# Negation

Negation
NOT
Definition${\displaystyle {\overline {x}}}$
Truth table${\displaystyle (10)}$
Logic gate
Normal forms
Disjunctive${\displaystyle {\overline {x}}}$
Conjunctive${\displaystyle {\overline {x}}}$
Zhegalkin polynomial${\displaystyle 1\oplus x}$
Post's lattices
0-preservingno
1-preservingno
Monotoneno
Affineyes
Self-dualyes

In logic, negation, also called the logical complement, is an operation that takes a proposition ${\displaystyle P}$ to another proposition "not ${\displaystyle P}$", written ${\displaystyle \neg P}$, which is interpreted intuitively as being true when ${\displaystyle P}$ is false, and false when ${\displaystyle P}$ 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 ${\displaystyle P}$ is the proposition whose proofs are the refutations of ${\displaystyle P}$.

## 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).[1]

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 ${\displaystyle P}$ is true, then ${\displaystyle \neg P}$ (pronounced "not P") would therefore be false; and conversely, if ${\displaystyle \neg P}$ is false, then ${\displaystyle P}$ would be true.

The truth table of ${\displaystyle \neg P}$ is as follows:

 ${\displaystyle P}$ ${\displaystyle \neg P}$ True False False True

Negation can be defined in terms of other logical operations. For example, ${\displaystyle \neg P}$ can be defined as ${\displaystyle P\rightarrow \bot }$ (where ${\displaystyle \rightarrow }$ is logical consequence and ${\displaystyle \bot }$ is absolute falsehood). Conversely, one can define ${\displaystyle \bot }$ as ${\displaystyle Q\land \neg Q}$ for any proposition ${\displaystyle Q}$ (where ${\displaystyle \land }$ 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, ${\displaystyle P\rightarrow Q}$ can be defined as ${\displaystyle \neg P\lor Q}$, where ${\displaystyle \lor }$ 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.

Other Languages
العربية: نفي (رياضيات)
čeština: Negace
dansk: Negation
Deutsch: Negation
eesti: Eitus
emiliàn e rumagnòl: Negasiòun (matemàtica)
Esperanto: Logika neo
فارسی: نقیض
हिन्दी: निषेध (तर्क)
hrvatski: Negacija
Bahasa Indonesia: Negasi
қазақша: Терістеу
Latina: Negatio
magyar: Negáció
македонски: Негација
Bahasa Melayu: Negasi
Nederlands: Logische negatie

norsk: Negasjon
ភាសាខ្មែរ: អវិជ្ចមានកម្ម
Piemontèis: Negassion
polski: Negacja
português: Negação
русский: Отрицание
shqip: Negacioni
Simple English: Logical negation
slovenčina: Negácia (logika)
slovenščina: Negacija
српски / srpski: Логичка негација
srpskohrvatski / српскохрватски: Logička negacija
svenska: Negation
ไทย: นิเสธ
тоҷикӣ: Инкор
українська: Заперечення