Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Part 5 - Revision history

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 With any system of logic, at least, that does not propose a purely syntactic rationale for itself, it is necessary to draw a distinction between the logical object that is denoted, expressed, or represented in thinking and the logical sign that denotes, expresses, or represents it.  Often one uses the contrast between ''proposition'' and ''expression'' or the shade of difference between ''statement'' and ''sentence'' to convey the distinction between the logical object signified and the syntactic assemblage that signifies it.  Another option is to let the division lie between a ''position'' and a ''proposition'', with the suggestion being that the function of a symbolic proposition is to indicate indifferently a plurality of logical positions.  In accord with my personal preference, I use the term ''proposition'' ambiguously, expecting context to resolve the question, and resorting to the term ''expression'' when it does not.
-'''Point 28.''' &nbsp;&nbsp;&nbsp; Adequate reasoning about the propositional constitution or the sentential representation of POVs and PODs requires a logical system that can work with ''higher order propositions'' (HOPs).
+'''Point 28.'''
 '''Point 29.'''

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 [[Category:Artificial Intelligence]]

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 Expressed in abstract fashion, the injunction of a reflective capacity and the injunction of a capacity limitation are recognized to impinge on each other in a way that brings to light a number of additional issues.  Expressed in more concrete detail, the experiential instances that lead to the formation of these two points in the first place, as organizing poles of topics explicitly noticed, and that continue to surround their particular arrangements, &hellip;
-'''Point 11.''' &nbsp;&nbsp;&nbsp; Computational models of intelligent agents are limited to the consideration of &ldquo;finitely informed constructions and computations&rdquo;, or as I more affectionately call them, &ldquo;finitely informed creatures&rdquo; (FICs).
+'''Point 11.''' &nbsp;&nbsp;&nbsp; Computational models of intelligent agents are limited to the consideration of &ldquo;finitely informed constructions and computations&rdquo;, or as I more affectionately call them, ''finitely informed creatures'' (FICs).
 This point arises as a specialization of the point about capacity limits, where the discussion is restricted to the kinds of interpretive agents and the models of interpretive faculties that are available in a computational framework.
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 As a minimum requirement, a capacity for reflection implies an ability to generate names for the elements, processes, and principles of thought.  Assuming the tenet of pragmatism that all thought takes place in signs, this is tantamount to having signs for signs, signs for sign processes, and signs for sign relations.  Further, each higher order sign that is generated in a process of reflection is required to take its place and to find its meaning within a correspondingly higher order sign relation.
-In this connection, the designation &ldquo;higher order&rdquo; (HO) can be used as a generic adjective to describe a sign of any object whose nature it is to involve signs as a part of its being.  The use of this adjective is subject to extension in natural ways to describe not only entire classes of signs but also the kinds of sign relations that involve them.
+In this connection, the designation ''higher order'' (HO) can be used as a generic adjective to describe a sign of any object whose nature it is to involve signs as a part of its being.  The use of this adjective is subject to extension in natural ways to describe not only entire classes of signs but also the kinds of sign relations that involve them.
 In order to reflect on signs themselves, it is necessary to have signs for signs, a necessary supply of which can be generated by quotation.  But reflection on sign processes requires a much larger supply of signs.  Initially, it requires a HO sign for each sign transition that actually occurs, that is, a name for each ordered pair of signs that is observed.  Eventually, it requires a HO sign for each sign sequence that actually appears in experience, that is, a name for each <math>k\!</math>-tuple of signs seen.  And reflection on sign relations requires an even larger stock of signs.  It requires, initially, a HO sign for each sign transaction of the form <math>(o, s, i)\!</math> that is observed in experience and, ultimately, a HO sign for each sign relation that is encountered in experience or contemplated in a hypothetical situation.
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 The combinatorial explosion engendered by reflection impinges on the capacity limitations of a FIC with such an impact that neither the standpoints of &ldquo;naive empiricism&rdquo; or &ldquo;naive intuitionism&rdquo; can continue to support viable forms of inquiry.
-This is what makes the mediation of a &ldquo;higher order hypothesis&rdquo; (HOH), a hypothesis about the qualifications of a hypothesis, or a hypothesis about what can count as a hypothesis, so essential to the life of a FIC.
+This is what makes the mediation of a ''higher order hypothesis'' (HOH), a hypothesis about the qualifications of a hypothesis, or a hypothesis about what can count as a hypothesis, so essential to the life of a FIC.
 The process of generating signs that refer to things already signs is incited by a syntactic operation that is commonly called a &ldquo;quotation&rdquo;.  Strictly speaking, the descriptive term &ldquo;quotation&rdquo; refers to generic class of syntactic functions, each of which maps one order of signs into the next higher order of signs.  A proper form of quotation function is required to map signs in a one to one or &ldquo;injective&rdquo; fashion, and thus associates each element of its source domain with a HO sign that denotes it and it alone.  In short, a quotation produces a unique &ldquo;name&rdquo; or a distinctive &ldquo;number&rdquo; to index each piece its source material.
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 But a level of genuine reflection on sign processes and sign relations exceeds the generative capacity of mere quotation.
-'''Point 13.''' &nbsp;&nbsp;&nbsp; A &ldquo;finitely informed creature&rdquo; (FIC), if it is reflective up to the point that it reflects on its own nature as such, crosses a singular threshold of reflection, whereupon it not only obeys its own capacity limitations, as it instinctively and necessarily must, but also observes and reflects on their character.
+'''Point 13.''' &nbsp;&nbsp;&nbsp; A ''finitely informed creature'' (FIC), if it is reflective up to the point that it reflects on its own nature as such, crosses a singular threshold of reflection, whereupon it not only obeys its own capacity limitations, as it instinctively and necessarily must, but also observes and reflects on their character.
 '''Point 14.''' &nbsp;&nbsp;&nbsp; Higher order sign relations tax the pragmatic resources of an interpretive agent to such a severe extent that they impinge on the practical limits of its representational capacity and computational ability.
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 '''Point 24.''' &nbsp;&nbsp;&nbsp; Reflective projects being partial, their refractory parts are likely to remain partial to their outward projections.
-An &ldquo;unreflective framework&rdquo; (UF), if it does not devolve into a condition of total confusion, and thus deserves to be called a framework at all, ordinarily maintains a clear separation between the objective and the interpretive parts of its organization.  This pragmatic division of labor coincides with a substantive distinction that is ordained to exist between the object system that is subject to observation or interpretation and the agent system that observes or interprets it.
+An ''unreflective framework'' (UF), if it does not devolve into a condition of total confusion, and thus deserves to be called a framework at all, ordinarily maintains a clear separation between the objective and the interpretive parts of its organization.  This pragmatic division of labor coincides with a substantive distinction that is ordained to exist between the object system that is subject to observation or interpretation and the agent system that observes or interprets it.
 But the goal of reflection is to make one's own conduct an object among other objects, something that can be critically evaluated as one choice among many and subsequently amended if found wanting.  In this aim a realistic project of reflection never sees more than partial success.  There is always a refractory residue of ongoing conduct that resists analysis and remains unreflected in any clear form of representation.  Thus, the actual effect of a reflective project is to represent only a part of one's interpretive conduct as a part of one's objective regard, in other words, to reconfigure a part of one's IF as a part of one's OF.

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 All in all, a RIF involves the close coordination of an OF and an IF, plus mechanisms for carrying out the so called ''reflective operations'' (ROs) that go to negotiate between the objective and the interpretive realms.  The work of ROing permits processes of interpretation, initially taking place largely in the IF and impinging on the OF only at isolated points, to be formalized and objectified, thereby becoming segments of the OF.  Taken over time the cumulative effect of this ROing motion gradually turns more and more of the IF into new sectors and layers of the OF.
-Point 27.
+'''Point 27.'''
 There is a portion of reasoning that consists in drawing distinctions, signifying the features thereby distinguished by means of logical terms, recognizing constraints on the conjoint occurrences of these features, expressing these constraints in the form of logical premisses, and then drawing the implications of these premisses as the occasion warrants.  This part of logic, in its formalizable aspects, is generally referred to as ''propositional calculus'' (PropC), ''sentential logic'' (SL), or sometimes as ''zeroth order logic'' (ZOL).
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 With any system of logic, at least, that does not propose a purely syntactic rationale for itself, it is necessary to draw a distinction between the logical object that is denoted, expressed, or represented in thinking and the logical sign that denotes, expresses, or represents it.  Often one uses the contrast between ''proposition'' and ''expression'' or the shade of difference between ''statement'' and ''sentence'' to convey the distinction between the logical object signified and the syntactic assemblage that signifies it.  Another option is to let the division lie between a ''position'' and a ''proposition'', with the suggestion being that the function of a symbolic proposition is to indicate indifferently a plurality of logical positions.  In accord with my personal preference, I use the term ''proposition'' ambiguously, expecting context to resolve the question, and resorting to the term ''expression'' when it does not.
-<pre>
+'''Point 28.''' &nbsp;&nbsp;&nbsp; Adequate reasoning about the propositional constitution or the sentential representation of POVs and PODs requires a logical system that can work with ''higher order propositions'' (HOPs).
-Point 29.
+'''Point 29.'''
-Finally, interlaced with the structures of the OF and the IF, there is a need for a structure that I call a "dynamic evaluative framework" (DEF).  This is intended to isolate the twin aspects of process and purpose that are observable on either side of the objective interpretive divide and to assist in formalizing the graded notions of directed change that are able to be actualized in the medium of a RIF.
+Finally, interlaced with the structures of the OF and the IF, there is a need for a structure that I call a ''dynamic evaluative framework'' (DEF).  This is intended to isolate the twin aspects of process and purpose that are observable on either side of the objective interpretive divide and to assist in formalizing the graded notions of directed change that are able to be actualized in the medium of a RIF.
 ----

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 All in all, a RIF involves the close coordination of an OF and an IF, plus mechanisms for carrying out the so called ''reflective operations'' (ROs) that go to negotiate between the objective and the interpretive realms.  The work of ROing permits processes of interpretation, initially taking place largely in the IF and impinging on the OF only at isolated points, to be formalized and objectified, thereby becoming segments of the OF.  Taken over time the cumulative effect of this ROing motion gradually turns more and more of the IF into new sectors and layers of the OF.
 Point 27.
-There is a portion of reasoning that consists in drawing distinctions, signifying the features thereby distinguished by means of logical terms, recognizing constraints on the conjoint occurrences of these features, expressing these constraints in the form of logical premisses, and then drawing the implications of these premisses as the occasion warrants.  This part of logic, in its formalizable aspects, is generally referred to as "propositional calculus" (PropC), "sentential logic" (SL), or sometimes as "zeroth order logic" (ZOL).
+There is a portion of reasoning that consists in drawing distinctions, signifying the features thereby distinguished by means of logical terms, recognizing constraints on the conjoint occurrences of these features, expressing these constraints in the form of logical premisses, and then drawing the implications of these premisses as the occasion warrants.  This part of logic, in its formalizable aspects, is generally referred to as ''propositional calculus'' (PropC), ''sentential logic'' (SL), or sometimes as ''zeroth order logic'' (ZOL).
-With any system of logic, at least, that does not propose a purely syntactic rationale for itself, it is necessary to draw a distinction between the logical object that is denoted, expressed, or represented in thinking and the logical sign that denotes, expresses, or represents it.  Often one uses the contrast between "proposition" and "expression" or the shade of difference between "statement" and "sentence" to convey the distinction between the logical object signified and the syntactic assemblage that signifies it.  Another option is to let the division lie between a "position" and a "proposition", with the suggestion being that the function of a symbolic proposition is to indicate indifferently a plurality of logical positions.  In accord with my personal preference, I use the term "proposition" ambiguously, expecting context to resolve the question, and resorting to the term "expression" when it does not.
+With any system of logic, at least, that does not propose a purely syntactic rationale for itself, it is necessary to draw a distinction between the logical object that is denoted, expressed, or represented in thinking and the logical sign that denotes, expresses, or represents it.  Often one uses the contrast between ''proposition'' and ''expression'' or the shade of difference between ''statement'' and ''sentence'' to convey the distinction between the logical object signified and the syntactic assemblage that signifies it.  Another option is to let the division lie between a ''position'' and a ''proposition'', with the suggestion being that the function of a symbolic proposition is to indicate indifferently a plurality of logical positions.  In accord with my personal preference, I use the term ''proposition'' ambiguously, expecting context to resolve the question, and resorting to the term ''expression'' when it does not.
 Point 28.	Adequate reasoning about the propositional constitution or the sentential representatation of POV's and POD's requires a logical system that can work with "higher order propositions" (HOP's).

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 In keeping with the spirit of a discussion based on concrete examples, the RIF to be improvised here is restrained to the scale of a minimal IF that can reflect on the scene of A and B, in this case, synthesizing a portion of the OFs and IFs suggested by the sign relations <math>A\!</math> and <math>B\!</math> into an integrated SOI.  While I do not plan to specify the additional constraints that would be needed to determine this RIF uniquely, even to say whether it is finite or infinite, it forms a convenient reference point for the rest of this section to designate the purported ideal as ''the RIF generated by <math>A\!</math> and <math>B\!</math>'' and to notate it as <math>\operatorname{RIF}(A, B).\!</math>
-<pre>
+In accord with the customary figure of speech, a RIF can be personified in the agency of a &ldquo;reflective interpreter&rdquo; that possesses the faculties to carry out its actions, and this agent is in turn characterized as the localized representative of a suitably reflective and situated process of interpretation.
-A reflective interpreter needs a capacity for referring to its own role in the process of interpretation, for conceptualizing each transition from sign to interpretant sign as occurring within a context of alternatives, and for noticing that each option has a potentially distinctive value with respect to a prevailing object or objective.  "Capacity", as used in this connection, is a word with both structural and functional connotations.  It implies the structural capacity that is required to articulate, record, and maintain data about observable forms of interpretive conduct, and it involves the functional capacity that is demanded to create and exploit this data, in effect, constituting a higher order of interpretive activity.
+A reflective interpreter needs a capacity for referring to its own role in the process of interpretation, for conceptualizing each transition from sign to interpretant sign as occurring within a context of alternatives, and for noticing that each option has a potentially distinctive value with respect to a prevailing object or objective.  ''Capacity'', as used in this connection, is a word with both structural and functional connotations.  It implies the structural capacity that is required to articulate, record, and maintain data about observable forms of interpretive conduct, and it involves the functional capacity that is demanded to create and exploit this data, in effect, constituting a higher order of interpretive activity.
 If one tries to understand the conduct of a reflective interpreter as a process of interpretation there are a number of questions that arise.  How can anything so ongoing as a process of interpretation refer to an object, and how can anything so fleeting as a process of interpretation be referred to as an object?
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 A process that refers to itself is not like a set that collects itself, or a collection that would enroll itself among its own elements, even if some attempts to process the reference and to lay it out in a literal account do try to dissect and explain it as such.  A sign that is elemental to a universe, perhaps by means of which one seeks to explain the universe, does not in fact collect, dominate, or encase the entire universe simply by referring to it, even if some interpretive interloper, at the risk of vitiating the whole account, is tempted to explain the elementary part in terms of the complex totality.
-One reason for introducing the distinction between OF's and IF's into the present discussion is to keep track of the complex relationships between object domains and sign domains, between the constitutions of objects and the constitutions of signs.  It is a frequent practice in mathematics to blur this distinction, often saying that an object is constituted as a set of further objects when one really means that the sign or information one has about the object is constituted as a set of further signs or further informations about the object, all of which can refer to further objects, but not always the sorts of objects that are literally intended as elementary constituents of the original object.  Furthermore, each use of the directive "further" in this description marks a place where a suitably reflective interpreter ought to ask whether "further" implies "simpler" or merely "other", and in turn whether "other" means essentially other or only otherwise appearing.
+One reason for introducing the distinction between OFs and IFs into the present discussion is to keep track of the complex relationships between object domains and sign domains, between the constitutions of objects and the constitutions of signs.  It is a frequent practice in mathematics to blur this distinction, often saying that an object is constituted as a set of further objects when one really means that the sign or information one has about the object is constituted as a set of further signs or further informations about the object, all of which can refer to further objects, but not always the sorts of objects that are literally intended as elementary constituents of the original object.  Furthermore, each use of the directive ''further'' in this description marks a place where a suitably reflective interpreter ought to ask whether ''further'' implies ''simpler'' or merely ''other'', and in turn whether ''other'' means essentially other or only otherwise appearing.
-But the distinction between object and sign, however important, is still a pragmatic distinction, involving a thing's use in a particular role, and not an essential distinction, fixing a thing's prior and eternal nature.  Of course, it can turn out that some objects will never serve as signs and that some signs will never be observed as objects, but these types of eventuality involve empirical questions and contingent facts, and their actualization depends on the kinds of circumstances that have to be discovered after the fact rather than dictated a priori.
+But the distinction between object and sign, however important, is still a pragmatic distinction, involving a thing's use in a particular role, and not an essential distinction, fixing a thing's prior and eternal nature.  Of course, it can turn out that some objects will never serve as signs and that some signs will never be observed as objects, but these types of eventuality involve empirical questions and contingent facts, and their actualization depends on the kinds of circumstances that have to be discovered after the fact rather than dictated ''a priori''.
-The construction of a RIF forces the discussion to a point where the OF's and IF's and the relationships between them suddenly become much more complex, and where confusion can arise precisely from the fact that the purpose of a RIF is to convert an IF into the sort of thing that can be referred to and reflected on as an object.  Developments like these make it all the more necessary to understand the exact character of the distinction between OF's and IF's.  In a complex IF signs do participate in constitutional relationships, with complex signs being constructed out of simpler signs.  But the relations involved in denotation and connotation are not limited to constitutional linkages of this sort, and thus they cannot be expected to generate by themselves the necessary sorts of analytic and synthetic hierarchies.
+The construction of a RIF forces the discussion to a point where the OFs and IFs and the relationships between them suddenly become much more complex, and where confusion can arise precisely from the fact that the purpose of a RIF is to convert an IF into the sort of thing that can be referred to and reflected on as an object.  Developments like these make it all the more necessary to understand the exact character of the distinction between OFs and IFs.  In a complex IF signs do participate in constitutional relationships, with complex signs being constructed out of simpler signs.  But the relations involved in denotation and connotation are not limited to constitutional linkages of this sort, and thus they cannot be expected to generate by themselves the necessary sorts of analytic and synthetic hierarchies.
-All in all, a RIF involves the close coordination of an OF and an IF, plus mechanisms for carrying out the so called "reflective operations" (RO's) that go to negotiate between the objective and the interpretive realms.  The work of ROing permits processes of interpretation, initially taking place largely in the IF and impinging on the OF only at isolated points, to be formalized and objectified, thereby becoming segments of the OF.  Taken over time the cumulative effect of this ROing motion gradually turns more and more of the IF into new sectors and layers of the OF.
+All in all, a RIF involves the close coordination of an OF and an IF, plus mechanisms for carrying out the so called ''reflective operations'' (ROs) that go to negotiate between the objective and the interpretive realms.  The work of ROing permits processes of interpretation, initially taking place largely in the IF and impinging on the OF only at isolated points, to be formalized and objectified, thereby becoming segments of the OF.  Taken over time the cumulative effect of this ROing motion gradually turns more and more of the IF into new sectors and layers of the OF.
 Point 27.

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 There is a piece of terminology that is often useful in this connection.  In set-theoretic contexts, either one of the phrases ''X collects Y'' or ''X encases Y'' can be used to mean the same thing as ''Y'' &isin; ''X''.  These formulations can be taken as abbreviated ways of saying that ''X'' enumerates ''Y'' among its cases.  Thus, they express the converse of the membership relation but manage to avoid the ambiguity of the phrase ''X contains Y'', a form that would otherwise have to be qualified on each occasion of its use by specifying whether one means ''contains as an element'' or ''contains as a subset'', as the case may be.
 To wrap up the development of this reflective project in a single line:  When the mind's original effort to catch itself at work seizes on the inventions of set theory to encapsulate its speculations, the ensuing breed of self reification that comes from mingling an unbridled capacity for self referent expressions with an unchecked propensity for creating abstract objects gives rise to the generation of set theoretic paradoxes.  As a result, it is incumbent on me to show how the concretely limited kinds of constructions that I have in mind can avoid a similar excess and steer clear of the corresponding difficulties.
 If formalized, a RIF would be an IF that can properly, if only partially, refer to itself as an OF.  Thus, as formalized, a RIF amounts to both a reflexive and a recursive SOI, one that can refer to itself as an object, to the extent that any formal system can.  As a reflexive SOI, a RIF has a sign that refers to itself.  As a recursive SOI, a RIF has a character that can be determined by invoking the record of signs that it uses to refer to simpler versions and earlier developments of itself.
-But more than all this, in order to be genuinely reflective a RIF's consideration of itself as a situated form of activity must extend to the consideration of alternative selves.  This means that a RIF must have references to other SOI's, not only those that are continuous with the space of its own potential conduct and correlated to the course of its own form of activity, but also those that are discontinuous from and independent of its own way of being.
+But more than all this, in order to be genuinely reflective a RIF's consideration of itself as a situated form of activity must extend to the consideration of alternative selves.  This means that a RIF must have references to other SOIs, not only those that are continuous with the space of its own potential conduct and correlated to the course of its own form of activity, but also those that are discontinuous from and independent of its own way of being.
-In keeping with the spirit of a discussion based on concrete examples, the RIF to be improvised here is restrained to the scale of a minimal IF that can reflect on the scene of A and B, in this case, synthesizing a portion of the OF's and IF's suggested by the sign relations A and B into an integrated SOI.  While I do not plan to specify the additional constraints that would be needed to determine this RIF uniquely, even to say whether it is finite or infinite, it forms a convenient reference point for the rest of this section to designate the purported ideal as "the RIF generated by A and B" and to notate it as "RIF (A, B)".
+In keeping with the spirit of a discussion based on concrete examples, the RIF to be improvised here is restrained to the scale of a minimal IF that can reflect on the scene of A and B, in this case, synthesizing a portion of the OFs and IFs suggested by the sign relations <math>A\!</math> and <math>B\!</math> into an integrated SOI.  While I do not plan to specify the additional constraints that would be needed to determine this RIF uniquely, even to say whether it is finite or infinite, it forms a convenient reference point for the rest of this section to designate the purported ideal as ''the RIF generated by <math>A\!</math> and <math>B\!</math>'' and to notate it as <math>\operatorname{RIF}(A, B).\!</math>
 In accord with the customary figure of speech, a RIF can be personified in the agency of a "reflective interpreter" that possesses the faculties to carry out its actions, and this agent is in turn characterized as the localized representative of a suitably reflective and situated process of interpretation.

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 Setting out initially on the positive track, I begin with the assumption that a RIF is a real possibility.  In order to conceive of a RIF being possible it is necessary to set aside a host of set theoretic difficulties that might be imagined to afflict any invocation of self referent themes.  No matter whether interpretation is presented in terms of a framework, a faculty, a process, a trajectory, or a hypostatic agent that is assumed to carry out its procedures, there is a problem about how anything so fleeting and so sweeping as an ongoing interpretation can refer to itself as a situated form of activity, in other words, as an objective system of interpretation that rests within a context of alternative interpretations.
-There is a piece of terminology that is often useful in this connection.  In set-theoretic contexts, either one of the phrases ''X collects Y''& or ''X encases Y'' can be used to mean the same thing as ''Y'' &isin; ''X''.  These formulations can be taken as abbreviated ways of saying that ''X'' enumerates ''Y'' among its cases.  Thus, they express the converse of the membership relation but manage to avoid the ambiguity of the phrase ''X contains Y'', a form that would otherwise have to be qualified on each occasion of its use by specifying whether one means ''contains as an element'' or ''contains as a subset'', as the case may be.
+There is a piece of terminology that is often useful in this connection.  In set-theoretic contexts, either one of the phrases ''X collects Y'' or ''X encases Y'' can be used to mean the same thing as ''Y'' &isin; ''X''.  These formulations can be taken as abbreviated ways of saying that ''X'' enumerates ''Y'' among its cases.  Thus, they express the converse of the membership relation but manage to avoid the ambiguity of the phrase ''X contains Y'', a form that would otherwise have to be qualified on each occasion of its use by specifying whether one means ''contains as an element'' or ''contains as a subset'', as the case may be.
 <pre>

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 Consequently, it must be possible to explain these apparent abilities in just one of two ways:  either to account for the faculties of reflection and selection by presenting a logical model of the processes involved, or else to dispel the illusion of each performance by showing what goes on in its place.  In either case, an inquiry into the virtues of critically reflective phenomena is called on to provide a plausible model for what is happening beneath the semblances of reflective and critical thought.  Whether the resulting resolution of a particular phenomenon preserves or dissolves its appearances is a matter that depends on the details of the case, and perhaps to a degree on personal taste.
-<pre>
+'''Point 26.''' &nbsp;&nbsp;&nbsp; This marks a branch point.  I tentatively assume that the apparent power of reflection is really more or less as it appears to be, at least, in the same spirit as it appears to be, and not some radically insidious self-deception arising on the part of its apparent agents.
 Setting out initially on the positive track, I begin with the assumption that a RIF is a real possibility.  In order to conceive of a RIF being possible it is necessary to set aside a host of set theoretic difficulties that might be imagined to afflict any invocation of self referent themes.  No matter whether interpretation is presented in terms of a framework, a faculty, a process, a trajectory, or a hypostatic agent that is assumed to carry out its procedures, there is a problem about how anything so fleeting and so sweeping as an ongoing interpretation can refer to itself as a situated form of activity, in other words, as an objective system of interpretation that rests within a context of alternative interpretations.
-There is a piece of terminology that is often useful in this connection.  In set theoretic contexts, I use either one of the phrases "X collects Y" or "X encases Y" to mean that Y C X.  These formulations can be taken as abbreviated ways of saying that "X enumerates Y among its cases".  Thus, they express the converse of the membership relation but manage to avoid the ambiguity of the phrase "X contains Y", a form that would otherwise have to be qualified on each occasion of its use by specifying whether one means "contains as an element" or "contains as a subset", as the case may be.
+There is a piece of terminology that is often useful in this connection.  In set-theoretic contexts, either one of the phrases <math>{}^{\backprime\backprime}X ~\operatorname{collects}~ Y{}^{\prime\prime}\!</math> or <math>{}^{\backprime\backprime}X ~\operatorname{encases}~ Y{}^{\prime\prime}\!</math> will be used to mean that <math>Y \in X.\!</math>  These formulations can be taken as abbreviated ways of saying that <math>{}^{\backprime\backprime}X ~\operatorname{enumerates}~ Y ~\operatorname{among~its~cases}{}^{\prime\prime}.\!</math>  Thus, they express the converse of the membership relation but manage to avoid the ambiguity of the phrase <math>{}^{\backprime\backprime}X ~\operatorname{contains}~ Y{}^{\prime\prime},\!</math> a form that would otherwise have to be qualified on each occasion of its use by specifying whether one means &ldquo;contains as an element&rdquo; or &ldquo;contains as a subset&rdquo;, as the case may be.
 To wrap up the development of this reflective project in a single line:  When the mind's original effort to catch itself at work seizes on the inventions of set theory to encapsulate its speculations, the ensuing breed of self reification that comes from mingling an unbridled capacity for self referent expressions with an unchecked propensity for creating abstract objects gives rise to the generation of set theoretic paradoxes.  As a result, it is incumbent on me to show how the concretely limited kinds of constructions that I have in mind can avoid a similar excess and steer clear of the corresponding difficulties.