Well-formed formulas: Quantifiers for predicates and functions, closed format

Question

Is it considered a well-formed formula (or even permissible) to have a quantifier operate on a predicate or function, such as $$ (∃f)f(X,Y) $$ Furthermore, does a well-formed formula require that the expression be a closed formula (contain no free variables)? As an example, $$ (∃X)(∀Y)h(X,Y,Z) $$ Z is a free variable in this instance.

Answer 1

Is it considered a well-formed formula (or even permissible) to have a qualifier operate on a predicate or function?

No for first-order logic, where function symbols and predicate symbols do not refer to objects in the domain (which the quantifiers range over). But yes for higher-order logic. Second-order logic allows quantification over functions and predicates in some formulations, and over subsets of the domain in others. To further quantify over functions/predicates of functions/predicates you go to third-order logic, and so on.

Furthermore, does a well-formed formula require that the expression be a closed formula (contain no free variables)?

No. A well-formed formula can have free variables. A sentence must have no free variables.