ONT Re: Inquiry Into Inquiry
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Subj: Inquiry & Analogy
Date: Mon, 21 May 2001 12:12:02 -0400
From: Jon Awbrey <jawbrey@oakland.edu>
To: Arisbe <arisbe@stderr.org>, SemioCom <semiocom@listbot.com>,
Standardize Unto Others <standard-upper-ontology@ieee.org>
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| Mark how readily each pliant string
| Prepares itself and as an off'ring
| The tribute of some gentle sound does bring.
| Then altogether in harmonious lays
| To the sublimest pitch themselves they raise,
| And loudly celebrate their Master's praise.
|
| Henry Purcell, 1685
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Inquiry SIG,
Here is another excerpt from my report on "Inquiry and Analogy".
This gives the source for the concept of "abductive inference",
that Peirce abducted from Aristotle's 'Prior Analytics'.
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| Document History:
|
| Project: Intelligent Systems Engineering
| Heading: Inquiry and Analogy
| Contact: Jon Awbrey <jawbrey@oakland.edu>
| Version: Draft 3.0
| Created: 1995 Feb 11
| Revised: 2001 May 20
| Faculty: F. Mili & M.A. Zohdy
| Setting: Oakland University, Rochester, MI
| Excerpt: "Aristotle's 'Apagogy'"
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.
1. Implications:
"AB" = "A <= B",
"BC" = "B <= C",
"AC" = "A <= C".
2. Subsumptions:
"AB" = "A subsumes B",
"BC" = "B subsumes C",
"AC" = "A subsumes C".
Here, "X subsumes Y" means that "X applies to all Y",
or that "X is predicated of all of Y". When there is
no danger of confusion, we may write this as "X >= Y".
| We have Reduction ['apagoge', or 'abduction']: (1) when it is obvious
| that the first term applies to the middle, but that the middle applies
| to the last term is not obvious, yet nevertheless is more probable or
| not less probable than the conclusion; or (2) if there are not many
| intermediate terms between the last and the middle; for in all such
| cases the effect is to bring us nearer to knowledge.
|
| (1) E.g., let A stand for "that which can be taught", B for "knowledge",
| and C for "morality". Then that knowledge can be taught is evident;
| but whether virtue is knowledge is not clear. Then if BC is not less
| probable or is more probable than AC, we have reduction; for we are
| nearer to knowledge for having introduced an additional term, whereas
| before we had no knowledge that AC is true.
|
| (2) Or again we have reduction if there are not many intermediate terms
| between B and C; for in this case too we are brought nearer to knowledge.
| E.g., suppose that D is "to square", E "rectilinear figure", and F "circle".
| Assuming that between E and F there is only one intermediate term -- that the
| circle becomes equal to a rectilinear figure by means of lunules -- we should
| approximate to knowledge.
|
| Aristotle, "Prior Analytics", Book 2, Chapter 25.
|'Aristotle, Volume 1', Translated by H.P. Cooke & H. Tredennick,
| Loeb Classical Library, William Heinemann, London, UK, 1938.
The method of abductive reasoning bears a close relation to the sense of reduction
in which we speak of one question reducing to another. The question being asked
is "Can virtue be taught?" The type of answer which develops is the following.
If virtue is a form of understanding, and if we are willing to grant that
understanding can be taught, then virtue can be taught. In this way
of approaching the problem, by detour and indirection, the form of
abductive reasoing is used to shift the attack from the original
question, whether virtue can be taught, to the hopefully easier
question, whether virtue is a form of understanding.
The logical structure of the process of hypothesis formation in
the first example follows the pattern of "abduction to a case",
whose abstract form is diagrammed and schematized as follows.
| T = Teachable
| o
| |\
| | \
| | \
| | \
| | \
| | \
| | \ R U L E
| | \
| | \
| F | \
| | \
| A | \
| | o U = Understanding
| C | /
| | /
| T | /
| | /
| | /
| | / C A S E
| | /
| | /
| | /
| | /
| | /
| |/
| o
| V = Virtue
|
| Figure Omega. Teachability, Understanding, Virtue
|
| T = Teachable (didacton),
| U = Understanding (epistemé),
| V = Virtue (areté).
|
| T is the Major term,
| U is the Middle term,
| V is the Minor term.
|
| TV = [T of V] = Fact in Question,
| TU = [T of U] = Rule in Donation,
| UV = [U of V] = Case in Question.
|
| Schema for Abduction to a Case:
|
| Fact: V => T?
| Rule: U => T.
| ----------------
| Case: V => U?
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