Predicate (logic)

{{Short description|Symbol representing a property or relation in logic}}

In logic, a '''predicate''' is a non-logical symbol that represents a property or a relation, though, formally, does not need to represent anything at all. For instance, in the first-order formula <math>P(a)</math>, the symbol <math>P</math> is a predicate that applies to the individual constant <math>a</math> which evaluates to either true or false. 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>. Predicates are considered a primitive notion of first-order, and higher-order logic and are therefore not defined in terms of other more basic concepts.

The term derives from the grammatical term "predicate", meaning a word or phrase that represents a property or relation.

In the semantics of logic, predicates are interpreted as relations. For instance, in a standard semantics for first-order logic, the formula <math>R(a,b)</math> would be true on an 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 symbols, 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.

Strictly speaking, a predicate does not need to be given any interpretation, so long as its syntactic properties are well-defined. For example, equality may be understood solely through its reflexive and substitution properties (cf. ''{{Slink|Equality (mathematics)|Axioms}}''). Other properties can be derived from these, and they are sufficient for proving theorems in mathematics. Similarly, set membership can be understood solely through the axioms of Zermelo–Fraenkel set theory.

== 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 formulas 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 is a non-logical relation symbol, which forms an atomic formula when applied to an appropriate number of terms. * In set theory with the law of excluded middle, predicates are understood to be characteristic functions or set indicator functions (i.e., 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.

==References== {{Reflist}}

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

Category:Predicate logic Category:Propositional calculus Category:Basic concepts in set theory Category:Fuzzy logic Category:Mathematical logic