Editing De Morgan's laws (section)

==Generalising De Morgan duality==
[[File:DeMorgan Logic Circuit diagram DIN.svg|thumb|De Morgan's Laws represented as a circuit with logic gates ([[International Electrotechnical Commission]] diagrams)]]

In extensions of classical propositional logic, the duality still holds (that is, to any logical operator one can always find its dual), since in the presence of the identities governing negation, one may always introduce an operator that is the De Morgan dual of another. This leads to an important property of logics based on [[classical logic]], namely the existence of [[negation normal form]]s: any formula is equivalent to another formula where negations only occur applied to the non-logical atoms of the formula. The existence of negation normal forms drives many applications, for example in [[digital circuit]] design, where it is used to manipulate the types of [[logic gate]]s, and in formal logic, where it is needed to find the [[conjunctive normal form]] and [[disjunctive normal form]] of a formula. Computer programmers use them to simplify or properly negate complicated [[Conditional (programming)|logical conditions]]. They are also often useful in computations in elementary [[probability theory]].

Let one define the dual of any propositional operator P(''p'', ''q'', ...) depending on elementary propositions ''p'', ''q'', ... to be the operator <math>\mbox{P}^d</math> defined by

:<math>\mbox{P}^d(p, q, ...) = \neg P(\neg p, \neg q, \dots).</math>