Contradictoriness, Paraconsistent Negation and Non-intended Models of Classical Logic

Given that, by definition, two statements are contradictories if and only if it is logically impossible for both to be true and logically impossible for both to be false, some authors have argued that the negation operators of certain paraconsistent logics are not “real” negations because they allow for a statement and its negation to be true together. In this paper we argue that the same kind of argument can be levelled against the negation operator of classical propositional logic. To this end, Carnap’s result that there are models of classical propositional logic with non-standard or non-normal interpretations of the connectives, and that one kind of those interpretations violate the semantical principle of non-contradiction which requires of a sentence and its negation that at least one of them be false can be used. We ponder the consequences of these arguments for the claims that paraconsistent negations are not genuine negations and that the negation of classical logic is a contradictory-forming operator and we consider the arguments that challenge the conflation between negation and contradiction.