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Inquiry Driven Systems : Part 3

MyWikiBiz, Author Your Legacy — Thursday January 18, 2018
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Editing Note. This Part is somewhat draftier than the previous two.

Contents

Part 3.

3.1. Outlook of the Project : All Ways Lead to Inquiry

I am using the word inquiry in a way that is roughly synonymous with the term scientific method. Use of inquiry is more convenient, aside from being the shorter term, because of the following advantages:

  1. It allows one to broaden the scope of investigation to include any form of proceeding toward knowledge that merely aims at such a method.
  2. It allows one to finesse the issue, for the time being, of how much "method" there is in science.

This Subdivision and the next deal with opposite aspects of inquiry. In many ways it might have been better to interlace the opposing points of comparison, taking them up in a parallel fashion, but this plan was judged to be too distracting for a first approach. In other ways, the negative sides of each topic are prior in point of time to the positive sides of the issue, but sensible people like to see the light at the end of the tunnel before they trouble themselves with the obscurities of the intervening journey. Thus, this Subdivision of the text emphasizes the positive features of inquiry and the positive qualities of its objective, while the next Subdivision is reserved to examine the negative aspects of each question.

In the order of nature, the absence of a feature naturally precedes the full development of its presence. In the order of discussion, however, positive terms must be proposed if it is desired to say anything at all.

The discussion in this Subdivision is placed to serve a primer, declaring at least the names of enough positive concepts to propose addressing the negative conditions of knowledge in which inquiry necessarily starts.

In this Subdivision I stand back once again from the problem of inquiry and allow myself take a more distant view of the subject, settling into what I think is a comfortable and a natural account of inquiry, the best that I have at my command, and attending to the task of describing its positive features in a positive light. I present my personal view of inquiry as I currently understand it, without stopping to justify every concept in detail or to examine every objection that might be made to this view. In the next Subdivision I discuss a few of the more obvious problems that stand in the way of this view and I try to remove a few of the more tractable obscurities that appear ready to be cleared up. The fact that I treat them as my "personal insights" does not mean that all of these ideas about inquiry originate with me, but only that I have come to adopt them for my personal use. There will be many occasions, the next time that I go over this ground, to point out the sources of these ideas, so far as I know them.

The reader may take my apology for this style of presentation to be implicit in its dogmatic character. It is done this way in a first approach for the sake of avoiding an immense number of distractions, each of which is not being slighted but demands to be addressed in its own good time. I want to convey the general drift of my current model, however conjectural, naive, uncritical, and unreflective it may seem.

3.1.1. The Matrix of Inquiry

Thus when mothers have children suffering from sleeplessness, and want to lull them to rest, the treatment they apply is to give them, not quiet, but motion, for they rock them constantly in their arms; and instead of silence, they use a kind of crooning noise; and thus they literally cast a spell upon the children (like the victims of a Bacchic frenzy) by employing the combined movements of dance and song as a remedy.

(Plato, Laws, VII, 790D).

Try as I might, I do not see a way to develop a theory of inquiry from nothing: To take for granted nothing more than is already given, to set out from nothing but absolutely certain beginnings, or to move forward with nothing but absolutely certain means of proceeding. In particular, the present inquiry into inquiry, \(y_0 = y \cdot y,\) ought not to be misconstrued as a device for magically generating a theory of inquiry from nothing. Like any other inquiry, it requires an agent to invest in a conjecture, to make a guess about the relevant features of the subject of interest, and to choose the actions, the aspects, and the attitudes with regard to the subject that are critical to achieving the objectives of the study.

I can sum all this up by saying that an inquiry requires an inquirer to suggest a hypothesis about the subject of interest and then to put that particular model of the subject to the test. This in turn requires one to devote a modicum of personal effort to the task of testing the chosen hypothesis, to put a quantum of personal interest at stake for the sake of finding out whether the model fits the subject, and, overall, to take the risk of being wrong. Any model that is feasible is also defeasible, at least, where it concerns a contingent subject of inquiry.

The first step, then, of an inquiry into inquiry, is to put forth a tentative model of inquiry, to make a hypothesis about the features of inquiry that are essential to explaining its experienced characteristics, and thus, in a sense, to make a guess at the very definition of inquiry. This requirement seems both obvious and outrageous at the same time. One is perfectly justified in objecting that there is much that precedes this so-called "first step", namely, the body of experience that prepares one to see it and the mass of observation that prompts one to take it. I can deal with this objection by making a distinction between mundane experience and olympian theory, and then by saying that the making of a conjecture is really the first "theoretical" step, but this is a hedge that covers the tracks of theory in a deceptive way, hiding how early in the empirical process the "cloven hoof" of theory actually enters.

Leaving behind the mythical conditions of pure experience and naive observation, and at least by the time that one comes to give a name to the subject of investigation, one's trek through the data is already half-shod, half-fettered by the connotations of the name, and in turn by all of the concepts that it invokes in its train. The name, the concepts that it suggests, and the tacit but vague definition of the subject that this complex of associations is already beginning to constellate, attract certain experiences to the complex and filter out other observations from having any bearing on the subject matter. By this point, one is already busy translating one's empirical acquaintance with the subject into an arrangement of concepts that is intended to define its essential nature.

An array of concepts that is set up to capture the essence of a subject is a provisional definition of it, an implicit model of the subject that contains the makings of an explicit theory. It amounts to a selection from the phenomenal aspects of the subject, expresses a guess about its relevant features, and constitutes a hypothesis in explanation of its experienced characteristics. This incipient order of model or theory is tantamount to a definition because it sets bounds on the "stretches" and the "holds" of a term — its extension, intension, and intention — but this is not the kind of definition that has to be taken on faith, or that constitutes the first and the last word on the subject. In other words, it is an empirical definition, one that is subject to being falsified in reference to its intended subject, by failing to indicate the necessary, the pertinent, or the relevant features that account for the presence of its phenomena or the persistence of its process.

If I reflect on the conduct of inquiry, seeking to fix it in a fitting image and trying to cast it in a positive light, the best I can do is this:

Inquiry is a process that aims at achieving belief or knowledge.

But even this simple a description already plunges the discussion deep into a number of obscurities. Most prominently, there is the disjunction between belief and knowledge that cries out to be explained or resolved. Stirring beneath the surface, and not quite fading into the background, many of the other terms that are invoked in the description are capable of hiding the entire contents of the original ignorance that the image as a whole is aimed to dispel. And yet, there is nothing that I can do in this avowedly positive context but to mark these points down as topics for future discussion.

There is already a model of inquiry that is implicit, at least partially, in the text of the above description. Let me see if I can tease out a few of its tacit assumptions.

3.1.1.1. Inquiry as Conduct

First of all, inquiry is conceived to be a form of conduct. This invokes the technical term conduct, referring to the species of prototypically human action that is both dynamic and deliberate, or conceived to fall under a form of purposeful control, usually conscious but possibly not. For the sake of clarity, it helps to seek a more formal definition of conduct, one that expresses the concept in terms of abstract features rather than trying to suggest it by means of typical examples.

Conduct is action with respect to an object. The distinction between action and conduct, reduced to the level of the most abstract formal relations that are involved, can be described in the following manner. Action is a matter of going from A to B, whereas conduct is matter of going from A to B in relation to C. In describing particular cases and types of conduct, the phrase “in relation to” can be filled out in more detail as “on account of”, “in the cause of”, “in order to bring about”, “for the sake of”, “in the interests of”, or in many other ways. Thus, action by itself has a dyadic character, involving transitions through pairs of states, while conduct has a triadic character, involving the kinds of transactions between states that relate throughout to an object.

With regard to this distinction, notice that action is used inclusively, to name the genus of which conduct names a species, and thus depicts whatever has the aspect of action, even if it is actually more complex. This creates the difficulty that the reputed “genus” is less than fully generative, generic, genetic, or even genuine — and so it is necessary to remain on guard against this source of misunderstanding.

What does this definition of conduct say about the temporal ordering of the object with respect to the states? The states are conceived to be ordered in time, but so far nothing has been said to pin down where in relation to these states the object must be conceived to fall in time. Nor does the definition make any particular specification necessary. This makes the question of relative time a secular parameter of the definition, allowing the consideration of the following options:

  1. If the object is thought to precede the action of the conduct, then it tends to be regarded as a creative act, an initial intention, an original stimulus, a principal cause, or a prime mover.
  2. If the object is thought to succeed the action of the conduct, then it tends to be regarded as an end, a goal, or a purpose, in other words, a state envisioned to be fulfilled.
  3. If the object is thought to be concurrent, immanent, or transcendent throughout the action of the conduct, then it tends to be regarded as falling under one of the following possibilities: a prevailing value, a controlling parameter, a universal system of effective forces, a pervasive field of potentials, a ruling law, or a governing principle.

A prevailing value or a controlling parameter, which guides the temporal development of a system, is a term that fits into a law or a principle, which governs the system at a higher level. The existence of a value or a law that rules a system, and the information that an agent of the system has about its parameters and its principles, are two different matters. Indeed, a major task of development for an inquiring agent is to inform itself about the values and the laws that form its own system. Thus, one of the objects of the conduct of inquiry is a description in terms of laws and values of the rules that govern and guide inquiry.

The elaboration of an object in terms of this rich vocabulary — as a cause, end, field, force, goal, intention, law, parameter, principle, purpose, system, or value — adds colorful detail and concrete sensation to the account, and it helps to establish connections with the arrays of terminology that are widely used to discuss these issues. From a formal and relational point of view, however, all of these concepts are simply different ways of describing, at possibly different levels of generality, the object of a form of conduct. With that in mind, I find it useful to return to the simpler form of description as often as possible.

This account of conduct brings to the fore a number of issues, some of them new and some of them familiar, but each of them allowing itself to be approached from a fresh direction by treating it as an implication of a critical thesis just laid down. I next examine these issues in accord with the tenets from which they stem.

1. Inquiry is a form of conduct.

This makes inquiry into inquiry a special case of inquiry into conduct. Certainly, it must be possible to reason about conduct in general, especially if forms of conduct need to be learned, examined, modified, and improved.

Placing the subject of inquiry within the subject of conduct and making the inquiry into inquiry a subordinate part of the inquiry into conduct does not automatically further the investigation, especially if it turns out that the general subject of conduct is more difficult to understand than the specialized subject of inquiry. But in those realms of inquiry where it is feasible to proceed hypothetically and recursively, stretching the appropriate sort of hypothesis over a wider subject area can act to prime the pump of mathematical induction all the more generously, and actually increase the power of the recursion. Of course, the use of a recursive strategy comes at the expense of having to establish a more extended result at the base.

2. The existence of an object that rules a form of conduct and the information that an agent of the conduct has about the object are two different matters.

This means that the exact specification of the object can require an order of information that the agent does not have available, at least, not for use in reflective action, or even an amount of information that the agent lacks the capacity to store. No matter how true it is that the actual course of the agent's conduct exactly reflects the influence of the object, and thus, in a sense, represents the object exactly, the question is whether the agent possesses the equivalent of this information in any kind of accessible, exploitable, reflective, surveyable, or usable form of representation, in effect, any mode of information that the agent can use to foresee, to modify, or to temper its own temporal course.

This issue may seem familiar as a repetition of the “meta” question. Once again, there is a distinction between (a) the properties of an action, agent, conduct, or system, as expressible by the agent that is engaged in the conduct, or as representable within the system that is undergoing the action, and (b) the properties of the same entities, as evident from an “external viewpoint”, or as statable by the equivalent of an “outside observer”.

3. Reflection is a part of inquiry. Reflection is a form of conduct.

The task of reflection on conduct is to pass from a purely interior view of one's own conduct to an outlook that is, effectively, an exterior view. What is sought is a wider perspective, one that is able to incorporate the sort of information that might be available to an outside observer, that ought to be evident from an external vantage point, or that one reasonably imagines might be obvious from an independent viewpoint. I am tempted to refer to such a view as a “quasi-objective perspective”, but only so long as it possible to keep in mind that there is no such thing as a “completely outside perspective”, at least, not one that a finite and mortal agent can hope to achieve, nor one that a reasonably socialized member of a community can wish to take up as a permanent station in life.

With these qualifications, reflection is a form of conduct that can serve inquiry into conduct. Inquiry and its component reflection, applied to a form of conduct, are intended to provide information that can be used to develop the conduct in question. The “reflective development” that occurs depends on the nature of the case. It can be the continuation, the correction, or the complete cessation of the conduct in question.

If it is to have the properties that it is commonly thought to have, then reflection must be capable of running in parallel, and not interfering too severely, with the conduct on which it reflects. If this turns out to be an illusion of reflection that is not really possible in actuality, then reflection must be capable, at the very least, of reviewing the memory record of the conduct in question, in ways that appear concurrent with a replay of its action. But these are the abilities that reflection is “pre-reflectively” thought to have, that is, before the reflection on reflection can get under way. If reflection is truly a form of conduct, then it becomes conceivable as a project to reflect on reflection itself, and this reflection can even lead to the conclusion that reflection does not have all of the powers that it is commonly portrayed to have.

3.1.1.2. Types of Conduct

The chief distinction that applies to different forms of conduct is whether the object is the same sort of thing as the states or whether it is something entirely different, a thing apart, of a wholly other order. Although I am using different words for objects and states, it is always possible that these words are indicative of different roles in a formal relation and not indicative of substantially different types of things. If objects and states are but formal points and naturally belong to the same domain, then it is conceivable that a temporal sequence of states can include the object in its succession, in other words, that a path through a state space can reach or pass through an object of conduct. But if a form of conduct has an object that is completely different from any one of its temporal states, then the role of the object in regard to the action cannot be like the end or goal of a temporal development.

What names can be given to these two orders of conduct?

3.1.1.3. Perils of Inquiry

Now suppose that making a hypothesis is a kind of action, no matter how covert, or that testing a hypothesis takes an action that is more overt. If entertaining a hypothesis in any serious way requires action, and if action is capable of altering the situation in which it acts, then what prevents this action from interfering with the subject of inquiry in a way that undermines, with positive or negative intentions, the very aim of inquiry, namely, to understand the situation as it is in itself?

That making a hypothesis is a type of action may seem like a hypothesis that is too far-fetched, but it appears to follow without exception from thinking that thinking is a form of conduct, in other words, an activity with a purpose or an action that wants an end. The justification of a hypothesis is not to be found in a rational pedigree, by searching back through a deductive genealogy, or determined by that which precedes it in the logical order, since a perfectly trivial tautology caps them all. Since a logical tautology, that conveys no empirical information, finds every proposition appearing to implicate it, in other words, since it is an ultimate implication of every proposition and a conceivable conclusion that is implicit in every piece of reasoning, it is obvious that seeking logical precedents is the wrong way to go for empirical content.

In making a hypothesis or choosing a model, one appears to select from a vaster number of conceivable possibilities than a finite agent could ever enumerate in complete detail or consider as an articulate totality. As the very nature of a contingent description and the very character of a discriminate action is to apply in some cases but not in others, there is no escaping the making of a risky hypothesis or a speculative interpretation, even in the realm of a purely mental action. Thus, all significant thought, even thinking to any purpose about thought itself, demands a guess at the subject or a grasp of the situation that is contingent, dubious, fallible, and uncertain.

If all this is true — if inquiry begins with doubt, if every significant hypothesis is itself a dubious proposition, if the making and the testing of a hypothesis are instances of equally doubtful actions, and if every action has the potential to alter the very situation and the very subject matter that are being addressed — then it leads to the critical question: How is the conduct of inquiry, that begins by making a hypothesis and that continues by testing this description in action, supposed to help with the situation of uncertainty that incites it in the first place and that is supposed to maintain its motivation until the end is reached? The danger is that the posing of a hypothesis may literally introduce an irreversible change in the situation or the subject matter in question. The fear is that this change might be one that too conveniently fulfills or too perversely subverts the very hypothesis that engenders it, that it may obstruct the hypothesis from ever being viewed with equanimity again, and thus prevent the order of reflection that is needed to amend or discard the hypothesis when the occasion to do so arises.

If one fears that merely contemplating a special hypothesis is enough to admit a spurious demonstration into the foundations of one's reasoning, even to allow a specious demon to subvert all one's hopes of a future rationality and to destroy all one's chances of a reasonable share of knowledge, then one is hardly in a state of mind that can tolerate the tensions of a full-fledged, genuine inquiry. If one is beset with such radical doubts, then all inquiry is no more comfort than pure enchoiry. Sometimes it seems like the best you can do is sing yourself a song that soothes your doubts. Perhaps it is even quite literally true that all inquiry comes back at last to a form of "enchoiry", the invocation of a nomos, a way of life, or a song and a dance. But even if this is the ultimate case, it does no harm and it does not seem like a bad idea to store up in this song one or two bits of useful lore, and to weave into its lyric a few suggestions of a practical character.

Let us now put aside these more radical doubts. This putting aside of doubts is itself a form of inquiry, that is, a way of allaying doubts. The fact that I appear to do this by fiat, and to beg for tacit assent, tends to make me suspect the validity of this particular tactic. Still, it is not too inanely dismissive, as its appeal is based on an argument, the argument that continuing to entertain this type of doubt leads to a paralysis of the reason, and that paralyzing the ability to think is not in the interests of the agent concerned. Thus, I adopt the hypothesis that the relationship between the world and the mind is not so perverse that merely making a hypothesis is enough to alter the nature of either. If, in future, I or anyone sees the need to reconsider this hypothesis, then I see nothing about making it that prevents anyone from doing so. Indeed, making it explicit only renders it more subject to reflection.

Of course, a finite person can only take up so many causes in a single lifetime, and so there is always the excuse of time for not chasing down every conceivable hypothesis that comes to mind.

3.1.1.4. Forms of Relations

The next distinguishing trait that I can draw out of this incipient treatise is its emphasis on the forms of relations. From a sufficiently formal and relational point of view, many of the complexities that arise from throwing intentions, objectives, and purposes into the mix of discussion are conceivably due to the greater arity of triadic relations over dyadic relations, and do not necessarily implicate any differences of essence inhering in the entities and the states invoked. As far as this question goes, whether a dynamic object is essentially different from a deliberate object, I intend to remain as neutral as possible, at least, until forced by some good reason to do otherwise. In the meantime, the factors that are traceable to formal differences among relations are ready to be investigated and useful to examine. With this in mind, it it useful to make the following definition:

A conduct relation is a triadic relation involving a domain of objects and two domains of states. When a shorter term is desired, I refer to a conduct relation as a conduit. A conduit is given in terms of its extension as a subset C c XxYxZ, where X is the object domain and where Y and Z are the state domains. Typically, Y = Z.

In general, a conduct relation serves as a model of conduct (MOC), not always the kind of model that is meant to be emulated, but the type of model that captures an aspect of structure in a form of conduct.

The question arises: What is the relationship between signs and states? On the assumption that signs and states are comparable in their levels of generality, consider the following possibilities:

  1. Signs are special cases of states.
  2. Signs and states are the same sorts of things.
  3. States are special cases of signs.

Depending on how one answers this question, one is also choosing among the following options:

  1. Sign relations are special cases of conduct relations.
  2. Sign relations and conduct relations are the same sorts of things.
  3. Conduct relations are special cases of sign relations.

I doubt if there is any hard and fast answer to this question, but think that it depends on particular interpreters and particular observers, to what extent each one interprets a state as a sign, and to what degree each one recognizes a sign as a component of a state.

3.1.1.5. Models of Inquiry

The value of a hypothesis, or the worth of a model, is not to be given a prior justification, as by a deductive proof, but has to be examined in practice, as by an empirical probation. It is not intended to be taken for granted or to go untested, but its meaning in practice has to be articulated before its usefulness can be judged. This means that the conceivable practical import of the hypothesis or the model has to be developed in terms of its predicted and its promised consequences, after which it is judged by the comparison of these speculative consequences with the actual results. But this is not the end of the matter, for it can be a useful piece of information to discover that a particular kind of conception fails a particular kind of comparison. Thus, the final justification for a hypothesis or a model is contained in the order of work that it leads one to do, and the value of this work is often the same whether or not its premiss is true. Indeed, the fruitfulness of a suggestion can lie in the work that proves it untrue.

My plan then has to be, rather than trying to derive a model of inquiry in a deductive fashion from a number of conditions like \(y_0 = y \cdot y,\) only to propose a plausible model, and then to test it under such conditions. Each of these tests is a two-edged sword, and the result of applying a particular test to a proposed model can have either one of two effects. If one believes that a particular test is a hard and fast rule of inquiry, or a condition that any inquiry is required to satisfy, then the failure of a model to live up to its standard tends only to rule out that model. If one has reason to believe that a particular model of inquiry covers a significant number of genuine examples, then the failure of these models to follow the prescribed rule can reflect badly on the test itself.

In order to prime the pump, therefore, let me offer the following account of inquiry in general, the whole of which can be taken as a plausible hypothesis about the nature of inquiry in general.

My observations of inquiry in general, together with a few suggestions that seem apt to me, have led me to believe that inquiry begins with a "surprise" or a "problem". The way I understand these words, they refer to departures, differences, or discrepancies among various modalities of experience, in particular, among "observations", "expectations", and "intentions".

  1. A surprise is a departure of an observation from an expectation, and thus it invokes a comparison between present experience and past experience, since expectations are based on the remembered disposition of past experience.
  2. A problem is a departure of an observation from an intention, and thus it invokes a comparison between present experience and future experience, since intentions choose from the envisioned disposition of future experience.

With respect to these

With respect to this hypothetical

I now test this model of inquiry under the conditions of an inquiry into inquiry, asking whether it is consistent in its application to itself. This leaves others to test the models they like best under the same conditions, should they ever see the need to do so.

Does the inquiry into inquiry begin with a surprise or a problem concerning the process or the conduct of inquiry? In other words, does the inquiry into inquiry start with one of the following forms of departure: (1) a surprising difference between what is expected of inquiry and what is observed about it, or (2) a problematic difference between what is observed about inquiry and what is intended for it?

3.1.2. The Moment of Inquiry

Every young man — not to speak of old men — on hearing or seeing anything unusual and strange, is likely to avoid jumping to a hasty and impulsive solution of his doubts about it, and to stand still; just as a man who has come to a crossroads and is not quite sure of his way, if he be travelling alone, will question himself, or if travelling with others, will question them too about the matter in doubt, and refuse to proceed until he has made sure by investigation of the direction of his path.

(Plato, Laws, VII, 799C).

Observe the paradox of this precise ambiguity: That both the occasion and the impulse of inquiry are instances of a negative moment. But the immediate discussion is aimed at the positive aspects of inquiry, and so I convert this issue into its corresponding positive form.

The positive aim of inquiry is a state of belief, certainty, or knowledge. There are distinctions that can be made in the use of these words, but the question remains as to what kind of distinctions these are. In my opinion, the differences that arise in practice have more to do with the purely grammatical distinctions of "case", "mood", "number", "person", and "voice", and thus raise the issues of plurality and point of view, as opposed to indicating substantial differences in the relevant features of state, as actually experienced by the agent concerned.

It is often claimed that there are signficant differences between the conditions of belief and knowledge, but the way that I understand the distinction is as follows. One says that a person "knows" something when that person believes exactly the same thing that one believes. When one is none other than the person in question, then one says that one "knows" exactly what one believes. Differences arise between the invocations of "belief" and "knowledge" only when more than one person is involved in the issue. Thus, there is no occasion for a difference between belief and knowledge unless there is more than one person that is being consulted about the matter in question, or else a single person in a divided state of opinion, in any case, when there is more than one impulse, moment, or occasion that currently falls under consideration.

In any case, belief or knowledge is the feature of state that an agent of inquiry lacks at the moment of setting out. Inquiry begins in a state of impoverishment, need, or privation, a state that is absent the quality of certainty. It is due to this feature that the agent is motivated, and it is on account of its continuing absence that the agent keeps on striving to achieve it, at least, with respect to the subject in question, and, at any rate, in sufficient measure to make action possible.

3.1.3. The Modes of Inquiry

Let the strange fact be granted, we say, that our hymns are now made into "nomes" (laws), just as the men of old, it would seem, gave this name to harp-tunes, — so that they, too, perhaps, would not wholly disagree with our present suggestion, but one of them may have divined it vaguely, as in a dream by night or a waking vision: anyhow, let this be the decree on the matter: — In violation of public tunes and sacred songs and the whole choristry of the young, just as in violation of any other "nome" (law), no person shall utter a note or move a limb in the dance.

(Plato, Laws, VII, 799E–800A).

In the present section, I am concerned with the kinds of reasoning that might be involved in the choice of a method, that is, in discovering a way to go about inquiry, in constructing a way to carry it through, and in justifying the way that one chooses. If the choice of a method can be established on the basis of reasoning, if it can be rationalized or reconstructed on grounds that are commonly thought to be sensible, or if it is likely to be affected or influenced in any way by a rational argument, then there is reason to examine the kinds of reasoning that go into this choice. All of this requires a minimal discussion of different modes of reasoning.

In this work as a whole, each instance of inquiry is analyzed in accord with various modes of reasoning, the prospective "elements of inquiry", and its structure as an object of inquiry is articulated, rationalized, and reconstructed with respect to the corresponding "form of analysis", "form of synthesis", or "objective genre" (OG).

According to my current understanding, the elements of inquiry can be found to rest on three types of steps, called "abductive", "deductive", and "inductive" modes of inference. As a result of this opinion, I do not believe that I can do any better at present than to articulate the structure of each instance of learning or reasoning according to these three types of motions of the mind. But since this work as a whole is nowhere near complete, I cannot dictate these steps in a dogmatic style, nor will it do for me to to call the tune of this form of analysis in a purely ritual or a wholly routine fashion.

Since the complexity of reasoning about different modes of reasoning is enough of a complication to occupy my attention at the present stage of development in this work, it is proably best to restrain this discussion along the majority of its other dimensions. A convenient way to do this is to limit its scope to simple examples and concrete situations, just enough to illustrate the selected modes of reasoning.

With all of these considerations in mind, the best plan that I can find for addressing the tasks of the present section is to proceed as follows: I make it my primary aim to examine only a few of the simplest settings in which these different modes of reasoning are able to appear, and I try to plot my path through this domain by way of concrete examples. Along the way, I discuss a few of the problems that are associated with reasoning about different modes of reasoning. Given the present stage of development, the majority of these issues have to be put aside almost as quickly as they are taken up. If they are ever going to be subject to resolution, it is not within reach of the present moment of discussion. In the body of this section, I therefore return to the initial strategy: to examine a few of the simplest cases and situations that can serve to illustrate the distinctions among the chosen modes of reasoning.

In trying to initiate a general discussion of the different modes of reasoning that might be available, and thus to motivate a model of this subject matter that makes an initial kind of sense to me, I meet once again with all the old "difficulties at the beginning", the kinds of obstructions that always seem to arise on trying to open up any new subject for discussion or in trying to introduce any new model of an old subject area. Much of this gratuitous bedevilment is probably due to the inherent conservatism of the human mind. Everything familiar is taken for granted, but each new picture of the situation is immediately subjected to the severest suspicions.

Now, I cannot reason with necessary force that the mind must use these particular modes of reasoning, any more than I can say that it must use a given language in order to express itself. But I can argue, relative to a particular model of thinking that must be proposed hypothetically, that certain modes of reasoning are available to the mind and are likely to be evident in its operation, if one only takes the trouble to look.

Ultimately, the model of thinking that I plan to propose makes use of the proposition that all thinking takes place in signs, and thus that inquiry is the transformation of a sign relation. Relative to this hypothesis, it would be possible to discharge the current assumptions about the basic modes of reasoning, that is, to derive the elementary modes of inquiry from a sign relational model of inquiry, and then to compare them with the current suggestions. Until this work is done, however, the assumption that these really are the most basic modes of reasoning has to be treated as a still more tentative hypothesis.

When a subject matter is so familiar that the logical connections between its parts are known both forwards and backwards, then it is reasonable and convenient to organize its presentation in an axiomatic fashion. This would not be such a bad idea, if it did not make it so easy to forget the nature of the reorganization that goes into a representation, and it would not constitute such a deceptive conception of the subject, if it did not mean that the exposition of the subject matter is just as often the falsification of its actual development and the covering up of its real excavation. Indeed, the logical order of axioms and theorems may have little to do with the original order of discovery and invention. In practice, the deepest axioms are often the last to come to light.

Once again, the structure of a reflective context means that each mode of reasoning is able to appear in a double role, once as an object and once as an instrument of the same extended discussion. And once again, the discussion runs into an array of obstructions, whose structures are becoming, if not more clear, at least, more familiar with each encounter. In particular, a description of different modes of reasoning involves a classification, and a classification presupposes a basis of distinctive features that cannot be treated as categorical, or objectively neutral, but has to be regarded as hypothetical, or potentially biased. In other words, the language that I use to describe different modes of reasoning may already have a particular model of reasoning built into it, and this disposition to a particular conception of logic may be lodged in such a way that it makes it nearly impossible to reflect on the operations and the limitations of this model.

Inquiry begins when a law is violated. It marks a time when a certain peace of mind is breached, it reigns all the while that a common accord is broken, disturbed, forgotten, or lost, and it rules right up until the time when a former condition of harmony is restored or until the moment when a new state of accord is established. Of course, the word "law" is a highly equivocal choice, especially to convey the sense of a founding principle. It renders not just its own meaning irrevocably subject to interpretation, but delivers into a similar subjection all the forms of understanding that depend on it. But the letter must release its hold on the spirit, if the word "law" is meant to evoke the requisite variety of connotations, and yet to maintain a sensible degree of order among their concrete meanings. Only in this way can it rise above the many different kinds of law that come into play.

There are descriptive laws, that organize experiences into expectations. There are prescriptive laws, that organize performances into intentions.

Other names for descriptive laws are "declarative" or "empirical" laws. Other names for prescriptive laws are "procedural" or "normative" laws.

Implicit in a descriptive law is the connection to be found or made, discovered or created, between past experience and present expectation. What one knows about these connections is kept in a descriptive model.

Implicit in a prescriptive law is the connection to be found or made, discovered or created, between current conduct and future experience. What one knows about these connections is kept in a prescriptive model.

A violation of an expectation, the contravention of a descriptive law, occurs when a present experience departs from a predicted experience, which is what a past expectation or description projected to be present. This is a "surprise", a state of affairs that calls for an explanation. An explanation points to other descriptions that better predict the actual experience, and suggests an alteration to the descriptive model that generated the expectation from a past experience.

A violation of an intention, the contravention of a prescriptive law, occurs when a present experience departs from a desired experience, which is what a past intention or prescription projected to be present. This is a "problem", a state of affairs that calls for a plan of action. , A plan of action points to other actions that better achieve the desired experience, and suggests an alteration to the prescriptive model that generated the conduct toward a prospective experience.

In the rest of this section, I treat the different modes of reasoning according to the forms that Aristotle gave them, collectively referred to as the "syllogistic" model. The discussion is kept within the bounds of propositional reasoning by considering only those "figures of syllogism" that are "purely universal", that is, the forms of argument all of whose premisses, and therefore all of whose conclusions, involve nothing but universal quantifications.

If it were only a matter of doing propositional reasoning as efficiently as possible, I would simply use the cactus language and be done with it, but there are several other reasons for revisiting the syllogistic model. Treating the discipline that is commonly called "logic" as a cultural subject with a rich and varied history of development, and attending to the thread of tradition in which I currently find myself, I observe what looks like a critical transition that occurs between the classical and the modern ages. Aside from supplying the barest essentials of a historical approach to the subject, a consideration of this elder standard makes it easier to appreciate the nature and the character of this transformation. In addition, and surprisingly enough to warrant further attention, there appear to be a number of cryptic relationships that exist between the syllogistic patterns of reasoning and the ostensibly more advanced forms of analysis and synthesis that are involved in the logic of relations.

3.1.3.1. Deductive Reasoning

In this subsection, I present a trimmed-down version of deductive reasoning in Aristotle, limiting the account to universal syllogisms, in effect, keeping to the level of propositional reasoning. Within these constraints, there are three basic "figures" of the syllogism.

In order to understand Aristotle's description of these figures, it is necessary to explain a few items of his technical terminology. In each figure of the syllogism, there are three "terms". Each term can be read as denoting either (1) a class of entities or (2) all of the members of a class of entities, depending on which interpretation the reader prefers. These terms are ranked in two ways: With respect to the "magnitudes" that they have in relation to each other, there are "major", "middle", and "minor" terms. With respect to the "positions" that they take up within the figure, there are "first", "intermediate", and "last" terms. The figures are distinguished by how the magnitudes correlate with the positions. However, the names for these rankings are not always used or translated in a rigorously systematic manner, so the reader has to be on guard to guess which type of ranking is meant.

In addition to this terminology, it is convenient to make use of the following nomenclature:

  1. The Fact is the proposition that applies the term in the first position to the term in the third or last position.
  2. The Case is the proposition that applies the term in the second or intermediate position to the term in the third or last position.
  3. The Rule is the proposition that applies the term in the first position to the term in the second or intermediate position.

Because the roles of Fact, Case, and Rule are defined with regard to positions rather than magnitudes they are insensitive to whether the proposition in question is being used as a premiss or is being drawn as a conclusion.

The first figure of the syllogism is explained as follows:

When three terms are so related to one another that the last is wholly contained in the middle and the middle is wholly contained in or excluded from the first, the extremes must admit of perfect syllogism. By "middle term" I mean that which both is contained in another and contains another in itself, and which is the middle by its position also; and by "extremes" (a) that which is contained in another, and (b) that in which another is contained. For if A is predicated of all B, and B of all C, A must necessarily be predicated of all C. … I call this kind of figure the First.

(Aristotle, Prior Analytics, 1.4).

For example, suppose A is "animal", B is "bird", and C is "canary". Then there is a deductive conclusion to be drawn in the first figure.

There is the Case:

"All canaries are birds." (C => B)

There is the Rule:

"All birds are animals." (B => A)

One deduces the Fact:

"All canaries are animals." (C => A)

The propositional content of this deduction is summarized on the right. Taken at this level of detail, deductive reasoning is nothing more than an application of the transitive rule for logical implications.

The second figure of the syllogism is explained as follows:

When the same term applies to all of one subject and to none of the other, or to all or none of both, I call this kind of figure the Second; and in it by the middle term I mean that which is predicated of both subjects; by the extreme terms, the subjects of which the middle is predicated; by the major term, that which comes next to the middle; and by the minor that which is more distant from it. The middle is placed outside the extreme terms, and is first by position.

(Aristotle, Prior Analytics, 1.5).

For example, suppose M is "mammal", N is "newt", and O is "opossum". Then there is a deductive conclusion to be drawn in the second figure.

There is the Fact:

"All opossums are mammals." (O => M)

There is the Rule:

"No newts are mammals." (N.M = 0)

One deduces the Case:

"No newts are opossums." (N.O = 0)

The propositional content of this deduction is summarized on the right. Expressed in terms of the corresponding classes, it says that if O c M and if N intersects M trivially, then N must also intersect O trivially. Here, I use a raised dot "." to indicate either the conjunction of two propositions or the intersection of two classes, and I use a zero "0" to indicate either the identically false proposition or the empty class, leaving the choice of interpretation to the option of the reader.

The third figure of the syllogism is explained as follows:

If one of the terms applies to all and the other to none of the same subject, or if both terms apply to all or none of it, I call this kind of figure the Third; and in it by the middle I mean that of which both the predications are made; by extremes the predicates; by the major term that which is [further from] the middle; and by the minor that which is nearer to it. The middle is placed outside the extremes, and is last by position.

(Aristotle, Prior Analytics, 1.6).

It appears that this passage is only meant to mark out the limiting cases of the type. From the examples that Aristotle gives it is clear that he includes many other kinds of logical situation under this figure. Perhaps the phrase "applies to all or none" is intended to specify that a term applies "affirmatively or negatively" to another term, but is not meant to require that it applies universally so.

For example, suppose P is "poem", R is "rhapsody", and S is "sonnet". Then there is deductive conclusion to be drawn in the third figure:

There is the Fact:

"All sonnets are poems." (S => P)

There is the Case:

"Some sonnets are rhapsodies." (S.R > 0)

One deduces the Rule:

"Some rhapsodies are poems." (R.P > 0)

The propositional content of this deduction is summarized on the right. Expressed in terms of the corresponding classes, it says that if S c P and if R intersects S non-trivially then R must intersect P non-trivially.

3.1.3.2. Inductive Reasoning

(Aristotle, Prior Analytics, 2.23).

3.1.3.3. Abductive Reasoning

A choice of method cannot be justified by deduction or by induction, at least, not wholly, but involves an element of hypothesis. In ancient times, this mode of inference to an explanatory hypothesis was described by the Greek word "apagoge", articulating an action or a process that "carries", "drives", or "leads" in a direction "away", "from", or "off". This was later translated into the Latin "abductio", and that is the source of what is today called "abduction" or "abductive reasoning". Another residue of this sense survives today in the terminology for "abductor muscles", those that "draw away (say, a limb or an eye) from a position near or parallel to the median axis of the body" (Webster's).

If an image is needed, one may think of Prometheus, arrogating for the sake of an earthly purpose the divine prerogative of the gods, and then drawing the fire of their heavenly ire for the presumption of this act. This seems to sum up pretty well, not only the necessity and the utility of hypotheses, but also the risks that one incurs in making conjectures. In other guises, abductive reasoning is the mode of inference that is used to diagnose a complex situation, one that originally presents itself under a bewildering array of signs and symptoms, and fixes it subject to the terms of a succinct "nomen" or a summary predicate. Finally, by way of offering a personal speculation, I think it is likely that this entire trio of terms, "abduction", "deduction", and "induction", have reference to a style of geometric diagrams that the Ancients originally used to illustrate their reasonings.

Abductive reasoning has also been called by other names. C.S. Peirce at times called it "presumption", perhaps because it puts a plausible assumption logically prior to the observed facts, and at other times referred to it as "retroduction", because it reasons backwards from the consequent to the antecedent of a logical implication.

In its simplest form, abductive reasoning proceeds from a "fact" that A is true, using a "rule" that B => A, to presume a "case" that B is true. Thus, if A is a surprising fact that one happens to observe, and B => A is a rule to the effect that if B is true then A necessarily follows, then guessing the case that B is true is an instance of abductive reasoning. This is a backward form of reasoning, and therefore extremely fallible, but when it works it has the effect of reducing the amount of surprise in the initial observation, and thus of partially explaining the fact.

In a slightly more complicated version, abduction proceeds from a fact that C => A, using a rule that B => A, to presume a case that C => B. This is an inessential complication, since the rule of modus ponens and the rule of transitivity are essentially equivalent in their logical force, but it is often convenient to imagine that C is the "common subject" or the "current situation" that is implicit throughout the argument, namely, the existing entity that substantiates or instantiates all of the other predicates that are invoked in its course.

Suppose I have occasion to reason as follows:

"It looks like a duck, so I guess it is a duck."

Or even more simply:

"It looks blue, therefore it is blue."

These are instances in which I am using abductive reasoning, according to the pattern of the following schema:

I observe a Fact:

"It looks like X." (X')

I have in the back of my mind a general Rule:

"If it is X, then it looks like X." (X => X')

I reason my way back from the observed Fact and the assumed Rule to assert what I guess to be the Case:

"It is X." (X)

The abduction is a hypothetical inference that results in a diagnostic conclusion, that is, a statement of opinion as to what is conjectured to be the case. In each case the operation of abductive reasoning starts from a complex configuration, involving a number of explicit observations in the foreground and a class of implicit assumptions in the background, and it offers a provisional statement about certain possibility, one that is typically less conspicuous, obvious, or prominent, but still potentially present in the situation, and hopefully serving to explain the surprising or the problematic aspects of the whole state of affairs.

What results from the abductive inference is a concept and possibly a term, for instance, "duck" or "blue". The concept attempts to grasp a vast complex of appearances within a unitary form, and the term that connotes the concept is used to put explicit bounds on what it conveys. Working in tandem, they express an approximation or a simplification, "a reduction of the manifold of phenomena to a unified conception". Finite minds cannot operate for very long with anything more than this.

The reader may have noticed some obvious distinctions between the two examples of abductive reasoning that I gave above, between the case of "looking like a duck" and the case of "looking blue". Just to mention the most glaring difference: Although a person is occasionally heard to reason out loud after the fashion of the former example, it is rare to hear anyone naturally reasoning along the lines of the latter example. Indeed, it is more likely that any appearance of doing so is always an artificial performance and a self-conscious reconstruction, if not a complete fabrication, and it is doubtful that the process of arriving at a perceptual judgment can follow this rule in just so literal a fashion.

This is true and important, but it is beside the point of the immediate discussion, which is only to identify the logical form of the inference, that is, to specify up to informational equivalence the class of conduct that is involved in each example. Thus, considering the inference as an information process, I do not care at this point whether the process is implemented by a literal-minded variety of rule-following procedure, so long as it "follows", "obeys", or "respects" these rules in the form of what it does. One can say that an information process "obeys" a set of rules in a "figurative" and a "formal" sense if the transformation that occurs in the state of information between the beginning and the end of the process has the form of a relation that can be achieved by literally following these rules with respect to the prospective class of materials.

The general drift of the strategy that is being mapped out here, the "abstract", the "formal", or the "functional" approach, is now evident. Conceptually, one partitions the space of processes into "effective", "informational", or "pragmatic" equivalence classes and then adopts the inditement of a sequence of rules as a symbolic "nomen" for the class of processes that all achieve the same class of effects. At this level of functional abstraction, the conception of a process is indifferent to the particulars of its implemenation, so long as it lives within the means of the indicated constraints. Moreover, unless there is a way to detect the nature of the "actual" process without interfering too severely with it, that is, a path-sensitive but still unobtrusive measure that can sort out a finer structure from these equivalence classes, then it is not possible to inquire any further into the supposedly "actual" details.

Similar remarks apply to every case where one attributes "law-abiding" or "rule-governed" behavior to oneself, to another person, or even to a physical process. Across this diverse spectrum of cases, it ranges from likely but not certain to unlikely but still conceivable that the action in question depends on the agent "knowing" the laws that abide or the rules that are effectively being obeyed. With this in mind, I can draw this digression on appearances to a conclusion: When I say that agents are acting according to a particular pattern of rules, it only means that it "looks like" they are. In other words, they are acting "as if" they are consciously following these rules, or they are acting just like I act when I conscientiously follow such rules. A concise way to sum all of this up is to say that a pattern of rules constitutes a model of conduct, one that I can deliberately emulate, or one that I can attribute to others by way of explaining their conduct. In attributing this model to others, or even in using it to account for my own less deliberate behavior, I am making an abductive inference.

One way to appreciate the pertinence of this point is to notice that this entire digression, concerned with explaining the similarities between "looking like a duck" and "looking blue", is itself a form of argument, making a case of abductive inference to a case of abductive inference. In short, I am reasoning according to the following pattern:

It appears to be the making of an abductive inference,

so I guess it is the making of an abductive inference.

Anyone who thinks that this style of reasoning is too chancy to be tolerated ought to observe that it is only the pattern of inference that one follows in attributing minds to others, solely on the evidence that they exhibit roughly the same array of external behaviors in reaction to various external conditions as one employs to express one's experience of roughly the same conditions.

It goes without saying that abductive reasoning is extremely fallible. The fact that it looks like a duck does not necessarily mean that it is a duck - it might be a decoy. Moreover, in most cases of actual practice the implicit rule that serves to catalyze the abductive inference is not an absolute rule or a necessary truth in its own right but may be only a contingent rule or a probable premiss. For instance, not every case of being blue presents the fact of looking blue - the conditions of observation may be trickier than that. This brings to the fore another mark that distinguishes the two examples, highlighting a potentially important difference between "looking like a duck" and "looking blue". This is the amount of oversight, or awareness and control, that an agent has with regard to an inference, in other words, the extent to which an inference really does "go without saying".

The abductive inference from "it looks blue" to "it is blue" and the abductive inference from "it looks like a duck" to "it is a duck" differ in the degrees to which they exhibit a complex of correlated properties. These variations are summed up in one sense by saying that the first, more perceptual inference is more automatic, compulsive, habitual, incorrigible, and inveterate. The correlations are summed up in the opposite sense by saying that the second, more conceptual inference is more aware, controllable, correctable, critical, deliberate, guarded, and reflective. From a fully pragmatic standpoint, these differences are naturally of critical importance. But from a purely logical standpoint, they have to be regarded as incidental aspects or secondary features of the underlying forms of inference.

There is one thing yet missing from this description of abductive reasoning, and that is its creative aspect. The description so far is likely to leave the impression that the posing of a hypothesis always takes place against a narrowly circumscribed background of established terms that are available for describing cases, and thus that it amounts to nothing more original than picking out the right label for the case. Of course, the forming of a hypothesis may be bound by the generative potential of the language that is ultimately in force, but that is a far cry from a prescriptively finite list of more or less obvious choices.

How does all of this bear on the choice of a method? In order to make a start toward answering that question, I need to consider the part that abductive reasoning plays in the inquiry into method, which is, after all, just another name for the inquiry into inquiry.

There are times when choosing a method looks more like discovering or inventing a method, a purely spontaneous creation of a novel way to proceed, but normally the choice of a path picks its way through a landscape of familiar options and mapped out opportunities, and this presupposes a description of previously observed forms of conduct and a classification of different paths from which to choose. Hence the etymology of the word "method", indicating a review of means or a study of ways.

I would now like to examine several types situations where a choice of method is involved, paying special attention to the way that abductive reasoning enters into the consideration.

Example 1.

Suppose I have occasion to reason along the following lines:

This situation looks like one in which this method will work, therefore I will proceed on the hypothesis that it will work.

The current situation (C) looks amenable (A') to this method, so I guess it really is amenable (A) to this method.

In this type of situation, my observations of the situation are reduced to a form of description that portrays it in the light of a given method, amounting to an estimate of whether the situation is a case to which the method applies. The form of the entire argument hinges on the question of whether the assurance of this application is apparent or actual.

I express my observations of the situation as a Fact:

"The current situation looks amenable." (C => A')

I have in the back of my mind a general Rule:

"If it is amenable, then it looks amenable." (A => A')

I reason my way back from the observed Fact and the assumed Rule to assert what I guess to be the Case:

"The current situation is amenable." (C => A)

As far as it goes, this style of reasoning follows the basic pattern of abductive inference. Its obvious facticity is due to the fact that the situation is being described solely in the light of a pre-selected method. That is a relatively specious way to go about describing a situation, in spite of the fact that it may be inevitable in many of the most ultimate and limiting cases. The overall effect is noticeably strained, perhaps because it results from dictating an artificial setting, attempting to reduce a situation to the patterns that one is prepared to observe, and trying to fit what is there to see into a precut frame. A more natural way to describe a situation is in terms of the freely chosen perceptual features that inform a language of affects, impressions, and sensations. But here a situation is forced to be described in terms of the prevailing operational features that constitute a language of actions, forcing the description to be limited by the actions that are available within a prescribed framework of methods.

Instead of describing a situation solely in terms of its reactive bearing, that is, wholly in terms of how it reacts to the application of a method, one can try to describe it in terms that appear to be more its own, its independent, natural, observational, perceptual, or "proper" features. What the "proper" or "object-oriented" features are and whether they can be distinguished in the end from "reactive" or "method-oriented" features are questions that cannot be answered in the early phases of an investigation.

Example 2.

Suppose I find myself reasoning as follows:

If the current world (C) is a blessed world (B),

then it is a world in which my method works (A).

Here, I call to mind an independent property of being, B, that a world or a situation can have, and I use it as a middle term to reason along the lines of the following scheme:

I express my inquiry by questioning the possibility of a certain Fact, that is, by interrogating the following statement:

"The current world is amenable." (C =?> A)

I have in the back of my mind a general Rule:

"What is blessed, is amenable." (B => A)

I reason my way back from the interrogated Fact and the assumed Rule to guess that I ought to contemplate the chances of the following Case:

"The current world is blessed." (C =?> B)

Altogether, the argument that underlies the current question of method falls into line with the following example of abductive reasoning:

I hope that C is A, so I guess I hope that C is B.

To proceed with the application of a given method on the basis of such a piece of reasoning is tantamount to the faith, the hope, or the wish that there is already the right kind of justice in the world that would make the prejudices of one's favorite method turn out to be right, that one is just lucky enough to be playing in accord with a pre-established harmony. If such a confidence is all that allows one to go on inquiring, then there is no harm in assuming it, so long as one reserves the right to question every particular of its grant, should the occasion arise.

If one abstracts from the specific content of this example and examines its underlying structure, it reveals itself as the pattern of abductive reasoning that occurs in relating complex questions to simpler questions or in reducing difficult problems to easier problems. Furthermore, the iteration of this basic kind of step motivates a downward recursion from questions of fact to questions of cases, in a hopeful search for a level of cases where most of the answers are already known.

The previous examples of inquiry into method are not very satisfactory. Indeed, their schematic forms have an absurdly sketchy character about them, and they fail to convey the realistic sorts of problems that are usually involved in reasoning about the choice of a method. The first example characterizes a situation wholly in terms of a selected method. The second example characterizes a situation in terms of a property that is nominally independent of the method chosen, but the ad hoc character of this property remains obvious. In order to reason "properly" about the choice of method, it is necessary to contemplate properties of the methods themselves, and not just the situations in which they are used.

Example 3.

If I reason that scientific method is wise because wise people use it, then I am making the hypothesis that they use it because they are wise. Here, my reasoning can be explained according to the following pattern:

I observe a fact:

"A certain conduct is done by wise people." (C => X)

I have in mind a rule:

"If a wise act, then done by wise people." (A => X)

I abduce the case:

"A certain conduct is a wise act." (C => A)

Example 4.

If I reason that scientific method is a good method on account of the fact that it works for now, then I am guessing that it works for now precisely because it is good.

I observe a fact:

"Scientific method works for now." (C => X)

I have in mind a rule:

"What is good, works for now." (A => X)

I abduce the case:

"Scientific method is good." (C => A)

As always, the abductive argument is extremely fallible. The fact that scientific method works for now can be one of its accidental features, and not due to any essential goodness that it might be thought to have.

Finally, it is useful to consider an important variation on this style of argument, one that exhibits its close relation to reasoning by analogy or inference from example. Suppose that the above argument is presented in the following manner:

Scientific method (C) has many of the features that a good method needs to have, for instance, it works for now (X), so I reason that it has all of the features of a good method, in short, that it is a good method (A).

So far, the underlying argument is exactly the same. In particular, it is important to notice that the abductive argument does not depend on the prior establishment of any known cases of good methods. As of yet, the phrase "good method" is a purely hypothetical description, a term that could easily turn out to be vacuous. One has in mind a number of properties that one thinks a good method ought to have, but who knows if there is any thing that would satisfy all of these requirements? There may be some sort of subtle contradiction that is involved in the very juxtaposition of the terms "good" and "method". In sum, it can happen that scientific method is the very first method that is being considered for membership in the class of good methods, and so it is still unknown whether the class labeled "good methods" is empty or not.

But what if an example of a good method is already known to exist, one that has all of the commonly accepted properties that appear to define what a good method ought to be? In this case, the abductive argument acquires the additional strength of an argument from analogy.

3.1.3.4. Analogical Reasoning

The classical treatment of analogical reasoning by Aristotle explains it as a combination of induction and deduction. More recently, C.S. Peirce gave two different ways of viewing the use of analogy, analyzing it into complex patterns of reasoning that involve all three types of inference. In the appropriate place, it will be useful to consider these alternative accounts of analogy in detail. At the present point, it is more useful to illustrate the different versions of analogical reasoning as they bear on the topic of choosing a method.

The next example, ostensibly concerned with reasoning about a choice of method, is still too artificial to be taken seriously for this purpose, but it does serve to illustrate Aristotle's analysis of analogical reasoning as a mixed mode of inference, involving inductive and deductive phases.

Example 5.

Suppose I reason as follows. I think I can establish it as a fact that scientific method is a good method by taking it as a case of a method that always works and by using a rule that what always works is good. I think I can establish this rule, in turn, by pointing to one or more examples of methods that share the criterial property of always working and that are already acknowledged to be good. In form, this pattern of reasoning works by noticing examples of good methods, by identifying a reason why they are good, in other words, by finding a property of the examples that seems sufficient to prove them good, and by noticing that the method in question is similar to these examples precisely in the sense that it has in common this cause, criterion, property, or reason.

In this situation, I am said to be reasoning by way of analogy, example, or paradigm. That is, I am drawing a conclusion about the main subject of discussion by way of its likeness to similar examples. These cases are like the main subject in the possession of a certain property, and the relation of this critical feature to the consequential feature of interest is assumed to be conclusive. The examples that exhibit the criterial property are sometimes known as "analogues" or "paradigms". For many purposes, one can imagine that the whole weight of evidence present in a body of examples is represented by a single example of the type, an exemplary or typical case, in short, an archetype or epitome. With this in mind, the overall argument can be presented as follows:

Suppose that there is an exemplary method (E) that I already know to be a good method (A). Then it pays to examine the other properties of the exemplary method, in hopes of finding a property (B) that explains why it is good. If scientific method (C) shares this property, then it can serve to establish that scientific method is good.

The first part of the argument is the induction of a rule:

I notice the case:

"The exemplary method always works." (E => B)

I observe the fact:

"The exemplary method is a good method." (E => A)

I induce the rule:

"What always works, is good." (B => A)

The second part of the argument is the deduction of a fact:

I notice the case:

"Scientific method always works." (C => B)

I recall the rule:

"What always works, is good." (B => A)

I deduce the fact:

"Scientific method is good." (C => A)

Example 6.

Example 7.

Suppose that several examples (S1, S2, S3) of a good method are already known to exist, ones that have a number of the commonly accepted properties (P1, P2, P3) that appear to define what a good method is. Then the abductive argument acquires the additional strength of an argument from analogy.

The first part of the argument is the abduction of a case:

I observe a set of facts:

"Scientific method is P1, P2, P3." (C => P)

I recall a set of rules:

"Bona fide inquiry is P1, P2, P3." (B => P)

I abduce the case:

"Scientific method is bona fide inquiry." (C => B)

The second part of the argument is the induction of a rule:

I notice a set of cases:

"S1, S2, S3 exemplify bona fide inquiry." (S => B)

I observe a set of facts:

"S1, S2, S3 exemplify good method." (S => A)

I induce the rule:

"Bona fide inquiry is good method." (B => A)

The third part of the argument is the deduction of a fact:

I recall the case:

"Scientific method is bona fide inquiry." (C => B)

I recall the rule:

"Bona fide inquiry is good method." (B => A)

I deduce the fact:

"Scientfic method is good method." (C => A)

Now, logically and rationally in the purest sense, the argument by analogy to an example has no more force than the abductive argument, but, empirically and existentially, the example serves, not only as a model of the method to be emulated, but as an object of experimental variation and a source of further experience.

It is time to ask the question: Why do these examples continue to maintain their unrealistic character, their comical and even ridiculous appearance, in spite of all my continuing attempts to reform them in a sensible way? It is not merely their simplicity. A simple example can be telling, if it grasps the essence of the problem, that is, so long as it captures even a single essential feature or highlights even a single critical property of the thing that one seeks to understand. It is more likely due to the circumstance that I am describing agents, methods, and situations all in one piece, that is, without any analysis, articulation, or definition of what exactly constitutes the self, the scientific method, or the world in question. It is not completely useless to consider cases of this type, since many forms of automatic, customary, and unreflective practice are underlain by arguments that are not much better that this. Of course, on reflection, their "commedius" character becomes apparent, and all deny or laugh off the suggestion that they ever think this way, but that is just the way of reflection.

In order to improve the character of the discussion on this score …

3.2. Obstacles to the Project : In the Way of Inquiry

The discussion in this Chapter addresses a set of conceptual and methodological obstacles that stand in the way of the current inquiry, threatening to undermine a reasonable level of confidence in the viability of its proceeding, all of which problems I think can be overcome.

Often the biggest obstacle to learning more is the need to feel that one already knows. And yet there are some things that a person knows, at least, in comparison to other things, and it makes sense to use what one already knows best in order to learn what one needs to know better. The question is, how does one know which is which? What test can tell what is known so well that it can be trusted in learning what is not?

One way to test a supposed knowledge is to try to formulate it in such a way that it can be taught to other people. A related test, harder in some ways but easier in others, is to try to formalize it so completely that even a computer could go through the motions that are supposed to be definitive of its practice. Both proposals for testing a supposition of knowledge invoke the critical notion of putting knowledge into a form that is communicable or transportable from one system or one medium of interpretation to another. If the knowledge is conceived to be residing already in one form or another, then this requirement simply points to a "reformation" or a "transformation" of knowledge, otherwise it demands a more radical metamorphosis, from a wholly disorganized condition to an incipiently communicable facility or an initially portable formulation.

3.2.1. The Initial Unpleasantness

Inquiry begins in doubt, a debit of certainty and a drought of information that is never a pleasant condition to acknowledge, and one of the primary obstacles to inquiry can be reckoned as owing to the onus that everyone feels on owning up to this debt. Human nature vastly prefers to revel in the positive features of the scientific knowledge it already possesses, and the mind defers as long as possible the revolt it feels arising on facing the uncertainties that still persist, the "nots" and "not yets" that as yet it cannot and ought not deny.

3.2.2. The Justification Trap

There is a particular type of "justification trap" that a person can fall into, of trying to prove the scientific method by solely deductive means, that is, of trying to show that the scientific method is a good method by starting from the simplest possible axioms, that everybody would accept, about what is good.

Often this happens, in spite of the fact that one really knows better, simply in the process of arranging one's thoughts in a rational order, say, from the most elementary and independent to the most complex and derivative, as if for the purposes of a logical and summary exposition. But when does this rearrangement cease to be a rational reconstruction and start to become a destructive rationalization, a distortion of the genuine article, and a falsification of the authentic inquiry that it attempts to recount?

Sometimes people express their recognition of this trap, and their appreciation of the factor that it takes to escape it, by saying that there is really no such thing as "the scientific method", that the very term "scientific method" is a misnomer and does not refer to any kind of method at all, in sum, that the development of knowledge cannot be reduced to any fixed method because it involves in an essential way such a large component of non-methodical activities. If one's idea of what counts as method is fixed on the ideal of a deductive procedure, then it is no wonder that one draws this conclusion.

3.2.3. A Formal Apology

Using "form" in the sense of "abstract structure", I can state that the focus of my interest in this research is limited to the formal properties of inquiry processes. Among their chief constituents these include all the thinking and unthinking processes that support the ability to learn and to reason.

This "formal apology", that is, the apologetics of declaring a decidedly formal intent, will be used on numerous occasions to beg off a host of material difficulties and thus to avoid the perceived necessity of meeting a multitude of conventional controversies.

The next several subsections enumerate a few of the ways that I plan to make use of this "formal apologetics".

3.2.3.1. Category Double-Takes

The first use of the formal apology is to rehabilitate certain classes of associations between concepts that would otherwise go down as category mistakes. This conversion can be achieved in each detailed case by flipping from one side of the concept's dual aspect to the other as the context demands. Thus it is possible in selected cases to reform the characters of category mistakes in the manner of categorical "retakes" or "double-takes".

3.2.3.2. Conceptual Extensions

The second use of the formal apology is to permit the tentative extension of concepts to novel areas, giving them experimental trial beyond the cases and domains where their use is already established in the precedents of accustomed habit and successful application.

This serves to dissipate the essential or "in principle" objection that any category distinction puts a prior constraint on the recognition of similar structure between materially dissimilar domains. As a result, it leaves this issue as a matter to be settled by a post hoc judgment, one that is directed to the question of what fits best "in practice".

3.2.3.3. Explosional Recombinations

Another obstacle to inquiry is posed by the combinatorial explosion of questions that can arise in complex cases. This embarrassment of riches is deceptively deadly to the ends of inquiry in the very measure that it seems so productive at first. The formalist strategy provides a way to manage this wealth of material diversity by identifying formal similarities among materially different domains, permitting the same formal answer to unify many contentious questions under a single roof, overall reducing the number of distinct topics that need to be covered.

3.2.3.4. Interpretive Frameworks

Iterations of this recombinatorial process will generate an alternative hierarchy of categories that helps to control the explosion of parts in the domain under inquiry. If by some piece of luck this alternative framework is uniquely suited to the natural ontology of the domain in question, then it would be advisable to reorganize the whole inquiry along the lines of its topic headings. However, a complex domain seldom falls out this neatly. The new interpretive framework will not preserve all the information in the object domain, but typically capture only another aspect of it. In order to take the maximal advantage of all the different frameworks that might be devised, it is best to quit depending on any one of them exclusively. Thus, a rigid reliance on a single hierarchy to define the ontology of a given domain passes over into a flexible application of interpretive frameworks to make contact with particular aspects of one's object domain.

3.2.4. A Material Exigency

On the other hand, I have cast this project as an empirical inquiry, proposing to represent experimental hypotheses in the form of computer programs. At the heart of this empirical attitude is a feeling that all formal theories should arise from and bear on experience.

Every season of growth in empirical knowledge begins with a rush to the sources of experience. Every fresh-thinking reed of intellect is raised to pipe up and chime in with the still-viable canons of inquiry in one glorious paen to the personal encounter with natural experience. But real progress in the community of inquiry depends on observers being able to orient themselves to objects of common experience, and so the uncontrolled exaltation of individual phenomenologies leads as a rule to the disappointment and the disillusionment that befalls the lot of unshared enthusiasms and fragmented impressions. Look again at the end of the season and see it faltering to a close, with every young scribe being rapped on the knuckles for departing from that uninspired identification with impersonal authority that expresses itself in third-person passive accounts of one's own experience.

It is easy to decry this turn of events, but anything that happens so often must have a cause, a force of reason to explain the dynamics of its recurring moment in the history of ideas. It appears that the heart of the development that transpires is not born on the sleeve of its first and last stages, where the initial explosion and final collapse march along their inevitable course in a lockstep fashion, but is embodied more naturally in the middle of the above narrative.

Experience exposes and explodes expectations. How can experiences impinge on expectations unless these two types of entities are both reflected in the same medium, for instance, and perhaps without loss of generality, in the form of representation that constitutes the domain of signs? However complex its world may be, internal or external to itself, or on the boundaries of its being, a finite creature can only describe it in a finite number of finite terms, or in a finite sketch of finite lines. Finite terms and lines are signs. What they indicate need not be finite, but what they are, must be.

The common sensorium.

The common sense and the senses of "common".

This is point where the empirical and the rational meet. I describe as "empirical" any method that exposes theoretical descriptions of an object to further experiences with that object.

3.2.5. A Reconciliation of Accounts

The reader may share with the author a feeling of discontent at this juncture, attempting to reconcile the formal intentions of this inquiry with the cardinal contentions of experience. Let me try to express this difficulty in the form of a question: What is the nature of the bond between form and content in experience, between the abstract formal categories and the concrete material contents that exist in experience?

Here is the tentative answer that I will entertain, seeking to test its usefulness in this work. I take there to be a primitive category of "form-in-experience" that presently lacks a compact name, but that from the standpoint of a given agent often passes from the "structure of experience" to the "experience of structure".

My personal definition of mathematical understanding has long been expressed in the chiasmatic figure of speech: "the form of experience and the experience of form". This is not the place to argue for the virtues of this concept, but I thought it would clarify a few points to share it here.

3.2.6. Objections to Reflexive Inquiry

Inquiry begins when an automatic routine or a normal course of activity is interrupted, when agents are thrown into a state of doubt about what is best for them to do next and what is really true of their situation. If this model applies at level of self-application, then an occasion for inquiry into inquiry arises when an ongoing activity of inquiry into any special area becomes obstructed and agents are obligated to initiate a new order of inquiry in order to obviate the problem. At such moments, agents must acknowledge the higher order of uncertainty that prevails and accept the interruption of a special inquiry in order to examine their accepted conventions and their antecedent convictions about the appropriate conduct of any inquiry at all. The new order demands that agents pause and reflect on the assumptions embodied in their previous inquiry, criticizing with a deliberate and reconstructive intent aspects of an activity that formerly proceeded through its paces untroubled by any articulate concern.

An agent may be able to articulate these apprehensions in the form of a pair of questions: What actions are advisable to adopt and apply in order to achieve the aim of the current activity? What assumptions among those accepted already are ready to be adapted or abandoned in order to advance this course of affairs?

Thus there is this negative aspect to the object of inquiry, a feature captured in the Greek word pragma for "object", whose manifold of senses and derivatives includes among its connotations the ideas of purposeful objectives and problematic objections, of booming inquiries and buzzing expositions.

The fact that an episode of inquiry has the style of an interlude, that it begins and ends "in medias res" with respect to a circumstantial action, is a feature of inquiry that is important to remember and a contingency of its process that has a couple of consequences:

First, it means that genuine inquiry does not touch on the inciting action at points of total doubt or absolute certainty. An incident of inquiry does not begin or end in absolute totalities but only in the differential and relative measures that actually occasion its departures and resolutions. Inquiry is not a process that demands an absolutely secure foundation from which to set out or any place to stand from which to judge the totality of onrushing events. It never requires more than it does in fact have at the outset: assumptions that were not in practice doubted just a moment before and a circumstance of conflict that will force the whole situation to be reviewed before returning to the normal course of affairs.

Second, the interruptive character or escapist interpretation of inquiry is especially significant when contemplating programs of inquiry with recursive definitions, like the motivating case of inquiry into inquiry. It means that the termination criterion for an inquiry subprocess is whatever allows continuation of the calling process.

3.2.7. Empirical Considerations

The use of computer programs to represent empirical hypotheses brings with it a number of novel considerations about the nature of hypotheses and the status of theories in relation to phenomena. It forces a re-examination of several issues whose traditional answers have long been taken for granted.

3.2.8. Computational Considerations

3.2.8.1. A Form of Recursion
3.2.8.2. A Power of Abstraction

Here's a scenario that often occurs. Inquiry begins with a question that leads to a number of further questions. After several iterations of this development a sense of despair sets in that the nominal progress of inquiry is doing more to multiply the tension of uncertainty than to clarify its issues. The only saving grace that rescues the effort comes from noticing that several groups of materially distinct questions have in fact similar forms.

3.3. Orientation of the Project : A Way Into Inquiry

I have used the word "inquiry" to signify my object, and acknowledged that mere use of the word does not indicate much knowledge of it.

Use of a word is adequate to the beginnings of inquiry if one can recognize a few examples of the concept and apply the term as a predicate of them.

This much tells me that one of the inputs to inquiry must be something in the nature of a sign, an argument to indicate the object of inquiry, and also that inquiry must be capable of operating on the vaguest of signs, the merest indications of a proposed object. This is required for two reasons: It goes with the job of inquiry to sort out meaningful from meaningless signs, and it cannot be expected of inquiry to start with a full comprehension of its object, such as that provided by a sign already perfect in its denotative effect or a text already complete on its descriptive power.

At first, the name "inquiry" serves as little more than a superficial heading or a topical adhesive, a label to stick on whatever file of thoughts collects beneath it, a license to collect a complex of ideas that become associated with each other in one's own mind, and in interaction with this personal concept, a tag to chase through the literature that other people's thinking has produced on what may or may not turn out to be the same object. But eventually, it is desired that the name should act as a pointer to a program, a concrete corpus of characters that codifies the contingencies of conduct that can be called on to carry out any conceivable inquiry.

By itself, a name points to nothing at all, or nothing beyond itself, at least. There must be adduced an interpretive framework (IF) if any bit of language is to get a meaning in practice. But there is nothing in the beginning that could distinguish one potential IF from all the others, and thus arises the issue of alternate paradigms and plural interpreters. Nothing in this excludes the possibility that a text of suitable character could perforce of its content lead any viable candidate interpreter from an otherwise vague initial state to the point of becoming the text's own ideal reader, but nothing makes this obviously so.

3.3.1. Initial Description of Inquiry

At first approach, I describe inquiry as a process that brings an agent, identical or analogous to a thinking human being, from a state of information that most people experience as doubt or uncertainty to a state of information that most people would describe as a condition of certainty and could express in the form of a determinate proposition, like those that represent settled beliefs or fixed items of knowledge.

I will refer to this as the "initial description" (ID) of inquiry, characterizing the relationship of the process of inquiry to the states of its systematic agents and to the symbolic expressions of these states. Simply to clarify the grounds on which the ID is stated will require me to address a number of constituent topics, seemingly trifling in the offing but hiding a host of stumbling blocks in their train.

3.3.2. Terms of Analysis

A sign, by itself, points to nothing at all — nothing beyond itself, at least. An interpretive framework (IF) must be adduced to give a sign a meaning in practice. So the simple name of "inquiry" and the complex expression of its ID indicate by themselves nothing beyond themselves. If an IF is adduced, then the ID can begin to take on meaning, at first by assigning tentative meanings to the ID's constituent terms and then testing the effects on the whole ID. But there is nothing unique about the interpretive framework that might be adduced, at least, nothing that can be distinguished at first. And so discussion is forced to take up the relations between interpretive frameworks that turn on the different senses of signs, the different directions they give to activity.

To start with, I need a working vocabulary that is adequate to address the kind of systematic agent that is invoked by the ID. As a minimal qualification, the agent must possess a "state of information" (SOI) and be able to move through a space of appreciably different information states, ranging from conditions of great confusion and uncertainty to greater levels of clarity and certainty. If this appreciation of information can be formalized in any way, then it may be possible to speak of "measurably" different information states and to delineate a "dimension" of information.

Readers who are past pursuing illusions of absolute certainty and neutral objectivity will not fail to miss that there are two notions of relativity involved in this ID.

  1. The measure of certainty that is being contemplated provides only a relative comparison between the agent's states of information at different times, but it does not lose a bit of its practical reality for all that.
  2. The kind of certainty that is intended here makes sense only in the context of a three-place relation or a triadic transaction, one that involves the systematic agent with an object system and also with a medium or a method of inquiry.

Altogether, the amount of uncertainty reduction at issue becomes well-defined only when parameters naming the object of inquiry, the precise conduct of inquiry, and the agent of inquiry are specified. Here, the "agent of inquiry" is regarded as encompassing both the "addressee" and the "terminus ad quem" of the inquiry in question. Properly speaking, the agent's uncertainty about an object system is changed by dint of the particular manner in which the inquiry is carried out.

It may be useful to expand on the current description of agent's role. A complete specification of the agent is required to indicate the whole "agent system" that is conceived to be involved in the inquiry. In its fullest sense, the role of the "agent" incorporates the whole complex of activity that is embodied by the inquiry. As a result, the function of the agent includes not only the parts of all the "actors" but the full participation of the "audience" that is involved in the play of inquiry, even if all of these personifications are filled by the same substantive entity or community. In doing so, the agent function assimilates the role of the ordinary agent or the "active" agency of inquiry along with both its "passive" addressee and its "terminus ad quem", the point to which the whole play of inquiry is directed. In a very real sense, the ultimate addressee of any inquiry is the all-purpose agent's intended "state of information" (SOI), a fixed and certain belief that hopefully embodies knowledge. This is the "end toward which" it tends, the goal and the object of the whole inquiry.

3.3.2.1. Digression on Signs

At this point, the discussion of inquiry makes contact with the pragmatic theory of signs. It appears that signs and inquiries have a lot in common. Signs and inquiries both act to affect the uncertainty that an agent has about an object. Moreover, taking these concepts in their positive senses, "good" signs and "good" inquiries always act to reduce that uncertainty.

The connection between signs and inquiry is a linkage that needs to be articulated further and developed in detail as the project proceeds. Indeed, a major portion of the conceptual analysis that is needed to implement inquiry processes is devoted to turning the intuitive picture of sign relations just given into a definitive model of their structure and a functional specification of their operation in practice. But these directions are largely tangential to the course this discussion needs to follow at the present point, and the immediate context can only afford the incursion of one further thought along these lines.

It is easy to imagine how one might have classified "signs", as embodied in the singular acts of observing a sign or receiving a signal, to be the simplest cases of "inquiries", incorporating arbitrarily complex protocols of sign-observations and symbol-manipulations, but the pragmatic theory of signs takes another tack. It analyzes inquiry into different kinds of logical proceedings that are called "arguments", classifies all arguments as being special forms of "symbols", and classifies all symbols as being special forms of "signs", namely, those that require the contribution of an interpretive agent in a significantly substantive way to constitute the resulting meaning of the sign.

3.3.2.2. Empirical Status of ID

Of all the descriptions of inquiry that live in memory and imagination, this ID is the one that comes forward in my own mind most frequently, having survived many tests of usefulness, both for its comprehensive coverage of established domains as well as for its fruitfulness in suggesting new avenues of exploration. It returns on a perennial basis at the outset of each new try at understanding inquiry. Perhaps it comes with the territory of inquiry or lies embedded in the equipment of thought itself. Perhaps it was designed to serve as a legend, a minimal set of instructions woven into the material basis of the mind's ability to map the world. Perhaps it is an archetyped script, an innate ID inscribed from the factory on an inobtrusive corner of the mind's otherwise blank state. Or perhaps it is only an ID fixe? But history is rife with comprehensive theories whose wealth of explanations was purchased at the price of inflating their descriptive economies to the point of indefeasibility. So this description continues to have the status of a hypothesis, or a summary of one, whose full account needs to be evaluated.

The task of evaluation begins by considering the theory of inquiry that develops out of this description and by examining how it performs against an array of questions. Does the theory point out relevant features of the phenomenon of inquiry, predicates that are decisive in explaining inquiry's manifestations, the how and why of its actual workings? Does the whole theory amount to nothing more than a logical tautology traveling incognito, the kind of thing that would continue to have the meaning it has even if there were no external objectives to be pursued and signified? Or does it attribute to the subject of inquiry a contingent predicate with empirical scope and definite limits? If the completed theory of inquiry is to have experimental content, what are the conceivable outcomes that could weigh against it?

After these preliminaries and others that might arise in the process have been satisfied, the synthetic mode of evaluation takes over. This stage of investigation requires a pause, to take stock of the accumulated theory of inquiry that has developed up that point, and a turn to the task of testing the theory in action, by constructing computational models that satify its fundamentals.

3.3.3. Expansion of Terms

In order to continue testing the aptness of the ID, its usefulness as a picture of inquiry, I need to expand its terms until they are clear enough to suggest computational models. Also, I need to develop my characterization and justification of the modeling methods chosen, and to say how their results are intended to be understood.

One thread of this description that needs to be disentangled from its context has to with the class of "agents" I have in mind, the ways they are "analogous" to the human kind, and the aspects of their relation to the realm of "most people" that are relevant to empirical inquiry.

3.3.3.1. Agency

As the word is used here, an "agent" is any kind of embodied activity. Thus, an electronic machine running a program is one sort of agent, but how it weighs in on the balance between a rock and a heart place of humane agency is another question.

3.3.3.2. Abstraction

An "abstract agent" refers to a general description or a functional specification of an agent, and allows that anything fitting the description or satisfying the specification will be recognized as an instance or model of that abstract description. The convenience of an abstraction, when it is apt, lies in our being able to reason about whole classes of entities from their stated properties alone. The usefulness of an abstraction, however, depends on our having identified the right list of properties for the purpose in mind.

3.3.3.3. Analogy

The models of an abstract description are all "analogues" of each other, since they all share the properties that have been identified as being relevant to a particular discussion. In the pragmatic theory of signs, an "icon" is any sign or symbolic entity that accomplishes its reference to its object by virtue of sharing one or more properties with it. Whenever a person builds a model of a phenomenon or process, whether fashioned in concrete materials or expressed in abstract calculi, the usefulness of the simulation is determined by the structural and functional properties that it has in common with its object.

3.3.3.4. Accuracy

The number and kinds of properties that a model is required to share with its object phenomenon can vary to a vast degree. Sometimes the only thing that matters is that the numbers generated at the end of a computation are the same, more or less, as the numbers that result from certain observations and measurements made on the phenomenon. These are usually called "numerical models" since they do little more than translate a quantitative mathematical model into a calculational form.

3.3.3.5. Authenticity

But there are times when the purpose of imitation is more sincere, and it cannot be satisfied with a flattering image or an inactive reduction of the original performance. If the intent of the model is fully authentic, and the aim of simulation is not to mock up a mere appearance but means to achieve a genuine result through the emulation of an actual performance, then the modeler must embody both more substantial and more instrumental properties of the target agency in order to reproduce the intended faculty.

The synthetic method employed in this project involves a reliance on computational models of abstractly specified agents. This practice implicitly extends the concept of an "agent" to include both human and machine forms of "agency", both concretely instantiated and abstractly formulated. This requires me to say how I understand the relations between these materially diverse categories of existence.

3.3.4. Anchoring Terms in Phenomena

To develop the ID of inquiry and articulate its terms in relation to actual experience, I need to pursue the phenomenology of the doubtful situation that initiates inquiry, and of its opposite pole, the condition of certainty that terminates inquiry. A couple of questions arise in this pursuit. How do the local poles that lie within an agent's finite compass of certainty line up with the global poles of information that pervade an encompassing field of inquiry? What are the modes of access that a finite agent has to information, certainty, knowledge, and belief?

To proceed with the conceptual analysis of the ID of inquiry, I need to examine the phenomena surrounding the situation of doubt and observe the conditions at its opposite pole, including the modes of access a finite agent has to information, certainty, knowledge, and belief.

To proceed with the conceptual analysis of the terms mentioned in the ID of inquiry, I need to pursue the phenomenology of the doubtful situation that initiates inquiry, and also to examine the conditions at its opposite pole, the situation of certainty that terminates inquiry.

Unless the very use of language will forever prejudice the question, I believe I can defer judgment on the existence of absolute poles for the field of inquiry. Therefore, the task is now to ask how the relative poles of an agent's local compass are related to the superrelative poles of a global field of inquiry.

3.3.4.1. A Mistaken ID
3.3.4.2. Phenomenology of Doubt
3.3.4.3. Modalities of Knowledge

This subsection enumerates several kinds of representation or "knowledge" that I shall speak of agents having of a concept or its objects.

3.3.5. Sets, Systems, and Substantive Agents

This project is heavily invested in the discussion of various mathematical objects and the ways that these objects can be said to be represented in minds and other media. Depending on the context of discussion and application, the same objects may be described from slightly different perspectives, most frequently in the alternative guises of sets, systems, or systematic agents.

To avoid a potential source of confusion, I need to address the issue of "shifting perspectives", that is, the various theoretical frameworks that I will need to take when contemplating the use of these mathematical objects as models of phenomena. From time to time, virtually the same phenomenal objects will be described as sets, as systems, and as the fields of activity of substantive agents.

If this were an abstract discussion in mathematics or philosophy, these differences would constitute little more than variant turns of phrase, selected for their momentary rhetorical advantages in a given context. But the requirements of a discussion aimed at applications in systems theory and systems engineering tenders these figures of speech as promissory notes, to be cashed in for real systems whose complete dynamics demands description and analysis, and real agents whose conduct needs to be in emulated, simulated, or implemented.

Retrospectively speaking, I have never known a mathematical object that I could not liken to a set. After the fact, when every object in sight was either familiar from its first acquaintance or has come to be known almost equally as well in the meantime, then it may well seem as though everything else to come will be described as a set eventually, but this mode of description only comes with adequate familiarity, and not until I come to know an objects very well indeed. But not every mathematical object presents itself from the first in this manner, with its elements arrayed between the braces of a set notation, or its features portrayed in the terms of a succinct formula.

From a post hoc point of view, it may well seem that everything thinkable can be conceptualized as a set, but this does not mean that every object is first or best conceived of in that fashion. Objects of long acquaintance and great familiarity can always be described, by virtue of that very richness of experience, in many alternative ways. After the defining features of an object and the positive examples of its concept have become well-known, a set-theoretic presentation often seems like the quickest way to single out the most relevant structures and ideas. However, even when a set-theoretic description can be recognized as the most efficient presentation of a conceptual object, there remains an important distinction to be made between the context of first discovery with the context of later description. One should not confuse the properties of the exposition that best describes a subject matter with the provisions of the expedition that best discovers it.

Intention = the state of holding something held in mind, the state or its object?

3.3.6. Interpretive Systems

This account of inquiry is intended to have a ...

If my account of inquiry is to have a reference to experience, then If I want this study of inquiry to arise from and have a bearing on experience

Experience is the basis of experimental science, and experiences are always the experiences of particular agents. That is to say, every particular experience is the experience of at least one particular agent. I did not say "one and only one", and this may demand an explanation.

My concept of a particular agent does not require that every experience be the exclusive possession of a unique, individual agent, and it allows that "shared experience" is a perfectly valid topic of discussion.

In my understanding, the concept of a particular agent does not require that every experience be the exclusive possession of a unique individual agent, and it is perfectly admissable to speak of a "shared experience". To clarify how this is possible, I need to introduce a technical distinction between "individuals" and "particulars", as regarded within the context of a particular discussion. An individual is a perfectly determinate entity, distinguished by at least one feature from every other individual in a particular account. A particular is a moderately determinate entity, distinguished only in the measure that it is actually described in a particular account.

These definitions have an optional and mutual recursive form. That is, if their introductory clauses are sufficient for the understanding of a particular interpreter, then their supplementary clauses can be ignored as redundant by that interpreter, otherwise the pursuit of a definition recurs on itself, and ultimately comes down to the naming of a particular account, which is only an alias for the particular interpreter or the interpretive framework that launched the original inquiry.

I begin by describing the process of inquiry as it presents itself in my own experience, concentrating on the kinds of salient and generic features that have a chance of being shared in the experience of others. Then I consider a variety of formal systems and mathematical objects that can serve to represent these phenomena in a deliberately organized fashion. I do not claim that these two stages, (1) the presentation of phenomena and (2) their representation in concepts, can ever be fully disentangled from each other in practice. It is simply convenient to take them up in this order.

It may be useful to emphasize this point. Making a formal distinction between the aspects of appearance and representation does not depend on a material separation of corresponding stages in actual practice. Indeed, I doubt if it is possible in an ongoing process of experience to separate (1) a phase of attending to phenomena and attempting to present them as they are "in and of themselves" from (2) a phase of gathering appearances under intellectual conceptions and representing them in terms of formal models.

Primarily, there is no mode of "pure experience" that delivers the contents of the given world at the "doors of perception" unstamped by any form of intellectual postage, free of all conceptual and conventional pre-arrangements. Conversely, there is no way of electing a conceptual representation solely on the basis of its formal qualifications, unbiased by the particular experiences of the elector. Individual choices of symbolic proxies are not only affected by irrelevant preferences and contentious distortions on the part of the chooser, but they are also subject to the risks of unaccountable autonomy and uncontrollable self-promotion on the part of the delegated representatives.

The motive of this work rests with understanding the formal properties of inquiry, which is regarded as a special kind of thinking process. Thus, the scope of the project is focused solely on formal features of inquiry and thought, initially as pointed out among scattered indications and circumscribed by various theoretical perspectives, and ultimately as they might be realized in autonomous and intelligent dynamical systems.

Ordinarily, the classification of inquiry as a style of thinking process raise no objection. An intuitive grasp of the concepts involved or a common sense induction from natural examples would make this an obvious inference, suggesting that inquiry is a controlled variety of thinking process. However, extending the concepts of inquiry and thinking to artificial systems, as I need to do in the present endeavor, could justly give occasion for pause and make it reasonable to review the immediacy of this inference.

Extending the concepts of inquiry and thinking to artificial systems, however, could break the link between them, if it were not for the explicit restriction of this discussion to the kinds of formal properties that can be held in common between natural and artificial systems, and thus shared between them in various forms of interaction.

I am taking it as a working provision of this project that the explicit restriction to formal properties, of the kind that can be shared by natural and artificial systems, equips discussion with a valid way of building a bridge over this material obstacle.

In regard to this general type of obstruction, there is nothing special about the distinction between natural and artificial. Exactly the same kind of objection arises every time a concept needs to be stretched and tested beyond its original domain.

The pragmatic theory of signs holds that all thought takes place in signs. This tenet is really much less surprising than it seems at first, especially after one comes to realize that pragmatists have deliberately made their concept of a sign so extensible, as befits the "purse" of fable, to cover anything that might conceivably enter into thought.

However, the concept is not indefinitely expandable — there is a definition, more or less, that puts a bound on its application. This does not mean that all one can think about is a sign relation. It means that everything one thinks about involves one in a sign relation, in a form of participation that makes one an interpreter of the ongoing relation.

This implies, quite literally, that a complete account of a thinking process could be represented a relational data base with exactly three columns, at each moment of the process listing the object, sign, and interpretant involved in that moment's transition.

3.3.6.1. Syntactic Systems

A notion of acceptability, or candidate potential for meaning, a dichotomy of meaningful versus meaningless expressions decidable on formal grounds alone. Implemented by means of programs called parsers.

3.3.6.2. Semantic Systems

A notion of meaning or sense, or equivalence and distinction of meaning, a partition of expressions into semantic equivalence classes.

Proofs and models.

The domain of expressions is not just a sign domain but also an object of experience, in other words, a domain of concrete experience about which it is possible to gain empirical knowledge. Thus, occasions of genuine phenomena can arise in formal domains, outcomes that surprise the interpretive agent as much as any occurring in the natural world. Rules of order. There are motions regarding the order of procedure and remarks addressing the character of the commenting on that are not themselves out of order but that fill the bill as items on the agenda.

Many propositions about the order are also propositions in the order.

A sufficient number of statements about the formal system are also represented by statements in the formal system.

Many propositions that comment on the proceedings are not themselves out of order but may find a proper place and receive due consideration as items on the agenda.

Propositions that comment on the proceedings are not for that reason out of order but may be entirely fitting and dutiful. If so, they ought to receive due consideration as items on the agenda and enjoy a proper place in the order of actions executed.

3.3.6.3. Pragmatic Systems

A notion of clarity, or quality of representation in a a semantic equivalence class.

Alternate refinements and extensions of semantic equivalence classes.

Pragmatic maxim as a strategy of representation: a regulative principle or heuristic slogan, a suggestion/ recommendation/ piece of advice for achieving maximal clarity in conceptual representations.

Pragmatic equivalence classes.

Consider what effects, that might conceivably have practical bearings, you conceive the object of your conception to have. Then, your conception of these effects is the whole of your conception of the object.

Deuteronomy, leading to self-knowledge. Addressed to a particular agent. Notice that the pragmatic maxim does not say anything about the meaning of a sign, in other words, it does not seek to define any kind of presumptively unique concept that must intervene between a sign and its object. It does not speak to any unique meaning, absolute conception, or neutral objective of signs and ideas but tells you how you can recognize your own particular concept of an object when you grasp it.

How to recognize the actual scope and practical limits of one's own particular grasp of an object when one encounters it.

Signs as effects? Causes and effects?

Practical bearing: under experienced conditions, actions lead to further experiences.

3.3.7. Inquiry Driven Systems

3.3.7.1. A Definition of Inquiry

John Dewey's lifetime of reflection on the nature and nurture of inquiry has tendered a definition of inquiry that remains unmatched in its clarity of depiction and its comprehension of the subject.

Inquiry is the controlled or directed transformation of an indeterminate situation into one that is so determinate in its constituent distinctions and relations as to convert the elements of the original situation into a unified whole.

(Dewey 1938, 108).

This definition of inquiry is abstracted from the conduct and reflection of living agents who carry on inquiry, deploying it to convert their situations, as they experience them, from a condition of uncertainty to a character of unity. With living agents, intelligent enough to fall into question, this transformation serves as a preparation for action. For an agent to engage in competent action, it relies on a power to resolve two kinds of uncertainties, those about the predicates of a current situation and those about the prerequisites of a desired situation.

3.3.7.2. The Faculty of Inquiry

It is conventional to assume that inquiring agents have a specialized faculty of inquiry that allows them to carry out the transformations required by the process. But even this simple postulate is not free from objection, and it should not be granted without a grain or two of critical reflection.

This "faculty of inquiry" may be nothing more than a figure of speech, a fiction imposed by the formalities of language that say a sentence should have a subject. As such, it should be regarded as a hypostatic construct, a reification of the living process of inquiry. The whole distribution of activities involved in inquiry may have nothing like the singular cohesion and the simple location suggested by a common noun. In any case, merely giving "a name and a local habitation" to the process and faculty of inquiry is a far cry from having a clue to its inner workings.

Thus, being in possession of a faculty of inquiry, in the sense of being able to exercise it, does not imply being in possession of this faculty as an object of knowledge, in the sense of being able to demonstrate a full acquaintance with its nature.

With the cautions noted, I will continue treating the faculty of inquiry as though it had a real identity, but whether it is trained or devolves from heaven is not a material question yet.

The next order of business is to analyze the structure and function of this faculty of inquiry with the aim of devising a program of inquiry.

3.3.7.3. A Definition of Determination
3.3.7.4. A Definition of Definition

3.4. Organization of the Project : A Way Through Inquiry

3.4.1. The Problem : Inquiry Found as an Object of Study

This project takes as its object the nature of inquiry, and contemplates its nurture in the computer medium.

3.4.2. The Method : Inquiry Found as a Means of Study

The subject is inquiry, and inquiry, too, is the method of approach. Can inquiry into inquiry be a valid form of inquiry, or is it only a vicious circle, a vacuous self-reference? The reader has a right to question whether this self-described project can possibly be taken seriously, or whether inquiry into inquiry is a mere form of words, a vapid prototype outlining an impossible bootstrap. Worse yet, it is not just the faculty of inquiry whose self-application can be called into question. It can be doubted whether it makes sense to apply any function at all to itself, or whether the very idea of self-application is itself vain, devoid of any consistent meaning or purpose.

3.4.2.1. Conditions for the Possibility of Inquiry into Inquiry

If inquiry into inquiry is to be found a sensible project, and not just a question that begs itself, a blank application to which nothing signs "nothing" as an empty formality, as arrogantly suspect as the proverbial agency that proposes to investigate its own improprieties, then I must examine the conditions for its possibility. But inquiry begins when one is uncertain about something, and it is clearly possible to be doubtful about the nature of inquiry, as I am for certain in setting out on this investigation.

If inquiry into inquiry is possible, and if all one knows about inquiry is that it begins with doubt and ends with belief, then doubt about doubt is possible, and one can actually be uncertain about the nature of uncertainty itself.

3.4.2.2. Conditions for the Success of Inquiry into Inquiry

For inquiry into inquiry to succeed it is only asked that one be able to use an ability before one has settled exactly how that ability is able to succeed, but this is clearly something that human beings do all the time. What it demands is a facility for carrying out the conceptual analysis of initially vague terms.

In pursuing the conceptual reconstruction of any process or faculty it is extremely beneficial to operationalize the ultimate terms of analysis. The operational analysis that results is no mere figment of individual ingenuity but can be tested in much the same manner as any empirical statement. In this connection computer models represent hypotheses as to how a thing might be done, that is, if the resources of the doer are limited to a specific collection of effective operations.

In viewing computer simulations as empirical hypotheses, however, an important qualification should be appreciated. Notice that these models can be modal or normative hypotheses, not necessarily descriptive ones. As such, they are especially useful in research enterprises where the goal is to extend a human capacity, not merely to describe its current level of functioning. And yet, because the extension is based on a principle of similarity with the original function, it can also have a measure of descriptive utility.

3.4.3. The Criterion : Inquiry in Search of a Sensible End

Inquiry comes to rest when the irritation of uncertainty that set it in motion is reduced to zero, that is, to a level of relative insignificance.

3.4.3.1. The Irritation of Doubt, and The Scratch Test

The first draft of this termination criterion is an idealized formulation, prescribing the sole end of inquiry to be relief from the irritation of doubt. But it treats the complex situation of inquiry as though each investigation is pursued in isolation from every other condition of the inquiring agent, and it operates under the doubtful assumption that every inquiry is eventually carried to completion. As a criterion for the end, goal, bounds, or purpose of inquiry there remains the need for a pragmatic definition, a touchstone that can serve in a hard-knocks test to mark each question's field of operation and to limit the term of its continuation. To do all this, the fundamental principle needs a few amendments (enabling provisos or stipulations) to make it useful in actual practice.

3.4.3.2. Enabling Provision 1 : The Scenes and Context of Inquiry

In a realistic setting, equilibrium with respect to a dimension of inquiry is reached when the issue raises a negligible effect in the contest of competing demands for attention. This need not mean that the question is stilled forever, but only that the net effects of its nettling causes have fallen for a spell under the threshold of other concerns. In sum, the issue of an issue may settle itself below the level of a noise that no longer annoys. The agent of inquiry becomes tolerant or habituated to anything less than the mean irritation it knows, and learns to ignore as making no sense many a nuisance that sinks beneath the main and the norm and the mundane sensation of its usual suspicions.

3.4.3.3. Enabling Provision 2 : The Stages and Content of Inquiry

The progress of inquiry is closely analogous to that of interpretation, a process of increasing determination that concentrates and precipitates gradients of richer and deeper meaning around the seeds provided by initially meaningless symbols. Indeed, "inquiry", or "thinking" in the best sense of the word, has been described as "a term denoting the various ways in which things acquire significance" (Dewey 1910, 38).

Formal systems are customarily interpreted according to a number of general principles, attaching relatively concrete meanings to the formal symbols involved. It is important to remember, however, that real-life interpretation proceeds in stages, settling the meaning of abstract symbols in layers of increasing concreteness, and that the level of concrete meaning determined at any stage may remain in the realm of relatively abstract mathematical objects.

It is not a requirement of successful interpretation to ever achieve a level of perfectly determinate objects. Interpreters may settle for a satisfactory power of resolution, just enough to distill the ambiguities blocking specific actions, seeking only to develop their pictures of a situation into clear choices for the success or failure of their active goals. What matters is lighting on a level of concrete meaning that is particulate enough to support discrete actions, injecting elements of determination into the categories of imperative that interpreters must negotiate if their contemplated decisions are to become consequential. In the mean time, if not in the end, interpreters are content to grasp any shade of determination that is solid enough to act on.

3.5. Objectives of the Project : Inquiry All the Way

This project has two objectives, one substantial and one instrumental.

3.5.1. Substantial Objective

The substantive aim is to undertake a comparative and developmental study of formal systems. This effort involves the following subprojects.

3.5.1.1. Objective 1a : The Propositions as Types Analogy

First, I investigate a correspondence that exists between two kinds of formal systems, called applicational calculi (AC's) and propositional calculi (PC's). This relationship is known as the "propositions as types" analogy, or PAT isomorphism. It has applications to the checking of type declarations in programming languages (AC's) and to the annotation of proofs in logical systems (PC's).

3.5.1.2. Objective 1b : The Styles of Proof Development

Second, I explore a variety of ways that proofs may be developed in different species of propositional calculus. The major dimension of interest is the contrast between classical and intuitionistic proof systems, but there are interesting questions also about the relationship of different proof styles among equivalent axiom systems.

3.5.1.3. Objective 1c : The Analysis of Interpreters, or A Problem with Authority

Finally, I advance an ulterior purpose for taking up the "comparative anatomy" and "developmental biology" of formal systems. Axiom systems and their associated proof styles can be taken as prototypes and precursors of a larger class of meaning-bearing calculi, called "interpretive systems". When it is convenient to describe systems in a substantive vein, the agents that implement interpretive systems are called "interpreters". With dynamic systems the agent is a local representative of the system that travels through succeeding points of its state space, but with formal systems the agent is regarded as a person or machine that executes the moves of the calculus according to its protocol of rules. It is one of the overarching goals of this project to seek a merger of these two perspectives, the dynamic and the formal.

3.5.2. Instrumental Objective

The instrumental goal is to provide software support for the design and analysis of complex formal systems, for example, programming languages and theorem provers, just to indicate the cases of ultimate interest. But first, a lot more work needs to be done understanding the functional and logical aspects of much simpler calculi.

3.5.3. Coordination of Objectives

These two efforts are intended to complement and support each other. The software implementation is an obvious way to catalyze the task of building theories about formal systems. Conversely, the best way of designing programs for any problem area is to build on a knowledge of structure in that domain. Discovering lawful relationships between applicational and propositional aspects of formal systems is one more source of structural constraints, information that can be exploited to improve the capabilities of software tools for the task.

3.5.4. Recapitulation — Da Capo, Al Segno

He who, prompted by some enigmatic desire, has, like me, long endeavored to think pessimism through to the bottom and to redeem it from the half-Christian, half-German simplicity and narrowness with which it finally presented itself to this century, namely in the form of the Schopenhaueran philosophy; he who has really gazed with an Asiatic and more than Asiatic eye down into the most world-denying of all possible modes of thought — beyond good and evil and no longer, like Buddha and Schopenhauer, under the spell and illusion of morality — perhaps by that very act, and without really intending to, may have had his eyes opened to the opposite ideal: to the ideal of the most exuberant, most living and most world-affirming man, who has not only learned to get on and treat with all that was and is but who wants to have it again as it was and is to all eternity, insatiably calling out da capo not only to himself but to the whole piece and play, and not only to a play but fundamentally to him who needs precisely this play — and who makes it necessary: because he needs himself again and again — and makes himself necessary — What? And would this not be — circulus vitiosus deus?

(Nietzsche, Beyond Good and Evil, 82).

The formula with which the initial part of this inquiry is annotated, \(y_0 = y \cdot y\), is intended to suggest that the present inquiry, \(y_0\!\), is the result of applying a generic inquiry, \(y\!\), to itself. A close inspection reveals, however, that this formula does not quite make sense, at least, not yet, and there are many things that lie in the way of its doing so.

For one thing, where is the system of interpretation that can make sense of these signs, not just the \({}^{\backprime\backprime} y {}^{\prime\prime}\) and the \({}^{\backprime\backprime} y_0 {}^{\prime\prime}\) but the \({}^{\backprime\backprime} \cdot {}^{\prime\prime}\) of an indicated application that is so often slighted to the point of omission? Neither the form of this application nor the medium of its transmission are likely to be so transparent, or quite so easily taken for granted.

For another thing, who really imagines that naming an ongoing process is all that it takes to fix it in mind, to make it a passive object of thought? The problem is that we never hold the inquiry itself within our grasp, but only the signs of it, and only a sample of these. When all is said and done, there is nothing but the protocol of an experiment, the record of a trial, or the text of an inquiry that we have to examine. In a sense, a statement to the effect that \(y_0 = {}^{\backprime\backprime} y {}^{\prime\prime} \cdot y\) makes for a better account of how this self-described self-application actually gets going, since it is only an arbitrary sign of the passing moment of inquiry that supplies the initial argument and indicates the ultimate object of the pressing moment of inquiry. In effect, the present inquiry, \(y_0\!\), is the result of calling up a generic faculty for inquiry, \(y\!\), and setting it to work on whatever is indicated by a purely conventional name for itself, that is to say, on the argument \({}^{\backprime\backprime} y {}^{\prime\prime}\).

All in all, the original formula serves well enough as a sigil, a wholly symbolic annotation, and so it can be allowed to rest at the top of the current phase of work, but if this semblance of an equation is intended to make sense in a less arbitrary, a more articulate, and a less occult fashion, then it will have to be given a more practical meaning, in terms that can guide the actual conduct of inquiry.

The work up to this point pushes the discussion of formalization as far as it can go in a certain direction, at least, for now. While maintaining this discussion, it is time to step back, to recall the broader context in which formalization takes place, and thus to enter on a discussion of the inquiry that this process of formalization is initially meant to serve.

\(y_0 = y \cdot y >\!\!= y \cdot \{ d, f \} >\!\!= y \cdot \{ d \}\)

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