Definition
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. Thus if statement is true, then (pronounced "not P") would then be false; and conversely, if is true, then would be false.
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 Q (where is logical conjunction). The idea here is that any contradiction is false, and while these ideas work in both classical and intuitionistic logic, they do not work in paraconsistent logic, where contradictions are not necessarily false. As a further example, negation can be defined in terms of NAND and can also be defined in terms of NOR.
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.
Notation
The negation of a proposition p is notated in different ways, in various contexts of discussion and fields of application. The following table documents some of these variants:
The notation is Polish notation.
In set theory, is also used to indicate 'not in the set of': is the set of all members of U that are not members of A.
Regardless how it is notated or symbolized, the negation can be read as "it is not the case that P", "not that P", or usually more simply as "not P".
Precedence
See also: Logical connective § Order of precedence
As a way of reducing the number of necessary parentheses, one may introduce precedence rules: ¬ has higher precedence than ∧, ∧ higher than ∨, and ∨ higher than →. So for example, is short for
Properties
Double negation
Within a system of classical logic, double negation, that is, the negation of the negation of a proposition , is logically equivalent to . Expressed in symbolic terms, . In intuitionistic logic, a proposition implies its double negation, but not conversely. This marks one important difference between classical and intuitionistic negation. Algebraically, classical negation is called an involution of period two.
However, in intuitionistic logic, the weaker equivalence does hold. This is because in intuitionistic logic, is just a shorthand for , and we also have . Composing that last implication with triple negation implies that .
As a result, in the propositional case, a sentence is classically provable if its double negation is intuitionistically provable. This result is known as Glivenko's theorem.
Distributivity
De Morgan's laws provide a way of distributing negation over disjunction and conjunction:
- , and
- .
Linearity
Let denote the logical xor operation. In Boolean algebra, a linear function is one such that:
If there exists , , for all .
Another way to express this is that each variable always makes a difference in the of the operation, or it never makes a difference. Negation is a linear logical operator.
Rules of inference
See also: Double negation
There are a number of equivalent ways to formulate rules for negation. One usual way to formulate classical negation in a natural deduction setting is to take as primitive rules of inference negation introduction (from a derivation of to both and , infer ; this rule also being called reductio ad absurdum), negation elimination (from and infer ; this rule also being called ex falso quodlibet), and double negation elimination (from infer ). One obtains the rules for intuitionistic negation the same way but by excluding double negation elimination.
Negation introduction states that if an absurdity can be drawn as conclusion from then must not be the case (i.e. is false (classically) or refutable (intuitionistically) or etc.). Negation elimination states that anything follows from an absurdity. Sometimes negation elimination is formulated using a primitive absurdity sign . In this case the rule says that from and follows an absurdity. Together with double negation elimination one may infer our originally formulated rule, namely that anything follows from an absurdity.
Typically the intuitionistic negation of is defined as . Then negation introduction and elimination are just special cases of implication introduction (conditional proof) and elimination (modus ponens). In this case one must also add as a primitive rule ex falso quodlibet.
Programming language and ordinary language
As in mathematics, negation is used in computer science to construct logical statements.
if (!(r == t))
{
/*...statements executed when r does NOT equal t...*/
}
The exclamation mark "!" signifies logical NOT in B, C, and languages with a C-inspired syntax such as C++, Java, JavaScript, Perl, and PHP. "NOT" is the operator used in ALGOL 60, BASIC, and languages with an ALGOL- or BASIC-inspired syntax such as Pascal, Ada, and Eiffel. Some languages (C++, Perl, etc.) provide more than one operator for negation. A few languages like PL/I and Ratfor use ¬ for negation. Most modern languages allow the above statement to be shortened from if (!(r == t)) to if (r != t), which allows sometimes, when the compiler/interpreter is not able to optimize it, faster programs.
References
- 1Virtually all Turkish high school math textbooks use p' for negation due to the books handed out by the Ministry of National Education representing it as p'.
- 2"NEGATION definition and meaning | Collins English Dictionary". www.collinsdictionary.com. 15 December 2025. Retrieved 20 December 2025.
{{cite web}}: CS1 maint: deprecated archival service (link) - 3Weisstein, Eric W. "Negation". mathworld.wolfram.com. Retrieved 2 September 2020.
- 4"Logic and Mathematical Statements - Worked Examples". www.math.toronto.edu. Retrieved 2 September 2020.
- 5Beall, Jeffrey C. (2010). Logic: the basics (1. publ ed.). London: Routledge. p. 57. ISBN 978-0-203-85155-5.
- 6Used as makeshift in early typewriter publications, e.g. Richard E. Ladner (January 1975). "The circuit value problem is log space complete for P". ACM SIGACT News. 7 (101): 18–20. doi:10.1145/990518.990519.
- 7O'Donnell, John; Hall, Cordelia; Page, Rex (2007), Discrete Mathematics Using a Computer, Springer, p. 120, .
Further reading
- Gabbay, Dov, and Wansing, Heinrich, eds., 1999. What is Negation?, Kluwer.
- Horn, L., 2001. A Natural History of Negation, University of Chicago Press.
- G. H. von Wright, 1953–59, "On the Logic of Negation", Commentationes Physico-Mathematicae 22.
- Wansing, Heinrich, 2001, "Negation", in Goble, Lou, ed., The Blackwell Guide to Philosophical Logic, Blackwell.
- Tettamanti, Marco; Manenti, Rosa; Della Rosa, Pasquale A.; Falini, Andrea; Perani, Daniela; Cappa, Stefano F.; Moro, Andrea (2008). "Negation in the brain: Modulating action representation". NeuroImage. 43 (2): 358–367. doi:10.1016/j.neuroimage.2008.08.004. hdl:11384/163588. PMID 18771737. S2CID 17658822.
In first-order logic, there are two quantifiers, one is the universal quantifier (means "for all") and the other is the existential quantifier (means "there exists"). The negation of one quantifier is the other quantifier ( and ). For example, with the predicate P as "x is mortal" and the domain of x as the collection of all humans, means "a person x in all humans is mortal" or "all humans are mortal". The negation of it is , meaning "there exists a person x in all humans who is not mortal", or "there exists someone who lives forever".
In C (and some other languages descended from C), double negation (!!x) is used as an idiom to convert x to a canonical Boolean, ie. an integer with a value of either 0 or 1 and no other. Although any integer other than 0 is logically true in C and 1 is not special in this regard, it is sometimes important to ensure that a canonical value is used, for example for printing or if the number is subsequently used for arithmetic operations.8
