Editing Negation (section)

==Definition==
''Classical negation'' is an [[logical operation|operation]] on one [[logical value]], typically the value of a [[proposition]], that produces a value of ''true'' when its operand is false, and a value of ''false'' when its operand is true. Thus if statement  <math>P</math> is true, then <math>\neg P</math> (pronounced "not P") would then be false; and conversely, if <math>\neg P</math> is true, then <math>P</math> would be false.

The [[truth table]] of <math>\neg P</math> is as follows:

:{| class="wikitable" style="text-align:center; background-color: #ddffdd;"
|- bgcolor="#ddeeff"
| <math> P </math> || <math> \neg P </math>
|-
| {{yes2|True}} || {{no2|False}}
|-
| {{no2|False}} || {{yes2|True}}
|}

Negation can be defined in terms of other logical operations. For example, <math>\neg P</math> can be defined as <math>P \rightarrow \bot</math> (where <math>\rightarrow</math> is [[logical consequence]] and <math>\bot</math> is [[False (logic)|absolute falsehood]]). Conversely, one can define <math>\bot</math> as <math>Q \land \neg Q</math> for any proposition {{mvar|Q}} (where <math>\land</math> is [[logical conjunction]]). The idea here is that any [[contradiction]] is false, and while these ideas work in both classical and intuitionistic logic, they do not work in [[paraconsistent logic]], where contradictions are not necessarily false. As a further example, negation can be defined in terms of NAND and can also be defined in terms of NOR.

{{anchor|Algebra}} Algebraically, classical negation corresponds to [[Complement (order theory)|complementation]] in a [[Boolean algebra (structure)|Boolean algebra]], and intuitionistic negation to pseudocomplementation in a [[Heyting algebra]]. These algebras provide a [[algebraic semantics (mathematical logic)|semantics]] for classical and intuitionistic logic.