Translate usual sentence into logical proposition

Question

Let $x$ is the notation of an element of argument domain.

Now $Ax$ and $Mx$ is predicate for the sentence that "$x$ is an American", and "x is a man", respectively.

I want to symbolize the sentence that "Some americans are men"

There are two ways. One is $\exists x (Ax \wedge Mx)$, the another logical syntax is $\exists x(Ax \rightarrow Mx)$.

In my textbook the only first syntax is right symbolization, second is not.

However, when we focus on the second syntax, it should be also true for an element $d \in D $, but $Ad$ is false.(vacously true), and I think it is well done.

Why does it happen?

Answer 1

Here's another way to think about it.

If you want to prove the statement $(\exists x)(Ax \to Mx)$, then you need to find an $x$ such that $Ax \to Mx$ is true. Well, if $Ax$ and $Mx$ are both false (for instance, $x$ is a woman from another country - say, Angela Merkel), then $Ax \to Mx$ is true.

But have you really proven that there is a man in America by pointing out that Angela Merkel is a German woman?

No. So something is wrong with your formulation.

Note, that as a rule, $(\forall x)$ usually has $\to$ in it, and $(\exists x)$ usually has an $\wedge$ in it.

Answer 2

$\exists x(Ax \to Mx)$ is equivalent to $\exists x(\neg Ax \lor Mx)$, and the latter is true if there exists something which isn't $A$.

So imagine a universe there all Americans are women, but there is someone who is not American -- can him Fidel. Then in this universe, it is false that (i) some Americans are men, yet true that (ii) $\exists x(Ax \to Mx)$. Hence (ii) cannot be an adequate translation of (i).