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Predicate (logic)

Symbol representing a property or relation in logic From Wikipedia, the free encyclopedia

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In logic, a predicate is a 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 , the symbol is a predicate that applies to the individual constant which evaluates to either true or false. Similarly, in the formula , the symbol is a predicate that applies to the individual constants and . 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 would be true on an interpretation if the entities denoted by and stand in the relation denoted by . 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. 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.

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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.

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See also

References

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