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In any of several studies that treat the use of sign
Sign (semiotics)

In semiotics, a sign is "something that stands for something else, to someone in some capacity". It may be understood as a discrete unit of Meaning , and includes words, images, gestures, scents, tastes, textures, sounds – essentially all of the ways in which information can be communicated as a message by any sentient, reasoning m...
s, for example in linguistics
Linguistics

Linguistics is the science study of natural language. Linguistics encompasses a number of sub-fields. An important topical division is between the study of language structure and the study of Meaning ....
, logic
Logic

Logic is the study of the principles of valid demonstration and inference. Logic is a branch of philosophy, a part of the classical Trivium . The word derives from Greek language ?????? , fem....
, mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, semantics
Semantics

Semantics is the study of meaning in communication. The word is derived from the Greek language word s??a?t???? , "significant", from s??a??? , "to signify, to indicate" and that from s??a , "sign, mark, token"....
, and semiotics
Semiotics

'Semiotics, also called semiotic studies or semiology, is the study of sign processes , or signification and communication, sign and symbols, both individually and grouped into sign systems....
, the extension of a concept, idea, or sign
Sign (semiotics)

In semiotics, a sign is "something that stands for something else, to someone in some capacity". It may be understood as a discrete unit of Meaning , and includes words, images, gestures, scents, tastes, textures, sounds – essentially all of the ways in which information can be communicated as a message by any sentient, reasoning m...
 consists of the things to which it applies, in contrast with its comprehension
Comprehension (logic)

In logic, the comprehension of an object is the totality of intensions, that is, attributes, characters, marks, properties, or qualities, that the object possesses, or else the totality of intensions that are pertinent to the context of a given discussion....
 or intension
Intension

Intension refers to the possible things a word or phrase could describe. It stands in contradistinction to extension , which refers to the actual things the word or phrase does describe....
, which consists very roughly of the ideas, properties, or corresponding signs that are implied or suggested by the concept in question.

In philosophical semantics
Semantics

Semantics is the study of meaning in communication. The word is derived from the Greek language word s??a?t???? , "significant", from s??a??? , "to signify, to indicate" and that from s??a , "sign, mark, token"....
 or the philosophy of language
Philosophy of language

Philosophy of language is the reasoned inquiry into the nature, origins, and usage of language. As a topic, the philosophy of language for Analytic philosophys is concerned with four central problems: the nature of Meaning , language use, language cognition, and the relationship between language and reality....
, the 'extension' of a concept or expression is the set of things it extends to, or applies to, if it is the sort of concept or expression that a single object by itself can satisfy.






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In any of several studies that treat the use of sign
Sign (semiotics)

In semiotics, a sign is "something that stands for something else, to someone in some capacity". It may be understood as a discrete unit of Meaning , and includes words, images, gestures, scents, tastes, textures, sounds – essentially all of the ways in which information can be communicated as a message by any sentient, reasoning m...
s, for example in linguistics
Linguistics

Linguistics is the science study of natural language. Linguistics encompasses a number of sub-fields. An important topical division is between the study of language structure and the study of Meaning ....
, logic
Logic

Logic is the study of the principles of valid demonstration and inference. Logic is a branch of philosophy, a part of the classical Trivium . The word derives from Greek language ?????? , fem....
, mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, semantics
Semantics

Semantics is the study of meaning in communication. The word is derived from the Greek language word s??a?t???? , "significant", from s??a??? , "to signify, to indicate" and that from s??a , "sign, mark, token"....
, and semiotics
Semiotics

'Semiotics, also called semiotic studies or semiology, is the study of sign processes , or signification and communication, sign and symbols, both individually and grouped into sign systems....
, the extension of a concept, idea, or sign
Sign (semiotics)

In semiotics, a sign is "something that stands for something else, to someone in some capacity". It may be understood as a discrete unit of Meaning , and includes words, images, gestures, scents, tastes, textures, sounds – essentially all of the ways in which information can be communicated as a message by any sentient, reasoning m...
 consists of the things to which it applies, in contrast with its comprehension
Comprehension (logic)

In logic, the comprehension of an object is the totality of intensions, that is, attributes, characters, marks, properties, or qualities, that the object possesses, or else the totality of intensions that are pertinent to the context of a given discussion....
 or intension
Intension

Intension refers to the possible things a word or phrase could describe. It stands in contradistinction to extension , which refers to the actual things the word or phrase does describe....
, which consists very roughly of the ideas, properties, or corresponding signs that are implied or suggested by the concept in question.

In philosophical semantics
Semantics

Semantics is the study of meaning in communication. The word is derived from the Greek language word s??a?t???? , "significant", from s??a??? , "to signify, to indicate" and that from s??a , "sign, mark, token"....
 or the philosophy of language
Philosophy of language

Philosophy of language is the reasoned inquiry into the nature, origins, and usage of language. As a topic, the philosophy of language for Analytic philosophys is concerned with four central problems: the nature of Meaning , language use, language cognition, and the relationship between language and reality....
, the 'extension' of a concept or expression is the set of things it extends to, or applies to, if it is the sort of concept or expression that a single object by itself can satisfy. Concepts and expressions of this sort are monadic or "one-place" concepts and expressions.

So the extension of the word "dog" is the set of all (past, present and future) dogs in the world: the set includes Fido, Rover, Lassie, Rex, and so on. The extension of the phrase "Wikipedia reader" includes each person who has ever read Wikipedia, including you.

The extension of a whole statement, as opposed to a word or phrase, is defined (since Frege 1892) as its logical truth value
Logical value

In logic and mathematics, a logical value, also called a truth value, is a value indicating the extent to which a proposition is truth.In classical logic, the only possible truth values are true and false....
. So the extension of "Lassie is famous" is the logical value 'true', since Lassie 'is' famous.

Some concepts and expressions are such that they don't apply to objects individually, but rather serve to relate objects to objects. For example, the words "before" and "after" do not apply to objects individually — it makes no sense to say "Jim is before" or "Jim is after" — but to one thing in relation to another, as in "The wedding is before the reception" and "The reception is after the wedding". Such "relational" or "polyadic" ("many-place") concepts and expressions have, for their extension, the set of all sequences of objects that satisfy the concept or expression in question. So the extension of "before" is the set of all (ordered) pairs of objects such that the first one is before the second one.

Mathematics


In mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, the 'extension' of a mathematical concept is the set that is specified by that concept.

For example, the extension of a function
Function (mathematics)

The mathematical concept of a function expresses dependence between two quantities, one of which is known and the other which is produced. A function associates a single output to each input element drawn from a fixed Set , such as the real numbers , although different inputs may have the same output....
 is a set of ordered pair
Ordered pair

In mathematics, an ordered pair is a collection of two distinguishable objects, one being the first coordinate system , and the other being the second coordinate ....
s that pair up the arguments and values of the function; in other words, the function's graph. The extension of an object in abstract algebra
Abstract algebra

Abstract algebra is the subject area of mathematics that studies algebraic structures, such as group , ring , field , module , vector spaces, and algebra over a field....
, such as a group
Group (mathematics)

In mathematics, a group is an algebraic structure consisting of a set together with an Binary operation that combines any two of its element to form a third element....
, is the underlying set of the object. The extension of a set is the set itself. That a set can capture the notion of the extension of anything is the idea behind the axiom of extensionality
Axiom of extensionality

In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom of extensionality, or axiom of extension, is one of the axioms of Zermelo-Fraenkel set theory....
 in axiomatic set theory.

This kind of extension is used so constantly in contemporary mathematics based on set theory
Set theory

Set theory is the branch of mathematics that studies Set , which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics....
 that it can be called an implicit assumption. It can mean different things in different cases, and there is no universal definition of the term "extension".

Computer science


In computer science
Computer science

Computer science is the study of the theoretical foundations of information and computation, and of practical techniques for their implementation and application in computer systems....
, some database
Database

A database is a structured collection of records or data that is stored in a computer system. The structure is achieved by organizing the data according to a database model....
 textbooks use the term 'intension' to refer to the schema
Logical schema

A Logical Schema is a data model problem domain expressed in terms of a particular data management technology. Without being specific to a particular database management product, it is in terms of either relational tables and columns, object-oriented classes, or XML tags....
 of a database, and 'extension' to refer to particular instances of a database.

Metaphysical implications


There is an ongoing controversy in metaphysics
Metaphysics

Metaphysics investigates principles of reality transcending those of any particular science. cosmology and ontology are traditional branches of metaphysics....
 about whether or not there are, in addition to actual, existing things, non-actual or nonexistent things. If there are--if, for instance, there are possible but non-actual dogs (dogs of some non-actual but possible species, perhaps) or nonexistent beings (like Sherlock Holmes, perhaps), then these things might also figure in the extensions of various concepts and expressions. If not, only existing, actual things can be in the extension of a concept or expression. Note that "actual" may not mean the same as "existing". Perhaps there exist things that are merely possible, but not actual. (Maybe they exist in other universes, and these universes are other "possible worlds
Possible Worlds

Possible Worlds may refer to:* Possible worlds, a concept in philosophy* Possible Worlds , by John Mighton** Possible Worlds , by Robert Lepage, based on the Mighton play...
"--possible alternatives to the actual world.) Perhaps some actual things are nonexistent. (Sherlock Holmes seems to be an 'actual' example of a fictional character; one might think there are many other characters Arthur Conan Doyle
Arthur Conan Doyle

Sir Arthur Ignatius Conan Doyle, Deputy Lieutenant was a Scotland author most noted for his stories about the Detective fiction Sherlock Holmes, which are generally considered a major innovation in the field of crime fiction, and for the adventures of Professor Challenger....
 'might' have invented, though he actually invented Holmes.)

A similar problem arises for objects that no longer exist. The extension of the term "Socrates", for example, seems to be a (currently) non-existent object. Free logic
Free logic

Free logic is a logic with no existential clause presuppositions. Alternatively, it is a logic whose theorems are valid in all domains, including the empty domain....
 is one attempt to avoid some of these problems.

General semantics


Some fundamental formulations in the field of general semantics
General Semantics

General Semantics is a non-Aristotelian educational discipline created by Alfred Korzybski during the years 1919 to 1933. General Semantics is distinct from semantics , a different subject....
 rely heavily on a valuation of extension over intension
Intension

Intension refers to the possible things a word or phrase could describe. It stands in contradistinction to extension , which refers to the actual things the word or phrase does describe....
. See for example extension
Extension

Extension may refer to:* A List of cheerleading stunts* The building of community capacity by outsiders, for instance agricultural extension* Extension , relating to the pulling apart of the Earth's crust and lithosphere...
, and the

See also


  • Enumerative definition
    Enumerative definition

    An enumerative definition of a concept or term is a special type of extensional definition that gives an explicit and exhaustive listing of all the object s that fall under the concept or term in question....
  • Extensional definition
    Extensional definition

    An extensional definition of a concept or term formulates its meaning by specifying its extension , that is, every object that falls under the definition of the concept or term in question....
  • Intensional definition
    Intensional definition

    In logic and mathematics, an intensional definition gives the Meaning of a term by specifying all the properties required to come to that definition, that is, the necessary and sufficient conditions for belonging to the set being defined....


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