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INTRO. Note 16

2.1. The Pragmatic Approach to Inquiry (cont.)

For our present purposes, the first feature to note in distinguishing
the three principal modes of reasoning from each other is whether each
of them is exact or approximate in character. In this light, deduction
is the only one the three types of reasoning that can be made exact,
in essence, always deriving true conclusions from true premisses,
while abduction and induction are unavoidably approximate in their
modes of operation, involving elements of fallible judgment in
practice and inescapable error in their application.

The reason for this is that deduction, in the ideal limit, can be
rendered a purely internal process of the reasoning agent, while
the other two modes of reasoning essentially demand a constant
interaction with the outside world, a source of phenomena and
problems that will no doubt continue to exceed the capacities
of any finite resource, human or machine, to master. Situated
in this larger reality, approximations can be judged appropriate
only in relation to their context of use and can be judged fitting
only with regard to a purpose in view.

A parallel distinction that is often made in this connection is to
call deduction a "demonstrative" form of inference, while abduction
and induction are classed as "non-demonstrative" forms of reasoning.
Strictly speaking, the latter two modes of reasoning are not properly
called inferences at all. They are more like controlled associations
of words or ideas that just happen to be successful often enough to be
preserved as useful heuristic strategies in the repertoire of the agent.
But non-demonstrative ways of thinking are inherently subject to error,
and must be constantly checked out and corrected as needed in practice.

Jon Awbrey

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INTRO. Note 17

2.1. The Pragmatic Approach to Inquiry (cont.)

In classical terminology, forms of judgment that require attention
to the context and the purpose of the judgment are said to involve
an element of "art", in a sense that is judged to distinguish them
from "science", and in their renderings as expressive judgments to
implicate arbiters in styles of rhetoric, as contrasted with logic.

In a figurative sense, this means that only deductive logic
can be reduced to an exact theoretical science, while the
practice of any empirical science will always remain to
some degree an art. This has important implications
for any attempt to support inquiry with automated
or computable procedures, constraining both the
manner and the degree of their likely success.
Among the more obvious consequences of this
contingency, we may observe the following:

1. Inquiry support software will need to be highly interactive,
   capable of being sensitive to the run-time conditions at
   at least two kinds of interfaces, those with its human
   users and those with the real world.
   
   
2. The main effect of automation, at least, in the beginning,
   will be to speed up and to strengthen deductive reasoning.
   
   
3. The chief assistance that computation can provide to induction
   is through measures of fit between the empirically gathered
   data sets and the theoretically conceived constructions.
   
   
4. The limited guidance that formal and computable methods
   can bring to abduction, diagnosis, and hypothesis generation
   is restricted to checking the partly logical properties of
   consistency or feasibility, and defeasibility or falsifiability,
   and to speeding up the process of evaluation that pursues
   the initial pretense of a hypothesis.
   
   
5. All of the above notwithstanding, on account of the
   circumstance that inquiry is an iterative cycle,
   improving the rate of performance at any
   critical bottleneck can serve to
   accelerate the entire process.
   

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INTRO. Note 18

2.1. The Pragmatic Approach to Inquiry (cont.)

As far as automating induction goes, we should not expect an
inductive program to make up the data for us, no matter how
sophisticated it gets! Inductive tests can provide measures
of how well a theoretical construct fits a set of data, but
no fit is perfect, nor is it ever intended to be. An inductive
concept is supposed to present a simplification of a complex
reality, otherwise it would serve no function over and above
just staring at the data. In gauging the slippage between
concept and data, the degree of tolerance acceptable in a
given situation is a matter of discretionary judgments that
have to be made under the actual conditions in the field.

When it comes to automating abductive reasoning, we should observe
the historical circumstance that it is often the most "unlikely" set
of hypotheses that turn out to form the correct conceptual framework,
at least when that likelihood has been judged from the standpoint
of the previous framework. Aside from their responsibilities to
the inquiry process, abductive hypotheses can be freely generated
in the most creative manner possible. Breaking the mind-set of the
problem as stated and reformulating data descriptions from radically
new perspectives are just some of the allowable strategies that are
frequently required for ultimate success.

Abductive reasoning is the mode of operation which is involved in
shifting from one paradigm to another. In order to reduce the overall
tension of uncertainty in a knowledge base, it is often necessary to
restructure our perspective on the data in radical ways, to change the
channel that parcels out information to us. But the true value of a
new paradigm is typically not appreciated from the standpoint of the
alternative or established models, that is, not until it has had
time to reorganize the knowledge base in ways that demonstrate
clear advantages to the entire community of inquiry concerned.

The preceding survey has introduced a model of inquiry and charted a
series of limits and obstacles to the prospects of automating a support
for inquiry. We should not let ourselves be too discouraged by the
acknowledgment of these limitations and obstructions. But we ought
to recognize that these constraints are not so much limits on the
computational extension of human inquiry as they are limits on the
instrumental nature of inquiry itself, being matters of the specific
adaptations of finite creatures to an infinite world. In effect, they
are nothing else but the familiar limits of the scientific method.
They are the limits that make it a method.

Jon Awbrey

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INTRO. Note 19

2.1. The Pragmatic Approach to Inquiry (concl.)

Inquiry is a form of reasoning process, in effect, a particular
way of conducting thought, and thus it can be said to institute
a specialized manner, style, or turn of thinking. Philosophers
of the school that is commonly called "pragmatic" hold that all
thought takes place in signs, where "sign" is the word they use
for the broadest conceivable variety of characters, expressions,
formulae, messages, signals, texts, ..., that might be imagined.
Indeed, even intellectual concepts and mental ideas are held to
be a special class of signs, corresponding to internal states
of the thinking agent that both issue in and result from the
interpretation of external signs.

The subsumption of inquiry within reasoning in general and the inclusion
of thinking within the class of sign processes allows us to approach the
subject of inquiry from two different perspectives:

1. The "syllogistic" approach treats inquiry as a logical species.
   
   
2. The "sign-theoretic" approach views inquiry as taking place
   within a more general setting of sign processes.
   

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INTRO. Note 20

2.2. The Syllogistic Approach

In this Division I discuss the syllogistic approach to inquiry,
considering it only insofar as the propositional or sentential
properties of the associated reasoning processes are concerned.

2.2.1. Terminology

In the case of propositional calculus or sentential logic,
deduction comes down to applications of the transitive law
for conditional implications and the approximate forms of
inference hang on the properties that derive from these.
In describing the various types of inference I will employ
a few old "terms of art" from classical logic that are still
of use in treating these kinds of simple problems in reasoning.

1. Expressed in these terms, Deduction takes a Case,
   the minor premiss X => Y, and combines it with a Rule,
   the major premiss Y => Z, to arrive at a Fact, namely,
   the demonstrative conclusion X => Z.
   
   
2. Contrasted with this pattern, Induction takes
   a Fact of the form X => Z and matches it with
   a Case of the form X => Y to guess that a Rule
   is possibly in play, one of the form Y => Z.
   
   
3. Cast on the same template, Abduction takes
   a Fact of the form X => Z and matches it with
   a Rule of the form Y => Z to guess that a Case
   is presently in view, one of the form X => Y.
   

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INTRO. Note 21

2.2.2. Analogy

The classic description of analogy in the syllogistic frame
comes from Aristotle, who called this form of inference by
the name "paradeigma", that is, reasoning by way of example
or through the parallel comparison of cases.

| We have an Example [analogy, 'paradeigma'] when the major extreme
| is shown to be applicable to the middle term by means of a term
| similar to the third. It must be known both that the middle
| applies to the third term and that the first applies to the
| term similar to the third.
|
| Aristotle, 'Prior Analytics', 2.24.

Aristotle illustrates this pattern of argument with the following
sample of reasoning. The setting is a discussion, taking place in
Athens, on the issue of going to war with Thebes. It is apparently
accepted that a war between Thebes and Phocis is or was a bad thing,
perhaps from the objectivity lent by non-involvement or perhaps as
a lesson of history.

| E.g., let A be "bad", B "to make war on neighbors",
| C "Athens against Thebes", and D "Thebes against Phocis".
| Then if we require to prove that war against Thebes is bad,
| we must be satisfied that war against neighbors is bad.
| Evidence of this can be drawn from similar examples, e.g.,
| that war by Thebes against Phocis is bad. Then since war
| against neighbors is bad, and war against Thebes is against
| neighbors, it is evident that war against Thebes is bad.
|
| Aristotle, 'Prior Analytics', 2.24.

We may analyze this argument as follows:

First, a Rule is induced from the consideration
of a similar Case and a relevant Fact:

Case: D => B, Thebes vs Phocis is war against neighbors.
Fact: D => A, Thebes vs Phocis is bad.
Rule: B => A, War against neighbors is bad.

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INTRO. Note 22

2.2.3. Inquiry

Getting back to our "Rainy Day" story, we find our
peripatetic hero presented with a surprising Fact:

• Fact: C => A, In the Current situation the Air is cool.
  

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INTRO. Note 23

2.2.3. Inquiry (cont.)

Figure 3 gives a graphical illustration of Dewey's example of inquiry,
isolating for the purposes of the present analysis the first two steps
in the more extended proceedings that go to make up the whole inquiry.

o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` A ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` D ` ` |
| ` ` `o` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `o` ` ` |
| ` ` ` \ * ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` * / ` ` ` |
| ` ` ` `\` * ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` * `/` ` ` ` |
| ` ` ` ` \ ` * ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` * ` / ` ` ` ` |
| ` ` ` ` `\` ` * ` ` ` ` ` ` ` ` ` ` ` ` ` * ` `/` ` ` ` ` |
| ` ` ` ` ` \ ` ` * ` ` ` ` ` ` ` ` ` ` ` * ` ` / ` ` ` ` ` |
| ` ` ` ` ` `\` `R u l e` ` ` ` ` ` `R u l e` `/` ` ` ` ` ` |
| ` ` ` ` ` ` \ ` ` ` * ` ` ` ` ` ` ` * ` ` ` / ` ` ` ` ` ` |
| ` ` ` ` ` ` `\` ` ` ` * ` ` ` ` ` * ` ` ` `/` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` \ ` ` ` ` * ` ` ` * ` ` ` ` / ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `\` ` ` ` ` * B * ` ` ` ` `/` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` `F a c t` ` ` ` o ` ` ` `F a c t` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `\` ` ` ` ` * ` ` ` ` `/` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` \ ` ` ` ` * ` ` ` ` / ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `\` ` ` ` * ` ` ` `/` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` \ ` ` ` * ` ` ` / ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` `\` `C a s e` `/` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` \ ` ` * ` ` / ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `\` ` * ` `/` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` \ ` * ` / ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` `\` * `/` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` \ * / ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` `\*/` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` C ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| A `=` the Air is cool ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| B `=` just Before it rains` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| C `=` the Current situation ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| D `=` a Dark cloud appears` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| A is a major term ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| B is a middle term` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| C is a minor term ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| D is a major term, associated with A` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
Figure 3. Dewey's "Rainy Day" Inquiry

In this analysis of the first steps of Inquiry,
we have a complex or a mixed form of inference
that can be seen as taking place in two steps:

1. The first step is an Abduction that abstracts a Case
   from the consideration of a Fact and a Rule.
   
   Fact: C => A, In the Current situation the Air is cool.
   Rule: B => A, Just Before it rains, the Air is cool.
   Case: C => B, The Current situation is just Before it rains.
   
   
2. The final step is a Deduction that admits this Case
   to another Rule and so arrives at a novel Fact.
   
   Case: C => B, The Current situation is just Before it rains.
   Rule: B => D, Just Before it rains, a Dark cloud will appear.
   Fact: C => D, In the Current situation, a Dark cloud will appear.
   

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INTRO. Note 24

2.2.3. Inquiry (concl.)

In order to comprehend the bearing of inductive reasoning
on the closing phases of inquiry there are a couple of
observations that we should make:

1. First, we need to recognize that smaller inquiries
   are typically woven into larger inquiries, whether
   we view the whole pattern of inquiry as carried on
   by a single agent or by a complex community.
   
   
2. Further, we need to consider the different ways
   in which the particular instances of inquiry can
   be related to ongoing inquiries at larger scales.
   Three modes of inductive interaction between the
   micro-inquiries and the macro-inquiries that are
   salient here can be described under the headings
   of the Learning, the Transfer, and the Testing
   of rules.
   

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INTRO. Note 25

2.2.3.1. Learning

Rules in a knowledge base, as far as their effective
content goes, can be obtained by any mode of inference.

For example, a rule like:

• Rule: B => A, Just Before it rains, the Air is cool,
  

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INTRO. Note 26

2.2.3.2. Transfer

What is it that gives a distinctively inductive character
to the acquisition of a knowledge base? It is evidently the
"analogy of experience" that underlies the useful application.
Whenever we find ourselves prefacing an argument with the phrase
"If past experience is any guide ..." then we can be sure that
this principle has come into play. We are invoking an analogy
between past experience, considered as a totality, and present
experience, considered as a point of application. What we mean
in practice is this: "If past experience is a fair sample of
possible experience, then the knowledge gained in it applies
to present experience". This is the mechanism that allows a
knowledge base to be carried across gulfs of experience that
are indifferent to the effective contents of its rules.

Here are the details of how this notion of transfer
works out in the case of the "Sign of Rain" example:

Let us consider a fragment, K_pres, of the reasoner's
knowledge base that is logically equivalent to the
conjunction of two rules:

• K_pres = (B => A) and (B => D).
  

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INTRO. Note 27

2.2.3.3. Testing

If the observer looks up and does not see dark clouds,
or if he runs for shelter but it does not rain, then
there is fresh occasion to question the validity of
his knowledge base.

I defer discussing the logical basis
of such a step until another occasion.

Jon Awbrey

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INTRO. Note 28

| NB. I am going to skip ahead, as Items 2.3 through 2.5 of the Outline need
| some work that I am not ready to do just yet, so I'll move on to Chapter 3.

3. Inquiry and Analogy

This Chapter 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
elementary 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.

3.1. Three Types of Reasoning

3.1.1. Types of Reasoning in Aristotle

[omitted]

3.1.2. Types of Reasoning in C.S. Peirce

Here we present one of Peirce's earliest treatments of
the three types of reasoning, from his Harvard Lectures
of 1865 "On the Logic of Science". It illustrates how
one and the same proposition might be reached from three
different directions, as the end result of an inference
in each of the three modes.

| We have then three different kinds of inference:
|
| Deduction or inference 'à priori',
|
| Induction or inference 'à particularis',
|
| Hypothesis or inference 'à posteriori'.
|
| C.S. Peirce, CE 1, p. 267.

| If I reason that certain conduct is wise
| because it has a character which belongs
| 'only' to wise things, I reason 'à priori'.
|
| If I think it is wise because it once turned out
| to be wise, that is, if I infer that it is wise on
| this occasion because it was wise on that occasion,
| I reason inductively ['à particularis'].
|
| But if I think it is wise because a wise man does it,
| I then make the pure hypothesis that he does it
| because he is wise, and I reason 'à posteriori'.
|
| C.S. Peirce, CE 1, p. 180.
|
| Charles Sanders Peirce, "Harvard Lectures 'On the Logic of Science' (1865)",
|'Writings of Charles S. Peirce: A Chronological Edition, Volume 1, 1857-1866',
| Peirce Edition Project, Indiana University Press, Bloomington, IN, 1982.

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INTRO. Note 29

3.1.3. Comparison of the Analyses

[omitted]

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

Peirce's notion of abductive reasoning was derived from Aristotle's treatment
of it in the 'Prior Analytics'. Aristotle's discussion begins with an example
that may appear incidental, but the question and its analysis are echoes of an
important investigation that was pursued in one of Plato's Dialogues, the 'Meno'.
This inquiry is concerned with the possibility of knowledge and the relationship
between knowledge and virtue, or between their objects, the true and the good.
It is not just because it forms a recurring question in philosophy, but because
it preserves a certain correspondence between its form and its content, that we
shall find this example increasingly relevant to our study.

A couple of notes on the reading may be helpful. The Greek text seems to
imply a geometric diagram, in which directed line segments AB, BC, AC are
used to indicate logical relations between pairs of the terms in A, B, C.
We have two options for reading these line labels, either as implications
or as subsumptions, as in the following two paradigms for interpretation.

• Read as Implications:
  
  "AB" = "A <= B",
  "BC" = "B <= C",
  "AC" = "A <= C".
  
  
• Read as Subsumptions:
  
  "AB" = "A subsumes B",
  "BC" = "B subsumes C",
  "AC" = "A subsumes C".
  

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INTRO. Note 30

3.2. Toward a Functional Conception of Quantificational Logic

Up till now quantification theory has been based on the assumption of
individual variables ranging over universal collections of perfectly
determinate elements. Merely to write down quantified formulas 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, however, these ideas begin to appear as problematic
hypotheses whose warrants are not beyond question, projects of
exhaustive determination that overreach the powers of finite
information and control to manage. Therefore, it is worth
considering how we might shift the scene of quantification
theory closer to familiar ground, toward the predicates
themselves that represent our continuing acquaintance
with phenomena.

Jon Awbrey

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