Free vs bound variables & scope
An occurrence of a variable is bound if it lies within the scope of a quantifier over that variable, and free otherwise. A formula with no free variables is a sentence and has a definite truth value; a formula with free variables is only true or false relative to an assignment. In ∀x P(x, y), the variable x is bound and y is free — so this expression says something about y, not a fixed fact.
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Method: how to approach it
The order below is what examiners expect to see, and each step carries its own marks.
- Mark each quantifier’s scopeThe scope of ∀x is the immediately following subformula. Bracket it explicitly before analysing anything else.
- Classify occurrence by occurrenceThe same letter can be free in one place and bound in another within one formula. Analyse occurrences, not letters.
- Check whether it is a sentenceNo free occurrences means the formula has a truth value once the structure is fixed. Otherwise it defines a relation.
- Rename before substitutingWhen substituting a term for a free variable, first rename any bound variable that would capture it. α-renaming a bound variable never changes meaning.
Worked example
In ∀x (P(x, y) → ∃y Q(y, z)), classify each variable occurrence.
- The scope of ∀x is the whole implication, so both occurrences of x are bound by it.
- The y in P(x, y) is outside the scope of ∃y, which covers only Q(y, z) — so that y is free.
- The y in Q(y, z) is inside ∃y, so it is bound.
- z is inside no quantifier over z, so it is free.
Answer. x: bound. First y: free. Second y: bound. z: free. The formula is not a sentence — it depends on the values of y and z.
Where marks get dropped
These are the specific errors that cost credit on free vs bound variables & scope questions — QED's rubric penalises each of them separately.
- Calling a variable "bound" because a quantifier over it appears somewhere in the formula. It must appear within that quantifier’s scope.
- Substituting into a formula without renaming, so a free variable becomes accidentally captured — substituting x for y in ∃x R(x, y) gives ∃x R(x, x), which says something different.
- Assigning a truth value to a formula with free variables. It has none until an assignment is supplied.
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Free vs bound variables & scope — frequently asked questions
Does renaming a bound variable change the formula?
No. ∀x P(x) and ∀w P(w) are the same statement, provided w does not clash with anything already in scope. This is exactly why renaming is safe before substitution.
Can a formula have the same variable free and bound?
Yes, and it is legal though bad style. P(x) ∧ ∀x Q(x) has a free first x and a bound second — most courses ask you to rename for clarity.
Why does variable capture matter?
Because it silently changes meaning. Automated provers and compilers implement careful renaming precisely to avoid this bug, and exams test that you spot it.
The rest of Predicate Logic
Quantifiers, predicates, binding, and validity. Each subtopic below has its own method, worked example and mark-losing traps.
- 1Universal & existential quantifiers
- 2Translating English with predicates
- 3Free vs bound variables & scope
- 4Negating quantified statements
- 5Validity & counter-models
- 6Nested quantifiers & quantifier order
- 7Prenex normal form
- 8Interpretations, structures & satisfaction
- 9Equality & uniqueness (∃!)
- 10Natural deduction with quantifier rules
- 11Proving ∀-statements with an arbitrary element
- 12Disproving with a single counterexample
- 13Skolemisation & clausal form
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