Wittgenstein also had problems with it. I'm not sure how well I can explain this, but I'll give it a go.
Crudely, his claim was that you can't talk about things that don't exist.
He says, in typically abstruse form:
I dissociate the concept of all from truth-functions.
Frege and Russell introduced generality in association with logical product or sum. This made it difficult to understand the propositions '$(\exists x).fx$' [there exists $x$ and the proposition $f$ holds for it] and $'(x).fx'$ [all $x$ and the proposition $f$ holds for it].
His difficulties are associated with treating "$\exists x$" and "$\forall x$" as valid statements on their own (which may or not be held true these days). The [brackets] are my attempt to interpret the statements in the way he thought was silly. He goes on to say:
If we are given objects then we are given all objects.
By which he basically means that $\forall$ is redundant and adds nothing to a statement, though he puts it in a very theory laden way. You can always write a statement without it.
But, it seems he would also have a problem with doing this:
$$\forall x : P(x) \rightarrow \nexists x: ¬P(x) $$
or in words, replace "for all $x$, $P$ is true" with "there is no $x$ for which $P$ does not hold". The two statements are mathematically the same (in first order logic). He would, I think, favor
which one might read as "$P$ is true", with the subtext of "for all $x$ that exist, obviously! How could we know if it is true for those things that don't?"