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 :

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