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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.
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.
<pre>
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. Elementary Structure and Terminology
</pre>
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 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.
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.
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.
<pre>
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
</pre>
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:
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:
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.
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.
<pre>
o-----------------------------------------------------------o
| |
| A D |
| o o |
| \ * * / |
| \ * * / |
| \ * * / |
| \ * * / |
| \ * * / |
| \ R u l e R u l e / |
| \ * * / |
| \ * * / |
| \ * * / |
| \ * B * / |
| F a c t o F a c t |
| \ * / |
| \ * / |
| \ * / |
| \ * / |
| \ C a s e / |
| \ * / |
| \ * / |
| \ * / |
| \ * / |
| \ * / |
| \*/ |
| o |
| C |
| |
| 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 |
| |
o-----------------------------------------------------------o
Figure 3. Dewey's "Rainy Day" Inquiry
</pre>
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:
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:
Figure 4 schematizes this way of viewing the "analogy of experience".
Figure 4 schematizes this way of viewing the "analogy of experience".
<pre>
o-----------------------------------------------------------o
| |
| K_pres |
| o |
| /|\ |
| / | \ |
| / | \ |
| / | \ |
| / Rule \ |
| / | \ |
| / | \ |
| / | \ |
| / E_poss \ |
| Fact / o \ Fact |
| / * * \ |
| / * * \ |
| / * * \ |
| / * * \ |
| / * * \ |
| / * Case Case * \ |
| / * * \ |
| / * * \ |
| /* *\ |
| o<<<---------------<<<---------------<<<o |
| E_past Analogy Morphism E_pres |
| More Known Less Known |
| |
o-----------------------------------------------------------o
Figure 4. Analogy of Experience
</pre>
In these terms, the "analogy of experience" proceeds by inducing a Rule about the validity of a current knowledge base and then deducing a Fact, its applicability to a current experience, as in the following sequence:
In these terms, the "analogy of experience" proceeds by inducing a Rule about the validity of a current knowledge base and then deducing a Fact, its applicability to a current experience, as in the following sequence:
The converging operation of all three reasonings is shown in Figure 5.
The converging operation of all three reasonings is shown in Figure 5.
<pre>
o---------------------------------------------------------------------o
| |
| D ("done by a wise man") |
| o |
| \* |
| \ * |
| \ * |
| \ * |
| \ * |
| \ * |
| \ * A ("a wise act") |
| \ o |
| \ /| * |
| \ / | * |
| \ / | * |
| . | o B ("benevolence", a certain character) |
| / \ | * |
| / \ | * |
| / \| * |
| / o |
| / * C ("contributes to charity", a certain conduct) |
| / * |
| / * |
| / * |
| / * |
| / * |
| /* |
| o |
| E ("earlier today", a certain occasion) |
| |
o---------------------------------------------------------------------o
Figure 5. A Thrice Wise Act
</pre>
The common proposition that concludes each argument is ''AC'', to wit, "contributing to charity is wise".
The common proposition that concludes each argument is ''AC'', to wit, "contributing to charity is wise".
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.
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>
==Toward a Functional Conception of Quantificational Logic==
==Toward a Functional Conception of Quantificational Logic==
Intuitively, the ''L''<sub>''uv''</sub> operators may be thought of as qualifying propositions according to the elements of the universe of discourse that each proposition positively values. Taken together, these measures provide us with the means to express many useful observations about the propositions in ''X''° = [''x'', ''y''], and so they mediate a subtext [''L''<sub>00</sub>, ''L''<sub>01</sub>, ''L''<sub>10</sub>, ''L''<sub>11</sub>] that takes place within the higher order universe of discourse ''X''°2 = [''X''°] = <nowiki>[[</nowiki>''x'', ''y''<nowiki>]]</nowiki>. Figure 12 summarizes the action of the ''L''<sub>''uv''</sub> on the ''f''<sub>''i''</sub> within ''X''°2.
Intuitively, the ''L''<sub>''uv''</sub> operators may be thought of as qualifying propositions according to the elements of the universe of discourse that each proposition positively values. Taken together, these measures provide us with the means to express many useful observations about the propositions in ''X''° = [''x'', ''y''], and so they mediate a subtext [''L''<sub>00</sub>, ''L''<sub>01</sub>, ''L''<sub>10</sub>, ''L''<sub>11</sub>] that takes place within the higher order universe of discourse ''X''°2 = [''X''°] = <nowiki>[[</nowiki>''x'', ''y''<nowiki>]]</nowiki>. Figure 12 summarizes the action of the ''L''<sub>''uv''</sub> on the ''f''<sub>''i''</sub> within ''X''°2.
<pre>
o-----------------------------------------------------------o
| |
| o |
| / \ |
| / \ |
| /x y\ |
| / o---o \ |
| o \ / o |
| / \ o / \ |
| / \ | / \ |
| / \ @ / \ |
| / x y \ / x y \ |
| o o---o o o---o o |
| / \ \ / \ / / \ |
| / \ @ / \ @ / \ |
| / \ / \ / \ |
| / y \ / \ / y \ |
| o @ o @ o o o |
| / \ / \ / \ | / \ |
| / \ / \ / \ @ / \ |
| / \ /x y\ / \ / \ |
| / x y \ / o o \ / x y \ / x y \ |
| o @ o \ / o o o o o o |
| |\ / \ o / \ | / \ \ / /| |
| | \ / \ | / \ @ / \ @ / | |
| | \ / \ @ / \ / \ / | |
| | \ / x \ / x y \ / x \ / | |
| | o @ o o---o o o o | |
| | |\ / \ \ / / \ | /| | |
| | | \ / \ @ / \ @ / | | |
| | | \ / \ / \ / | | |
| |L_11| \ / o y \ / x o \ / |L_00| |
| o---------o | o | o---------o |
| | \ x @ / \ @ y / | |
| | \ / \ / | |
| | \ / \ / | |
| |L_10 \ / o \ / L_01| |
| o---------o | o---------o |
| \ @ / |
| \ / |
| \ / |
| \ / |
| o |
| |
o-----------------------------------------------------------o
Figure 12. Higher Order Universe of Discourse [L_uv] c [[x, y]]
</pre>
===Application of Higher Order Propositions to Quantification Theory===
===Application of Higher Order Propositions to Quantification Theory===
==Document History==
==Document History==
<pre>
| Introduction to Inquiry Driven Systems
|
| Author: Jon Awbrey
| Version: Draft 12.03
| Created: 01 Aug 1996
| Revised: 20 Aug 2002
Amalgamates the following:
Amalgamates the following:
| Inquiry and Analogy
|
| Author: Jon Awbrey
| Version: Draft 3.24
| Created: 01 Jan 1995
| Revised: 28 Jul 2002
| Aspects of Inquiry
|
| Author: Jon Awbrey
| Version: Draft 11.30
| Created: 04 Aug 1996
| Revised: 31 Oct 2001
| Approaches to Inquiry
|
| Author: Jon Awbrey
| Version: Draft 6.30
| Created: 20 Aug 1996
| Revised: 26 Jul 2002
</pre>
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{{aficionados}}
