  Perfect paradefinite algebras are De Morgan algebras expanded with an
operation that allows for the full behavior of classical negation to be
restored. They form a variety that is term-equivalent to the variety of
involutive Stone algebras. Their associated multiple-conclusion (Set-Set) and
single-conclusion (Set-Fmla) order-preserving logics are non-algebraizable
self-extensional logics of formal inconsistency and undeterminedness determined
by a six-valued matrix, studied in depth by Gomes et al. (2022) from both the
algebraic and the proof-theoretical perspectives. In the present paper, we
continue that study by investigating directions for conservatively expanding
these logics with an implication connective (essentially, one that admits the
deduction-detachment theorem). We first consider logics given by very simple
and manageable non-deterministic semantics whose implication (in isolation) is
classical. These, nevertheless, fail to be self-extensional. We then consider
the implication realized by the relative pseudo-complement over the six-valued
perfect paradefinite algebra. Our strategy is to expand the language of the
latter algebra with this connective and study the (self-extensional) Set-Set
and Set-Fmla order-preserving and top-assertional logics of the variety induced
by the resulting algebra. We provide axiomatizations for such new variety and
for such logics, drawing parallels with the class of symmetric Heyting algebras
and with Moisil's 'symmetric modal logic'. For the Set-Set logic, in
particular, the axiomatization we obtain is analytic. We close by studying
interpolation properties for these logics and concluding that the new variety
has the Maehara amalgamation property.
