf{D}$ and use $\mathrm{■}$ to define $\bullet$ and $\mathrm{▼}$ 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 $\mathrm{■}$. We equip these modalities with Kripke-frame-based semantics and construct a sound and complete analytic cut system for ${\mathsf{B}\mathsf{D}}^{\mathrm{■}}$ and ${\mathsf{B}\mathsf{D}}^{\mathbf{I}}$ – the expansions of $\mathsf{B}\mathsf{D}$ with $\mathrm{■}$ and $\mathbf{I}$. In addition, we show that $\mathrm{\square }$ as it is customarily defined in $\mathsf{B}\mathsf{D}$ cannot define any of the introduced modalities, nor, conversely, neither $\mathrm{■}$ nor $\mathbf{I}$ can define $\mathrm{\square }$. We also demonstrate that $\mathrm{■}$ and $\mathbf{I}$ are not interdefinable and establish the definability of several important classes of frames using $\mathrm{■}$.