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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.

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For your first $\exists x [P(x) \rightarrow Q(x)]$ is true if there is some $x$ for which $P(x)$ is false. It is still true if $P(x)$ is always false. Then there is not a $P$, so there cannot be a $P$ that is $Q$. Similarly for the second $\forall x [P(x) \land Q(x)]$ requires that $P(x)$ and $Q(x)$ are true for all $x$. You only want to require $Q(x)$ when $P(x)$ is true. If our universe is all naturals and $P(x)$ is $x=1$ while $Q(x)$ is $x \lt 10$ it is true that all $P$ are $Q$, but your translation is false because $x$ could be $11$

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