Negating statements with quantifiers

Question

I understand that when we want to negate a statement with universal quantifier, that quantifier changes to existential quantifier, and vice versa. For example,

negation of $(\exists x\in\Bbb N)(x+1=8)$ would be $(\forall x\in\Bbb N)(x+1\neq 8)$ and the negation of $(\forall x\in\Bbb N)(x<8)$ would be ($\exists x\in\Bbb N)x\geq 8$.

However, for some reason I don't understand how negating statements with both universal and existential quantifiers works. For instance,

$(\forall x\in\Bbb R)(\exists y\in\Bbb R) x^{2}+y^{2}\geq 4$ should be negated as $(\exists x\in\Bbb R)(\forall y\in\Bbb R) x^{2}+y^{2}<4$. $(*)$

I'm not even sure what I don't understand, I know that

$\neg(\forall x \in\Bbb R)=(\exists x\in\Bbb R)$

$\neg(\exists y\in\Bbb R)=(\forall y\in\Bbb R)$

$\neg(x^{2}+y^{2}\geq 4)=x^{2}+y^{2}<4$

I guess that when reading those statements $(*)$, I don't understand how one negates the other.

Answer 1

When you negate a quantifier, you 'bring the negation inside', e.g. $\neg \forall x P(x)$ is equivalent to $\exists x \: \neg P(x)$, where P(x) is some claim about $x$.

If you have two quantifiers, that still works the same way, e.g. $\neg \forall x \exists y P(x,y)$ is equivalent to $\exists x \neg \exists y P(x,y)$, which in turn is equivalent to $\exists x \forall y \neg P(x,y)$. And once you see that, you can understand that you can move a negation through a series of any number of quantifiers, as long as you change the quantifier: each $\forall$ becomes a $\exists$ and vice versa.

Also, since these are all equivalences, you can also bring negations outside, if that's what you ever wanted to, again as long as you change each quantifier that you move the negation through. For this reason, this is sometimes called the 'dagger rule': you can 'stab' a dagger (the negation) all the way through a quantifier, thereby changing the quantifier.