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In classical logic the rule of noncontradiction states $\neg(A\wedge\neg A)$ In the Paraconsistent logic, this rule doesn't apply, at least in some case. Now, in the classical logic, if we avoid to state this rule, we can't avoid the 'ex falso sequitur quodlibet', that means: $$(A\wedge\neg A)\to B$$ So, under what conditions we can accept the noncontradiction rule to be non valid in order to avoid the $(A\wedge\neg A)\to B$? Or, alternatively is there in this logic a rule that allows to accept this 'explosion'?

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Relevant: Can the principle of explosion be removed from constructive logic? My conclusion after that question was that Graham Priest's book In Contradiction has the answer to pretty much any elementary question I wanted to ask about paraconsistent logic, and that I should just reread that; I recommend that you do the same. –  MJD May 20 '13 at 17:27

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Paraconsistent logics are not concerned with invalidating one formula or another, but rather with disallowing the inference $A,\neg A\vdash B$. There are plenty of paraconsistent logics in which $\neg(A\land\neg A)$ is a valid formula, and there are a few paraconsistent logics in which the formula $(A\land\neg A)\to B$ is valid. An early example of the first variety was proposed by Asenjo (Notre Dame Journal of Formal Logic, 1966), and similar examples abound in the literature on 3-valued logics ; an early example of the second variety was proposed by Jaskowski (Studia Societatis Scientiarum Torunensis, 1948), in an apparently incidental kind of approach later to be known as 'non-adjunctive'.

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The principle of explosion is not valid in paraconsistent logic. Somewhat loosely, the meaning of an implication $p\to q$ is such that if $p$ is false, then $p\to q$ has not truth value. It's a bit like saying "don't ask me about such nonsense of whether $q$ follows from $p$ with $p$ is not even true".

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