Michael wrote:

> 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.