In
logic, a predicate is a symbol that represents a property or a relation. For instance, in the
first-order formula, the symbol is a predicate that applies to the
individual constant. Similarly, in the formula , the symbol is a predicate that applies to the individual constants and .
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 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.
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
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.