> It's kind of like the discovery of zero but for logic.
Exactly! The analogy goes quite deep. I don't want to go to far into it right now, but the fact that "for all \\(x \in S\\), \\(P(x)\\) is true" is automatically true when \\(S\\) is empty is analogous to the fact that the product of a collection of numbers is automatically equal to \\(1\\) when that collection is empty. You may not have thought much about what happens when you multiply an empty collection of numbers, but you know examples, like
$$ 3^0 = 1 $$
which means that if you multiply no 3's at all, you get 1.