In logics and mathematics, negationMathworldPlanetmath (from Latin negare ‘to deny’) is the unary operation “¬” which swaps the truth value of any operand to the truth value.  So, if the statement P is true then its negated statement ¬P is false, and vice versa.

Note 1.  The negated statement ¬P (by Heyting) has been denoted also with -P (Peano), P (Russell), P¯ (Hilbert) and NP (by the Polish notation).

Note 2.¬P may be expressed by implicationMathworldPlanetmath as


where means any contradictory statement.

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


Analogical results concern the quantifierMathworldPlanetmath statements:


These all are known as de Morgan’s laws.

Note 4.  Many mathematical relationMathworldPlanetmath statements, expressed with such special relation symbols as  =,,,,,,  are negated by using in the symbol an additional cross line:   ,,,,,.

Title negation
Canonical name Negation
Date of creation 2015-04-25 17:44:13
Last modified on 2015-04-25 17:44:13
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 8
Author pahio (2872)
Entry type Definition
Classification msc 03B05
Synonym logical not
Related topic SetMembership