[Inquiry] Re: Extension x Comprehension = Information

[Jon Awbrey]

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ECI.  Note 10

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I continue with the out lay of my incidental musings
on the theme of "approximal inference rules" (AIR's).

Chorus:

| For this purpose, I must call your attention to
| the differences there are in the manner in which
| different representations stand for their objects.
|
| In the first place there are likenesses or copies -- such as
| 'statues', 'pictures', 'emblems', 'hieroglyphics', and the like.
| Such representations stand for their objects only so far as they
| have an actual resemblance to them -- that is agree with them in
| some characters.  The peculiarity of such representations is that
| they do not determine their objects -- they stand for anything
| more or less;  for they stand for whatever they resemble and
| they resemble everything more or less.
|
| The second kind of representations are such as are set up
| by a convention of men or a decree of God.  Such are 'tallies',
| 'proper names', &c.  The peculiarity of these 'conventional signs'
| is that they represent no character of their objects.  Likenesses
| denote nothing in particular;  'conventional signs' connote nothing
| in particular.
|
| The third and last kind of representations are 'symbols' or general
| representations.  They connote attributes and so connote them as to
| determine what they denote.  To this class belong all 'words' and
| all 'conceptions'.  Most combinations of words are also symbols.
| A proposition, an argument, even a whole book may be, and
| should be, a single symbol.
|
| CSP, CE 1, pages 467-468.
|
| 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.

Aside from Aristotle, the influence of Kant on Peirce
is very strongly marked in these earliest expositions.
The invocations of "conceptions of the understanding",
the "use" of concepts and thus of symbols in reducing
the manifold of extension, and the not so subtle hint
of the synthetic à priori in Peirce's discussion, not
only of natural kinds, but of the kinds of signs that
lead up to genuine symbols, can all be recognized as
being reprises of dominant, pervasive Kantian themes.

In order to draw out these themes, and to see how Peirce
was led and often inspired to develop their main motives,
let us bring together our previous Figures, abstracting
away from all of those distractingly ephemeral details
about defunct stockyards full of imaginary beasts, and
see if we can see what is really going to go on here.

Figure 3 shows an abductive step of inquiry,
as it is taken on the cue of an iconic sign.

o-----------------------------------------------------------o
|                                                           |
|  t_1   t_2         t_3   t_4                              |
|   o     o           o     o                               |
|     .    .         .    .                                 |
|       .   .       .   .                                   |
|         .  .     .  .                                     |
|           . .   . .                                       |
|             .. ..                                         |
|               o z = icon?                                 |
|               * *                                         |
|               *   *   Rule                                |
|               *     * y=>z                                |
|               *       *                                   |
|               *         *                                 |
|          Fact *           o y = object?                   |
|          x=>z *         *                                 |
|               *       *                                   |
|               *     * Case                                |
|               *   *   x=>y                                |
|               * *                                         |
|               o                                           |
|               x = subject                                 |
|                                                           |
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Figure 3.  Conjunctive Predicate z, Abduction of Case (x (y))

Figure 4 depicts an inductive step of inquiry,
as it is taken on the cue of an indicial sign.

o-----------------------------------------------------------o
|                                                           |
|               w = predicate                               |
|               o                                           |
|               * *     Rule                                |
|               *   *   v=>w                                |
|               *     *                                     |
|               *       *                                   |
|               *         *                                 |
|          Fact *           o v = object?                   |
|          u=>w *         *                                 |
|               *       *                                   |
|               *     * Case                                |
|               *   *   u=>v                                |
|               * *                                         |
|               o u = index?                                |
|             .. ..                                         |
|           . .   . .                                       |
|         .  .     .  .                                     |
|       .   .       .   .                                   |
|     .    .         .    .                                 |
|   o     o           o     o                               |
|  s_1   s_2         s_3   s_4                              |
|                                                           |
o-----------------------------------------------------------o
Figure 4.  Disjunctive Subject u, Induction of Rule (v (w))

I have up to this point followed Peirce's suggestions somewhat
unthinkingly, but I can tell you now that previous unfortunate
experience has led me concurrently to remain suspicious of all
attempts to conflate the types of signs and the roles of terms
in arguments quite so facilely, so I will keep that as a topic
for future inquiry.

Jon Awbrey

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