Describes any deductive system $\text{L}$ that is not explosive, that is, for a unary connective $\neg$ construed as a negation, there exist formulae $\phi$ and $\psi$ such that

• • $\phi ,\neg \phi {⊬}_{\text{L}}\psi$

Equivalently, any deductive system $\text{L}$ such that there exists an $\text{L}$-theory $T$ that is both negation-inconsistent and non-trivial. There are many independent motivations that lead to a rejection of explosion. An example is the case of supporters of relevant logic, who insist that a valid entailment requires relevance between the premisses and conclusion. The validity of explosion clearly is contrary to the criterion of relevance, e.g., in the sentence

• • That both $2+2=4$ and $2+2\ne 4$ entails that the moon is made of green cheese.

There is no clear sense in which the consequent is relevant to the antecedent. Reasons for investigating paraconsistent logic can be more or less metaphysically loaded. While, for example, one who embraces dialetheism—i.e., the metaphysical position according to which there are true contradictions—may insist on paraconsistency to avoid a commitment to trivialism, paraconsistent logic may be useful merely to give an account of what can be inferred from a database in which there are inconsistent entries.