Predicate (logic)

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 ${\displaystyle R(a,b)}$ would be true on an interpretation if the entities denoted by ${\displaystyle a}$ and ${\displaystyle b}$ stand in the relation denoted by ${\displaystyle R}$. 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.

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 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.