Logic predicate syntax

Question

$ \forall x \exists y F \implies \exists y \forall x F$

My assumption is that F is a predicate F(x, y), so swapping $\forall x$ and $\exists y$ doesn't change the formula, so it's a tautology.

Please correct me if I'm wrong!

Answer 1

I'll correct you!

Consider: $F = P(x,y)$ where $P(x,y)$: '$y$ is the parent of $x$' and domain is people.

Everyone has a parent, sure, but is there a single person that is the parent of everyone? No.

Another example:

$F = x<y$ with domain numbers

For every number we can find a greater number, yes, but is there a number greater than all others? No.