[Inquiry] Re: Extension x Comprehension = Information

[Jon Awbrey]

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ECI.  Commentary Note 7

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At this point it will help to jump ahead a bit in time,
and to pick up the more systematic account of the same
material from Peirce's "New List of Categories" (1867).

| I shall now show how the three conceptions of reference to a ground,
| reference to an object, and reference to an interpretant are the
| fundamental ones of at least one universal science, that of logic.
|
| 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.

We will have occasion to consider this paragraph in detail later,
but for the present purpose let's hurry on down to the end of it.

| 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.  In deductive argument, the conclusion is
| represented by the premisses as by a general sign under which it is contained.
| In hypotheses, something 'like' the conclusion is proved, that is, the premisses
| form a likeness of the conclusion.  Take, for example, the following argument:
| 
| [Abduction of a Case]
|
| M is, for instance, P_1, P_2, P_3, and P_4;
|
| S is P_1, P_2, P_3, and P_4:
|
| Therefore, S is M.
|
| Here the first premiss amounts to this, that "P_1, P_2, P_3, and P_4"
| is a likeness of M, and thus the premisses are or represent a likeness
| of the conclusion.  That it is different with induction another example
| will show:
|
| [Induction of 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.

Peirce's analysis of the patterns of abductive argument
can be understood according to the following paraphrase:

 Fact:  S => P_1,  S => P_2,  S => P_3,  S => P_4
 Rule:  M => P_1,  M => P_2,  M => P_3,  M => P_4
---------------------------------------------------
 Case:  S => M

If  X  =>  each of A, B, C, D, ...,

then we have the following equivalents:

1.  X  =>  the "greatest lower bound" (GLB) of A, B, C, D, ...

2.  X  =>  the logical conjunction A & B & C & D & ...

3.  X  =>  Q  =  A & B & C & D & ...

More succinctly, letting Q = P_1 & P_2 & P_3 & P_4,
the argument is summarized by the following scheme:

Abduction of a Case:

 Fact:  S => Q
 Rule:  M => Q
---------------.
 Case:  S => M

In this piece of Abduction, it is the GLB or the conjunction
of the ostensible predicates that is the operative predicate
of the argument, that is, it is the predicate that is common
to both the Fact and the Rule of the inference.

Finally, the reason why one can say that Q is an iconic sign
of the object M is that Q can be taken to denote M by virtue
of the qualities that they share, namely, P_1, P_2, P_3, P_4. 

Notice that the iconic denotation is symmetric, at least in principle,
that is, icons are icons of each other as objects, at least potentially,
whether or not a particular interpretive agent is making use of their
full iconicity during a particular phase of semeiosis.

The situation is diagrammed in Figure 1.

o-------------------------------------------------o
|                                                 |
|           P_1   P_2         P_3   P_4           |
|            o     o           o     o            |
|             \*    \         /    */|            |
|              \ *   \       /   * / |            |
|               \  *  \     /  *  /  |            |
|                \   * \   / *   /   |            |
|                 \    *\ /*    /    |            |
|                  .     Q     .     |            |
|                  |     | *   |     |            |
|                  |     |   * |     |            |
|                  |     |     |     |            |
|                  |     |     | *   |            |
|                  |     |     |   * |            |
|                  .     |     .     M            |
|                   \    |    /    *              |
|                    \   |   /   *                |
|                     \  |  /  * Case             |
|                      \ | / *   S=>M             |
|                       \|/*                      |
|                        o                        |
|                        S                        |
|                                                 |
o-------------------------------------------------o
| Figure 1.  Abduction of the Case S => M         |
o-------------------------------------------------o

In a diagram like this, even if one does not bother to
show all of the implicational or the subject-predicate
relationships by means of explicit lines, then one may
still assume the "transitive closure" of the relations
that are actually shown, along with any that are noted
in the text that accompanies it.

Peirce's analysis of the patterns of inductive argument
can be understood according to the following paraphrase:

 Case:  S_1 => M,  S_2 => M,  S_3 => M,  S_4 => M
 Fact:  S_1 => P,  S_2 => P,  S_3 => P,  S_4 => P
---------------------------------------------------
 Rule:  M => P

If  X  <=  each of A, B, C, D, ...,

then we have the following equivalents:

1.  X  <=  the "least upper bound" (LUB) of A, B, C, D, ...

2.  X  <=  the logical disjunction A v B v C v D v ...

3.  X  <=  L  =  A v B v C v D v ...

More succinctly, letting L = P_1 v P_2 v P_3 v P_4,
the argument is summarized by the following scheme:

Induction of a Rule:

 Case:  L => M
 Fact:  L => P
---------------.
 Rule:  M => P

In this bit of Induction, it is the LUB or the disjunction
of the ostensible subjects that is the operative subject
of the argument, to wit, the subject that is common
to both the Case and the Fact of the inference.

Finally, the reason why one can say that L is an indexical sign
of the object M is that L can be taken to denote M by virtue of
the instances that they share, namely, S_1, S_2, S_3, S_4.

Notice that the indexical denotation is symmetric, at least in principle,
that is, indices are indices of each other as objects, at least potentially,
whether or not a particular interpretive agent is making use of their full
indiciality during a particular phase of semeiosis.

The situation is diagrammed in Figure 2.

o-------------------------------------------------o
|                                                 |
|                        P                        |
|                        o                        |
|                       /|\*    Rule              |
|                      / | \ *  M=>P              |
|                     /  |  \  *                  |
|                    /   |   \   *                |
|                   /    |    \    *              |
|                  .     |     .     M            |
|                  |     |     |   * |            |
|                  |     |     | *   |            |
|                  |     |     |     |            |
|                  |     |   * |     |            |
|                  |     | *   |     |            |
|                  .     L     .     |            |
|                 /    */ \*    \    |            |
|                /   * /   \ *   \   |            |
|               /  *  /     \  *  \  |            |
|              / *   /       \   * \ |            |
|             /*    /         \    *\|            |
|            o     o           o     o            |
|           S_1   S_2         S_3   S_4           |
|                                                 |
o-------------------------------------------------o
| Figure 2.  Induction of the Rule M => P         |
o-------------------------------------------------o

Jon Awbrey

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