Negation normal form

Negation normal form is an elementary canonical form in mathematical logic. There are similar requirements for negation normal form in different logic fragments.

In predicate logic, a logical formula is in negation normal form if negation occurs only immediately above elementary propositions, and {$\lnot, \lor,\land$} are the only allowed Boolean connectives. In classical logic each formula can be brought into this form by replacing implications and equivalences by their definitions, using De Morgan's laws to push negation inside, and eliminating double negations. This process can be represented using the following rewrite rules:

$\lnot (\forall x. G) \to \exists x. \lnot G$

$\lnot (\exists x. G) \to \forall x. \lnot G$

$\lnot \lnot G \to G$

$\lnot (G_1 \land G_2) \to (\lnot G_1) \lor (\lnot G_2)$

$\lnot (G_1 \lor G_2) \to (\lnot G_1) \land (\lnot G_2)$

A formula in negation normal form can be put into the stronger conjunctive normal form or disjunctive normal form by applying the distributivity laws.

External links

• Java applet for converting logical formula to Negation Normal Form, showing laws used