Assuming that the account is correct, to know that an inference is deductively valid is to know that there are no situations in which the premisses are true and the conclusion is not.
This is an excellent point brought by Priest. Consider the modus ponens:
(A → B) ∧ A ⊢ B
The implication (A → B) ∧ A ⊢ B is obviously true and we don't need to scan the whole universe, or the past and the future, to see that it is true. We only need to look at it and exercise our supreme wits. However, consider now that we may want to apply the modus ponens to real-world situations. Obviously, we understand that the modus ponens will be true of all real-world situations. In fact, it is true even of all imaginary world situations. No worries here.
Now consider the implication A → B which is nestled within the modus ponens. It may be true and it may be false, and we don't know of any a priori why it should be true. Whether it is true or not is not going to affect the truth of the modus ponens, but it remains that we have there, nestled within the modus ponens, an implication which may be true or false. We don't need to know whether it is true or false to decide on the truth of the modus ponens, but we sure need to know whenever we will want to apply the modus ponens to real cases.
When we do that, we are not interested in proving the truth of the modus ponens, we already know it is true. We are only interested in applying it to a concrete situation. However, what we do need to know in this case is whether B is true, and to be able to decide on that, we need to know if the two terms A → B and A are true. Concerning A, this is possibly trivial given that we are supposed to be considering a concrete situation. If A for example means "Trump lost the election", we will probably assume that A is true. In any case, we will not need to scan the entire universe and beyond to decide. We just look at the situation we are considering.
However, the implication A → B is something else entirely. Suppose A means "x is a man". This seems easy enough to handle. For instance, if x is Trump, we obtain "Trump is a man", and we can easily decide that A is true, even those of us who will want to add some disparaging qualifications. And again, no need to scour the entire creation. However, we have a problem, with A → B. If A → B means "If x is a man, then x is mortal", then do we know that? Even if we assume that only the Earth harbours humans, we don't really know if all humans in the past were mortals and if all humans in the future will still be mortal. I guess this is what Priest is after.
To be clear, this isn't a logical problem at all. This is 100% an empirical problem. In fact, the problem already appears with things like "Trump is a man" because we don't actually know that. All we can do is feel confident that this is true, or even just true on balance of probabilities. However, once we admit doing it for "Trump is a man", there is not justification for not also doing it for "If x is a man, then x is mortal". Maybe this is false, but we will nonetheless trust that this is true. And logic does not require that we commit no mistake in this respect. All be need to do is reason logically, trust our perception and our common sense, and hope for the best. This seems to work well. So Priest made a good point, but this is no logical problem. This is yet again the eternal problem that we don't actually know anything about material world.