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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 In logic, the semantics of logic or formal semantics is the study of the semantics, or interpretations, of formal and (idealizations of) natural languages usually trying to capture the pre-theoretic notion of entailment. Overview The truth cond ...
, 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 Interpretation may refer to: Culture * Aesthetic interpretation, an explanation of the meaning of a work of art * Allegorical interpretation, an approach that assumes a text should not be interpreted literally * Dramatic Interpretation, an event ...
if the entities denoted by $a$ and $b$ stand in the relation denoted by $R$. Since predicates are
non-logical symbol In logic, the formal languages used to create expressions consist of symbols, which can be broadly divided into constants and variables. The constants of a language can further be divided into logical symbols and non-logical symbols (sometimes al ...
s, 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 quantifie ...
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 b ...
,
atomic formula In mathematical logic, an atomic formula (also known as an atom or a prime formula) is a formula with no deeper propositional structure, that is, a formula that contains no logical connectives or equivalently a formula that has no strict subformu ...
s 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 quantifie ...
, 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 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 noncontradi ...
, 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\i ...
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 pro ...
).
Set-builder notation In set theory and its applications to logic, mathematics, and computer science, set-builder notation is a mathematical notation for describing a set by enumerating its elements, or stating the properties that its members must satisfy. Defining ...
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 noncontradi ...
, 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 Fuzzy logic is a form of many-valued logic in which the truth value of variables may be any real number between 0 and 1. It is employed to handle the concept of partial truth, where the truth value may range between completely true and completel ...
, 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 phenomenon ...
. That is, the strict true/false valuation of the predicate is replaced by a quantity interpreted as the degree of truth.

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Classifying topos In mathematics, a classifying topos for some sort of structure is a topos ''T'' such that there is a natural equivalence between geometric morphisms from a cocomplete topos ''E'' to ''T'' and the category of models for the structure in ''E''. Exam ...
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Free variables and bound variables In mathematics, and in other disciplines involving formal languages, including mathematical logic and computer science, a free variable is a notation (symbol) that specifies places in an expression where substitution may take place and is not ...
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Multigrade predicate In mathematics and logic, plural quantification is the theory that an individual variable x may take on ''plural'', as well as singular, values. As well as substituting individual objects such as Alice, the number 1, the tallest building in London ...
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Opaque predicate In computer programming, an opaque predicate is a predicate, an expression that evaluates to either "true" or "false", for which the outcome is known by the programmer ''a priori'', but which, for a variety of reasons, still needs to be evaluated at ...
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Predicate functor logic In mathematical logic, predicate functor logic (PFL) is one of several ways to express first-order logic (also known as predicate logic) by purely algebraic means, i.e., without quantified variables. PFL employs a small number of algebraic device ...
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Predicate variable In mathematical logic, a predicate variable is a predicate letter which functions as a "placeholder" for a relation (between terms), but which has not been specifically assigned any particular relation (or meaning). Common symbols for denoting predi ...
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Truthbearer A truth-bearer is an entity that is said to be either true or false and nothing else. The thesis that some things are true while others are false has led to different theories about the nature of these entities. Since there is divergence of o ...
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Well-formed formula In mathematical logic, propositional logic and predicate logic, a well-formed formula, abbreviated WFF or wff, often simply formula, is a finite sequence of symbols from a given alphabet that is part of a formal language. A formal language can ...