$P(x,y)$ means that for given $x$ and $y$, the property $P(x,y)$ is true
a.(∃x∀yP(x,y))⇒(∀y∃xP(x,y))
b.(∀x∃yP(x,y))⇒(∃y∀xP(x,y))
Please someone explain which one is true?
I am confused between these two. It seems to me both are correct. Please clarify.
a. is valid, but b. is not.
Just a simple example to illustrate why a. is valid:
Suppose that we are talking about persons, and that $P(x,y)$ means ''x likes y'.
Then $\exists x \forall y P(x,y)$ means that there is a really friendly person: someone who likes everyone! .... let's call this really friendly person 'Bob'.
Now, $\forall y \exists x P(x,y)$ means that everyone is liked by at least one person. Is that true? Yes! Everyone is liked by Bob!
This is of course one possible interpretation, but if you think about, the logic works the same for any interpretation of $P(x,y)$. So, a. is valid.
b. on the other is not valid. Let's take the same interpretation of $P(x,y)$:
$\forall x \exists y P(x,y)$ means that everyone likes at least one person (possibly themselves). OK, does that mean that $\exists y \forall x P(x,y)$, i.e. that there is a super likable person: a person liked by everyone?! No, maybe all the people are self-centered, and they only like themselves. Assuming there is more than 1 person, then there is no one liked by everyone. So, b. is invalid.