# In ${\forall}x(P)$, is $P$ any WFF or specifically an open one?

One of the rules of formation for the language of set theory is

If $x$ is a variable and $P$ is a ${\square}$, then ${\forall}x(P)$ is a WFF

The reason I wrote ${\square}$ is that I have heard two versions of this rule: that $P$ is any WFF and that $P$ is an open formula.

So which version is the correct one?

The standard definition of well-formed formula (short: wff) for first-order language is:

• every atomic formula is a wff

• if $$\varphi, \psi$$ are wff, then $$(\lnot \varphi), (\varphi \lor \psi), (\varphi \to \psi), (\varphi \land \psi)$$ are wff

• if $$\varphi$$ is a wff and $$x_i$$ a variable, then $$(\forall x_i \varphi)$$ and $$(\exists x_i \varphi)$$ are wff.

Usual conventions for omitting parentheses are straightforward.

Note In order to "apply" a quantifier to a formula $$\varphi$$, it is not mandatory that the quantified variable is free in $$\varphi$$. (Of course, quantifying a variable that it is not in the scope of the quantifier has no "practical" effects, i.e. it does not affect the "meaning" of the formula.)

See :