Predicate (mathematical logic)

In logic, a predicate is a symbol which represents a property or a relation. For instance, the first order formula $P(a)$ , the symbol $P$ is a predicate which applies to the individual constant $a$ . Similarly, in the formula $R(a,b)$ the predicate $R$ is a predicate which applies to the individual constants $a$ and $b$ .

In the semantics of logic, predicates are interpreted as relations. For instance, in a standard semantics for first-order logic, the formula $R(a,b)$ would be true on an interpretation if the entities denoted by $a$ and $b$ stand in the relation denoted by $R$ . Since predicates are non-logical symbol, they can denote different relations depending on the interpretation used to interpret them. While first-order logic only includes predicates which apply to individual constants, other logics may allow predicates which apply to other predicates.

Predicates in different systems[]

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 predicates forms an atomic formula when applied to an appropriate number of terms.
In set theory with 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, predicates are the characteristic functions of a probability distribution. That is, the strict true/false valuation of the predicate is replaced by a quantity interpreted as the degree of truth.

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.

External links[]

Introduction to predicates

[lavrov-1] Lavrov, Igor Andreevich; Maksimova, Larisa (2003). Problems in Set Theory, Mathematical Logic, and the Theory of Algorithms. New York: Springer. p. 52. ISBN 0306477122.

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Predicate (mathematical logic)

Contents

Predicates in different systems[]

See also[]

References[]

External links[]