Paraconsistent logics

A paraconsistent logic is a non-trivial logic which allows inconsistencies. More specifically, it allows both a statement and its negation to be asserted, without absurdity following. In standard logics, anything can be derived from an inconsistency; this is known as ex contradictione quodlibet (ECQ), also known as the principle of explosion. A paraconsistent logic is then a logical system in which ECQ does not hold.

Paraconsistent logic can be used in modelling belief systems which are inconsistent, and yet from which not anything can be inferred. In standard logics, care has to be taken to not allow such statements as the liar paradox to be formed; paraconsistent logics can be much simplified in that they do not have to excise such statements (though they still have to excise Curry's paradox). Additionally, a paraconsistent logic can potentially overcome the limitation of arithmetic that Gödel's incompleteness theorem implies, and be complete.

Approaches to paraconsistent logic include:

• Relevant logics
• Dialetheism
• Many-valued logics
• Non-adjunctive logics