Symbol representing a property or relation in 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 $b$.

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

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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.