Extension (semantics)

From Academic Kids

The extension of an idea or (linguistic) expression consists of the things that it applies to; it contrasts with intension. This general notion is from semantics, and has been applied to some other fields.

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In mathematics

In mathematics, 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 pairs 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, of course, 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".

Computer science

In computer science, some database textbooks use the term intension to refer to the schema of a database, and extension to refer to particular instances of a database.

Semantics

In philosophical semantics or 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 the 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 (by convention) as its truth-value. So the extension of "Lassie is famous" is the truth-value truth, 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.

Metaphysical implications

There is an ongoing controversy in 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 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.)

General semantics

Some fundamental formulations in the field of general semantics rely heavily on a valuation of extension over intension. See for example the extensional devices (http://esgs.free.fr/de/ext.htm).

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