The most natural paradefinite logic relative to classical logic

TL;DR

The paper investigates paradefinite logics—logics suitable for inconsistent or incomplete theories—by examining expansions of Belnap-Dunn logic relative to classical logic. It argues that the most natural such logic is the expansion $BD^{\supset,\mathsf{F}}$, which adds a falsity connective and an implication with the standard deduction theorem. The work develops formal preliminaries for propositional logics, defines the languages $CL^{\supset,\mathsf{F}}$ and $BD^{\supset,\mathsf{F}}$ on a common alphabet, and presents implicit definitions of their consequence relations along with corresponding sequent-calculus soundness and completeness results. It shows that removing the law of non-contradiction and the law of the excluded middle from classical logic yields a natural paraconsistent/paracomplete baseline, with $BD^{\supset,\mathsf{F}}$ embedded as the closest relative to $CL^{\supset,\mathsf{F}}$, supporting BD as the natural underlying logic for inconsistent or incomplete theories.

Abstract

A paradefinite logic is a logic that can serve as the underlying logic for theories that are inconsistent or incomplete. A well-known paradefinite logic is Belnap-Dunn logic. Various expansions of Belnap-Dunn logic have been studied in the literature. In this note, it is argued that the most natural paradefinite logic relative to classical logic is the expansion of Belnap-Dunn logic with a falsity connective and an implication connective for which the standard deduction theorem holds.