Encyclopedia > NOT

  Article Content

Negation

Redirected from NOT

In logic, logical negation is a unary logical operator that reverses the truth value of its operand.

The negation of the statement p is written in various ways:

  • ~p;
  • ¬p;
  • p (which is p with a bar over it);
  • NOT p;
  • !p
It is read as "It is not the case that p", or simply "not p".

~p is true if and only if p is false. For instance, if p denotes the statement "today is Saturday", then its negation ~p is the statement "today is not Saturday".

In classical logic, double negation means affirmation; i.e., the statements p and ~(~p) are logically equivalent. In intuitionistic logic, however, ~~p is a weaker statement than p. Nevertheless, ~~~p and ~p are logically equivalent even intuitionistically.

Logical negation can be defined in terms of other logical operations. For example, ~p can be defined as pF, where → is material implication and F is absolute falsehood. Conversely, one can define F as p & ~p for any proposition p, where & is logical conjunction. The idea here is that any contradiction is false. While these ideas work in both classical and intuitionistic logic, they don't work in Brazilian logic[?], where contradictions are not necessarily false. But in classical logic, we get a further identity: pq can be defined as ~pq, where ∨ is logical disjunction.

Algebraically, logical negation corresponds to the complement in a Boolean algebra (for classical logic) or a Heyting algebra[?] (for intuitionistic logic).



All Wikipedia text is available under the terms of the GNU Free Documentation License

 
  Search Encyclopedia

Search over one million articles, find something about almost anything!
 
 
  
  Featured Article
Museums in England

... Armouries Museum[?], Leeds West Yorkshire Folk Museum[?] Yorkshire Bronte Museum[?], Haworth Cannon Hall Museum[?] Castle Museum, York[?] Nation ...

 
 
 
This page was created in 57 ms