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===1.4. Aristotle's "Apagogy" : Abductive Reasoning as Problem Reduction===
===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.
{|
| width=36 | || Read as Implications:
|}
{|
| width=36 | || "''AB''" || = || "''A'' <= ''B''",
|-
| || "''BC''" || = || "''B'' <= ''C''",
|-
| || "''AC''" || = || "''A'' <= ''C''".
|}
<br>
{|
| width=36 | || Read as Subsumptions:
|}
{|
| width=36 | || "''AB''" || = || "''A'' subsumes ''B''",
|-
| || "''BC''" || = || "''B'' subsumes ''C''",
|-
| || "''AC''" || = || "''A'' subsumes ''C''".
|}
<br>
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''".
{| cellpadding=2 width=90%
| width=36 |
| 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" 2.25, in ''Aristotle, Volume 1'', H.P. Cooke and H. Tredennick (trans.), 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 reasoning 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 in Figure 6.
<pre>
o-------------------------------------------------o
| |
| T = Teachable |
| o |
| ^^ |
| | \ |
| | \ |
| | \ |
| | \ |
| | \ R U L E |
| | \ |
| | \ |
| F | \ |
| | \ |
| A | \ |
| | o U = Understanding |
| C | ^ |
| | / |
| T | / |
| | / |
| | / |
| | / C A S E |
| | / |
| | / |
| | / |
| | / |
| |/ |
| o |
| V = 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 Evidence |
| UV = "U of V" = Case in Question |
| |
| Schema for Abduction to a Case: |
| |
| Fact: V => T? |
| Rule: U => T. |
| ---------------- |
| Case: V => U? |
o-------------------------------------------------o
Figure 6. Teachability, Understanding, Virtue
</pre>
===1.5. Aristotle's "Paradigm" : Reasoning by Analogy or Example===
===1.5. Aristotle's "Paradigm" : Reasoning by Analogy or Example===