  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
$\blacksquare$.
