Editing Paraconsistent logic (section)

==Definition==
In [[classical logic]] (as well as [[intuitionistic logic]] and most other logics), contradictions [[Entailment|entail]] everything. This feature, known as the [[principle of explosion]] or ''ex contradictione sequitur quodlibet'' ([[Latin]], "from a contradiction, anything follows")<ref>{{cite journal|last1=Carnielli |first1=W. |author-link1=Walter Carnielli |last2=Marcos |last3=J. |year=2001 |url=https://dimap.ufrn.br/~jmarcos/papers/JM/01-CM-ECNSQL.pdf |title=Ex contradictione non sequitur quodlibet |journal=Bulletin of Advanced Reasoning and Knowledge |volume=1 |pages=89–109}}</ref> can be expressed formally as

{| class="wikitable" style="width:400px;"
|-
| 1
| align="center" | <math>P \land\neg P</math>
| colspan="2" | Premise
|-
| 2
| align="center" | <math>P\,</math>
| [[Conjunction elimination]]
| align="center" | from 1
|-
| 3
| align="center" | <math>P \lor A</math>
| [[Disjunction introduction]]
| align="center" | from 2
|-
| 4
| align="center" | <math>\neg P\,</math>
| [[Conjunction elimination]] 
| align="center" | from 1
|-
| 5
| align="center" | <math>A\,</math>
| [[Disjunctive syllogism]]
| align="center" | from 3 and 4
|}

Which means: if ''P'' and its negation ¬''P'' are both assumed to be true, then of the two claims ''P'' and (some arbitrary) ''A'', at least one is true.  Therefore, ''P'' or ''A'' is true.  However, if we know that either ''P'' or ''A'' is true, and also that ''P'' is false (that ¬''P'' is true) we can conclude that ''A'', which could be anything, is true. Thus if a [[theory (logic)|theory]] contains a single inconsistency, the theory is [[trivialism|trivial]] – that is, it has every sentence as a theorem.

The characteristic or defining feature of a paraconsistent logic is that it rejects the principle of explosion. As a result, paraconsistent logics, unlike classical and other logics, can be used to formalize inconsistent but non-trivial theories.