[Arisbe] Re: Inquiry Into Inquiry
Jon Awbrey
arisbe@stderr.org
Tue, 21 Aug 2001 16:44:02 -0400
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Howard, John, Mishtu, Stan, & Other OCAmists, ...
I keep returning to this juncture in our conversation about inquiry,
in part because I feel like I said at least one thing as clearly as
I have ever said it -- don't say it! -- but that the point I tried
to make is still missing from our discussions since that time --
but I am nothing if not persistent, so I will try another way.
Moreover, with Autumn accelerating toward me like a Falling locomotive,
I continue to ponder the problem of how I will ever get my K-page opus
into a form from which anybody might accidentally benefit someday, and
so I appreciate the online exam that you have been visiting on me here.
As I carry out one of my periodic post mortem investigations
of the many communication failures that I have made over the
last two or three or ten years, I suspect that one of the most
serious of them may be that I have consistently failed to convey
the aptness of the propositional and syllogistic levels of analysis
to the task of understanding the logical and informational structure
of the inquiry process. So I will endeavor to work on that for a while.
Back to the grindstone ...
Cos I know no elsewise ...
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Howard Pattee wrote (HP):
Jon Awbrey wrote (JA):
JA: Initial Invitation To Inquiry Into Inquiry:
http://suo.ieee.org/ontology/msg02959.html
JA: Approaches To Inquiry -- Divisions 1 and 2:
http://suo.ieee.org/ontology/msg02961.html
HP quoting AE:
| The whole of science is nothing more than a refinement of everyday thinking.
| It is for this reason that the critical thinking of the physicist cannot
| possibly be restricted to the examination of concepts of his own special
| field. He cannot proceed without considering critically a much more
| difficult problem, the problem of analyzing the nature of everyday
| thinking.
|
| Einstein, "Physics and Reality", 1936.
HP quoting AE:
| What, precisely, is 'thinking'? When, at the reception of sense impressions,
| memory-pictures emerge, this is not yet 'thinking'. And when such pictures form
| series, each member of which calls forth another, this too is not 'thinking'. When,
| however, a certain image turns up in many such series, then -- precisely through such
| return -- it becomes an ordering element for such a series, in that it connects series
| that are in themselves unconnected. Such an element becomes an instrument, a concept.
| I think that the transition from free association or 'dreaming' to thinking is
| characterized by the more or less dominating role which the concept plays in it.
| It is by no means necessary that the concept must be connected with sensorily
| cognizable and reproducible sign (word); but when this is the case,
| thinking becomes, by means of that fact, communicable.
|
| Einstein, "Autobiographical Notes", 1944.
HP: It is now generally accepted, by physicists at least, that the relation between
sense impressions and such unifying concepts cannot be articulated by means of
any method or logic, but arises as a type of aesthetic or "vague instinct that
must be felt" (Poincare). I don't "think" it's going to rain the logical way
Dewey and Jon do. I feel it's going to rain.
JA: We seem to be about the business of marking explicitly what's given implicitly.
But is it a mark against our ephemeris that celestial bodies do not consult it
for their itineraries, nor have the eyes in their orbs to read our fine prints,
that the planets are enlightened by the sun on their courses and their destiny
by some adumbration other than differential equations in plain black and white?
JA: The point here is that a different order of being comes into play
when being begins relating itself to itself via the automediation
of signs. Being a material and a natural thinker, I see this all
taking place within the order of matter and nature, as an utterly
internal development, differentiation, and "ontologogenesis" of a
cosmos, if you catch my drift, but nothing about saying this does
anything to diminish the import of the formal aspect of its being.
JA: I hope this acts to reduce the metaphor of senseless impressions to a eusemy.
I will keep this phyle intact and try to develop its phylogogenesis as we go.
Well, it seems it did not.
What comes to mind in this connection is the analogy between
the inquiry ability and language development in mental growth.
Think of a person who will one day grow up to become a linguist
at three different stages along life's way:
1. The neonate who is born with an innate competence
for learning and for using any language whatever
of the human kind to which he or she is exposed.
2. The person who demonstrates an adequate level of performance
in the comprehension and the use of his or her native language.
3. The expert in languages or specialist in linguistics who has achieved
a "comparative and reflective theoretical acquaintance" (CARTA) with
many diverse languages types, and who can write out something that
approaches a formal grammar for rulier portions of these languages.
I think that it is useful to contemplate three similar
moments in the development of our ability for inquiry:
First, when we are merely granted the benefit of a doubt
that we might have the use of an inquiring mind; Second,
when we actually show our ability to carry out successful
inquiries in practice; Third, when we can reflect on the
conduct of inquiries, successful or otherwise, enough to
provide a rational reconstruction of the procedure, say,
as a test, enough to teach others how the thing is done.
I have to break here, but if you preserve this text for a few turns
I am going to make a concerted effort to hash out all of the issues.
Jon Awbrey
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HP: As Polanyi ["Personal Knowledge"] expressed it: "I believe that by now three things
have been established beyond reasonable doubt: the power of intellectual beauty to
reveal truth about nature; the vital importance of distinguishing this beauty from
merely formal attractiveness, and the delicacy of the test between them, so difficult
that it may baffle the most penetrating scientific minds."
HP: I have more confidence in empirical approaches to inquiry.
The analytic Peircian pragmatic "canonical" approach that
Jon describes is certainly closer to what we have been
indoctrinated with in our western culture. But that is
not what actually goes on in the brain. My conversion
from the canonical approach evolved from teaching a course,
"The Psychology of Problem Solving," over a period of 25 years.
There is now much empirical evidence of how inquiry actually takes
place from many quarters:
HP: 1. the introspection of creative scientists [e.g., Hadamard,
'The Psychology of Invention in the Mathematical Field', Polanyi,
'Personal Knowledge', Ghiselin, 'The Creative Process', Miller,
'Imagery in Scientific Thought', Lakatos, 'History of Science and
its Rational Reconstructions', Feyerabend, 'Against Method', etc.],
HP: 2. the more recent models of distributed, concurrent networks and evolved behavior --
including agent-based approaches to artificial life and artificial intelligence [e.g.,
Brooks, 'Cambrian Intelligence', Hinton and Sejnowski, eds., 'Unsupervised Learning',
Arkins, 'Behavior-Based Robotics', Mitchell, 'An Introduction to Genetic Algorithms', etc.],
HP: 3. on empirical knowledge of how brains actually integrate their
evolved instincts, senses, and individual experiences [e.g.,
Abbott and Sejnowski, eds., 'Neural Codes and Distributed
Representations', Rugg, ed., 'Cognitive Neuroscience'].
HP: What is now evident is that by the time we are using words and logics of any type
where thinking is explicit and communicable, we are no longer in the creative mode in
which images and concepts emerge from our instincts, memories, and sense impressions.
Furthermore, the creative mode is by its nature a sub-symbolic mode, or more precisely,
a sub-thinking mode. (Whether it is still explicit enough to be called symbolic or
a sign activity is only a matter of definition.) The brain's activities in even the
simplest pattern recognition or one-bit decision involves hundreds of millions of
neurons in which instinct, memory, models, and sensory inputs are concurrently
seeking some kind of metastability. This network dynamic activity is so
complex, diffused, and delicate that any attempt to impose rules, methods,
and logic would only disturb and limit the emergence of novel ideas.
JA: In the pragmatic way of thinking everything has a purpose,
and the purpose of each thing is the first thing we should
try to note about it. The purpose of inquiry is to reduce
doubt and lead to a state of belief, which a person in that
state will usually call knowledge or certainty.
HP: This is not the case for many physicists. The purpose of models is to reduce ambiguity,
not doubt. Doubt should always be a dominant emotion since it is the primary check
against overenthusiasm and error. The state of "belief" is especially dangerous,
since no model is complete, and very likely will be replaced. Belief is for the
religious. What physicists seek first in their models is clarity, elegance,
and empirical decidability.
JA: For our present purposes, the first feature to note in
distinguishing these modes of reasoning is whether they
are exact or approximate in character. Deduction is the
only type of reasoning that can be made exact, always
deriving true conclusions from true premisses, while
induction and abduction are unavoidably approximate
in their mode of operation, involving elements of
fallible judgment and inescapable error in their
application.
HP: Paraphrasing Einstein:
Insofar as deductive reasoning ("the propositions of mathematics")
is exact ("certain") it does not apply to reality; and insofar
as it applies to reality it is not exact ("certain").
JA: Yes, I think I said that.
JA: 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 another model, that is, not
until it has had time to reorganize the knowledge base in ways
that demonstrate clear advantages to the community of inquiry
concerned.
HP: Abduction, as I understand it, is not reasoning. It is sub-rational, and I would
say sub-symbolic. Computers lack the knowledge base acquired from 4 billion years
of surviving in a complex environment as well as the vast distributed network, senses,
and body actions necessary to efficiently integrate this mass of experience. Most of
this was acquired by natural selection and integrated into our metabolism, hormonal
and motor controls, senses, pattern recognition, perceptions, motivations, brains,
thoughts, imagination -- the whole organism. It's not likely we can pull this off
in silicon except for simple, closed domains.
JA: I fully understand, I guess, why anybody would want to say this,
especially if they judge ab-apodictic, approximate, contingent,
and non-demonstrative manners of inference against the bar of
deductive reasoning. But it seems fairly clear when you come
to examine it that some patterns of abduction and induction
just plain work better than others in the long run, and so
our concern is one of accounting for how this can be so.
I have been posting a bunch of stuff on these topics
and will be interested in your reactions to them.
¤~~~~~~~~~¤~~~~~~~~~¤~~ESSAY~~¤~~~~~~~~~¤~~~~~~~~~¤
| Document History:
|
| Subject: Approaches To Inquiry
| Contact: Jon Awbrey <jawbrey@oakland.edu>
| Version: Draft 6.0
| Created: 20-Aug-1996
| Revised: 01-Sep-2000
| Revised: 20-Aug-2001
| Advisor: M.A. Zohdy
| Faculty: Lipman, Mili, Windeknecht
| Setting: Oakland University, Rochester, Michigan
| Excerpt: Division 1 (Introduction),
| Division 2 (Syllogistic Approach).
Approaches To Inquiry
In this article I lay out the pragmatic theory of inquiry that
I will use in my study of inquiry driven systems. In Division 1
I introduce the basic features of a canonical model of inquiry
processes. After this, I outline two different approaches to the
functional structure of inquiry. Finally, I discuss a collection
of computational routines that I have implemented to study various
aspects of this model.
1. Introduction
The pragmatic theory or model of inquiry was extracted by
C.S. Peirce from basic materials in classical logic and
refined in parallel with the development of symbolic logic
to address problems about the nature of scientific reasoning.
Borrowing concepts from Aristotle, Peirce identified three
fundamental modes of reasoning, called deductive, inductive,
and abductive inference. In rough terms, "abduction" is what
we use to generate a likely hypothesis or initial diagnosis in
response to a phenomenon of interest or a problem of concern,
while "deduction" is used to clarify and to derive relevant
consequences of our hypotheses, and "induction" is used to
test the sum of our predictions against the sum of the data.
These three processes typically operate in a cyclic fashion,
systematically reducing the uncertainties and difficulties
which initiate inquiry, and thereby lead to an increase
in knowledge.
In the pragmatic way of thinking everything has a purpose,
and the purpose of each thing is the first thing we should
try to note about it. The purpose of inquiry is to reduce
doubt and lead to a state of belief, which a person in that
state will usually call knowledge or certainty. As they
contribute to the purpose of inquiry, we should appreciate
that the three kinds of inference form a cycle that can only
be understood as a whole, and none of them makes complete sense
in isolation from the others. For instance, the purpose of
abduction is to generate guesses of a kind that deduction can
explicate and induction can evaluate. This places a mild but
meaningful constraint on the production of hypotheses, since
it is not just any wild guess at explanation that submits itself
to reason and bows out when defeated in a match with reality.
In a similar fashion, each of the other types of inference
realizes its purpose only in accord with its role in the
cycle of inquiry. No matter how much it may be necessary
to study these processes in abstraction from each other,
the integrity of inquiry places strong limitations on
the effective modularity of its components.
For our present purposes, the first feature to note in
distinguishing these modes of reasoning is whether they
are exact or approximate in character. Deduction is the
only type of reasoning that can be made exact, always
deriving true conclusions from true premisses, while
induction and abduction are unavoidably approximate
in their mode of operation, involving elements of
fallible judgment 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 that will no doubt
keep exceeding the capacities of any finite resource, human
or machine. Embedded in this larger reality, approximations
can only be judged appropriate in relation to a context of
use and a purpose in view.
A parallel distinction 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 types
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. But non-demonstrative ways of thinking
are inherently subject to error, and must be constantly
checked out in practice.
In classical terminology, forms of judgment that require
attention to context and purpose are said to involve elements
of art, as compared with science, and to be styles of rhetoric,
as contrasted with logic. In a figurative sense, this means
that only deductive logic can be reduced to an exact science,
while the practice of empirical science will always remain
to some degree an art. This fact has important implications
for any attempt to support inquiry with automated procedures,
constraining both the manner and degree of their likely success.
It means that inquiry software will need to be highly interactive,
sensitive to run-time conditions at two kinds of interfaces,
those with its human users and those with the real world.
Further, it means that the main effect of automation will
be to speed up and strengthen deductive reasoning. The
chief assistance that computation provides to induction
is through measures of fit between theoretical constructs
and empirical data sets. The limited guidance that formal
methods can bring to hypothesis generation is restricted
to checking the partly logical property of falsifiability
and speeding up the subsequent evaluation process. However,
because inquiry is an iterative cycle, improving the rate of
performance at any critical bottleneck can serve to accelerate
the entire process.
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, or even 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 field conditions.
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 new perspectives are just
some of the allowable strategies that are required for 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 another model, that is, not
until it has had time to reorganize the knowledge base in ways
that demonstrate clear advantages to the community of inquiry
concerned.
The preceding survey has introduced a model of inquiry and
charted a series of limits on the prospects for automating
inquiry. We should not be too discouraged by the acknowledgment
of these limits. But we ought to notice 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 the specific adaptation of a finite creature to
an infinite world. In other words, these are only the familiar
limits of the scientific method. They are the limits that make
it a method.
I now return to discussing the pragmatic theory of inquiry,
treating its positive features in more depth. In the rest
of this introduction, I will examine the theory along the
lines of a classic example that serves to illustrates many
generic aspects of the inquiry process. In the process of
doing this I will continue to introduce basic terminology
and issues for the larger discussion of inquiry.
Inquiry is a form of reasoning process, and therefore a particular
manner of thinking. Pragmatist philosophers hold that all thought
takes place in "signs", which is the word they use for the most
general class of signals, messages, symbolic expressions, texts,
and so on that might be imagined. Even ideas and concepts are
held to be a special class of signs, namely, internal states
of the thinking agent that 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. The "syllogistic" approach
treats inquiry as a logical species. The "sign-theoretic"
approach views inquiry as taking place within a more general
setting of sign processes.
The best point of departure that I know for both approaches
to inquiry is the following story of inquiry activities in
everyday life, as told by John Dewey.
| A man is walking on a warm day. The sky was clear the
| last time he observed it; but presently he notes, while
| occupied primarily with other things, that the air is cooler.
| It occurs to him that it is probably going to rain; looking up,
| he sees a dark cloud between him and the sun, and he then quickens
| his steps. What, if anything, in such a situation can be called
| thought? Neither the act of walking nor the noting of the cold
| is a thought. Walking is one direction of activity; looking and
| noting are other modes of activity. The likelihood that it will
| rain is, however, something 'suggested'. The pedestrian 'feels'
| the cold; he 'thinks of' clouds and a coming shower.
|
| John Dewey, 'How We Think', 1910, pages 6-7
I now proceed to analyze this example from the standpoints of
the syllogistic and the sign-theoretic approaches. The ultimate
task before us is to understand the relation between these two
perspectives as they are unified in a single, coherent subject.
2. The Syllogistic Approach
In this Division I discuss the syllogistic approach to inquiry,
considering it only so far as the propositional or sentential
aspects of the reasoning process are concerned.
2.1 Terminology
In the case of propositional logic, deduction comes down to
applications of the transitive law for conditional implications.
Employing a few old "terms of art" from classical logic that are
still useful in treating these kinds of problems, 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, the demonstrative
conclusion X => Z.
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.
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.
In its original usage a statement of Fact has to do with
a deed done or a record made, that is, a type of event that
is openly observable and not riddled with speculation as to
its very occurrence. In contrast, a statement of Case may
refer to a hidden or a hypothetical cause, that is, a type
of event that is not immediately observable to all concerned.
Obviously, the distinction is a rough one and the question
of which mode applies can depend on the points of view that
different observers adopt over time. Finally, a statement
of a Rule is called that because it states a regularity or
a regulation that governs a whole class of situations, and
not because of its syntactic form. So far in this discussion,
all three types of constraint are expressed in the form of
conditional propositions, but this is not a fixed requirement.
In practice, these modes of statement are distinguished by
the roles that they play within an argument, not by their
style of expression. When the time comes to branch out from
the syllogistic framework, we will find that propositional
constraints can be discovered and represented in arbitrary
syntactic forms.
In the normal course of inquiry, the fundamental types
of inference proceed in the order: abduction, deduction,
induction. However, the same building blocks can be assembled
in other ways to yield different kinds of complex inferences.
Of particular importance for our purposes, reasoning by analogy
can be analyzed as a combination of induction and deduction,
in other words, as the abstraction and application of a rule.
Because a complicated pattern of analogical inference will be
used in our example of a complete inquiry, it will help to
prepare the ground if we first stop to consider an example
of analogy in its simplest form.
For ease of reference, Figure 1 and the Legend beneath it
summarize the classical terminology for the three types
of inference and the relationships among them.
| Z
| o
| |\
| | \
| | \
| | \
| | \
| | \
| | \ R U L E
| | \
| | \
| F | \
| | \
| A | \
| | o Y
| C | /
| | /
| T | /
| | /
| | /
| | / C A S E
| | /
| | /
| | /
| | /
| | /
| |/
| o
| X
|
| Figure 1. Basic Structure & Terminology
|
| Deduction takes a Case, the minor premiss of the form X => Y,
| matches it with a Rule, the major premiss of the form Y => Z,
| then adverts to a Fact, the bound outcome of the form X => Z.
|
| Induction takes a Case of the form X => Y,
| matches it with a Fact of the form X => Z,
| then adverts to a Rule of the form Y => Z.
|
| Abduction takes a Fact of the form X => Z,
| matches it with a Rule of the form Y => Z,
| then adverts to a Case of the form X => Y.
|
| Even more succinctly:
|
| Abduction Deduction Induction
|
| Premiss: Fact Rule Case
| Premiss: Rule Case Fact
| Outcome: Case Fact Rule
OK, that will suffice to renew everybody's acquaintance
with this quaint old manner of speaking about syllogism.
Now it is time to give the Illustrated Classics Edition
of my own two favorite examples of analogy and inquiry.
2.2 Analogy
(Omitted)
2.3 Inquiry
Returning to the "Rainy Day" story, we find our hero presented with
a surprising Fact,
(Fact) C => A, "in the Current situation the Air is cool".
Responding to an intellectual reflex of puzzlement about the situation,
his resource of common knowledge about the world is impelled to seize
on an approximate Rule,
(Rule) B => A, "just Before it rains, the Air is cool".
This Rule can be recognized as having a potential relevance to
the situation because it matches the surprising Fact, C => A,
in its consequential feature A. All of this suggests that the
present Case may be one in which it is just about to rain,
(Case) C => B, "the Current situation is just Before it rains".
The whole mental performance, however automatic and semi-conscious
it may be, that leads up from a problematic Fact and a knowledge base
of Rules to the plausible suggestion of a Case description, is what we
are calling abductive inference.
The next phase of inquiry uses deductive inference to expand
the implied consequences of the abductive hypothesis, with the
aim of testing its truth. For this purpose, the inquirer needs
to think of other things that would follow from the consequence
of his precipitate explanation. Thus, he now reflects on the
Case just assumed,
(Case) C => B, "the Current situation is just Before it rains".
He looks up to scan the sky, perhaps in a random search for further
information, but since the sky is a logical place to look for details
of an imminent rainstorm, symbolized in our story by the letter B,
we may safely suppose that our reasoner has already detached the
consequence of the abductive Case, C => B, and has begun to expand
on its further implications. So let us imagine that the up-looker
has a more deliberate purpose in mind, and that his search for new
data is driven by the new-found, determinate Rule,
(Rule) B => D, "just Before it rains, Dark clouds appear".
Contemplating the assumed Case in combination with this new Rule would
lead him by an immediate deduction to predict an additional Fact,
(Fact) C => D, "in the Current situation Dark clouds appear".
The reconstructed picture of reasoning assembled in this second phase
of inquiry is true to the pattern of deductive inference.
Whatever the case, our subject observes a Dark cloud, just as he would
expect on the basis of the new hypothesis. The explanation of imminent
rain removes the discrepancy between observations and expectations and
thereby reduces the shock of surprise that made this inquiry necessary.
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.
| A D
| o o
| \ * * /
| \ * * /
| \ * * /
| \ * * /
| \ * * /
| \ * * /
| \ * * /
| \ * * /
| \ * * /
| \ * B * /
| \ o /
| \ * /
| \ * /
| \ * /
| \ * /
| \ * /
| \ * /
| \ * /
| \ * /
| \ * /
| \ * /
| \*/
| o
| C
|
| Figure 3. Dewey's "Rainy Day" Inquiry
|
| 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.
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 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.
One last thing ought to be noticed here, the formal duality
between this portion of inquiry and the argument from analogy.
To understand the relevance of inductive reasoning to the closing phases
of inquiry there are a couple of observations we should make. First, we
need to recognize that smaller inquiries are woven into larger inquiries,
whether we view the whole pattern of inquiry as carried on by single agents
or complex communities. Next, we need to consider three distinct ways in
which particular instances of inquiry can relate to an ongoing inquiry at
a larger scale. These inductive modes of interaction between inquiries
may be referred to as the learning, transfer, and testing of rules.
Throughout inquiry the reasoner makes use of rules that have to be
transported across intervals of experience, from masses of experience
where they are learned to moments of experience where they are used.
Inductive reasoning is involved in the learning and transfer of these
rules, both in accumulating a knowledge base and in carrying it through
the times between acquisition and application.
Thus, the first way that induction contributes to an ongoing inquiry is
through the learning of rules, that is, by creating each of the rules in
the knowledge base that gets used along the way. The second way is through
the use of analogy, a two-step combination of induction and deduction, to
transfer rules from one context to another. Finally, every inquiry making
use of a knowledge base constitutes a "field test" of its accumulated contents.
If the knowledge base fails to serve any live inquiry in a satisfactory manner,
then there may be reason to reconsider some of its rules.
I will now detail how these principles of learning, transfer, and testing
apply to the "Rainy Day" example.
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",
is usually induced from a consideration of many past events, as follows:
(Case) C => B, "in Certain events, it is just Before it rains".
(Fact) C => A, "in Certain events, the Air is cool".
(Rule) B => A, "just Before it rains, the Air is cool".
However, the same proposition could also be abduced as an explanation
of a singular occurrence or deduced as a conclusion of a prior theory.
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 works out in the "Rainy Day" example.
Let us consider a fragment K(0) of the reasoner's knowledge base that
is logically equivalent to the conjunction of two rules:
K(0) = (B => A) and (B => D).
It is convenient to have the option of expressing all logical statements in terms
of their models, that is, in terms of the primitive circumstances or the elements
of experience over which they hold true. Let C(-) be a chosen set of experiences,
or the circumstances we have in mind when we refer to "past experience". Let C(+)
be a collective set of experiences, or the projective total of possible circumstances.
Let C(0) be a current experience, or the circumstances present to the reasoner. If we
think of the knowledge base K(0) as referring to the "regime of experience" over which
it is valid, then all of these sets of models can be compared by the simple relations
of set inclusion or logical implication.
In these terms, the "analogy of experience" proceeds by inducing
a Rule about the validity of a current knowledge base and then
deducing its applicability to a current experience.
(Case) C(-) => C(+), "Chosen events fairly sample Collective events".
(Fact) C(-) => K(0), "Chosen events support the Knowledge regime".
(Rule) C(+) => K(0), "Collective events support the Knowledge regime".
(Case) C(0) => C(+), "Current events fairly sample Collective events".
(Fact) C(0) => K(0), "Current events support the Knowledge regime".
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.
To be continued ...
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