Concept in mathematical logic
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 computer science, a Boolean domain is usually written as {0, 1}, or formula_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 computer science, a Boolean variable is a variable that takes values in some Boolean domain. Some programming languages feature reserved words or symbols for the elements of the Boolean domain, for example codice_1 and codice_2. However, many programming languages do not have a Boolean data type in the strict sense. In C or BASIC, for example, falsity is represented by the number 0 and truth is represented by the number 1 or −1, and all variables that can take these values can also take any other numerical values.
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, negation (NOT) is replaced with formula_2 conjunction (AND) is replaced with multiplication (formula_3), and disjunction (OR) is defined via De Morgan's law to be formula_4.
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.
References.
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