The textbook says that to express "There is a $P$ that is $Q$", one can write the following preposition:
$$ \exists x [P(x) \land Q(x)\ ] $$
However, I am unsure why I can't write like this:
$$ \exists x [P(x) \rightarrow Q(x)] $$
For the case of universial quantifier, I have the opposite question. The textbook says "All $P$ is $Q$" can be written as below:
$$ \forall x [P(x) \rightarrow Q(x)] $$
and I am unsure why I can't write like this:
$$ \forall x [P(x) \land Q(x)] $$
I do understand that $P \land Q$ and $P \rightarrow Q$ are different in that $P \rightarrow Q$ also accepts vacuously true. But I am curious why $\forall$ can accept vacously true while $\exists$ cannot.