ONT Re: Extension x Comprehension = Information
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
Note 95
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
Here the New List text about indices and induction:
| In an argument, the premisses form a representation of
| the conclusion, because they indicate the interpretant
| of the argument, or representation representing it to
| represent its object. The premisses may afford a
| likeness, index, or symbol of the conclusion. ...
|
| [Induction to a Rule]
|
| S_1, S_2, S_3, and S_4 are taken as samples of the collection M;
|
| S_1, S_2, S_3, and S_4 are P:
|
| Therefore, All M is P.
|
| Hence the first premiss amounts to saying that "S_1, S_2, S_3, and S_4"
| is an index of M. Hence the premisses are an index of the conclusion.
|
| CSP, CP 1.559, CE 2, 58.
And here is my picture, so far as it goes:
o-----------------------------o-----------------------------o
| Objective Framework | Interpretive Framework |
o-----------------------------o-----------------------------o
| |
| |
| P <------------@------------ "P" |
| ^^^ ^^^ |
| | \ | \ |
| . | .\ . | .\ |
| | \ | \ |
| . | . \ . | . \ |
| | M <------@--------------|--- "M" |
| . | .= . . | .^ # |
| | = | / |
| . | = . . | / . |
| | = . | / # |
| . |= . . |/ . |
| S <------------@------------ "S" |
| . .. .. . . . * .. .# . # |
| . . . . * . . . # |
| . . . . . . * . . . . # . |
| .. . . ... * .. . . #.# |
| o o o o o o o o |
| m h k w "m" "h" "k" "w" |
| S_1 S_2 S_3 S_4 "S_1" "S_2" "S_3" "S_4" |
| |
o-----------------------------------------------------------o
| Disjunctive Subject "S" and Inductive Rule "M => P" |
o-----------------------------------------------------------o
I got as far as sketching a few readings of the penultimate sentence:
| Hence the first premiss amounts to saying that
| "S_1, S_2, S_3, and S_4" is an index of M.
Uncertain as my comprehension remains at this point, I will
have to leave it in suspension for the time being. But let
me make an initial pass at the final sentence, so as not to
leave an utterly incomplete impression of the whole excerpt.
| Hence the premisses are an index of the conclusion.
The first premiss is this:
| S_1, S_2, S_3, and S_4 are taken as samples of the collection M.
We gather that it says that "S_1, S_2, S_3, S_4" is an index of M.
Taking this very literally, I would guess that it holds by
way of the path from "S_1, S_2, S_3, S_4" to |_| S_j to M.
The second premiss is this:
| S_1, S_2, S_3, and S_4 are P.
Together these premisses form an index of the conclusion, namely:
| All M is P.
And all of this is said to be so because:
| In an argument, the premisses form a representation of
| the conclusion, because they indicate the interpretant
| of the argument, or representation representing it to
| represent its object.
And that is a bit that I will need
to try to think about a bit before
I even try to draw a picture of it.
But just so this won't be one of those retrospective shows
for which the season is so justly notorious, I will advance
a few words in prospect of how I plan to address the problem
of objectivity. Although I've been using the bipartite scheme
of objective and interpretive frameworks, this is only a matter
of convenient organization, and embodies nothing like a claim to
the invariant status of either objects or signs, as we have seen
plenty of examples already of just how shifty these roles can be.
As I suggested in my last note, one sort of evidence, the amassing
of which tends to make me assign a matter to the objective side of
my experience, is the possibility of viewing it from many diverse
angles, of being able to describe it from manifold points of view,
and being able to relate these angles and views in a sensible way.
In the mathematical perspective known as "category theory",
the question of objectivity is handled by way of what are
derivatively enough nomenclated as "universal properties".
Working within a given a category, the things that are potentially worth
caring about, called "objects" or "spaces", along with the corresponding
metamorphoses among them, called "arrows" or "morphisms", can be treated
as having an objective status to the extent that there are many distinct
"views" of them, called "functors", that relate to each other in natural
and especially nice ways called "natural transformations". That's it in
a nutshell, very roughly, and I have forced a few details in prospect of
the ways that I will have to change the setting a little for the sake of
better accommodating semiotics in a suitably re-modelled category theory.
Jon Awbrey
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o