Predicate Logic · step 7 of 13

Prenex normal form

A formula is in prenex normal form when all quantifiers sit at the front, followed by a quantifier-free matrix: Q₁x₁ … Qₙxₙ M. Every first-order formula has a prenex equivalent. The form matters because it exposes the quantifier alternation pattern (∀∃∀ …), which drives Skolemisation, decidability classifications and complexity results.

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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.

Remove → and ↔ from the matrixRewrite implications as ¬ ∨, which makes the movement rules apply uniformly.
Rename bound variables apartGive every quantifier a distinct variable. This step is not optional — pulling quantifiers out without it causes capture.
Push negations through quantifiers¬∀x A becomes ∃x ¬A and ¬∃x A becomes ∀x ¬A, until negations sit on atoms.
Pull quantifiers to the frontSince x is not free in B, (∀x A) ∧ B ≡ ∀x (A ∧ B), and similarly for ∨ and ∃. Move them out one at a time, preserving relative order.

Worked example

Convert ∀x P(x) → ∃x Q(x) to prenex normal form.

Rename apart: ∀x P(x) → ∃y Q(y).
Remove the implication: ¬∀x P(x) ∨ ∃y Q(y).
Push the negation in: ∃x ¬P(x) ∨ ∃y Q(y).
Pull both quantifiers out: ∃x ∃y (¬P(x) ∨ Q(y)).

Answer. ∃x ∃y (¬P(x) ∨ Q(y)).

Where marks get dropped

These are the specific errors that cost credit on prenex normal form questions — QED's rubric penalises each of them separately.

Pulling a quantifier out over a subformula in which its variable occurs free elsewhere — this is capture, and it changes the meaning.
Forgetting that a quantifier in the antecedent of an implication flips when the implication is expanded. ∀x in the antecedent becomes ∃x at the front.
Assuming the prenex form is unique. Different legal orders of extraction give different, equally correct prenex formulas.

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Prenex normal form — frequently asked questions

Is prenex form unique?

No. When quantifiers can be pulled out in more than one order, several prenex forms are equivalent to the original. Any correct one is acceptable unless the question fixes an order.

Why does the antecedent flip the quantifier?

Because A → B is ¬A ∨ B, and the negation on A turns ∀ into ∃ before extraction. This is the single most common source of errors in prenex questions.

What comes after prenex form?

Usually Skolemisation: existentials are replaced by functions of the preceding universals, giving a universally quantified clause set suitable for resolution.

The rest of Predicate Logic

Quantifiers, predicates, binding, and validity. Each subtopic below has its own method, worked example and mark-losing traps.

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