TITLE: What's the point of allowing only quantification of variables in first-order logic.
QUESTION [11 upvotes]: In first-order languages, ${\forall}$ is allowed to quantify only over variables, so that ${\forall}v(P)$, where $v$ is some variable and $P$ is a WFF is the only kind of a WFF concering universal quantifiers allowed in such languages. 
Why is that? No texts I read on this subject (which due to the limitations I have due to the place I live have only been sites like Wikipedia, 
Proofwiki or Wolfram MathWorld) provided an explanation why quantifying over WFFs is not allowed. I don't see any advantages to this and it leads to annoying problems like ZFC being an infinite list of axioms, which don't avoid the concept of ${\forall}P(Q)$, $P$ and $Q$ being WFFs, just phrase it in another way.
So what's the justification of not quantifying over WFFs? What are the advantages of such approach?

REPLY [2 votes]: You can actually have formulas like $\forall P. \textbf{Apply}(P,x)$ (where $\textbf{Apply}$ is your predicate symbol that you want to denote the application of the predicate $P$ to $x$) in a first order logic with two type of variables. The problem is there can be nothing in the rules implying that the variable $P$ must range on fbf's, strings, sets or numbers: it will depend on your model.