Boolean domain should not be confused with Binary 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 computer science, a Boolean domain is usually written as, or
B.
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 false and true. 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.
The Boolean domain can be replaced by the unit interval, 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
1-x,
xy
1-(1-x)(1-y)=x+y-xy
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.
de:Bernd Steinbach
. Recent Progress in the Boolean Domain . . Newcastle upon Tyne, UK . Freiberg, Germany . 1 . 2014-04-01 . 2013-09-25 . 978-1-4438-5638-6 . 2019-08-04. https://m.tau.ac.il/~ilia1/publications/rpbd_book.pdf draft version --> (xxx+428 pages) https://web.archive.org/web/20200131123408/http://www.informatik.tu-freiberg.de/prof2/ws_bp10/index.html (NB. Contains extended versions of the best manuscripts from the 10th International Workshop on Boolean Problems held at the Technische Universität Bergakademie Freiberg, Germany on 2012-09-19/21.)de:Bernd Steinbach
. Problems and New Solutions in the Boolean Domain . . Newcastle upon Tyne, UK . Freiberg, Germany . 1 . 2016-05-01 . 978-1-4438-8947-6 . 2019-08-04. (xxxv+1+445+1 pages) https://web.archive.org/web/20200112133809/http://www.informatik.tu-freiberg.de/prof2/ws_bp11/index.html (NB. Contains extended versions of the best manuscripts from the 11th International Workshop on Boolean Problems held at the Technische Universität Bergakademie Freiberg, Germany on 2014-09-17/19.)de:Bernd Steinbach
. Further Improvements in the Boolean Domain . . Newcastle upon Tyne, UK . Freiberg, Germany . 1 . 2018-01-01 . 978-1-5275-0371-7 . 2019-08-04. https://www.cambridgescholars.com/download/sample/64246 (xli+1+494 pages) https://web.archive.org/web/20200112133313/http://www.informatik.tu-freiberg.de/prof2/ws_bp12/index.html (NB. Contains extended versions of the best manuscripts from the 12th International Workshop on Boolean Problems held at the Technische Universität Bergakademie Freiberg, Germany on 2016-09-22/23.)de:Bernd Steinbach
. Advances in the Boolean Domain . . Newcastle upon Tyne, UK . Freiberg, Germany . 1 . 2022-09-29 . 978-1527-58872-1 . 2024-07-15. (xxii+231+1 pages)