Changes

There is a thought that forms the theme of the present inquiry, indeed, as a chorus to a lyric are its evocations to the text that records this inquiry, and I find myself returning to its expressions on a constantly recurring basis, however much I strive to introduce variations for the sake of developing its implications and reflecting on its meanings from a fresh angle.  So let me give the current rendition:

There is a thought that forms the theme of the present inquiry, indeed, as a chorus to a lyric are its evocations to the text that records this inquiry, and I find myself returning to its expressions on a constantly recurring basis, however much I strive to introduce variations for the sake of developing its implications and reflecting on its meanings from a fresh angle.  So let me give the current rendition:

The present inquiry, <math>y_0,\!</math> portraying itself as an inquiry into inquiry, <math>y \cdot y,\!</math> proceeds on the premiss that a generic inquiry, <math>y,\!</math> can generally inquire into a generic inquiry, <math>y,\!</math> thereby achieving a settled result, one that awaits a mere determination to be signified by the name <math>{}^{\backprime\backprime} y \cdot y {}^{\prime\prime}.</math>  Thus the present inquiry, acting on the pretext of a ''formal posability'', that is, a poetic license, a verbal permission, or a written suggestion, being motivated and justified by no more authority than these connote, is led to define itself in terms that appose its own term to its own term, and so it is led to take on a recursive, a reflective, or a reflexive cast.

The present inquiry, <math>y_0,\!</math> portraying itself as an inquiry into inquiry, <math>y \cdot y,\!</math> proceeds on the premiss that a generic inquiry, <math>y,\!</math> can generally inquire into a generic inquiry, <math>y,\!</math> thereby achieving a settled result, one that awaits a mere determination to be signified by the name <math>{}^{\backprime\backprime} y \cdot y {}^{\prime\prime}.\!</math>  Thus the present inquiry, acting on the pretext of a ''formal posability'', that is, a poetic license, a verbal permission, or a written suggestion, being motivated and justified by no more authority than these connote, is led to define itself in terms that appose its own term to its own term, and so it is led to take on a recursive, a reflective, or a reflexive cast.

The terms of this description need to be inquired into, and their implications pursued in greater detail.

The terms of this description need to be inquired into, and their implications pursued in greater detail.

The present inquiry, <math>y_0,\!</math> portraying itself as an inquiry into inquiry, <math>y \cdot y,\!</math> proceeds on the premiss that a generic inquiry, <math>y,\!</math> can generally inquire into a generic inquiry, <math>y,\!</math> and thereby achieve a settled result, and that this result awaits nothing other than its determination by the present inquirer to confer an objective significance on the name <math>{}^{\backprime\backprime} y \cdot y {}^{\prime\prime}.</math>  All of this is summed up in the formula: <math>{}^{\backprime\backprime} y_0 = y \cdot y {}^{\prime\prime}.</math>

The present inquiry, <math>y_0,\!</math> portraying itself as an inquiry into inquiry, <math>y \cdot y,\!</math> proceeds on the premiss that a generic inquiry, <math>y,\!</math> can generally inquire into a generic inquiry, <math>y,\!</math> and thereby achieve a settled result, and that this result awaits nothing other than its determination by the present inquirer to confer an objective significance on the name <math>{}^{\backprime\backprime} y \cdot y {}^{\prime\prime}.\!</math>  All of this is summed up in the formula: <math>{}^{\backprime\backprime} y_0 = y \cdot y {}^{\prime\prime}.\!</math>

Thus the present inquiry, acting on the pretext of a ''formal posability'', namely, the circumstance that the rules of a prospective formal grammar allow one to write the expression <math>{}^{\backprime\backprime} y \cdot y {}^{\prime\prime}</math> and to inquire after its meaning, is led to define itself in terms that apply to its own case as argument, since the present inquiry, <math>y_0,\!</math> must be an example of whatever genus, <math>Y,\!</math>, that a generic inquiry, <math>y,\!</math> is selected to represent.  As a consequence, the present inquiry is forced to pursue the development of its own case in terms that appose its own actions to its own motives, and so is led to take on a recursive, a reflective, or a reflexive cast.

Thus the present inquiry, acting on the pretext of a ''formal posability'', namely, the circumstance that the rules of a prospective formal grammar allow one to write the expression <math>{}^{\backprime\backprime} y \cdot y {}^{\prime\prime}\!</math> and to inquire after its meaning, is led to define itself in terms that apply to its own case as argument, since the present inquiry, <math>y_0,\!</math> must be an example of whatever genus, <math>Y,\!</math>, that a generic inquiry, <math>y,\!</math> is selected to represent.  As a consequence, the present inquiry is forced to pursue the development of its own case in terms that appose its own actions to its own motives, and so is led to take on a recursive, a reflective, or a reflexive cast.

=====5.1.2.2. The Symbolic Object=====

=====5.1.2.2. The Symbolic Object=====

To paraphrase, the present inquiry acts on the pretense that an inquiry can inquire into other inquiries, perhaps even those that are presently ongoing, and even inquire into itself, in sum, being entitled to inquire into the full genus of inquiry, <math>Y,\!</math> a class that includes <math>y_0\!</math> as a member.  But these representations, under cross examination, lead to a number of unanswered questions, like:  Just what is a &ldquo;generic inquiry&rdquo;, anyway?  Even more critically, their close and repeated examination leads to a host of &ldquo;unquestioned answers&rdquo;, answers already accepted as adequate, but whose appearances as answers need to be questioned again.

To paraphrase, the present inquiry acts on the pretense that an inquiry can inquire into other inquiries, perhaps even those that are presently ongoing, and even inquire into itself, in sum, being entitled to inquire into the full genus of inquiry, <math>Y,\!</math> a class that includes <math>y_0\!</math> as a member.  But these representations, under cross examination, lead to a number of unanswered questions, like:  Just what is a &ldquo;generic inquiry&rdquo;, anyway?  Even more critically, their close and repeated examination leads to a host of &ldquo;unquestioned answers&rdquo;, answers already accepted as adequate, but whose appearances as answers need to be questioned again.

The ''formal posability'' of a self application, for example, as expressed by the term <math>{}^{\backprime\backprime} y \cdot y {}^{\prime\prime},</math> especially when the formal calculus that is called on to make sense of these applications is still merely prospective and still highly speculative, ought to arouse a lot of suspicion from the purely formal point of view.  Indeed, I cannot justify this way of proceeding, beginning in the middle of things and without stopping to establish a well defined formal system ahead of time, except to say that something very like it is unavoidable in a large number of natural circumstances, and so one ought to find a way of getting used to it.  A way of getting used to the natural situation of inquiry is one of the things that the present inquiry hopes to find.

The ''formal posability'' of a self application, for example, as expressed by the term <math>{}^{\backprime\backprime} y \cdot y {}^{\prime\prime},\!</math> especially when the formal calculus that is called on to make sense of these applications is still merely prospective and still highly speculative, ought to arouse a lot of suspicion from the purely formal point of view.  Indeed, I cannot justify this way of proceeding, beginning in the middle of things and without stopping to establish a well defined formal system ahead of time, except to say that something very like it is unavoidable in a large number of natural circumstances, and so one ought to find a way of getting used to it.  A way of getting used to the natural situation of inquiry is one of the things that the present inquiry hopes to find.

If it appears that this allows the present inquiry an unlimited scope or an excessive freedom, it has to be remembered that a ''formal posability'' is barely enough of a formal subsistence to begin an inquiry, but rarely enough to finish it.  It can be invaluable as the provisional ''grubstake'' for a prospecting expedition, supplying the initial overhead it takes to ''prime the pump'' of subsequent exploration, but it is not sufficient to continue very far with an investigation.  In essence, it is nothing more substantial than a grammatical allowance or a syntactic hypothesis, in effect, a poetic license, a verbal permission, or a written suggestion.  Taking all of these cautions into account, it leaves the present inquiry motivated and justified by no more authority than their titles connote, and it obliges the precocity of what is written to be atoned for with all the critical benevolence of afterthought that can be mustered after the fact, to wit, through the diligent application of that turn of mind that allows one to write first and only later to think on the meaning.

If it appears that this allows the present inquiry an unlimited scope or an excessive freedom, it has to be remembered that a ''formal posability'' is barely enough of a formal subsistence to begin an inquiry, but rarely enough to finish it.  It can be invaluable as the provisional ''grubstake'' for a prospecting expedition, supplying the initial overhead it takes to ''prime the pump'' of subsequent exploration, but it is not sufficient to continue very far with an investigation.  In essence, it is nothing more substantial than a grammatical allowance or a syntactic hypothesis, in effect, a poetic license, a verbal permission, or a written suggestion.  Taking all of these cautions into account, it leaves the present inquiry motivated and justified by no more authority than their titles connote, and it obliges the precocity of what is written to be atoned for with all the critical benevolence of afterthought that can be mustered after the fact, to wit, through the diligent application of that turn of mind that allows one to write first and only later to think on the meaning.

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The present inquiry acts on the purely formal suggestion that a generic inquiry can inquire into other inquiries, perhaps even those that remain ongoing, moreover, that a particular inquiry can even inquire into itself.  Interpolating the appropriate symbols, the present inquiry, referring to itself as <math>{}^{\backprime\backprime} y_0 {}^{\prime\prime},</math> acts on the instance of a purely formal possibility, one that it expresses as a premiss in the formula <math>{}^{\backprime\backprime} y_0 = y \cdot y {}^{\prime\prime},</math> intending this to be interpreted to the effect that an inquiry can inquire into a class of inquiries that includes itself as a member, and this is a hypothesis that is based on little more authority than the fact of its expression a prospective formal language, in other words, one whose interpretation is still a largely prospective matter.

The present inquiry acts on the purely formal suggestion that a generic inquiry can inquire into other inquiries, perhaps even those that remain ongoing, moreover, that a particular inquiry can even inquire into itself.  Interpolating the appropriate symbols, the present inquiry, referring to itself as <math>{}^{\backprime\backprime} y_0 {}^{\prime\prime},\!</math> acts on the instance of a purely formal possibility, one that it expresses as a premiss in the formula <math>{}^{\backprime\backprime} y_0 = y \cdot y {}^{\prime\prime},\!</math> intending this to be interpreted to the effect that an inquiry can inquire into a class of inquiries that includes itself as a member, and this is a hypothesis that is based on little more authority than the fact of its expression a prospective formal language, in other words, one whose interpretation is still a largely prospective matter.

Stepping back and reflecting on the situation, one needs to ask how in general and how in particular does one fall so blithely into these forms and into these manners of representation.  Once that process is better understood then it becomes possible to evaluate in a fairer way whether this direction of fall is tantamount to a happy accident of the natural intuition or whether it constellates a disastrous catastrophe that needs to be remedied through the application of a severer style of reasoning.  Generally speaking, the point at which intellectual developments like these begin to take on an automatic character is when the intention is formed of devising a formal calculus, in the present case, a prospective calculus of ''applications'' or ''appositions'' of the form <math>f \cdot g,\!</math> the terms of which are intended to be capable of referring to processes potentially as complex as inquiries.  The project of an ''appositional calculus'' (AC) is what formalizes the intuitive possibility of an inquiry into inquiry and continues to suggest the formal possibility that any inquiry can be applied to itself, at least, any inquiry that can be symbolized in this calculus.

Stepping back and reflecting on the situation, one needs to ask how in general and how in particular does one fall so blithely into these forms and into these manners of representation.  Once that process is better understood then it becomes possible to evaluate in a fairer way whether this direction of fall is tantamount to a happy accident of the natural intuition or whether it constellates a disastrous catastrophe that needs to be remedied through the application of a severer style of reasoning.  Generally speaking, the point at which intellectual developments like these begin to take on an automatic character is when the intention is formed of devising a formal calculus, in the present case, a prospective calculus of ''applications'' or ''appositions'' of the form <math>f \cdot g,\!</math> the terms of which are intended to be capable of referring to processes potentially as complex as inquiries.  The project of an ''appositional calculus'' (AC) is what formalizes the intuitive possibility of an inquiry into inquiry and continues to suggest the formal possibility that any inquiry can be applied to itself, at least, any inquiry that can be symbolized in this calculus.

The present situation, as far as it goes, is a suitable subject for being investigated along the lines of the pragmatic theory of sign relations.

The present situation, as far as it goes, is a suitable subject for being investigated along the lines of the pragmatic theory of sign relations.

Since <math>{}^{\backprime\backprime} x {}^{\prime\prime}</math> is a sign, it has the potential to denote an object <math>x,\!</math> if and when there is determined to be a signified object, and one with a power to impress itself on the mind of the operative interpreter of that sign.  Likewise, since <math>{}^{\backprime\backprime} y_0 = y \cdot y {}^{\prime\prime}</math> is a sign, it has the potential to denote an object, one that syntactic compunctions stop me from saying is <math>y_0 = y \cdot y,</math> that is, if I want to avoid a definite risk of failing to be understood.  But what is this object, if it exists?  At any rate, what sort of object is the receiver of the sign thereby entitled to expect it to be, whether or not the object that it foreshadows ever does come to be actualized?

Since <math>{}^{\backprime\backprime} x {}^{\prime\prime}\!</math> is a sign, it has the potential to denote an object <math>x,\!</math> if and when there is determined to be a signified object, and one with a power to impress itself on the mind of the operative interpreter of that sign.  Likewise, since <math>{}^{\backprime\backprime} y_0 = y \cdot y {}^{\prime\prime}\!</math> is a sign, it has the potential to denote an object, one that syntactic compunctions stop me from saying is <math>y_0 = y \cdot y,\!</math> that is, if I want to avoid a definite risk of failing to be understood.  But what is this object, if it exists?  At any rate, what sort of object is the receiver of the sign thereby entitled to expect it to be, whether or not the object that it foreshadows ever does come to be actualized?

In order to have a variety of more convenient names for referring to the object potentially denoted by the sign <math>{}^{\backprime\backprime} y_0 = y \cdot y {}^{\prime\prime},</math> I refer to the expression <math>{}^{\backprime\backprime} y_0 = y \cdot y {}^{\prime\prime}</math> as ''&ldquo;The Initial Equation&rdquo;'', or as ''&ldquo;TIE&rdquo;'', for short.  Although it is not strictly necessary for such a small piece of text as <math>{}^{\backprime\backprime} y_0 = y \cdot y {}^{\prime\prime},</math> I here obey the rule that the titles of texts are italicized.  Furthermore, the object, situation, or state that satisfies ''TIE'', to the effect that <math>y_0 = y \cdot y,\!</math> and is therefore potentially denoted by ''TIE'', can also be referred to as &ldquo;the intended state&rdquo;, or as &ldquo;TIS&rdquo;, for short.

In order to have a variety of more convenient names for referring to the object potentially denoted by the sign <math>{}^{\backprime\backprime} y_0 = y \cdot y {}^{\prime\prime},\!</math> I refer to the expression <math>{}^{\backprime\backprime} y_0 = y \cdot y {}^{\prime\prime}\!</math> as ''&ldquo;The Initial Equation&rdquo;'', or as ''&ldquo;TIE&rdquo;'', for short.  Although it is not strictly necessary for such a small piece of text as <math>{}^{\backprime\backprime} y_0 = y \cdot y {}^{\prime\prime},\!</math> I here obey the rule that the titles of texts are italicized.  Furthermore, the object, situation, or state that satisfies ''TIE'', to the effect that <math>y_0 = y \cdot y,\!</math> and is therefore potentially denoted by ''TIE'', can also be referred to as &ldquo;the intended state&rdquo;, or as &ldquo;TIS&rdquo;, for short.

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Are virtues the species and teachings the genus, or perhaps vice versa?  Or do virtues and teachings form domains that are essentially distinct?  Whether one is a species of the other or whether the two are essentially different, what are the features that apparently distinguish the one from the other?

Are virtues the species and teachings the genus, or perhaps vice versa?  Or do virtues and teachings form domains that are essentially distinct?  Whether one is a species of the other or whether the two are essentially different, what are the features that apparently distinguish the one from the other?

Let me begin by assuming a situation that is plausibly general enough, that some virtues can be taught, symbolized as <math>V \land T</math>, and that some cannot, symbolized as <math>V \land \lnot T</math>.  I am not trying to say yet whether both kinds of cases actually occur, but merely wish to consider what follows from the likely alternatives.  Then the question as to what distinguishes virtues from teachings has two senses:

Let me begin by assuming a situation that is plausibly general enough, that some virtues can be taught, symbolized as <math>V \land T\!</math>, and that some cannot, symbolized as <math>V \land \lnot T\!</math>.  I am not trying to say yet whether both kinds of cases actually occur, but merely wish to consider what follows from the likely alternatives.  Then the question as to what distinguishes virtues from teachings has two senses:

# Among virtues that are special cases of teachings, <math>V \land T</math>, the features that distinguish virtues from teachings are known as ''specific differences''.  These qualities serve to mark out virtues for special consideration from amidst the common herd of teachings and tend to distinguish the more exemplary species of virtues from the more inclusive genus of teachings.

# Among virtues that are special cases of teachings, <math>V \land T\!</math>, the features that distinguish virtues from teachings are known as ''specific differences''.  These qualities serve to mark out virtues for special consideration from amidst the common herd of teachings and tend to distinguish the more exemplary species of virtues from the more inclusive genus of teachings.

# Among virtues that transcend the realm of teachings, <math>V \land \lnot T</math>, the features that distinguish virtues from teachings are aptly called ''exclusionary exemptions''.  These properties place the reach of virtues beyond the grasp of what is attainable through any order of teachings and serve to remove the orbit of virtues a discrete pace from the general run of teachings.

# Among virtues that transcend the realm of teachings, <math>V \land \lnot T\!</math>, the features that distinguish virtues from teachings are aptly called ''exclusionary exemptions''.  These properties place the reach of virtues beyond the grasp of what is attainable through any order of teachings and serve to remove the orbit of virtues a discrete pace from the general run of teachings.

In either case it can always be said, though without contributing anything of substance to the understanding of the problem, that it is their very property of ''virtuosity'' or their very quality of ''excellence'' that distinguishes the virtues from the teachings, whether this character appears to do nothing but add specificity to what can be actualized through learning alone, or solely through teaching, or whether it requires a nature that transcends the level of what can be achieved through any learning or teaching at all.  But this sort of answer only begs the question.  The real question is whether this mark is apparent or real, and how it ought to be analyzed and construed.

In either case it can always be said, though without contributing anything of substance to the understanding of the problem, that it is their very property of ''virtuosity'' or their very quality of ''excellence'' that distinguishes the virtues from the teachings, whether this character appears to do nothing but add specificity to what can be actualized through learning alone, or solely through teaching, or whether it requires a nature that transcends the level of what can be achieved through any learning or teaching at all.  But this sort of answer only begs the question.  The real question is whether this mark is apparent or real, and how it ought to be analyzed and construed.

<ol style="list-style-type:decimal">

<ol style="list-style-type:decimal">

<li>Did Socrates assert or believe that virtue can be taught, or not?<br>In symbols, did he assert or believe that <math>V \Rightarrow T</math>, or not?</li>

<li>Did Socrates assert or believe that virtue can be taught, or not?<br>In symbols, did he assert or believe that <math>V \Rightarrow T\!</math>, or not?</li>

<li>Did he think that:</li>

<li>Did he think that:</li>

<ol style="list-style-type:lower-latin">

<ol style="list-style-type:lower-latin">

<li>knowledge is virtue, in the sense that <math>U \Rightarrow V</math>?</li>

<li>knowledge is virtue, in the sense that <math>U \Rightarrow V\!</math>?</li>

<li>virtue is knowledge, in the sense that <math>U \Leftarrow V</math>?</li>

<li>virtue is knowledge, in the sense that <math>U \Leftarrow V\!</math>?</li>

<li>knowledge is virtue, in the sense that <math>U \Leftrightarrow V</math>?</li></ol>

<li>knowledge is virtue, in the sense that <math>U \Leftrightarrow V\!</math>?</li></ol>

<li>Did he teach or try to teach that knowledge can be taught?<br>In symbols, did he teach or try to teach that <math>U \Rightarrow T</math>?</li></ol>

<li>Did he teach or try to teach that knowledge can be taught?<br>In symbols, did he teach or try to teach that <math>U \Rightarrow T\!</math>?</li></ol>

My current understanding of the record that is given to us in Plato's Socratic Dialogues can be summarized as follows:

My current understanding of the record that is given to us in Plato's Socratic Dialogues can be summarized as follows:

At one point Socrates seems to assume the rule that knowledge can be taught, <math>U \Rightarrow T</math>, but simply in order to pursue the case that virtue is knowledge, <math>V \Rightarrow U</math>, toward the provisional conclusion that virtue can be taught, <math>V \Rightarrow T</math>.  This seems straightforward enough, if it were not for the good chance that all of this reasoning is taking place under the logical aegis of an indirect argument, a reduction to absurdity, designed to show just the opposite of what it has assumed for the sake of initiating the argument.  The issue is further clouded by the circumstance that the full context of the argument most likely extends over several Dialogues, not all of which survive, and the intended order of which remains in question.

At one point Socrates seems to assume the rule that knowledge can be taught, <math>U \Rightarrow T\!</math>, but simply in order to pursue the case that virtue is knowledge, <math>V \Rightarrow U\!</math>, toward the provisional conclusion that virtue can be taught, <math>V \Rightarrow T\!</math>.  This seems straightforward enough, if it were not for the good chance that all of this reasoning is taking place under the logical aegis of an indirect argument, a reduction to absurdity, designed to show just the opposite of what it has assumed for the sake of initiating the argument.  The issue is further clouded by the circumstance that the full context of the argument most likely extends over several Dialogues, not all of which survive, and the intended order of which remains in question.

At other points Socrates appears to claim that knowledge and virtue are neither learned nor taught, in the strictest senses of these words, but can only be ''divined'', ''recollected'', or ''remembered'', that is, recalled, recognized, or reconstituted from the original acquaintance that a soul, being immortal, already has with the real idea or the essential form of each thing in itself.  Still, this leaves open the possibility that one person can help another to guess a truth or to recall what both of them already share in knowing, as if locked away in one or another partially obscured or temporarily forgotten part of their inmost being.  And it is just this freer interpretation of ''learning'' and ''teaching'', whereby one agent catalyzes not catechizes another, that a liberal imagination would yet come to call ''education''.  Therefore, the real issue at stake, both with regard to the aim and as it comes down to the end of this inquiry, is not so much whether knowledge and virtue can be learned and taught as what kind of education is apt to achieve their actualization in the individual and is fit to maintain their realization in the community.

At other points Socrates appears to claim that knowledge and virtue are neither learned nor taught, in the strictest senses of these words, but can only be ''divined'', ''recollected'', or ''remembered'', that is, recalled, recognized, or reconstituted from the original acquaintance that a soul, being immortal, already has with the real idea or the essential form of each thing in itself.  Still, this leaves open the possibility that one person can help another to guess a truth or to recall what both of them already share in knowing, as if locked away in one or another partially obscured or temporarily forgotten part of their inmost being.  And it is just this freer interpretation of ''learning'' and ''teaching'', whereby one agent catalyzes not catechizes another, that a liberal imagination would yet come to call ''education''.  Therefore, the real issue at stake, both with regard to the aim and as it comes down to the end of this inquiry, is not so much whether knowledge and virtue can be learned and taught as what kind of education is apt to achieve their actualization in the individual and is fit to maintain their realization in the community.

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Of course, there is much that is open to interpretation about the maxim "knowledge is virtue".  In particular, does the copula "is" represent a necessary implication (<math>\Rightarrow</math>), a sufficient reduction ("is only", <math>\Leftarrow</math>), or a necessary and sufficient identification (<math>\Leftrightarrow</math>)?

Of course, there is much that is open to interpretation about the maxim "knowledge is virtue".  In particular, does the copula "is" represent a necessary implication (<math>\Rightarrow\!</math>), a sufficient reduction ("is only", <math>\Leftarrow\!</math>), or a necessary and sufficient identification (<math>\Leftrightarrow\!</math>)?

====5.2.9. Principle of Rational Action====

====5.2.9. Principle of Rational Action====

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How does this ancient issue, concerning the relation of reason, to action, to the good that is overall desired or intended, transform itself through the medium of intellectual history onto the modern scene?  In particular, what bearing does it have on the subjects of artificial intelligence and systems theory, and on the object of the present inquiry?  As it turns out, in classical cybernetics and in systems theory, and especially in the parts of AI and cognitive science that have to do with heuristic reasoning, the transformations of the problem have tarried so long in the vicinity of a singular triviality that the original form of the question is nearly unmistakable in every modern version.  The transposition of the theme <math>(\text{Reason}, \text{Action}, \text{Good})</math> into the mode of <math>(\text{Intelligence}, \text{Operation}, \text{Goal})</math> can make for an interesting variation, but it does not alter the given state of accord or discord among its elements and does nothing to turn the lock into its key.

How does this ancient issue, concerning the relation of reason, to action, to the good that is overall desired or intended, transform itself through the medium of intellectual history onto the modern scene?  In particular, what bearing does it have on the subjects of artificial intelligence and systems theory, and on the object of the present inquiry?  As it turns out, in classical cybernetics and in systems theory, and especially in the parts of AI and cognitive science that have to do with heuristic reasoning, the transformations of the problem have tarried so long in the vicinity of a singular triviality that the original form of the question is nearly unmistakable in every modern version.  The transposition of the theme <math>(\text{Reason}, \text{Action}, \text{Good})\!</math> into the mode of <math>(\text{Intelligence}, \text{Operation}, \text{Goal})\!</math> can make for an interesting variation, but it does not alter the given state of accord or discord among its elements and does nothing to turn the lock into its key.

How do these questions bear on the present inquiry?  Suppose that one is trying to understand something like an agency of life, a capacity for inquiry, a faculty of intelligence, or a power of learning and reasoning.  For starters, ''something like'' is a little vague, so let me suggest calling the target class of agencies, capacities, faculties, or powers that most hold my interest here by the name of ''virtues'', thereby invoking as an offstage direction the classical concepts of ''anima'' and ''arete'' that seem to prompt them all.  What all of these virtues have in common is their appearance, whether it strikes one on first impression or only develops in one's appreciation through a continuing acquaintance over time, of transcending or rising infinitely far beyond all of one's attempts to construct them from or reduce them to the sorts of instrumentalities that are much more basic, familiar, mundane, ordinary, simpler, in short, the kinds of abilities that one already understands well enough and is granted to have well under one's command or control.  For convenience, I dub this class of abilities, that a particular agent has a thorough understanding of and a complete competency in, as the ''resources'' of that agent.

How do these questions bear on the present inquiry?  Suppose that one is trying to understand something like an agency of life, a capacity for inquiry, a faculty of intelligence, or a power of learning and reasoning.  For starters, ''something like'' is a little vague, so let me suggest calling the target class of agencies, capacities, faculties, or powers that most hold my interest here by the name of ''virtues'', thereby invoking as an offstage direction the classical concepts of ''anima'' and ''arete'' that seem to prompt them all.  What all of these virtues have in common is their appearance, whether it strikes one on first impression or only develops in one's appreciation through a continuing acquaintance over time, of transcending or rising infinitely far beyond all of one's attempts to construct them from or reduce them to the sorts of instrumentalities that are much more basic, familiar, mundane, ordinary, simpler, in short, the kinds of abilities that one already understands well enough and is granted to have well under one's command or control.  For convenience, I dub this class of abilities, that a particular agent has a thorough understanding of and a complete competency in, as the ''resources'' of that agent.

In accord with this analysis of inquiry as a process of clarification, and of clarification in turn as a process that operates on sign relations, the next few paragraphs consider various interpretations of the clarification task, initiating the process of comparing and contrasting their elements, and ultimately seeking to classify their variety.  This discussion notices one general feature that all types of clarification process appear to have in common and it discerns another general feature that splits the genus of clarification processes into a couple of broad moieties or species.

In accord with this analysis of inquiry as a process of clarification, and of clarification in turn as a process that operates on sign relations, the next few paragraphs consider various interpretations of the clarification task, initiating the process of comparing and contrasting their elements, and ultimately seeking to classify their variety.  This discussion notices one general feature that all types of clarification process appear to have in common and it discerns another general feature that splits the genus of clarification processes into a couple of broad moieties or species.

Inquiry, considered as a process of clarification, is the chief way that a sign relation can grow and develop in service to the life of its agent.  If not assured as the principal way, at least, while the jurisdictions of automatic adaptation, oblique evolution, and random ramification are yet uncharted and unassessed, it is probably still the most principled way that sign relations have of adapting and evolving to meet the objectives of interpretive agents in their given environments of needs and objects.  By way of a general comparison, then, all reasonable interpretations of the clarification task involve the augmentation of sign relations by the addition of ''elementary sign relations'', that is, ordered triples of the form <math>(o, s, i).</math>

Inquiry, considered as a process of clarification, is the chief way that a sign relation can grow and develop in service to the life of its agent.  If not assured as the principal way, at least, while the jurisdictions of automatic adaptation, oblique evolution, and random ramification are yet uncharted and unassessed, it is probably still the most principled way that sign relations have of adapting and evolving to meet the objectives of interpretive agents in their given environments of needs and objects.  By way of a general comparison, then, all reasonable interpretations of the clarification task involve the augmentation of sign relations by the addition of ''elementary sign relations'', that is, ordered triples of the form <math>(o, s, i).\!</math>

Treating the process of clarification as one that affects the growth and development of a sign relation, even if constrained to the medium of its syntactic domain, there is, of course, an overwhelming diversity of ways that one can imagine an arbitrary sign relation as growing through time.  No matter whether it restrains its labors to the monotonic annexation of ever more triples <math>(o, s, i)</math> to the masses of data already accumulated or whether it liberates the full deliberations of a discursive process, thus invoking the ebb and flow of corrective, editorial, reflective, remedial, and reversible processes, not every mode of growth or development that can occur in a sign relation has a bearing on reducing the uncertainty of an agent about an object or has the effect of promoting the clarity of the given signs.

Treating the process of clarification as one that affects the growth and development of a sign relation, even if constrained to the medium of its syntactic domain, there is, of course, an overwhelming diversity of ways that one can imagine an arbitrary sign relation as growing through time.  No matter whether it restrains its labors to the monotonic annexation of ever more triples <math>(o, s, i)\!</math> to the masses of data already accumulated or whether it liberates the full deliberations of a discursive process, thus invoking the ebb and flow of corrective, editorial, reflective, remedial, and reversible processes, not every mode of growth or development that can occur in a sign relation has a bearing on reducing the uncertainty of an agent about an object or has the effect of promoting the clarity of the given signs.

With regard to inquiry as clarification, and clarification in turn as the evolution of a sign relation, it does not matter whether one views it as a process of exploration and discovery, taking place in a preconceived cartesian space <math>O \times S \times I</math> and seeking to find clearer signs for each known object, or whether one views it as a process of creation and invention, staking out the syntactic parts of elementary sign relations <math>(o, s, i),</math> following the directions of transient clarity to the signs of maximal achievable clarity, making and testing novel combinations with an eye toward present objects, and picking out the clearest indications for inclusion in one's current sign relation.

With regard to inquiry as clarification, and clarification in turn as the evolution of a sign relation, it does not matter whether one views it as a process of exploration and discovery, taking place in a preconceived cartesian space <math>O \times S \times I\!</math> and seeking to find clearer signs for each known object, or whether one views it as a process of creation and invention, staking out the syntactic parts of elementary sign relations <math>(o, s, i),\!</math> following the directions of transient clarity to the signs of maximal achievable clarity, making and testing novel combinations with an eye toward present objects, and picking out the clearest indications for inclusion in one's current sign relation.

To review:  Inquiry depends on clarification, and clarification depends on the augmentation or the evolution of sign relations in various ways.  In order to stay within the realms of possibility that are accessible to computational processes and covered by computational models, it is best to look for varieties of clarification process that are tantamount to recursive forms of development in sign relations, those that one can contemplate being carried out by a recursively defined growth process.  Even working under these constraints, there is still an amazingly large variety of different ways that the ''eking out'' of initial sign relations and the ''imping out'' of fledgling sign relations can proceed.

To review:  Inquiry depends on clarification, and clarification depends on the augmentation or the evolution of sign relations in various ways.  In order to stay within the realms of possibility that are accessible to computational processes and covered by computational models, it is best to look for varieties of clarification process that are tantamount to recursive forms of development in sign relations, those that one can contemplate being carried out by a recursively defined growth process.  Even working under these constraints, there is still an amazingly large variety of different ways that the ''eking out'' of initial sign relations and the ''imping out'' of fledgling sign relations can proceed.

Some of the most important general features that mark out the pragmatic theory of sign relations from its original material are instrumental in character and arise largely due to changes in the technological base, formally speaking, between the ancient and the present times, that is, by innovations in the formal languages and the technical methods that are made available for carrying out the discussion.  Three of these general instrumental features are taken up next.

Some of the most important general features that mark out the pragmatic theory of sign relations from its original material are instrumental in character and arise largely due to changes in the technological base, formally speaking, between the ancient and the present times, that is, by innovations in the formal languages and the technical methods that are made available for carrying out the discussion.  Three of these general instrumental features are taken up next.

# In conformity with the modern facility for thinking of relations in general in extensional terms, as collections of ordered n tuples of domain components that belong to the relation in question, current versions of the theory of signs render it easiest to think of each given sign relation as a particular collection of ordered triples.  Elements of a sign relation are called ''elementary sign relations'' (ESRs), and the data of each given element of the sign relation can be represented as an ordered triple, of the form <math>(o, s, i),</math> that names its object, sign, and interpretant, respectively.

# In conformity with the modern facility for thinking of relations in general in extensional terms, as collections of ordered n tuples of domain components that belong to the relation in question, current versions of the theory of signs render it easiest to think of each given sign relation as a particular collection of ordered triples.  Elements of a sign relation are called ''elementary sign relations'' (ESRs), and the data of each given element of the sign relation can be represented as an ordered triple, of the form <math>(o, s, i),\!</math> that names its object, sign, and interpretant, respectively.

# Among the other props on the modern stage, the pragmatic theory of sign relations can make especially good use of the bounteous ''logics'' of relations and ''algebras'' of relative terms that are currently available, as expressed in any one of several symbolic calculi with approximately the power of predicate logic.  Indeed, many of these algebras, calculi, and logics of relations received their first &ldquo;modern&redquo; formulations in the work of C.S.&nbsp;Peirce, and in the very process of trying to deal with the problems presented by the classical theory of signs.  As it happens, this coincidence of origins and this parallelism of derivations may help to account for the appearance of a quality of pre-established harmony that is presently manifested between the general subject of relations and the special subject of sign relations.

# Among the other props on the modern stage, the pragmatic theory of sign relations can make especially good use of the bounteous ''logics'' of relations and ''algebras'' of relative terms that are currently available, as expressed in any one of several symbolic calculi with approximately the power of predicate logic.  Indeed, many of these algebras, calculi, and logics of relations received their first &ldquo;modern&redquo; formulations in the work of C.S.&nbsp;Peirce, and in the very process of trying to deal with the problems presented by the classical theory of signs.  As it happens, this coincidence of origins and this parallelism of derivations may help to account for the appearance of a quality of pre-established harmony that is presently manifested between the general subject of relations and the special subject of sign relations.

# Developments in other fields in the intervening times have caused the prevailing paradigms to shift a number of times.  For starters, the lately recognized inescapability of participatory observation, and the multitude of constraints on knowledgeable action that the necessity of this contingency implies, that ought to have always been clear in marking the horizons of anthropology, economics, politics, psychology, and sociology, and the phenomenological consequences of this unavoidability that have recently forced themselves to the status of physical principles and tardily made their appearance in the symbolic rites of the attendant formalities, against all the fields of reluctance that physics can generate, and in spite of the full recalcitrance that its occasional ancillary, mathematics, can bring to heel.  These cautions leave even the casual observer nowadays much more suspicious about declaring the self evident independence of diverse aspects and axes of experience, whether assuming the disentanglement of different features of experiential quality or presuming on the orthogonality of their coordinate dimensions of formal quantity, for instance, as represented by the aspects of particles versus waves, or the axes of space versus time.  Features and dimensions of experience that appear as relevant or arise into salience at one level of action, exchange, or observation can disappear from the scene of relevant regard at other stages of participation and weigh imponderably on other scales of transaction.  In relation to one another, aspects and axes of experience that appear unrelated just so long as they are considered at one level of interaction and perception may not preserve their appearance of indifference and independence if the scales of participation under consideration are radically shifted, whether up or down in their order of magnitude.  As a result, the sort of consideration that makes a line of experience conspicuous as it falls on one plane of existence is seldom enough to draw it through every plane of being.  In a related fashion, the brand of consideration whose bearing on an intermediate scale of treatment causes one to regard two features or dimensions of experience as ''moderately independent'' or as ''relatively orthogonal'' is rarely ever relevant to all levels of regard and is almost never enough to justify one's calling these aspects ''absolutely independent'' or to support one's calling these axes "perfectly orthogonal".

# Developments in other fields in the intervening times have caused the prevailing paradigms to shift a number of times.  For starters, the lately recognized inescapability of participatory observation, and the multitude of constraints on knowledgeable action that the necessity of this contingency implies, that ought to have always been clear in marking the horizons of anthropology, economics, politics, psychology, and sociology, and the phenomenological consequences of this unavoidability that have recently forced themselves to the status of physical principles and tardily made their appearance in the symbolic rites of the attendant formalities, against all the fields of reluctance that physics can generate, and in spite of the full recalcitrance that its occasional ancillary, mathematics, can bring to heel.  These cautions leave even the casual observer nowadays much more suspicious about declaring the self evident independence of diverse aspects and axes of experience, whether assuming the disentanglement of different features of experiential quality or presuming on the orthogonality of their coordinate dimensions of formal quantity, for instance, as represented by the aspects of particles versus waves, or the axes of space versus time.  Features and dimensions of experience that appear as relevant or arise into salience at one level of action, exchange, or observation can disappear from the scene of relevant regard at other stages of participation and weigh imponderably on other scales of transaction.  In relation to one another, aspects and axes of experience that appear unrelated just so long as they are considered at one level of interaction and perception may not preserve their appearance of indifference and independence if the scales of participation under consideration are radically shifted, whether up or down in their order of magnitude.  As a result, the sort of consideration that makes a line of experience conspicuous as it falls on one plane of existence is seldom enough to draw it through every plane of being.  In a related fashion, the brand of consideration whose bearing on an intermediate scale of treatment causes one to regard two features or dimensions of experience as ''moderately independent'' or as ''relatively orthogonal'' is rarely ever relevant to all levels of regard and is almost never enough to justify one's calling these aspects ''absolutely independent'' or to support one's calling these axes "perfectly orthogonal".

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In the discussion that follows, I am going to use the letters <math>C, L, M</math> to stand for three generic features or classes of properties, yet to be fully analyzed or completely specified, that are commonly appreciated, desired, or valued as virtues of signs and expressions.  For now, a list of adjectives appropriate to each class can give a sufficient indication of their intended characters, even though it is easily possible and eventually necessary to find important distinctions that exist among the items in each given list of exemplary properties.

In the discussion that follows, I am going to use the letters <math>C, L, M\!</math> to stand for three generic features or classes of properties, yet to be fully analyzed or completely specified, that are commonly appreciated, desired, or valued as virtues of signs and expressions.  For now, a list of adjectives appropriate to each class can give a sufficient indication of their intended characters, even though it is easily possible and eventually necessary to find important distinctions that exist among the items in each given list of exemplary properties.

# The class <math>C</math> is suggested by the adjectives ''certain'', ''cogent'', ''compelling'', or ''convincing'', and, in some of their senses, by ''apparent'', ''evident'', ''obvious'', or ''patent''.

# The class <math>C\!</math> is suggested by the adjectives ''certain'', ''cogent'', ''compelling'', or ''convincing'', and, in some of their senses, by ''apparent'', ''evident'', ''obvious'', or ''patent''.

# The class <math>L</math> is suggested by the adjectives ''clear'', ''lucid'', ''perspicuous'', ''plain'', ''relevant'', or ''vivid''.  To the geometric imagination, these terms suggest a ''bluntness'' (of surfaces) or a ''sharpness'' (of edges).

# The class <math>L\!</math> is suggested by the adjectives ''clear'', ''lucid'', ''perspicuous'', ''plain'', ''relevant'', or ''vivid''.  To the geometric imagination, these terms suggest a ''bluntness'' (of surfaces) or a ''sharpness'' (of edges).

# The class <math>M</math> is suggested by the adjectives ''distinct'', ''decided'', ''defined'', ''definite'', ''determinate'', ''different'', ''differentiated'', or ''discrete'', and, within a stretch of the imagination, by ''acute'', ''conspicuous'', ''eminent'', ''manifest'', ''poignant'', ''salient'', or ''striking''.  To the geometric imagination, these terms suggest a ''pointedness''.

# The class <math>M\!</math> is suggested by the adjectives ''distinct'', ''decided'', ''defined'', ''definite'', ''determinate'', ''different'', ''differentiated'', or ''discrete'', and, within a stretch of the imagination, by ''acute'', ''conspicuous'', ''eminent'', ''manifest'', ''poignant'', ''salient'', or ''striking''.  To the geometric imagination, these terms suggest a ''pointedness''.

In this frame of thought, it needs to be understood that the intended sense of these last two classes excludes the common usage of words like ''clear'', ''clearly'', and so on, or ''distinct'', ''distinctly'', and so on, as elliptic figures of speech that are intended to be taken in a more literal way to mean ''clearly true'', and so on, or ''distinctly true'', and so on.

In this frame of thought, it needs to be understood that the intended sense of these last two classes excludes the common usage of words like ''clear'', ''clearly'', and so on, or ''distinct'', ''distinctly'', and so on, as elliptic figures of speech that are intended to be taken in a more literal way to mean ''clearly true'', and so on, or ''distinctly true'', and so on.

In this connection, when I mention one of these properties it is only meant as a representative of its class.  Also, as they are used in this context, these terms are intended only in what is diversely called their ''impressionistic'', ''nominal'', ''subjective'', ''superficial'', or ''topical'' sense, implying the sorts of qualities that one can judge &ldquo;by inspection&rdquo; of the expression and its immediate situation, and without the need of a prolonged investigation.  Thus, none of their intentions is damaged for this purpose by prefacing their proposal with an attitude of ''seeming''.  For all one cares in these concerns, <math>{}^{\backprime\backprime} \operatorname{seems}~X {}^{\prime\prime} = {}^{\backprime\backprime} X {}^{\prime\prime},</math> for <math>X = C, L, M.</math>  This makes the judgment of these qualities a matter of ''seeming syntax'' and ''seeming semantics'', involving only the sorts of decision that are commonly and easily made without carrying out complex computations or without delving into the abstruse equivalence classes of expressions.

In this connection, when I mention one of these properties it is only meant as a representative of its class.  Also, as they are used in this context, these terms are intended only in what is diversely called their ''impressionistic'', ''nominal'', ''subjective'', ''superficial'', or ''topical'' sense, implying the sorts of qualities that one can judge &ldquo;by inspection&rdquo; of the expression and its immediate situation, and without the need of a prolonged investigation.  Thus, none of their intentions is damaged for this purpose by prefacing their proposal with an attitude of ''seeming''.  For all one cares in these concerns, <math>{}^{\backprime\backprime} \operatorname{seems}~X {}^{\prime\prime} = {}^{\backprime\backprime} X {}^{\prime\prime},\!</math> for <math>X = C, L, M.\!</math>  This makes the judgment of these qualities a matter of ''seeming syntax'' and ''seeming semantics'', involving only the sorts of decision that are commonly and easily made without carrying out complex computations or without delving into the abstruse equivalence classes of expressions.

People frequently use the adverbs ''immediately'' or ''intuitively'' to get this sense across, and even though these terms have technical meanings that prevent me from using them in this way in anything but a casual setting, they can do for the moment.  Still, when I use ''immediately'' in this sense it is meant in contrast only to ''ultimately'', and more or less synonymous to ''mediately'', suggesting that which holds in the meantime.  In a pinch, a determination of seeming certainty or seeming clarity is enough to put an inquiry on hold for a time being, but the distinction between ''seeming so to me, for now'' and ''seeming so to all, forever'' still holds, with only the latter deserving the title of ''being so''.

People frequently use the adverbs ''immediately'' or ''intuitively'' to get this sense across, and even though these terms have technical meanings that prevent me from using them in this way in anything but a casual setting, they can do for the moment.  Still, when I use ''immediately'' in this sense it is meant in contrast only to ''ultimately'', and more or less synonymous to ''mediately'', suggesting that which holds in the meantime.  In a pinch, a determination of seeming certainty or seeming clarity is enough to put an inquiry on hold for a time being, but the distinction between ''seeming so to me, for now'' and ''seeming so to all, forever'' still holds, with only the latter deserving the title of ''being so''.

These observations on im/mediate, intuitive, or meantime determinations of certainty, clarity, and distinctness have a bearing on the styles of mathematical formulation and the modes of computational implementation that are candidates for mediating a natural style of inquiry, in other words, the sort of inquiry that a human being can relate to.  Because a decision that a sign or expression has one of the virtues <math>C, L, M,</math> even to a mediate, a moderate, or a modest degree, is often enough to end an inquiry on a temporary basis, it becomes necessary to recognize a form of recursive foundation that also rests on a temporal basis.  And yet, because these modes of judgment are all the while fallible and subject to change, it is possible that deeper foundations remain to be found.

These observations on im/mediate, intuitive, or meantime determinations of certainty, clarity, and distinctness have a bearing on the styles of mathematical formulation and the modes of computational implementation that are candidates for mediating a natural style of inquiry, in other words, the sort of inquiry that a human being can relate to.  Because a decision that a sign or expression has one of the virtues <math>C, L, M,\!</math> even to a mediate, a moderate, or a modest degree, is often enough to end an inquiry on a temporary basis, it becomes necessary to recognize a form of recursive foundation that also rests on a temporal basis.  And yet, because these modes of judgment are all the while fallible and subject to change, it is possible that deeper foundations remain to be found.

What does this mean for the topic of reflection?  Well, reflection is precisely that mode of thinking that is capable of beginning with the axioms and working backward, that is, of searching out the more basic forms that conceivably underlie one's received formulations.

What does this mean for the topic of reflection?  Well, reflection is precisely that mode of thinking that is capable of beginning with the axioms and working backward, that is, of searching out the more basic forms that conceivably underlie one's received formulations.

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On reflection, the observation that appeared just before these last questions arose can be seen to make a very broad claim about a certain class of properties affecting expressions, namely, all those properties that can be analogous to the ordered measures of expressive quality.  For future reference, let me call this the ''monotone assumption'' (MA).  This generatrix of so many future and specious assumptions takes for granted a sweeping claim about the ways that an order of analysis of expressions translates into an order of comparison of their measures under one of these properties.  But this entire and previously unstated assumption is itself just another manner of working hypothesis for the mental procedure or the process of inquiry that makes use of it, and its proper understanding is perhaps better served if it is rephrased as a question:  Can the <math>X</math> of a claim or a concept be greater than the <math>X</math> of the subordinate claims and concepts that it calls on, where <math>{}^{\backprime\backprime} X {}^{\prime\prime}</math> stands for ''certainty'', ''clarity'', or any one of the corresponding class of measures, orders, properties, qualities, or virtues?

On reflection, the observation that appeared just before these last questions arose can be seen to make a very broad claim about a certain class of properties affecting expressions, namely, all those properties that can be analogous to the ordered measures of expressive quality.  For future reference, let me call this the ''monotone assumption'' (MA).  This generatrix of so many future and specious assumptions takes for granted a sweeping claim about the ways that an order of analysis of expressions translates into an order of comparison of their measures under one of these properties.  But this entire and previously unstated assumption is itself just another manner of working hypothesis for the mental procedure or the process of inquiry that makes use of it, and its proper understanding is perhaps better served if it is rephrased as a question:  Can the <math>X\!</math> of a claim or a concept be greater than the <math>X\!</math> of the subordinate claims and concepts that it calls on, where <math>{}^{\backprime\backprime} X {}^{\prime\prime}\!</math> stands for ''certainty'', ''clarity'', or any one of the corresponding class of measures, orders, properties, qualities, or virtues?

Rather than taking this claim for granted, suppose I go looking for any properties, that might be similar to certainty or clarity, for which the measure of a whole expression is capable of exceeding the measure of its parts.  Is there an order property that is dependent on the constitution of the whole expression and a function of its analytic constituents but not necessarily tied down to monotonically conservative relationships like the sum, the average, or the lowest common denominator of the measures affecting its syntactic elements?  Once I take the trouble to formulate the question in explicit terms, any number of familiar examples are free to come to mind as fitting its requirements.  Indeed, since the notions of dependency and independence that accompany the use of mathematical functions and mathematical forms of decomposition do not by themselves implicate the more constrained types of dependency and the more radical types of independence that arise in relation and in reaction to the MA, it is rather easy to think of many that will do.

Rather than taking this claim for granted, suppose I go looking for any properties, that might be similar to certainty or clarity, for which the measure of a whole expression is capable of exceeding the measure of its parts.  Is there an order property that is dependent on the constitution of the whole expression and a function of its analytic constituents but not necessarily tied down to monotonically conservative relationships like the sum, the average, or the lowest common denominator of the measures affecting its syntactic elements?  Once I take the trouble to formulate the question in explicit terms, any number of familiar examples are free to come to mind as fitting its requirements.  Indeed, since the notions of dependency and independence that accompany the use of mathematical functions and mathematical forms of decomposition do not by themselves implicate the more constrained types of dependency and the more radical types of independence that arise in relation and in reaction to the MA, it is rather easy to think of many that will do.

====5.3.1. Looking Back====

====5.3.1. Looking Back====

Let me review the developments that bring me to this point.  I began by describing my present inquiry, <math>y_0,\!</math> as an inquiry into inquiry, <math>y \cdot y.</math>  Then I focussed on the activities of discussion, <math>d,\!</math> and formalization, <math>f,\!</math> as two components of the faculty or the process of inquiry, <math>y >\!\!= \{ d , f \}.</math>  This led me to the present discussion of formalization, <math>f \cdot d.</math>  Considered as classes of activities, the collective instances of formalization, <math>F,</math> appeared to be encompassed by the collective instances of discussion, <math>D,</math> thereby yielding the relationship <math>F \subseteq D.</math>

Let me review the developments that bring me to this point.  I began by describing my present inquiry, <math>y_0,\!</math> as an inquiry into inquiry, <math>y \cdot y.\!</math>  Then I focussed on the activities of discussion, <math>d,\!</math> and formalization, <math>f,\!</math> as two components of the faculty or the process of inquiry, <math>y >\!\!= \{ d , f \}.\!</math>  This led me to the present discussion of formalization, <math>f \cdot d.\!</math>  Considered as classes of activities, the collective instances of formalization, <math>F,\!</math> appeared to be encompassed by the collective instances of discussion, <math>D,\!</math> thereby yielding the relationship <math>F \subseteq D.\!</math>

I initially characterized discussion and formalization, in regard to each other, as being an &ldquo;actively instrumental&rdquo; versus a &ldquo;passively objective&rdquo; aspect, component, or &ldquo;face&rdquo; of the inquiry <math>y >\!\!= \{ d , f \}.</math>  In casting them this way I clearly traded on the ambiguity of &ldquo;-ionized&rdquo; terms to force the issue a bit.  In other words, I used the flexibility that is freely available within their &ldquo;-ionic&rdquo; construals, as processes or as products, to cast discussion and formalization into sundry molds, drawing out the patent energies that are manifested by the active process of discussion and placing them in contrast with the latent inertias that are immanent in the dormant product of formalization.  In this partially arbitrary way, I decided on the one hand to treat discussion in respect of its ongoing process, the only thing that it has any assurance of accomplishing, but I decided on the other hand to treat formalization in respect of its end product, the abstract image or the formal model that constitutes its chief qualification and thus becomes the mark of what it is.

I initially characterized discussion and formalization, in regard to each other, as being an &ldquo;actively instrumental&rdquo; versus a &ldquo;passively objective&rdquo; aspect, component, or &ldquo;face&rdquo; of the inquiry <math>y >\!\!= \{ d , f \}.\!</math>  In casting them this way I clearly traded on the ambiguity of &ldquo;-ionized&rdquo; terms to force the issue a bit.  In other words, I used the flexibility that is freely available within their &ldquo;-ionic&rdquo; construals, as processes or as products, to cast discussion and formalization into sundry molds, drawing out the patent energies that are manifested by the active process of discussion and placing them in contrast with the latent inertias that are immanent in the dormant product of formalization.  In this partially arbitrary way, I decided on the one hand to treat discussion in respect of its ongoing process, the only thing that it has any assurance of accomplishing, but I decided on the other hand to treat formalization in respect of its end product, the abstract image or the formal model that constitutes its chief qualification and thus becomes the mark of what it is.

By casting inquiry into the form <math>y >\!\!= \{ d , f \},</math> I made it more likely that my development of its self application <math>y \cdot y >\!\!= \{ d , f \}\{ d , f \}</math> would first take up the application of discussion to formalization, <math>f \cdot d,</math> and only later get around to the application of formalization to discussion, <math>d \cdot f,</math> that brings the active side of the formalization process into a greater prominence.  But the bias that I exploited in these readings does not seem at present to be a property of the incipient algebra that would determine the sense of the applications and the decompositions envisioned here.  Thus, if I initially saw a difference between the two presentations <math>\{ d , f \}</math> and <math>\{ f , d \},</math> then it must have been a purely interpretive and not a substantial one, and the task of giving explicit notice to these interpretive distinctions and working out their algebra or calculus yet remains to be carried out in any sort of convincing fashion.

By casting inquiry into the form <math>y >\!\!= \{ d , f \},\!</math> I made it more likely that my development of its self application <math>y \cdot y >\!\!= \{ d , f \}\{ d , f \}\!</math> would first take up the application of discussion to formalization, <math>f \cdot d,\!</math> and only later get around to the application of formalization to discussion, <math>d \cdot f,\!</math> that brings the active side of the formalization process into a greater prominence.  But the bias that I exploited in these readings does not seem at present to be a property of the incipient algebra that would determine the sense of the applications and the decompositions envisioned here.  Thus, if I initially saw a difference between the two presentations <math>\{ d , f \}\!</math> and <math>\{ f , d \},\!</math> then it must have been a purely interpretive and not a substantial one, and the task of giving explicit notice to these interpretive distinctions and working out their algebra or calculus yet remains to be carried out in any sort of convincing fashion.

Still, the casting of discussion and formalization as active and inert, respectively, was not entirely out of character with their distinctive natures, since a process that has an end is more naturally suited to be represented by its result than a process that conceivably never ends.  And whereas a ''discussion'' was allowed to be a form of discourse that does not need to have an end, with the possible exception of itself, a ''formalization'' was sensed to be a form of discourse that has, needs, seeks, or wants a distinct end, not just any end but a form of product that is preferred to satisfy a general description, and one that most likely resides outside the form of a vacuous vanity that simply refers, in a reflexive but hollow echo, to the entirety of its own proceedings.

Still, the casting of discussion and formalization as active and inert, respectively, was not entirely out of character with their distinctive natures, since a process that has an end is more naturally suited to be represented by its result than a process that conceivably never ends.  And whereas a ''discussion'' was allowed to be a form of discourse that does not need to have an end, with the possible exception of itself, a ''formalization'' was sensed to be a form of discourse that has, needs, seeks, or wants a distinct end, not just any end but a form of product that is preferred to satisfy a general description, and one that most likely resides outside the form of a vacuous vanity that simply refers, in a reflexive but hollow echo, to the entirety of its own proceedings.

In this merely penultimate analysis, and to the extent that the question of ends has been analyzed up to the present, it needs to be noted that more than a bit of ambiguity yet remains.  When one speaks of a form of discourse each of whose instances necessarily has an end, does one mean that the definition of the form requires each instance to have an end, and does one then mean that each valid instance actually achieves its end, or does one only mean that each instance of some empirically given class of discourses actually reaches some end or another?

In this merely penultimate analysis, and to the extent that the question of ends has been analyzed up to the present, it needs to be noted that more than a bit of ambiguity yet remains.  When one speaks of a form of discourse each of whose instances necessarily has an end, does one mean that the definition of the form requires each instance to have an end, and does one then mean that each valid instance actually achieves its end, or does one only mean that each instance of some empirically given class of discourses actually reaches some end or another?

The word &ldquo;reflection&rdquo; first entered this discussion in what seemed like a purely incidental and instrumental way, as a part of the definition of a &ldquo;meditation&rdquo; as &ldquo;a discourse intended to express its author's reflections or to guide others in contemplation&rdquo; (Webster's).  I converted this term to my own use as a name for a particular class of activities, describing the class of ''meditations'', <math>M,\!</math> as a brand of ''measured'' and ''motivated'' discussions that can serve to mediate formalizations within the realm of discussions at large.  Thus, I borrowed the term for no better reason than that of interposing a middle term between formalized discussions and discussions in general, thereby yielding the relationship <math>F \subseteq M \subseteq D.</math>

The word &ldquo;reflection&rdquo; first entered this discussion in what seemed like a purely incidental and instrumental way, as a part of the definition of a &ldquo;meditation&rdquo; as &ldquo;a discourse intended to express its author's reflections or to guide others in contemplation&rdquo; (Webster's).  I converted this term to my own use as a name for a particular class of activities, describing the class of ''meditations'', <math>M,\!</math> as a brand of ''measured'' and ''motivated'' discussions that can serve to mediate formalizations within the realm of discussions at large.  Thus, I borrowed the term for no better reason than that of interposing a middle term between formalized discussions and discussions in general, thereby yielding the relationship <math>F \subseteq M \subseteq D.\!</math>

In this respect, it seems to be instructive that the issue of reflection first arrived on the present scene, quietly enough, under the aegis of a borrowed term, imported without deliberate design among the components and the connotations of its associated sample of discourse, and involved in a process that seeks to negotiate the conflicting claims that arise between formal and casual discourse.  In the simplest sense of the word, an activity of reflection implies only that an agent thinks quietly and calmly about a matter, the etymology of the word suggesting the actions of bending, bounding, casting, folding, giving, turning, throwing, or yielding back again, and hence a pause, a return, or a review.  In this regard, the word "reflection" barely alludes to the idea that what the agent turns back to is something that involves itself, its own patterns of activity, and thus the word only hints as yet at the complicities of self reference and self application that are involved in an agent turning back to view its past, its present, or its ongoing forms of conduct.

In this respect, it seems to be instructive that the issue of reflection first arrived on the present scene, quietly enough, under the aegis of a borrowed term, imported without deliberate design among the components and the connotations of its associated sample of discourse, and involved in a process that seeks to negotiate the conflicting claims that arise between formal and casual discourse.  In the simplest sense of the word, an activity of reflection implies only that an agent thinks quietly and calmly about a matter, the etymology of the word suggesting the actions of bending, bounding, casting, folding, giving, turning, throwing, or yielding back again, and hence a pause, a return, or a review.  In this regard, the word "reflection" barely alludes to the idea that what the agent turns back to is something that involves itself, its own patterns of activity, and thus the word only hints as yet at the complicities of self reference and self application that are involved in an agent turning back to view its past, its present, or its ongoing forms of conduct.

When one crosses a critical threshold or a threshold of decision, ...

When one crosses a critical threshold or a threshold of decision, ...

A notion of reflection, in a more authentically reflexive sense, was implicitly involved in the application of inquiry to itself, <math>y_0 = y \cdot y,</math> and was eventually encountered on a recurring basis in the application of each newly recognized component of inquiry to itself:  <math>y \cdot y >\!\!= d \cdot d, f \cdot f.</math>  In a more substantial role, the option of a capacity for reflection was already noticed as a significant parameter in the constitution of an IF.

A notion of reflection, in a more authentically reflexive sense, was implicitly involved in the application of inquiry to itself, <math>y_0 = y \cdot y,\!</math> and was eventually encountered on a recurring basis in the application of each newly recognized component of inquiry to itself:  <math>y \cdot y >\!\!= d \cdot d, f \cdot f.\!</math>  In a more substantial role, the option of a capacity for reflection was already noticed as a significant parameter in the constitution of an IF.

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The relationships among the activities and faculties of discussion, contemplation, formalization, meditation, and reflection need to be explored in more detail.  In particular, the relationship between formalization and reflection is especially relevant to the task of constructing a RIF.

The relationships among the activities and faculties of discussion, contemplation, formalization, meditation, and reflection need to be explored in more detail.  In particular, the relationship between formalization and reflection is especially relevant to the task of constructing a RIF.

Unlike a discussion of discussion, <math>d \cdot d,</math> which is easy to start and hard to put an end to once it gets going, it is difficult for a reflection on reflection, <math>r \cdot r,</math> to get itself going with nothing to reflect on but itself.  I have just illustrated one way of doing this, namely, by leading a text to reflect on itself, as long as you understand this figure of speech to mean that it leads its interpreters, its writer and reader, without whose agency there would be no reflection at all, to reflect on how it reflects on itself.  But I obviously need other ways than this to demonstrate the functions and properties of reflection in anything like their full variety.

Unlike a discussion of discussion, <math>d \cdot d,\!</math> which is easy to start and hard to put an end to once it gets going, it is difficult for a reflection on reflection, <math>r \cdot r,\!</math> to get itself going with nothing to reflect on but itself.  I have just illustrated one way of doing this, namely, by leading a text to reflect on itself, as long as you understand this figure of speech to mean that it leads its interpreters, its writer and reader, without whose agency there would be no reflection at all, to reflect on how it reflects on itself.  But I obviously need other ways than this to demonstrate the functions and properties of reflection in anything like their full variety.

Toward this end, it can also help to illustrate the action of reflection if I find it some material besides itself to reflect off of, in other words, if I supply it with an independently generated and concretely finished text as an argument to exercise its powers of reflection on.  Accordingly, in the next part of this discussion I will interleave my text with &hellip;

Toward this end, it can also help to illustrate the action of reflection if I find it some material besides itself to reflect off of, in other words, if I supply it with an independently generated and concretely finished text as an argument to exercise its powers of reflection on.  Accordingly, in the next part of this discussion I will interleave my text with &hellip;

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If one considers the formula that characterizes an inquiry into inquiry, <math>y_0 = y \cdot y,</math> and examines the term <math>y \cdot y</math> that factors <math>y_0\!</math> along the lines of an ostensible self-application, it is evident that any power invoked on the right is instantly echoed on the left and so required to survive the application or else be revoked.  If the use of a given power of inquiry, working from the right and serving in the role of an operator, leads to a prospective description of inquiry, worked on the left from the role of an operand to the role of a result, and if the proffered characterization of inquiry is found to be out of accord with significant instances of its actual practice, then either the depiction of inquiry, as it is mediately improvised in progress, or the performance of inquiry, as it is actually conducted in practice, can turn out to be at fault.

If one considers the formula that characterizes an inquiry into inquiry, <math>y_0 = y \cdot y,\!</math> and examines the term <math>y \cdot y\!</math> that factors <math>y_0\!</math> along the lines of an ostensible self-application, it is evident that any power invoked on the right is instantly echoed on the left and so required to survive the application or else be revoked.  If the use of a given power of inquiry, working from the right and serving in the role of an operator, leads to a prospective description of inquiry, worked on the left from the role of an operand to the role of a result, and if the proffered characterization of inquiry is found to be out of accord with significant instances of its actual practice, then either the depiction of inquiry, as it is mediately improvised in progress, or the performance of inquiry, as it is actually conducted in practice, can turn out to be at fault.

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Returning to the formula of an inquiry into inquiry, <math>y_0 = y \cdot y,</math> it is possible to derive a few of its consequences for the character of the operation that is to be called &ldquo;reflection&rdquo;.  In general, a formula like <math>f = g \cdot h</math> constitutes a movement of conceptual reorganization, one whose resultant syntactic structure may or may not reflect an objective form of being, that is, an aspect of structure in the being that constitutes its object.  If there is a similarity of structure to be found between the formula and the object, then one has what is called an ''iconic formula'', but this is not always the case, and even this special situation requires the proper interpretation to tell in exactly what respect the form of the sign and the form of the object are alike.  Whatever the case, the role of the formula as a sign should not be confused with the role of the object in reality, no matter how similar their forms may be.

Returning to the formula of an inquiry into inquiry, <math>y_0 = y \cdot y,\!</math> it is possible to derive a few of its consequences for the character of the operation that is to be called &ldquo;reflection&rdquo;.  In general, a formula like <math>f = g \cdot h\!</math> constitutes a movement of conceptual reorganization, one whose resultant syntactic structure may or may not reflect an objective form of being, that is, an aspect of structure in the being that constitutes its object.  If there is a similarity of structure to be found between the formula and the object, then one has what is called an ''iconic formula'', but this is not always the case, and even this special situation requires the proper interpretation to tell in exactly what respect the form of the sign and the form of the object are alike.  Whatever the case, the role of the formula as a sign should not be confused with the role of the object in reality, no matter how similar their forms may be.

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If one reads the form <math>y \cdot y</math> according to the convention adopted, where a latently but actively instrumentalized inquiry on the right applies to a patently but patiently objectified inquiry on the left, almost as if they were two distinct agencies, faculties, or processes, then it is clear that an inquiry into inquiry can begin with little more than a nominal object, taking the name of &ldquo;inquiry&rdquo; in its sights to yield a clue in name only, while it can reserve all the power of an established capacity for inquiry to conduct its review, of which no account, no prescribed code, nor any catalog of procedure has to be given at the outset of its investigation.

If one reads the form <math>y \cdot y\!</math> according to the convention adopted, where a latently but actively instrumentalized inquiry on the right applies to a patently but patiently objectified inquiry on the left, almost as if they were two distinct agencies, faculties, or processes, then it is clear that an inquiry into inquiry can begin with little more than a nominal object, taking the name of &ldquo;inquiry&rdquo; in its sights to yield a clue in name only, while it can reserve all the power of an established capacity for inquiry to conduct its review, of which no account, no prescribed code, nor any catalog of procedure has to be given at the outset of its investigation.

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But it is important to remember that the full intention of this factious formulation is more analogous to an interpretive doubling of vision, an amplification of resolving power and a coordination of perspectives, than it is to an objective division of being, a substantial disconnection of essentials or a disintegration of being.  Even when the factions of the term <math>y \cdot y</math> are conceived in practice to be implemented by substantially different parts of the same agency, constitutionally they embody but a single power.

But it is important to remember that the full intention of this factious formulation is more analogous to an interpretive doubling of vision, an amplification of resolving power and a coordination of perspectives, than it is to an objective division of being, a substantial disconnection of essentials or a disintegration of being.  Even when the factions of the term <math>y \cdot y\!</math> are conceived in practice to be implemented by substantially different parts of the same agency, constitutionally they embody but a single power.

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The form of inquiry into inquiry, <math>y \cdot y,</math> requires that any power assumed on the part of the right is open to be indicted on the part of the left.  This entails that any power arrogated for the ends of inquiry has to be given a name, not only under which it is invoked as an executive power, but also by which it is entered on the agenda of issues to inquire into, and finally through which it is indicted for submission to all the powers of inquiry that be.  This combination of ''appellation'' and ''supplication'', or ''nomination for'' and ''submission to'' the jurisdiction of a reflexive application, makes up a large part of what is usually called ''reflection''.

The form of inquiry into inquiry, <math>y \cdot y,\!</math> requires that any power assumed on the part of the right is open to be indicted on the part of the left.  This entails that any power arrogated for the ends of inquiry has to be given a name, not only under which it is invoked as an executive power, but also by which it is entered on the agenda of issues to inquire into, and finally through which it is indicted for submission to all the powers of inquiry that be.  This combination of ''appellation'' and ''supplication'', or ''nomination for'' and ''submission to'' the jurisdiction of a reflexive application, makes up a large part of what is usually called ''reflection''.

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