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arXiv · 2402.11249

Non-contingecy in a paraconsistent setting

Daniil Kozhemiachenko·Liubov Vashentseva
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Abstract

We study an extension of First Degree Entailment (FDE) by Dunn and Belnap with a non-contingency operator $\blacktriangleϕ$ which is construed as "$ϕ$ has the same value in all accessible states" or "all sources give the same information on the truth value of $ϕ$". 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}.

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Daniil Kozhemiachenko, Liubov Vashentseva. 2024-02-17. Non-contingecy in a paraconsistent setting. https://doi.org/10.1093/jigpal%2Fjzac081

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