# On the strongest three-valued paraconsistent logic contained in classical logic and its dual

## TL;DR

This paper investigates the unique properties of a specific three-valued paraconsistent logic within classical logic, distinguishing it from many others using logical equivalence properties, and explores its paracomplete analogue.

## Contribution

It identifies the strongest three-valued paraconsistent logic contained in classical logic and characterizes its unique logical equivalence properties among 8192 such logics.

## Key findings

- LP^{ ext{supset}, ext{F}} is uniquely distinguished by its logical equivalence relation.
- Only 32 of 8192 three-valued paraconsistent logics satisfy key algebraic laws.
- The paracomplete analogue of LP^{ ext{supset}, ext{F}} also exhibits comparable properties.

## Abstract

LP$^{\supset,\mathsf{F}}$ is a three-valued paraconsistent propositional logic which is essentially the same as J3. It has most properties that have been proposed as desirable properties of a reasonable paraconsistent propositional logic. However, it follows easily from already published results that there are exactly 8192 different three-valued paraconsistent propositional logics that have the properties concerned. In this paper, properties concerning the logical equivalence relation of a logic are used to distinguish LP$^{\supset,\mathsf{F}}$ from the others. As one of the bonuses of focussing on the logical equivalence relation, it is found that only 32 of the 8192 logics have a logical equivalence relation that satisfies the identity, annihilation, idempotent, and commutative laws for conjunction and disjunction. For most properties of LP$^{\supset,\mathsf{F}}$ that have been proposed as desirable properties of a reasonable paraconsistent propositional logic, its paracomplete analogue has a comparable property. In this paper, properties concerning the logical equivalence relation of a logic are also used to distinguish the paracomplete analogue of LP$^{\supset,\mathsf{F}}$ from the other three-valued paracomplete propositional logics with those comparable properties.