*NOT* [quantifier]. Why is *NOT* [for every] equivalent to [there is]? Given a proposition such as, "For every real number $x \ge 2$, $x^2 + x - 6 \ge 0$", I am told that the negation, "NOT [For every real number $x \ge 2$, $x^2 + x - 6 \ge 0$]", would be "There is a real number $x \ge 2$ such that $x^2 + x - 6 < 0$".
I am specifically confused with regards to why NOT [for every] is equivalent to [there is] rather than [for none]? It seems logical to me that the negation of everything (for all) should actually be equivalent to nothing?
I would greatly appreciate it if someone could please take the time to clarify this concept.

As an addendum to Casper's answer: there are logics in which $\neg (\forall x) P(x)$ is not equivalent to $(\exists x) [\neg P(x)]$.

For example (and you might like to look up "constructive mathematics" for more on this), take $(\exists x)$ to mean "there is $x$ and moreover we can in principle compute an explicit such $x$", rather than the usual "there is $x$". Let $P(x)$ (where $x$ is restricted to be an integer between $0$ and $9$) be the statement that the digit $x$ appears only finitely often in the decimal expansion of $\pi$.
Then the statement $\neg (\forall x) P(x)$ is certainly true - if not, $\pi$ would have a finite decimal expansion. But $(\exists x) [\neg P(x)]$ is not currently known to be true, because we don't know of any digit which definitely appears infinitely often in $\pi$'s decimal expansion.
