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 .
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 constants, other logics may allow predicates that apply to 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.