This paper is devoted to a consequence relation combining the negation of Classical Logic ) and a paraconsistent negation based on Graham Priest’s Logic of Paradox ). We give a number of natural desiderata for a logic \ that combines both negations. They are motivated by a particular property-theoretic perspective on paraconsistency and are all about warranting that the combining logic has the same characteristics as the combined logics, without giving up on the radically paraconsistent nature of the paraconsistent negation. (...) We devise the logic \ by means of an axiomatization and three equivalent semantical characterizations. By showing that this logic is maximally paraconsistent, we prove that \ is the only logic satisfying all postulated desiderata. Finally we show how the logic’s infinite-valued semantics permits defining different types of entailment relations. (shrink)