1. Describes a deductive system with a negation connective such for all sets of formulae and formulae and , the following holds:
That is, negation-inconsistent -theory is trivial. A deductive system that is not explosive is known as paraconsistent. A weaker notion that is a hallmark of the paraconsistent C-systems or logics of formal inconsistency is that of a logic’s being gently explosive. This is defined so that there exists a set of formulae in which the atomic formula appears such that there exist formulae and such that
That is, there exists a uniform means to express the consistency of in such a way that it is only when conjoined to the assertion that is consistent that a contradiction entails every arbitrary formula in the respective language.
2. Describes a negation connective with respect to which some deductive system is explosive in the earlier sense.
3. With respect to a deductive system , describes an -theory such that for all formulae , the deductive closure of is trivial, that is, if for arbitrary .