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

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 $R(a,b)$ 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 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 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.