In the appendix about mathematical logic of his book "Analysis 1", Terence Tao explains quantifiers via gaming metaphors. For example he writes
In the first game, the opponent gets to pick what x is, and then you have to prove P(x); if you can always win this game, then you have proven that P(x) is true for all x.
Isn't he confusing "truth" with "provability" here? Because being able to always win this game means that for each x, there is an individual proof of the fact that P(x). But proving “for all x, we have P(x)” is something different than proving “for each x, there is a proof of P(x)”. (To speak about provability of statements, one normally gives an exact definition of what “(first-order) terms”, “well-formed (first-order) formulas” are and then specifies a formal calculus consisting of inference rules and logical axioms, in order to then define what it means for a formula to “be deducible from a set of axioms”. But even if one has done this, one shouldn’t confuse “deducible formulas” with “true formulas”.)
The other gaming metaphors are:
In the second game, you get to choose what x is, and then you prove P(x); if you can win this game, you have proven that P(x) is true for some x.
To continue the gaming metaphor, suppose you play a game where your opponent first picks a positive number x, and then you pick a positive number y. You win the game if y 2 = x. If you can always win the game regardless of what your opponent does, then you have proven that for every positive x, there exists a positive y such that y 2 = x.
He gives us the exercise to think of such gaming metaphors for other statements, for example:
There exists a positive number x such that for every positive number y, we have y^2 = x.
What would be the corresponding gaming metaphor? I would say it’s something like this: I choose a positive number x. Then, my opponent gives me a positive y. I win iff y^2 = x. The statement is true iff there is a fixed positive number x with which I can always win. Is that correct? I somehow don’t really understand the purpose of these gaming metaphors. What is it? In this example, thinking about what the corresponding gaming metaphor could be, doesn’t help me understand the statement more.