[Inquiry] Re: Information = Comprehension x Extension -- Commentary

Mon Nov 22 12:30:12 CST 2004

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

ICE.  Commentary Note 6

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

Before we return to Peirce's description of a near duality
between icons and indices, involving a reciprocal symmetry
between intensions and extensions of concepts that becomes
perturbed to the breaking and yet the growing point by the
receipt of a fresh bit of information, I think that it may
help to recall a few pieces of technical terminology that
Peirce introduced into this discussion.

Passage 1 --

| 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'.
|
| C.S. Peirce, 'Chronological Edition', CE 1, 459-460.
| ICE 12.  http://stderr.org/pipermail/inquiry/2004-November/001926.html

Passage 2 --

| The highest terms are therefore broadest and
| the lowest terms the narrowest.  We can take
| a term so broad that it contains all other
| spheres under it.  Then it will have no
| content whatever.  There is but one
| such term -- with its synonyms --
| it is 'Being'.  We can also take a
| term so low that it contains all other
| content within it.  Then it will have no
| sphere whatever.  There is but one such term --
| it is 'Nothing'.
|
| o------------------------o------------------------o
| | ` ` ` ` ` ` ` ` ` ` ` `| ` ` ` ` ` ` ` ` ` ` ` `|
| | `Being` ` ` ` ` ` ` ` `| `Nothing` ` ` ` ` ` ` `|
| | ` ` ` ` ` ` ` ` ` ` ` `| ` ` ` ` ` ` ` ` ` ` ` `|
| | `All breadth` ` ` ` ` `| `All depth` ` ` ` ` ` `|
| | ` ` ` ` ` ` ` ` ` ` ` `| ` ` ` ` ` ` ` ` ` ` ` `|
| | `No depth ` ` ` ` ` ` `| `No breadth ` ` ` ` ` `|
| | ` ` ` ` ` ` ` ` ` ` ` `| ` ` ` ` ` ` ` ` ` ` ` `|
| o------------------------o------------------------o
|
| We can conceive of terms so narrow that they are next to nothing,
| that is have an absolutely individual sphere.  Such terms would be
| innumerable in number.  We can also conceive of terms so high that
| they are next to 'being', that is have an entirely simple content.
| Such terms would also be innumerable.
|
| o------------------------o------------------------o
| | ` ` ` ` ` ` ` ` ` ` ` `| ` ` ` ` ` ` ` ` ` ` ` `|
| | `Simple terms ` ` ` ` `| `Individual terms ` ` `|
| | ` ` ` ` ` ` ` ` ` ` ` `| ` ` ` ` ` ` ` ` ` ` ` `|
| o------------------------o------------------------o
|
| But such terms though conceivable in one sense --
| that is intelligible in their conditions --
| are yet impossible.
| You never can narrow down to an individual.
| Do you say Daniel Webster is an individual?
| He is so in common parlance,
| but in logical strictness he is not.
| We think of certain images in our memory --
| a platform and a noble form uttering convincing and patriotic words --
| a statue --
| certain printed matter --
| and we say that which
| that speaker and the
| man whom that statue
| was taken for and the
| writer of this speech --
| that which these are in
| common is Daniel Webster.
| Thus, even the proper name
| of a man is a general term or
| the name of a class, for it names
| a class of sensations and thoughts.
| The true individual term the absolutely
| singular 'this' & 'that' cannot be reached.
| Whatever has comprehension must be general.
|
| In like manner, it is impossible to find any simple term.
| This is obvious from this consideration.  If there is
| any simple term, simple terms are innumerable for in
| that case all attributes which are not simple are
| made up of simple attributes.  Now none of these
| attributes can be affirmed or denied universally
| of whatever has any one.  For let 'A' be one
| simple term and 'B' be another.  Now suppose
| we can say All 'A' is 'B';  then 'B' is
| contained in 'A'.  If, therefore, 'A'
| contains anything but 'B' it is
| a compound term, but 'A' is
| different from 'B', and is
| simple;  hence it cannot
| be that All 'A' is 'B'.
| Suppose No 'A' is 'B', then
| not-'B' is contained in 'A';
| if therefore 'A' contains anything
| besides not-'B' it is not a simple term;
| but if it is the same as not-'B', it is not a
| simple term but is a term relative to 'B'.  Now it is a
| simple term and therefore Some 'A' is 'B'.  Hence if we take
| any two simple terms and call one 'A' and the other 'B' we have
|
|       Some 'A' is 'B'
|
| and   Some 'A' is not 'B'
|
| or in other words the universe will contain every possible kind of thing
| afforded by the permutation of simple qualities.  Now the universe does not
| contain all these things;  it contains no 'well-known green horse'.  Hence the
| consequence of supposing a simple term to exist is an error of fact.  There
| are several other ways of showing this besides the one that I have adopted.
| They all concur to show that whatever has extension must be composite.
|
| C.S. Peirce, 'Chronological Edition', CE 1, 460-461.
| ICE 13.  http://stderr.org/pipermail/inquiry/2004-November/001927.html
| ICE 14.  http://stderr.org/pipermail/inquiry/2004-November/001928.html
| ICE 15.  http://stderr.org/pipermail/inquiry/2004-November/001929.html

Passage 3 --

| The moment, then, that we pass from nothing and the vacuity of being
| to any content or sphere, we come at once to a composite content and
| sphere.  In fact, extension and comprehension -- like space and time --
| are quantities which are not composed of ultimate elements;  but
| every part however small is divisible.
|
| The consequence of this fact is that when we wish to enumerate the
| sphere of a term -- a process termed 'division' -- or when we wish
| to run over the content of a term -- a process called 'definition' --
| since we cannot take the elements of our enumeration singly but must
| take them in groups, there is danger that we shall take some element
| twice over, or that we shall omit some.  Hence the extension and
| comprehension which we know will be somewhat indeterminate.  But
| we must distinguish two kinds of these quantities.  If we were to
| subtilize we might make other distinctions but I shall be content
| with two.  They are the extension and comprehension relatively to
| our actual knowledge, and what these would be were our knowledge
| perfect.
|
| Logicians have hitherto left the doctrine of extension
| and comprehension in a very imperfect state owing to the
| blinding influence of a psychological treatment of the
| matter.  They have, therefore, not made this distinction
| and have reduced the comprehension of a term to what it
| would be if we had no knowledge of fact at all.  I mention
| this because if you should come across the matter I am now
| discussing in any book, you would find the matter left in
| quite a different state.
|
| C.S. Peirce, 'Chronological Edition', CE 1, 462.
| ICE 16.  http://stderr.org/pipermail/inquiry/2004-November/001930.html

Jon Awbrey

---------------------------------------------------------------------------------

In: ICE 00.  http://forum.wolframscience.com/showthread.php?threadid=609
Re: ICE 12.  http://forum.wolframscience.com/showthread.php?postid=2008#post2008
Re: ICE 13.  http://forum.wolframscience.com/showthread.php?postid=2009#post2009
Re: ICE 14.  http://forum.wolframscience.com/showthread.php?postid=2010#post2010
Re: ICE 15.  http://forum.wolframscience.com/showthread.php?postid=2011#post2011
Re: ICE 16.  http://forum.wolframscience.com/showthread.php?postid=2012#post2012

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
inquiry e-lab: http://stderr.org/pipermail/inquiry/
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o