By the way, Michael Hong, until quite recently a lot of logicians believed this axiom:

\[ \forall x \, P(x) \implies \exists x \, P(x) \]

This says

If every \\(x\\) has property \\(P\\) then some \\(x\\) has property \\(P\\).

But we know realizes this axiom is bad because it fails in the cases where the set of \\(x\\)'s is empty! This is the tricky case that was confusing you. If I'm a bachelor, _everyone who is my wife is a millionaire_ does not imply _there exists a wife of mine who is a millionaire_.

So, you're not the only one who was confused by this. It turns out that dropping the above axiom, which allows the empty set of \\(x\\)'s to work just as well as any other set, is crucial to making the rules of logic work smoothly.