Logic

Why does inconsistency entail validity?

Comments

Let me spell out the claim I think you have in mind. Any argument whose premises form an inconsistent set of sentences is a valid argument.

To understand why this is so, let's be clear about what "inconsistent set of sentences" means and what "valid argument" means in this context. To say that a set of sentences is inconsistent is to say (roughly) that it is not possible that all the sentences in that set are true: they could all be false, some could be true and some false, but there is no way that all could be true. To say that an argument is valid is to say that it's not possible for all its premises to be true and its conclusion at the same time to be false; sometimes people express this by saying that the truth of the premises forces the truth of the conclusion.

But now think about it: If you have an argument whose premises are inconsistent, then it's certainly not possible that all its premises be true and its conclusion false - since it's already not possible for all its premises to be true! Hence, any argument whose premises are inconsistent is (trivially) a valid argument.

Without disagreeing with anything Alex has said, let me just add one more thing: There are logicians who sympathize with this sort of question, and so who would deny that an argument with inconsistent premises is always valid. There are logics, that is to say, that do NOT validate all inferences of the form: A & ~A, therefore B, for arbitrary B. Such logics are called "paraconsistent, and if you'd like to read about them I'd recommend the Stanford Encyclopedia article as a start.

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