Predicate (mathematical logic)

In logic, a predicate is a non-logical 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 \(P(a)\), the symbol \(P\) is a predicate that applies to the individual constant \(a\) which evaluates to either true or false. Similarly, in the formula \(R(a,b)\), the symbol \(R\) is a predicate that applies to the individual constants \(a\) and \(b\). 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 \(R(a,b)\) 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.

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.

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 is a non-logical relation symbol, which 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.