What is the correct usage of conditionals in universal and existential quantifiers?

Question

Recently I were tasked to translate the sentence "All humans has a father." into predicate logic. I gave this answer:

H__ : __ is a human
F__ __ : __ is the father of __

∀x∃y(Hx∧Fyx)

Which apparently is wrong, as the correct answer is:

∃x(Hx->Fyx)

This is clearly right, but I fail to see how my answer is incorrect. What would the translation of my answer into English be?

Edit: The "correct" answer I gave is wrong, but I will leave it in so that the replies to my question makes sense.

Answer 1

Your answer would translate as:

For all objects $x$ there exists an object $y$ such that $x$ is human and $y$ their father.

You may note that this implies in particular that all objects are human, which makes the claim obviously wrong.

Btw, $\forall x(Hx\to Fyx)$ is also not correct. It should be $\forall x(Hx\to \exists y(Fyx))$

As a general rule, remember $$ \forall x\in A\;\phi(x)\iff \forall x\,(x\in A\to \phi(x))$$ and $$ \exists x\in A\;\phi(x)\iff \exists x\,(x\in A\land \phi(x))$$

Answer 2

Your answer says "For every thing A, there is a thing B such that A is human and B is A's father." In particular, it implies "Every thing is human" (this is the "$\forall xHx$" part of the sentence you write).

But that's of course nonsense. A rock is not human, and moreover has no father. What you really want to say is:

Anything that is human, has a father.

The most natural way to write this would be

$\forall x(Hx\implies \exists y Fyx)$,

which says "For every thing, if it is human, then it has a father" or - in better English - "Every thing which is human has a father."

In general, "Every $[blah]$ has the property $[foo]$" (e.g. $[blah]$=human, [foo]=has a father) is expressed as $$\forall x([blah](x)\implies $[foo]$(x)).$$ Here "$[blah](x)$" is "$Hx$", and "$[foo](x)$" is "$\exists yHxy$".