SUO: Re: Building Ontologies Through Signs And Inquiries (BOTSAI)
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Here is one of my last best attempts to articulate the relations
among the three ductions within the cycle of inquiry, at least,
in so far as the anatomy of inquiry can be made out from its
zeroth order projection.
| Document History:
|
| Subject: Approaches To Inquiry
| Contact: Jon Awbrey <jawbrey@oakland.edu>
| Version: Draft 6.1
| Created: 20-Aug-1996
| Revised: 28-Oct-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 ...
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:
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.
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 ...
2.1 Terminology
In the case of propositional logic, deduction comes down to
applications of the transitive law for conditional implications.
In describing the various kinds of inference I will employ a few
old "terms of art" from classical logic that are still of use in
treating these kinds of problems.
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, 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.
2.2 Analogy ...
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.
a. Let C(-) be a chosen set of experiences,
or the circumstances that we have in mind
when we refer to "past experience".
b. Let C(+) be a collective set of experiences,
or the projective total of possible circumstances.
c. Let C(0) be a current experience, or the circumstances
that are present to the reasoner at the current moment.
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.
Jon Awbrey
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