SUO: Re: Abstraction, Analogy, Example, Icon, Metaphor, Model, Morphism, Paradigm, Prototype, Simulation
- To: John F Sowa <sowa@bestweb.net>
- Subject: SUO: Re: Abstraction, Analogy, Example, Icon, Metaphor, Model, Morphism, Paradigm, Prototype, Simulation
- From: Jon Awbrey <jawbrey@oakland.edu>
- Date: Sun, 08 Oct 2000 21:48:14 -0400
- CC: Stand Up Ontology <standard-upper-ontology@ieee.org>
- References: <39DCC4ED.796C7A74@oakland.edu>
- Sender: owner-standard-upper-ontology@ieee.org
SUO Group:
Here is the historical introduction that I promised you last time.
I start as near to the beginning as I can get and present Aristotle's
original account of the form of reasoning that he called "paradeigma".
The name suggests a "side-show", or a parallel comparison of cases, and
it is usually translated as reasoning by analogy, example, or paradigm.
In this context, it is useful to revisit once again the story
of a simple instance of inquiry that I presented a while back:
<http://ltsc.ieee.org/logs/suo/msg00676.html>
It is interesting to note that this story is really a narrative cycle
in miniature, with the cycle of inquiry proceeding through all of its
typical phases in the typical order: Abduction, Deduction, Induction.
Regarding the two examples together in this way, one can make make the
curious observation that the first two steps of inquiry, taken in tandem,
present a type of lattice-theoretic dual to the two steps that are spanned
in reasoning by analogy. I don't really know what to make of this yet --
I just think that it's curious.
To the PieMan,
Jon
¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤
Complex and Mixed Forms of Reasoning (CAMFOR's)
In discussing complex and mixed forms of reasoning,
like those that are typically involved in forming
analogies and conducting inquiries, it helps to
begin with some maximally simple examples, and
this will lead us to look under the lamp-post
of "apophantic" reasoning, that is, among
propositional, sentential, or syllogistic
styles of reasoning. This is alright,
so long as we do not imagine that we
can remain in this charmed circle
forever.
Examples of Cyclic, Parallel, and Serial Syllogisms
The discussion of analogy and inquiry cries out for a couple
of good examples -- models of inference that will be concrete,
intuitive, simple enough to cut our teeth on at the early phases
of our reflective meta-morphosis, but solid enough to also serve
in sharpening the tools of our meta-logical reasoning capacities.
Along these lines, the best points of departure that I have yet
encountered are:
1. For analogy, Aristotle's incipient but incisive account
of its anatomy and function, treating it as a species
of hybrid inference.
2. For inquiry, John Dewey's deceptively simple but
richly suggestive story of inquiry activities in
everyday life.
In order to allow readers an opportunity to contemplate these
examples on their own, before I truck up the grounds with my
own way of laying out their constructions, I will cite them
here first, then set out a traditional array of terminology
that I will need, and then proceed to the dissecting room.
Aristotle's "War Against Neighbors" Example
| 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
John Dewey's "Sign of Rain" Example
| 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
A good start at analyzing these examples, surprisingly enough,
can be made within a primitive propositional and syllogistic
framework. In order to carry this out, I will first outline
a few terms of art from classical logic that can be used to
articulate the procedural stages of a generic inquiry process,
and then I will present my analysis of what is taking place,
logically speaking, to move this particular inquiry forward.
Basic Terminology
In the case of propositional logic, deduction comes down to an
application of the transitive law for conditional implications.
Contemplated on the scheme of Figure 1, deduction takes
a Case, the minor premiss X => Y, and puts 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 of the form Y => Z is possibly in play.
Cast on this 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 of the form X => Y is presently in view.
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 a typical inquiry, the three basic 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 is analyzed
as a combination of induction and deduction, in other words,
as the abstraction and application of a Rule.
For ease of reference, Figure 1 and
the Legend beneath it summarize the
classical terminology for the three
types of inference and the relations
that can be observed among them.
---------------------------------------------------------------------
Z
•
|\
| \
| \
| \
| \ Rule
| \
| \
| A > \
| \ / \
Fact | <-¤-D • Y
| / \ /
| I > /
| /
| /
| / Case
| /
| /
| /
|/
•
X
Figure 1. Basic Structure & Terminology
---------------------------------------------------------------------
Legend 1.
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.
---------------------------------------------------------------------
Aristotle's Illustration of Reasoning by Analogy, Example, Paradigm
Here is the original statement again:
| 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
Figure 2 gives a graphical illustration of Aristotle's
example of "Example", that is, the form of reasoning
that proceeds by Analogy or according to a Paradigm.
---------------------------------------------------------------------
A
•
/*\
/ * \
/ * \
/ * \
/ * \
/ * \
/ R u l e \
/ * \
/ * \
/ * \
/ * \
F a c t B F a c t
/ * * \
/ * * \
/ * * \
/ * * \
/ C a s e C a s e \
/ * * \
/ * * \
/ * * \
/ * * \
/ * * \
• •
C D
Figure 2. Aristotle's "War Against Neighbors" Example
---------------------------------------------------------------------
Legend 2.
A = Atrocious, Adverse to All, A bad thing.
B = Belligerent Battle Between Brethren.
C = Contest of Athens against Thebes.
D = Debacle of Thebes against Phocis.
A is a major term,
B is a middle term,
C is a minor term,
D is a minor term, similar to C.
---------------------------------------------------------------------
In this analysis of reasoning by Analogy,
it is a complex or a mixed form of inference
that can be seen as taking place in two steps:
1. The first step is an Induction that abstracts a Rule
from a Case and a Fact.
(Case) D => B, Thebes vs Phocis is a battle between neighbors.
(Fact) D => A, Thebes vs Phocis is adverse to all.
(Rule) B => A, A battle between neighbors is adverse to all.
2. The final step is a Deduction that applies this Rule
to a Case to arrive at a Fact.
(Case) C => B, Athens vs Thebes is a battle between neighbors.
(Rule) B => A, A battle between neighbors is adverse to all.
(Fact) C => A, Athens vs Thebes is adverse to all.
Analysis of Dewey's Example of Inquiry
Returning to the "Rainy Day" story, we find our peripatetic
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 from a problematic Fact and a knowledge base
of Rules to the plausible suggestion of a Case description, is what
is usually called an 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 abduced Case, C => B,
and has begun to expand on its further implications. So let us
imagine that our up-looker has a more deliberate purpose in mind,
and that his search for additional 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
leads 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
• •
\ * * /
\ * * /
\ * * /
\ * * /
\ * * /
\ R u l e R u l e /
\ * * /
\ * * /
\ * * /
\ * * /
F a c t B F a c t
\ * /
\ * /
\ * /
\ * /
\ C a s e /
\ * /
\ * /
\ * /
\ * /
\ * /
\*/
•
C
Figure 3. Dewey's "Rainy Day" Inquiry
---------------------------------------------------------------------
Legend 3.
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 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 next 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.
This is nowhere near a complete analysis of the Rainy Day inquiry,
even insofar as it might be carried out within the constraints of
the syllogistic framework, and it covers only the first two steps
of the relevant inquiry process, but maybe it will do for a start.
¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤
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