nLab boolean domain

The boolean domain or boolean field $\mathbb{B}$ is a $2$-element set, say $\mathbb{B} = \{ 0, 1 \}$ or $\mathbb{B} = \{ \bot, \top \}$, whose elements may be interpreted as truth values. Note that $\mathbb{B}$ is the set of all truth values in classical logic, but this cannot be assumed in all logics. If we think of $\mathbb{B}$ as a pointed set equipped with the true element, then there is an effectively unique boolean domain.

A boolean variable $x$ is a variable that takes its value in a boolean domain, as $x \in \mathbb{B}$. If this variable depends on parameters, then it is (or defines) a Boolean-valued function, that is a function whose target is $\mathbb{B}$.

An element of $\mathbb{B}$ is a binary digit, or bit.

As a boolean algebra or ring

Note that the term ‘boolean field’ (or just ‘field’, depending on the context) is sometimes used more generally for any boolean algebra. In fact, the boolean domain is the initial boolean algebra. If we interpret a boolean algebra as a boolean ring, then the boolean domain is the finite field with $2$ elements.