PlanetMath: negation

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negation

(Definition)

In logics and mathematics, negation (from Latin negare `to deny') is the unary operation “ $\lnot$ ” which swaps the truth value of any operand to the opposite truth value. So, if the statement is true then its negated statement $\lnot P$ is false, and vice versa.

Note 1. The negated statement $\lnot P$ (by Heyting) has been denoted also with (Peano), $\sim\! P$ (Russell), $\overline{P}$ (Hilbert) and (by the Polish notation).

Note 2. $\lnot P$ may be expressed by implication as

$\displaystyle P\to\curlywedge$

where $\curlywedge$ means any contradictory statement.

Note 3. The negation of logical or and logical and give the results

$\displaystyle \lnot(P\lor Q) \equiv \lnot P \land \lnot Q,\;\;\; \lnot(P\land Q) \equiv \lnot P \lor \lnot Q.$

Analogical results concern the quantifier statements:

$\displaystyle \lnot (\exists x)P(x) \equiv (\forall x)\lnot P(x),\;\;\; \lnot (\forall x)P(x) \equiv (\exists x)\lnot P(x).$

These all are known as de Morgan's laws.

Note 4. Many mathematical relation statements, expressed with such special relation symbols as $=,\, \subseteq,\, \in,\, \cong,\, \parallel,\, \mid$ , are negated by using in the symbol an additional cross line: $\neq,\, \nsubseteq,\, \notin,\, \ncong,\, \nparallel,\, \nmid$ .

"negation" is owned by pahio.

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Other names:

logical not

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Cross-references: relation symbols, relation, de Morgan's laws, quantifier, logical and, logical or, contradictory statement, implication, Polish notation, operation, unary, logics
There are 27 references to this entry.

This is version 4 of negation, born on 2006-12-12, modified 2006-12-12.
Object id is 8619, canonical name is Negation.
Accessed 1651 times total.

Classification:

AMS MSC:

03B05 (Mathematical logic and foundations :: General logic :: Classical propositional logic)

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