Definition:Logical Complement

Boolean Algebra

Let $f: \Bbb B^k \to \Bbb B$ be a boolean function.

The (logical) complement of $f$ is the function $f'$ (or $\overline f$) defined as:

$\forall p \in \operatorname{Dom} \left({f}\right): f' \left({p}\right) = \neg \left({f \left({p}\right)}\right)$

Logic

The (logical) complement of a statement $p$ is the negation of $p$, that is, $\neg p$.

From Double Negation, it follows that the complement of $\neg p$ is $p$.