Extension (semantics)

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In any of several fields of study that treat the use of signs — for example, in
linguistics Linguistics is the scientific study of human language. It is called a scientific study because it entails a comprehensive, systematic, objective, and precise analysis of all aspects of language, particularly its nature and structure. Linguis ...
,
logic Logic is the study of correct reasoning. It includes both Mathematical logic, formal and informal logic. Formal logic is the science of Validity (logic), deductively valid inferences or of logical truths. It is a formal science investigating h ...
,
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
,
semantics Semantics (from grc, wikt:σημαντικός, σημαντικός ''sēmantikós'', "significant") is the study of reference, Meaning (philosophy), meaning, or truth. The term can be used to refer to subfields of several distinct discipline ...
,
semiotics Semiotics (also called semiotic studies) is the systematic study of sign processes (semiosis) and meaning making. Semiosis is any activity, conduct, or process that involves Sign (semiotics), signs, where a sign is defined as anything that commun ...
, and
philosophy of language In analytic philosophy, philosophy of language investigates the nature of language and the relations between language, language users, and the world. Investigations may include inquiry into the nature of Meaning (philosophy of language), meanin ...
— the extension of a concept, idea, or sign consists of the things to which it applies, in contrast with its comprehension or
intension In any of several fields of study that treat the use of signs — for example, in linguistics Linguistics is the scientific study of human language. It is called a scientific study because it entails a comprehensive, systematic, objectiv ...
, 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 (from grc, wikt:σημαντικός, σημαντικός ''sēmantikós'', "significant") is the study of reference, Meaning (philosophy), meaning, or truth. The term can be used to refer to subfields of several distinct discipline ...
or the
philosophy of language In analytic philosophy, philosophy of language investigates the nature of language and the relations between language, language users, and the world. Investigations may include inquiry into the nature of Meaning (philosophy of language), meanin ...
, 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
Gottlob Frege Friedrich Ludwig Gottlob Frege (; ; 8 November 1848 – 26 July 1925) was a German philosopher, Mathematical logic, logician, and mathematician. He was a mathematics professor at the University of Jena, and is understood by many to be the fath ...
's " On Sense and Reference") 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
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the 'extension' of a mathematical concept $C$ is the set that is specified by $C$. (That set might be empty, currently.) For example, the extension of a function is a set of
ordered pair In mathematics, an ordered pair (''a'', ''b'') is a pair of objects. The order in which the objects appear in the pair is significant: the ordered pair (''a'', ''b'') is different from the ordered pair (''b'', ''a'') unless ''a'' = ''b''. (In con ...
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 In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathematics), fields, module (mathe ...
, 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 and the branches of logic Logic is the study of correct reasoning. It includes both Mathematical logic, formal and informal logic. Formal logic is the science of Validity (logic), deductively valid inferences or of l ...
in
axiomatic 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 Mathematic ...
. This kind of extension is used so constantly in contemporary mathematics based on
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 mathematics, ...
that it can be called an implicit assumption. A typical effort in mathematics evolves out of an observed
mathematical object A mathematical object is an abstract concept arising in mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantit ...
requiring description, the challenge being to find a characterization for which the object becomes the extension.

# Computer science

In
computer science Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to Applied science, practical discipli ...
, some
database In computing, a database is an organized collection of Data (computing), data stored and accessed electronically. Small databases can be stored on a file system, while large databases are hosted on computer clusters or cloud storage. The Databas ...
textbooks use the term 'intension' to refer to the
schema The word schema comes from the Greek word ('), which means ''shape'', or more generally, ''plan''. The plural is ('). In English, both ''schemas'' and ''schemata'' are used as plural forms. Schema may refer to: Science and technology * SCHEMA ...
of a database, and 'extension' to refer to particular instances of a database.

# Metaphysical implications

There is an ongoing controversy in
metaphysics Metaphysics is the branch of philosophy that studies the fundamental nature of reality, the first principles of being, identity and change, space and time, causality, necessity, and possibility. It includes questions about the nature of conscio ...
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.

# General semantics

Some fundamental formulations in the field of general semantics rely heavily on a valuation of extension over
intension In any of several fields of study that treat the use of signs — for example, in linguistics Linguistics is the scientific study of human language. It is called a scientific study because it entails a comprehensive, systematic, objectiv ...
. See for example extension, and the extensional devices.

* Enumerative definition * Extensional definition * Extensional logic *
Generalization A generalization is a form of abstraction whereby common properties of specific instances are formulated as general concepts or claims. Generalizations posit the existence of a domain or set theory, set of elements, as well as one or more commo ...
* Sense and reference * Semantic property * Type–token distinction

Towards a Reference Terminology for Ontology Research and Development

{{Formal semantics Semantics Set theory Concepts in logic Definition Formal semantics (natural language)