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I think the answer is yes but I'm not quite sure.

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    Yes; P, therefore not-P is invalid. Oct 13 at 12:50
  • Do you mean a valid argument where the premises + the conclusion are inconsistent? This is a bit more interesting; the answer is still yes, but now it needs to be the case that the premises by themselves are already inconsistent (this is essentially what "validity" is about in the first place - a valid argument never introduces new inconsistencies). Oct 13 at 17:09
  • @NoahSchweber "but now it needs to be the case that the premises by themselves are already inconsistent" No. If we have an argument which consists of a single false premise or a collection of only false premises, and a true conclusion, we have a valid argument. The premises are not inconsistent. But, the premises are inconsistent with the conclusion. And validity can introduce new inconsistencies, if we have false premises. Oct 13 at 19:07
  • @DougSpoonwood If the premises of a valid argument are inconsistent with their conclusion then they are already inconsistent - by definition, a set of sentences is inconsistent iff it can be used to derive falsum. I'm not sure what you're getting at. Oct 13 at 19:50
  • @NoahSchweber "by definition, a set of sentences is inconsistent iff it can be used to derive falsum" If that's your definition of inconsistency, I certainly don't agree. A set of sentences is inconsistent if and only if one sentence is the negation of the other. I. E. A set of inconsistent sentences has the form {p, Np} (or {Np, p}, which is the same set). If we only have false premises though, there is no true premise, so we cannot form a set of inconsistent sentences. There is no inconsistency in only having false premises. Such premises are consistently false. Oct 13 at 20:16

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