Let $E/F$ be an algebraic extension.
Then, $E/F$ is normal iff $E$ is a splitting field of a family of polynomials in $F[X]$.
So does this mean that if $E$ is a splliting field of a given one family of polynomials in $F[X]$? Or splitting field for every family of polynomials?
( That is, $\exists$-family or $\forall$-family?)
The answer is "$\exists$". $E$ need not be a splitting field of all families of polynomials (that would be algebraic closure), nor need it be all splitting fields of a family of polynomials.
For example $E:=\mathbb Q[\sqrt 2]$ is (a, or upto isomorphism the) splitting field for the one-element family of polynomials $X^2-2$ over $F:=\mathbb Q$. On the other hand, $E:=\mathbb Q[\sqrt[3]2]$ is not a spliting field (over $F:=\mathbb Q$): The obvious candidate polynomial $X^3-2$ splits only into $(X-\sqrt[3]2)(X^2+\sqrt[3]2 X+\sqrt[3]4)$, not completely.