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# Inquiry Driven Systems : Part 4

## Part 4. Discussion of Inquiry

The subject matter under review is assigned the name “inquiry”, a name that is presently both general and vague. The generality is essential, marking the actual extension and the eventual coverage that the name is intended to have. The vagueness is incidental, hinging on the personal concept of the subject matter and the prevailing level of comprehension that relates an interpreter of the name to the object of its indication. As the investigation proceeds, it is hoped that the name can become as general as it is meant to be, but not forever remain so vague.

In regarding and presenting this subject, I need the freedom to adopt any one of several views. These are the perspectives that I call the “classical” or syllogistic, the “pragmatic” or sign theoretic, and the “dynamical” or system theoretic points of view. Each perspective or point of view is supported by a corresponding framework or a supply of resources, the intellectual tools that allow a person taking up such a view to develop and to share a picture of what can be seen from it.

The situation with respect to these different views can be described as follows. The part of the subject that can be seen in each view appears to make sense, if taken by itself, but it does not appear to be entirely consistent with all that is obvious from the other perspectives, and all of these pictures put together are almost certainly incomplete in regard to the subject they are meant to depict. Although the various pictures can be presented in the roughly historical order of their development, it is a mistaken view of their progression to think that the later views can simply replace the former pictures. In particular, if the classical view is taken as an initial approximation to its subject, then it can be regarded as relatively complete with respect to this intention, but both of the later frameworks, that build on and try to reform this basis, are very much works in progress and far from being completed projects. Taken in order of historical development, each succeeding view always promises to keep all that is good from the preceding points of view, but the current state of development is such that these claims have to be taken as promissory notes, not yet matured and not yet due.

In accord with this situation, I would like to be able to take up any one of these views at any point in the discussion and to move with relative ease among the different pictures of inquiry that they frequently show. Along with this ability, I need to have ways to put their divergent and varying views in comparison with each other, to reflect on the reasons for their distinctions and variations, and overall to have some room to imagine how these views might be corrected, extended, reconciled, and integrated into a coherent and a competent picture of the subject.

This part of the discussion cannot be formal or systematic. It is not intended to argue for any particular point of view, but only to introduce some of the language, ideas, and issues that surround the topic of inquiry. No amount of forethought or premeditation that I can muster would be sufficient to impose an alien organization on this array. The most that I can hope to bring to this forum is to seek clarification of some the terms and to pursue the consequences of some of the axioms that are in the air. Nor do I expect the reader to be disarmed by this apology, at least, not yet, and never unilaterally, since it is obvious that my own point of view, however unformed and inarticulate it may be, cannot help but to affect the selection of problems, subtopics, and suggested angles of approach.

Taking all of these factors into account, the best plan that I can arrange for presenting the subject matter of inquiry is as follows:

First, I describe the individual perspectives and their corresponding frameworks in very general terms, but only insofar as they bear on the subject of inquiry, so that each way of looking at the subject is made available as a resource for the discussion that follows.

Next, I discuss the context of inquiry, treating it as a general field of observation in which the task is to describe an interesting phenomenon or a problematic process. There are a number of dilemmas that arise in this context, especially when it comes to observing and describing the process of inquiry itself. Since these difficulties threaten to paralyze the basic abilities to observe, to describe, and to reflect, a reasonable way around their real obstacles or through their apparent obstructions has to be found before proceeding.

Afterwards, I consider concrete examples of inquiry activities, using any combination of views that appears to be of service at a given moment of discussion, and pointing out problems that call for further investigation.

### 4.1. Approaches to Inquiry

Try as I might, I do not see a way to develop a theory of inquiry from nothing: To begin from a point where there is nothing to question, to strike out in a given direction without putting anything of consequence at stake, and to trace an unbroken, forward course by following steps that are never unsure. Acquiring a theory of inquiry is not, in short, a purely deductive exercise.

At the risk of being wrong, I am ready to venture that a theory of inquiry is not to be gained for nothing. If I try to base this claim on the evidence of all previous attempts, in both my own and others' trials, having already failed, then it is certain only that no positive proof can arise from so negative a recommendation. Acquiring a theory of inquiry is not, in sum, a purely inductive exercise.

#### 4.1.2. The Pragmatic Framework : Sign-Theoretic Approaches

I would like to introduce a pair of ideas from pragmatism that can help to address the issues of knowledge and inquiry in an integrated way.

The first idea is that knowledge is a product of inquiry. The impact of this idea is that one's interest in knowledge shifts to an interest in the process of inquiry that is capable of yielding knowledge as a result. In the pragmatic perspective, the theory of knowledge, or epistemology, is incorporated within a generative theory of inquiry. The result is a theory of inquiry that treats it as a general form of conduct, that is, as a dynamic process with a deliberate purpose.

The second idea is that all thought takes place in signs. This means that all thinking occurs within a general representational setting that is called a "sign relation". As a first approximation, a sign relation can be thought of as a triadic relation or a three place transaction that exists among the various domains of objects, signs, and ideas that are involved in a given situation. For example, suppose that there is a duck on the lake (this is an object); one refers to it by means of the word "duck" (this is a sign) and one has has an image of the duck in one's mind (this is an idea).

Since an inquiry is a special case of a thought process, an activity that operates on sign and ideas in respect of certain objects, this means that the theory of inquiry and the theory of sign relations are very tightly integrated within this point of view, and are almost indistinguishable. Putting the idea that knowledge is a product of inquiry together with the idea that inquiry takes place within a sign relation, one can even say that the inquiry itself, or the production of knowledge, is just the transformation of a sign relation.

Generally speaking, a transformation of a sign relation allows any numbers of objects, signs, and ideas that are involved in a given situation to be engaged in process of change. For example, adding a new word to one's vocabulary, such as the word "mallard" for that which one formerly called a "duck", is just one of many ways that a sign relation can be transformed.

Constant references to "transformations of sign relations", or else to "sign relational transformations", can eventually become a bit unwieldy, and so I assign them the briefer name of "pragmatic transformations". Considered in their full generality, the potential array of pragmatic transformations that one might find it necessary to consider can be very general indeed, exibiting an overwhelming degree of complexity. To deal with this level of complexity, one needs to find strategies for approaching it in stages. Two common tactics are: (1) to classify special types of pragmatic transformations in terms of which kinds of entities are changing the most, and (2) to focus on special cases of pragmatic transformations in which one class of entities is fixed.

If one intends to study processes of development that are every bit as general as "cultural transformations", in which all of the artifacts, symbols, and values are capable of being thrown into a state of flux, then I suggest that pragmatic transformations are a relatively generic but a reasonably well defined form of intermediate case, in other words, a suitable type of transitional object.

What I just gave was the popular version of the theory of signs. This much was already evident in Aristotle's work On Interpretation and was probably derived from Stoic sources. It is still the most natural and intuitive way to approach the idea of a sign relation. But within the frame of pragmatism proper, a number of changes need to be worked on the idea of a sign relation, in order to make it a more exact and more flexible instrument of thought.

From a pragmatic perspective, ideas are taken to be signs in the mind. In this role they come to serve as special cases of "interpretant signs", those that follow other signs in the ongoing process of interpretation. As far as their essential qualities go, signs and ideas can be classed together, though a sign and its interpretant can still be distinguished by their roles in relation to each other. At this point, the reader is probably itching to ask: Where is the interpreter in all of this? Ultimately, signs and ideas can be recognized as features that affect or indirectly characterize the state of the interpretive agent, and their specifications can even be sharpened up to point that one can say it is the states of the interpreter that are the real signs and interpretants in the process. This observation, that Peirce summed up by saying that the person is a sign, has consequences for bringing about a synthesis between the theory of sign relations and the theory of dynamic systems.

#### 4.1.3. The Dynamical Framework : System-Theoretic Approaches

“Inquiry” 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 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.

##### 4.1.3.1. Inquiry and Computation

Three questions immediately arise in the connection between inquiry and computation. As they reflect on the concept of inquiry, these questions have to do with its integrity, its effectiveness, and its complexity.

1. Integrity. Do all the activities and all the processes that are commonly dubbed "inquiry" have anything essential in common?
2. Effectiveness. Can any useful parts of these so called inquiries be automated in practice?
3. Complexity. Just how deep is the analysis, the disassembly, or the "takedown" of inquiry that is required to reach the level of routine steps?

The issues of effectiveness and complexity are discussed throughout the remainder of this text, but the problem of integrity must be dealt with immediately, since doubts about it can interfere with the very ability to use the word "inquiry" in this discussion.

Thus, I must examine the integrity, or well-definedness, of the very idea of inquiry, in other words, "inquiry" as a general concept rather than a catch all term. 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 carried out on computers, inquiry includes everything from database searches, through dynamic simulation and statistical reasoning, to mathematical theorem proving. Insofar as these tasks constitute specialized efforts, each one needs software that is tailored to the individual purpose. To the extent that these different modes of investigation contribute to larger inquiries, present methods for coordinating their separate findings are mostly ad hoc and still a matter of human skill. Thus, one can question whether the very name "inquiry" 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 this alternative 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.

The focus of my work will narrow in three steps. First, I will concentrate on the design of intelligent software systems that support inquiry. Then, I will select mathematical systems theory as an indispensable tool, both for the analysis of inquiry itself and for the design of programs to support it. Finally, I will develop a theory of qualitative differential equations, implement methods for their computation and solution, and apply the resulting body of techniques to two kinds of recalcitrant problems, (1) those where an inquiry must begin with too little information to justify quantitative methods, and (2) those where a complete logical analysis is necessary to identify critical assumptions.

##### 4.1.3.2. Inquiry Driven Systems

The stages of work just described lead me to introduce the concept of an "inquiry driven system". In rough terms, this type of system is designed to integrate the functions of a data driven adaptive system and a rule driven intelligent system. The idea is to have a system whose adaptive transformations are determined, not just by learning from observations alone, or else by reasoning from concepts alone, but by the interactions that occur between these two sources of knowledge. A system which combines different contributions to its knowledge base, much less the mixed modes of empirical and rational types of knowledge, will find its next problem lies in reconciling the mismatches that arise between these sources. Thus, one arrives at the concept of an adaptive knowledge-base whose changes over time are driven by the differences that it encounters between what is observed in the data it gathers and what is predicted by the laws it knows. This sounds, at the proper theoretical distance, like an echo of an error-controlled cybernetic system. Moreover, it falls in line with the general description of scientific inquiry. Finally, it raises the interesting possibility that good formulations of these "differences of opinion" might allow one to find differential laws for the time evolution of inquiry processes.

There are several implications of my approach that I need to emphasize. Many distractions can be avoided if I continue to guide my approach by the two questions raised above, of principles and extensions, and if I guard against confusing what they ask and do not ask. The issues that surround these points, concerning the actual nature and the possible nurture of the capacity for inquiry, can be taken up shortly. But first I need to deal with a preliminary source of confusion. This has to do with the two vocabularies, the language of the application domain, that talks about the higher order functions and intentions of software users, and the language of the resource domain, that describes the primitive computational elements to which software designers must try to reduce the problems that confront them. I am forced to use, or at least to mention, both of these terminologies in my effort to bridge the gap between them, but each of them plays a different role in the work.

In studies of formal specifications the designations "reduced language" and "reducing language" are sometimes used to discuss the two roles of language that are being encountered here. It is a characteristic of some forms of reductionism to call a language "reduced" simply because it is intended to be reduced and long before it is actually reduced, but aside from that this language of "reduced" and "reducing" can still be useful. The reduced language, or the language that is targeted to be reduced, is the language of the application, practice, or target domain. The reducing language, or the language that is intended to provide the sources of explanation and the resources for reduction, is the language of the resource, method, or base domain. I will use all of these terms, with the following two qualifications.

First, I need to note a trivial caution. One's sense of "source" and "target" will often get switched depending on one's direction of work. Further, these terms are reserved in category theory to refer to the domain and the codomain of a function, mapping, or transformation. This will limit their use, in the above sense, to informal contexts.

Now, I must deal with a more substantive issue. In trying to automate even a fraction of such grand capacities as intelligence and inquiry, it is seldom that we totally succeed in reducing one domain to the other. The reduction attempt will usually result in our saying something like this: That we have reduced the capacity A in the application domain to the sum of the capacity B in our base domain plus some residue C of unanalyzed abilities that must be called in from outside the basic set. In effect, the residual abilities are assigned to the human side of the interface, that is, they are attributed to the conscious observation, the common sense, or the creative ingenuity of users and programmers.

In the theory of recursive functions, this situation is expressed by saying that A is a "relatively computable" function, more specifically, that A is computable relative to the assumption of an "oracle" for C. For this reason, I usually speak of "relating" a task A to a method B, rather than fully "reducing" it. A measure of initial success is often achieved when a form of analysis relates or connects an application task to a basic method, and this usually happens long before a set of tasks are completely reduced to a set of methods. The catch is whether the basic set of resources is already implemented, or is just being promised, and whether the residual ability has a lower complexity than the original task, or is actually more difficult.

At this point I can return to the task of analyzing and extending the capacity for inquiry. In order to enhance a human capacity it is first necessary to understand its process.

To extend a human capacity we need to know the critical functions which support that ability, and this involves us in a theory of the practice domain. This means that most of the language describing the target functions will come from sources outside the areas of systems theory and software engineering. The first thoughts that we take for our specs will come from the common parlance that everyone uses to talk about learning and reasoning, and the rest will come from the special fields which study these abilities, from psychology, education, logic and the philosophy of science. This particular hybrid of work easily fits under the broad banner of artificial intelligence, yet I need to repeat that my principal aim is not to build any kind of autonomous intelligence, but simply to amplify our own capacity for inquiry.

There are many well-reasoned and well-respected paradigms for the study of learning and reasoning, any one of which I might have chosen as a blueprint for the architecture of inquiry. The model of inquiry that works best for me is one with a solid standing in the philosophy of science and whose origins are entwined with the very beginnings of symbolic logic. Its practical applications to education and social issues have been studied in depth, and aspects of it have received attention in the AI literature (Refs 1-8). This is the pragmatic model of inquiry, formulated by C.S. Peirce from his lifelong investigations of classical logic and experimental reasoning. For my purposes, all this certification means is that the model has survived many years of hard knocks testing, and is therefore a good candidate for further trial. Since we are still near the beginning of efforts to computerize inquiry, it is not necessary to prove that this is the best of all possible models. At this early stage, any good ideas would help.

My purpose in looking to the practical arena of inquiry and to its associated literature is to extract a body of tasks that are in real demand and to start with a stock of plausible suggestions for ways to meet their requirements. Some of what one finds depicted in current pictures of learning and reasoning may turn out to be inconsistent or unrealizable projections, beyond the scope of any present methods or possible technology to implement. This is the very sort of thing that one ought to be interested in finding out! It is one of the benefits of submitting theories to trial by computer that we obtain this knowledge. Of course, the fact that no one can presently find a way to render a concept effectively computable does not prove that it is unworkable, but it does place the idea in a different empirical class.

This should be enough to say about why it is sometimes necessary to cite the language of other fields and to critically reflect on the associated concepts in the process of doing work within the disciplines of systems theory and software engineering. To sum it up, it is not a question of entering another field or absorbing its materials, but of finding a good standpoint on one's own grounds from which to tackle the problems that the outside world presents.

Sorting out which procedures are effective in inquiry and finding out which functions are feasible to implement is a job can be done better in the hard light demanded by formalized programs. But there is nothing wrong in principle with a top down approach, so long as one does come down, that is, so long as one eventually descends from a level of purely topical reasoning. I will follow the analogy of a recursive program that progresses down discrete steps to its base, stepwise refining the topics of higher level specifications to arrive at their more concrete details. The best reinforcement for such a program is to maintain a parallel effort that builds up competencies from fundamentals.

Once I have addressed the question of what the principles are that enable human inquiry it brings me to the question of how I would set out to improve the human capacity for inquiry by computational means.

Within the field of AI there are many ways of simulating and supporting learning and reasoning that would not involve me in systems theory proper, that is, in reflecting on mathematically defined systems or in considering the dynamics that automata trace out through abstract state spaces. However, I have chosen to take the system-theoretic route for several reasons, which I will now discuss.

First, if we succeed in understanding intelligent inquiry in terms of system-theoretic properties and processes, it equips this knowledge with the greatest degree of transferability between comparable systems. In short, it makes our knowledge robust, and not narrowly limited to a particular instantiation of the target capacity.

Second, if we organize our thinking in terms of a coherent system or integrated agent which carries out inquiries, it helps to manage the complexity of the design problem by splitting it into discrete stages. This strategy is especially useful in dealing with the recursive or reflexive quality that bedevils all such inquiries into inquiry itself. This aspect of self-application in the problem is probably unavoidable, due to the following facts. Human beings are complex agents, and any system likely to support significant inquiry is bound to surpass the complexity of most systems we can fully analyze today. Research into complex systems is one of the jobs that will depend on intelligent software tools to advance in the future. For this we need programs that can follow the drift of inquiry and perhaps even scout out fruitful directions of exploration. Programs to do this will need to acquire a heuristic model of the inquiry process they are designed to assist. And so it goes. Programs for inquiry will pull themselves up by their own bootstraps.

Taking as given the system-theoretic approach from now on, I can focus and rephrase my question about the technical enhancement of inquiry. How can we put computational foundations under the theoretical models of inquiry, at least, the ones we discover to be accessible? In more detail, what is the depth and content of the task analysis that we need to relate the higher order functions of inquiry with the primitive elements given in systems theory and software engineering? Connecting the requirements of a formal theory of inquiry with the resources of mathematical systems theory has led me to the concept of inquiry driven systems.

The concept of an inquiry driven system is intended to capture the essential properties of a broad class of intelligent systems, and to highlight the crucial processes which support learning and reasoning in natural and cultural systems. The defining properties of inquiry driven systems are discussed in the next few paragraphs. I then consider what is needed to supply these abstractions with operational definitions, concentrating on the terms of mathematical systems theory as a suitable foundation. After this, I discuss my plans to implement a software system which is designed to help analyze the qualitative behavior of complex systems, inquiry driven systems in particular.

An inquiry driven system has components of state, accessible to the system itself, which characterize the norms of its experience. The idea of a norm has two meanings, both of which are useful here. In one sense, we have the descriptive regularities which are observed in summaries of past experience. These norms are assumed to govern the expectable sequences of future states, as determined by natural laws. In another sense, we have the prescriptive policies which are selected with an eye to future experience. These norms govern the intendable goals of processes, as controlled by deliberate choices. Collectively, these components constitute the knowledge base or intellectual component of the system.

An inquiry driven system, in the simplest cases worth talking about, requires at least three different modalities of knowledge component, referred to as the expectations, observations, and intentions of the system. Each of these components has the status of a theory, that is, a propositional code which the agent of the system carries along and maintains with itself through all its changes of state, possibly updating it as the need arises in experience. However, all of these theories have reference to a common world, indicating under their varying lights more or less overlapping regions in the state space of the system, or in some derivative or extension of the basic state space.

The inquiry process is driven by the nature and extent of the differences existing at any time among its principal theories, for example, its expectations, observations, and intentions. These discrepancies are evidenced by differences in the sets of models which satisfy the separate theories. Normally, human beings experience a high level of disparity among these theories as a dissatisfying situation, a state of cognitive discord. For people, the incongruity of cognitive elements is accompanied by an unsettled affective state, in Peirce's phrase, the "irritation of doubt". A person in this situation is usually motivated to reduce the annoying disturbance by some action, all of which activities we may classify under the heading of inquiry processes.

Without insisting on strict determinism, we can say that the inquiry process is lawful if there is any kind of informative relationship connecting the state of cognitive discord at each time with the ensuing state transitions of the system. Expressed in human terms, a difference between expectations and observations is experienced as a surprise to be explained, a difference between observations and intentions is experienced as a problem to be solved. We begin to understand a particular example of inquiry when we can describe the relation between the intellectual state of its agent and the subsequent action that the agent undertakes.

These simple facts, the features of inquiry outlined above, already raise a number of issues, some of which are open problems that my research will have to address. Given the goal of constructing supports for inquiry on the grounds of systems theory, each of these difficulties is an obstacle to progress in the chosen direction, to understanding the capacity for inquiry as a systems property. In the next few paragraphs I discuss a critical problem to be solved in this approach, indicating its character to the extent I can succeed at present, and I suggest a reasonable way of proceeding.

In human inquiry there is always a relation between cognitive and affective features of experience. We have a sense of how much harmony or discord is present in a situation, and we rely on the intensity of this sensation as one measure of how to proceed with inquiry. This works so automatically that we have trouble distinguishing the affective and cognitive aspects of the irritating doubt that drives the process. In the artificial systems we build to support inquiry, what measures can we take to supply this sense or arrange a substitute for it? If the proper measure of doubt cannot be formalized, then all responsibility for judging it will have to be assigned to the human side of the interface. This would greatly reduce the usefulness of the projected software.

The unsettled state which instigates inquiry is characterized by a high level of uncertainty. The settled state of knowledge at the end of inquiry is achieved by reducing this uncertainty to a minimum. Within the framework of information theory we have a concept of uncertainty, the entropy of a probability distribution, as being something we can measure. Certainly, how we feel about entropy does not enter the equation. Can we form a connection between the kind of doubt that drives human inquiry and the kind of uncertainty that is measured on scales of information content? If so, this would allow important dynamic properties of inquiry driven systems to be studied in abstraction from the affective qualities of the disagreements which drive them. With respect to the measurable aspects of uncertainty, inquiry driven systems could be taken as special types of control systems, where the variable to be controlled is the total amount of disparity or dispersion in the knowledge base of the system.

The assumption of modularity, that the affective and intellectual aspects of inquiry can be disentangled into separate components of the system, is a natural one to make. Whenever it holds, even approximately, it simplifies the task of understanding and permits the analyst or designer to assign responsibility for these factors to independent modules of the simulation or implementation.

However, this assumption appears to be false in general, or true only in approaching certain properties of inquiry. Other features of inquiry are not completely understandable on this basis. To tackle the more refractory properties, I will be forced to examine the concept of a measure which separates the affective and intellectual impacts of disorder. To the extent that this issue can be resolved by analysis, I believe that it hinges on the characters that make a measure objective, that is, an impression whose value is invariant over many different perspectives and interpretations, as opposed to being the measure of a purely subjective moment, that is, an impression whose value is limited to a special interpretation or perspective.

The preceding discussion has indicated a few of the properties that are attributed to inquiry and its agents and has initiated an analysis of their underlying principles. Now I engage the task of giving these processes operational definitions in the framework of mathematical systems theory.

Consider the inquiry driven system as described by a set of variables:

x1, ... , xn, a1, ... , ar.

The xi are regarded as ordinary state variables and the aj are regarded as variables codifying the state of knowledge with respect to a variety of different questions. Many of the parameters aj will simply anticipate or echo the transient features of state that are swept out in reality by the ordinary variables xi. This puts these information variables subject to fairly direct forms of interpretation, namely, as icons and indices of the ordinary state of the system. However, in order for the system to have a knowledge base which takes a propositional stance with respect to its own state space, other information variables among the aj will have to be used in less direct ways, in other words, made subject to more symbolic interpretations. In particular, some of them will be required to serve as the signs of logical operators.

The most general term that I can find to describe the informational parameters aj is to call them "signs". These are the syntactic building blocks that go into constructing the various knowledge bases of the inquiry driven system. Although these variables can be employed in a simple analogue fashion to represent information about remembered, observed, or intended states, ultimately it is necessary for the system to have a formal syntax of expressions in which propositions about states can be represented and manipulated. I have already implemented a fairly efficient way of doing this, using only three arbitrary symbols beyond the set that is used to echo the ordinary features of state.

A task that remains for future work is to operationalize a suitable measure of difference between alternative propositions about the world, that is, to sort out competing statements about the state space of the system. A successful measure will gauge the differences in objective models and not be overly sensitive to unimportant variations in syntax. This means that its first priority is to recognize logical equivalence classes of expressions, in other words, to discriminate between members of different equivalence classes, but to respond in equal measure to every member of the same equivalence class. This requirement brings the project within the fold of logical inquiry. Along with finding an adequate measure of difference between propositions, it is necessary to specify how these differences can determine, in some measure, the state transitions of an inquiry driven system. At this juncture, a variety of suggestive analogies arise, connecting the logical or qualitative problem just stated with the kinds of questions that are commonly treated in differential geometry and in geometric representations of dynamics.

### 4.2. The Context of Inquiry

#### 4.2.1. The Field of Observation

My discussion of these questions will be organized in the following way. I use the term "pragmatic object" to describe any form of conduct or any pattern of activity that attracts one's attention and that comes to form the focus of a deliberate consideration. This usage is intended to reflect the meaning of the Greek word "pragma", that covers the sphere of senses from objects and objectives to purposes, issues, and concerns. In fact, I consistently use the word "object" in precisely this sense, regarding even the ordinary sorts of physical objects as special cases of processes that appear to be going on in the external world and that seem to demand the specialized activities of orientation on the part of their observers. The forms of conduct that commonly and spontaneously attract one's interest are considered to take place within a catch all setting that I call the "informal context". This name simply reflects the fact that most of the things that come to one's attention are capable of attracting one's interest long before one is capable of understanding them or even carefully describing them.

Whenever one reflects on a form of conduct or observes a pattern of activity in the informal context, and is successfully motivated to articulate one's reflections and observations, then one generates a description, a text or an image that depicts something going on in the world. Depending on the angle of approach, the attitude of observation, the point of view, or the line of sight that one takes up with respect to a pragmatic object, one will be able to observe different aspects of it. Of course, I am using this visual terminology in ways that are partly metaphorical, and there are times when it will be necessary to reflect on whether this style of spectator language is leading us into illusions.

Operating on the tacit assumption that one is able to articulate one's observations and reflections, the end of the process of observation or reflection is to generate a description of the object under consideration. This description represents a first step toward a formalization of the conduct in question, and so I can place the end result of this entire process within a conceptual area that I call the "formal context". From this point on, it is convenient to use the name "formalization" to sum up all of the effects of observation, reflection, articulation, and description that produce an image in the formal context. A number of signficant features of this situation need to be noted at this point.

The line between the formal context and the informal context is not intended to be hard and fast. It merely marks a relative difference in the character of a description.

The formal context is wholly contained within the informal context. Remember, I did call the informal context a "catch all" setting. Thus, the acts of demarcation that one uses to distinguish the formal context, along with the acts of formalization that lead into it, are just the kinds of intentional activities that are likely to form the most interesting sorts of pragmatic objects.

Descriptions are necessarily partial. This statement is conditioned on the fact that the typical object of interest is most likely well beyond anyone's capacity to describe completely, at least, while I continue to talk about finite agents and mortal observers. This means that different observers, or the same observers under different conditions, are likely to end up describing different aspects of the same activity.

At the level of communities these differences and partialities tend to become embodied and institutionalized into specialized disciplines, represented by areas within the formal context that I describe as "formal arenas". A typical formal arena is dedicated to a particular angle of approach or a special attitude of observation toward whatever activities in the informal context happen to fall under its purview. Participants in a formal arena devote themselves to sharpening their view of the aspect in question. In time, a formal arena will usually develop a specialized language, and even a peculiar interpretation of selected terms in the common language, all the better to focus on its chosen aspect of the activity in question.

At first for reasons of simple efficiency, and later by dint of inveterate and unreflective habits, formal arenas gradually develop the structures of bunkers, duck blinds, ivory towers, and silos, with opaque barriers coming to grow up or being erected around its perimeter that obstruct the view of the informal context from other directions than along their favorite and habitual lines of vision. When the relationship between a formal arena and the informal context reaches this stage of development, the boundaries are no longer permeable but appear to be cast in stone.

Analysis has acquired a privileged role in this process of differentiation. But if it succeeds to the throne of knowledge and comes to dominate the field of observation, it is only by default, on account of the absence of a balanced attention being accorded to the process of integration, due to the notion that synthesis can be left to the last, and justified by the fond hope that all disharmonies that are generated in the meantime can be atoned for in the end. It is a feature of modernism that it produces an overemphasis on the analytic aspect of the process of description while marginalizing or trivializing the synthetic component.

Out of this problematic situation one can see emerging three important issues. The first arises as a simplifying assumption but hardens into the character of a dogmatic thesis, the "triviality of integration" hypothesis. The next concerns the tension between the informal context and the formal context, along with the tensions that develop among the various formal arenas. The last is the problematic of communication that is created by differing styles of mental models and preserved through a lack of appreciation for how the several aspects of representation are themselves meant to be integrated into a coherent and competent whole.

In the attempt to reflect on our own theories of knowledge, to inquire into the nature of inquiry, and to form a coherent theory that serves as a competent guide for how to improve our performance in learning and reasoning, we encounter a couple of extremely bewildering difficulties: One problem threatens to keep us from getting started in any sensible direction at all. The other problem finds us in the middle of a mass of indications, that we typically collect in the meantime, and leaves us at a loss about how to sum up and how to draw even a tentative conclusion.

It is necessary to distinguish these skeptical questions of methodology from the more substantive issues that I described at the outset of this discussion. Because these problems of process are so elusive, defeating even the attempt to give them initial names or to find them final terms, I resort to calling them the "alpha problem" and the "omega problem", respectively. The first can be roughly characterized as involving a problem of reflection, while the last can be roughly characterized as involving a problem of reconstruction. In essence, each problem is more like a generic source of problems, or a general problem area from which a multitude of further difficulties can be found to arise. Each of these problem areas contains an especially acute instance, that I abstractly refer to as the "initial dilemma" and the "final dilemma", respectively.

The initial dilemma can be recognized as a dilemma of reflective inquiry. It is typically resolved only by making a plausible hypothesis and then moving on from there to see how one's guesses fare. The final dilemma can be described as a dilemma of critical democracy. It arises from a confrontation between two persistent factors, opposing one of the most favored notions of the modern point of view with the sheer unlikelihood that it is really true. The notion underlying the dilemma is the same "triviality of integration" hypothesis that modernism received from its classical foundations and extended with little change of direction up to the present time. The unlikelihood of its truth that is becoming more apparent, forces us to find ways, not simply of denying it in principle, but of generating constructive and livable alternative to its habitual structures, those that are built into our present cultural institutions. These problems and dilemmas are discussed in the next two subsections, after which I return to the main concern, namely, to remedy the consequences of the long held notion that integration is trivial.

#### 4.2.2. The Problem of Reflection

 Tell me, good Brutus, can you see your face? No, Cassius, for the eye sees not itself But by reflection, by some other things. — Julius Caesar, 1.2.53–55

The faculty of reflection is the capacity to reflect on one's own conduct. This ability is commonly agreed to be an important ingredient in all of the efforts to improve conduct that are otherwise known as "learning". In this way, one comes to the questions: (1) whether a particular form of conduct is naturally reflective in and of itself, (2) whether it can be rendered reflective through the application of appropriate means, and (3) whether an individual or an organization can become more reflective, and thus more capable of criticizing and improving its own performance. Not too surprisingly, these questions are critical to the enterprises of achieving "reflective practice" and building "learning organizations", in essence, of studying and designing "self aware" and "self organizing" systems.

Reflection is an act of self observation that parallels the observation of others. This simple statement already conceals a host of difficulties. An observation can be simple act or a complex process, taking place at a single point in spacetime or extending over a multitude of dimensions. Even the term "observation" is equivocal, referring in a single breath to both the act and its articulation. With this much leeway in our speech, reflection can incorporate the observation of others and even the kinds of observation that are said to occur in the imagination, as when one speaks of "reflecting on a situation" to mean observing or imagining a situation that one is merely a part of, or only intends to be a party to.

Any form of observation, if it is articulated, issues in a description. A description is a verbal text or a visual image that can be judged according to how well it captures or conveys the nature of what lies under observation. Whether reflection on oneself is easier or harder than the analogous process of observing others — this is another question altogether. My present focus is on the role of reflection in learning or improving conduct.

When we observe a form of conduct in others that we desire to emulate, our task is to describe it well enough to ourselves that we are capable of reproducing the performance, at least, moderately well and more or less in accordance with our own style and taste. When we reflect on a form of conduct in ourselves that we wish to examine, to criticize, and to improve, the question is whether we can judge this performance with the same degree of detachment that we usually take in regard to others. But the first task is the same in either case, to arrive at a description that is clear enough to serve the purpose of improving conduct.

But how does the skill of reflection itself arise? Is it innate or is it learned, and if it is acquired in a succession of stages, then how can it be improved, except through reflection on itself? To put the question more generally, if inquiry is the form of conduct that occupies our interest, and if reflection is an integral part of inquiry, then how is inquiry made reflective, if not through reflection on itself and inquiry into itself? These questions lead to the dilemma of reflective inquiry.

This problem arises from the question: How do we know that our methods of inquiry are any good, that they lead to knowledge as a result? The horns of the dilemma are these:

a. If we say that our methods of inquiry are justified on the basis of authority, then we invite the charge of hypocrisy, and we are guilty of this charge if we continue to maintain that inquiry and authority are fundamentally different ways of deciding questions. Unless we make it clear that all pretence of inquiry reduces to a matter of authority, then we are merely dissembling a question that is already decided and posing it under the guise of a misleading name.

b. If we say that our methods of inquiry are justified on the basis of inquiry, then we invite the charge of begging the question, and we even run the risk of falling into an infinite regress. Unless we have hopes that the recursion of inquiry to itself is not one of those forms of self application that leads to paradox, then there is no good reason to choose the path of inquiry.

The presence of this dilemma at the roots of our reflective tradition and the influence of the various answers to it, as options that fill out the background of our common reflective field — all of these features are adequately illustrated by the way that Aristotle asks the question:

 One might raise the question: if the mind is a simple thing, and not liable to be acted upon, and has nothing in common with anything else, ... how will it think, if thinking is a form of being acted upon? For it is when two things have something in common that we regard one as acting and the other as acted upon. And our second problem is whether the mind itself can be an object of thought. (Aristotle, On the Soul, III.iv.429b24–28, p. 169)

Of the two choices, I confess to favoring the application of inquiry to itself, since I know that it is possible for a properly constructed recursive procedure to terminate with a determinate result and thus to render an account of itself that is ultimately well founded in the end. According to this strategy, which operates in the meantime more as a hope or a regulative principle than as an item of certified knowledge, but without which it is impossible to proceed at all, one acts as if the methods of inquiry can themselves be justified on the basis of inquiry. In order for this to be possible, methods of inquiry that come under suspicion need to be subject to examination by means of an inquiry into their workings, and those that are valid need to be validated through a study that compares their actual effects with their intended ends. Whatever the case, it seems that the sheer self consistency of inquiry as a way of life demands that its principles and methods can themselves be the subjects of inquiry, and unless this form of consistency is discovered to be an illusion then it deserves to be pursued.

#### 4.2.3. The Problem of Reconstruction

 Tell me where is fancy bred, Or in the heart, or in the head? How begot, how nourishèd? … It is engendered in the eyes, With gazing fed; and fancy dies In the cradle where it lies. — Merchant of Venice, 3.2.63–69

The faculty of integration is the capacity to reconstruct the splintered images that are fashioned with regard to an object of interest, to reform them into a coherent picture that captures the essence of the original, and to preserve a sense of vision that continues to inspire the desire to know more. This ability is argued here to be an critically important, but frequently neglected ingredient in the efforts to improve conduct that all the world calls "learning". In the aim to give this task the attention it deserves and to take its demands seriously, one comes up against, not just the prevailing notion that the whole exercise is not worth the candle, but the new difficulty of how to deny this founding notion in a positive way, and thus to devise an alternative that is genuinely worth having.

In this way, one comes to the following question: If common sense, a faculty that is neither necessary nor possible to educate, does not suffice to integrate the senses, then what can be found to do the job, and how are we to train this faculty, as train it we must?

In the process of denying the triviality of integration I come to an especially acute instance of the reconstruction problem. This final dilemma is the dilemma of critical democracy.

This dilemma is evident in both the classical and the modern situation, but it seems to have become more acute with the passage of time and to form an especially troubling issue at the present juncture. It arises from a problematic thesis that is already well expressed in classical sources, but one that was, surprisingly enough, transmitted with only a passing challenge into the axioms of the modern tradition.

If the faculty of integration is not adequately covered by common sense, and if we need this faculty to set wise goals, to make wise choices of the means toward these ends, and overall to direct our conduct toward goals worth having, then how is the power of choice to be acquired by a person capable of learning, and how should the power to choose a common course be distributed throughout a democratic organization?

These questions lead to the dilemma of critical democracy. This is the problem of how to constitute a society on a principle of equality, not just taking the mode of common opinion or the mean point of view, and thus achieving the facile coherence of a superficial solidarity, but to form a truly coherent collective that is competent to deal with reality. The manifestations of this question are most clearly reflected in the public sphere, but analogous issues also arise in the considerations of "dispersed leadership" and "learning organizations".

One way to deal with the problem of reconstruction is simply to ignore it, to blithely wave one's hand, and summarily, if inanely to dismiss it. The sources of this particular response appear to go back at least as far as Aristotle, but …

The formal materials that one needs to resolve this final dilemma, if only in principle, are already present in Aristotle's teaching. One need only apply these principles to the received assumptions about common sense. To say how it is possible, in principle, for a wisdom to arise that does not reduce to common sense, or to say how such a state could exist as a critical democracy without trivializing the difficulty of achieving it, I can utilize a couple of distinctions that Aristotle himself makes: the first between "potentiality" and "actuality", and the second within the category of actuality between "possession" and "exercise".

 Matter is potentiality (dynamis), while form is realization or actuality (entelecheia), and the word actuality is used in two senses, illustrated by the possession of knowledge (episteme) and the exercise of it (theorein). (Aristotle, 1936, 67).

Using these distinctions, it is fair to say that just about everyone has the potential for wisdom, or possesses the capacity for this highest level of integration in one's total conduct, but that not everyone will take the trouble to actualize it, or to go through the exercise of developing their full potential. This is a pretty solution, but it only solves the problem in principle. To say how it is possible, in practice, for such a wisdom or such a democracy to come about — this is clearly another matter.

#### 4.2.4. The Trivializing of Integration

The roots of this "modern" structure appear to be traceable to Aristotle and Descartes.

The presence of this assumption can be detected in three fundamental texts, the implications of which are well worth the time to examine.

The first passage occurs in Aristotle's treatise On Interpretation, where he articulates his understanding of the fundamental relationship that exists among objects or objectives in the world, signs and images in the various media of communication, and ideas or "affective impressions" in the mind. Due to the complexity of this relationship, Aristotle is forced to make a number of simplifying assumptions. This is a reasonable way to begin, but the fixing of these assumptions into the form of a dogma led many subsequent generations of thinkers to ignore the full potential of this relationship.

 Words spoken are symbols or signs (symbola) of affections or impressions (pathemata) of the soul (psyche); written words are the signs of words spoken. As writing, so also is speech not the same for all races of men. But the mental affections themselves, of which these words are primarily signs (semeia), are the same for the whole of mankind, as are also the objects (pragmata) of which those affections are representations or likenesses, images, copies (homoiomata). (Aristotle, On Interpretation, i.16a4–9, p. 115)

Aristotle's account contains two claims of constancy or uniformity, asserting that objects and ideas are the same, respectively, for all human interpreters. This ignores the plurality and the mutability of interpretation, issues that we cannot afford to trivialize in the general consideration of diverse perspectives, if only with respect to the human potential for creative variation, or else in the application to education, where the whole idea is to learn new interpretations. In their effects, these assumptions lead to the idea that every diversity among observers is merely a disagreement about words.

The second passage occurs in Aristotle's discussion of psychology, where he argues that the mind has an inherent capacity to integrate the data of the senses. Aristotle's doctrine of the common sense faculty, or sensus communis, is ably summarized by W.S. Hett in his introduction to Aristotle's treatise On the Soul, where he glosses this term in the following way:

 Book III is chiefly concerned with other vital faculties, but some accessories to the theory of sensation overflow into its opening chapters. The connecting link is formed by the problem: What is it that unifies (or distinguishes) the data of sense? Sensus Communis. The solution given is that there is a common sense faculty (located in or near the heart …) which receives and co ordinates the stimuli passed on to it from the various sense organs. This same faculty also directly perceives the "common sensibles" (i.e., those attributes, such as shape, size, number, etc., which are perceptible by more than one sense), among which Aristotle includes movement and time …. It also accounts for our consciousness of sensation, and it is responsible for the process of imagination. (Hett, in (Aristotle, 1936), p. 5)

Without trying to answer Shakespeare's question, that appears to be an echo of this very issue, I can stop to make the following observations. The question of a common sense, that compares and contrasts the data of the senses, can be put in relation to the question of interpretation by recognizing that the data of the senses are particular kinds of signs that naturally refer to objects in the world.

The third passage that I offer for examination comes from Descartes' Discourse on the Method of Properly Conducting One's Reason and of Seeking the Truth in the Sciences.

 Good sense is the most evenly shared thing in the world, for each of us thinks he is so well endowed with it that even those who are the hardest to please in all other respects are not in the habit of wanting more than they have. It is unlikely that everyone is mistaken in this. It indicates rather that the capacity to judge correctly and to distinguish the true from the false, which is properly what one calls common sense or reason, is naturally equal in all men, and consequently that the diversity of our opinions does not spring from some of us being more able to reason than others, but only from our conducting our thoughts along different lines and not examining the same things. … As far as reason or good sense is concerned, … I am ready to believe that it is complete and entire in each one of us, … that there are degrees only between accidents and not between the forms or natures of the individuals of a given specie. (Descartes, 1968, 27 28)

Surprisingly enough, for all the passage of time that intervenes between the two accounts, and for all the other contingencies that are commonly imagined to have changed, this passage closely echoes in all of its main respects the doctrine of Aristotle concerning the common sense.

In contemplating these texts and trying to assess their impact on the contemporary scene, one can view them as expressions of underlying assumptions, maintaining their force in our culture whether or not individual members of the culture have ever heard them rendered explicit in precisely these terms before. These ideas form the basis for an especially refractory modernist thesis, one that I am calling the "triviality of integration". This is the idea that nothing is lost in taking things apart, as in the processes of a selective observation or a reductive analysis, because "just about anyone" can put things back together. In other words, common sense suffices to achieve the necessary synthesis or the subsequent reconstruction.

Aristotle's thesis that the senses are integrated by a common sense has served as a metaphor for the relation of the special sciences to the overall unity of science and it has informed the relations that are instituted between the specialized disciplines and the whole realm of knowledge that is presided over by the university. In exploiting this template, it has been taken for granted that the disciplines bear the same automatic relationship to the whole of knowledge that the senses bear to common sense. This underlying belief leads to the problematic assumption that the integration of the disciplines is a trivial matter. If anyone can do it, then it is not incumbent on us as educators to develop this skill in our students, and experts in any discipline are automatically well equipped to interpret and synthesize the knowledge that is delivered to them by disciplines in which they are novices.

#### 4.2.5. Tensions in the Field of Observation

Two kinds of tension in the field of observation were recognized to arise from the pressure toward articulate and analytic description. There is a tension between the informal context and the formal context and there are tensions that develop as a consequence among the various formal arenas.

Properly considered, each of these tensions ought to be recognized as a positive force. Each one serves as a nagging reminder that something important has been omitted from our descriptions, and a sensitivity to the directions of their tugging and nudging can act to draw us back toward wholeness.

#### 4.2.6. Problems of Representation and Communication

Another pair of closely linked issues were seen to arise from the assumption that integration is trivial. One is the problematic of communications that is created by differing styles of mental models, in other words, by the tendency to form internally coherent but externally disparate systems of mental images. The other is the disjunction that this axiom permits to occur between the denotative aspects and the connotative aspects in the full representation of reality. Those who specialize in either aspect tend to ignore the importance of the other, and even if they do appreciate that both are necessary they tend to take the union for granted, rather than recognizing the complex nature of the complementarities and the dualities that are actually involved.

Aristotle's assumption that objects and their mental impressions are the same for everybody and that only their signs are different for different language communities makes it seem like all problems of communication reduce to problems of translation rather than constituting appreciably different ways of perceiving and interpreting the world.

### 4.3. The Conduct of Inquiry

In this section I lay out the pragmatic theory of inquiry that I will use in my study of inquiry driven systems. In the first section I introduce the basic features of a canonical model of inquiry processes. After this, I outline two different approaches to the functional structure of inquiry. Finally, I discuss a collection of computational routines that I have implemented to study various aspects of this model.

#### 4.3.1. Introduction

The pragmatic theory or model of inquiry was extracted by C.S. Peirce from basic materials in classical logic and refined in parallel with the historical development of symbolic logic to address problems about the nature of scientific reasoning. Borrowing on concepts from Aristotle, Peirce identified three fundamental modes of reasoning, called deductive, inductive, and abductive inference. In rough terms, "abduction" is what one uses to generate a likely hypothesis or initial diagnosis in response to a phenomenon or a problem of interest, while "deduction" is used to clarify and derive relevant consequences of one's hypotheses, and where "induction" is used to test the sum of one's predictions against the sum of the data that is gleaned from experience. Generally speaking, these three processes operate in a cyclic fashion, systematically reducing the uncertainties and the difficulties which initiate inquiry, and thereby leading to an increase in knowledge.

In the pragmatic way of thinking everything has a purpose, and the purpose of each thing is the first thing we should try to note about it. The purpose of inquiry is to reduce doubt and lead to a state of belief, which a person in that state will usually call knowledge or certainty. As they contribute to the purpose of inquiry, we should appreciate that the three kinds of inference form a cycle that can only be understood as a whole, and none of them makes complete sense in isolation from the others. For instance, the purpose of abduction is to generate guesses of a kind that deduction can explicate and induction can evaluate. This places a mild but meaningful constraint on the production of hypotheses, since it is not just any wild guess at explanation that submits itself to reason and bows out when defeated in a match with reality. In a similar fashion, each of the other types of inference realizes its purpose only in accord with its role in the cycle of inquiry. No matter how much it may be necessary to study these processes in abstraction from each other, the integrity of inquiry places strong limitations on the effective modularity of its components.

For our present purposes, the first feature to note in distinguishing these modes of reasoning is whether they are exact or approximate in character. Deduction is the only type of reasoning that can be made exact, always deriving true conclusions from true premisses, while induction and abduction are unavoidably approximate in their mode of operation, involving elements of fallible judgment and inescapable error in their application. The reason for this is that deduction, in the ideal limit, can be rendered a purely internal process of the reasoning agent, while the other two modes of reasoning essentially demand a constant interaction with the outside world, a source of phenomena that will no doubt keep exceeding any finite resource, human or machine. Embedded in this larger reality, approximations can only be judged appropriate in relation to a context of use and a purpose in view.

A parallel distinction made in this connection is to call deduction a demonstrative inference, while abduction and induction are classed as non demonstrative forms of reasoning. Strictly speaking, the latter types of reasoning are not properly called inferences at all. They are more like controlled associations of words or ideas that just happen to be successful often enough to be preserved. But non demonstrative ways of thinking are inherently subject to error, and must be checked out in practice.

In classical terminology, forms of judgment that require attention to context and purpose are said to involve elements of art, as compared with science, and rhetoric, as contrasted with logic. In a figurative sense, this means that only deductive logic can be reduced to an exact science, while the practice of empirical science will always remain to some degree an art. This fact has important implications for any attempt to support inquiry with automated procedures, constraining both the manner and degree of likely success. It means that inquiry software will need to be highly interactive, sensitive to run time conditions at two kinds of interfaces, those with its human users and those with the real world. Further, it means that the main effect of automation will be to speed up and strengthen deductive reasoning. The chief assistance that computation provides to induction is through measures of fit between theoretical constructs and empirical data sets. The limited guidance that formal methods can bring to hypothesis generation is restricted to checking the partly logical property of falsifiability and speeding up the subsequent evaluation process. However, because inquiry is an iterative cycle, improving the rate of performance at any bottleneck can serve to accelerate the entire process.

As far as automating induction goes, we should not expect a program to make up the data for us, no matter how sophisticated it gets! Inductive tests can provide measures of how well a theoretical construct fits a set of data, but no fit is perfect, or intended to be. An inductive concept is supposed to present a simplification of a complex reality, otherwise it would serve no function over and above just staring at the data. In gauging the slippage between concept and data, the degree of tolerance acceptable in a given situation is a matter of discretionary judgments that have to be made under field conditions.

When it comes to automating abductive reasoning, we should observe the historical circumstance that it is often the most "unlikely" set of hypotheses that turn out to form the correct conceptual framework, at least when that likelihood has been judged from the standpoint of the previous framework. Aside from their responsibilities to the inquiry process, abductive hypotheses can be freely generated in the most creative manner possible. Breaking the mind-set of the problem as stated and reformulating data descriptions from new perspectives are just some of the allowable strategies that are required for success.

Abductive reasoning is the mode of operation which is involved in shifting from one paradigm to another. In order to reduce the overall tension of uncertainty in a knowledge base, it is often necessary to restructure our perspective on the data in radical ways, to change the channel that parcels out information to us. But the true value of a new paradigm is typically not appreciated from the standpoint of another model, that is, not until it has had time to reorganize the knowledge base in ways that demonstrate clear advantages to the community of inquiry concerned.

The preceding survey has introduced a model of inquiry and charted a series of limits on the automation of inquiry. We should not be too discouraged by the acknowledgement of these limits. But we ought to notice that these constraints are not so much limits on the computational extension of human inquiry as they are limits on the instrumental nature of inquiry itself, being the specific adaptation of a finite creature to an infinite world. In other words, these are only the familiar limits of the scientific method. They are the limits that make it a method.

I now return to discussing the pragmatic theory of inquiry, treating its positive features in more depth. I will examine the theory in terms of a canonical model that illustates generic aspects of inquiry processes. My plan for the remainder of this section is to introduce basic terminology and issues.

Inquiry is a form of reasoning process, and therefore a manner of thinking. Pragmatist philosophers hold that all thought takes place in "signs", which is the word they use for the most general class of signals, messages, symbolic expressions, etc. that might be imagined. Even ideas and concepts are held to be a special class of signs, namely, internal states of the thinking agent that result from the interpretation of external signs. The subsumption of inquiry within reasoning and of thinking within sign processes allows us to approach the subject of inquiry from two perspectives. The "syllogistic approach" views inquiry as a logical species. The "sign-theoretic" approach views inquiry within a more general setting of sign processes.

The best point of departure I know for both approaches to inquiry is the following story of inquiry activities in everyday life, as told by John Dewey.

 A man is walking on a warm day. The sky was clear the last time he observed it; but presently he notes, while occupied primarily with other things, that the air is cooler. It occurs to him that it is probably going to rain; looking up, he sees a dark cloud between him and the sun, and he then quickens his steps. What, if anything, in such a situation can be called thought? Neither the act of walking nor the noting of the cold is a thought. Walking is one direction of activity; looking and noting are other modes of activity. The likelihood that it will rain is, however, something suggested. The pedestrian feels the cold; he thinks of clouds and a coming shower. (Dewey, 1910, 6–7)

I now proceed to analyze this example from the standpoints of the syllogistic and sign-theoretic approaches. The ultimate task before us is to understand the relation between these two perspectives as they are unified in a single, coherent subject.

#### 4.3.2. The Types of Reasoning

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.

Case, Fact, Rule

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

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 $$X \Rightarrow Y,$$ and combines it with a Rule, the major premiss $$Y \Rightarrow Z,$$ to arrive at a Fact, the demonstrative conclusion $$X \Rightarrow Z.$$

##### 4.3.2.2. Induction

Contrasted with this pattern, induction takes a Fact of the form $$X \Rightarrow Z$$ and matches it with a Case of the form $$X \Rightarrow Y$$ to guess that a Rule is possibly in play, one of the form $$Y \Rightarrow Z.$$

##### 4.3.2.3. Abduction

Cast on the same template, abduction takes a Fact of the form $$X \Rightarrow Z$$ and matches it with a Rule of the form $$Y \Rightarrow Z$$ to guess that a Case is presently in view, one of the form $$X \Rightarrow Y.$$

#### 4.3.3. Hybrid Types of Inference

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

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 example or by a parallel comparison of cases.

 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.

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.

 $$D \Rightarrow B,$$ "Thebes vs Phocis is war against neighbors". (Case) $$D \Rightarrow A,$$ "Thebes vs Phocis is bad". (Fact) $$B \Rightarrow A,$$ "War against neighbors is bad". (Rule)

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

 $$C \Rightarrow B,$$ "Athens vs Thebes is war against neighbors". (Case) $$B \Rightarrow A,$$ "War against neighbors is bad". (Rule) $$C \Rightarrow A,$$ "Athens vs Thebes is bad". (Fact)

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

Returning to the “Rainy Day” story, we find our hero presented with a surprising Fact:

 $$C \Rightarrow A,$$ "in the Current situation the Air is cool". (Fact)

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:

 $$B \Rightarrow A,$$ "just Before it rains, the Air is cool". (Rule)

This Rule can be recognized as having a potential relevance to the situation because it matches the surprising Fact, $$C \Rightarrow A,$$ in its consequential feature $$A.\!$$ All of this suggests that the present Case may be one in which it is just about to rain:

 $$C \Rightarrow B,$$ "the Current situation is just Before it rains". (Case)

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.

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:

 $$C \Rightarrow B,$$ "the Current situation is just Before it rains". (Case)

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 $$B,\!$$ we may safely suppose that our reasoner has already detached the consequence of the abductive Case, $$C \Rightarrow B,$$ 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:

 $$B \Rightarrow D,$$ "just Before it rains, Dark clouds appear". (Rule)

Contemplating the assumed Case in combination with this new Rule would lead him by an immediate deduction to predict an additional Fact:

 $$C \Rightarrow D,$$ "in the Current situation Dark clouds appear". (Fact)

The reconstructed picture of reasoning assembled in this second phase of inquiry is true to the pattern of deductive inference.

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

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.

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.

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 “field test” 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.

I will now detail how these principles of learning, transfer, and testing apply to the Rainy Day example.

##### 4.3.4.1. Learning

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:

 $$B \Rightarrow A,$$ "just Before it rains, the Air is cool".

Such a proposition is usually induced from a consideration of many past events, as follows.

 $$C \Rightarrow B,$$ "in Certain events, it is just Before it rains". (Case) $$C \Rightarrow A,$$ "in Certain events, the Air is cool". (Fact) $$B \Rightarrow A,$$ "just Before it rains, the Air is cool". (Rule)

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

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, “If past experience is any guide … ” 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: “If past experience is a fair sample of possible experience, then the knowledge gained in it applies to present experience.” 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.

Here are the details of how this works out in the Rainy Day example. Let us consider a fragment $$K\!$$ of the reasoner's knowledge base that is logically equivalent to the conjunction of two rules.

 $$K \Leftrightarrow (B \Rightarrow A) \land (B \Rightarrow D).$$

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 $$C^-\!$$ be a chosen set of experiences, or the circumstances we have in mind when we refer to "past experience". Let $$C^+\!$$ be a collective set of experiences, or the projective total of possible circumstances. Let $$C\!$$ be a current experience, or the circumstances present to the reasoner. If we think of the knowledge base $$K\!$$ 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.

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.

 $$C^- \Rightarrow C^+,$$ "Chosen events fairly sample Collective events". (Case) $$C^- \Rightarrow K,$$ "Chosen events support the Knowledge regime". (Fact) $$C^+ \Rightarrow K,$$ "Collective events support the Knowledge regime". (Rule) $$C \Rightarrow C^+,$$ "Current events fairly sample Collective events". (Case) $$C \Rightarrow K,$$ "Collective events support the Knowledge regime". (Fact)
##### 4.3.4.3. Testing

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.