Paraclassical Logics with Two Negations

We study three paraclassical propositional logics with two negations, CLoaN, CLouN, and CLauN, obtained by pairwise combining the one-negation systems CLuN, CLaN, and CLoN.CLuN's negation ∼ u is glut-tolerant and paraconsistent, CLaN's negation ∼ a is gap-tolerant and paracomplete, and CLoN's negation ∼ o is both paraconsistent and paracomplete.We provide sound and complete two-valued non-deterministic semantics for each of the two-negations systems.We then focus on CLauN, the system containing the primitive dual pair (∼ a , ∼ u ).In this setting, we show deductive relationships between different negations expressible in CLauN; we show two diferent ways to recapture classical logic by means of translations, locally and globally; and we show a duality results for the multiple conclusion consequence relation of the logic.