Paraconsistent logic

A logic is paraconsistent if it does not validate the principle that from a pair of contradictory sentences, A and ∼A, everything follows, as most orthodox logics do. If a theory has a paraconsistent underlying logic, it may be inconsistent without being trivial (that is, entailing everything). Sustained work in formal paraconsistent logics started in the early 1960s. A major motivating thought was that there are important naturally occurring inconsistent but non-trivial theories. Some logicians have gone further and claimed that some of these theories may be true. By the mid-1970s, details of the semantics and proof-theories of many paraconsistent logics were well understood. More recent research has focused on the applications of these logics and on their philosophical underpinnings and implications.