# Is a vacuously true argument a valid argument? [duplicate]

From what I know, given some argument, the argument is valid when it has true premises that lead to a true conclusion. Now, what if the premises were false? I mean, the conclusion would be vacuously true, but does that make the argument valid? It sounds rather odd, since taking the argument to be valid is essentially accepting some ridiculous statement. But I don't know, I'm just a bit confused.

## 2 Answers

the argument is valid when it has true premises that lead to a true conclusion

Not quite: An argument is valid iff every way of making all the premises true also makes the conclusion true, or equivalently, there is no situation in which all the premises are true but the conclusion is false.
Sentences may have different truth values in different situations, and not all the premises need to be true in the real world in order for the argument to be valid. Instead, to check the validity of an argument, we need to check every combination of truth values the sentences in the argument can take, and verify that if all premises are true in this situation, then conclusion is true in that situation as well (if not all premsies are true, then the conclusion may or may not be true). Situations which render at least one of the premises false count positive towards the validity of the argument, and if there is no situation at all that can make all the premises true at the same time, then trivially there is no situation that makes all the premises true but the conclusion false, and hence, the argument is (vacuously) valid. So the answer to your question

Is a vacuously true argument a valid argument?

is yes.

What you may be looking for is the notion of a sound argument: An argument is sound iff it is valid and in addition the premises are true in the actual world.

The argument

Vancouver is in Canada and Vancouver is not in Canada
∴ Pigs can fly

is vacuously valid, because the premise is in itself contradictory, and thus there trivially can be no way to make the premise true but the conclusion false. It is not sound, because the premise is false in every situation and thus also in the real world.

The argument

Pigs are animals
Animals can fly
∴ Pigs can fly

is also valid, but not vacuously. The premises are not contradictory -- in terms of the logical structure of the sentence, there is nothing that prevents "Animals can fly" to hypothetically become true under an appropriate interpretation of the predicates; but every situation that makes both premises true will necessarily also make the conclusion true, so the argument is valid. However, the argument is, again, not sound, because the second premise is false in the real world.

An argument is valid if its form guarantees that the conclusion is true upon true premises; that does not prevent you from instantiating it with false premises, of course. So the validity of your argument does not (per se) guarantee the truth of your conclusion, which is bound to be no more plausible than the premises are.

Edit: Upon re-reading, I am unsure whether I got your question right. But surely, cases containing false premises are irrelevant to assessing the validity of the argument at hand, regardless of whether the conclusion is true or not.