[Inquiry] Re: Extension x Comprehension = Information
Jon Awbrey
jawbrey at oakland.edu
Tue Apr 1 16:02:22 CST 2003
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ECI. Commentary Note 9
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In the process of rationalizing Peirce's account of induction to myself
I find that I have now lost sight of the indexical sign relationships,
so let me go back to the drawing board one more time to see if I can
get the indexical and the inductive aspects of the situation back
into the very same picture. Here is how we left off last time:
o-----------------------------o-----------------------------o
| Objective Framework | Interpretive Framework |
o-----------------------------o-----------------------------o
| |
| P <------------ at ------------ "P" |
| |\ |\ |
| | | |
| | \ | \ |
| | | |
| | \ | \ |
| | M <------ at --------------|--- "M" |
| | = | / |
| | = | / |
| | = | / |
| | = | / |
| |= |/ |
| S <------------ at ------------ "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
In this double-entry account we are more careful to distinguish
the signs that belong to the "interpretive framework" (IF) from
the objects that belong to the "objective framework" (OF). One
benefit of this scheme is that it immediately resolves many of
the conceptual puzzles that arise from confusing the roles of
objects and the roles of signs in the relevant sign relation.
For example, we observe the distinction between
the objects S, M, P and the signs "S", "M", "P".
The objects may be regarded as extensive classes
or as intensive properties, as the context demands.
The signs may be regarded as sentences or as terms,
in accord with the application and the ends in view.
It is as if we collected a stratified sample S of the disjoint
type "man, horse, kangaroo, whale" from the class M of mammals,
and observed the property P to hold true of each of them. Now
we know that this could be a statistical fluke, in other words,
that S is just an arbitrary subset of the relevant universe of
discourse, and that the very next M you pick from outside of S
might not have the property P. But that is not very likely if
the sample was "fairly" or "randomly" drawn. So the objective
domain is not a lattice like the power set of the universe but
something more constrained, of a kind that makes induction and
learning possible, a lattice of "natural kinds", you might say.
In the natural kinds lattice, then, the LUB of S is close to M.
Now that I have this much of the picture assembled in one frame,
it occurs to me that I might be confusing myself about what are
the sign relations of actual interest in this situation. After
all, samples and signs are closely related, as evidenced by the
etymological connection between them that goes back at least as
far as Hippocrates.
I need not mention any further the more obvious sign relations
that we use just to talk about the objects in the example, for
the signs and the objects in these relations of denotation are
organized according to their roles in the diptych of objective
and interpretive frames. But there are, outside the expressly
designated designations, the ways that samples of species tend
to be taken as signs of their genera, and these sign relations
are discovered internal to the previously marked object domain.
| 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.
|
| C.S. Peirce, 'Collected Papers', CP 1.559.
|
|"On a New List of Categories" (1867),
|'Chronological Edition', CE 2, 49-59,
|'Collected Papers', CP 1.545-567.
Let us look to Peirce's "New List" of the next year for guidance.
There we see an abstract example with the same logical structure
and almost precisely the same labeling. It is a premiss of this
argument that "S_1, S_2, S_3, S_4" is an index of M. But we are
left wondering if he means the objective class M or the sign "M".
If we take the quotation marks of "S_1, S_2, S_3, S_4" as giving
the disjunctive term equal to "S", then we have the next picture:
o-----------------------------o-----------------------------o
| Objective Framework | Interpretive Framework |
o-----------------------------o-----------------------------o
| |
| P <------------ at ------------ "P" |
| |\ |\ |
| | \ | \ |
| | \ | \ |
| | \ | \ |
| | \ | \ |
| | M <------ at --------------|--- "M" |
| | = * | / # |
| | = | / |
| | = | / |
| | = * | / # |
| |= |/ |
| S <------------ at ------------ "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
So we have two readings of what Peirce is saying:
1. The interpretation where "S" is an index of M
by virtue of "S" being a property of each S_j,
literally a generic sign of each of them, and
by virtue of each S_j being an instance of M.
The "S" to S_4 to M linkage is painted * * *.
2. The interpretation where "S" is an index of "M"
by virtue of "S" being a property of each "S_j",
literally an implicit sign of each of them, and
by dint of each "S_j" being an instance of "M".
The "S" to "S_4" to "M" link is drawn as # # #.
On third thought, there is still the possibility
of a sense in which S is literally an index of M,
that is, we might regard a fair sample from S as
nothing less than a representative sample from M.
Jon Awbrey
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