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| A '''boolean-valued function''', in some usages a '''predicate''' or a '''proposition''', is a function of the type |
A '''boolean-valued function''', in some usages a '''predicate''' or a '''proposition''', is a ] of the type f : X → '''B''', where X is an arbitrary set, where '''B''' is a generic 2-element set, typically '''B''' = {0, 1}, and where the latter is frequently interpreted for logical applications as having 0 = false and 1 = true. | ||
| A '''boolean domain''' '''B''' is a generic 2-element ], say, '''B''' = {0, 1}, whose elements are interpreted as ]s, typically 0 = false and 1 = true. | A '''boolean domain''' '''B''' is a generic 2-element ], say, '''B''' = {0, 1}, whose elements are interpreted as ]s, typically 0 = false and 1 = true. | ||
Revision as of 05:15, 31 July 2006
A boolean-valued function, in some usages a predicate or a proposition, is a function of the type f : X → B, where X is an arbitrary set, where B is a generic 2-element set, typically B = {0, 1}, and where the latter is frequently interpreted for logical applications as having 0 = false and 1 = true.
A boolean domain B is a generic 2-element set, say, B = {0, 1}, whose elements are interpreted as logical values, typically 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 (volume).
See also
Equivalent concepts
- Characteristic function
- Indicator function
- Predicate, in some senses.
- Proposition, in some senses.