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

Abstract

|$\textrm{LP}^{\mathbin{\supset },{\mathsf{F}}}$| is a three-valued paraconsistent propositional logic that is essentially the same as J3. It has the 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 |$\textrm{LP}^{\mathbin{\supset },{\mathsf{F}}}$| from the others. As one of the bonuses of focusing 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 |$\textrm{LP}^{\mathbin{\supset },{\mathsf{F}}}$| 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 |$\textrm{LP}^{\mathbin{\supset },{\mathsf{F}}}$| from the other three-valued paracomplete propositional logics with those comparable properties.