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==6. Reflective Interpretive Frameworks==
 
==6. Reflective Interpretive Frameworks==
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<pre>
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??? The rest of this Section ???, continuing the discussion of formalization in terms of concrete examples and extending over the next 50 ??? Subsections ???, details the construction of a "reflective interpretive framework" (RIF).  This is a special type of sign theoretic setting, illustrated in the present case as based on the sign relations A and B, but intended more generally to constitute a fully developed environment of objective and interpretive resources, in the likes of which an "inquiry into inquiry" can reasonably be expected to find its home.
 +
 +
This Subdivision of the text begins by presenting an outline of the developments ahead, working through the motivation, the construction, and the application of a RIF that is broad enough to moderate the dialogue of A and B.
 +
 +
The first fifteen Sections (§§ 1 15) deal with a selection of preliminary topics and techniques that are involved in approaching the construction of a RIF.  The topics of these sections are described in greater detail as follows:
 +
 +
The first section (S 1) takes up the phenomenology of reflection.  The next three sections (§§ 2 4) are allotted to surveying the site of the planned construction, presenting it from three different points of view.  An introductory discussion (S 2) presents the main ideas that lead up to the genesis of a RIF.  These ideas are treated at first acquaintance in an informal manner, located within a broader cultural context, and put in relation to the ways that intelligent agents can come to develop characterictic belief systems and communal perspectives on the world.  The next section (S 3) points out a specialized mechanism that serves to make inobvious types of observation of a reflective character.  The last section (S 4) takes steps to formalize the concepts of a "point of view" (POV) and a "point of development" (POD).  These ideas characterize the outlooks, perspectives, world views, and other systems of belief, knowledge, or opinion that are employed by agents of inquiry, with especial regard to the ways that these outlooks develop over time.
 +
 +
A further discussion (S 5), in preparation for the task of reflection, identifies three styles of linguistic usage that deploy increasing grades of formalization in their approaches to any given subject matter.
 +
 +
In the next three sections (§§ 6 8), the features that distinguish each style of usage are taken up individually and elaborated in detail.  This is done by presenting the basic ideas of three theoretical subjects that develop under the corresponding points of view and that exemplify their respective ideals.  The next three sections (§§ 9 11) take up the classes of higher order sign relations that play an important role in reflexive inquiries and then apply the battery of concepts arising with higher order sign relations to an example that anticipates many features of a realistic interpreter.  In the light of the experience gained with the foregoing styles and subjects, the next three sections (§§ 12 14) are able to take up important issues regarding the status of theoretical entities that are needed in this work.
 +
 +
Finally (S 15), the relevance of these styles, subjects, and issues is made concrete by bringing their various considerations to bear on a single example of a formal system that serves to integrate their concerns, namely, "propositional calculus".
 +
 +
A point by point outline follows:
 +
 +
§ 1. An approach to the phenomenology of reflective experience, as it bears on the conduct of reflective activity, is given its first explicit discussion.
 +
 +
§ 2. The main ideas leading up to the development of a RIF are presented, starting from the bare necessity of applying inquiry to itself.  I introduce the idea of a "point of view" (POV) in an informal way, as it arises from natural considerations about the relationship of an immanent "system of interpretation" (SOI) to a generated "text of inquiry" (TOI).  In this connection, I pursue the idea of a "point of development" (POD), that captures a POV at a particular moment of its own proper time.
 +
 +
§ 3. A Projective POV
 +
 +
§ 4. The idea of a POV, as manifested from moment to moment in a series of POD's, is taken up in greater detail. 
 +
 +
A formalization for talking about a diversity of POV's and their development through time is introduced and its consequences explored.  Finally, this formalization is applied to an issue of pressing concern for the present project, namely, the status of the distinction between dynamic and symbolic aspects of intelligent systems.
 +
 +
§ 5. The symbolic forms employed in the construction of a RIF are found at the nexus of several different interpretive influences.  This section picks out three distinctive styles of usage that this work needs to draw on throughout its progress, usually without explicit notice, and discusses their relationships to each other in general terms.  These three styles of usage, distinguished according to whether they encourage an "ordinary language" (OL), a "formal language" (FL), or a "computational language" (CL) approach, have their relevant properties illustrated in the next three sections (§§ 26 28), each style being exemplified by a theoretical subject that thrives under its guidance.
 +
 +
§ 6. For ease of reference, the basic ideas of group theory used in this project are separated out and presented in this section.  Throughout this work as a whole, the subject of group theory serves in both illustrative and instrumental roles, providing, besides a rough stock of exemplary materials to work on, a ready array of precision tools to work with.
 +
 +
Group theory, as a methodological subject, is used to illustrate the "mathematical language" (ML) approach, which ordinarily takes it for granted that signs denote something, if not always the objects intended.  It is therefore recognizable as a special case of the OL style of usage.
 +
 +
To the basic assumption of the OL approach the ML style adds only the faith that every object one desires to name has a unique proper name to do it with, and thus that all the various expressions for an object can be traded duty free and without much ado for a suitably compact name to denote it.  This means that the otherwise considerable work of practical computation, that is needed to associate arbitrarily obscure expressions with their clearest possible representatives, is not taken seriously as a feature that deserves theoretical attention, and is thus ignored as a factor of theoretical concern.  This is appropriate to the mathematical level, which abstracts away from pragmatic factors and is intended precisely to do so.
 +
 +
More instrumentally to the aims of this investigation, and not entirely accidentally, group theory is one of the most adaptable of mathematical tools that can be used to understand the relation between general forms and particular instantiations, in other words, the relationship between abstract commonalities and their concrete diversities.
 +
 +
§ 7. The basic notions of formal language theory are presented.  Not surprisingly, formal language theory is used to illustrate the FL style of usage.  Instrumentally, it is one of the most powerful tools available to clear away both the understandable confusions and the unjustifiable presuppositions of informal discourse.
 +
 +
§ 8. The notion of computation that makes sense in this setting is one of a process that replaces an arbitrary sign with a better sign of the same object.  In other words, computation is an interpretive process that improves the indications of intentions.  To deal with computational processes it is necessary to extend the pragmatic theory of signs in a couple of new but coordinated directions.  To the basic conception of a sign relation is added a notion of progress, which implies a notion of process together with a notion of quality.
 +
 +
§ 9. This section introduces "higher order" sign relations, which are used to formalize the process of reflection on interpretation.  The discussion is approaching a point where multiple levels of signs are becoming necessary, mainly for referring to previous levels of signs as the objects of an extended sign relation, and thereby enabling a process of reflection on interpretive conduct.  To begin dealing with this issue, I take advantage of a second look at A and B to introduce the use of "raised angle brackets" (< >), also called "supercilia" or "arches", as quotation marks.  Ordinary quotation marks (" ") have the disadvantage, for formal purposes, of being used informally for many different tasks.  To get around this obstacle, I use the "arch" operator to formalize one specific function of quotation marks in a computational context, namely, to create distinctive names for syntactic expressions, or what amounts to the same thing, to signify the generation of their godel numbers.
 +
 +
§ 10. Returning to the sign relations A and B, various kinds of HO signs are exemplified by considering a selection of HO sign relations that are based on these two examples.
 +
 +
§ 11. In this section the tools that come with the theory of higher order sign relations are applied to an illustrative exercise, roughing out the shape of a complex form of interpreter.
 +
 +
The next three sections (§§ 32 34) discuss how the identified styles of usage bear on three important issues in the usage of a technical language, namely, the respective theoretical statuses of "signs", "sets", and "variables".
 +
 +
§ 12. The Status of Signs
 +
 +
§ 13. The Status of Sets
 +
 +
§ 14. At this point the discussion touches on an topic, concerning the being of a so called "variable", that issues in many unanswered questions.  Although this worry over the nature and use of a variable may seem like a trivial matter, it is not.  It needs to be remembered that the first adequate accounts of formal computation, Schonfinkel's combinator calculus and Church's lambda calculus, both developed out of programmes intended to clarify the concept of a variable, indeed, even to the point of eliminating it altogether as a primitive notion from the basis of mathematical logic (van Heijenoort, 355 366).
 +
 +
The pragmatic theory of sign relations has a part of its purpose in addressing these same questions about the natural utility of variables, and even though its application to computation has not enjoyed the same level of development as these other models, it promises in good time to have a broader scope.  Later, I will illustrate its potential by examining a form of the combinator calculus from a sign relational point of view.
 +
 +
§ 15. There is an order of logical reasoning that is typically described as "propositional" or "sentential" and represented in a type of formal system that is commonly known as a "propositional calculus" or a "sentential logic" (SL).  Any one of these calculi forms an interesting example of a formal language, one that can be used to illustrate all of the preceding issues of style and technique, but one that can also serve this inquiry in a more instrumental fashion.  This section presents the elements of a calculus for propositional logic that I described in earlier work (Awbrey, 1989 & 1994).  The imminent use of this calculus is to construct and analyze logical representations of sign relations, and the treatment here focuses on the concepts and notation that are most relevant to this task.
 +
 +
The next four sections (§§ 16 19) treat the theme of self reference that is invoked in the overture to a RIF.  To inspire confidence in the feasibility and the utility of well chosen reflective constructions and to allay a suspicion of self reference in general, it is useful to survey the varieties of self reference that arise in this work and to distinguish the forms of circular referrals that are likely to vitiate consistent reasoning from those that are relatively innocuous and even beneficial.
 +
 +
§ 16. Recursive Aspects
 +
 +
§ 17. Patterns of Self Reference
 +
 +
§ 18. Practical Intuitions
 +
 +
§ 19. Examples of Self Reference
 +
 +
The intertwined themes of logic and time will occupy center stage for the next eight sections (§§ 20 27).
 +
 +
§ 20. First, I discuss three distinct ways that the word "system" is used in this work, reflecting the variety of approaches, aspects, or perspectives that present themselves in dealing with what are often the same underlying objects in reality.
 +
 +
§ 21. There is a general set of situations where the task arises to "build a bridge" between significantly different types of representation.  In these situations, the problem is to translate between the signs and expressions of two formal systems that have radically different levels of interpretation, and to do it in a way that makes appropriate connections between diverse descriptions of the same objects.  More to the point of the present project, formal systems for mediating inquiry, if they are intended to remain viable in both empirical and theoretical uses, need the capacity to negotiate between an "extensional representation" (ER) and an "intensional representation" (IR) of the same domain of objects.  It turns out that a cardinal or pivotal issue in this connection is how to convert between ER's and IR's of the same objective domain, working all the while within the practical constraints of a computational medium and preserving the equivalence of information.  To illustrate the kinds of technical issues that are involved in these considerations, I bring them to bear on the topic of representing sign relations and their dyadic projections in various forms.
 +
 +
The next four sections (§§ 22-25) give examples of ER's and IR's, indicate the importance of forming a computational bridge between them, and discuss the conceptual and technical obstacles that will have to be faced in doing so.
 +
 +
§ 22. For ease of reference, this section collects previous materials that are relevant to discussing the ER's of the sign relations A and B, and explicitly details their dyadic projections.
 +
 +
§ 23. This section discusses a number of general issues that are associated with the IR's of sign relations.  Because of the great degree of freedom there is in selecting among the potentially relevant properties of any real object, especially when the context of relevance to the selection is not known in advance, there are many different ways, perhaps an indefinite multitude of ways, to represent the sign relations A and B in terms of salient properties of their elementary constituents.  In this connection, the next two sections explore a representative sample of these possibilities, and illustrate several different styles of approach that can be used in their presentation.
 +
 +
§ 24. A transitional case between ER's and IR's of sign relations is found in the concept of a "literal intensional representation" (LIR).
 +
 +
§ 25. A fully fledged IR is one that accomplishes some measure of analytic work, bringing to the point of salient notice a selected array of implicit and otherwise hidden features of its object.  This section presents a variety of these "analytic intensional representations" (AIR's) for the sign relations A and B.
 +
 +
Note for future reference.  The problem so naturally encountered here, due to the "embarassment of riches" that presents itself in choosing a suitable IR, and tracing its origin to the wealth of properties that any real object typically has, is a precursor to one of the deepest issues in the pragmatic theory of inquiry:  "the problem of abductive reasoning".  This topic will be discussed at several later stages of this investigation, where it typically involves the problem of choosing, among the manifold aspects of an objective phenomenon or a problematic objective, only the features that are:  (1) relevant to explaining a present fact, or (2) pertinent to achieving a current purpose.
 +
 +
§ 26. Differential Logic & Directed Graphs
 +
 +
§ 27. Differential Logic & Group Operations
 +
 +
§ 28. The Bridge:  From Obstruction to Opportunity
 +
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§ 29. Projects of Representation
 +
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§ 30. Connected, Integrated, Reflective Symbols
 +
 +
The next seven sections (§§ 31 37) are designed to incrementally motivate the idea that a language as simple as propositional calculus, remarkably enough, can be used to articulate significant properties of n place relations.  The course of the discussion will proceed as follows:
 +
 +
§ 31. First, I introduce concepts and notation designed to expand and generalize the orders of relations that are available to be discussed in an adequate fashion.
 +
 +
§ 32. Second, I elaborate a particular mode of abstraction, that is, a systematic strategy for generalizing the collections of formal objects that are initially given to discussion.  This dimension of abstraction or direction of generalization will be described under the thematic heading of "partiality".
 +
 +
§ 33. Third, I present an alternative approach to the issue of "degenerate", "defective", or "fragmentary" n place relations, proceeding by way of generalized objects known as "n place relational complexes".  Illustrating these ideas with respect to their bearing on sign relations the discussion arrives at a notion of "sign relational complexes", or "sign complexes".
 +
 +
In the next three sections (§§ 34 36) I consider a collection of "identification tasks" for n place relations.  Of particular interest is the extent to which the determination of an n place relation is constrained by a particular type of data, namely, by the specification of lower arity relations that occur as its projections.  This topic is often treated as a question about a relation's "reducibility" or "irreduciblity" with respect to its projections.  For instance, if the identity of an n place relation is completely determined by the data of its k place projections, then R is said to be "identifiable by", "reducible to", or "reconstructible from" its k place components, otherwise R is said to be "irreducible" with respect to its k place projections.
 +
 +
§ 34. First, I consider a number of set theoretic operations that can be utilized in discussing these "identification", "reducibility", or "reconstruction" questions.  Once a level of general discussion has been surveyed enough to make a start, these tools can be specialized and applied to concrete examples in the realm of sign relations and also applied in the neighborhood of closely associated triadic relations.
 +
 +
§ 35. This section considers the positive case of reducibility, presenting examples of triadic relations that can be reconstructed from their dyadic projections.  In fact, it happens that the sign relations A and B fall into this category of dyadically reducible triadic relations.
 +
 +
§ 36. This section considers the negative case of reduciblity, presenting examples of "irreducibly triadic relations", or triadic relations that cannot be reconstructed from their lower dimensional projections or "faces".
 +
 +
§ 37. Finally, the discussion culminates in an exposition of the so called "propositions as types" (PAT) analogy, outlining a formal system of "type expressions" or "type formulas" that bears a strong resemblance to propositional calculus.  Properly interpreted, the resulting "calculus of propositional types" (COPT) can be used as a language for talking about well formed types of n place relations.
 +
 +
§ 38. Considering the Source
 +
 +
§ 39. Prospective Indices : Pointers to Future Work
 +
 +
§ 40. Interlaced with the structural and reflective developments that go into the OF and the IF is a conceptual arrangement called the "dynamic evaluative framework" (DEF).  This utility works to isolate the aspects of process and purpose that are observable on either side of the objective interpretive divide and helps to organize the graded notions of directed change that can be actualized in the RIF.
 +
 +
§ 41. Elective and Motive Forces
 +
 +
§ 42. Sign Processes : A Start
 +
 +
§ 43. Reflective Extensions
 +
 +
§ 44. Reflections on Closure
 +
 +
§ 45. Intelligence => Critical Reflection
 +
 +
§ 46. Looking Ahead:  The Meta Issue
 +
 +
§ 47. Mutually Intelligible Codes
 +
 +
§ 48. Discourse Analysis : Ways and Means
 +
 +
§ 49. Combinations of Sign Relations
 +
 +
§ 50. Revisiting the Source
 +
</pre>
  
 
===6.1. The Phenomenology of Reflection===
 
===6.1. The Phenomenology of Reflection===

Revision as of 21:44, 13 August 2011

• Contents • Part 1 • Part 2 • Part 3 • Part 4 • Part 5 • Part 6 • Appendices • References • Document History •

6. Reflective Interpretive Frameworks

??? The rest of this Section ???, continuing the discussion of formalization in terms of concrete examples and extending over the next 50 ??? Subsections ???, details the construction of a "reflective interpretive framework" (RIF).  This is a special type of sign theoretic setting, illustrated in the present case as based on the sign relations A and B, but intended more generally to constitute a fully developed environment of objective and interpretive resources, in the likes of which an "inquiry into inquiry" can reasonably be expected to find its home.

This Subdivision of the text begins by presenting an outline of the developments ahead, working through the motivation, the construction, and the application of a RIF that is broad enough to moderate the dialogue of A and B.

The first fifteen Sections (§§ 1 15) deal with a selection of preliminary topics and techniques that are involved in approaching the construction of a RIF.  The topics of these sections are described in greater detail as follows:

The first section (S 1) takes up the phenomenology of reflection.  The next three sections (§§ 2 4) are allotted to surveying the site of the planned construction, presenting it from three different points of view.  An introductory discussion (S 2) presents the main ideas that lead up to the genesis of a RIF.  These ideas are treated at first acquaintance in an informal manner, located within a broader cultural context, and put in relation to the ways that intelligent agents can come to develop characterictic belief systems and communal perspectives on the world.  The next section (S 3) points out a specialized mechanism that serves to make inobvious types of observation of a reflective character.  The last section (S 4) takes steps to formalize the concepts of a "point of view" (POV) and a "point of development" (POD).  These ideas characterize the outlooks, perspectives, world views, and other systems of belief, knowledge, or opinion that are employed by agents of inquiry, with especial regard to the ways that these outlooks develop over time.

A further discussion (S 5), in preparation for the task of reflection, identifies three styles of linguistic usage that deploy increasing grades of formalization in their approaches to any given subject matter.

In the next three sections (§§ 6 8), the features that distinguish each style of usage are taken up individually and elaborated in detail.  This is done by presenting the basic ideas of three theoretical subjects that develop under the corresponding points of view and that exemplify their respective ideals.  The next three sections (§§ 9 11) take up the classes of higher order sign relations that play an important role in reflexive inquiries and then apply the battery of concepts arising with higher order sign relations to an example that anticipates many features of a realistic interpreter.  In the light of the experience gained with the foregoing styles and subjects, the next three sections (§§ 12 14) are able to take up important issues regarding the status of theoretical entities that are needed in this work.

Finally (S 15), the relevance of these styles, subjects, and issues is made concrete by bringing their various considerations to bear on a single example of a formal system that serves to integrate their concerns, namely, "propositional calculus".

A point by point outline follows:

§ 1.	An approach to the phenomenology of reflective experience, as it bears on the conduct of reflective activity, is given its first explicit discussion.

§ 2.	The main ideas leading up to the development of a RIF are presented, starting from the bare necessity of applying inquiry to itself.  I introduce the idea of a "point of view" (POV) in an informal way, as it arises from natural considerations about the relationship of an immanent "system of interpretation" (SOI) to a generated "text of inquiry" (TOI).  In this connection, I pursue the idea of a "point of development" (POD), that captures a POV at a particular moment of its own proper time.

§ 3.	A Projective POV

§ 4.	The idea of a POV, as manifested from moment to moment in a series of POD's, is taken up in greater detail.  

A formalization for talking about a diversity of POV's and their development through time is introduced and its consequences explored.  Finally, this formalization is applied to an issue of pressing concern for the present project, namely, the status of the distinction between dynamic and symbolic aspects of intelligent systems.

§ 5.	The symbolic forms employed in the construction of a RIF are found at the nexus of several different interpretive influences.  This section picks out three distinctive styles of usage that this work needs to draw on throughout its progress, usually without explicit notice, and discusses their relationships to each other in general terms.  These three styles of usage, distinguished according to whether they encourage an "ordinary language" (OL), a "formal language" (FL), or a "computational language" (CL) approach, have their relevant properties illustrated in the next three sections (§§ 26 28), each style being exemplified by a theoretical subject that thrives under its guidance.

§ 6.	For ease of reference, the basic ideas of group theory used in this project are separated out and presented in this section.  Throughout this work as a whole, the subject of group theory serves in both illustrative and instrumental roles, providing, besides a rough stock of exemplary materials to work on, a ready array of precision tools to work with.

Group theory, as a methodological subject, is used to illustrate the "mathematical language" (ML) approach, which ordinarily takes it for granted that signs denote something, if not always the objects intended.  It is therefore recognizable as a special case of the OL style of usage.

To the basic assumption of the OL approach the ML style adds only the faith that every object one desires to name has a unique proper name to do it with, and thus that all the various expressions for an object can be traded duty free and without much ado for a suitably compact name to denote it.  This means that the otherwise considerable work of practical computation, that is needed to associate arbitrarily obscure expressions with their clearest possible representatives, is not taken seriously as a feature that deserves theoretical attention, and is thus ignored as a factor of theoretical concern.  This is appropriate to the mathematical level, which abstracts away from pragmatic factors and is intended precisely to do so.

More instrumentally to the aims of this investigation, and not entirely accidentally, group theory is one of the most adaptable of mathematical tools that can be used to understand the relation between general forms and particular instantiations, in other words, the relationship between abstract commonalities and their concrete diversities.

§ 7.	The basic notions of formal language theory are presented.  Not surprisingly, formal language theory is used to illustrate the FL style of usage.  Instrumentally, it is one of the most powerful tools available to clear away both the understandable confusions and the unjustifiable presuppositions of informal discourse.

§ 8.	The notion of computation that makes sense in this setting is one of a process that replaces an arbitrary sign with a better sign of the same object.  In other words, computation is an interpretive process that improves the indications of intentions.  To deal with computational processes it is necessary to extend the pragmatic theory of signs in a couple of new but coordinated directions.  To the basic conception of a sign relation is added a notion of progress, which implies a notion of process together with a notion of quality.

§ 9.	This section introduces "higher order" sign relations, which are used to formalize the process of reflection on interpretation.  The discussion is approaching a point where multiple levels of signs are becoming necessary, mainly for referring to previous levels of signs as the objects of an extended sign relation, and thereby enabling a process of reflection on interpretive conduct.  To begin dealing with this issue, I take advantage of a second look at A and B to introduce the use of "raised angle brackets" (< >), also called "supercilia" or "arches", as quotation marks.  Ordinary quotation marks (" ") have the disadvantage, for formal purposes, of being used informally for many different tasks.  To get around this obstacle, I use the "arch" operator to formalize one specific function of quotation marks in a computational context, namely, to create distinctive names for syntactic expressions, or what amounts to the same thing, to signify the generation of their godel numbers.

§ 10.	Returning to the sign relations A and B, various kinds of HO signs are exemplified by considering a selection of HO sign relations that are based on these two examples.

§ 11.	In this section the tools that come with the theory of higher order sign relations are applied to an illustrative exercise, roughing out the shape of a complex form of interpreter.

The next three sections (§§ 32 34) discuss how the identified styles of usage bear on three important issues in the usage of a technical language, namely, the respective theoretical statuses of "signs", "sets", and "variables".

§ 12.	The Status of Signs

§ 13.	The Status of Sets

§ 14.	At this point the discussion touches on an topic, concerning the being of a so called "variable", that issues in many unanswered questions.  Although this worry over the nature and use of a variable may seem like a trivial matter, it is not.  It needs to be remembered that the first adequate accounts of formal computation, Schonfinkel's combinator calculus and Church's lambda calculus, both developed out of programmes intended to clarify the concept of a variable, indeed, even to the point of eliminating it altogether as a primitive notion from the basis of mathematical logic (van Heijenoort, 355 366).

The pragmatic theory of sign relations has a part of its purpose in addressing these same questions about the natural utility of variables, and even though its application to computation has not enjoyed the same level of development as these other models, it promises in good time to have a broader scope.  Later, I will illustrate its potential by examining a form of the combinator calculus from a sign relational point of view.

§ 15.	There is an order of logical reasoning that is typically described as "propositional" or "sentential" and represented in a type of formal system that is commonly known as a "propositional calculus" or a "sentential logic" (SL).  Any one of these calculi forms an interesting example of a formal language, one that can be used to illustrate all of the preceding issues of style and technique, but one that can also serve this inquiry in a more instrumental fashion.  This section presents the elements of a calculus for propositional logic that I described in earlier work (Awbrey, 1989 & 1994).  The imminent use of this calculus is to construct and analyze logical representations of sign relations, and the treatment here focuses on the concepts and notation that are most relevant to this task.

The next four sections (§§ 16 19) treat the theme of self reference that is invoked in the overture to a RIF.  To inspire confidence in the feasibility and the utility of well chosen reflective constructions and to allay a suspicion of self reference in general, it is useful to survey the varieties of self reference that arise in this work and to distinguish the forms of circular referrals that are likely to vitiate consistent reasoning from those that are relatively innocuous and even beneficial.

§ 16.	Recursive Aspects

§ 17.	Patterns of Self Reference

§ 18.	Practical Intuitions

§ 19.	Examples of Self Reference

The intertwined themes of logic and time will occupy center stage for the next eight sections (§§ 20 27).

§ 20.	First, I discuss three distinct ways that the word "system" is used in this work, reflecting the variety of approaches, aspects, or perspectives that present themselves in dealing with what are often the same underlying objects in reality.

§ 21.	There is a general set of situations where the task arises to "build a bridge" between significantly different types of representation.  In these situations, the problem is to translate between the signs and expressions of two formal systems that have radically different levels of interpretation, and to do it in a way that makes appropriate connections between diverse descriptions of the same objects.  More to the point of the present project, formal systems for mediating inquiry, if they are intended to remain viable in both empirical and theoretical uses, need the capacity to negotiate between an "extensional representation" (ER) and an "intensional representation" (IR) of the same domain of objects.  It turns out that a cardinal or pivotal issue in this connection is how to convert between ER's and IR's of the same objective domain, working all the while within the practical constraints of a computational medium and preserving the equivalence of information.  To illustrate the kinds of technical issues that are involved in these considerations, I bring them to bear on the topic of representing sign relations and their dyadic projections in various forms.

The next four sections (§§ 22-25) give examples of ER's and IR's, indicate the importance of forming a computational bridge between them, and discuss the conceptual and technical obstacles that will have to be faced in doing so.

§ 22.	For ease of reference, this section collects previous materials that are relevant to discussing the ER's of the sign relations A and B, and explicitly details their dyadic projections.

§ 23.	This section discusses a number of general issues that are associated with the IR's of sign relations.  Because of the great degree of freedom there is in selecting among the potentially relevant properties of any real object, especially when the context of relevance to the selection is not known in advance, there are many different ways, perhaps an indefinite multitude of ways, to represent the sign relations A and B in terms of salient properties of their elementary constituents.  In this connection, the next two sections explore a representative sample of these possibilities, and illustrate several different styles of approach that can be used in their presentation.

§ 24.	A transitional case between ER's and IR's of sign relations is found in the concept of a "literal intensional representation" (LIR).

§ 25.	A fully fledged IR is one that accomplishes some measure of analytic work, bringing to the point of salient notice a selected array of implicit and otherwise hidden features of its object.  This section presents a variety of these "analytic intensional representations" (AIR's) for the sign relations A and B.

Note for future reference.  The problem so naturally encountered here, due to the "embarassment of riches" that presents itself in choosing a suitable IR, and tracing its origin to the wealth of properties that any real object typically has, is a precursor to one of the deepest issues in the pragmatic theory of inquiry:  "the problem of abductive reasoning".  This topic will be discussed at several later stages of this investigation, where it typically involves the problem of choosing, among the manifold aspects of an objective phenomenon or a problematic objective, only the features that are:  (1) relevant to explaining a present fact, or (2) pertinent to achieving a current purpose.

§ 26.	Differential Logic & Directed Graphs

§ 27.	Differential Logic & Group Operations

§ 28.	The Bridge:  From Obstruction to Opportunity

§ 29.	Projects of Representation

§ 30.	Connected, Integrated, Reflective Symbols

The next seven sections (§§ 31 37) are designed to incrementally motivate the idea that a language as simple as propositional calculus, remarkably enough, can be used to articulate significant properties of n place relations.  The course of the discussion will proceed as follows:

§ 31.	First, I introduce concepts and notation designed to expand and generalize the orders of relations that are available to be discussed in an adequate fashion.

§ 32.	Second, I elaborate a particular mode of abstraction, that is, a systematic strategy for generalizing the collections of formal objects that are initially given to discussion.  This dimension of abstraction or direction of generalization will be described under the thematic heading of "partiality".

§ 33.	Third, I present an alternative approach to the issue of "degenerate", "defective", or "fragmentary" n place relations, proceeding by way of generalized objects known as "n place relational complexes".  Illustrating these ideas with respect to their bearing on sign relations the discussion arrives at a notion of "sign relational complexes", or "sign complexes".

In the next three sections (§§ 34 36) I consider a collection of "identification tasks" for n place relations.  Of particular interest is the extent to which the determination of an n place relation is constrained by a particular type of data, namely, by the specification of lower arity relations that occur as its projections.  This topic is often treated as a question about a relation's "reducibility" or "irreduciblity" with respect to its projections.  For instance, if the identity of an n place relation is completely determined by the data of its k place projections, then R is said to be "identifiable by", "reducible to", or "reconstructible from" its k place components, otherwise R is said to be "irreducible" with respect to its k place projections.

§ 34.	First, I consider a number of set theoretic operations that can be utilized in discussing these "identification", "reducibility", or "reconstruction" questions.  Once a level of general discussion has been surveyed enough to make a start, these tools can be specialized and applied to concrete examples in the realm of sign relations and also applied in the neighborhood of closely associated triadic relations.

§ 35.	This section considers the positive case of reducibility, presenting examples of triadic relations that can be reconstructed from their dyadic projections.  In fact, it happens that the sign relations A and B fall into this category of dyadically reducible triadic relations.

§ 36.	This section considers the negative case of reduciblity, presenting examples of "irreducibly triadic relations", or triadic relations that cannot be reconstructed from their lower dimensional projections or "faces".

§ 37.	Finally, the discussion culminates in an exposition of the so called "propositions as types" (PAT) analogy, outlining a formal system of "type expressions" or "type formulas" that bears a strong resemblance to propositional calculus.  Properly interpreted, the resulting "calculus of propositional types" (COPT) can be used as a language for talking about well formed types of n place relations.

§ 38.	Considering the Source

§ 39.	Prospective Indices : Pointers to Future Work

§ 40.	Interlaced with the structural and reflective developments that go into the OF and the IF is a conceptual arrangement called the "dynamic evaluative framework" (DEF).  This utility works to isolate the aspects of process and purpose that are observable on either side of the objective interpretive divide and helps to organize the graded notions of directed change that can be actualized in the RIF.

§ 41.	Elective and Motive Forces

§ 42.	Sign Processes : A Start

§ 43.	Reflective Extensions

§ 44.	Reflections on Closure

§ 45.	Intelligence => Critical Reflection

§ 46.	Looking Ahead:  The Meta Issue

§ 47.	Mutually Intelligible Codes

§ 48.	Discourse Analysis : Ways and Means

§ 49.	Combinations of Sign Relations

§ 50.	Revisiting the Source

6.1. The Phenomenology of Reflection

6.2. A Candid Point of View

6.3. A Projective Point of View

6.4. A Formal Point of View

6.5. Three Styles of Linguistic Usage

6.6. Basic Notions of Group Theory

6.7. Basic Notions of Formal Language Theory

6.8. A Perspective on Computation

6.9. Higher Order Sign Relations : Introduction

6.10. Higher Order Sign Relations : Examples

6.11. Higher Order Sign Relations : Application

6.12. Issue 1. The Status of Signs

6.13. Issue 2. The Status of Sets

6.14. Issue 3. The Status of Variables

6.15. Propositional Calculus

6.16. Recursive Aspects

6.17. Patterns of Self-Reference

6.18. Practical Intuitions

6.19. Examples of Self Reference

6.20. Three Views of Systems

6.21. Building Bridges Between Representations

6.22. Extensional Representations of Sign Relations

6.23. Intensional Representations of Sign Relations

6.24. Literal Intensional Representations

6.25. Analytic Intensional Representations

6.26. Differential Logic and Directed Graphs

6.27. Differential Logic and Group Operations

6.28. The Bridge : From Obstruction to Opportunity

6.29. Projects of Representation

6.30. Connected, Integrated, Reflective Symbols

6.31. Generic Orders of Relations

6.32. Partiality : Selective Operations

6.33. Sign Relational Complexes

6.34. Set Theoretic Constructions

6.35. Reducibility of Sign Relations

6.36. Irreducibly Triadic Relations

6.37. Propositional Types

6.38. Considering the Source

6.39. Prospective Indices : Pointers to Future Work

6.40. Dynamic and Evaluative Frameworks

6.41. Elective and Motive Forces

6.42. Sign Processes : A Start

6.43. Reflective Extensions

6.44. Reflections on Closure

6.45. Intelligence => Critical Reflection

6.46. Looking Ahead

6.47. Mutually Intelligible Codes

6.48. Discourse Analysis : Ways and Means

6.49. Combinations of Sign Relations

6.50. Revisiting the Source

• Contents • Part 1 • Part 2 • Part 3 • Part 4 • Part 5 • Part 6 • Appendices • References • Document History •


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