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Predicate (logic)

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Symbol representing a property or relation in logic

For other uses, seePredicate (disambiguation) § Logic.

Inlogic, apredicate is anon-logical symbol that represents aproperty or arelation, though, formally, does not need to represent anything at all. For instance, in thefirst-order formula $P(a)$ , the symbol $P {\displaystyle P}$ is a predicate that applies to theindividual constant $a {\displaystyle a}$ which evaluates to eithertrue or false. Similarly, in the formula $R(a,b)$ , the symbol $R {\displaystyle R}$ is a predicate that applies to the individual constants $a {\displaystyle a}$ and $b {\displaystyle b}$ . Predicates are considered aprimitive notion of first-order, and higher-order logic and are therefore not defined in terms of other more basic concepts.

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

In thesemantics of logic, predicates are interpreted asrelations. For instance, in a standard semantics for first-order logic, the formula $R(a,b)$ would be true on aninterpretation if the entities denoted by $a {\displaystyle a}$ and $b {\displaystyle b}$ stand in the relation denoted by $R {\displaystyle R}$ . Since predicates arenon-logical symbols, they can denote different relations depending on the interpretation given to them. Whilefirst-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.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 ofZermelo–Fraenkel set theory.

Predicates in different systems

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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.

Inpropositional logic,atomic formulas are sometimes regarded as zero-place predicates.^[1] In a sense, these are nullary (i.e. 0-arity) predicates.
Infirst-order logic, a predicate forms an atomic formula when applied to an appropriate number ofterms.
Inset theory with thelaw of excluded middle, predicates are understood to becharacteristic functions or setindicator functions (i.e.,functions from a set element to atruth value).Set-builder notation makes use of predicates to define sets.
Inautoepistemic logic, which rejects the law of excluded middle, predicates may be true, false, or simplyunknown. In particular, a given collection of facts may be insufficient to determine the truth or falsehood of a predicate.
Infuzzy logic, the strict true/false valuation of the predicate is replaced by a quantity interpreted as the degree of truth.

References

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^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

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Introduction to predicates

Mathematical logic

General

Theorems (list)
and paradoxes

Logics

Traditional	Classical logic Logical truth Tautology Proposition Inference Logical equivalence Consistency Equiconsistency Argument Soundness Validity Syllogism Square of opposition Venn diagram
Propositional	Boolean algebra Boolean functions Logical connectives Propositional calculus Propositional formula Truth tables Many-valued logic 3 finite ∞
Predicate	First-order list Second-order Monadic Higher-order Fixed-point Free Quantifiers Predicate Monadic predicate calculus

Set theory

Set hereditary Class (Ur-)Element Ordinal number Extensionality Forcing Relation equivalence partition Set operations: intersection union complement Cartesian product power set identities
Types ofsets	Countable Uncountable Empty Inhabited Singleton Finite Infinite Transitive Ultrafilter Recursive Fuzzy Universal Universe constructible Grothendieck Von Neumann
Maps and cardinality	Function/Map domain codomain image In/Sur/Bi-jection Schröder–Bernstein theorem Isomorphism Gödel numbering Enumeration Large cardinal inaccessible Aleph number Operation binary
Set theories	Zermelo–Fraenkel axiom of choice continuum hypothesis General Kripke–Platek Morse–Kelley Naive New Foundations Tarski–Grothendieck Von Neumann–Bernays–Gödel Ackermann Constructive