Difference of these two First Order Logic statements

Question

1) $(\forall x)(\exists y)x{\le}y$

2) $(\exists y)(\forall x)x{\le}y$

Assume that the domain of the variable is $D={0,1,2,...,99}$

These two statements says two things in natural language. I just cannot distinguish two translations. Can some one help me?

Answer 1

"$\forall x\ ( P(x) )$" means "For any $x$, it is true that $P(x)$", equivalently "$P(x)$ is true for every $x$". If you claim this statement, then you're effectively claiming that:

No matter what $x$ I give you, you can show me that $P(x)$ is true.

"$\exists x\ ( P(x) )$" means "For some $x$, it is true that $P(x)$", equivalently "$P(x)$ is true for at least one $x$". If you claim this statement, then you're effectively claiming that:

You can give me some $x$ and then show me that $P(x)$ is true.

What you have asked about is better written in the form:

$\forall x\ ( \exists y\ ( P(x,y) ) )$.

$\exists y\ ( \forall x\ ( P(x,y) ) )$.

It should be clear from my definition that they say different things. You should slowly expand the quantifiers one by one to see why. For example the first one expands to:

For any $x$, it is true that $\exists y\ ( P(x,y) )$.

and then:

For any $x$, it is true that ( for some $y$, it is true that $P(x,y)$ ).

which if you claim it means that:

No matter what $x$ I give you, you can give me some $y$ and then show me that $P(x,y)$ is true.

Note that the $y$ that you give me is after I give you the $x$, so you are free to choose a different $y$ if I give you a different $x$.

In contrast the second expands to:

For some $y$, it is true that $\forall x\ ( P(x,y) )$.

and then:

For some $y$, it is true that ( for every $x$, it is true that $P(x,y)$ ).

which if you claim it means that:

You can give me some $y$ and then show me that, no matter what $x$ I give you, you can show me that $P(x,y)$ is true.

Note that the $y$ that you give me is before I give you the $x$, so that single $y$ must work for every $x$ that I give you.