Minimally Nonstandard K3 and FDE

Graham Priest has formulated the minimally inconsistent logic of paradox (MiLP), which is paraconsistent like Priest’s logic of paradox (LP), while staying closer to classical logic. We present logics that stand to (the propositional fragments of) strong Kleene logic (K3) and the logic of first-degree entailment (FDE) as MiLP stands to LP. That is, our logics share the paracomplete and the paraconsistent-cum-paracomplete nature of K3 and FDE, respectively, while keeping these features to a minimum in order to stay closer to classical logic. We give semantic and sequent-calculus formulations of these logics, and we highlight some reasons why these logics may be interesting in their own right.