The driving thought of paraconsistency is that there are situations in which information, or legal, scientific, or philosophical principles (and so on) are inconsistent, but in which people want to draw conclusions in a sensible fashion. Clearly, if one uses a logical consequence relation in which contradictions imply everything—that is, in which A,¬A ⊦ B, for all A and B—this is not possible: a person would have to conclude everything (triviality). This motivates the definition of a paraconsistent logic. The principle of inference that contradictions entail everything is called explosion (or ex falso quodlibet sequitur). A paraconsistent logic is one in which explosion is not valid.

Paraconsistent logics are not new. As Aristotle (An. Pr. 63^b31–64^a16) points out, syllogistic is paraconsistent. The idea that explosion is a correct principle of inference seems to have arisen in the twelfth century, with the discovery of...