The most natural paradefinite logic relative to classical logic

TL;DR

This paper proposes that the most natural paradefinite logic relative to classical logic is an expansion of Belnap-Dunn logic with specific connectives, maintaining the deduction theorem, to better handle inconsistent or incomplete theories.

Contribution

It introduces a new expansion of Belnap-Dunn logic with a falsity and implication connective that preserves the deduction theorem, enhancing paradefinite logic frameworks.

Findings

The proposed logic maintains the deduction theorem.

It extends Belnap-Dunn logic with new connectives.

It is argued to be 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.