Predicate (mathematical logic)

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. 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.