Negation normal form

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