# When can $A \wedge \neg A$ be true?

This may seem like a silly question but I know that for Boolean logic $A\wedge \neg A$ is always false. I was wondering if there is a form of logic where this could be true. If so what is it and when does it occur?

• en.wikipedia.org/wiki/Paraconsistent_logic – Michael McGovern Feb 28 '17 at 0:09
• [Insert political joke here] – Bobbie D Feb 28 '17 at 0:10
• See also Dialetheism : "A dialetheia is a sentence, $A$, such that both it and its negation, $¬A$, are true. Therefore, dialetheism amounts to the claim that there are true contradictions. As such, dialetheism opposes the so-called Law of Non-Contradiction." – Mauro ALLEGRANZA Feb 28 '17 at 7:05
• You can see Paraconsistent Logic : "The contemporary logical orthodoxy has it that, from contradictory premises, anything can be inferred. Let $⊨$ be a relation of logical consequence, defined either semantically or proof-theoretically. Call $⊨$ explosive if it validates $\{ A , ¬A \} ⊨ B$ for every $A$ and $B$ (Ex Contradictione Quodlibet). Classical logic, and most standard ‘non-classical’ logics too such as intuitionist logic, are explosive. Paraconsistent logic challenges this orthodoxy." 1/2 – Mauro ALLEGRANZA Feb 28 '17 at 11:12
• ... "A logical consequence relation is said to be paraconsistent if it is not explosive. Thus, if $⊨$ is paraconsistent, then even if we are in certain circumstances where the available information is inconsistent, the inference relation does not explode into triviality. [...] Nevertheless, many paraconsistent logics validate the Law of Non-Contradiciton ($⊨ ¬(A ∧ ¬A)$) even though they invalidate ECQ." Thus, also with this approach, $(A ∧ ¬A)$ is false. 2/2 – Mauro ALLEGRANZA Feb 28 '17 at 11:14