Functional Logic

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FL.  Note 1

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Inquiry and Analogy

| Version:  Draft 3.25
| Created:  01 Jan 1995
| Revised:  24 Dec 2001
| Revised:  12 Mar 2004

Abstract

This report discusses C.S. Peirce's treatment of analogy,
placing it in relation to his overall theory of inquiry.
The first order of business is to introduce the three
fundamental types of reasoning that Peirce adopted
from classical logic.  In Peirce's analysis both
inquiry and analogy are complex programs of
reasoning which develop through stages of
these three types, although normally in
different orders.

1.  Three Types of Reasoning

1.1.  Types of Reasoning in Aristotle

1.2.  Types of Reasoning in C.S. Peirce

1.3.  Comparison of the Analyses

1.4.  Aristotle's "Apagogy":  Abductive Reasoning as Problem Reduction

1.5.  Aristotle's "Paradigm":  Reasoning by Analogy or Example

1.6.  Peirce's Formulation of Analogy

1.7.  Dewey's "Sign of Rain":  An Example of Inquiry

2.  Functional Conception of Quantification Theory

Up till now quantification theory has been based on the assumption of
individual variables ranging over universal collections of perfectly
determined elements.  Merely to write down quantified notations like
"(For All)_(x in X) F(x)" and "(For Some)_(x in X) F(x)" involves a
subscription to such notions, as shown by the membership relations
invoked in their indices.  Reflected on pragmatic and constructive
principles, these ideas begin to appear as problematic hypotheses
whose warrants to be granted are not beyond question, as projects
of exhaustive determination that overreach the powers of finite
information and control to manage.  Consequently, it is worth
considering how we might shift the medium of quantification
theory closer to familiar ground, toward the predicates
themselves that represent our continuing acquaintance
with phenomena.

2.1.  Higher Order Propositional Expressions

By way of equipping this inquiry with a bit of concrete material, I begin
with a consideration of "higher order propositional expressions" (HOPE's),
in particular, those that stem from the propositions on 1 and 2 variables.

2.1.1.  Higher Order Propositions and Logical Operators (n = 1)

A "higher order" proposition is, very roughly speaking, a proposition about propositions.
If the original order of propositions is a class of indicator functions F : X -> B, then
the next higher order of propositions consists of maps of the type m : (X -> B) -> B.

For example, consider the case where X = B.  Then there are exactly four
propositions F : B -> B, and exactly sixteen higher order propositions 
that are based on this set, all bearing the type m : (B -> B) -> B.

Table 10 lists the sixteen higher order propositions about propositions on
one boolean variable, organized in the following fashion:  Columns 1 & 2
form a truth table for the four F : B -> B, turned on its side from the
way that one is most likely accustomed to see truth tables, with the
row leaders in Column 1 displaying the names of the functions F_i,
for i = 1 to 4, while the entries in Column 2 give the values of
each function for the argument values that are listed in the
corresponding column head.  Column 3 displays one of the
more usual expressions for the proposition in question.
The last sixteen columns are topped by a collection of
conventional names for the higher order propositions,
also known as the "measures" m_j, for j = 0 to 15,
where the entries in the body of the Table record
the values that each m_j assigns to each F_i.

Table 10.  Higher Order Propositions (n = 1)
o------o-----o-----o---o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o
|  \ x | 1 0 |  F  | m |m |m |m |m |m |m |m |m |m |m |m |m |m |m |m |
| F \  |     |     | 00|01|02|03|04|05|06|07|08|09|10|11|12|13|14|15|
o------o-----o-----o---o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o
|      |     |     |                                                |
| F_0  | 0 0 |  0  | 0  1  0  1  0  1  0  1  0  1  0  1  0  1  0  1 |
|      |     |     |                                                |
| F_1  | 0 1 | (x) | 0  0  1  1  0  0  1  1  0  0  1  1  0  0  1  1 |
|      |     |     |                                                |
| F_2  | 1 0 |  x  | 0  0  0  0  1  1  1  1  0  0  0  0  1  1  1  1 |
|      |     |     |                                                |
| F_3  | 1 1 |  1  | 0  0  0  0  0  0  0  0  1  1  1  1  1  1  1  1 |
|      |     |     |                                                |
o------o-----o-----o---o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o

I am going to put off explaining Table 11, that presents a sample of
what I call "Interpretive Categories for Higher Order Propositions",
until after we get beyond the 1-dimensional case, since these lower
dimensional cases tend to be a bit "condensed" or "degenerate" in
their structures, and a lot of what is going on here will almost
automatically become clearer as soon as we get even two logical
variables into the mix.

Table 11.  Interpretive Categories for Higher Order Propositions (n = 1)
o-------o----------o------------o------------o----------o----------o-----------o
|Measure| Happening| Exactness  | Existence  | Linearity|Uniformity|Information|
o-------o----------o------------o------------o----------o----------o-----------o
| m_0   | nothing  |            |            |          |          |           |
|       | happens  |            |            |          |          |           |
o-------o----------o------------o------------o----------o----------o-----------o
| m_1   |          |            | nothing    |          |          |           |
|       |          | just false | exists     |          |          |           |
o-------o----------o------------o------------o----------o----------o-----------o
| m_2   |          |            |            |          |          |           |
|       |          | just not x |            |          |          |           |
o-------o----------o------------o------------o----------o----------o-----------o
| m_3   |          |            | nothing    |          |          |           |
|       |          |            | is x       |          |          |           |
o-------o----------o------------o------------o----------o----------o-----------o
| m_4   |          |            |            |          |          |           |
|       |          | just x     |            |          |          |           |
o-------o----------o------------o------------o----------o----------o-----------o
| m_5   |          |            | everything | F is     |          |           |
|       |          |            | is x       | linear   |          |           |
o-------o----------o------------o------------o----------o----------o-----------o
| m_6   |          |            |            |          | F is not | F is      |
|       |          |            |            |          | uniform  | informed  |
o-------o----------o------------o------------o----------o----------o-----------o
| m_7   |          | not        |            |          |          |           |
|       |          | just true  |            |          |          |           |
o-------o----------o------------o------------o----------o----------o-----------o
| m_8   |          |            |            |          |          |           |
|       |          | just true  |            |          |          |           |
o-------o----------o------------o------------o----------o----------o-----------o
| m_9   |          |            |            |          | F is     | F is not  |
|       |          |            |            |          | uniform  | informed  |
o-------o----------o------------o------------o----------o----------o-----------o
| m_10  |          |            | something  | F is not |          |           |
|       |          |            | is not x   | linear   |          |           |
o-------o----------o------------o------------o----------o----------o-----------o
| m_11  |          | not        |            |          |          |           |
|       |          | just x     |            |          |          |           |
o-------o----------o------------o------------o----------o----------o-----------o
| m_12  |          |            | something  |          |          |           |
|       |          |            | is x       |          |          |           |
o-------o----------o------------o------------o----------o----------o-----------o
| m_13  |          | not        |            |          |          |           |
|       |          | just not x |            |          |          |           |
o-------o----------o------------o------------o----------o----------o-----------o
| m_14  |          | not        | something  |          |          |           |
|       |          | just false | exists     |          |          |           |
o-------o----------o------------o------------o----------o----------o-----------o
| m_15  | anything |            |            |          |          |           |
|       | happens  |            |            |          |          |           |
o-------o----------o------------o------------o----------o----------o-----------o

Jon Awbrey

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