TITLE: Intuitive Reason that Quantifier Order Matters
QUESTION [15 upvotes]: Is there some understandable rationale why $\forall x\, \exists y\, P(x,y) \not \equiv \exists y\, \forall x\, P(x,y)$?
I'm looking for a sentence I can explain to students, but I am failing every time I try to come up with one.
Example
Let $P(x,y)$ mean that $x$ is greater than $y$.

$\forall x\, \exists y\, P(x,y)$ means that for all $x$, there is a number $y$, such that $x$ is greater than $y$.
$\exists y\, \forall x\, P(x,y)$ means that there is some $y$, that every number $x$ is greater than.

These don't seem to mean different things to me. Is this perhaps an example where they do mean the same thing or am I just translating to English incorrectly?

REPLY [3 votes]: A well-formed formula in prenex normal form can be interpreted as a game. The classical logical connectives can also be interpreted as a game: $\lor$ is a choice that the player owns and $\lnot$ exchanges the roles of the players.
I think it's less confusing though to just consider statements in prenex normal forms and not assign a game interpretation to connectives, though.
So, we have a game. We're trying to make the statement surrounded by the quantifiers true and the opponent is trying to make it false.
$ \exists x \cdots $ can be read as we choose $x$. Using I choose conveys the same intuition, but is confusing when speaking to people.
$ \forall x \cdots $ can be read as they choose $x$. Using you choose likewise conveys the same intuition, but is confusing when speaking.
$ \exists x \forall y \mathop. P(x, y)$ is intuitively a more difficult game to win than $\forall y \exists x \mathop. P(x, y)$. In the first game, we have to move first and the opponent can react to whatever we did. In the second game, the opponent moves first, so we only need to make a move that beats the move that the opponent happened to pick and we don't need a single strategy that always works.