  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.
