Let us consider the statement $\exists x P(x)$ - translated into English, "there exists an $x$ in our universe of discourse such that $P(x)$ is true." In writing the negation of this, we are taught to switch quantifiers ($\exists \leftrightarrow \forall$) and to negate that statement $P(x)$.
$$\neg (\exists x P(x)) = \forall x (\neg P(x))$$
However, humor me for a second. Let's consider what negation is - it is the "logical complement." The negation of a statement is always false when the statement is true, and vice versa. In that light, why would we not say the following is also the negation?
$$\neg (\exists x P(x)) = \not \exists x P(x)$$
Or, taking this a bit further, why would we not write this as well?
$$\forall x (\neg P(x)) = \not \exists x P(x)$$
Both seem to imply the same thing: there does not exist an $x$ such that $P(x)$ is true (and thus for all $x$, $P(x)$ is false, or, rather, $\neg P(x)$ is true).
So is there some underlying reason why we don't do negations in this way? As far as I can tell, they mean the same thing, yet I always have seen the $\forall$ version as above. Looking around MSE, I've only seen some posts which have $\neg \exists$ (basically the same as $\not \exists$), but they're only in the context of simplifying a logical expression.
So I guess, if indeed these are logically equivalent, my follow-up question would be - why is $\forall$ considered a simplification of $\neg \exists$ or $\not \exists$?
My only guess is that "$\not \exists$" isn't a standard notation, or so I recall from some notes my complex analysis professor gave us last semester. Or perhaps to say "for all $x$, this is false" more immediately is understood (or a more "direct" way of saying it) than "there does not exist $x$ such this is true?"
(Footnote of note: I haven't had much education in predicate logic and such. We went over it for a little while in one of my classes so I understand some basics like the above but we never went into much detail. So I apologize if this question is poorly framed or worded.)