Using $\forall$ and $\exists$ with multiple variables. Does order matter?

Question

I am trying to understand what $\forall$ and $\exists$ mean. Please consider the following expression. $$ \forall x \, \exists y \, \forall z \, \left( z = \frac{x-y}{3} \right) $$ I claim that the above statement is false because when $x = 2$ is selected and $y = 0$ is selected we have: $$ \forall z \, \left( z = \frac{2}{3} \right) $$ which is not true for all $z$ Also consider the following expression. $$ \forall x \, \forall z \, \exists y \, \left( z = \frac{x-y}{3} \right) $$ I claim this is true because we can always set $y = x - 3z$. It appears to me that the order of the variables does matter. That is, $\forall x \, \exists y \, \forall z$ has a different meaning than $ \forall x \, \forall z \, \exists y \, $. Am I right?

Answer 1

One way to think of it is as a game where if you assert a sentence is true, I get to pick all the variables that have $\forall$ in an effort to make the sentence false and you get to choose the $\exists$ variables in an effort to make the sentence true. We pick in the order the quantifiers come. In your example, if the quantifiers are $\forall x \exists y \forall z$, I pick $x$, then you pick $y$ and finally I pick $z$. For your sentence, it is easy to see I can always pick a $z$ that makes it false. If the quantifiers are $\forall x \forall z \exists y$ I have to pick $x$ and $z$ before you pick $y$. You can rearrange the sentence to $y=x-3z$, so you can make it true no matter which $x,z$ I pick. The first version is false, the second is true, so clearly the order of quantifiers matters.

Answer 2

Here's a simple way of understanding that quantifier order matters. Let $Lxy$ mean $x$ loves $y$. This is, sadly, not a symmetric relation. Consider the difference between:

• $\forall x\exists y Lxy$
• $\exists y \forall x Lxy$

The first of these says everybody loves somebody, the second of these says there is somebody who is loved by everybody. Clearly different claims! If Hermia and Lysander love each other, and Demetrius and Helena love each other, (and none of them love anyone else) then the first is true and the second is false.

Note that the order of the same quantifier doesn't matter, so if you have a string of universal quantifiers, you can switch them around with impunity.