[Inquiry] Re: Introduction to Inquiry Driven Systems

[Jon Awbrey]

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INT.  Note 21

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2.2.2.  Analogy

The classic description of analogy in the syllogistic frame
comes from Aristotle, who called this form of inference by
the name "paradeigma", that is, reasoning by way of example
or through the parallel comparison of cases.

| We have an Example [analogy, 'paradeigma'] when the major extreme
| is shown to be applicable to the middle term by means of a term
| similar to the third.  It must be known both that the middle
| applies to the third term and that the first applies to the
| term similar to the third.
|
| Aristotle, 'Prior Analytics', 2.24.

Aristotle illustrates this pattern of argument with the following
sample of reasoning.  The setting is a discussion, taking place in
Athens, on the issue of going to war with Thebes.  It is apparently
accepted that a war between Thebes and Phocis is or was a bad thing,
perhaps from the objectivity lent by non-involvement or perhaps as
a lesson of history.

| E.g., 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.

We may analyze this argument as follows:

First, a Rule is induced from the consideration
of a similar Case and a relevant Fact:

   Case:  D => B,  Thebes vs Phocis is war against neighbors.
   Fact:  D => A,  Thebes vs Phocis is bad.
   Rule:  B => A,  War against neighbors is bad.

Next, the Fact to be proved is deduced from the application
of this Rule to the present Case:

   Case:  C => B,  Athens vs Thebes is war against neighbors.
   Rule:  B => A,  War against neighbors is bad.
   Fact:  C => A,  Athens vs Thebes is bad.

In practice, of course, it would probably take a mass of
comparable cases to establish a rule.  As far as the logical
structure goes, however, this quantitative confirmation only
amounts to "gilding the lily".  Perfectly valid rules can be
guessed on the first try, abstracted from a single experience
or adopted vicariously with no personal experience.  Numerical
factors only modify the degree of confidence and the strength
of habit that govern the application of previously learned rules.

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.

o-----------------------------------------------------------o
|                                                           |
|                             A                             |
|                             o                             |
|                            /*\                            |
|                           / * \                           |
|                          /  *  \                          |
|                         /   *   \                         |
|                        /    *    \                        |
|                       /     *     \                       |
|                      /   R u l e   \                      |
|                     /       *       \                     |
|                    /        *        \                    |
|                   /         *         \                   |
|                  /          *          \                  |
|              F a c t        o        F a c t              |
|                /          * B *          \                |
|               /         *       *         \               |
|              /        *           *        \              |
|             /       *               *       \             |
|            /   C a s e            C a s e    \            |
|           /     *                       *     \           |
|          /    *                           *    \          |
|         /   *                               *   \         |
|        /  *                                   *  \        |
|       / *                                       * \       |
|      o                                             o      |
|     C                                               D     |
|                                                           |
| 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                           |
|                                                           |
o-----------------------------------------------------------o
Figure 2.  Aristotle's "War Against Neighbors" Example

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.

Jon Awbrey

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