Extension (semantics)
Encyclopedia
In any of several studies that treat the use of sign
s - for example, in linguistics
, logic
, mathematics
, semantics
, and semiotics
- the extension of a concept, idea, or sign
consists of the things to which it applies, in contrast with its comprehension
or intension
, which consists very roughly of the ideas, properties, or corresponding signs that are implied or suggested by the concept in question.
In philosophical semantics
or the philosophy of language
, 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 truth value. 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.
, the 'extension' of a mathematical concept is the set that is specified by that concept.
For example, the extension of a function
is a set of ordered pair
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
, such as a group
, 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
in axiomatic set theory.
This kind of extension is used so constantly in contemporary mathematics based on set theory
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".
, some database
textbooks use the term 'intension' to refer to the schema
of a database, and 'extension' to refer to particular instance
s of a database.
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 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
'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
is one attempt to avoid some of these problems.
rely heavily on a valuation of extension over intension
. See for example extension, and the extensional devices.
Sign (semiotics)
A sign is understood as a discrete unit of meaning in semiotics. It is defined as "something that stands for something, to someone in some capacity" It includes words, images, gestures, scents, tastes, textures, sounds – essentially all of the ways in which information can be...
s - for example, in linguistics
Linguistics
Linguistics is the scientific study of human language. Linguistics can be broadly broken into three categories or subfields of study: language form, language meaning, and language in context....
, logic
Logic
In philosophy, Logic is the formal systematic study of the principles of valid inference and correct reasoning. Logic is used in most intellectual activities, but is studied primarily in the disciplines of philosophy, mathematics, semantics, and computer science...
, mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, semantics
Semantics
Semantics is the study of meaning. It focuses on the relation between signifiers, such as words, phrases, signs and symbols, and what they stand for, their denotata....
, and semiotics
Semiotics
Semiotics, also called semiotic studies or semiology, is the study of signs and sign processes , indication, designation, likeness, analogy, metaphor, symbolism, signification, and communication...
- the extension of a concept, idea, or sign
Sign (semiotics)
A sign is understood as a discrete unit of meaning in semiotics. It is defined as "something that stands for something, to someone in some capacity" It includes words, images, gestures, scents, tastes, textures, sounds – essentially all of the ways in which information can be...
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
In linguistics, logic, philosophy, and other fields, an intension is any property or quality connoted by a word, phrase or other symbol. In the case of a word, it is often implied by the word's definition...
, 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. It focuses on the relation between signifiers, such as words, phrases, signs and symbols, and what they stand for, their denotata....
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 philosophers is concerned with four central problems: the nature of meaning, language use, language cognition, and the relationship between language...
, 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
Monad (Greek philosophy)
Monad , according to the Pythagoreans, was a term for Divinity or the first being, or the totality of all beings, Monad being the source or the One meaning without division....
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
Lassie
Lassie is a fictional collie dog character created by Eric Knight in a short story expanded to novel length called Lassie Come-Home. Published in 1940, the novel was filmed by MGM in 1943 as Lassie Come Home with a dog named Pal playing Lassie. Pal then appeared with the stage name "Lassie" in six...
, Rex, and so on. The extension of the phrase "Wikipedia reader" includes each person who has ever read Wikipedia, including you
You
You is the second-personpersonal pronoun, both singular and plural, and both nominative and objective case, in Modern English. The oblique/objective form you functioned originally as both accusative and dative)...
.
The extension of a whole statement, as opposed to a word or phrase, is defined (since Frege 1892) as its truth value. 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 mathematicsMathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the 'extension' of a mathematical concept is the set that is specified by that concept.
For example, the extension of a function
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
is a set of ordered pair
Ordered pair
In mathematics, an ordered pair is a pair of mathematical objects. In the ordered pair , the object a is called the first entry, and the object b the second entry of the pair...
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 groups, rings, fields, modules, vector spaces, and algebras...
, such as a group
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
, 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.- Formal statement :...
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 sets, 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 scienceComputer science
Computer science or computing 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 an organized collection of data for one or more purposes, usually in digital form. The data are typically organized to model relevant aspects of reality , in a way that supports processes requiring this information...
textbooks use the term 'intension' to refer to the schema
Logical schema
A Logical Schema is a data model of a specific 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 instance
Row (database)
In the context of a relational database, a row—also called a record or tuple—represents a single, implicitly structured data item in a table. In simple terms, a database table can be thought of as consisting of rows and columns or fields...
s of a database.
Metaphysical implications
There is an ongoing controversy in metaphysicsMetaphysics
Metaphysics is a branch of philosophy concerned with explaining the fundamental nature of being and the world, although the term is not easily defined. Traditionally, metaphysics attempts to answer two basic questions in the broadest possible terms:...
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 Worlds , by Peter Porter...
"--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 DL was a Scottish physician and writer, most noted for his stories about the detective Sherlock Holmes, generally considered a milestone 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
A free logic is a logic with fewer existential presuppositions than classical logic. Free logics may allow for terms that do not denote any object. Free logics may also allow models that have an empty domain...
is one attempt to avoid some of these problems.
General semantics
Some fundamental formulations in the field of general semanticsGeneral Semantics
General semantics is a program begun in the 1920's that seeks to regulate the evaluative operations performed in the human brain. After partial program launches under the trial names "human engineering" and "humanology," Polish-American originator Alfred Korzybski fully launched the program as...
rely heavily on a valuation of extension over intension
Intension
In linguistics, logic, philosophy, and other fields, an intension is any property or quality connoted by a word, phrase or other symbol. In the case of a word, it is often implied by the word's definition...
. See for example extension, and the extensional devices.
See also
- Enumerative definitionEnumerative definitionAn enumerative definition of a concept or term is a special type of extensional definition that gives an explicit and exhaustive listing of all the objects that fall under the concept or term in question...
- Extensional definitionExtensional definitionAn 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 definitionIntensional definitionIn 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....
- Sense and referenceSense and referenceSinn and bedeutung are usually translated, respectively, as sense and reference. Two different aspects of some terms' meanings, a term's reference is the object that the term refers to, while the term's sense is the way that the term refers to that object.Sinn and bedeutung were introduced by...