No agreement exists as to the possibility of defining negation, as to its logical status, function, and meaning, as to its field of applicability..., and as to the interpretation of the negative judgment, (F.H. Heinemann 1944).
Classical negation is an 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. So, if statement is true, then (pronounced "not P") would therefore be false; and conversely, if is false, then would be true.
The truth table of is as follows:
Negation can be defined in terms of other logical operations. For example, can be defined as (where is logical consequence and is absolute falsehood). Conversely, one can define as for any proposition (where is logical conjunction). The idea here is that any contradiction is false. While these ideas work in both classical and intuitionistic logic, they do not work in paraconsistent logic, where contradictions are not necessarily false. In classical logic, we also get a further identity, can be defined as , where is logical disjunction.
Algebraically, classical negation corresponds to complementation in a Boolean algebra, and intuitionistic negation to pseudocomplementation in a Heyting algebra. These algebras provide a semantics for classical and intuitionistic logic respectively.