The most natural paradefinite logic relative to classical logic

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.