Negation
In logic, negation, also called the logical complement, is an operation that takes a proposition to another proposition "not ", written, which is interpreted intuitively as being true when is false, and false when is true. Negation is thus a unary 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 is the proposition whose proofs are the refutations of.
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,.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 would therefore be false; and conversely, if is false, then would be true.
The truth table of is as follows:
| True | False |
| False | True |
Negation can be defined in terms of other logical operations. For example, can be defined as . Conversely, one can define as for any proposition . 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.
Notation
The negation of a proposition is notated in different ways in various contexts of discussion and fields of application. Among these variants are the following:| Notation | Plain Text | Vocalization |
| ¬p | Not p | |
| ~p | Not p | |
| -p | Not p | |
| Np | En p | |
| p' | ||
| ̅p | ||
| !p |
The notation Np is Łukasiewicz notation.
In set theory is also used to indicate 'not member of': is the set of all members of that are not members of.
No matter how it is notated or symbolized, the negation can be read as "it is not the case that ", "not that ", or usually more simply as "not ".
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 we do have the equivalence of. Moreover, 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
provide a way of distributing negation over disjunction and conjunction :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 truth-value of the operation or it never makes a difference. Negation is a linear logical operator.
Self dual
In Boolean algebra a self dual function is one such that:for all
Negation is a self dual logical operator.
Negations of quantifiers
In first-order logic, there are two quantifiers, one is the universal quantifier and the other is the existential quantifier . The negation of one quantifier is the other quantifier. 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".Rules of inference
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, negation elimination, and double negation elimination. 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 or refutable. 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 and elimination. 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
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, Eiffel and Seed7. Some languages provide more than one operator for negation. A few languages like PL/I and Ratfor use ¬ for negation. Some modern computers and operating systems will display ¬ as ! on files encoded in ASCII. Most modern languages allow the above statement to be shortened from if to if , which allows sometimes, when the compiler/interpreter is not able to optimize it, faster programs.In computer science there is also bitwise negation. This takes the value given and switches all the binary 1s to 0s and 0s to 1s. See bitwise operation. This is often used to create ones' complement or "
~" in C or C++ and two's complement as it basically creates the opposite or mathematical complement of the value.To get the absolute value of a given integer the following would work as the "
-" changes it from negative to positive unsigned int abs
To demonstrate logical negation:
unsigned int abs
Inverting the condition and reversing the outcomes produces code that is logically equivalent to the original code, i.e. will have identical results for any input.
This convention occasionally surfaces in ordinary written speech, as computer-related slang for not. For example, the phrase
!voting means "not voting". Another example is the phrase !clue which is used as a synonym for "no-clue" or "clueless".