Nonmonotone laws

Statement

The complement operation is defined by the following two laws.

$$\begin{aligned} &\text{Complementation 1} & x \wedge \neg x & = 0 \\ &\text{Complementation 2} & x \vee \neg x & = 1 \end{aligned}$$

All properties of negation including the laws below follow from the above two laws alone.

In both ordinary and Boolean algebra, negation works by exchanging pairs of elements, hence in both algebras it satisfies the double negation law (also called involution law)

$$\begin{aligned} &\text{Double negation} & \neg{(\neg{x})} & = x \end{aligned}$$

But whereas ordinary algebra satisfies the two laws

$$\begin{aligned} (-x)(-y) & = xy \\ (-x) + (-y) & = -(x + y) \end{aligned}$$

Boolean algebra satisfies De Morgan's laws:

$$\begin{aligned} &\text{De Morgan 1} & \neg x \wedge \neg y & = \neg{(x \vee y)} \\ &\text{De Morgan 2} & \neg x \vee \neg y & = \neg{(x \wedge y)} \end{aligned}$$