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Is an argument whose premises logically cannot be true at the same time a valid one? It seems it would be "vacuously" valid but I'm not sure about the precise definitions.

Ie, is the following a valid argument:

P1. Dogs are cats.
P2. Dogs are not cats.
C. Cats are blue.

Or symbolically:

P1. p
P2. ¬p
C. q

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    Yes. If we define an argument to be valid just in case it's impossible to make its premises true and its conclusion false, you can see that since it's impossible to make p and ¬p both true, it will "vacuously" be impossible to make all premises of an argument that contains p and ¬p true and its conclusion false, whatever that conclusion may be. That's the standard story. – Hunan Rostomyan Apr 18 '14 at 4:25
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An argument is deductively valid if and only if it's impossible for all its premises to be true and its conclusion to be false at the same time. If it's impossible for its premises to be true at the same time, then that is itself sufficient to meet that definition, and make the argument valid, independent of the truth-value of the conclusion.

Whether this state of affairs should count as a feature or a bug of the standard predicate calculus depends on one's other interests in logic. It depends on what deductive logic is for or what it's supposed to capture. It depends on whether you think there is a single true account of logical consequence or whether different accounts of logical consequence—including, for instance, accounts which would prohibit everything being derivable from a contradiction—might be appropriate in different circumstances. (See the final section of this SEP article on logical consequence for discussion.)

One way to think of it as a feature is to think of the predicate calculus as having hard-coded a prejudice against contradictions. Once you accept as true two sentences which deny one another, you're in la-la land, it implicitly suggests. You might think of the system itself as expressing in its assumptions: “if that can be true, anything goes, and all bets are off. Truth preservation no longer works.” And there's a good rationale for this prejudice. Socrates' arguments set up a tradition in western philosophy of seeing refutation by exposing a contradiction as a gold standard for argumentation against any position. That anything follows from a contradiction is simply a reflection of the absolute inadmissibility of contradictions into a reasonable set of beliefs. But of there are other views about contradictions, and whether they are ever reasonable to accept. See the SEP articles on contradiction and dialetheism for more.

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Yes it is valid. The definition of validity is that, an argument is valid if and only if its conclusion is true if all its premises are true. In logical explanation, V <=> (P1 ^ P2 ^ ... ^ P3 => C). As P1 ^ P2 ^ ... ^ P3 is false, then no matter C is, the right hand side of the iff is true, then the left hand side should be true as well.

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