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Negation

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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).



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