Negation

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

~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 p → F, 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: p → q can be defined as ~p ∨ q, 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).