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{{DISPLAYTITLE:Introduction to Inquiry Driven Systems}}

{{DISPLAYTITLE:Introduction to Inquiry Driven Systems}}

('''Note to Self.''' Some parts of this page need to be updated.  — JA, Sep 2010)

'''Author: [[User:Jon Awbrey|Jon Awbrey]]''' <br>

The following essay is intended to provide readers with background on the pragmatic theory of [[inquiry]] and its relationship to the pragmatic [[theory of signs]].

The following essay is intended to provide readers with background on the pragmatic theory of inquiry and its relationship to the pragmatic theory of signs.

==Aspects of Inquiry==

==Aspects of Inquiry==

"Inquiry" is a word in common use for a process that resolves doubt and creates knowledge.  [[Computer]]s are involved in inquiry today, and are likely to become more so as time goes on.  The aim of my research is to improve the service that computers bring to inquiry.  I plan to approach this task by analyzing the nature of inquiry processes, with an eye to those elements that can be given a computational basis.

&ldquo;Inquiry&rdquo; is a word in common use for a process that resolves doubt and creates knowledge.  Computers are involved in inquiry today, and are likely to become more so as time goes on.  The aim of my research is to improve the service that computers bring to inquiry.  I plan to approach this task by analyzing the nature of inquiry processes, with an eye to those elements that can be given a computational basis.

I am interested in the kinds of inquiries which human beings carry on in all the varieties of learning and reasoning from everyday life to scientific practice.  I would like to design [[computer software|software]] that people could use to carry their inquiries further, higher, faster.  Needless to say, this could be an important component of all intelligent software systems in the future.  In any application where a knowledge base is maintained, it will become more and more important to examine the processes that deliver the putative knowledge.

I am interested in the kinds of inquiries which human beings carry on in all the varieties of learning and reasoning from everyday life to scientific practice.  I would like to design software that people could use to carry their inquiries further, higher, faster.  Needless to say, this could be an important component of all intelligent software systems in the future.  In any application where a knowledge base is maintained, it will become more and more important to examine the processes that deliver the putative knowledge.

===Preliminary Questions===

===Preliminary Questions===

Three questions immediately arise in the connection between inquiry and computation.  As they reflect on the very idea of inquiry, they have to do with its integrity, its effectiveness, and its complexity.  These questions ask in their turn whether all such processes that are dubbed "inquiry" have anything essential in common, whether any useful parts of these processes can be automated in practice, and just how deep is the takedown needed to reach the level of routine steps.  The issues of effectiveness and complexity will be discussed throughout the rest of this work, but the problem of integrity must be dealt with immediately, since doubts about it may interfere with my ability to exercise this title to "inquiry".

Three questions immediately arise in the connection between inquiry and computation.  As they reflect on the very idea of inquiry, they have to do with its integrity, its effectiveness, and its complexity.  These questions ask in their turn whether all such processes that are dubbed &ldquo;inquiry&rdquo; have anything essential in common, whether any useful parts of these processes can be automated in practice, and just how deep is the takedown needed to reach the level of routine steps.  The issues of effectiveness and complexity will be discussed throughout the rest of this work, but the problem of integrity must be dealt with immediately, since doubts about it may interfere with my ability to exercise this title to &ldquo;inquiry&rdquo;.

Thus, we must examine the integrity, or well-definedness, of the very idea of inquiry, that is, "inquiry" as a general concept rather than a catch-all word.  Is the faculty of inquiry a principled capacity, leading to a disciplined form of conduct, or is it only a disjointed collection of unrelated skills?  As it is currently being carried out on computers today, inquiry includes everything from database searches, through dynamic simulation and statistical reasoning, to mathematical theorem proving.  Insofar as these tasks constitute specialized efforts, each of them demands software that is tailored to its individual purpose.  Insofar as these different modes of investigation contribute to larger inquiries, our present methods for coordinating their separate findings are mostly ad hoc and still a matter of human skill.  Thus, we might question whether the very name "inquiry" succeeds in referring to a coherent and independent process.

Thus, we must examine the integrity, or well-definedness, of the very idea of inquiry, that is, &ldquo;inquiry&rdquo; as a general concept rather than a catch-all word.  Is the faculty of inquiry a principled capacity, leading to a disciplined form of conduct, or is it only a disjointed collection of unrelated skills?  As it is currently being carried out on computers today, inquiry includes everything from database searches, through dynamic simulation and statistical reasoning, to mathematical theorem proving.  Insofar as these tasks constitute specialized efforts, each of them demands software that is tailored to its individual purpose.  Insofar as these different modes of investigation contribute to larger inquiries, our present methods for coordinating their separate findings are mostly ad hoc and still a matter of human skill.  Thus, we might question whether the very name &ldquo;inquiry&rdquo; succeeds in referring to a coherent and independent process.

Do all the varieties of inquiry have something in common, a structure or a function that defines the essence of inquiry itself?  I will say "yes".  One advantage of this answer is that it brings the topic of inquiry within human scope, and also within my capacity to research.  Without this, the field of inquiry would be impossible for any one human being to survey, because a person would have to cover the union of all the areas that employ inquiry.  By grasping what is shared by all inquiries, I can focus on the intersection of their generating principles.  Another benefit of opting for this answer is that it promises a common medium for inquiry, one in which the many disparate pieces of our puzzling nature may be bound together in a unified whole.

Do all the varieties of inquiry have something in common, a structure or a function that defines the essence of inquiry itself?  I will say &ldquo;yes&rdquo;.  One advantage of this answer is that it brings the topic of inquiry within human scope, and also within my capacity to research.  Without this, the field of inquiry would be impossible for any one human being to survey, because a person would have to cover the union of all the areas that employ inquiry.  By grasping what is shared by all inquiries, I can focus on the intersection of their generating principles.  Another benefit of opting for this answer is that it promises a common medium for inquiry, one in which the many disparate pieces of our puzzling nature may be bound together in a unified whole.

When I look at other examples of instruments that people have used to extend their capacities, I see that two questions must be faced.  First, what are the principles that enable human performance?  Second, what are the principles that can be augmented by available technology?  I will refer to these two issues as the question of original principles and the question of technical extensions, respectively.  Following this model leads me to examine the human capacity for inquiry, asking which of its principles can be reflected in the computational medium, and which of its faculties can be sharpened in the process.  It is not likely that everybody with the same interests and applications would answer these questions the same way, but I will describe how I approach them, what has resulted so far, and what directions I plan to explore next.

When I look at other examples of instruments that people have used to extend their capacities, I see that two questions must be faced.  First, what are the principles that enable human performance?  Second, what are the principles that can be augmented by available technology?  I will refer to these two issues as the question of original principles and the question of technical extensions, respectively.  Following this model leads me to examine the human capacity for inquiry, asking which of its principles can be reflected in the computational medium, and which of its faculties can be sharpened in the process.  It is not likely that everybody with the same interests and applications would answer these questions the same way, but I will describe how I approach them, what has resulted so far, and what directions I plan to explore next.

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.

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Figure 1.  Elementary Structure and Terminology

Figure 1.  Elementary Structure and Terminology

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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.

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Figure 2.  Aristotle's "War Against Neighbors" Example

Figure 2.  Aristotle's "War Against Neighbors" Example

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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.

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Figure 3.  Dewey's "Rainy Day" Inquiry

Figure 3.  Dewey's "Rainy Day" Inquiry

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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".

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Figure 4.  Analogy of Experience

Figure 4.  Analogy of Experience

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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.

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Figure 5.  A Thrice Wise Act

Figure 5.  A Thrice Wise Act

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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.

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Figure 6.  Teachability, Understanding, Virtue

Figure 6.  Teachability, Understanding, Virtue

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==Toward a Functional Conception of Quantificational Logic==

==Toward a Functional Conception of Quantificational Logic==

In order to get a handle on the space of higher order propositions and eventually to carry out a functional approach to quantification theory, it serves to construct some specialized tools.  Specifically, I define a higher order operator &Upsilon;, called the "umpire operator", which takes up to three propositions as arguments and returns a single truth value as the result.  Formally, this so-called "multi-grade" property of <math>\Upsilon\!</math> can be expressed as a union of function types, in the following manner:

In order to get a handle on the space of higher order propositions and eventually to carry out a functional approach to quantification theory, it serves to construct some specialized tools.  Specifically, I define a higher order operator &Upsilon;, called the "umpire operator", which takes up to three propositions as arguments and returns a single truth value as the result.  Formally, this so-called "multi-grade" property of <math>\Upsilon\!</math> can be expressed as a union of function types, in the following manner:

: <math>\Upsilon : \cup^{m = 1, 2, 3}((\mathbb{B}^k \to \mathbb{B})^m \to \mathbb{B}).</math>

: <math>\Upsilon : \cup^{m = 1, 2, 3}((\mathbb{B}^k \to \mathbb{B})^m \to \mathbb{B}).\!</math>

In contexts of application the intended sense can be discerned by the number of arguments that actually appear in the argument list.  Often, the first and last arguments appear as indices, the one in the middle being treated as the main argument while the other two arguments serve to modify the sense of the operation in question.  Thus, we have the following forms:

In contexts of application the intended sense can be discerned by the number of arguments that actually appear in the argument list.  Often, the first and last arguments appear as indices, the one in the middle being treated as the main argument while the other two arguments serve to modify the sense of the operation in question.  Thus, we have the following forms:

Intuitively, the ''L''_''uv'' 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''₀₀, ''L''₀₁, ''L''₁₀, ''L''₁₁] 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''_''uv'' on the ''f''_''i'' within ''X''°2.

Intuitively, the ''L''_''uv'' 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''₀₀, ''L''₀₁, ''L''₁₀, ''L''₁₁] 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''_''uv'' on the ''f''_''i'' within ''X''°2.

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Figure 12.  Higher Order Universe of Discourse [L_uv] c [[x, y]]

Figure 12.  Higher Order Universe of Discourse [L_uv] c [[x, y]]

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===Application of Higher Order Propositions to Quantification Theory===

===Application of Higher Order Propositions to Quantification Theory===

| Revised:  26 Jul 2002

| Revised:  26 Jul 2002

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[[Category:Artificial Intelligence]]

[[Category:Artificial Intelligence]]