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In logic, a predicate is a symbol that represents a property or a relation. For instance, in the first-order formula P ( a ) {\displaystyle P(a)} , the symbol P {\displaystyle P} is a predicate that applies to the individual constant a {\displaystyle a} . Similarly, in the formula R ( a , b ) {\displaystyle R(a,b)} , the symbol R {\displaystyle R} is a predicate that applies to the individual constants a {\displaystyle a} and b {\displaystyle b} .

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

In the semantics of logic, predicates are interpreted as relations. For instance, in a standard semantics for first-order logic, the formula R ( a , b ) {\displaystyle R(a,b)} would be true on an interpretation if the entities denoted by a {\displaystyle a} and b {\displaystyle b} stand in the relation denoted by R {\displaystyle R} . 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.

• Predicates in different systems

• See also

• References

• External links

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.cite-ref-lavrov-1-0[1] 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 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.

• Classifying topos

• Free variables and bound variables

• Multigrade predicate

• Opaque predicate

• Predicate functor logic

• Predicate variable

• Truthbearer

• Truth value

• Well-formed formula

cite-note-lavrov-11. ↑ citereflavrovmaksimova2003Lavrov, Igor Andreevich; Maksimova, Larisa (2003). Problems in Set Theory, Mathematical Logic, and the Theory of Algorithms. New York: Springer. p. 52. ISBN 0306477122.

• Introduction to predicates