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copy text from [http://www.opencycle.net/ OpenCycle] of which Jon Awbrey is the sole author
A '''boolean-valued function''', in some usages a '''predicate''' or a '''proposition''', is a [[function (mathematics)|function]] of the type ''f'' : ''X'' → '''B''', where ''X'' is an arbitrary [[set]] and where '''B''' is a [[boolean domain]].
A '''boolean domain''' '''B''' is a generic 2-element set, say, '''B''' = {0, 1}, whose elements are interpreted as [[logical value]]s, for example, 0 = false and 1 = true.
In the [[formal science]]s, [[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 [[semantics|formal semantic]] theories of [[truth]], a '''truth predicate''' is a predicate on the [[sentence]]s 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==
* [[Frank Markham Brown|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.
* [[Zvi Kohavi|Kohavi, Zvi]] (1978), ''Switching and Finite Automata Theory'', 1st edition, McGraw–Hill, 1970. 2nd edition, McGraw–Hill, 1978.
* [[Robert R. Korfhage|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.
* [[Marvin L. Minsky|Minsky, Marvin L.]], and [[Seymour A. Papert|Papert, Seymour, A.]] (1988), ''[[Perceptrons]], An Introduction to Computational Geometry'', MIT Press, Cambridge, MA, 1969. Revised, 1972. Expanded edition, 1988.
==See also==
{|
| valign=top |
* [[Boolean algebra]]
* [[Boolean domain]]
* [[Boolean logic]]
| valign=top |
* [[Propositional calculus]]
* [[Truth table]]
* [[Zeroth order logic]]
|}
===Equivalent concepts===
* [[Characteristic function]]
* [[Indicator function]]
* [[Predicate]], in some senses.
* [[Proposition]], in some senses.
===Related concepts===
* [[Finitary boolean function]]
A '''boolean domain''' '''B''' is a generic 2-element set, say, '''B''' = {0, 1}, whose elements are interpreted as [[logical value]]s, for example, 0 = false and 1 = true.
In the [[formal science]]s, [[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 [[semantics|formal semantic]] theories of [[truth]], a '''truth predicate''' is a predicate on the [[sentence]]s 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==
* [[Frank Markham Brown|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.
* [[Zvi Kohavi|Kohavi, Zvi]] (1978), ''Switching and Finite Automata Theory'', 1st edition, McGraw–Hill, 1970. 2nd edition, McGraw–Hill, 1978.
* [[Robert R. Korfhage|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.
* [[Marvin L. Minsky|Minsky, Marvin L.]], and [[Seymour A. Papert|Papert, Seymour, A.]] (1988), ''[[Perceptrons]], An Introduction to Computational Geometry'', MIT Press, Cambridge, MA, 1969. Revised, 1972. Expanded edition, 1988.
==See also==
{|
| valign=top |
* [[Boolean algebra]]
* [[Boolean domain]]
* [[Boolean logic]]
| valign=top |
* [[Propositional calculus]]
* [[Truth table]]
* [[Zeroth order logic]]
|}
===Equivalent concepts===
* [[Characteristic function]]
* [[Indicator function]]
* [[Predicate]], in some senses.
* [[Proposition]], in some senses.
===Related concepts===
* [[Finitary boolean function]]