Order of three or more quantifiers This is my understanding of meaning of order of two quantifiers:
$\forall  x \forall y$ is true if the scope of quantifiers is true no matter which values from universe for x and y you choose.
$\forall x \exists y$ is true if any value from universe can be combined with at least one element to make the scope of quantifier true.
$\exists x \forall y$ is true if there is at least one element from universe which outputs true when combined with any element.
$\exists x \exists y$ is true if there is at least one combination of elements which will output true.
But what I don't understand is what things like $\forall x \exists y \forall z$ mean. How to read this and how to know for which values (in what relation to other variables, like the difference in relation of $x$ and $y$ in $\forall x \exists y$ and $\exists x \forall y$) $x$, $y$, $z$, ... are true.
In other words, how does order and meaning of quantifiers affect truth value of wffs with 3+ quantifiers?
 A: You shouldn't really think of "three quantifiers" as a separate case from "two quantifiers". Think of it one quantifier at a time.
$\exists x \varphi(x)$ means that there is some $x$ that makes $\varphi(x)$ true.
$\forall y \exists x \varphi(x,y)$ means that no matter which $y$ you pick, $\exists x\varphi(x,y)$ is true; so no matter which $y$ you pick, there is some $x$ that makes $\varphi(x,y)$ true.
$\exists z\forall y\exists x\varphi(x,y,z)$ means that there is some particular $z$ so that the statement $\forall y\exists x\varphi(x,y,z)$ is true. So there is some particular $z$ so that no matter which $y$ you pick, there is some $x$ that makes $\varphi(x,y,z)$ true.
You can tell what depends on other things based on the order: in this last example, $y$ depends on nothing, but the choice of $x$ may depend on both $y$ and $z$. As a general rule, variables bound by $\forall$ quantifiers don't depend on anything, because they aren't being "chosen" (you're talking about all possibilities for them). Variables bound by $\exists$ quantifiers depend on all variables that preceded them.
I like to think of heavily-quantified statements as games, played between me and an opponent. I'm trying to make my statement true, while my opponent is trying to make it false. Whenever we see an $\exists$ quantifier, I get to pick the value of the variable; whenever we see an $\forall$, my opponent gets to pick. The statement is true if I have a winning strategy - that is, if there's a way for me to win no matter how my opponent plays.
For example, say we're looking at the sentence $\forall x\exists y\forall z(z \leq x \vee y \leq z)$, and all quantifiers are over the set of natural numbers $\mathbb{N}$. This statement is true, and here's why: first, my opponent gets to pick $x$. Whatever $x$ he picked, I'll choose $y = x + 1$. Now, no matter which $z$ he picks, $z$ will be either at most $x$ or at least $x + 1$ (because there aren't any numbers between $x$ and $x + 1$). Because I can always win this way, the original statement is true. And because we saw why this is a winning strategy, you can also see what the original statement was "really" saying: "for every $x$, there is a $y$ so that nothing is bigger than $x$ but smaller than $y$"; or, put more succinctly, "every number has a next number".
A: $$\forall x \; \exists y \;\forall z $$
means that for each element $x $, there exist an element $y $ which depends on $x $ , such that for all the elements $z $ ....
If we write $$\forall x \;\forall z\; \exists y ,$$ it means that $y $ depends on both $x $ and $z $.
