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 ?
 A: You're not quite right on the first one. Here's a transition from "predicate English" to "proper English":


*

*For all $x$, there exists a $y$, such that "x is an American" implies ("y is a dream" and "x has y")

*For all $x$, if $x$ is an American then there exists a $y$ such that $y$ is a dream and $x$ has $y$.

*For all Americans $x$, there is a dream $y$ that $x$ has.

*Every American has a dream.


As for the second, here's what it looks like at the start:


*

*For all $x$, there exists a $y$, such that ("x is an American") and ("y is a dream") implies "x has y".

*For all $x$, there exists a $y$, such that if $x$ is an American and $y$ is a dream, then $x$ has $y$.


This is where it gets weird. Because it's saying that for every $x$ there's a $y$ that makes the inner part true. But the inner part can be true if:


*

*$x$ is not American.

*$y$ is not a dream.

*$x$ has $y$.


In particular, it says that for every American $x$, there's something $y$ that, if $y$ is a dream, then $x$ has. But it's fine if it's just not a dream, then it doesn't matter if $x$ has it or not. If we use the equivalence of $A \rightarrow B$ and $\lnot(A \land \lnot B)$, then we get something like:


*

*For every American $x$, there is something $y$ that, if $x$ doesn't have it, isn't a dream.


And that's incredibly hard to put into "real" English. Something like:


*

*For every American there's something that, if it's a dream, they have it.


And that's about as good as I can get.
A: 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 $(\neg A(x) \vee \neg D(y) \vee H(x,y))$.
A: What ∀x∃y(A(x)∧D(y)→H(x,y)) literally means in English, according to your given information is that for all Americans, there exists a dream y such that if every American x has at least one dream y, then they have their own dreams. In other words, every American with a dream sets has their own set of dreams.
