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In
logic Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the study of deductively valid inferences or logical truths. It examines how conclusions follow from premises based on the structure o ...
, 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. According to
Gottlob Frege Friedrich Ludwig Gottlob Frege (; ; 8 November 1848 – 26 July 1925) was a German philosopher, logician, and mathematician. He was a mathematics professor at the University of Jena, and is understood by many to be the father of analytic philos ...
, the meaning of a predicate is exactly a function from the domain of objects to the
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''). Truth values are used in ...
s "true" and "false". In the
semantics of logic In logic, the semantics of logic or formal semantics is the study of the meaning and interpretation of formal languages, formal systems, and (idealizations of) natural languages. This field seeks to provide precise mathematical models tha ...
, 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 objects, other logics may allow predicates that apply to collections of objects defined by 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 In logic, mathematics, and computer science, arity () is the number of arguments or operands taken by a function, operation or relation. In mathematics, arity may also be called rank, but this word can have many other meanings. In logic and ...
) predicates. * In first-order logic, 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 Set (mathematics), 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 mathema ...
with the law of 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 , then the indicator functio ...
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''). Truth values are used in ...
). 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 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 completely ...
, the strict true/false valuation of the predicate is replaced by a quantity interpreted as the degree of truth.


See also

* Classifying topos *
Free variables and bound variables In mathematics, and in other disciplines involving formal languages, including mathematical logic and computer science, a variable may be said to be either free or bound. Some older books use the terms real variable and apparent variable for f ...
* Multigrade predicate * Opaque predicate * Predicate functor logic * Predicate variable * Truthbearer *
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''). Truth values are used in ...
*
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. The abbreviation wf ...


References


External links


Introduction to predicates
{{Authority control Predicate logic Propositional calculus Basic concepts in set theory Fuzzy logic Mathematical logic