negation

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

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

Note 2.  $\mathrm{\neg }P$ may be expressed by implication as

$$P\to ⋏$$

where $⋏$ means any contradictory statement.

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

$$\mathrm{\neg }(P\vee Q)\equiv \mathrm{\neg }P\wedge \mathrm{\neg }Q,\mathrm{\neg }(P\wedge Q)\equiv \mathrm{\neg }P\vee \mathrm{\neg }Q.$$

Analogical results concern the quantifier statements:

$$\mathrm{\neg }(\exists x)P(x)\equiv (\forall x)\mathrm{\neg }P(x),\mathrm{\neg }(\forall x)P(x)\equiv (\exists x)\mathrm{\neg }P(x).$$

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:  $\ne ,⊈,\notin ,\ncong ,\nparallel ,\nmid$.