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First-Order Logic: normal forms revisited

Module by: Ian Barland, John Greiner, Phokion Kolaitis, Moshe Vardi, Matthias Felleisen

Summary: Using CNF and DNF with first-order-logic.

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CNF and DNF revisited (Optional)

In first-order logic, normal forms are still useful for providing a notion of a canonical form. However, their other benefit of corresponding closely to truth tables does not apply here, since truth tables aren't useful for first-order logic.

A formula in Prenex Conjunctive Normal Form, or Prenex CNF, has a body in CNF preceded by a series of quantifiers. Similarly, a formula in Prenex Disjunctive Normal Form, or Prenex DNF, has a body in DNF preceded by a series of quantifiers.

Example 1

Assuming φφ is in CNF, then the following are each in prenex CNF. On the other hand, if φφ is in DNF, these are in prenex DNF.

  • φφ
  • ∀x. φ∀x.φ
  • ∃x. ∀y. ∃z. φ∃x.∀y.∃z.φ

Every formula has an equivalent prenex CNF formula and equivalent prenex CNF formula. For brevity, we'll skip proving this.

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