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 ($\mathsf{BD}$, 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 $\blacksquare$

Non-contingency in a Paraconsistent Setting

International audienceAbstract We study an extension of first-degree entailment (FDE) by Dunn and Belnap with a non-contingency operator $\blacktriangle \phi$ which is construed as ‘$\phi$ has the same value in all accessible states’ or ‘all sources give the same information on the truth value of $\phi$’. We equip this logic dubbed $\textbf {K}^\blacktriangle _{\textbf {FDE}}$ with frame semantics and show how the bi-valued models can be interpreted as interconnected networks of Belnapian databases with the $\blacktriangle$ operator modelling search for inconsistencies in the provided information. We construct an analytic cut system for the logic and show its soundness and completeness. We prove that $\blacktriangle$ is not definable via the necessity modality $\Box$ of $\textbf {K}_{\textbf{FDE}}$. Furthermore, we prove that in contrast to the classical non-contingency logic, reflexive, $\textbf {S4}$ and $\textbf {S5}$ (among others) frames are definable