Intension refers to the logical or definitional conditions that specify the set of all possible things a word or phrase could describe, while extension refers to the set of all actual things the word or phrase describes.
In linguistics, logic, philosophy, and other fields, an intension is any property or quality or state of affairs connoted by a word, phrase or other symbol. In case of a word, it is often implied by its definition. The term may also refer to the complete set of meanings or properties that are implied by a concept, although the term comprehension is technically more correct for this.
Intension is generally discussed with regard to extension (or denotation). For example, the intension of a car is the all-inclusive concept of a car, including, for example, mile-long cars made of chocolate that may not actually exist. But the extension of a car is all actual instances of cars (past, present, and future), which will amount to millions or billions of cars, but probably does not include any mile-long cars made of chocolate.
Intension is analogous to the signified extension to the referent. The intension thus links the signifier to the sign's extension. Without intension of some sort, words can have no meaning.
Intension and intensionality (the state of having intension) should not be confused with intention and intentionality, which are pronounced the same and occasionally arise in the same philosophical context. Where this happens, the letter 's' or 't' is sometimes italicized to emphasize the distinction.
In any of several studies that treat the use of signs, for example, 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.
In the context of formal logic, the extension of a whole statement, as opposed to a word or phrase, is sometimes defined (arguably by convention) as its logical value. So, in that view, the extension of "Lassie is famous" is the logical value true, since Lassie is famous.
Some concepts and expressions are such that they do not 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 (precedes) the second one.
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 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."
Russell's paradox, discovered by Bertrand Russell, specified as "the set of all sets that are not members of themselves," was an interesting case of a specification of a set (a supposed intension) that could not be satisfied—it could have no extension—because the intension specified in the definition of that set led to a contradiction. The result of the discovery of Russell's paradox was to show that the so-called naive set theory of Gottlob Frege required revision because Frege had thought that any specifiable condition (intension) should be able to define a set (extension), but this assumption was shown by Russell to be false. This required a revision of the axioms of set theory so that they would not permit such contradictory membership conditions (such contradictory intensions) to be specified within the system. Russell and Whitehead's solution (in their work Principia Mathematica) was to set up a theory of types, in which membership was restricted to a given type, and there were different levels (or types) of membership. Other set theories have coped with the problem in different ways.
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. The distinction, however, is the same: intension is the logical specification of something, whereas extension is the set of objects or other things that satisfy the conditions of the logical specification given in the intension.
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 extension of the term "Socrates," for example, seems to be a (currently) non-existent object. Bertrand Russell dealt with this problem by means of his theory of definite descriptions. He used as an example the statement, "The present king of France is bald." But there is no present king of France. How then shall we deal with such statements, that purport to be about someone or something, but that purported someone or something does not, in fact, exist? Are such statements true? False? Meaningless?
Russell, proposed that when we say "the present King of France is bald," we are making three separate assertions:
Since assertion 1 is plainly false, and our statement is the conjunction of all three assertions, our statement is false.
Free logic is another attempt to avoid some of these problems.
Some fundamental formulations in the field of general semantics rely heavily on a valuation of extension over intension.
All links retrieved March 4, 2018.
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