Non-contingecy in a paraconsistent setting

We study an extension of First Degree Entailment (FDE) by Dunn and Belnap with a non-contingency operator $\mathrm{▲}\phi$ 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 ${\mathbf{K}}_{\mathbf{F}\mathbf{D}\mathbf{E}}^{\mathrm{▲}}$ with frame semantics and show how the bi-valued models can be interpreted as interconnected networks of Belnapian databases with the $\mathrm{▲}$ 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 $\mathrm{▲}$ is not definable via the necessity modality $\mathrm{\square }$ of ${\mathbf{K}}_{\mathbf{F}\mathbf{D}\mathbf{E}}$. Furthermore, we prove that in contrast to the classical non-contingency logic, reflexive, $\mathbf{S}\mathbf{4}$, and $\mathbf{S}\mathbf{5}$ (among others) frames \emph{are definable}.