Extension (semantics)

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.

= In mathematics =

In mathematics, the extension of a mathematical concept is the set (mathematics) that is specified by that concept.

For example, the extension of a function (mathematics) 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 (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 (by convention) as its truth-value. So the extension of Lassie is famous is the truth-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.

= 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.)

A similar problem arises for objects that no longer exist. The extention 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. See for example extension, and the [http://esgs.free.fr/de/ext.htm extensional devices.]