In grammar, logic, and mathematics, negation is an operation on truth values, for instance, the truth value of a proposition, that sends true to false and false to true.
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 (which is p with a bar over it) ~p ¬p NOT p !p
It is read as "It is not the case that p", or simply "not p". Truth Table A not A F T T F
~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.
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.