Changes

==Work Area==

==Work Area==

===1.6.  Orientation of the Project : A Way Into Inquiry===

===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.

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.

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.

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.

====1.6.1.  Initial Description of Inquiry====

====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.

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.

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.

====1.6.2.  Terms of Analysis====

====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.

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.

# 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.

# 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.

# 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.

# 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.

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.

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.

=====1.6.2.1.  Digression on Signs=====

=====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.

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.

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.

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.

=====1.6.2.2.  Empirical Status of ID=====

=====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.

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?

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.

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.

====1.6.3.  Expansion of Terms====

====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.

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.

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.

=====1.6.3.1.  Agency=====

=====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.

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.

=====1.6.3.2.  Abstraction=====

=====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.

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.

=====1.6.3.3.  Analogy=====

=====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.

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.

=====1.6.3.4.  Accuracy=====

=====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.

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.

=====1.6.3.5.  Authenticity=====

=====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.

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.

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.

====1.6.4.  Anchoring Terms in Phenomena====

====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 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?

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.

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.

=====1.6.4.1.  A Mistaken ID=====

=====3.3.4.1.  A Mistaken ID=====

=====1.6.4.2.  Phenomenology of Doubt=====

=====3.3.4.2.  Phenomenology of Doubt=====

=====1.6.4.3.  Modalities of Knowledge=====

=====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.

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

====1.6.5.  Sets, Systems, and Substantive Agents====

====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.

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.

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.

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.

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.

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?

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

====1.6.6.  Interpretive Systems====

====3.3.6.  Interpretive Systems====

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

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

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.

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.

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.

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.

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.

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.

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.

=====1.6.6.1.  Syntactic Systems=====

=====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.

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.

=====1.6.6.2.  Semantic Systems=====

=====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.

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

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.

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.

=====1.6.6.3.  Pragmatic Systems=====

=====3.3.6.3.  Pragmatic Systems=====

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

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

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

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

====1.6.7.  Inquiry Driven Systems====

====3.3.7.  Inquiry Driven Systems====

=====1.6.7.1.  A Definition of Inquiry=====

=====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.

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.

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.

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.

=====1.6.7.2.  The Faculty of Inquiry=====

=====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.

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.

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.

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.

=====1.6.7.3.  A Definition of Determination=====

=====3.3.7.3.  A Definition of Determination=====

=====1.6.7.4.  A Definition of Definition=====

=====3.3.7.4.  A Definition of Definition=====

===1.7.  Organization of the Project : A Way Through Inquiry===

===3.4.  Organization of the Project : A Way Through Inquiry===

====1.7.1.  The Problem : Inquiry Found as an Object of Study====

====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.

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

====1.7.2.  The Method : Inquiry Found as a Means of Study====

====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.

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.

=====1.7.2.1.  Conditions for the Possibility of Inquiry into Inquiry=====

=====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 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.

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.

=====1.7.2.2.  Conditions for the Success of Inquiry into Inquiry=====

=====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.

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 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.

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.

====1.7.3.  The Criterion : Inquiry in Search of a Sensible End====

====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.

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.

=====1.7.3.1.  The Irritation of Doubt, and The Scratch Test=====

=====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.

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.

=====1.7.3.2.  Enabling Provision 1 : The Scenes and Context of Inquiry=====

=====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.

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.

=====1.7.3.3.  Enabling Provision 2 : The Stages and Content of Inquiry=====

=====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).

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).

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.

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.

===1.8.  Objectives of the Project : Inquiry All the Way===

===3.5.  Objectives of the Project : Inquiry All the Way===

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

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

====1.8.1.  Substantial Objective====

====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.

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

=====1.8.1.1.  Objective 1a : The Propositions as Types Analogy=====

=====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).

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).

=====1.8.1.2.  Objective 1b : The Styles of Proof Development=====

=====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.

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.

=====1.8.1.3.  Objective 1c : The Analysis of Interpreters, or A Problem with Authority=====

=====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.

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.

====1.8.2.  Instrumental Objective====

====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.

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.

====1.8.3.  Coordination of Objectives====

====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.

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.

====1.8.4.  Recapitulation — Da Capo, Al Segno====

====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?

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).

(Nietzsche, Beyond Good and Evil, 82).

The formula with which the initial part of this inquiry is annotated, y0 = y.y, is intended to suggest that the present inquiry, y0, 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.

The formula with which the initial part of this inquiry is annotated, y0 = y.y, is intended to suggest that the present inquiry, y0, 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 "y" and the "y0" but the "." 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 one thing, where is the system of interpretation that can make sense of these signs, not just the "y" and the "y0" but the "." 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 y0 = "y".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, y0, 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 "y".

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 y0 = "y".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, y0, 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 "y".

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.

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.  y0 = y?y >= y?{d, f} >= y?{d}.

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.  y0 = y?y >= y?{d, f} >= y?{d}.