# Predicate (mathematical logic)

__: Symbol representing a property or relation in logic__

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

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 [math]\displaystyle{ R(a,b) }[/math] would be true on an interpretation if the entities denoted by [math]\displaystyle{ a }[/math] and [math]\displaystyle{ b }[/math] stand in the relation denoted by [math]\displaystyle{ 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 constants, other logics may allow predicates that apply to 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 formulas are sometimes regarded as zero-place predicates.
^{[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.

## See also

- Classifying topos
- Free variables and bound variables
- Multigrade predicate
- Opaque predicate
- Predicate functor logic
- Predicate variable
- Truthbearer
- Well-formed formula

## References

- ↑ Lavrov, Igor Andreevich; Maksimova, Larisa (2003).
*Problems in Set Theory, Mathematical Logic, and the Theory of Algorithms*. New York: Springer. p. 52. ISBN 0306477122. https://books.google.com/books?id=zPLjjjU1C9AC.

## External links

Original source: https://en.wikipedia.org/wiki/Predicate (mathematical logic).
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