Non-contingecy in a paraconsistent setting

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 \(\mathbf{K}^\blacktriangle_\mathbf{FDE}\) 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 \(\mathbf{K_{FDE}}\). Furthermore, we prove that in contrast to the classical non-contingency logic, reflexive, \(\mathbf{S4}\), and \(\mathbf{S5}\) (among others) frames \emph{are definable}.