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For other uses, see Negation (disambiguation) and NOT gate.
In logic, negation, also called the logical complement, is an operation that takes a proposition p to another proposition "not p", written ¬p, which is interpreted intuitively as being true when p is false, and false when p is true. Negation is thus a unary (single-argument) logical connective. It may be applied as an operation on notions, propositions, truth values, or semantic values more generally.
In classical logic, negation is normally identified with the truth function that takes truth to falsity and vice versa.
In intuitionistic logic, according to the Brouwer–Heyting–Kolmogorov interpretation, the negation of a proposition p is the proposition whose proofs are the refutations of p.
Contents
1 Definition
2 Notation
Definition
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 A is true, then ¬A (pronounced "not A") would therefore be false; and conversely, if ¬A is false, then A would be true.
The truth table of ¬p is as follows:
Truth table of ¬p
p ¬p
True False
False True
Classical negation can be defined in terms of other logical operations.
For example, ¬p can be defined as p → F, where "→" is logical consequence 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 do not work in paraconsistent 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: "not p, or q".
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.
Notation
The negation of a proposition p is notated in different ways in various contexts of discussion and fields of application.
Among these variants are the following:
Notation Vocalization
~p Not p
−p Not p
Np En p
¬p Not p
In set theory \ is also used to indicate 'not member of': U \ A is the set of all members of U that are not members of A.
No matter how it is notated or symbolized, the negation ¬p / −p can be read as "it is not the case that p", "not that p", or usually more simply as "not p".
