TITLE: Why is $x$ free but $\forall x$ is not?
QUESTION [6 upvotes]: For example $F(x)$ is considered open because $x$ can be anything but then $\forall x F(x)$ is considered bound even though to my eye "can be anything" and "for all" seem kind of like saying the same thing.
What's the difference / why does it matter?

REPLY [4 votes]: True or false?
$$\forall x. x+1 > x$$
"True, of course. There is no counter example."
True or false? 
$$\forall x. x\ge 0$$
"False, of course. $x$ could be $-17$, which is a counter example."
True or false?
$$x\ge 0$$
"How should I know? You didn't say what $x$ is."