Intension

In linguistics, logic, philosophy, and other fields, an intension is any property or quality connoted by a word, phrase, or another symbol.[1] In the case of a word, the word's definition often implies an intension. For instance, the intensions of the word plant include properties such as "being composed of cellulose", "alive", and "organism", among others. A comprehension is the collection of all such intensions.

This article is about referential context. For the identically pronounced term referring to a future commitment, see Intention. For the state of having intension, see intentionality. For the song "Intension" by Tool, see 10,000 Days (Tool album).

The term "intension" and "extension" were introduced by Rudolf Carnap.[2]

Overview

The meaning of a word can be thought of as the bond between the idea the word means and the physical form of the word. Swiss linguist Ferdinand de Saussure (1857–1913) contrasts three concepts:

  1. the signifier – the "sound image" or the string of letters on a page that one recognizes as the form of a sign
  2. the signified – the meaning, the concept or idea that a sign expresses or evokes
  3. the referent – the actual thing or set of things a sign refers to. See Dyadic signs and Reference (semantics).

Without intension of some sort, a word has no meaning.[3] For instance, the terms rantans or brillig have no intension and hence no meaning. Such terms may be suggestive, but a term can be suggestive without being meaningful. For instance, ran tan is an archaic onomatopoeia for chaotic noise or din and may suggest to English speakers a din or meaningless noise, and brillig though made up by Lewis Carroll may be suggestive of 'brilliant' or 'frigid'. Such terms, it may be argued, are always intensional since they connote the property 'meaningless term', but this is only an apparent paradox and does not constitute a counterexample to the claim that without intension a word has no meaning. Part of its intension is that it has no extension Intension is analogous to the signified in the Saussurean system, extension to the referent.

In philosophical arguments about dualism versus monism, it is noted that thoughts have intensionality and physical objects do not (S. E. Palmer, 1999), but rather have extension in space and time.

Statement forms

A statement-form is simply a form obtained by putting blanks into a sentence where one or more expressions with extensions occur—for instance, "The quick brown ___ jumped over the lazy ___'s back." An instance of the form is a statement obtained by filling the blanks in.

Intensional statement form

An intensional statement-form is a statement-form with at least one instance such that substituting co-extensive expressions into it does not always preserve logical value. An intensional statement is a statement that is an instance of an intensional statement-form. Here co-extensive expressions are expressions with the same extension.

That is, a statement-form is intensional if it has, as one of its instances, a statement for which there are two co-extensive expressions (in the relevant language) such that one of them occurs in the statement, and if the other one is put in its place (uniformly, so that it replaces the former expression wherever it occurs in the statement), the result is a (different) statement with a different logical value. An intensional statement, then, is an instance of such a form; it has the same form as a statement in which substitution of co-extensive terms fails to preserve logical value.

Examples
  1. Everyone who has read Huckleberry Finn knows that Mark Twain wrote it.
  2. It is possible that Aristotle did not tutor Alexander the Great.
  3. Aristotle was pleased that he had a sister.

To see that these are intensional, make the following substitutions: (1) "Mark Twain" → "The author of 'Corn-pone Opinions'"; (2) "Aristotle" → "the tutor of Alexander the Great"; (3) can be seen to be intensional given "had a sister" → "had a female sibling."

The intensional statements above feature expressions like "knows", "possible", and "pleased". Such expressions always, or nearly always, produce intensional statements when added (in some intelligible manner) to an extensional statement, and thus they (or more complex expressions like "It is possible that") are sometimes called intensional operators. A large class of intensional statements, but by no means all, can be spotted from the fact that they contain intensional operators.

Extensional statement form

An extensional statement is a non-intensional statement. Substitution of co-extensive expressions into it always preserves logical value. A language is intensional if it contains intensional statements, and extensional otherwise. All natural languages are intensional.[4] The only extensional languages are artificially constructed languages used in mathematical logic or for other special purposes and small fragments of natural languages.

Examples
  1. Mark Twain wrote Huckleberry Finn.
  2. Aristotle had a sister.

Note that if "Samuel Clemens" is put into (1) in place of "Mark Twain", the result is as true as the original statement. It should be clear that no matter what is put for "Mark Twain", so long as it is a singular term picking out the same man, the statement remains true. Likewise, we can put in place of the predicate any other predicate belonging to Mark Twain and only to Mark Twain, without changing the logical value. For (2), likewise, consider the following substitutions: "Aristotle" → "The tutor of Alexander the Great"; "Aristotle" → "The author of the 'Prior Analytics'"; "had a sister" → "had a sibling with two X-chromosomes"; "had a sister" → "had a parent who had a non-male child".

Significance

Intensional languages cannot be given an adequate semantics in terms of the extensions of expressions in them, since the extensions themselves do not suffice to determine a logical value. (If they did, then one could not change the logical value by substituting co-extensive expressions.) On the other hand, for the first half of the 20th century the only known systems of formal semantics worked by assigning extensions to expressions and used a Tarski-style truth-definition of statements constructed from the primitive expressions of the language under analysis. Hence, these semantical methods were pathetically inadequate for understanding the semantics of any but a few small artificial languages or mutilated fragments of natural languages.

This situation changed in the 1960s with the invention of possible-world or "intensional" semantics, the main form of which is due to Saul Kripke. Though this has enabled improvements in the semantic modelling of natural languages, much work remains to be done.

gollark: To clarify, do you mean how fast you can type/read/whatever or more general stuff?
gollark: Not that I'm some sort of engineered superhuman, but still.
gollark: I generally don't find myself running at anywhere near my maximum I/O data rate.
gollark: I have heard it said that Google and such aren't that efficiently run, but just have money-printers operating somewhere.
gollark: GPUs can go up to many tens of TFLOP/s but only have a few tens of gigabytes of memory, which is 3 OOM off.

See also

Notes
  1. Antony Flew (1979). Dictionary of Philosophy. p. 117.
  2. Fitting, Melvin. "Intensional logic". In Zalta, Edward N. (ed.). Stanford Encyclopedia of Philosophy.
  3. Putnam, Hilary (1973). "Meaning and Reference". The Journal of Philosophy. 70 (19): 699–711. doi:10.2307/2025079. JSTOR 2025079.
  4. Carnap, Rudolf (April 1955). "Meaning and synonymy in natural languages". Philosophical Studies. 6 (3): 33–47. doi:10.1007/BF02330951. ISSN 0031-8116.

References
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