Why is the negation of the statement $\exists x P(x)$ given by $\forall x (\neg P(x))$ and not $\not \exists x P(x)$?

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 $$P(x)$$.

Thus,

$$\neg (\exists x P(x)) = \forall x (\neg P(x))$$

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.)

• The idea of "there exists no $x$ such that $P(x)$ is true" is equivalent to "for all $x$, $P(x)$ is not true". I think it is just easier to work with something like "for all $x$, $P(x)$ is not true" rather than saying "there is no $x$ such that $P(x)$ is true". – Dave Jan 1 at 0:11
• Okay, that resolves one doubt of mine regarding their equivalence. Though it then raises my follow up question - why is the latter phrasing favored? – Eevee Trainer Jan 1 at 0:12
• I edited my comment to say that I think the "for all" statement is easier to work with than "there exists no..". I'm not entirely sure why one is used over the other, but in fact I know some people who often phrase it as "there exists no...". – Dave Jan 1 at 0:14

The two statements are equivalent, the reason why we write it that way it is because it is easier to deal/prove it. It is irrelevant which way it is easier to say it, the important thing is being able to use.

Just consider the statement $$\exists x \in \mathbb Z, x^2+(x+1)^2 \mbox{ is even}$$. This is not true, now try to show that this is false.

Try to prove separately each of the following two statements:

• $$\not\exists x \in \mathbb Z, x^2+(x+1)^2 \mbox{ is even}$$
• $$\forall x \in \mathbb Z, x^2+(x+1)^2 \mbox{ is odd}$$

It is easier to deal with the second one.

• Ahhhh, I see, I can't believe I looked over something so basic. Thanks for your insight, I appreciate it. ^_^ – Eevee Trainer Jan 1 at 0:20
• Good answer, +1. This is basically what I was trying to say with the "for all" statements being easier to work with. – Dave Jan 1 at 0:26

Which form to consider simpler is basically a matter of taste and convention.

There are some accounts of predicate logic that consider $$\exists$$ the only primitive quantifier and treat $$\forall x\,\varphi$$ as an abbreviation for $$\neg\exists x\neg\varphi$$.

My impression is that this is something of a minority option these days, but it is not wrong as such.