How is a closed well-formed formula different from a formula with free variables?

In formal logic, specifically within the predicate calculus, the distinction between a closed well-formed formula (wff) and a formula with free variables lies in whether the individual variables are bound by quantifiers.1

• Free Variables: An occurrence of an individual variable is considered free if it is not bound by a quantifier. In a formula containing free variables, those variables act as placeholders that must be replaced by specific names of individuals to obtain a complete proposition. For example, in the formula $\varphi x$, the variable $x$ is free, requiring a substitution to form a meaningful statement like "Socrates is wise".12
• Closed Wffs: A closed wff is a formula that contains no free individual variables. Because all individual variables are bound by quantifiers, they do not require replacement to form a proposition. Instead, the bound variables indicate the scope of the quantifier, such as in $(\exists x)\varphi x$, where replacing the predicate $\varphi$ with "is wise" yields the complete proposition "Something is wise".1

In short, while free variables represent points requiring replacement, bound variables indicate the logical scope of the quantifier.1