ONT Re: Extension x Comprehension = Information
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| It is important to distinguish between the two functions of a word:
| 1st to denote something -- to stand for something, and 2nd to mean
| something -- or as Mr. Mill phrases it -- to 'connote' something.
|
| What it denotes is called its 'Sphere'.
| What it connotes is called its 'Content'.
| Thus the 'sphere' of the word 'man' is for
| me every man I know; and for each of you it
| is every man you know. The 'content' of 'man'
| is all that we know of all men, as being two-
| legged, having souls, having language, &c., &c.
| It is plain that both the 'sphere' and the
| 'content' admit of more and less. ...
|
| Now the sphere considered as a quantity is called the Extension;
| and the content considered as quantity is called the Comprehension.
| Extension and Comprehension are also termed Breadth and Depth. So that
| a wider term is one which has a greater extension; a narrower one is
| one which has a less extension. A higher term is one which has a
| less Comprehension and a lower one has more.
|
| The narrower term is said to be contained under the wider one;
| and the higher term to be contained in the lower one.
|
| We have then:
|
| o-----------------------------o-----------------------------o
| | | |
| | What is 'denoted' | What is 'connoted' |
| | | |
| | Sphere | Content |
| | | |
| | Extension | Comprehension |
| | | |
| | ( wider | ( lower |
| | Breadth < | Depth < |
| | ( narrower | ( higher |
| | | |
| | What is contained 'under' | What is contained 'in' |
| | | |
| o-----------------------------o-----------------------------o
|
| The principle of explicatory or deductive reasoning then is that:
|
| Every part of a word's Content belongs to
| every part of its Sphere,
|
| or:
|
| Whatever is contained 'in' a word belongs to
| whatever is contained under it.
|
| Now this maxim would not be true if the Extension and Comprehension
| were directly proportional to one another; this is to say if the
| Greater the one the greater the other. For in that case, though
| the whole Content would belong to the whole Sphere; yet only
| a particular part of it would belong to a part of that Sphere
| and not every part to every part. On the other hand if the
| Comprehension and Extension were not in some way proportional
| to one another, that is if terms of different spheres could
| have the same content or terms of the same content different
| spheres; then there would be no such fact as a content's
| 'belonging' to a sphere and hence again the maxim would
| fail. For the maxim to be true, then, it is absolutely
| necessary that the comprehension and extension should
| be inversely proportional to one another. That is
| that the greater the sphere, the less the content.
|
| Now this evidently true. If we take the term 'man' and increase
| its 'comprehension' by the addition of 'black', we have 'black man'
| and this has less 'extension' than 'man'. So if we take 'black man'
| and add 'non-black man' to its sphere, we have 'man' again, and so
| have decreased the comprehension. So that whenever the extension
| is increased the comprehension is diminished and 'vice versa'.
|
| CSP, CE 1, pages 459-460.
|
| Charles Sanders Peirce,
|"The Logic of Science, or, Induction and Hypothesis",
| Lowell Institute Lectures of 1866, pages 357-504 in:
|
|'Writings of Charles S. Peirce: A Chronological Edition',
|'Volume 1, 1857-1866', Peirce Edition Project,
| Indiana University Press, Bloomington, IN, 1982.
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