# Boolean-valued function

A **boolean-valued function** (sometimes called a predicate or a proposition) is a function of the type f : X → **B**, where X is an arbitrary set and where **B** is a boolean domain, i.e. a generic two-element set, (for example **B** = {0, 1}), whose elements are interpreted as logical values, for example, 0 = false and 1 = true.

In the formal sciences, mathematics, mathematical logic, statistics, and their applied disciplines, a boolean-valued function may also be referred to as a characteristic function, indicator function, predicate, or proposition. In all of these uses it is understood that the various terms refer to a mathematical object and not the corresponding semiotic sign or syntactic expression.

In formal semantic theories of truth, a **truth predicate** is a predicate on the sentences of a formal language, interpreted for logic, that formalizes the intuitive concept that is normally expressed by saying that a sentence is true. A truth predicate may have additional domains beyond the formal language domain, if that is what is required to determine a final truth value.

## References

- Brown, Frank Markham (2003),
*Boolean Reasoning: The Logic of Boolean Equations*, 1st edition, Kluwer Academic Publishers, Norwell, MA. 2nd edition, Dover Publications, Mineola, NY, 2003. - Kohavi, Zvi (1978),
*Switching and Finite Automata Theory*, 1st edition, McGraw–Hill, 1970. 2nd edition, McGraw–Hill, 1978. - Korfhage, Robert R. (1974),
*Discrete Computational Structures*, Academic Press, New York, NY. - Mathematical Society of Japan,
*Encyclopedic Dictionary of Mathematics*, 2nd edition, 2 vols., Kiyosi Itô (ed.), MIT Press, Cambridge, MA, 1993. Cited as EDM. - Minsky, Marvin L., and Papert, Seymour, A. (1988),
*Perceptrons, An Introduction to Computational Geometry*, MIT Press, Cambridge, MA, 1969. Revised, 1972. Expanded edition, 1988.