Non-contingency in a Paraconsistent Setting

Abstract

Abstract 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.