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Classical logic says that from a contradiction you can derive anything. I think that depends on how you define a contradiction. If you have two opposing truth values with respect to A, A is true and A is false what can we infer about the truth status of A? Well in one way to look at it you could say that to assert a contradiction means we hold that both statements about A are true regardless of whether they contradict each other. A is true regardless of the contrary position that A is false. Likewise A is false regardless of the contrary position that A is true. If we define a contradiction in this manner then we can separately infer both truth values of A. Given A is true and A is false we can conclude A is true and given A is true and A is false we can conclude that A is false. If you infer A is true from the contradiction then A or B is true. If A or B is true then if A is false then B is true. A is true regardless of whether A is false therefor we can not conclude an explosion occurs. It seems that for Classical logic to make sense of a contradiction in such a way that it leads to explosion that it must define what it means to hold a contradiction in a particular way. I don't know which way it defines a contradiction but wouldn't it be defined in some arbitrary way that forces us into the "explosion" scenario?

Classical logic says that from a contradiction you can derive anything. I think that depends on how you define a contradiction. If you have two opposing truth values with respect to A, A is true and A is false what can we infer about the truth status of A? Well in one way to look at it you could say that to assert a contradiction means we hold that both statements about A are true regardless of whether they contradict each other. A is true regardless of the contrary position that A is false. Likewise A is false regardless of the contrary position that A is true. If we define a contradiction in this manner then we can separately infer both truth values of A. Given A is true and A is false we can conclude A is true and given A is true and A is false we can conclude that A is false. If you infer A is true from the contradiction then A or B is true. If A or B is true then if A is false then B is true. A is true regardless of whether A is false therefor we can not conclude an explosion occurs. It seems that for Classical logic to make sense of a contradiction in such a way that it leads to explosion that it must define what it means to hold a contradiction in a particular way. I don't know which way it defines a contradiction but wouldn't it be defined in some arbitrary way that forces us into the "explosion" scenario?

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You wrote: (i) "It seems that for classical logic to make sense of a contradiction in such a way that it leads to explosion...it must define what it means to hold a contradiction in a particular way" and (ii) "[W]ouldn't it be defined in some arbitrary way that forces us into the 'explosion' scenario?"

Regarding (i): If the assertion "A is true and A is false" means anything, then surely it implies that A is true and implies that A is false. I can't think of another way to construe the assertion. Are you suggesting that a conjunction doesn't imply each of its conjuncts?

Regarding (ii): How is it arbitrary to infer the truth of A and the falsity of A from the assertion "A is true and A is false"? Again, I can't think of another way to understand the assertion.

As far as I know, paraconsistent logicians tend to object to inferring B from (A or B) and not-A: they point out that the inference relies on the implicit assumption that not-A rules out A, an assumption they reject.