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→‎4.2.3. The Problem of Reconstruction: accent [nourishèd]
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• [[Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Part 4|Part 4]]
 
• [[Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Part 4|Part 4]]
 
• [[Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Part 5|Part 5]]
 
• [[Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Part 5|Part 5]]
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• [[Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Part 6|Part 6]]
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• [[Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Part 7|Part 7]]
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• [[Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Part 8|Part 8]]
 
• [[Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Appendices|Appendices]]
 
• [[Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Appendices|Appendices]]
 
• [[Directory:Jon Awbrey/Papers/Inquiry Driven Systems : References|References]]
 
• [[Directory:Jon Awbrey/Papers/Inquiry Driven Systems : References|References]]
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<p>Tell me where is fancy bred,<br>
 
<p>Tell me where is fancy bred,<br>
 
Or in the heart, or in the head?<br>
 
Or in the heart, or in the head?<br>
−
How begot, how nourished?<br>
+
How begot, how nourishèd?<br>
 
&hellip;<br>
 
&hellip;<br>
 
It is engendered in the eyes,<br>
 
It is engendered in the eyes,<br>
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====4.3.2. The Types of Reasoning====
 
====4.3.2. The Types of Reasoning====
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In this section I discuss the syllogistic approach to inquiry, considering it only so far as the propositional or sentential aspects of the reasoning process are concerned.
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Case, Fact, Rule
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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 Rule is called that because it states a regularity or a regulation that governs a situation, not because of its syntactic form.  At present, all three constraints are expressed in the form of conditional propositions, but this is not a fixed requirement.  In practice, the different modes of statement are distinguished by the roles 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.
    
=====4.3.2.1. Deduction=====
 
=====4.3.2.1. Deduction=====
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In the case of propositional logic, deduction comes down to applications of the transitive law for conditional implications.  Employing a few "terms of art" from classical logic that are still useful in treating these kinds of problems, deduction takes a Case, the minor premiss <math>X \Rightarrow Y,</math> and combines it with a Rule, the major premiss <math>Y \Rightarrow Z,</math> to arrive at a Fact, the demonstrative conclusion <math>X \Rightarrow Z.</math>
    
=====4.3.2.2. Induction=====
 
=====4.3.2.2. Induction=====
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Contrasted with this pattern, induction takes a Fact of the form <math>X \Rightarrow Z</math> and matches it with a Case of the form <math>X \Rightarrow Y</math> to guess that a Rule is possibly in play, one of the form <math>Y \Rightarrow Z.</math>
    
=====4.3.2.3. Abduction=====
 
=====4.3.2.3. Abduction=====
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Cast on the same template, abduction takes a Fact of the form <math>X \Rightarrow Z</math> and matches it with a Rule of the form <math>Y \Rightarrow Z</math> to guess that a Case is presently in view, one of the form <math>X \Rightarrow Y.</math>
    
====4.3.3. Hybrid Types of Inference====
 
====4.3.3. Hybrid Types of Inference====
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In the normal course of inquiry, the fundamental types of inference proceed in the order:  abduction, deduction, induction.  However, the same building blocks can be assembled in other ways to yield different kinds of complex inferences.  Of particular importance for our purposes, reasoning by analogy can be analyzed as a combination of induction and deduction, in other words, as the abstraction and 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.
    
=====4.3.3.1. Analogy=====
 
=====4.3.3.1. Analogy=====
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The classic description of analogy in the syllogistic frame comes from Aristotle, who called this form of inference by the name &ldquo;paradeigma&rdquo;, that is, reasoning by example or by a parallel comparison of cases.
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{| align="center" cellpadding="0" cellspacing="0" width="90%"
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|
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<p>We have an Example (''paradeigma'', or analogy) 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.</p>
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|}
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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.
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{| align="center" cellpadding="0" cellspacing="0" width="90%"
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|
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<p>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.</p>
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|-
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| align="right" | (Aristotle, ''Prior Analytics'', 2.24)
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|}
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We may analyze this argument as follows.  First, a Rule is induced from the consideration of a similar Case and a relevant Fact.
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{| align="center" cellpadding="0" cellspacing="0" width="90%"
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| width="20%" | <math>D \Rightarrow B,</math>
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| width="60%" | "Thebes vs Phocis is war against neighbors".
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| width="20%" | (Case)
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|-
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| <math>D \Rightarrow A,</math>
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| "Thebes vs Phocis is bad".
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| (Fact)
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|-
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| <math>B \Rightarrow A,</math>
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| "War against neighbors is bad".
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| (Rule)
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|}
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Next, the Fact to be proved is deduced from the application of this Rule to the present Case.
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{| align="center" cellpadding="0" cellspacing="0" width="90%"
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| width="20%" | <math>C \Rightarrow B,</math>
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| width="60%" | "Athens vs Thebes is war against neighbors".
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| width="20%" | (Case)
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|-
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| <math>B \Rightarrow A,</math>
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| "War against neighbors is bad".
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| (Rule)
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|-
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| <math>C \Rightarrow A,</math>
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| "Athens vs Thebes is bad".
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| (Fact)
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|}
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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.
    
=====4.3.3.2. Inquiry=====
 
=====4.3.3.2. Inquiry=====
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Returning to the &ldquo;Rainy Day&rdquo; story, we find our hero presented with a surprising Fact:
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{| align="center" cellpadding="0" cellspacing="0" width="90%"
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| width="20%" | <math>C \Rightarrow A,</math>
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| width="60%" | "in the Current situation the Air is cool".
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| width="20%" | (Fact)
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|}
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Responding to an intellectual reflex of puzzlement about the situation, his resource of common knowledge about the world is impelled to seize on an approximate Rule:
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{| align="center" cellpadding="0" cellspacing="0" width="90%"
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| width="20%" | <math>B \Rightarrow A,</math>
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| width="60%" | "just Before it rains, the Air is cool".
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| width="20%" | (Rule)
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|}
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This Rule can be recognized as having a potential relevance to the situation because it matches the surprising Fact, <math>C \Rightarrow A,</math> in its consequential feature <math>A.\!</math>  All of this suggests that the present Case may be one in which it is just about to rain:
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{| align="center" cellpadding="0" cellspacing="0" width="90%"
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| width="20%" | <math>C \Rightarrow B,</math>
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| width="60%" | "the Current situation is just Before it rains".
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| width="20%" | (Case)
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|}
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The whole mental performance, however automatic and semi conscious it may be, that leads up from a problematic Fact and a knowledge base of Rules to the plausible suggestion of a Case description, is what we are calling abductive inference.
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The next phase of inquiry uses deductive inference to expand the implied consequences of the abductive hypothesis, with the aim of testing its truth.  For this purpose, the inquirer needs to think of other things that would follow from the consequence of his precipitate explanation.  Thus, he now reflects on the Case just assumed:
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{| align="center" cellpadding="0" cellspacing="0" width="90%"
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| width="20%" | <math>C \Rightarrow B,</math>
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| width="60%" | "the Current situation is just Before it rains".
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| width="20%" | (Case)
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|}
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He looks up to scan the sky, perhaps in a random search for further information, but since the sky is a logical place to look for details of an imminent rainstorm, symbolized in our story by the letter <math>B,\!</math> we may safely suppose that our reasoner has already detached the consequence of the abductive Case, <math>C \Rightarrow B,</math> and has begun to expand on its further implications.  So let us imagine that the up looker has a more deliberate purpose in mind, and that his search for new data is driven by the new found, determinate Rule:
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{| align="center" cellpadding="0" cellspacing="0" width="90%"
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| width="20%" | <math>B \Rightarrow D,</math>
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| width="60%" | "just Before it rains, Dark clouds appear".
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| width="20%" | (Rule)
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|}
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Contemplating the assumed Case in combination with this new Rule would lead him by an immediate deduction to predict an additional Fact:
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{| align="center" cellpadding="0" cellspacing="0" width="90%"
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| width="20%" | <math>C \Rightarrow D,</math>
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| width="60%" | "in the Current situation Dark clouds appear".
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| width="20%" | (Fact)
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|}
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The reconstructed picture of reasoning assembled in this second phase of inquiry is true to the pattern of deductive inference.
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Whatever the case, our subject observes a Dark cloud, just as he would expect on the basis of the new hypothesis.  The explanation of imminent rain removes the discrepancy between observations and expectations and thereby reduces the shock of surprise that made this inquiry necessary.
    
====4.3.4. Details of Induction====
 
====4.3.4. Details of Induction====
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To understand the relevance of inductive reasoning to the closing phases of inquiry there are a couple of observations we should make.  First, we need to recognize that smaller inquiries are woven into larger inquiries, whether we view the whole pattern of inquiry as carried on by single agents or complex communities.  Next, we need to consider three distinct ways in which particular instances of inquiry can relate to an ongoing inquiry at a larger scale.  These inductive modes of interaction between inquiries may be referred to as the learning, transfer, and testing of rules.
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Throughout inquiry the reasoner makes use of rules that have to be transported across intervals of experience, from masses of experience where they are learned to moments of experience where they are used.  Inductive reasoning is involved in the learning and transfer of these rules, both in accumulating a knowledge base and in carrying it through the times between acquisition and application.
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Thus, the first way that induction contributes to an ongoing inquiry is through the learning of rules, that is, by creating each of the rules in the knowledge base that gets used along the way.  The second way is through the use of analogy, a two step combination of induction and deduction, to transfer rules from one context to another.  Finally, every inquiry making use of a knowledge base constitutes a &ldquo;field test&rdquo; of its accumulated contents.  If the knowledge base fails to serve any live inquiry in a satisfactory manner, then there may be reason to reconsider some of its rules.
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I will now detail how these principles of learning, transfer, and testing apply to the ''Rainy Day'' example.
    
=====4.3.4.1. Learning=====
 
=====4.3.4.1. Learning=====
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Rules in a knowledge base, as far as their effective content goes, can be obtained by any mode of inference.  For example, consider a proposition like the following:
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{| align="center" cellpadding="0" cellspacing="0" width="90%"
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| width="20%" | <math>B \Rightarrow A,</math>
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| width="60%" | "just Before it rains, the Air is cool".
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| width="20%" | &nbsp;
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|}
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Such a proposition is usually induced from a consideration of many past events, as follows.
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{| align="center" cellpadding="0" cellspacing="0" width="90%"
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| width="20%" | <math>C \Rightarrow B,</math>
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| width="60%" | "in Certain events, it is just Before it rains".
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| width="20%" | (Case)
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|-
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| <math>C \Rightarrow A,</math>
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| "in Certain events, the Air is cool".
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| (Fact)
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|-
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| <math>B \Rightarrow A,</math>
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| "just Before it rains, the Air is cool".
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| (Rule)
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|}
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However, the same proposition could also be abduced as an explanation of a singular occurrence or deduced as a conclusion of a prior theory.
    
=====4.3.4.2. Transfer=====
 
=====4.3.4.2. Transfer=====
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What really gives a distinctively inductive character to the acquisition of a knowledge base is the "analogy of experience" that underlies its useful application.  Whenever we find ourselves prefacing an argument with the phrase, &ldquo;If past experience is any guide&nbsp;&hellip;&nbsp;&rdquo; we can be sure this principle has come into play.  We are invoking an analogy between past experience, considered as a totality, and present experience, considered as a point of application.  What we mean in practice is this:  &ldquo;If past experience is a fair sample of possible experience, then the knowledge gained in it applies to present experience.&rdquo;  This is the mechanism that allows a knowledge base to be carried across gulfs of experience that are indifferent to the effective contents of its rules.
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Here are the details of how this works out in the ''Rainy Day'' example.  Let us consider a fragment <math>K\!</math> of the reasoner's knowledge base that is logically equivalent to the conjunction of two rules.
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{| align="center" cellpadding="0" cellspacing="0" width="90%"
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| <math>K \Leftrightarrow (B \Rightarrow A) \land (B \Rightarrow D).</math>
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|}
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It is convenient to have the option of expressing all logical statements in terms of their models, that is, in terms of the primitive circumstances or the elements of experience over which they hold true.  Let <math>C^-\!</math> be a chosen set of experiences, or the circumstances we have in mind when we refer to "past experience".  Let <math>C^+\!</math> be a collective set of experiences, or the projective total of possible circumstances.  Let <math>C\!</math> be a current experience, or the circumstances present to the reasoner.  If we think of the knowledge base <math>K\!</math> as referring to the "regime of experience" over which it is valid, then all of these sets of models can be compared by simple relations of set inclusion or logical implication.
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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 its applicability to a current experience.
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{| align="center" cellpadding="0" cellspacing="0" width="90%"
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| width="20%" | <math>C^- \Rightarrow C^+,</math>
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| width="60%" | "Chosen events fairly sample Collective events".
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| width="20%" | (Case)
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|-
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| <math>C^- \Rightarrow K,</math>
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| "Chosen events support the Knowledge regime".
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| (Fact)
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|-
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| <math>C^+ \Rightarrow K,</math>
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| "Collective events support the Knowledge regime".
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| (Rule)
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|-
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| <math>C \Rightarrow C^+,</math>
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| "Current events fairly sample Collective events".
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| (Case)
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|-
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| <math>C \Rightarrow K,</math>
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| "Collective events support the Knowledge regime".
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| (Fact)
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|}
    
=====4.3.4.3. Testing=====
 
=====4.3.4.3. Testing=====
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If the observer looks up and does not see dark clouds, or if he runs for shelter but it does not rain, then there is fresh occasion to question the validity of his knowledge base.
    
====4.3.5. The Stages of Inquiry====
 
====4.3.5. The Stages of Inquiry====
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&bull; [[Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Part 4|Part 4]]
 
&bull; [[Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Part 4|Part 4]]
 
&bull; [[Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Part 5|Part 5]]
 
&bull; [[Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Part 5|Part 5]]
 +
&bull; [[Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Part 6|Part 6]]
 +
&bull; [[Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Part 7|Part 7]]
 +
&bull; [[Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Part 8|Part 8]]
 
&bull; [[Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Appendices|Appendices]]
 
&bull; [[Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Appendices|Appendices]]
 
&bull; [[Directory:Jon Awbrey/Papers/Inquiry Driven Systems : References|References]]
 
&bull; [[Directory:Jon Awbrey/Papers/Inquiry Driven Systems : References|References]]
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</div>
 
</div>
 
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<br><sharethis />
      
[[Category:Artificial Intelligence]]
 
[[Category:Artificial Intelligence]]
Jon Awbrey
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