[Inquiry] Re: Introduction to Inquiry Driven Systems

[Jon Awbrey]

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INT.  Note 20

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2.2.  The Syllogistic Approach

In this Subdivision I discuss the syllogistic approach to
inquiry, considering it only so far as the propositional or
sentential properties of the reasoning process are concerned.

2.2.1.  Terminology

In the case of propositional calculus or sentential logic,
deduction comes down to applications of the transitive law
for conditional implications and the approximate forms of
inference hang on the properties that derive from these.
In describing the various types of inference I will employ
a few old "terms of art" from classical logic that are still
of use in treating these kinds of simple problems in reasoning.

1.  Expressed 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, namely,
    the demonstrative conclusion X => Z.

2.  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.

3.  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.

For ease of reference, Figure 1 and the Legend beneath it
summarize the classical terminology for the three types
of inference and the relationships among them.

o-------------------------------------------------o
|                                                 |
|                  Z                              |
|                  o                              |
|                  |\                             |
|                  | \                            |
|                  |  \                           |
|                  |   \                          |
|                  |    \                         |
|                  |     \   R U L E              |
|                  |      \                       |
|                  |       \                      |
|              F   |        \                     |
|                  |         \                    |
|              A   |          \                   |
|                  |           o Y                |
|              C   |          /                   |
|                  |         /                    |
|              T   |        /                     |
|                  |       /                      |
|                  |      /                       |
|                  |     /   C A S E              |
|                  |    /                         |
|                  |   /                          |
|                  |  /                           |
|                  | /                            |
|                  |/                             |
|                  o                              |
|                  X                              |
|                                                 |
| Deduction takes a Case of the form X => Y,      |
| matches it with a Rule of the form Y => Z,      |
| then adverts to a Fact 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         |
|                                                 |
o-------------------------------------------------o
Figure 1.  Basic Structure and Terminology

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 elementary types of inference
proceed in the order:  Abduction, Deduction, Induction.  However,
the same building blocks can be assembled in other ways to yield
different types of complex inferences.  Of particular importance,
reasoning by analogy can be analyzed as a combination of induction
and deduction, in other words, as the abstraction and the 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.

Jon Awbrey

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