Non-contingecy in a paraconsistent setting

TL;DR

It is proved that in contrast to the classical non-contingency logic, reflexive, $\mathbf{S4}$, and $\mathbf{S5}$ (among others) frames \emph{are definable}.

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}.

Figures (14)

• Figure 1: All variables have the same values exemplified by $p$.
• Figure 2: $p$ is true and not-false at $w_0$ but is both true and false at $w_1$. Thus, $\blacktriangle p$ is false and not-true at $w_0$.
• Figure 3: $w$ is the investigator; $w_1$ and $w_2$ stand for the accounts of the witnesses.
• Figure 4: $w_c$ is the computer at the central office that the auditor is looking into; $w_h$ is the database at the warehouse, and $w_s$ is the database in the store.
• Figure 5: Here, $\blacktriangle p$ is both true and false at $w_0$ and neither true nor false at $w'_0$.

Theorems & Definitions (35)

• Remark 1
• Definition 1: Semantics
• Definition 2
• Remark 2
• Remark 3
• Example 1
• Example 2
• Remark 4
• Definition 3: Dual models
• Lemma 1