Boolean domain

Boolean not needed

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 $1-x,$ conjunction (AND) is replaced with multiplication ($xy$), and disjunction (OR) is defined via De Morgan's law to be $1-(1-x)(1-y)$.

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.

See also

• Boolean-valued function