First order logic to English statement? Assume
$A(x) = x$ is an American.
$D(y) = y$ is a dream.
$H(x,y) = x$ has $y$.

Then, Convert below first order logic to English statements :

*

*$∀x ∃y \left ( A(x)\rightarrow D(y) ∧ H(x,y) \right )$

I tried to translate this as "Every American has his own set of dreams".



*

*$∀x ∃y \left ( A(x) ∧ D(y) \rightarrow  H(x,y) \right )$

Not getting how is this pronounced ?
For this, I guess it is like "For all x if x is an American and there exists some y, such that y is a dream then x has y".


How much both of them are correct ?

The first one is pretty close. I would say "Every American has a dream," because we only know that there exists one $y$.
The second one is weird. If there exists any y that is not a dream, then it is vacuously true. Literally it's something like "For every American there is a thing y such that if y is a dream, then the American has that dream." But y could just not be a dream.
Edit: I think the second one is logically equivalent to "Either there exists something that is not a dream, or every American has a dream."
Formally, ($A(x) \wedge D(Y) \implies H(x,y)$) is equivalent to $(\neg(A(x) \wedge D(y)) \vee H(x,y))$, and subsequently $(\neg A(x) \vee \neg D(y) \vee H(x,y))$.
