Boolean domain

https://en.wikipedia.org/wiki/Boolean_domain

In mathematics and abstract algebra, a Boolean domain is a set consisting of exactly two elements whose interpretations include false and true.

In logic, mathematics and theoretical CS, a Boolean domain is usually written as 𝔹 = {0, 1}.

The algebraic structure that naturally builds on a Boolean domain is the Boolean algebra with two elements.

The initial object in the category of bounded lattices is a Boolean domain.

In CS, a Boolean variable is a variable that takes values in some Boolean domain. Some PLs have reserved keywords for the elements of the Boolean domain (false and true). However, many don't have a Boolean datatype in the strict sense; in C, falsity is represented by the number 0 and truth is represented by any other number (although commonly, 1 or −1), but due to the C's poor type system, a variable that ranges over the "Boolean" values can also take any other numerical value.

Generalizations

The Boolean domain, {0, 1}, can be replaced by the unit interval, [0,1], in which case rather than only taking values 0 or 1, any value between and including 0 and 1 can be assumed.

Algebraically, logical operations are replaced with:

• negation is replaced with 1 - x

• conjunction with multiplication as xy

• disjunction is defined via De Morgan's law as 1 - (1 - x)(1 - y)

Interpreting these values as logical truth values yields a multi-valued logic, which forms the basis for fuzzy logic and probabilistic logic. In these interpretations, a value is interpreted as the "degree" of truth - to what extent a proposition is true, or the probability that the proposition is true.