# Predicate (mathematical logic)

**predicatepredicatespredicationlogical predicatepredicativeLogical notionsone-place predicatepredicate functionpredicate letterpredicate logic**

In mathematical logic, a predicate is commonly understood to be a Boolean-valued function P: X→ {true, false}, called the predicate on X.wikipedia

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### Boolean-valued function

**booleanboolean valuelogic '0' state**

In mathematical logic, a predicate is commonly understood to be a Boolean-valued function P: X→ {true, false}, called the predicate on X.

A Boolean-valued function (sometimes called a predicate or a proposition) is a function of the type f : X → B, where X is an arbitrary set and where B is a Boolean domain, i.e. a generic two-element set, (for example B = {0, 1}), whose elements are interpreted as logical values, for example, 0 = false and 1 = true, i.e., a single bit of information.

### Indicator function

**characteristic functionmembership functionindicator**

So, for example, when a theory defines the concept of a relation, then a predicate is simply the characteristic function (otherwise known as the indicator function) of a relation. In set theory, 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 other contexts, such as computer science, this would more often be described as a boolean predicate function (to test set inclusion).

### Proposition

**propositionspropositionalclaim**

Here, P(x) is referred to as the predicate, and x the placeholder of the proposition.

Aristotelian logic identifies a proposition as a sentence which affirms or denies a predicate of a subject with the help of a 'Copula'.

### Propositional function

**predicates**

Sometimes, P(x) is also called a (template in the role of) propositional function, as each choice of the placeholder x produces a proposition.

As a mathematical function, A(x) or A(x, x, ..., x), the propositional function is abstracted from predicates or propositional forms.

### Arity

**nullarymonadicarities**

In propositional logic, atomic formulas are called propositional variables. In a sense, these are nullary (i.e. 0-arity) predicates.

The arity of a relation (or predicate) is the dimension of the domain in the corresponding Cartesian product.

### First-order logic

**predicate logicfirst-orderpredicate calculus**

In first-order logic, an atomic formula consists of a predicate symbol applied to an appropriate number of terms.

While propositional logic deals with simple declarative propositions, first-order logic additionally covers predicates and quantification.

### Set-builder notation

**set builder notationbuild the setsabstraction**

In set theory, 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 this form, set builder notation has three parts: a variable, a colon or vertical bar separator, and a logical predicate.

### Atomic formula

**atomatomicatomic expressions**

The precise semantic interpretation of an atomic formula and an atomic sentence will vary from theory to theory.

that is, a proposition is recursively defined to be an n-ary predicate P whose arguments are terms t k, or an expression composed of logical connectives (and, or) and quantifiers (for-all, there-exists) used with other propositions.

### Propositional variable

In propositional logic, atomic formulas are called propositional variables. In a sense, these are nullary (i.e. 0-arity) predicates.

Propositional variables are represented as nullary predicates in first order logic.

### Propositional calculus

**propositional logicpropositionalsentential logic**

In propositional logic, atomic formulas are called propositional variables. In a sense, these are nullary (i.e. 0-arity) predicates.

First-order logic (a.k.a. first-order predicate logic) results when the "atomic sentences" of propositional logic are broken up into terms, variables, predicates, and quantifiers, all keeping the rules of propositional logic with some new ones introduced.

### Opaque predicate

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 run time.

### Mathematical logic

**formal logicsymbolic logiclogic**

In mathematical logic, a predicate is commonly understood to be a Boolean-valued function P: X→ {true, false}, called the predicate on X.

### Set theory

**axiomatic set theoryset-theoreticset**

In set theory, 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. However, not all theories have relations, or are founded on set theory, and so one must be careful with the proper definition and semantic interpretation of a predicate.

### Property (philosophy)

**propertypropertiesattributes**

Predicates are also commonly used to talk about the properties of objects, by defining the set of all objects that have some property in common.

### Template processor

**template enginetemplatingtemplating language**

Sometimes, P(x) is also called a (template in the role of) propositional function, as each choice of the placeholder x produces a proposition.

### Boolean expression

**Booleanboolean "orboolean expressions**

A simple form of predicate is a Boolean expression, in which case the inputs to the expression are themselves Boolean values, combined using Boolean operations.

### Function (mathematics)

**functionfunctionsmathematical function**

In set theory, 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.

### Truth value

**truth-valuelogical valuetruth values**

In set theory, 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.

### Autoepistemic logic

**crisp known/unknown values**

In autoepistemic logic, which rejects the law of excluded middle, predicates may be true, false, or simply unknown; i.e. a given collection of facts may be insufficient to determine the truth or falsehood of a predicate.

### Law of excluded middle

**law of the excluded middleexcluded middletertium non datur**

In autoepistemic logic, which rejects the law of excluded middle, predicates may be true, false, or simply unknown; i.e. a given collection of facts may be insufficient to determine the truth or falsehood of a predicate.

### Fuzzy logic

**fuzzyfuzzy logicsfuzziness**

In fuzzy logic, predicates are the characteristic functions of a probability distribution. That is, the strict true/false valuation of the predicate is replaced by a quantity interpreted as the degree of truth.

### Characteristic function (probability theory)

**characteristic functioncharacteristic functionscharacteristic function:**

In fuzzy logic, predicates are the characteristic functions of a probability distribution. That is, the strict true/false valuation of the predicate is replaced by a quantity interpreted as the degree of truth.

### Probability distribution

**distributioncontinuous probability distributiondiscrete probability distribution**

In fuzzy logic, predicates are the characteristic functions of a probability distribution. That is, the strict true/false valuation of the predicate is replaced by a quantity interpreted as the degree of truth.

### Free variables and bound variables

**free variablesbound variablebound**

Free variables and bound variables