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 - xconjunction with multiplication as
xydisjunction 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.
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