Free variable

A free variable is a variable occurrence not governed by a quantifier for that variable. Its interpretation depends on an assignment supplied from outside the expression.

Bound variable: A variable occurrence governed by a quantifier or other binding construct. It contrasts with a free occurrence because a binder fixes its scope.
Free-variable function: A function that returns the set of variables occurring free in a formula or term. It formalizes which variables remain unbound in an expression.
First-order logic: A formal system whose formulas quantify over objects and express relations among them. Its formulas use free variables to express claims relative to assignments.
Binding operation: A syntactic construct that introduces a variable and governs specified occurrences within its scope. Quantification is one instance of the broader pattern that defines bound occurrences.
Variable scope: The part of an expression in which a variable binding applies. Scope determines which occurrences a quantifier governs.
Capture-avoiding substitution: Substitution performed so inserted variables do not become bound unintentionally. Replacing a free variable can cause capture unless binders are handled carefully.
Model theory: The study of mathematical structures and the formulas true in them. Satisfaction for formulas with free variables is defined relative to assignments.
Lambda abstraction: A lambda-calculus expression that defines a function by binding a parameter in a body. Its body may contain free variables not bound by the abstraction's parameter.
Quantifier: An operator that specifies how many objects in a domain satisfy a predicate. Quantifiers bind variables within their scope.
Alpha conversion: Renaming a bound variable without changing the meaning of a lambda expression or formula. Renaming binders can prevent them from capturing free variables.

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