Editing Predicate (logic)

{{Short description|Symbol representing a property or relation in logic}}
{{Other uses|Predicate (disambiguation)#Logic}}
In [[mathematical logic|logic]], a '''predicate''' is a symbol that represents a property or a relation. For instance, in the [[first order logic|first-order formula]] <math>P(a)</math>, the symbol <math>P</math> is a predicate that applies to the [[individual constant]] <math>a</math>. Similarly, in the formula <math>R(a,b)</math>, the symbol <math>R</math> is a predicate that applies to the individual constants <math>a</math> and <math>b</math>. 

According to [[Gottlob Frege]], the meaning of a predicate is exactly a function from the domain of objects to the [[truth value]]s "true" and "false".

In the [[semantics of logic]], predicates are interpreted as [[relation (mathematics)|relation]]s. For instance, in a standard semantics for first-order logic, the formula <math>R(a,b)</math> would be true on an [[interpretation (logic)|interpretation]] if the entities denoted by <math>a</math> and <math>b</math> stand in the relation denoted by <math>R</math>. Since predicates are [[non-logical symbol]]s, they can denote different relations depending on the interpretation given to them. While [[first-order logic]] only includes predicates that apply to individual objects, other logics may allow predicates that apply to collections of objects defined by other predicates.

== Predicates in different systems ==
A predicate is a statement or mathematical assertion that contains variables, sometimes referred to as predicate variables, and may be true or false depending on those variables’ value or values.
* In [[propositional logic]], [[atomic formula]]s are sometimes regarded as zero-place predicates.<ref name=lavrov>{{cite book|last1=Lavrov|first1=Igor Andreevich|first2=Larisa|last2=Maksimova|author2-link= Larisa Maksimova |title=Problems in Set Theory, Mathematical Logic, and the Theory of Algorithms|year=2003|publisher=Springer|location=New York|isbn=0306477122|page=52|url=https://books.google.com/books?id=zPLjjjU1C9AC}}</ref> In a sense, these are nullary (i.e. 0-[[arity]]) predicates.
* In [[first-order logic]], a predicate forms an atomic formula when applied to an appropriate number of [[term (logic)|term]]s. 
* In [[set theory]] with the [[law of excluded middle]], predicates are understood to be [[Indicator function|characteristic functions]] or set [[indicator function]]s (i.e., [[function (mathematics)|functions]] from a set element to a [[truth value]]). [[Set-builder notation]] makes use of predicates to define sets.
* In [[autoepistemic logic]], which rejects the law of excluded middle, predicates may be true, false, or simply ''unknown''. In particular, a given collection of facts may be insufficient to determine the truth or falsehood of a predicate.
* In [[fuzzy logic]], the strict true/false valuation of the predicate is replaced by a quantity interpreted as the degree of truth.

==See also==
*[[Classifying topos]]
* [[Free variables and bound variables]]
* [[Multigrade predicate]]
* [[Opaque predicate]]
* [[Predicate functor logic]]
* [[Predicate variable]]
* [[Truthbearer]]
* [[Truth value]]
* [[Well-formed formula]]

==References==
{{Reflist}}

==External links==
*[http://cs.odu.edu/~toida/nerzic/content/logic/pred_logic/predicate/pred_intro.html Introduction to predicates]

{{Mathematical logic}}
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[[Category:Predicate logic]]
[[Category:Propositional calculus]]
[[Category:Basic concepts in set theory]]
[[Category:Fuzzy logic]]
[[Category:Mathematical logic]]