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 computer science, a Boolean domain is usually written as {0, 1},[1][2][3][4][5] or [6][7]
The
bounded lattices
is a Boolean domain.
In
, 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 conjunction (AND) is replaced with multiplication (), and disjunction (OR) is defined via
De Morgan's law
to be .
Interpreting these values as logical
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.
See also
References
- ^ Dirk van Dalen, Logic and Structure. Springer (2004), page 15.
- ^ David Makinson, Sets, Logic and Maths for Computing. Springer (2008), page 13.
- Richard C. Jeffrey, Computability and Logic. Cambridge University Press (1980), page 99.
- ^ Elliott Mendelson, Introduction to Mathematical Logic (4th. ed.). Chapman & Hall/CRC (1997), page 11.
- Eric C. R. Hehner, A Practical Theory of Programming. Springer (1993, 2010), page 3.
- ISBN 978-0-262-16148-0.
- ISBN 978-3-540-43152-7.
Further reading
- (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.)
- (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.)
- ISBN 978-1-5275-0371-7. Retrieved 2019-08-04. [4] Archived 2019-08-04 at the Wayback Machine (536 pages) [5](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.)
- (NB. Contains extended versions of the best manuscripts from the 13th International Workshop on Boolean Problems (IWSBP 2018) held in Bremen, Germany on 2018-09-19/21.)
- ISBN 978-3-030-68070-1. (204 pages) [7] (NB. Contains extended versions of the best manuscripts from the 14th International Workshop on Boolean Problems (IWSBP 2020) held virtuallyon 2020-09-24/25.)