CONTRADICTION AND CONTRARIETY. PRIEST ON NEGATION

Although it is not younger than other areas of non-classical logic, paraconsistent logic has received full recognition only in recent years, largely due to the work of, among others, Newton da Costa, Graham Priest, Diderik Batens, and Jerzy Perzanowski. A logical system ? is paraconsistent if there is a set of ?-formulas ? ? {A} such that (i) in ? one may derive from ? both A and its negation, and (ii) the deductive closure of ? with respect to ? is different from the set of all formulas. If from ? one may derive a formula and its negation, ? is said to be syntactically inconsistent. But is every syntactically inconsistent set of formulas contradictory? In classical logic and many non-classical logics, every syntactically inconsistent set is unsatisfiable, that is, semantically inconsistent. If contradictoriness means semantical inconsistency, there is, up to logical equivalence, only one contradiction. In paraconsistent logics, there are usually many non-equivalent formulas representing the semantically unique contradiction in a non-paraconsistent logic. So when is a formula the contradiction of another formula, and, moreover, how does the notion of contradiction relate to the notions of contrariety and negation?