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What is the definition of validity under possible world semantics?

Please excuse my parentheses; I hope they don't obscure my answer. As I understand it, an inference is (deductively) valid if and only if there's no possible world in (or at) which the premise(s) is (are) true and the conclusion is false. So "Socrates is a man; therefore, there's at least one man" is a deductively valid inference, since there's no possible world in (or at) which the premise is true and the conclusion is false. Ditto for the inference "Socrates exists; Socrates doesn't exist; therefore, snow is green": Barring equivocation, there's no possible world in (or at) which the premises are both true, and so there's no possible world in (or at) which the premises are both true and the conclusion is false.

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