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In
logic Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from prem ...
, a predicate is a symbol which represents a property or a relation. For instance, in the first order formula $P\left(a\right)$, the symbol $P$ is a predicate which applies to the individual constant $a$. Similarly, in the formula $R\left(a,b\right)$, $R$ is a predicate which 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 $R\left(a,b\right)$ 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 used to interpret them. While
first-order logic First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantifi ...
only includes predicates which apply to individual constants, other logics may allow predicates which apply to other predicates.

# Predicates in different systems

* In
propositional logic Propositional calculus is a branch of logic. It is also called propositional logic, statement logic, sentential calculus, sentential logic, or sometimes zeroth-order logic. It deals with propositions (which can be true or false) and relations ...
, atomic formulas are sometimes regarded as zero-place predicates In a sense, these are nullary (i.e. 0-
arity Arity () is the number of arguments or operands taken by a function, operation or relation in logic, mathematics, and computer science. In mathematics, arity may also be named ''rank'', but this word can have many other meanings in mathematics. ...
) predicates. * In
first-order logic First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantifi ...
, a predicate forms an atomic formula when applied to an appropriate number of terms. * In
set theory Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concern ...
with excluded middle, predicates are understood to be characteristic functions or set
indicator function In mathematics, an indicator function or a characteristic function of a subset of a set is a function that maps elements of the subset to one, and all other elements to zero. That is, if is a subset of some set , one has \mathbf_(x)=1 if x\ ...
s (i.e., functions from a set element to a
truth value In logic and mathematics, a truth value, sometimes called a logical value, is a value indicating the relation of a proposition to truth, which in classical logic has only two possible values (''true'' or '' false''). Computing In some progra ...
). Set-builder notation makes use of predicates to define sets. * In
autoepistemic logic The autoepistemic logic is a formal logic for the representation and reasoning of knowledge about knowledge. While propositional logic can only express facts, autoepistemic logic can express knowledge and lack of knowledge about facts. The stable ...
, which rejects the
law of excluded middle In logic, the law of excluded middle (or the principle of excluded middle) states that for every proposition, either this proposition or its negation is true. It is one of the so-called three laws of thought, along with the law of noncon ...
, 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, predicates are the characteristic functions of a
probability distribution In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomeno ...
. That is, the strict true/false valuation of the predicate is replaced by a quantity interpreted as the degree of truth.