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?
When can $A \wedge \neg A $ be true?
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0@vadim123 neither of those allows $ A \wedge \neg A $ to be true. – 2017-02-28
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6https://en.wikipedia.org/wiki/Paraconsistent_logic – 2017-02-28
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5[Insert political joke here] – 2017-02-28
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0If your logic is inconsistent, for most logics, $A\land \neg A$ will (also) be true. – 2017-02-28
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2See also [Dialetheism](https://plato.stanford.edu/entries/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*." – 2017-02-28
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1You can see [Paraconsistent Logic](https://plato.stanford.edu/entries/logic-paraconsistent/) : "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 – 2017-02-28
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1... "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 – 2017-02-28
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0See also [Inconsistent Mathematics](https://plato.stanford.edu/entries/mathematics-inconsistent/). – 2017-02-28
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0The more important question is why you would want a logic where such statements could be true. What purpose would it serve? Once you can answer that question properly with a meaningful answer, then you might have the proper motivation for such a logic. Otherwise it's just useless. – 2017-02-28
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0@user218022 the reason why I was asking is because I know that there are some forms of logic where proof by contradiction, and proof by contrapositive don't work. Furthermore proof by contradiction while still a powerful tool we use, doesn't translate as well as direct proofs. So I wanted further understanding into interesting forms of reasoning that dealt with not being able to use those tools. Situations where $A\wedge \neg A$ are both true seemed like a logical place to start. – 2017-02-28
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0@Sentinel135: You typed the wrong username but I saw your comment anyway. You're actually looking for the wrong thing, because practical logic systems never allow a pure contradiction to be true, not to say provable. Intuitionistic logic for example does not assume LEM (law of excluded middle) but still treats contradiction as always false and by the principle of explosion you can prove anything from a contradiction. There is a tight connection between intuitionistic logic and programs via the BHK (Brouwer-Heyting-Kolmogorov) interpretation. – 2017-02-28
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1For a different kind of non-boolean logic, you may be interested in Kleene's 3-valued system that Kripke used in a striking way in his theory of truth (see https://plato.stanford.edu/entries/self-reference). In this 3-valued system propositions may be true, false or undefined, and LEM does not hold in general, but still "$A \lor \neg A$" is never false and "$A \land \neg A$" is never true. – 2017-02-28
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1@BobbieD That's the most hilarious (and correct) comment I've read all week! Thanks. – 2017-03-11
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0Nice comment, @user21820. Thank you for adding it! – 2017-03-11