[Inquiry] Re: Extension x Comprehension = Information

[Jon Awbrey]

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ECI.  Commentary Note 4

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I'm going to make yet another try at following the links that Peirce makes
among conventions, disjunctive terms, indexical signs, and inductive rules.
For this purpose, I'll break the text up into smaller pieces, and pick out
just those parts of it that have to do with the indexical aspect of things.

Before I can get to this, though, I will need to deal with the uncertainty
that I am experiencing over the question as to whether a "connotation" is
just another "notation", and thus belongs to the interpretive framework,
that is, the SI-plane, or whether it is an objective property, a quality
of objects of terms.  I have decided to finesse the issue by forcing my
own brand of interpretation on the next text, where the trouble starts:

| 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.

I am going to treat Peirce's use of the "quantity consideration"
as a significant operator that transforms its argument from the
syntactic domain S |_| I to the objective domain O.

| Now the sphere considered as a quantity is called the Extension;
| and the content considered as quantity is called the Comprehension.

Taking this point of view, then, I will consider the Extensions of terms
and the Comprehensions of terms, to be "quantities", in effect, objective
formal elements that are subject to being compared with one another within
their respective domains.  In particular, I will view them as elements of
partially ordered sets.  On my reading of Peirce's text, the word "content"
is still ambiguous from context of use to context of use, but I will simply
let that be as it may, hoping that it will suffice to fix the meaning of the
more technical term "comprehension".

This is still experimental -- I'll just have to see how it works out.

Jon Awbrey

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