PrenexForm

Prenex form is a standardized way of writing formulas in first‑order logic in which all quantifiers are placed at the front of the expression, followed by a quantifier‑free matrix. A formula in prenex form has the shape Q₁x₁ Q₂x₂ … Qₙxₙ M, where each Qᵢ is either ∀ or ∃, the variables xᵢ are distinct, and M is a quantifier‑free combination of atomic predicates and logical connectives.

The conversion of an arbitrary formula to prenex form proceeds by eliminating implications, moving negations inward using De Morgan’s laws, and repeatedly applying quantifier‑shifting equivalences such as ∀x (P ∧ Q) ≡ (∀x P) ∧ Q when x does not occur free in Q. Existential quantifiers are handled analogously. Renaming bound variables avoids accidental capture during the transformation.

Prenex form is useful because it isolates the logical structure of quantification from the propositional content, facilitating meta‑theoretical analysis, automated theorem proving, and transformations to other normal forms. In particular, Skolem normal form is obtained from a prenex formula by eliminating existential quantifiers through Skolemization, after which the formula consists solely of universal quantifiers followed by a quantifier‑free part.

Not all logical systems admit a prenex form; the definition relies on classical first‑order logic with standard quantifier rules. Nevertheless, most textbooks on logic present prenex normal form as a fundamental tool for studying decidability, completeness, and proof‑search strategies.