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Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Part 6 (view source)
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A point by point outline follows:
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 PODs, 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.
A formalization for talking about a diversity of POVs 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.
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.
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.
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".
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.
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.
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).
The intertwined themes of logic and time will occupy center stage for the next eight sections (§§&nbdsp;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 ERs and IRs 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.
The next four sections (§§ 22–25) give examples of ERs and IRs, 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 ERs 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 IRs 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 ERs and IRs 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" (AIRs) 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.
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:
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.
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
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3. There is a feature that C.S. Peirce called "tuity", acknowledging the aspect of "thouness" or the prospect of a second person POV that is brought into play whenever one self addresses another. Along with the perspective of a genuine other, this recognizes all the referrals and deferrals that an interpretive agent can make to a past, present, or potential self.
3. There is a feature that C.S. Peirce called "tuity", acknowledging the aspect of "thouness" or the prospect of a second person POV that is brought into play whenever one self addresses another. Along with the perspective of a genuine other, this recognizes all the referrals and deferrals that an interpretive agent can make to a past, present, or potential self.
All of these dimensions of concern focus on the circumstance that signs, especially written or recorded signs, moderate a complexly integrated sort of relationship between self and other, or between "first person" and "second person" POV's, in such a way that they render the paired categories of each scheme inextricably involved in one another.
All of these dimensions of concern focus on the circumstance that signs, especially written or recorded signs, moderate a complexly integrated sort of relationship between self and other, or between "first person" and "second person" POVs, in such a way that they render the paired categories of each scheme inextricably involved in one another.
There are well known dangers of paradox, but not so well acknowledged risks of distortion, that arise in the interrogation of any reflection. Although its outward signs are obvious, the source of the difficulty is remarkably difficult to trace. Perhaps it can be approached as follows. Without trying to say what consciousness is, I can still speak sensibly of its contents, and talk of their structures in relation to each other. These contents, whether percepts or concepts or whatever, are all signs. And so I can study the effects of reflection in the medium of its texts and develop a model of reflection as a process that evolves these texts.
There are well known dangers of paradox, but not so well acknowledged risks of distortion, that arise in the interrogation of any reflection. Although its outward signs are obvious, the source of the difficulty is remarkably difficult to trace. Perhaps it can be approached as follows. Without trying to say what consciousness is, I can still speak sensibly of its contents, and talk of their structures in relation to each other. These contents, whether percepts or concepts or whatever, are all signs. And so I can study the effects of reflection in the medium of its texts and develop a model of reflection as a process that evolves these texts.
Two Gentlemen of Verona: Speed—2.1.127–132
Two Gentlemen of Verona: Speed—2.1.127–132
When I write out my thinking in the form of a text, a critical thing happens: It faces me as the thought of another, and I start to think of what it says as though another person had said it. Almost unwittingly, a critical process comes into play. In regarding the text as expressing the thought of another, I begin to see it from different POV's than the one that led to its writing. As I find my own inquiry reflected in one or another TOI, it addresses me afresh as the question of another and I encounter it again as a novel line of investigation. This time around, though, the topic of concern and the style of expression become subject to directions of criticism that would probably not occur to me otherwise, since the angles of attack permitting them do not open up on their own, neither on first thinking nor ever, most likely, while merely speaking. This can be the beginning of critical reflection, but it can also stir up destructive forms of interference that inhibit and obstruct the very flow of thought itself.
When I write out my thinking in the form of a text, a critical thing happens: It faces me as the thought of another, and I start to think of what it says as though another person had said it. Almost unwittingly, a critical process comes into play. In regarding the text as expressing the thought of another, I begin to see it from different POVs than the one that led to its writing. As I find my own inquiry reflected in one or another TOI, it addresses me afresh as the question of another and I encounter it again as a novel line of investigation. This time around, though, the topic of concern and the style of expression become subject to directions of criticism that would probably not occur to me otherwise, since the angles of attack permitting them do not open up on their own, neither on first thinking nor ever, most likely, while merely speaking. This can be the beginning of critical reflection, but it can also stir up destructive forms of interference that inhibit and obstruct the very flow of thought itself.
If I can be granted the license to continue saying that a text says this or that about itself when what I really mean is that a person or process employs its text to say the corresponding thing about itself or its text, then I can begin to introduce a variety of descriptive terms and logical tools into this text that can be used to talk about what this or another TOI "thinks" or "believes" at various points in its development, that is, in order to detail what I or its proper author thinks or believes at the corresponding points of discussion.
If I can be granted the license to continue saying that a text says this or that about itself when what I really mean is that a person or process employs its text to say the corresponding thing about itself or its text, then I can begin to introduce a variety of descriptive terms and logical tools into this text that can be used to talk about what this or another TOI "thinks" or "believes" at various points in its development, that is, in order to detail what I or its proper author thinks or believes at the corresponding points of discussion.
Robert Burns, A Sonnet Upon Sonnets
Robert Burns, A Sonnet Upon Sonnets
One of the main problems that the present TOI has to address is how a TOI can address the problems of self reference that an inquiry into inquiry involves. If a sonnet can say something true about sonnets, then a TOI, far less limited in the number and measure of its lines, ought to be able to say something true about TOI's in general, unless the removal of these limitations takes away the only things whereof and whereby it has to speak, the ends and means of its own form of speech.
One of the main problems that the present TOI has to address is how a TOI can address the problems of self reference that an inquiry into inquiry involves. If a sonnet can say something true about sonnets, then a TOI, far less limited in the number and measure of its lines, ought to be able to say something true about TOIs in general, unless the removal of these limitations takes away the only things whereof and whereby it has to speak, the ends and means of its own form of speech.
Using the pragmatic theory of signs, the forms of self reference that have to be addressed in this project can be divided into two kinds, or classified in accord with two dimensions of referential involvement. Roughly speaking, reference in the broader sense can suggest either a denotative reference to an object or a connotative reference to a sense. Therefore, a projected self reference can be classified according to the ways that its components of reference propose to recur on themselves: how much pretends to be a self description along denotative lines and how much purports to be a self address in the connotative direction.
Using the pragmatic theory of signs, the forms of self reference that have to be addressed in this project can be divided into two kinds, or classified in accord with two dimensions of referential involvement. Roughly speaking, reference in the broader sense can suggest either a denotative reference to an object or a connotative reference to a sense. Therefore, a projected self reference can be classified according to the ways that its components of reference propose to recur on themselves: how much pretends to be a self description along denotative lines and how much purports to be a self address in the connotative direction.
In this setting, it is possible to bring about an accommodation between the mathematical and the psychological concepts of "projection" and to reconcile their discordant uses of the term within a concerted paradigm. For example, in dealing with the joint configuration space of a multiple agent system, one considers this "yoked extension space" (YES) to fall within a "common extension" (CE) of all the single agent state spaces. Each agent involved in such a system "projects", in a geometric sense, the total action of the system on its own "section" of the whole CE, its "local outlook", "mental plane", personal "frame of reference" (FOR), or "point of view" (POV).
In this setting, it is possible to bring about an accommodation between the mathematical and the psychological concepts of "projection" and to reconcile their discordant uses of the term within a concerted paradigm. For example, in dealing with the joint configuration space of a multiple agent system, one considers this "yoked extension space" (YES) to fall within a "common extension" (CE) of all the single agent state spaces. Each agent involved in such a system "projects", in a geometric sense, the total action of the system on its own "section" of the whole CE, its "local outlook", "mental plane", personal "frame of reference" (FOR), or "point of view" (POV).
What does the POV of an agent consist in? Generally speaking, agents are not dumb. They are not limited to a single view of their situation, nor are they restricted to a single scenario for its ongoing development. They can entertain many different possibilities as candidates for the so called and partly self describing "objective situation" and they can envision many different ways that these potential situations might be developing, both before and after their passage through the moment in question. Furthermore, under circumstances favorable to reflection, agents can invoke POV's that help them to contemplate many different possible developments in the constitution of these very same POV's.
What does the POV of an agent consist in? Generally speaking, agents are not dumb. They are not limited to a single view of their situation, nor are they restricted to a single scenario for its ongoing development. They can entertain many different possibilities as candidates for the so called and partly self describing "objective situation" and they can envision many different ways that these potential situations might be developing, both before and after their passage through the moment in question. Furthermore, under circumstances favorable to reflection, agents can invoke POVs that help them to contemplate many different possible developments in the constitution of these very same POVs.
Now, it is conceivable that all the POV's entertained by a single agent are predetermined as having the same collection of generic characters, and thus that this invariant constitution is what really limits the range of all possible POV's for the agent in question. If so, it leads to the idea that this invariant constitution defines a "uniquely general POV", a "highest order meta POV", or a "consummate POV" of the agent involved. Still, the only points of access and the only paths of approach that an agent can have to its own consummate POV, if indeed such a goal does make sense, are through the agency and the medium of whatever POV's it happens to have at each passing moment in its developmental history. Consequently, a persistent enough search for a good POV opens up the investigation of each agent's prevailing "point of develoment" (POD).
Now, it is conceivable that all the POVs entertained by a single agent are predetermined as having the same collection of generic characters, and thus that this invariant constitution is what really limits the range of all possible POVs for the agent in question. If so, it leads to the idea that this invariant constitution defines a "uniquely general POV", a "highest order meta POV", or a "consummate POV" of the agent involved. Still, the only points of access and the only paths of approach that an agent can have to its own consummate POV, if indeed such a goal does make sense, are through the agency and the medium of whatever POVs it happens to have at each passing moment in its developmental history. Consequently, a persistent enough search for a good POV opens up the investigation of each agent's prevailing "point of develoment" (POD).
In the best of all possible worlds, then, being under the influence of one POV does not render an agent incapacitated for considering others. Of course, there are practical limitations that affect both the capacity and the flexibility of a particular POV, and there can be found in force both logical constraints and resource constraints that leave a POV with a narrowly fixed and impoverished character, one that the agent opting for it can fail to represent reflectively enough within the scope of this POV itself. In particular, the "finite information constructions" (FIC's) that are accessible from a computational standpoint are especially limited in the kinds of POV's they are able to attain.
In the best of all possible worlds, then, being under the influence of one POV does not render an agent incapacitated for considering others. Of course, there are practical limitations that affect both the capacity and the flexibility of a particular POV, and there can be found in force both logical constraints and resource constraints that leave a POV with a narrowly fixed and impoverished character, one that the agent opting for it can fail to represent reflectively enough within the scope of this POV itself. In particular, the "finite information constructions" (FICs) that are accessible from a computational standpoint are especially limited in the kinds of POVs they are able to attain.
This means that POV's and POD's have recursive constitutions and recursive involvements with one another, calling on and referring to other POV's and POD's, both for the exact definitions that are needed and also for the more illuminating elaborations that might be possible, both those belonging to the same agent, reflexively, and those possessed by other agents, vicariously. A large part of the task of building a RIF is taken up with formalizing POV's and POD's, in part by analyzing their intuitive notions in terms of their implicit recursive structures and their referential involvements with each other, and in part by exploring their potential relationships with the previously formalized concepts of "objective concerns" (OC's).
This means that POVs and PODs have recursive constitutions and recursive involvements with one another, calling on and referring to other POVs and PODs, both for the exact definitions that are needed and also for the more illuminating elaborations that might be possible, both those belonging to the same agent, reflexively, and those possessed by other agents, vicariously. A large part of the task of building a RIF is taken up with formalizing POVs and PODs, in part by analyzing their intuitive notions in terms of their implicit recursive structures and their referential involvements with each other, and in part by exploring their potential relationships with the previously formalized concepts of "objective concerns" (OCs).
In settings where recursion is contemplated, it is possible to conceive of a distinction between "well founded" recursions, that lead to determinate definitions of the entities in question, and "buck passing" recursions, that lead one down the "garden path" to an interminable "run around". The catch, of course, is that it is not always possible to implement an effective procedure that can accomplish what it is possible to conceive. Thus, there are cases where the imagined distinction does not apply and times when the putative difference is not always detectable in practice.
In settings where recursion is contemplated, it is possible to conceive of a distinction between "well founded" recursions, that lead to determinate definitions of the entities in question, and "buck passing" recursions, that lead one down the "garden path" to an interminable "run around". The catch, of course, is that it is not always possible to implement an effective procedure that can accomplish what it is possible to conceive. Thus, there are cases where the imagined distinction does not apply and times when the putative difference is not always detectable in practice.
1. What makes a POV or a POD "well founded"?
1. What makes a POV or a POD "well founded"?
2. Can "buck passing" POV's and POD's be tolerated?
2. Can "buck passing" POVs and PODs be tolerated?
3. How should they be treated and regulated, if tolerated?
3. How should they be treated and regulated, if tolerated?
If all thought takes place in signs, as a tenet of pragmatism holds, then mental space is a space of signs and their interpretants, in other words, it is a connotative realm.
If all thought takes place in signs, as a tenet of pragmatism holds, then mental space is a space of signs and their interpretants, in other words, it is a connotative realm.
In this perspective, that is to say, in the POV of the present project and in the current opinion of its author, a POV is associated with an abstractly defined, but concretely embodied and frequently distributed, "section of memory" (SOM), where the signs constituting it are stored. In this rendition, a SOM is a curve, surface, volume, or more general subspace of the total memory space, in other words, a subset of memory that can be treated, under the appropriate change of coordinates, as being swept out by a set of variables, and ultimately addressed as being generated by a list of binary variables or bits. Working under the assumption that agents can engage in non trivial developments, it must be granted that they have the ability to change their POV's in significant ways between the successive POD's in their progress, and thus to move or jump from one SOM to another, as dictated by will or as constrained by habit.
In this perspective, that is to say, in the POV of the present project and in the current opinion of its author, a POV is associated with an abstractly defined, but concretely embodied and frequently distributed, "section of memory" (SOM), where the signs constituting it are stored. In this rendition, a SOM is a curve, surface, volume, or more general subspace of the total memory space, in other words, a subset of memory that can be treated, under the appropriate change of coordinates, as being swept out by a set of variables, and ultimately addressed as being generated by a list of binary variables or bits. Working under the assumption that agents can engage in non trivial developments, it must be granted that they have the ability to change their POVs in significant ways between the successive PODs in their progress, and thus to move or jump from one SOM to another, as dictated by will or as constrained by habit.
In this comparison, what is visualized as the geometric structure of a "cone" is commonly implemented through the data structure of a "tree", that is, a set of memory addresses (along with their associated contents) that are accessible from a single location, namely, the "root" of the tree, or the literal "point" of the POV.
In this comparison, what is visualized as the geometric structure of a "cone" is commonly implemented through the data structure of a "tree", that is, a set of memory addresses (along with their associated contents) that are accessible from a single location, namely, the "root" of the tree, or the literal "point" of the POV.
Typically, but not infallibly, an agent can reduce the complexity of what is projected on its personal POV by employing a reductive hypothesis or a simplifying assumption. Often, but not always, this idealization is arrived at by picking one agent to treat as "nominal", in other words, whose actions and perceptions are regarded as "natural", "normal", or otherwise unproblematic. Usually, especially if one is a "mature" agent, this nominal agent is just oneself, but a "novice" agent, unsure of what to do in a novel situation, can chose another agent to fill the role of a nominal guide and to serve as a reference point.
Typically, but not infallibly, an agent can reduce the complexity of what is projected on its personal POV by employing a reductive hypothesis or a simplifying assumption. Often, but not always, this idealization is arrived at by picking one agent to treat as "nominal", in other words, whose actions and perceptions are regarded as "natural", "normal", or otherwise unproblematic. Usually, especially if one is a "mature" agent, this nominal agent is just oneself, but a "novice" agent, unsure of what to do in a novel situation, can chose another agent to fill the role of a nominal guide and to serve as a reference point.
It would be nice if one could ignore the sharper edge of knowledge that is brought to light at this point, and fret but lightly over the smooth and middling courses that gloss the conformal plateaus of established knowledge. However, it is the nature of the inquiry into inquiry that one cannot forever restrict one's attention to the generic, nominal, or unexceptional case, well away from the initial conditions of learning and the boundary conditions of reasoning. Still, for the purposes of a first discussion of POV's and POD's, I limit my concern to the nominal case, where the reductive strategy indicated is useful to some degree and where the nominal agent of choice is none other than oneself.
It would be nice if one could ignore the sharper edge of knowledge that is brought to light at this point, and fret but lightly over the smooth and middling courses that gloss the conformal plateaus of established knowledge. However, it is the nature of the inquiry into inquiry that one cannot forever restrict one's attention to the generic, nominal, or unexceptional case, well away from the initial conditions of learning and the boundary conditions of reasoning. Still, for the purposes of a first discussion of POVs and PODs, I limit my concern to the nominal case, where the reductive strategy indicated is useful to some degree and where the nominal agent of choice is none other than oneself.
Under default conditions of operation, then, each one's POV embodies the reductive assumption that one's own particular actions and perceptions are "nominal", that is, natural, normal, or otherwise "not a problem". Relative to this ordinary setting, each one's POV is normally configured for tracking the more problematic courses of other agents and the drift of the residual system as a whole. Therefore, the natural setting of a POV can be pictured in terms of the perceptual "gestalt" it facilitates.
Under default conditions of operation, then, each one's POV embodies the reductive assumption that one's own particular actions and perceptions are "nominal", that is, natural, normal, or otherwise "not a problem". Relative to this ordinary setting, each one's POV is normally configured for tracking the more problematic courses of other agents and the drift of the residual system as a whole. Therefore, the natural setting of a POV can be pictured in terms of the perceptual "gestalt" it facilitates.
Beyond the dim inkling of an underlying influence, a sufficiently critical level of reflection on a POV requires a language that is articulate and analytic enough to transform each thesis posed in it into the form of a question. A deliberately reflective technology is needed to bring the prevailing, prejudicial, and hypocritical underpinnings of a POV to light, since biases due to assumptions obscurely held are seldom automatically revealed. This highlights the need for a critical apparatus that can be applied to the typical TOI, supplying its interpreter with the technical means to take up a critical POV with respect to it.
Beyond the dim inkling of an underlying influence, a sufficiently critical level of reflection on a POV requires a language that is articulate and analytic enough to transform each thesis posed in it into the form of a question. A deliberately reflective technology is needed to bring the prevailing, prejudicial, and hypocritical underpinnings of a POV to light, since biases due to assumptions obscurely held are seldom automatically revealed. This highlights the need for a critical apparatus that can be applied to the typical TOI, supplying its interpreter with the technical means to take up a critical POV with respect to it.
A logical calculus cannot initiate reflection on a text, but it can help to support and maintain it. The raw ability to perceive selected features of an ongoing text and the basic language of primitive terms, that allow one to mark the presence and note the passing of these features, have to be supplied from outside the calculus at the outset of its calculations. In the present text, the means to support critical reflection on its own POV and others are implemented in the form of a propositional calculus. Given the raw ability of a perceptive interpreter to form glosses on the text and to reflect on the contents of its current POV, a logical calculus can serve to augment the text and assist its critique by catalyzing the consideration of alternative POV's and facilitating reasoning about the wider implications of any particular POV.
A logical calculus cannot initiate reflection on a text, but it can help to support and maintain it. The raw ability to perceive selected features of an ongoing text and the basic language of primitive terms, that allow one to mark the presence and note the passing of these features, have to be supplied from outside the calculus at the outset of its calculations. In the present text, the means to support critical reflection on its own POV and others are implemented in the form of a propositional calculus. Given the raw ability of a perceptive interpreter to form glosses on the text and to reflect on the contents of its current POV, a logical calculus can serve to augment the text and assist its critique by catalyzing the consideration of alternative POVs and facilitating reasoning about the wider implications of any particular POV.
The discussion so far has dwelt at length on a particular scene, returning periodically to the fragmentary but concrete situation of a dialogue between A and B, poring over the formal setting and teasing out the casual surroundings of a circumscribed pair of sign relations. If the larger inquiry into inquiry is ever to lift itself off from these concrete and isolated grounds, then there is need for a way to extract the lessons of this exercise for reuse on other occasions. If items of knowledge with enduring value are to be found in this arena, then they ought to be capable of application to broader areas of interest and to richer domains of inquiry, and this demands ways to test their tentative findings in analogous and alternative situations of a more significant stripe. One way to do this is to identify properties and details of the selected examples that can be varied within the bounds of a common theme and treated as parameters whose momentary values convey the appearance of complete individuality to each particular case.
The discussion so far has dwelt at length on a particular scene, returning periodically to the fragmentary but concrete situation of a dialogue between A and B, poring over the formal setting and teasing out the casual surroundings of a circumscribed pair of sign relations. If the larger inquiry into inquiry is ever to lift itself off from these concrete and isolated grounds, then there is need for a way to extract the lessons of this exercise for reuse on other occasions. If items of knowledge with enduring value are to be found in this arena, then they ought to be capable of application to broader areas of interest and to richer domains of inquiry, and this demands ways to test their tentative findings in analogous and alternative situations of a more significant stripe. One way to do this is to identify properties and details of the selected examples that can be varied within the bounds of a common theme and treated as parameters whose momentary values convey the appearance of complete individuality to each particular case.
In an effort to gradually begin formalizing these issues, I introduce the concept of a "point of development" (POD). This notion is intended to capture a particular moment in the history of a system or its agent, as it is reflected in the systems of propositions associated with each POD. Relative to a particular POD there can be distinguished, though neither exclusively nor exhaustively, two types of propositions that are said to be "associated" with it. Roughly speaking, these types of propositions reflect the thoughts that are "applied" to a POD and the thoughts that are "attached" to a POD, respectively.
In an effort to gradually begin formalizing these issues, I introduce the concept of a "point of development" (POD). This notion is intended to capture a particular moment in the history of a system or its agent, as it is reflected in the systems of propositions associated with each POD. Relative to a particular POD there can be distinguished, though neither exclusively nor exhaustively, two types of propositions that are said to be "associated" with it. Roughly speaking, these types of propositions reflect the thoughts that are "applied" to a POD and the thoughts that are "attached" to a POD, respectively.
1. A proposition that "applies" to a POD can be formulated in more detail as a "proposition about or on a POD" (PAO'POD). This describes the corresponding POD as though observed from an outside perspective, stating features that locate it within a space of dynamic configurations or that place it in relation to some other medium of common description. This manner of associating propositions with POD's is tantamount to adopting a third person POV on the system or its agent, and it is commonly used to convey an impression of objectivity, no matter whether this standpoint is well taken or not.
1. A proposition that "applies" to a POD can be formulated in more detail as a "proposition about or on a POD" (PAO'POD). This describes the corresponding POD as though observed from an outside perspective, stating features that locate it within a space of dynamic configurations or that place it in relation to some other medium of common description. This manner of associating propositions with PODs is tantamount to adopting a third person POV on the system or its agent, and it is commonly used to convey an impression of objectivity, no matter whether this standpoint is well taken or not.
2. A proposition that "attaches" to a POD can be formalized in more detail as a "proposition at or in a POD" (PAI'POD). This represents what an agent thinks or believes, entertains or maintains, in sum, what an agent is aware of or willing to assert at a particular POD. By way of filling out the formula, this type of proposition expresses thoughts and is expressed in signs that are likewise regarded as "attached" to the POD in question. In general, propositions at a POD can be formed to express every conceivable modality. Collectively, they can state anything that an agent notes or thinks, observes or imagines at a given moment of its developmental history. They can reflect any aspect of an agent's awareness, belief, conjecture, doubt, expectation, intention, observation, or any other latitude of thought that is actively considered or faithfully preserved throughout the moment in question, and in this sense they are considered to be attached to, bound to, contained in, or localized at a particular POD.
2. A proposition that "attaches" to a POD can be formalized in more detail as a "proposition at or in a POD" (PAI'POD). This represents what an agent thinks or believes, entertains or maintains, in sum, what an agent is aware of or willing to assert at a particular POD. By way of filling out the formula, this type of proposition expresses thoughts and is expressed in signs that are likewise regarded as "attached" to the POD in question. In general, propositions at a POD can be formed to express every conceivable modality. Collectively, they can state anything that an agent notes or thinks, observes or imagines at a given moment of its developmental history. They can reflect any aspect of an agent's awareness, belief, conjecture, doubt, expectation, intention, observation, or any other latitude of thought that is actively considered or faithfully preserved throughout the moment in question, and in this sense they are considered to be attached to, bound to, contained in, or localized at a particular POD.
In one sense, propositions about a POD are potentially the general case, since propositions at a POD can be incorporated within their formulation. That is, a proposition about a POD is allowed to make assertions about the propositions at that POD, plus assertions about their relation to propositions at other POD's. But propositions whose references are this involved, articulated as "propositions about propositions at a POD", for instance, are classed as "higher order propositions" (HOP's) and need to be inferred through processes of hypothesis and experiment, conjecture and confirmation, instead of being observed outright. In another sense, propositions at a POD are intrinsically the prototype, since it is from their data that every other type must be constructed.
In one sense, propositions about a POD are potentially the general case, since propositions at a POD can be incorporated within their formulation. That is, a proposition about a POD is allowed to make assertions about the propositions at that POD, plus assertions about their relation to propositions at other PODs. But propositions whose references are this involved, articulated as "propositions about propositions at a POD", for instance, are classed as "higher order propositions" (HOPs) and need to be inferred through processes of hypothesis and experiment, conjecture and confirmation, instead of being observed outright. In another sense, propositions at a POD are intrinsically the prototype, since it is from their data that every other type must be constructed.
Propositions about POD's naturally collect into theories about POD's, and at the next level of aggregation these constitute the familiar sorts of dynamic theories that are used to describe the state spaces of systems and the trajectories of agents through them. Concentrating on these types of propositions leads to the kinds of theories about systems where a "neutral observer", not involved in the system itself, is postulated or fancied to stand outside the dynamics of the "observable object system": where this "objective reasoner" is supposedly able to theorize about the observable system without essentially becoming a part of its operations or necessarily being involved as a participant in its actual workings, and where the same "passive agent" never finds itself forced to interact in an irreversible or irrevocable manner with the autonomous course of the object system's action.
Propositions about PODs naturally collect into theories about PODs, and at the next level of aggregation these constitute the familiar sorts of dynamic theories that are used to describe the state spaces of systems and the trajectories of agents through them. Concentrating on these types of propositions leads to the kinds of theories about systems where a "neutral observer", not involved in the system itself, is postulated or fancied to stand outside the dynamics of the "observable object system": where this "objective reasoner" is supposedly able to theorize about the observable system without essentially becoming a part of its operations or necessarily being involved as a participant in its actual workings, and where the same "passive agent" never finds itself forced to interact in an irreversible or irrevocable manner with the autonomous course of the object system's action.
The thoughts attached to a POD, the things an agent thinks or believes, entertains or maintains at one POD, in relation to what the agent thinks or believes, is aware of or willing to assert at another POD, is the very form of subject matter that is bound to come to light and bound to fall into play whenever one studies the development of a reflective system, whether the focus of interest is the course of a particular inquiry or the emergence of a generic intelligence.
The thoughts attached to a POD, the things an agent thinks or believes, entertains or maintains at one POD, in relation to what the agent thinks or believes, is aware of or willing to assert at another POD, is the very form of subject matter that is bound to come to light and bound to fall into play whenever one studies the development of a reflective system, whether the focus of interest is the course of a particular inquiry or the emergence of a generic intelligence.
It may be thought that there is an important distinction between belief and knowledge that ought to be recognized in the modes of maintaining propositions at or in a POD. Given the pragmatic definition of belief, however, there is no local mark that can tell belief and knowledge apart. That is, there is no practical difference that can be sustained, in the propositions attached to a single POD, between those that reflect items of contingent belief and those that reflect items of certain knowledge. Even if the propositions at or in a POD are artificially marked in ways that can later be reliably detected, the problem of constantly updating so fleeting a form of distinction makes the accumulating profusion of ephemeral distinctions as immaterial and unenlightening as every other genre of eracist obliterature.
It may be thought that there is an important distinction between belief and knowledge that ought to be recognized in the modes of maintaining propositions at or in a POD. Given the pragmatic definition of belief, however, there is no local mark that can tell belief and knowledge apart. That is, there is no practical difference that can be sustained, in the propositions attached to a single POD, between those that reflect items of contingent belief and those that reflect items of certain knowledge. Even if the propositions at or in a POD are artificially marked in ways that can later be reliably detected, the problem of constantly updating so fleeting a form of distinction makes the accumulating profusion of ephemeral distinctions as immaterial and unenlightening as every other genre of eracist obliterature.
A distinction between belief and knowledge appears to arise only in the interactions and comparisons that can be made between different POD's, either those enjoyed by a single agent in the history of a single system or those passed through by ostensibly different agents and systems. The sense of the distinction can be sustained only if the order of its relational context continues to be recognized, which means that the mark of the distinction cannot be strained to the point of being an absolute. In this context, different systems and their agents are said to be "at" comparable POD's precisely to the degree and exactly to the extent that the propositions "at" and "about" them, respectively, can be compared. In many respects, the comparison of propositions at different POD's is equally complex and problematic whether it is one agent or several that is being considered.
A distinction between belief and knowledge appears to arise only in the interactions and comparisons that can be made between different PODs, either those enjoyed by a single agent in the history of a single system or those passed through by ostensibly different agents and systems. The sense of the distinction can be sustained only if the order of its relational context continues to be recognized, which means that the mark of the distinction cannot be strained to the point of being an absolute. In this context, different systems and their agents are said to be "at" comparable PODs precisely to the degree and exactly to the extent that the propositions "at" and "about" them, respectively, can be compared. In many respects, the comparison of propositions at different PODs is equally complex and problematic whether it is one agent or several that is being considered.
With all this in mind, I can give a formulation of what the practical difference between belief and knowledge consists in. Roughly speaking, an agent says that an agent "knows" something if and only if the one believes what the other believes. More precisely, an agent at one POD has reason to say that an agent at another POD (possibly a former self) knows something about something (or knew something about something) if and only if the one believes what the other believes about it, all things being relative to the POD's that the agents are at.
With all this in mind, I can give a formulation of what the practical difference between belief and knowledge consists in. Roughly speaking, an agent says that an agent "knows" something if and only if the one believes what the other believes. More precisely, an agent at one POD has reason to say that an agent at another POD (possibly a former self) knows something about something (or knew something about something) if and only if the one believes what the other believes about it, all things being relative to the PODs that the agents are at.
Propositions associated with a POD are often found in organized bodies, forming more or less logical systems of more or less logical statements. Whatever their type or modality with respect to a POD, "propositions of a feather gather together". That is, they tend to collect into organized bodies of propositions that share compatible types of association and comparable modes of assertion. In logic, an arbitrary collection of propositions is called a "theory", no matter how coherent, complete, or consistent it turns out to be when subjected in time to critical review. Taking up this liberalized notion of a theory in the present setting, a bunch of PAI'POD's forms a "theory at or in a POD" (TAI'POD), while a bunch of PAO'POD's forms a "theory about or on a POD" (TAO'POD).
Propositions associated with a POD are often found in organized bodies, forming more or less logical systems of more or less logical statements. Whatever their type or modality with respect to a POD, "propositions of a feather gather together". That is, they tend to collect into organized bodies of propositions that share compatible types of association and comparable modes of assertion. In logic, an arbitrary collection of propositions is called a "theory", no matter how coherent, complete, or consistent it turns out to be when subjected in time to critical review. Taking up this liberalized notion of a theory in the present setting, a bunch of PAI'PODs forms a "theory at or in a POD" (TAI'POD), while a bunch of PAO'PODs forms a "theory about or on a POD" (TAO'POD).
A reasonably organized system is amenable to having its propositions sorted further, forming collections of propositions that are intended to be interpreted in the same light, and constellating theories that bear on single modes of contemplation or declaration among their propositions.
A reasonably organized system is amenable to having its propositions sorted further, forming collections of propositions that are intended to be interpreted in the same light, and constellating theories that bear on single modes of contemplation or declaration among their propositions.
2. There is the moment of desire or difficulty that is countenanced in a problematic situation, providing an impulse for the component of inquiry that seeks a plan of action to resolve the trouble. This factor driving inquiry can be analyzed as deriving from the differences that occur between one's intentions and one's observations.
2. There is the moment of desire or difficulty that is countenanced in a problematic situation, providing an impulse for the component of inquiry that seeks a plan of action to resolve the trouble. This factor driving inquiry can be analyzed as deriving from the differences that occur between one's intentions and one's observations.
It should be obvious that these conceptions represent another attempt to formalize the relationship between dynamic and symbolic approaches to intelligent systems. Once again, the paradigms that are established for dealing with propositions at or about POD's are typically specialized to consider one or the other but seldom both. This leads to the familiar sorts of dichotomies being imposed on a subject matter where the types are more complementary and generative than exclusive and exhaustive. Thus, one finds methodologies in the field that can work well either from an "external" (dynamic, model theoretic, empirical) perspective or from an "internal" (symbolic, proof theoretic, rational) perspective, but that are seldom able to incorporate both technologies into an integrated methodology.
It should be obvious that these conceptions represent another attempt to formalize the relationship between dynamic and symbolic approaches to intelligent systems. Once again, the paradigms that are established for dealing with propositions at or about PODs are typically specialized to consider one or the other but seldom both. This leads to the familiar sorts of dichotomies being imposed on a subject matter where the types are more complementary and generative than exclusive and exhaustive. Thus, one finds methodologies in the field that can work well either from an "external" (dynamic, model theoretic, empirical) perspective or from an "internal" (symbolic, proof theoretic, rational) perspective, but that are seldom able to incorporate both technologies into an integrated methodology.
The concept of a POD in the history of a system, with its associated division of propositions into those that apply exterior to it and those that attach interior to it, is yet another way of approaching a recurring subject, "the being and the role of the interpreter", that the general concept of an "objective concern" (OC), broached at an earlier point of development in this text, is also intended to capture. Advancing as if from a pair of complementary and convergent directions, the notion of a POD, in the way it supplies a footing to the propositions about or on it and serves to encapsulate the propositions at or in it, equips a growing SOI with all the pivotal, trophic, and vital functions that the notion of an "objective motif" (OM) realized in an "interpretive moment" (IM) is likewise meant to provide.
The concept of a POD in the history of a system, with its associated division of propositions into those that apply exterior to it and those that attach interior to it, is yet another way of approaching a recurring subject, "the being and the role of the interpreter", that the general concept of an "objective concern" (OC), broached at an earlier point of development in this text, is also intended to capture. Advancing as if from a pair of complementary and convergent directions, the notion of a POD, in the way it supplies a footing to the propositions about or on it and serves to encapsulate the propositions at or in it, equips a growing SOI with all the pivotal, trophic, and vital functions that the notion of an "objective motif" (OM) realized in an "interpretive moment" (IM) is likewise meant to provide.
The relationship between a POD and an OM at an IM can be understood as follows. ...
The relationship between a POD and an OM at an IM can be understood as follows. ...
In order to continue formalizing the discussion of POV's and POD's within the text that uses them, I introduce the following notations:
In order to continue formalizing the discussion of POVs and PODs within the text that uses them, I introduce the following notations:
j : x | y, x |j y, x | y : j,
j : x | y, x |j y, x | y : j,
In practice, it does not matter whether one regards x and y as logical features or as boolean variables, so long as the full set of positive and negative features { x, (x), y, (y) } is initially available to classify the relevant space of object perceptions or interpretive actions. Analogous to its role in the staging relations { < , > }, the label "j" indicates the active interpreter, that is, the system and moment of interpretation or the state of the interpretive system that is held to be responsible for finding, making, testing, or following through the consequences of posing the contemplated distinctions.
In practice, it does not matter whether one regards x and y as logical features or as boolean variables, so long as the full set of positive and negative features { x, (x), y, (y) } is initially available to classify the relevant space of object perceptions or interpretive actions. Analogous to its role in the staging relations { < , > }, the label "j" indicates the active interpreter, that is, the system and moment of interpretation or the state of the interpretive system that is held to be responsible for finding, making, testing, or following through the consequences of posing the contemplated distinctions.
Dual to the statements of momentary interpretive distinctions (MID's) are the respective statements of momentary interpretive coincidences (MIC's):
Dual to the statements of momentary interpretive distinctions (MIDs) are the respective statements of momentary interpretive coincidences (MICs):
j : x = y, x =j y, x = y : j,
j : x = y, x =j y, x = y : j,
Unfortunately, the customary and habitual classification of a problem as "insoluble", even when justified, can work against the recognition of methods that are available to ameliorate its more objectionable impacts. When it comes to the relationship of logic to time, I believe that the resources are currently available that could advance our understanding of this issue in new directions. All it would take is the will to reconfigure those resources in the appropriate ways.
Unfortunately, the customary and habitual classification of a problem as "insoluble", even when justified, can work against the recognition of methods that are available to ameliorate its more objectionable impacts. When it comes to the relationship of logic to time, I believe that the resources are currently available that could advance our understanding of this issue in new directions. All it would take is the will to reconfigure those resources in the appropriate ways.
To expand the formula: The realm of "logic" is typified by rational concepts regarding invariant patterns, virtually, by ideas about forms, while the rule of "time" is filled out by realistic experiences with changing qualities, ultimately, by feelings of content and discontent. The application of the integrative effort to intelligent systems in general and to "inquiry driven systems" (IDS's) in particular only sharpens the question of logic and time to the point of self application.
To expand the formula: The realm of "logic" is typified by rational concepts regarding invariant patterns, virtually, by ideas about forms, while the rule of "time" is filled out by realistic experiences with changing qualities, ultimately, by feelings of content and discontent. The application of the integrative effort to intelligent systems in general and to "inquiry driven systems" (IDSs) in particular only sharpens the question of logic and time to the point of self application.
Considerations like these, as old and as constant as the hills, and as much over our heads as the eternally renewed and inconstant weather, are deserving of occasional notice, yet their relevance to the work of the moment is doomed by their very quality of necessity to fade into the backgound of present concerns, and their saliency as problematic phenomena quickly recedes from the scope of any perspective so bent on immediate application as that falling within my present focus.
Considerations like these, as old and as constant as the hills, and as much over our heads as the eternally renewed and inconstant weather, are deserving of occasional notice, yet their relevance to the work of the moment is doomed by their very quality of necessity to fade into the backgound of present concerns, and their saliency as problematic phenomena quickly recedes from the scope of any perspective so bent on immediate application as that falling within my present focus.
3. Most constrained of all is the "computational language" (CL) context, which incorporates the interests of computational linguistics along with the aims of implementing and using programming languages. There are many styles of programming languages and many more styles of putting them to use. I concentrate here on a particular version of the Pascal language and describe the particular ways I have chosen to implement the concepts I need with the constructs it makes available.
3. Most constrained of all is the "computational language" (CL) context, which incorporates the interests of computational linguistics along with the aims of implementing and using programming languages. There are many styles of programming languages and many more styles of putting them to use. I concentrate here on a particular version of the Pascal language and describe the particular ways I have chosen to implement the concepts I need with the constructs it makes available.
Next I need to consider the complex of relationships that exists among these three styles of usage, along with the corresponding relationships that exist among their associated perspectives and contexts. In regard to the questions raised by these three NOS's, the pragmatic theory of sign relations is intended to help reflective interpreters, and other students of language, maintain all the advantages of taking up abstract and isolated perspectives on language use, but to achieve this without losing a sense of the connection that each peculiar outlook has to the richly interwoven pattern of a larger unity.
Next I need to consider the complex of relationships that exists among these three styles of usage, along with the corresponding relationships that exist among their associated perspectives and contexts. In regard to the questions raised by these three NOSs, the pragmatic theory of sign relations is intended to help reflective interpreters, and other students of language, maintain all the advantages of taking up abstract and isolated perspectives on language use, but to achieve this without losing a sense of the connection that each peculiar outlook has to the richly interwoven pattern of a larger unity.
In many places these variegated styles of usage express themselves not so much in isolated domains of influence or distinctive layers of context as in different perspectives on the same text. But different lights on a developing picture can cause different figures and patterns to emerge, and different ways of treating a developing text can lead it to grow in different directions. Thus, discrepant points of view on the emergence of a literature can stimulate different works to vie for its canon, and discriminating angles of approach to what seems like a level plain and a unified field of language can harvest a wealth of alternate appreciations. And so different styles of writing arise in correspondence with different styles of reading, and each rising style of readership engenders a new style of authorship in its wake.
In many places these variegated styles of usage express themselves not so much in isolated domains of influence or distinctive layers of context as in different perspectives on the same text. But different lights on a developing picture can cause different figures and patterns to emerge, and different ways of treating a developing text can lead it to grow in different directions. Thus, discrepant points of view on the emergence of a literature can stimulate different works to vie for its canon, and discriminating angles of approach to what seems like a level plain and a unified field of language can harvest a wealth of alternate appreciations. And so different styles of writing arise in correspondence with different styles of reading, and each rising style of readership engenders a new style of authorship in its wake.
Even though the styles of usage at the three degrees of formalization use overlapping vocabularies of technical terms, the interpretations that they put on some of these terms, together with the working attitudes that they promote toward the corresponding concepts, are tantamount in practice to the possession of distinct concepts for the very same terms.
Even though the styles of usage at the three degrees of formalization use overlapping vocabularies of technical terms, the interpretations that they put on some of these terms, together with the working attitudes that they promote toward the corresponding concepts, are tantamount in practice to the possession of distinct concepts for the very same terms.
Three issues of linguistic usage on which these three NOS's get most out of joint are on the questions of (1) signs and their significance, (2) the utilization of set theory and set theoretic constructions, and (3) the ontological or pragmatic status of variables. The rest of this section makes a cursory survey of the bearings that the three NOS's take toward these issues, in preparation for more detailed treatments in later sections.
Three issues of linguistic usage on which these three NOSs get most out of joint are on the questions of (1) signs and their significance, (2) the utilization of set theory and set theoretic constructions, and (3) the ontological or pragmatic status of variables. The rest of this section makes a cursory survey of the bearings that the three NOSs take toward these issues, in preparation for more detailed treatments in later sections.
In each perspective that an observer takes up, the natural attitude is to focus on a particular class of objects, to remain less aware of the signs being used to denote them, and to remain even less aware that these objects and signs can take up other roles in the same or other sign relations. In constantly shifting from one perspective to another, however, the transparent uses of signs and the ulterior circumstances that determine how objects and signs are cast start to become visible. Altogether, the interaction between casual and formal styles of usage is like an exchange carried on between radically different economies, where commodities and utilities that are freely traded in one kind of market are severely taxed in the other.
In each perspective that an observer takes up, the natural attitude is to focus on a particular class of objects, to remain less aware of the signs being used to denote them, and to remain even less aware that these objects and signs can take up other roles in the same or other sign relations. In constantly shifting from one perspective to another, however, the transparent uses of signs and the ulterior circumstances that determine how objects and signs are cast start to become visible. Altogether, the interaction between casual and formal styles of usage is like an exchange carried on between radically different economies, where commodities and utilities that are freely traded in one kind of market are severely taxed in the other.
A group X = <X, +, 0> is "cyclic" if and only if there is an element g C X such that every x C X can be written as x = ng for some n C Z. In this case, an element such as g is called a "generator" of the group.
A group X = <X, +, 0> is "cyclic" if and only if there is an element g C X such that every x C X can be written as x = ng for some n C Z. In this case, an element such as g is called a "generator" of the group.
Mathematical systems, like the R's and X's encountered above, are seldom comprehended in perfect isolation, but need to be viewed in relation to each other, as belonging to families of comparable systems. Systems are compared by finding or making correspondences between them, and this can be formalized as a task of setting up and probing various types of mappings between the sundry appearances of their objective structures. This requires techniques for exploring the spaces of mappings that exist between families of systems, for inquiring into and demonstrating the existence of specified types of functions between them, plus technical concepts for classifying and comparing their diverse representations. Therefore, in order to compare the structures of different objective systems and to recognize the same objective structure when it appears in different phenomenal or syntactic disguises, it helps to develop general forms of comparison that can organize the welter of possible associations between systems and single out those that represent a preservation of the designated forms.
Mathematical systems, like the Rs and Xs encountered above, are seldom comprehended in perfect isolation, but need to be viewed in relation to each other, as belonging to families of comparable systems. Systems are compared by finding or making correspondences between them, and this can be formalized as a task of setting up and probing various types of mappings between the sundry appearances of their objective structures. This requires techniques for exploring the spaces of mappings that exist between families of systems, for inquiring into and demonstrating the existence of specified types of functions between them, plus technical concepts for classifying and comparing their diverse representations. Therefore, in order to compare the structures of different objective systems and to recognize the same objective structure when it appears in different phenomenal or syntactic disguises, it helps to develop general forms of comparison that can organize the welter of possible associations between systems and single out those that represent a preservation of the designated forms.
The next series of definitions develops the mathematical concepts of "homomorphism" and "isomorphism", special types of mappings between systems that serve to formalize the intuitive notions of structural analogy and abstract identity, respectively. In very rough terms, a "homomorphism" is a "structure preserving mapping" between systems, but only in the sense that it preserves some part or some aspect of the structure mapped, whereas an "isomorphism" is a correspondence that preserves all of the relevant structure.
The next series of definitions develops the mathematical concepts of "homomorphism" and "isomorphism", special types of mappings between systems that serve to formalize the intuitive notions of structural analogy and abstract identity, respectively. In very rough terms, a "homomorphism" is a "structure preserving mapping" between systems, but only in the sense that it preserves some part or some aspect of the structure mapped, whereas an "isomorphism" is a correspondence that preserves all of the relevant structure.
<x1, ..., xn> C P => h(<x1, ..., xn>) C Q.
<x1, ..., xn> C P => h(<x1, ..., xn>) C Q.
Applying this definition to the case of two binary operations or LOC's, say *1 on X1 and *2 on X2, which are special kinds of triadic relations, say *1 c X13 and *2 c X23, one obtains:
Applying this definition to the case of two binary operations or LOCs, say *1 on X1 and *2 on X2, which are special kinds of triadic relations, say *1 c X13 and *2 c X23, one obtains:
<x, y, z> C *1 => h(<x, y, z>) C *2.
<x, y, z> C *1 => h(<x, y, z>) C *2.
h(x *1 y) = h(x) *2 h(y).
h(x *1 y) = h(x) *2 h(y).
To sum up the development so far in a general way: A "homomorphism" is a mapping from a system to a system that preserves an aspect of systematic structure, usually one that is relevant to an understood purpose or context. When the pertinent aspect of structure for both the source and the target system is a binary operation or a LOC, then the condition that the LOC's be preserved in passing from the pre image to the image of the mapping is frequently expressed by stating that "the image of the product is the product of the images". That is, if h : X1 >X2 is a homomorphism from X1 = <X1, *1> to X2 = <X2, *2>, then for every x, y C X1 the following condition holds:
To sum up the development so far in a general way: A "homomorphism" is a mapping from a system to a system that preserves an aspect of systematic structure, usually one that is relevant to an understood purpose or context. When the pertinent aspect of structure for both the source and the target system is a binary operation or a LOC, then the condition that the LOCs be preserved in passing from the pre image to the image of the mapping is frequently expressed by stating that "the image of the product is the product of the images". That is, if h : X1 >X2 is a homomorphism from X1 = <X1, *1> to X2 = <X2, *2>, then for every x, y C X1 the following condition holds:
h(x *1 y) = h(x) *2 h(y).
h(x *1 y) = h(x) *2 h(y).
Next, the concept of a homomorphism or "structure preserving map" is specialized to the different kinds of structure of interest here.
Next, the concept of a homomorphism or "structure preserving map" is specialized to the different kinds of structure of interest here.
A "semigroup homomorphism" from a semigroup X1 = <X1, *1> to a semigroup X2 = <X2, *2> is a mapping between the underlying sets that preserves the structure appropriate to semigroups, namely, the LOC's. This makes it a map h : X1 >X2 whose induced action on the LOC's is such that it takes every element of *1 to an element of *2. That is:
A "semigroup homomorphism" from a semigroup X1 = <X1, *1> to a semigroup X2 = <X2, *2> is a mapping between the underlying sets that preserves the structure appropriate to semigroups, namely, the LOCs. This makes it a map h : X1 >X2 whose induced action on the LOCs is such that it takes every element of *1 to an element of *2. That is:
<x, y, z> C *1 => h(<x, y, z>) = <h(x), h(y), h(z)> C *2.
<x, y, z> C *1 => h(<x, y, z>) = <h(x), h(y), h(z)> C *2.
A "monoid homomorphism" from a monoid X1 = <X1, *1, e1> to a monoid X2 = <X2, *2, e2> is a mapping between the underlying sets, h : X1 >X2, that preserves the structure appropriate to monoids, namely, the LOC's and the identity elements. This means that the map h is a semigroup homomorphism from X1 to X2, where these are considered as semigroups, but with the extra condition that h takes e1 to e2.
A "monoid homomorphism" from a monoid X1 = <X1, *1, e1> to a monoid X2 = <X2, *2, e2> is a mapping between the underlying sets, h : X1 >X2, that preserves the structure appropriate to monoids, namely, the LOCs and the identity elements. This means that the map h is a semigroup homomorphism from X1 to X2, where these are considered as semigroups, but with the extra condition that h takes e1 to e2.
A "group homomorphism" from a group X1 = <X1, *1, e1> to a group X2 = <X2, *2, e2> is a mapping between the underlying sets, h : X1 >X2, that preserves the structure appropriate to groups, namely, the LOC's, the identity elements, and the inverse elements. This means that the map h is a monoid homomorphism from X1 to X2, where these are viewed as monoids, with the extra condition that h(x 1) = h(x) 1 for all x C X1. As it happens, the inverse elements are automatically preserved if the LOC's and the identity elements are, so a monoid homomorphism suffices to constitute a group homomorphism for a monoid that is also a group. To see why this is so, consider the following chain of equalities:
A "group homomorphism" from a group X1 = <X1, *1, e1> to a group X2 = <X2, *2, e2> is a mapping between the underlying sets, h : X1 >X2, that preserves the structure appropriate to groups, namely, the LOCs, the identity elements, and the inverse elements. This means that the map h is a monoid homomorphism from X1 to X2, where these are viewed as monoids, with the extra condition that h(x 1) = h(x) 1 for all x C X1. As it happens, the inverse elements are automatically preserved if the LOCs and the identity elements are, so a monoid homomorphism suffices to constitute a group homomorphism for a monoid that is also a group. To see why this is so, consider the following chain of equalities:
h(x) *2 h(x 1) = h(x *1 x 1) = h(e1) = e2.
h(x) *2 h(x 1) = h(x *1 x 1) = h(e1) = e2.
Then q makes a PIR to X in R if and only if X c De(q, R). Of course, this includes the limiting case where X is a singleton, say X = {o}. When this happens then the reference is neither plural nor indefinite, properly speaking, but q denotes o uniquely.
Then q makes a PIR to X in R if and only if X c De(q, R). Of course, this includes the limiting case where X is a singleton, say X = {o}. When this happens then the reference is neither plural nor indefinite, properly speaking, but q denotes o uniquely.
The proper exploitation of PIR's in sign relations makes it possible to finesse the distinction between HI signs and HU signs, in other words, to provide a ready means of translating between the two kinds of signs that preserves all the relevant information, at least, for many purposes. This is accomplished by connecting the sides of the distinction in two directions. First, a HI sign that makes a PIR to many triples of the form <o, s, i> can be taken as tantamount to a HU sign that denotes the corresponding sign relation. Second, a HU sign that denotes a singleton sign relation can be taken as tantamount to a HI sign that denotes its single triple. The relation of one sign being "tantamount to" another is not exactly a full fledged semantic equivalence, but it is a reasonable approximation to it, and one that serves a number of practical purposes.
The proper exploitation of PIRs in sign relations makes it possible to finesse the distinction between HI signs and HU signs, in other words, to provide a ready means of translating between the two kinds of signs that preserves all the relevant information, at least, for many purposes. This is accomplished by connecting the sides of the distinction in two directions. First, a HI sign that makes a PIR to many triples of the form <o, s, i> can be taken as tantamount to a HU sign that denotes the corresponding sign relation. Second, a HU sign that denotes a singleton sign relation can be taken as tantamount to a HI sign that denotes its single triple. The relation of one sign being "tantamount to" another is not exactly a full fledged semantic equivalence, but it is a reasonable approximation to it, and one that serves a number of practical purposes.
In particular, it is not absolutely necessary for a sign relation to contain a HU sign in order for it to contain a description of itself or another sign relation. As long the sign relation is "content" to maintain its reference to the object sign relation in the form of a constant name, then it suffices to use a HI sign that makes a PIR to all of its triples.
In particular, it is not absolutely necessary for a sign relation to contain a HU sign in order for it to contain a description of itself or another sign relation. As long the sign relation is "content" to maintain its reference to the object sign relation in the form of a constant name, then it suffices to use a HI sign that makes a PIR to all of its triples.
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This section considers an issue that affects the status of signs and their mode of significance, as it appears under each of the three NOS's. The concerns that arise with respect to this issue can be divided into two sets of questions. The first type of question has to do with the default assumptions that are made about the meanings of signs and the strategies that are used to deal with signs that fail to have meanings. The second type of question has to do with higher order signs, or signs that involve signs among their objects.
This section considers an issue that affects the status of signs and their mode of significance, as it appears under each of the three NOSs. The concerns that arise with respect to this issue can be divided into two sets of questions. The first type of question has to do with the default assumptions that are made about the meanings of signs and the strategies that are used to deal with signs that fail to have meanings. The second type of question has to do with higher order signs, or signs that involve signs among their objects.
Only certain types of signs are able to make their appearance in a given medium or a particular style of text, while many others are not. But a sign is a sign by virtue of the fact that it is interpreted as a sign, and thus plays the role of a sign in a sign relation, and not of necessity because it has any special construction other than that of being construed as a sign.
Only certain types of signs are able to make their appearance in a given medium or a particular style of text, while many others are not. But a sign is a sign by virtue of the fact that it is interpreted as a sign, and thus plays the role of a sign in a sign relation, and not of necessity because it has any special construction other than that of being construed as a sign.
The theory of formal languages, as pursued under the FL perspective, is closely related to the theory of semigroups, as pursued under the IL perspective, in the sense that arbitrary formal languages can be studied as subsets of the semigroups that embody the primitive concatenation of linguistic symbols within their algebraic laws of composition. Thus, in staging any discussion of formal languages, the theory of semigroups is often taken for a neutral, indifferent, or undifferentiated background, but the wisdom of using this setting is contingent on understanding the distinct outlooks of the casual and formal NOS's. What divides the two styles and their favorite subjects in practice is a certain difference in attitude toward the status and role of their subject materials. Namely, it turns on the question of whether their primitive and derived elements are valued as terminal objects in and of themselves or whether these syntactic objects and constructions are interpreted as mere signs and sundry expressions whose true value lies elsewhere.
The theory of formal languages, as pursued under the FL perspective, is closely related to the theory of semigroups, as pursued under the IL perspective, in the sense that arbitrary formal languages can be studied as subsets of the semigroups that embody the primitive concatenation of linguistic symbols within their algebraic laws of composition. Thus, in staging any discussion of formal languages, the theory of semigroups is often taken for a neutral, indifferent, or undifferentiated background, but the wisdom of using this setting is contingent on understanding the distinct outlooks of the casual and formal NOSs. What divides the two styles and their favorite subjects in practice is a certain difference in attitude toward the status and role of their subject materials. Namely, it turns on the question of whether their primitive and derived elements are valued as terminal objects in and of themselves or whether these syntactic objects and constructions are interpreted as mere signs and sundry expressions whose true value lies elsewhere.
In taking up the IL attitude toward any mathematical system, semigroups in particular, one assumes that signs are available for denoting a class of formal objects, but the issue of how these notational matters come to be constellated is considered to be peripheral, lacking in a substantive weight of concern and enjoying a purely marginal interest.
In taking up the IL attitude toward any mathematical system, semigroups in particular, one assumes that signs are available for denoting a class of formal objects, but the issue of how these notational matters come to be constellated is considered to be peripheral, lacking in a substantive weight of concern and enjoying a purely marginal interest.
What can be done with the signs in question? Apparently, signs viewed as objects in the formal arena, temporarily cut off from their usual associations, treated as terminal values in themselves, and put under review to suggest explanations for themselves, can still be discussed. Doing this involves the use of other signs for denoting the signs in question. These extra signs, whose sense and use are not in question at the moment in question, are called into play as "higher order" (HO) signs, and it is their very meaningfulness and effectiveness that one must rely on to carry out the investigation of the "lower order" (LO) signs that are in question.
What can be done with the signs in question? Apparently, signs viewed as objects in the formal arena, temporarily cut off from their usual associations, treated as terminal values in themselves, and put under review to suggest explanations for themselves, can still be discussed. Doing this involves the use of other signs for denoting the signs in question. These extra signs, whose sense and use are not in question at the moment in question, are called into play as "higher order" (HO) signs, and it is their very meaningfulness and effectiveness that one must rely on to carry out the investigation of the "lower order" (LO) signs that are in question.
An apt and proper discussion of a set of signs in question requires the ability to classify the tokens of these signs according to their types. Doing this calls on the use of other HO signs to denote these "tokens", the transient instances of signs, and their "types", the propertied classes of tokens that correspond to what is typically valued as a sign. The invocation of HO signs can be iterated in a succession of HO's that extends as far as one pleases, but no matter how much of this order is progressively formalized one eventually must resort to signs of such a high order that they are taken for granted as resting, for the moment, in an informal context of interpretation.
An apt and proper discussion of a set of signs in question requires the ability to classify the tokens of these signs according to their types. Doing this calls on the use of other HO signs to denote these "tokens", the transient instances of signs, and their "types", the propertied classes of tokens that correspond to what is typically valued as a sign. The invocation of HO signs can be iterated in a succession of HOs that extends as far as one pleases, but no matter how much of this order is progressively formalized one eventually must resort to signs of such a high order that they are taken for granted as resting, for the moment, in an informal context of interpretation.
What is the sense and use of such a proceeding? Evidently, the signs in question, as a class, must present the inquirer with phenomena that are somehow simpler than, and yet convey instructive information about, the phenomenon known as the "whole objective world" (WOW). If their orders of complexity and perplexity are just as great as the world at large, then their investigation affords no advantage over the general empirical problem of trying to account for the WOW. If they enjoy no informative connection with the greater wonders of why the world is the way it is, and therefore fail to present a significant representation of the original question, then their isolated inquiry can serve no larger purpose in the world.
What is the sense and use of such a proceeding? Evidently, the signs in question, as a class, must present the inquirer with phenomena that are somehow simpler than, and yet convey instructive information about, the phenomenon known as the "whole objective world" (WOW). If their orders of complexity and perplexity are just as great as the world at large, then their investigation affords no advantage over the general empirical problem of trying to account for the WOW. If they enjoy no informative connection with the greater wonders of why the world is the way it is, and therefore fail to present a significant representation of the original question, then their isolated inquiry can serve no larger purpose in the world.
It is the language being discussed that is the formal one, to be treated initially as an object, while the language that is used to carry out the discussion tries to maintain its informal viability, expecting in effect to be taken on faith as not undermining or vitiating the effort at inquiry due to unexamined flaws of its own. Nevertheless, if inquiry in general is expected to be self correcting, then a continuing series of failures to conclude inquiries by means of a given arrangement, that is, an inability to resolve uncertainties through a particular division of labor between FL and IL contexts, must lead to the grounds of attack being shifted.
It is the language being discussed that is the formal one, to be treated initially as an object, while the language that is used to carry out the discussion tries to maintain its informal viability, expecting in effect to be taken on faith as not undermining or vitiating the effort at inquiry due to unexamined flaws of its own. Nevertheless, if inquiry in general is expected to be self correcting, then a continuing series of failures to conclude inquiries by means of a given arrangement, that is, an inability to resolve uncertainties through a particular division of labor between FL and IL contexts, must lead to the grounds of attack being shifted.
In working out compromises between the FL and IL styles of usage one faces all the problems usually associated with integrating different "frameworks of interpretation" (FOI's), but compounded by the additional factors (1) that this conflict of attitudes, or its practical importance, is seldom openly acknowledged, and (2) that the frameworks in and of the negotiation to be transacted are rarely capable of being formalized, or even of being made conscious, to the same degree at the same time. These circumstances make the consequences of the underlying conflict difficult to address, and thus they continue to obstruct the desired implementation of a common CL environment that could serve as a resource for work on both sides of the frame.
In working out compromises between the FL and IL styles of usage one faces all the problems usually associated with integrating different "frameworks of interpretation" (FOIs), but compounded by the additional factors (1) that this conflict of attitudes, or its practical importance, is seldom openly acknowledged, and (2) that the frameworks in and of the negotiation to be transacted are rarely capable of being formalized, or even of being made conscious, to the same degree at the same time. These circumstances make the consequences of the underlying conflict difficult to address, and thus they continue to obstruct the desired implementation of a common CL environment that could serve as a resource for work on both sides of the frame.
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That the word "set" is being used indiscriminately for completely different notions and that this is the source of the apparent paradoxes of this young branch of science, that, moreover, set theory itself can no more dispense with axiomatic assumptions than can any other exact science and that these assumptions, just as in other disciplines, are subject to a certain arbitrariness, even if they lie much deeper here — I do not want to represent any of this as something new. (Julius Konig, 1905).
That the word "set" is being used indiscriminately for completely different notions and that this is the source of the apparent paradoxes of this young branch of science, that, moreover, set theory itself can no more dispense with axiomatic assumptions than can any other exact science and that these assumptions, just as in other disciplines, are subject to a certain arbitrariness, even if they lie much deeper here — I do not want to represent any of this as something new. (Julius Konig, 1905).
Set theory is not as young as it used to be, and not half as naive as it was when this statement was originally made, but the statement itself is just as apt in its application to the present scene and just as fresh in its lack of novelty as it was then. In the current setting, though, I am not so concerned with potentially different theoretical notions of a set that are represented by conventionally different axiom systems as I am with the actual diversity of practical notions that are used to deal with sets under each of the three NOS's identified.
Set theory is not as young as it used to be, and not half as naive as it was when this statement was originally made, but the statement itself is just as apt in its application to the present scene and just as fresh in its lack of novelty as it was then. In the current setting, though, I am not so concerned with potentially different theoretical notions of a set that are represented by conventionally different axiom systems as I am with the actual diversity of practical notions that are used to deal with sets under each of the three NOSs identified.
Even though all three NOS's use set theoretic constructions, the implicit theories of sets that are involved in their different uses are so varied in their assumptions and intentions that it amounts to a major source of friction between the casual and formal styles to try to pretend that the same subject is being invoked in every case. In particular, it makes a huge difference whether these sets are treated objectively, as belonging to the OF, or treated syntactically, as belonging to the IF.
Even though all three NOSs use set theoretic constructions, the implicit theories of sets that are involved in their different uses are so varied in their assumptions and intentions that it amounts to a major source of friction between the casual and formal styles to try to pretend that the same subject is being invoked in every case. In particular, it makes a huge difference whether these sets are treated objectively, as belonging to the OF, or treated syntactically, as belonging to the IF.
In practical terms it makes all the difference in the world whether a set is viewed as a set of objects or whether it is viewed as a set of signs. The same set can be contemplated in each type of placement, but it does not always fit as well into both types of role. A set of objects is properly a part of the OF, and this is intended in its typical parts to model those realities whose laws and vagaries can extend outside the means of an agent's control. A set of signs is properly part of the IF, and this is constructed in its typical parts so that its variations and selections are subject to control for the ends of interpretive indication. The relevant variable is one of control, and the measure of it tells how well matched are the proper placements and the typical assignments that a given set is given.
In practical terms it makes all the difference in the world whether a set is viewed as a set of objects or whether it is viewed as a set of signs. The same set can be contemplated in each type of placement, but it does not always fit as well into both types of role. A set of objects is properly a part of the OF, and this is intended in its typical parts to model those realities whose laws and vagaries can extend outside the means of an agent's control. A set of signs is properly part of the IF, and this is constructed in its typical parts so that its variations and selections are subject to control for the ends of interpretive indication. The relevant variable is one of control, and the measure of it tells how well matched are the proper placements and the typical assignments that a given set is given.
According to one way of understanding the term, there is no object called a "variable" unless that object is a sign, and so the name "variable name" is redundant. Variables, if they are anything at all, are analogous to numerals, not numbers, and thus they fall within the broad class of signs called "identifiers", more specifically, as "indices". In the case of variables, the advice of nominalism, not to confuse a variable name with the name of a variable, seems to be well taken.
According to one way of understanding the term, there is no object called a "variable" unless that object is a sign, and so the name "variable name" is redundant. Variables, if they are anything at all, are analogous to numerals, not numbers, and thus they fall within the broad class of signs called "identifiers", more specifically, as "indices". In the case of variables, the advice of nominalism, not to confuse a variable name with the name of a variable, seems to be well taken.
If the world of elements appropriate to this discussion is organized into objective and syntactic domains, then there are fundamentally just two different ways of regarding variables, as objects or as signs. One can say that a variable is a fictional object that is contrived to provide a variable name with a form of objective referent, or one can say that a variable is a sign itself, the same thing as a variable name. In the present setting, it is convenient to arrange these broad approaches to variables under the NOS's where one finds them most often pursued.
If the world of elements appropriate to this discussion is organized into objective and syntactic domains, then there are fundamentally just two different ways of regarding variables, as objects or as signs. One can say that a variable is a fictional object that is contrived to provide a variable name with a form of objective referent, or one can say that a variable is a sign itself, the same thing as a variable name. In the present setting, it is convenient to arrange these broad approaches to variables under the NOSs where one finds them most often pursued.
1. The IL approach to the question takes the "objective construal" of variables as its most commonly chosen default. The IL style that is used in ordinary mathematical discussion associates a variable with a determinate set, one that the variable is regarded as "ranging over". As a result, this NOS is forced to invoke a version of set theory, usually naive, to account for its use of variables.
1. The IL approach to the question takes the "objective construal" of variables as its most commonly chosen default. The IL style that is used in ordinary mathematical discussion associates a variable with a determinate set, one that the variable is regarded as "ranging over". As a result, this NOS is forced to invoke a version of set theory, usually naive, to account for its use of variables.
The first order of business is to comment on the logical significance of the rhetorical distinctions that appear to prevail among these objects. My reason for introducing these distinctions is not to multiply the number of entities beyond necessity but merely to summarize the variety of entities that have been used historically, to figure out a series of conversions between them, and to integrate suitable analogues of them within a unified system.
The first order of business is to comment on the logical significance of the rhetorical distinctions that appear to prevail among these objects. My reason for introducing these distinctions is not to multiply the number of entities beyond necessity but merely to summarize the variety of entities that have been used historically, to figure out a series of conversions between them, and to integrate suitable analogues of them within a unified system.
For many purposes the distinction between a property and a proposition does not affect the structural aspects of the domains being considered. Both properties and propositions are tantamount to fictional objects, made up to supply general signs with singular denotations, and serving as indirect ways to explain the "plural indefinite references" (PIR's) of general signs to the multitudes of their ultimately denoted objects. A property is signfied by a sign called a "term" that achieves by a form of indirection a PIR to all the elements in a class of "things". A proposition is signified by a sign called a "sentence" that achieves by a form of indirection a PIR to all the elements in a class of "situations". But "things" are any objects of discussion and thought, in other words, a perfectly general category, and "situations" are just special cases of these "things".
For many purposes the distinction between a property and a proposition does not affect the structural aspects of the domains being considered. Both properties and propositions are tantamount to fictional objects, made up to supply general signs with singular denotations, and serving as indirect ways to explain the "plural indefinite references" (PIRs) of general signs to the multitudes of their ultimately denoted objects. A property is signfied by a sign called a "term" that achieves by a form of indirection a PIR to all the elements in a class of "things". A proposition is signified by a sign called a "sentence" that achieves by a form of indirection a PIR to all the elements in a class of "situations". But "things" are any objects of discussion and thought, in other words, a perfectly general category, and "situations" are just special cases of these "things".
There is still something left to the logical distinction between properties and propositions, but it is largely immaterial to the order of reasoning that is found reflected in propositional logic. When it is useful to emphasize their commonalities, properties and propositions can both be referred to as "Props". As a handle on the aspects of structure that are shared between these two domains and as a mechanism for ignoring irrelevant distinctions, it also helps to have a single term for a "domain of properties" (DOP) and a "domain of propositions" (DOP).
There is still something left to the logical distinction between properties and propositions, but it is largely immaterial to the order of reasoning that is found reflected in propositional logic. When it is useful to emphasize their commonalities, properties and propositions can both be referred to as "Props". As a handle on the aspects of structure that are shared between these two domains and as a mechanism for ignoring irrelevant distinctions, it also helps to have a single term for a "domain of properties" (DOP) and a "domain of propositions" (DOP).
In setting out the plan of a full scale RIF there is an unspoken promise to justify eventually the thematic motives that experimentally tolerate its indulgence in self reference, and it seems that this implicit hope for a full atonement in time is a key to the tensions of the work being borne. This section, in order to inspire confidence in the prospects of a RIF being achievable, and by way of allaying widespread suspicions about all types of self reference, examines several forms of circular referral and notes that not all contemplation of self reference is incurably vicious.
In setting out the plan of a full scale RIF there is an unspoken promise to justify eventually the thematic motives that experimentally tolerate its indulgence in self reference, and it seems that this implicit hope for a full atonement in time is a key to the tensions of the work being borne. This section, in order to inspire confidence in the prospects of a RIF being achievable, and by way of allaying widespread suspicions about all types of self reference, examines several forms of circular referral and notes that not all contemplation of self reference is incurably vicious.
In this section I consider signs, expressions, sign relations, and systems of interpretation (SOI's) that involve forms of self reference. Because it is the abstract forms of self reference that constitute the chief interest of this study, I collect this whole subject matter under the heading "patterns of self reference" (POSR's). With respect to this domain I entertain the classification of POSR's in two different ways.
In this section I consider signs, expressions, sign relations, and systems of interpretation (SOIs) that involve forms of self reference. Because it is the abstract forms of self reference that constitute the chief interest of this study, I collect this whole subject matter under the heading "patterns of self reference" (POSRs). With respect to this domain I entertain the classification of POSRs in two different ways.
In this section I take notice of a broad family of formal structures that I refer to as "patterns of self reference" (POSR's), because they seem to have in common the proposed description of a formal object by means of recursive or circular references. In their basic characters, POSR's range from the familiar to the strange, from the obvious to the problematic, and from the legitimate to the spurious. Often a POSR is best understood as a formal object in its own right, or as a formal sign that foreshadows a definite object, but occasionally a POSR can only be interpreted as something in the character of a syntactic pattern, one that goes into the making of a questionable specification and represents merely a dubious attempt to indicate or describe an object. All in all, POSR's range from the kinds of functions and objects, or programs and data structures, that are successfully defined by recursion to the sorts of vitiating circles that doom every attempt to define an unknown term in terms of itself.
In this section I take notice of a broad family of formal structures that I refer to as "patterns of self reference" (POSRs), because they seem to have in common the proposed description of a formal object by means of recursive or circular references. In their basic characters, POSRs range from the familiar to the strange, from the obvious to the problematic, and from the legitimate to the spurious. Often a POSR is best understood as a formal object in its own right, or as a formal sign that foreshadows a definite object, but occasionally a POSR can only be interpreted as something in the character of a syntactic pattern, one that goes into the making of a questionable specification and represents merely a dubious attempt to indicate or describe an object. All in all, POSRs range from the kinds of functions and objects, or programs and data structures, that are successfully defined by recursion to the sorts of vitiating circles that doom every attempt to define an unknown term in terms of itself.
Because POSR's span the spectrum from the moderately straightforward to the deliberately misleading, there is a need for ways to tell them apart, at least, before pursuing their consequences too far. Of course, if one cannot rest without having all computable functions at one's command, then no program can tell all the good and bad programs apart. But if one can be satisfied with a somewhat more modest domain, then there is hope for a way, an experimental, fallible, and incremental way, but a way nonetheless, that eventually leads one to know the good and ultimately keeps one away from the bad.
Because POSRs span the spectrum from the moderately straightforward to the deliberately misleading, there is a need for ways to tell them apart, at least, before pursuing their consequences too far. Of course, if one cannot rest without having all computable functions at one's command, then no program can tell all the good and bad programs apart. But if one can be satisfied with a somewhat more modest domain, then there is hope for a way, an experimental, fallible, and incremental way, but a way nonetheless, that eventually leads one to know the good and ultimately keeps one away from the bad.
When it comes to their propriety, POSR's are found on empirical grounds to fall into two varieties: the "exculpable" and the "indictable" kinds. Thus, it is reasonable to attempt an empirical distinction, proposing to let experience mark each POSR as an "excusable self reference" (ESR) or an "improper self reference" (ISR), as the case may be. But empirical grounds can be a hard basis to fall back on, since a recourse to actual experience with POSR's can risk an agent's participation in pretended sign relations and promissory representations that amount in the end to nothing more than forms of interpretive futility. Therefore, one seeks an arrangement of methods in general or an ordering of options in these special cases that makes the empirical trial a court of last resort and that avoids resorting to the actual experience of interpretation as a routine matter of course.
When it comes to their propriety, POSRs are found on empirical grounds to fall into two varieties: the "exculpable" and the "indictable" kinds. Thus, it is reasonable to attempt an empirical distinction, proposing to let experience mark each POSR as an "excusable self reference" (ESR) or an "improper self reference" (ISR), as the case may be. But empirical grounds can be a hard basis to fall back on, since a recourse to actual experience with POSRs can risk an agent's participation in pretended sign relations and promissory representations that amount in the end to nothing more than forms of interpretive futility. Therefore, one seeks an arrangement of methods in general or an ordering of options in these special cases that makes the empirical trial a court of last resort and that avoids resorting to the actual experience of interpretation as a routine matter of course.
First, I recognize an "empirical distinction" that seems to exist between the less problematic and the more problematic varieties of self reference, allowing POSR's to be sorted according to the consequential features that they have in actual experience. There are the "good" sorts, those cleared up to the limits of accumulated experience as innocuous usages and even as probable utilities, and then there are the "bad" sorts, those marked by hard experience as definitely problematic.
First, I recognize an "empirical distinction" that seems to exist between the less problematic and the more problematic varieties of self reference, allowing POSRs to be sorted according to the consequential features that they have in actual experience. There are the "good" sorts, those cleared up to the limits of accumulated experience as innocuous usages and even as probable utilities, and then there are the "bad" sorts, those marked by hard experience as definitely problematic.
Next, I search for an "intuitive distinction" that can be supposed to exist between the good and the bad sorts of POSR's, invoking a formal character or computable predicate of a POSR whose prior inspection can provide interpreters with a definitive indication or a decisive piece of information as to whether a POSR is good or bad, without forcing them to undergo the consequences of its actual use.
Next, I search for an "intuitive distinction" that can be supposed to exist between the good and the bad sorts of POSRs, invoking a formal character or computable predicate of a POSR whose prior inspection can provide interpreters with a definitive indication or a decisive piece of information as to whether a POSR is good or bad, without forcing them to undergo the consequences of its actual use.
Before I can pin down what is involved in finding these intuitive characters and distinctions, it is necessary to discuss the concept of "intuition" that is relevant here. This issue requires a substantial digression and is taken up in the next section. After that, the concrete examples I take to be acceptable POSR's are presented.
Before I can pin down what is involved in finding these intuitive characters and distinctions, it is necessary to discuss the concept of "intuition" that is relevant here. This issue requires a substantial digression and is taken up in the next section. After that, the concrete examples I take to be acceptable POSRs are presented.
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For ease of repeated reference, I introduce the following terminology. With respect to the empirical dimension, a "good" POSR is described as an "exculpable self reference" (ESR) while a "bad" POSR is described as an "indictable self reference" (ISR). With respect to the intuitive dimension, a "good" POSR is depicted as an "explicative self reference" (ESR) while a "bad" POSR is depicted as an "implicative self reference" (ISR). Here, underscored acronyms are used to mark the provisionally settled, hypothetically tentative, or "status quo" status of these casually intuitive categories.
For ease of repeated reference, I introduce the following terminology. With respect to the empirical dimension, a "good" POSR is described as an "exculpable self reference" (ESR) while a "bad" POSR is described as an "indictable self reference" (ISR). With respect to the intuitive dimension, a "good" POSR is depicted as an "explicative self reference" (ESR) while a "bad" POSR is depicted as an "implicative self reference" (ISR). Here, underscored acronyms are used to mark the provisionally settled, hypothetically tentative, or "status quo" status of these casually intuitive categories.
These categories of POSR's can be discussed in greater detail as follows:
These categories of POSRs can be discussed in greater detail as follows:
1. There is an "empirical distinction" that appears to impose itself on the varieties of self reference, separating the forms that lead to trouble in thought and communication from the forms that do not. And there is a pragmatic reason for being interested in this distinction, the motive being to avoid the corresponding types of trouble in reflective thinking. Whether this apparent distinction can hold up under close examination is a good question to consider at a later point. But the real trouble to be faced at the moment is that an empirical distinction is a post hoc mark, a difference that makes itself obvious only after the possibly unpleasant facts to be addressed are already present in experience. Consequently, its certain recognition comes too late to avert the adverse portions of those circumstances that its very recognition is desired to avoid.
1. There is an "empirical distinction" that appears to impose itself on the varieties of self reference, separating the forms that lead to trouble in thought and communication from the forms that do not. And there is a pragmatic reason for being interested in this distinction, the motive being to avoid the corresponding types of trouble in reflective thinking. Whether this apparent distinction can hold up under close examination is a good question to consider at a later point. But the real trouble to be faced at the moment is that an empirical distinction is a post hoc mark, a difference that makes itself obvious only after the possibly unpleasant facts to be addressed are already present in experience. Consequently, its certain recognition comes too late to avert the adverse portions of those circumstances that its very recognition is desired to avoid.
According to the form of this empirical distinction, a POSR can be classified either as an "exculpable self reference" (ESR) or as an "indictable self reference" (ISR). The distinction and the categories to either side of it are intended to sort out the POSR's that are safe and effective to use in thought and communication from the POSR's that can be hazardous to the health of inquiry.
According to the form of this empirical distinction, a POSR can be classified either as an "exculpable self reference" (ESR) or as an "indictable self reference" (ISR). The distinction and the categories to either side of it are intended to sort out the POSRs that are safe and effective to use in thought and communication from the POSRs that can be hazardous to the health of inquiry.
More explicitly, the distinction between ESR's and ISR's is intended to capture the differences that exist between the following cases:
More explicitly, the distinction between ESRs and ISRs is intended to capture the differences that exist between the following cases:
a. ESR's are POSR's that cause no apparent problems in thought or communication, often appearing as practiaclly useful in many contexts and even as logically necessary in some contexts.
a. ESRs are POSRs that cause no apparent problems in thought or communication, often appearing as practiaclly useful in many contexts and even as logically necessary in some contexts.
b. ISR's are POSR's that lead to various sorts of trouble in the attempt to reason with them or to reason about them, that is, to use them consistently or even to decide for or against their use.
b. ISRs are POSRs that lead to various sorts of trouble in the attempt to reason with them or to reason about them, that is, to use them consistently or even to decide for or against their use.
I refer to this as an "empirical distinction", in spite of the fact that the domain of experience in question is decidedly a formal one, because it rests on the kinds of concrete experiences and grows through the kinds of unforseen developments that are ever the hallmark of experimental knowledge.
I refer to this as an "empirical distinction", in spite of the fact that the domain of experience in question is decidedly a formal one, because it rests on the kinds of concrete experiences and grows through the kinds of unforseen developments that are ever the hallmark of experimental knowledge.
In summary, one has the following problem. There is found an empirical distinction between different kinds of self reference, one that becomes evident and is easy to judge after the onset of their effects has begun to set in, between the kinds of self reference that lead to trouble and the kinds that do not. But what kinds of intuitive features, properties that one could recognize before the fact, would serve to distinguish the immanent and imminent empirical categories before one has gone through the trouble of suffering their effects?
In summary, one has the following problem. There is found an empirical distinction between different kinds of self reference, one that becomes evident and is easy to judge after the onset of their effects has begun to set in, between the kinds of self reference that lead to trouble and the kinds that do not. But what kinds of intuitive features, properties that one could recognize before the fact, would serve to distinguish the immanent and imminent empirical categories before one has gone through the trouble of suffering their effects?
Thus, one has the problem of translating between a given collection of empirical categories and a suitable collection of intuitive categories, the latter being of a kind that can be judged before the facts of experience have become inevitable, hoping thereby to correlate the two dimensions in such a way that the categories of intuition about POSR's can foretell the categories of experience with POSR's.
Thus, one has the problem of translating between a given collection of empirical categories and a suitable collection of intuitive categories, the latter being of a kind that can be judged before the facts of experience have become inevitable, hoping thereby to correlate the two dimensions in such a way that the categories of intuition about POSRs can foretell the categories of experience with POSRs.
2. In a tentative approach to the subject of self reference, I notice a principled distinction between two varieties of self reference, that I call "constitutional", "implicative", or "intrinsic self reference" (ISR) and "extra constitutional", "explicative", or "extrinsic self reference" (ESR), respectively.
2. In a tentative approach to the subject of self reference, I notice a principled distinction between two varieties of self reference, that I call "constitutional", "implicative", or "intrinsic self reference" (ISR) and "extra constitutional", "explicative", or "extrinsic self reference" (ESR), respectively.
In the rest of this section I put aside the question of defining a thing, symbol, or concept in terms of itself, which promises to be an exercise in futility, and consider only the possibility of explaining, explicating, or elaborating a thing, symbol, or concept in terms of itself. In this connection I attach special importance to a particular style of exposition, one that reformulates one's initial idea of an object in terms of the active implications or the effective consequences that its presence in a situation or its recognition and use in an application constitutes for the practical agent concerned. This style of "pragmatic reconstruction" can serve a useful purpose in clarifying the information one possesses about the object, sign, or idea of concern. Properly understood, it is marks the effective reformulation of ideas in ways that are akin to the more reductive sorts of "operational definition", but overall is both more comprehensive and more pointedly related to the pragmatic agent, or the actual interpreter of the symbols and concepts in question.
In the rest of this section I put aside the question of defining a thing, symbol, or concept in terms of itself, which promises to be an exercise in futility, and consider only the possibility of explaining, explicating, or elaborating a thing, symbol, or concept in terms of itself. In this connection I attach special importance to a particular style of exposition, one that reformulates one's initial idea of an object in terms of the active implications or the effective consequences that its presence in a situation or its recognition and use in an application constitutes for the practical agent concerned. This style of "pragmatic reconstruction" can serve a useful purpose in clarifying the information one possesses about the object, sign, or idea of concern. Properly understood, it is marks the effective reformulation of ideas in ways that are akin to the more reductive sorts of "operational definition", but overall is both more comprehensive and more pointedly related to the pragmatic agent, or the actual interpreter of the symbols and concepts in question.
The pending example of a POSR is, of course, the system composed of a pair of sign relations {A, B}, where the nouns and pronouns in each sign relation refer to the hypostatic agents A and B that are known solely as embodiments of the sign relations A and B. But this example, as reduced as it is, already involves an order of complexity that needs to be approached in more discrete stages than the ones enumerated in the current account. Therefore, it helps to take a step back from the full variety of sign relations and to consider related classes of POSR's that are typically simpler in principle.
The pending example of a POSR is, of course, the system composed of a pair of sign relations {A, B}, where the nouns and pronouns in each sign relation refer to the hypostatic agents A and B that are known solely as embodiments of the sign relations A and B. But this example, as reduced as it is, already involves an order of complexity that needs to be approached in more discrete stages than the ones enumerated in the current account. Therefore, it helps to take a step back from the full variety of sign relations and to consider related classes of POSRs that are typically simpler in principle.
1. The first class of POSR's I want to consider is diverse in form and content and has many names, but the feature that seems to unite all its instances is a "self commenting" or "self documenting" character. Typically, this means a "partially self documenting" (PSD) character. As species of formal structures, PSD data structures are rife throughout computer science, and PSD developmental sequences turn up repeatedly in mathematics, logic, and proof theory. For the sake of euphony and ease of reference I collect this class of PSD POSR's under the name of "auto graphs" (AG's).
1. The first class of POSRs I want to consider is diverse in form and content and has many names, but the feature that seems to unite all its instances is a "self commenting" or "self documenting" character. Typically, this means a "partially self documenting" (PSD) character. As species of formal structures, PSD data structures are rife throughout computer science, and PSD developmental sequences turn up repeatedly in mathematics, logic, and proof theory. For the sake of euphony and ease of reference I collect this class of PSD POSRs under the name of "auto graphs" (AGs).
The archetype of all auto graphs is perhaps the familiar model of the natural numbers N as a sequence of sets, each of whose successive sets collects all and only the previous sets of the sequence:
The archetype of all auto graphs is perhaps the familiar model of the natural numbers N as a sequence of sets, each of whose successive sets collects all and only the previous sets of the sequence:
For future reference, I dub this "model of natural numbers" as "MON". The very familiarity of this MON means that one reflexively proceeds from reading the signs of its set notation to thinking of its sets as mathematical objects, with little awareness of the sign relation that mediates the process, or even much reflection after the fact that is independent of the reflections recorded. Thus, even though this MON documents a process of reflective develoment, it need inspire no extra reflection on the acts of understanding needed to follow its directions.
For future reference, I dub this "model of natural numbers" as "MON". The very familiarity of this MON means that one reflexively proceeds from reading the signs of its set notation to thinking of its sets as mathematical objects, with little awareness of the sign relation that mediates the process, or even much reflection after the fact that is independent of the reflections recorded. Thus, even though this MON documents a process of reflective develoment, it need inspire no extra reflection on the acts of understanding needed to follow its directions.
In order to render this MON instructive for the development of a RIF, something intended to be a deliberately "self conscious" construction, it is important to remedy the excessive lucidity of this MON's reflections, the confusing mix of opacity and transparency that comes in proportion to one's very familiarity with an object and that is compounded by one's very fluency in a language. To do this, it is incumbent on a proper analysis of the situation to slow the MON down, to interrupt one's own comprehension of its developing intent, and to articulate the details of the sign process that mediates it much more carefully than is customary.
In order to render this MON instructive for the development of a RIF, something intended to be a deliberately "self conscious" construction, it is important to remedy the excessive lucidity of this MONs reflections, the confusing mix of opacity and transparency that comes in proportion to one's very familiarity with an object and that is compounded by one's very fluency in a language. To do this, it is incumbent on a proper analysis of the situation to slow the MON down, to interrupt one's own comprehension of its developing intent, and to articulate the details of the sign process that mediates it much more carefully than is customary.
These goals can be achieved by singling out the formal language that is used by this MON to denote its set theoretic objects. This involves separating the object domain O = OMON from the sign domain S = SMON, paying closer attention to the naive level of set notation that is actually used by this MON, and treating its primitive set theoretic expressions as a formal language all its own.
These goals can be achieved by singling out the formal language that is used by this MON to denote its set theoretic objects. This involves separating the object domain O = OMON from the sign domain S = SMON, paying closer attention to the naive level of set notation that is actually used by this MON, and treating its primitive set theoretic expressions as a formal language all its own.
2. Reflection principles in propositional calculus. Many statements about the order are also statements in the order. Many statements in the order are already statements about the order.
2. Reflection principles in propositional calculus. Many statements about the order are also statements in the order. Many statements in the order are already statements about the order.
3. Next, I consider a class of POSR's that turns up in group theory./ The next class of POSR's I want to discuss is one that arises in group theory.
3. Next, I consider a class of POSRs that turns up in group theory./ The next class of POSRs I want to discuss is one that arises in group theory.
Although it is seldom recognized, a similar form of self reference appears in the study of "group representations", and more generally, in the study of homomorphic representations of any mathematical structure. In particular, this type of ESR arises from the "regular representation" of a group in terms of its action on itself, that is, in the collection of effects that each element has on the all the individual elements of the group.
Although it is seldom recognized, a similar form of self reference appears in the study of "group representations", and more generally, in the study of homomorphic representations of any mathematical structure. In particular, this type of ESR arises from the "regular representation" of a group in terms of its action on itself, that is, in the collection of effects that each element has on the all the individual elements of the group.
There is a standard form of disclaimer that needs to be attached to this scheme of categories, qualifying any claim that it might be interpreted as making about the ontological status of the proposed distinctions. As often as not, the three categories of systems identified above do not correspond to materially different types of underlying entities so much as different stages in their development, or only in the development of discussions about them. As always, these distinctions do not reveal the essential categories and the substantial divergences of real systems so much as they reflect different ways of viewing them.
There is a standard form of disclaimer that needs to be attached to this scheme of categories, qualifying any claim that it might be interpreted as making about the ontological status of the proposed distinctions. As often as not, the three categories of systems identified above do not correspond to materially different types of underlying entities so much as different stages in their development, or only in the development of discussions about them. As always, these distinctions do not reveal the essential categories and the substantial divergences of real systems so much as they reflect different ways of viewing them.
The need for a note of caution at this point is due to a persistent but unfortunate tendency of the symbol using mentality, one that forms a potentially deleterious side effect to the necessary analytic capacity. Namely, having once discovered the many splendored facets of each real object worth looking into, the mind never ceases from trying to force its imagined "categories of descriptive expressions" (CODE's) down into the original "categories of real entities" (CORE's). In spite of every contrary impression, the deeper lying substrate of existence is solely responsible for funding the phenomenal appearances of the world.
The need for a note of caution at this point is due to a persistent but unfortunate tendency of the symbol using mentality, one that forms a potentially deleterious side effect to the necessary analytic capacity. Namely, having once discovered the many splendored facets of each real object worth looking into, the mind never ceases from trying to force its imagined "categories of descriptive expressions" (CODEs) down into the original "categories of real entities" (COREs). In spite of every contrary impression, the deeper lying substrate of existence is solely responsible for funding the phenomenal appearances of the world.
Out of this tendency of the symbol using mentality arises a constant difficulty with every theory of every reality. Namely, every use of a "theoretical framework" (TF) to view an "underlying reality" (UR) leads the user to forget, temporarily, that the reality is "anything but" (AB) its appearance, image, or representation in that framework. Logically speaking, there is an inalienable spectre of negation involved in every form of apparition, imagination, or representation. This AB negation would be complete if it were not for the possibility held out that some UR's may nevertheless be capable of representing themselves over time.
Out of this tendency of the symbol using mentality arises a constant difficulty with every theory of every reality. Namely, every use of a "theoretical framework" (TF) to view an "underlying reality" (UR) leads the user to forget, temporarily, that the reality is "anything but" (AB) its appearance, image, or representation in that framework. Logically speaking, there is an inalienable spectre of negation involved in every form of apparition, imagination, or representation. This AB negation would be complete if it were not for the possibility held out that some URs may nevertheless be capable of representing themselves over time.
The relationship of objects in an UR to their images in a TF is a topic that this discussion will return to repeatedly as the work progresses. In sum, for now, all of the following statements are approximations to the truth. At any given moment, the image is usually not the object. At times, it can almost be anything but the object. It is even entirely possible, oddly enough, that the image is nothing but the negation of the object, but as often as not it enjoys a more complex relationship than that of sheer opposition. Over time, in some instances, the image can become nearly indistinguishable from its object, but whether this is a good thing or not, in the long run, I cannot tell. The sense of the resulting identification, the bearing of the image on its object, depends on "exactly how" and "how exactly" this final coincidence comes about.
The relationship of objects in an UR to their images in a TF is a topic that this discussion will return to repeatedly as the work progresses. In sum, for now, all of the following statements are approximations to the truth. At any given moment, the image is usually not the object. At times, it can almost be anything but the object. It is even entirely possible, oddly enough, that the image is nothing but the negation of the object, but as often as not it enjoys a more complex relationship than that of sheer opposition. Over time, in some instances, the image can become nearly indistinguishable from its object, but whether this is a good thing or not, in the long run, I cannot tell. The sense of the resulting identification, the bearing of the image on its object, depends on "exactly how" and "how exactly" this final coincidence comes about.
An object system may seem little more than a set, the barest attempt to unify a manifold of interesting phenomena under a common concept, but no object system becomes an object of discussion and thought without invoking the informal precursors of formal systems, in other words, systems of practice, casually taken up, that reflection has the power to formalize in time. And any formal system, put to work in practice, has a temporal and dynamic aspect, especially in the transitions taking place from sign to interpretant sign that fill out its connotative component. Thus, a formal system implicitly involves a temporal system, even if its own object system is not itself temporal in nature but rests in a stable, a static, or an abstract state.
An object system may seem little more than a set, the barest attempt to unify a manifold of interesting phenomena under a common concept, but no object system becomes an object of discussion and thought without invoking the informal precursors of formal systems, in other words, systems of practice, casually taken up, that reflection has the power to formalize in time. And any formal system, put to work in practice, has a temporal and dynamic aspect, especially in the transitions taking place from sign to interpretant sign that fill out its connotative component. Thus, a formal system implicitly involves a temporal system, even if its own object system is not itself temporal in nature but rests in a stable, a static, or an abstract state.
Formal systems and their SOP's are subject to conversion into object systems, becoming the objects of higher order formal systems through the operation of a critical intelectual step usually called "reflection".
Formal systems and their SOPs are subject to conversion into object systems, becoming the objects of higher order formal systems through the operation of a critical intelectual step usually called "reflection".
Using the pragmatic theory of sign relations, I regard every OS in the context of a particular FS. I take these two as one, for now, because an FS and its OS are defined in relation to each other and are not really separable in practice. Later, I will discuss a form of independence that can exist between the two, but only in the derivative sense that many FS's can be brought to bear on what turn out to be equivalent OS's.
Using the pragmatic theory of sign relations, I regard every OS in the context of a particular FS. I take these two as one, for now, because an FS and its OS are defined in relation to each other and are not really separable in practice. Later, I will discuss a form of independence that can exist between the two, but only in the derivative sense that many FSs can be brought to bear on what turn out to be equivalent OSs.
Any physical system, subject to recognizably lawful constraints, can generally be turned to use as a channel of communication, contingent only on the limitations imposed by its inherent informational capacity. Therefore, any OS of sufficient capacity that resides under an agent's interpretive control can used as a medium for language and converted to convey the more specialized FS.
Any physical system, subject to recognizably lawful constraints, can generally be turned to use as a channel of communication, contingent only on the limitations imposed by its inherent informational capacity. Therefore, any OS of sufficient capacity that resides under an agent's interpretive control can used as a medium for language and converted to convey the more specialized FS.
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On the way to integrating dynamic and symbolic approaches to systems there is one important watershed that has to be crossed and recrossed, time and time again. This is a form of continental divide that decides between two alternative and exclusive "modes of description" (MOD's) or "categories of representation" (COR's), and marks a writer's moment to moment selection of "extensional representation" (ER), on the one side, or "intensional representation" (IR), on the other. To apply the theme, in this section I address the task of building conceptual bridges between two different ways of describing or representing sign relations: (1) the ER that describes a sign relation in terms of its instances, and (2) the IR that describes a sign relation in terms of its properties.
On the way to integrating dynamic and symbolic approaches to systems there is one important watershed that has to be crossed and recrossed, time and time again. This is a form of continental divide that decides between two alternative and exclusive "modes of description" (MODs) or "categories of representation" (CORs), and marks a writer's moment to moment selection of "extensional representation" (ER), on the one side, or "intensional representation" (IR), on the other. To apply the theme, in this section I address the task of building conceptual bridges between two different ways of describing or representing sign relations: (1) the ER that describes a sign relation in terms of its instances, and (2) the IR that describes a sign relation in terms of its properties.
It is best to begin the work of bridge building on informal grounds, using concrete examples of ER's and IR's and taking advantage of basic ideas about their relationship that are readily available to every reader. After the overall scheme of construction is roughed out in this fashion, I plan to revisit the concept of representation in a more formal style, examining the balance of its in and ex "tensions" with a sharper eye to the relevant details and a greater chance of compassing the depths of form that arise between the two points of view.
It is best to begin the work of bridge building on informal grounds, using concrete examples of ERs and IRs and taking advantage of basic ideas about their relationship that are readily available to every reader. After the overall scheme of construction is roughed out in this fashion, I plan to revisit the concept of representation in a more formal style, examining the balance of its in and ex "tensions" with a sharper eye to the relevant details and a greater chance of compassing the depths of form that arise between the two points of view.
The task of building this bridge is not trivial. In places, the basic elements of construction are yet to be forged from the available stocks, in others, the needed materials still lie in their ores, awaiting a suitable process to extract them, refine them, and bring them to a usable state. Due to the difficulties of this task and the length of time it will take to carry it out, I think it is advisable to establish two points of reference before setting to work.
The task of building this bridge is not trivial. In places, the basic elements of construction are yet to be forged from the available stocks, in others, the needed materials still lie in their ores, awaiting a suitable process to extract them, refine them, and bring them to a usable state. Due to the difficulties of this task and the length of time it will take to carry it out, I think it is advisable to establish two points of reference before setting to work.
Terms referring to properties of sign relations make it possible to formulate propositions about sign relations, either as occasioned by a clear and present example or in abstraction from any concrete instance. In turn, this makes it possible to carry on chains of reasoning about the properties of sign relations in detachment from the presence of actual cases that may or may not come to mind in the immediate present. This mode of abstraction, invoking the kind of IR that is involved in mediating every form of propositional reasoning, gives logic its wings and can lead to theories of great conceptual power, but it incurs the risk of leading reasoning astray into realms of irreferent pretension, eventually degenerating into spurious sounds that signify nothing.
Terms referring to properties of sign relations make it possible to formulate propositions about sign relations, either as occasioned by a clear and present example or in abstraction from any concrete instance. In turn, this makes it possible to carry on chains of reasoning about the properties of sign relations in detachment from the presence of actual cases that may or may not come to mind in the immediate present. This mode of abstraction, invoking the kind of IR that is involved in mediating every form of propositional reasoning, gives logic its wings and can lead to theories of great conceptual power, but it incurs the risk of leading reasoning astray into realms of irreferent pretension, eventually degenerating into spurious sounds that signify nothing.
It is only by means of an IR that logical reasoning, properly speaking, is able to begin. The stringency of this precept, if it is taken too strictly as a starting condition and applied solely in absolute terms, would be correctly perceived as demanding a provision that is jarring to every brand of good sense. But it was never meant to be taken this severely. In practice, the starkness of this tentative stipulation is moderated by the degree of fuzziness that still continues to reside in the interpretive distinction between ER's and IR's.
It is only by means of an IR that logical reasoning, properly speaking, is able to begin. The stringency of this precept, if it is taken too strictly as a starting condition and applied solely in absolute terms, would be correctly perceived as demanding a provision that is jarring to every brand of good sense. But it was never meant to be taken this severely. In practice, the starkness of this tentative stipulation is moderated by the degree of fuzziness that still continues to reside in the interpretive distinction between ERs and IRs.
The alleged distinction between ER's and IR's, when it is projected to have a global application, remains arbitrary so long as it is taken at that level of abstraction, and it comes to take on the semblance of a definition only in relation to the interpretive conduct of a particular arbiter. No representation in actual practice is purely of one sort or the other, nor fails to have the characters of both types as a part of its mix. In other words, extensions and intensions are only abstractions from a profounder "tension" that is logically prior but functionally intermediate to them both, and every representation of any use will have its aspect of extensional particularity permeated by its aspect of intensional generality.
The alleged distinction between ERs and IRs, when it is projected to have a global application, remains arbitrary so long as it is taken at that level of abstraction, and it comes to take on the semblance of a definition only in relation to the interpretive conduct of a particular arbiter. No representation in actual practice is purely of one sort or the other, nor fails to have the characters of both types as a part of its mix. In other words, extensions and intensions are only abstractions from a profounder "tension" that is logically prior but functionally intermediate to them both, and every representation of any use will have its aspect of extensional particularity permeated by its aspect of intensional generality.
Toward the end of this construction I hope it will become clear that this bridge is a project intermediate in scale between the elementary linkage of signs to interpretants that is built into every sign relation and all the courses of conduct that go to span the gulf and build communication between vastly different systems of interpretation. In the meantime, there are strong analogies that make the architecture of this bridge parallel in form to the structures existing at both ends of the scale, shaping it in congruence with patterns of action that reside at both the micro and the macro levels. Observing these similarities and their lines of potential use as they arise will serve to guide the current work.
Toward the end of this construction I hope it will become clear that this bridge is a project intermediate in scale between the elementary linkage of signs to interpretants that is built into every sign relation and all the courses of conduct that go to span the gulf and build communication between vastly different systems of interpretation. In the meantime, there are strong analogies that make the architecture of this bridge parallel in form to the structures existing at both ends of the scale, shaping it in congruence with patterns of action that reside at both the micro and the macro levels. Observing these similarities and their lines of potential use as they arise will serve to guide the current work.
A sign relation is a complex object and its representations, insofar as they faithfully preserve its structure, are complex signs. Accordingly, the problems of translating between ER's and IR's of sign relations, of detecting when representations alleged to be of sign relations do indeed represent objects of the specified character, and of recognizing whether different representations do or do not represent the same sign relation as their common object — these are the familiar questions that would be asked of the signs and interpretants in a simple sign relation, but this time asked at a higher level, in regard to the complex signs and complex interpretants that are posed by the different stripes of representation. At the same time, it should be obvious that these are also the natural questions to be faced in building a bridge between representations.
A sign relation is a complex object and its representations, insofar as they faithfully preserve its structure, are complex signs. Accordingly, the problems of translating between ERs and IRs of sign relations, of detecting when representations alleged to be of sign relations do indeed represent objects of the specified character, and of recognizing whether different representations do or do not represent the same sign relation as their common object — these are the familiar questions that would be asked of the signs and interpretants in a simple sign relation, but this time asked at a higher level, in regard to the complex signs and complex interpretants that are posed by the different stripes of representation. At the same time, it should be obvious that these are also the natural questions to be faced in building a bridge between representations.
How many different sorts of entities are conceivably involved in translating between ER's and IR's of sign relations? To address this question it helps to introduce a system of type notations that can be used to keep track of the various sorts of things, or the varieties of objects of thought, that are generated in the process of answering it. Table 47.1 summarizes the basic types of things that are needed in this pursuit, while the rest can be derived by constructions of the form "X of Y", notated "X(Y)" or just "XY", for any basic types X and Y. The constructed types of things involved in the ER's and IR's of sign relations are listed in Tables 47.2 and 47.3, respectively.
How many different sorts of entities are conceivably involved in translating between ERs and IRs of sign relations? To address this question it helps to introduce a system of type notations that can be used to keep track of the various sorts of things, or the varieties of objects of thought, that are generated in the process of answering it. Table 47.1 summarizes the basic types of things that are needed in this pursuit, while the rest can be derived by constructions of the form "X of Y", notated "X(Y)" or just "XY", for any basic types X and Y. The constructed types of things involved in the ERs and IRs of sign relations are listed in Tables 47.2 and 47.3, respectively.
Table 47.1 Basic Types for ER's & IR's of Sign Relations
Table 47.1 Basic Types for ERs & IRs of Sign Relations
Type Symbol
Type Symbol
Property P
Property P
Underlying Element U
Underlying Element U
Table 47.2 Derived Types for ER's of Sign Relations
Table 47.2 Derived Types for ERs of Sign Relations
Type Symbol Construction
Type Symbol Construction
Relation R S(T(U))
Relation R S(T(U))
Table 47.3 Derived Types for IR's of Sign Relations
Table 47.3 Derived Types for IRs of Sign Relations
Type Symbol Construction
Type Symbol Construction
Relation P(R) P(S(T(U)))
Relation P(R) P(S(T(U)))
Let S be the type of signs, S the type of sets, T the type of triples, and U the type of underlying objects. Now consider the various sorts of things, or the varieties of objects of thought, that are invoked on each side, annotating each type as it is mentioned:
Let S be the type of signs, S the type of sets, T the type of triples, and U the type of underlying objects. Now consider the various sorts of things, or the varieties of objects of thought, that are invoked on each side, annotating each type as it is mentioned:
1. ER's of sign relations describe them as sets (S's) of triples (T's) of underlying elements (U's). This makes for three levels of objective structure that must be put in coordination with each other, a task that is projected to be carried out in the appropriate OF of sign relations. Corresponding to this aspect of structure in the OF, there is a parallel aspect of structure in the IF of sign relations. Namely, the accessory sign relations that are used to discuss a targeted sign relation need to have signs for sets (SS's), signs for triples (ST's), and signs for the underlying elements (SU's). This accounts for three levels of syntactic structure in the IF of sign relations that must be coordinated with each other and also with the targeted levels of objective structure.
1. ERs of sign relations describe them as sets (Ss) of triples (Ts) of underlying elements (Us). This makes for three levels of objective structure that must be put in coordination with each other, a task that is projected to be carried out in the appropriate OF of sign relations. Corresponding to this aspect of structure in the OF, there is a parallel aspect of structure in the IF of sign relations. Namely, the accessory sign relations that are used to discuss a targeted sign relation need to have signs for sets (SSs), signs for triples (STs), and signs for the underlying elements (SUs). This accounts for three levels of syntactic structure in the IF of sign relations that must be coordinated with each other and also with the targeted levels of objective structure.
2. IR's of sign relations describe them in terms of properties (P's) that are taken as primitive entities in their own right. / refer to properties (P's) of transactions (T's) of underlying elements (U's).
2. IRs of sign relations describe them in terms of properties (Ps) that are taken as primitive entities in their own right. / refer to properties (Ps) of transactions (Ts) of underlying elements (Us).
2. IR's of sign relations refer to properties of sets (PS's), properties of triples (PT's), and properties of underlying elements (PU's). This amounts to three more levels of objective structure in the OF of the IR that need to be coordinated with each other and interlaced with the OF of the ER if the two are to be brought into the same discussion, possibly for the purpose of translating either into the other. Accordingly, the accessory sign relations that are used to discuss an IR of a targeted sign relation need to have SPS's, SPT's, and SPU's.
2. IRs of sign relations refer to properties of sets (PSs), properties of triples (PTs), and properties of underlying elements (PUs). This amounts to three more levels of objective structure in the OF of the IR that need to be coordinated with each other and interlaced with the OF of the ER if the two are to be brought into the same discussion, possibly for the purpose of translating either into the other. Accordingly, the accessory sign relations that are used to discuss an IR of a targeted sign relation need to have SPSs, SPTs, and SPUs.
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Up to this point, the concept of a sign relation has been discussed largely in terms of ER's. The sign relations A and B were initially described as collections of transactions among three participants and formalized as sets of triples of underlying elements.
Up to this point, the concept of a sign relation has been discussed largely in terms of ERs. The sign relations A and B were initially described as collections of transactions among three participants and formalized as sets of triples of underlying elements.
Other examples of ER's are widely distributed throughout the foregoing discussion of A and B. The extensional mode of description is prevalent, not only in the presentation of sign relations by means of relational data tables, but also in the presentation of dyadic projections by means of digraphs. This manner of presentation follows the natural order of acquaintance with abstract relations, since the extensional mode of description is the category of representation that usually prevails whenever it is necessary to provide a detailed treatment of simple examples or an exhaustive account of individual instances.
Other examples of ERs are widely distributed throughout the foregoing discussion of A and B. The extensional mode of description is prevalent, not only in the presentation of sign relations by means of relational data tables, but also in the presentation of dyadic projections by means of digraphs. This manner of presentation follows the natural order of acquaintance with abstract relations, since the extensional mode of description is the category of representation that usually prevails whenever it is necessary to provide a detailed treatment of simple examples or an exhaustive account of individual instances.
Starting from a standpoint in concrete constructions, the easiest way to begin developing an explicit treatment of ER's is to gather the relevant materials in the forms already presented, to fill out the missing details and expand the abbreviated contents of these forms, and to review their full structures in a more formal light. Consequently, this section inaugurates the formal discussion of ER's by taking a second look at the interpreters A and B, recollecting the Tables of their sign relations and finishing up the Tables of their dyadic components. Since the form of the sign relations A and B no longer presents any novelty, I can exploit their second presentation as a first opportunity to examine a selection of finer points, previously overlooked. Also, in the process of reviewing this material it is useful to anticipate a number of incidental issues that are reaching the point of becoming critical within this discussion and to begin introducing the generic types of technical devices that are needed to deal with them.
Starting from a standpoint in concrete constructions, the easiest way to begin developing an explicit treatment of ERs is to gather the relevant materials in the forms already presented, to fill out the missing details and expand the abbreviated contents of these forms, and to review their full structures in a more formal light. Consequently, this section inaugurates the formal discussion of ERs by taking a second look at the interpreters A and B, recollecting the Tables of their sign relations and finishing up the Tables of their dyadic components. Since the form of the sign relations A and B no longer presents any novelty, I can exploit their second presentation as a first opportunity to examine a selection of finer points, previously overlooked. Also, in the process of reviewing this material it is useful to anticipate a number of incidental issues that are reaching the point of becoming critical within this discussion and to begin introducing the generic types of technical devices that are needed to deal with them.
The next set of Tables summarizes the ER's of A and B. For ease of reference, Tables 48.1 and 49.1 repeat the contents of Tables 1 and 2, respectively, the only difference being that appearances of ordinary quotation marks ("...") are transcribed as invocations of the so called "arch operator" (<...>). The reason for this slight change of notation will be explained shortly. The denotative components Den A and Den B are shown in the first two columns of Tables 48.2 and 49.2, respectively, while the third column gives the transition from sign to object as an ordered pair <s, o>. The connotative components Con A and Con B are shown in the first two columns of Tables 48.3 and 49.3, respectively, while the third column gives the transition from sign to interpretant as an ordered pair <s, i>.
The next set of Tables summarizes the ERs of A and B. For ease of reference, Tables 48.1 and 49.1 repeat the contents of Tables 1 and 2, respectively, the only difference being that appearances of ordinary quotation marks ("...") are transcribed as invocations of the so called "arch operator" (<...>). The reason for this slight change of notation will be explained shortly. The denotative components Den A and Den B are shown in the first two columns of Tables 48.2 and 49.2, respectively, while the third column gives the transition from sign to object as an ordered pair <s, o>. The connotative components Con A and Con B are shown in the first two columns of Tables 48.3 and 49.3, respectively, while the third column gives the transition from sign to interpretant as an ordered pair <s, i>.
Table 48.1 ER (A): Extensional Representation of A
Table 48.1 ER (A): Extensional Representation of A
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The next three sections beginning with this one consider how the ER's of A and B can be translated into a variety of different IR's. For the purposes of this introduction, only "faithful" translations between the different categories of representation are contemplated. This means that the conversion from ER to IR is intended to convey what is essentially the same information about A and B, to preserve all the relevant structural details that are implied by their various modes of description, but to do it in a way that brings selected aspects of their objective forms to light. General considerations surrounding the task of translation are taken up in this section, while the next two sections lay out different ways of carrying it through.
The next three sections beginning with this one consider how the ERs of A and B can be translated into a variety of different IRs. For the purposes of this introduction, only "faithful" translations between the different categories of representation are contemplated. This means that the conversion from ER to IR is intended to convey what is essentially the same information about A and B, to preserve all the relevant structural details that are implied by their various modes of description, but to do it in a way that brings selected aspects of their objective forms to light. General considerations surrounding the task of translation are taken up in this section, while the next two sections lay out different ways of carrying it through.
The larger purpose of this discussion is to serve as an introduction, not just to the special topic of devising IR's for sign relations, but to the general issue of producing, using, and comprehending IR's for any kind of relation or any domain of formal objects. It is hoped that a careful study of these simple IR's can inaugurate a degree of insight into the broader arenas of formalism of which they occupy an initial niche and into the wider landscapes of discourse of which they inhabit a natural corner, in time progressing up to the axiomatic presentation of formal theories about combinatorial domains and other mathematical objects.
The larger purpose of this discussion is to serve as an introduction, not just to the special topic of devising IRs for sign relations, but to the general issue of producing, using, and comprehending IRs for any kind of relation or any domain of formal objects. It is hoped that a careful study of these simple IRs can inaugurate a degree of insight into the broader arenas of formalism of which they occupy an initial niche and into the wider landscapes of discourse of which they inhabit a natural corner, in time progressing up to the axiomatic presentation of formal theories about combinatorial domains and other mathematical objects.
For the sake of maximum clarity and reusability of results, I begin by articulating the abstract skeleton of the paradigm structure, treating the sign relations A and B as sundry aspects of a single, unitary, but still uninterpreted object. Then I return at various successive stages to differentiate and individualize the two interpreters, to arrange more functional flesh on the basis provided by their structural bones, and to illustrate how their bare forms can be arrayed in many different styles of qualitative detail.
For the sake of maximum clarity and reusability of results, I begin by articulating the abstract skeleton of the paradigm structure, treating the sign relations A and B as sundry aspects of a single, unitary, but still uninterpreted object. Then I return at various successive stages to differentiate and individualize the two interpreters, to arrange more functional flesh on the basis provided by their structural bones, and to illustrate how their bare forms can be arrayed in many different styles of qualitative detail.
In building connections between ER's and IR's of sign relations the discussion turns on two types of partially ordered sets, or "posets". Suppose that R is one of the sign relations A or B, and let ER (R) be an ER of R.
In building connections between ERs and IRs of sign relations the discussion turns on two types of partially ordered sets, or "posets". Suppose that R is one of the sign relations A or B, and let ER (R) be an ER of R.
In the sign relations A and B, both of their ER's are based on a common world set:
In the sign relations A and B, both of their ERs are based on a common world set:
W = { A, B, "A", "B", "i", "u"}.
W = { A, B, "A", "B", "i", "u"}.
There is a foundational issue arising in this context that I do not pretend to fully understand and cannot attempt to finally dispatch. What I do understand I will try to express in terms of an aesthetic principle: On balance, it seems best to regard extensional elements and intensional features as independently given entities. This involves treating points and properties as fundamental realities in their own rights, placing them on an equal basis with each other, and seeking their relation to each other, but not trying to reduce one to the other.
There is a foundational issue arising in this context that I do not pretend to fully understand and cannot attempt to finally dispatch. What I do understand I will try to express in terms of an aesthetic principle: On balance, it seems best to regard extensional elements and intensional features as independently given entities. This involves treating points and properties as fundamental realities in their own rights, placing them on an equal basis with each other, and seeking their relation to each other, but not trying to reduce one to the other.
The discussion is now specialized to consider the IR's of the sign relations A and B, their denotative projections as the digraphs Den (A) and Den (B), and their connotative projections as the digraphs Con (A) and Con (B). In doing this I take up two different strategies of representation:
The discussion is now specialized to consider the IRs of the sign relations A and B, their denotative projections as the digraphs Den (A) and Den (B), and their connotative projections as the digraphs Con (A) and Con (B). In doing this I take up two different strategies of representation:
1. The first strategy is called the "literal coding", because it sticks to obvious features of each syntactic element to contrive its code, or the "O(n) coding", because it uses a number on the order of n logical features to represent a domain of n elements.
1. The first strategy is called the "literal coding", because it sticks to obvious features of each syntactic element to contrive its code, or the "O(n) coding", because it uses a number on the order of n logical features to represent a domain of n elements.
Fragments
Fragments
In the formalized examples of IR's to be presented in this work, I will keep to the level of logical reasoning that is usually referred to as "propositional calculus" or "sentential logic" (SL).
In the formalized examples of IRs to be presented in this work, I will keep to the level of logical reasoning that is usually referred to as "propositional calculus" or "sentential logic" (SL).
The constrast between ER's and IR's is strongly correlated with another dimension of interest in the study of inquiry, namely, the tension between empirical and rational modes of inquiry.
The constrast between ERs and IRs is strongly correlated with another dimension of interest in the study of inquiry, namely, the tension between empirical and rational modes of inquiry.
This section begins the explicit discussion of ER's by taking a second look at the sign relations A and B. Since the form of these examples no longer presents any novelty, this second presentation of A and B provides a first opportunity to introduce some new material. In the process of reviewing this material, it is useful to anticipate a number of incidental issues that are on the point of becoming critical, and to begin introducing the generic types of technical devices that are needed to deal with them.
This section begins the explicit discussion of ERs by taking a second look at the sign relations A and B. Since the form of these examples no longer presents any novelty, this second presentation of A and B provides a first opportunity to introduce some new material. In the process of reviewing this material, it is useful to anticipate a number of incidental issues that are on the point of becoming critical, and to begin introducing the generic types of technical devices that are needed to deal with them.
Therefore, the easiest way to begin an explicit treatment of ER's is by recollecting the Tables of the sign relations A and B and by finishing the corresponding Tables of their dyadic components. Since the form of the sign relations A and B no longer presents any novelty, I can use the second presentation of these examples as a first opportunity to examine a selection of their finer points, previously overlooked.
Therefore, the easiest way to begin an explicit treatment of ERs is by recollecting the Tables of the sign relations A and B and by finishing the corresponding Tables of their dyadic components. Since the form of the sign relations A and B no longer presents any novelty, I can use the second presentation of these examples as a first opportunity to examine a selection of their finer points, previously overlooked.
Starting from this standpoint, the easiest way to begin developing an explicit treatment of ER's is to gather the relevant materials in the forms already presented, to fill out their missing details and expand the abbreviated contents of these forms, and to review their full structures in a more formal light.
Starting from this standpoint, the easiest way to begin developing an explicit treatment of ERs is to gather the relevant materials in the forms already presented, to fill out their missing details and expand the abbreviated contents of these forms, and to review their full structures in a more formal light.
Because of the perfect parallelism that the literal coding contrives between individual signs and grammatical categories, this arrangement illustrates not so much a code transformation as a re interpretation of the original signs under different headings.
Because of the perfect parallelism that the literal coding contrives between individual signs and grammatical categories, this arrangement illustrates not so much a code transformation as a re interpretation of the original signs under different headings.
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In this section I prepare the grounds for building bridges between ER's and IR's of sign relations. To establish an initial foothold on either side of the distinction and to gain a first march on connecting the two sites of the intended construction, I introduce an intermediate mode of description called a "literal intensional representation" (LIR).
In this section I prepare the grounds for building bridges between ERs and IRs of sign relations. To establish an initial foothold on either side of the distinction and to gain a first march on connecting the two sites of the intended construction, I introduce an intermediate mode of description called a "literal intensional representation" (LIR).
Any LIR is a nominal form of IR that has exactly the same level of detail as an ER, merely shifting the interpretation of primitive terms from an extensional to an intensional modality, namely, from a frame of reference terminating in "points", "atomic elements", "elementary objects", or "real particulars" to a frame of reference terminating in "qualities", "basic features", "fundamental properties", or "simple propositions". This modification, that translates the entire set of elementary objects in an ER into a parallel set of fundamental properties in a LIR, constitutes a form of modulation that can be subtle or trivial, depending on one's point of view. Regarded as trivial, it tends to go unmarked, leaving it up to the judgment of the interpreter to decide whether the same sign is meant to denote a point, a particular, a property, or a proposition. An interpretive variance that goes unstated tends to be treated as final. It is always possible to bring in more signs in an attempt to signify the variants intended, but it needs to be noted that every effort to control the interpretive variance by means of these epithets and expletives only increases the level of liability for accidental errors, if not the actual probability of misinterpretation. For the sake of this introduction, and in spite of these risks, I treat the distinction between extensional and intensional modes of interpretation as worthy of note and deserving of an explicit notation.
Any LIR is a nominal form of IR that has exactly the same level of detail as an ER, merely shifting the interpretation of primitive terms from an extensional to an intensional modality, namely, from a frame of reference terminating in "points", "atomic elements", "elementary objects", or "real particulars" to a frame of reference terminating in "qualities", "basic features", "fundamental properties", or "simple propositions". This modification, that translates the entire set of elementary objects in an ER into a parallel set of fundamental properties in a LIR, constitutes a form of modulation that can be subtle or trivial, depending on one's point of view. Regarded as trivial, it tends to go unmarked, leaving it up to the judgment of the interpreter to decide whether the same sign is meant to denote a point, a particular, a property, or a proposition. An interpretive variance that goes unstated tends to be treated as final. It is always possible to bring in more signs in an attempt to signify the variants intended, but it needs to be noted that every effort to control the interpretive variance by means of these epithets and expletives only increases the level of liability for accidental errors, if not the actual probability of misinterpretation. For the sake of this introduction, and in spite of these risks, I treat the distinction between extensional and intensional modes of interpretation as worthy of note and deserving of an explicit notation.
u s3 w6
u s3 w6
If the world of A and B, the set W = {A, B, "A", "B", "i", "u"}, is viewed abstractly, as an arbitrary set of six atomic points, then there are exactly 26 = 64 "abstract properties" (AP's) or "potential attributes" (PA's) that might be applied to or recognized in these points. The extensions of these AP's are the subsets of W, otherwise known as members of the "power set" Pow (W). In order to make this way of talking about properties consistent with the previous definition of reality, it is necessary to say that one potential property is never realized, since no point has it, and its extension is the empty set {}. All the "natural" properties of points that one observes in a concrete situation, properties whose extensions are known as "natural kinds", can be recognized among the "abstract", "arbitrary", or "set theoretic" properties that are systematically generated in this way. Typically, however, many of these abstract properties will not be recognized as falling among the more natural kinds.
If the world of A and B, the set W = {A, B, "A", "B", "i", "u"}, is viewed abstractly, as an arbitrary set of six atomic points, then there are exactly 26 = 64 "abstract properties" (APs) or "potential attributes" (PAs) that might be applied to or recognized in these points. The extensions of these APs are the subsets of W, otherwise known as members of the "power set" Pow (W). In order to make this way of talking about properties consistent with the previous definition of reality, it is necessary to say that one potential property is never realized, since no point has it, and its extension is the empty set {}. All the "natural" properties of points that one observes in a concrete situation, properties whose extensions are known as "natural kinds", can be recognized among the "abstract", "arbitrary", or "set theoretic" properties that are systematically generated in this way. Typically, however, many of these abstract properties will not be recognized as falling among the more natural kinds.
Tables 54.1, 54.2, and 54.3 show three different ways of representing the elements of the world set W as vectors in the coordinate space W and as singular propositions in the universe of discourse W[]. Altogether, these Tables present the "literal" codes for the elements of W and W[] in their "mnemonic", "pragmatic", and "abstract" versions, respectively. In each Table, Column 1 lists the element w C W, while Column 2 gives the corresponding coordinate vector w C W in the form of a bit string. The next two Columns represent each w C W as a proposition in W[], in effect, reconstituting it as a function w : W >B. Column 3 shows the propositional expression of each element in the form of a conjunct term, in other words, as a logical product of positive and negative features. Column 4 gives the compact code for each element, using a conjunction of positive features in subscripted angle brackets to represent the singular proposition corresponding to each element.
Tables 54.1, 54.2, and 54.3 show three different ways of representing the elements of the world set W as vectors in the coordinate space W and as singular propositions in the universe of discourse W[]. Altogether, these Tables present the "literal" codes for the elements of W and W[] in their "mnemonic", "pragmatic", and "abstract" versions, respectively. In each Table, Column 1 lists the element w C W, while Column 2 gives the corresponding coordinate vector w C W in the form of a bit string. The next two Columns represent each w C W as a proposition in W[], in effect, reconstituting it as a function w : W >B. Column 3 shows the propositional expression of each element in the form of a conjunct term, in other words, as a logical product of positive and negative features. Column 4 gives the compact code for each element, using a conjunction of positive features in subscripted angle brackets to represent the singular proposition corresponding to each element.
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In this section the ER's of A and B are translated into a variety of different IR's that actually accomplish some measure of analytic work. These are referred to as "analytic intensional representations" (AIR's). This strategy of representation is also called the "structural coding" or the "sensitive coding", because it pays attention to the structure of its object domain and attends to the nuances of each sign's interpretation to fashion its code, or the "log(n) coding", because it uses roughly log2(n) binary features to represent a domain of n elements.
In this section the ERs of A and B are translated into a variety of different IRs that actually accomplish some measure of analytic work. These are referred to as "analytic intensional representations" (AIRs). This strategy of representation is also called the "structural coding" or the "sensitive coding", because it pays attention to the structure of its object domain and attends to the nuances of each sign's interpretation to fashion its code, or the "log(n) coding", because it uses roughly log2(n) binary features to represent a domain of n elements.
For the domain O = {A, B} of two elements one needs to use a single logical feature. It is often convenient to use an object feature that is relative to the interpreter using it, for instance, telling whether the object described is the self or the other.
For the domain O = {A, B} of two elements one needs to use a single logical feature. It is often convenient to use an object feature that is relative to the interpreter using it, for instance, telling whether the object described is the self or the other.
The symbol "O ", read "thus", "then", or "yields" can be used to mark sequential inferences, allowing for expressions like "x & dx O (x)". In each case, a suitable context of temporal moments <t, t'> is understood to underlie the inference.
The symbol "O ", read "thus", "then", or "yields" can be used to mark sequential inferences, allowing for expressions like "x & dx O (x)". In each case, a suitable context of temporal moments <t, t'> is understood to underlie the inference.
A "sequential inference constraint" (SIC) is a logical condition that applies to a temporal system, providing information about the kinds of SI that apply to the system in a hopefully large number of situations. Typically, a SIC is formulated in intensional terms and expressed by means of a collection of SI rules or schemas that tell what SI's apply to the system in particular situations. Since it has the status of logical theory about an empirical system, a SIC is subject to being reformulated in terms of its set theoretic extension, and it can be established as existing in the customary sort of dual relationship with this extension. Logically, it determines, and, empirically, it is determined by the corresponding set of "SI triples", the <x, y, z> such that x & y O z. The set theoretic extension of a SIC is thus a certain triadic relation, generically denoted by "O", where O c X.dX.X is defined as follows:
A "sequential inference constraint" (SIC) is a logical condition that applies to a temporal system, providing information about the kinds of SI that apply to the system in a hopefully large number of situations. Typically, a SIC is formulated in intensional terms and expressed by means of a collection of SI rules or schemas that tell what SIs apply to the system in particular situations. Since it has the status of logical theory about an empirical system, a SIC is subject to being reformulated in terms of its set theoretic extension, and it can be established as existing in the customary sort of dual relationship with this extension. Logically, it determines, and, empirically, it is determined by the corresponding set of "SI triples", the <x, y, z> such that x & y O z. The set theoretic extension of a SIC is thus a certain triadic relation, generically denoted by "O", where O c X.dX.X is defined as follows:
O = {<x, y, z> C X.dX.X : x & y O z}.
O = {<x, y, z> C X.dX.X : x & y O z}.
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There are many reasons for using IR's to describe formal objects, especially as the size and complexity of these objects grows beyond the bounds of finite information capacities to represent in practical terms. This is extremely pertinent to the progress of the present discussion. As often happens, when a top down investigation of complex families of formal objects actually succeeds in arriving at examples that are simple enough to contemplate in extensional terms, it can be difficult to see the relation of such impoverished examples to the cases of original interest, all of them typically having infinite cardinality and indefinite complexity. In short, once a discussion is brought down to the level of its smallest cases it can be nearly impossible to bring it back up to the level of its intended application. Without invoking IR's of sign relations there is little hope that this discussion can rise far beyond its present level, eternally elaborating the subtleties of cases as elementary as A and B.
There are many reasons for using IRs to describe formal objects, especially as the size and complexity of these objects grows beyond the bounds of finite information capacities to represent in practical terms. This is extremely pertinent to the progress of the present discussion. As often happens, when a top down investigation of complex families of formal objects actually succeeds in arriving at examples that are simple enough to contemplate in extensional terms, it can be difficult to see the relation of such impoverished examples to the cases of original interest, all of them typically having infinite cardinality and indefinite complexity. In short, once a discussion is brought down to the level of its smallest cases it can be nearly impossible to bring it back up to the level of its intended application. Without invoking IRs of sign relations there is little hope that this discussion can rise far beyond its present level, eternally elaborating the subtleties of cases as elementary as A and B.
There are many obstacles to building this bridge, but if these forms of obstruction are understood in the proper fashion, it is possible to use them as stepping stones, to capitalize on their redoubtable structures, and to convert their recalcitrant materials into a formal calculus that can serve the aims and means of instruction.
There are many obstacles to building this bridge, but if these forms of obstruction are understood in the proper fashion, it is possible to use them as stepping stones, to capitalize on their redoubtable structures, and to convert their recalcitrant materials into a formal calculus that can serve the aims and means of instruction.
Inverse projections are often referred to as "extensions", in spite of the conflict this creates with the "extensions" of terms, concepts, and sets.
Inverse projections are often referred to as "extensions", in spite of the conflict this creates with the "extensions" of terms, concepts, and sets.
One of the standard turns of phrase that finds use in this setting, not only for translating between ER's and IR's, but for converting both into computational forms, is to associate any set S contained in a space X with two other types of formal objects: (1) a logical proposition pS known as the characteristic, indicative, or selective proposition of S, and (2) a binary valued function fS: X >B known as the characteristic, indicative, or selective function of S.
One of the standard turns of phrase that finds use in this setting, not only for translating between ERs and IRs, but for converting both into computational forms, is to associate any set S contained in a space X with two other types of formal objects: (1) a logical proposition pS known as the characteristic, indicative, or selective proposition of S, and (2) a binary valued function fS: X >B known as the characteristic, indicative, or selective function of S.
Strictly speaking, the logical entity pS is the IR of the tribe, presiding at the highest level of abstraction, while fS and S are its concrete ER's, rendering its concept in functional and geometric materials, respectively. Whenever it is possible to do so without confusion, I try to use identical or similar names for the corresponding objects and species of each type, and I generally ignore the distinctions that otherwise set them apart. For instance, in moving toward computational settings, fS makes the best computational proxy for pS, so I commonly refer to the mapping fS: X >B as a "proposition" on X.
Strictly speaking, the logical entity pS is the IR of the tribe, presiding at the highest level of abstraction, while fS and S are its concrete ERs, rendering its concept in functional and geometric materials, respectively. Whenever it is possible to do so without confusion, I try to use identical or similar names for the corresponding objects and species of each type, and I generally ignore the distinctions that otherwise set them apart. For instance, in moving toward computational settings, fS makes the best computational proxy for pS, so I commonly refer to the mapping fS: X >B as a "proposition" on X.
Regarded as logical models, the elements of the contension P&Q satisfy the proposition referred to as the "conjunction of extensions" P' and Q'.
Regarded as logical models, the elements of the contension P&Q satisfy the proposition referred to as the "conjunction of extensions" P' and Q'.
As a way of resolving the discriminated "tensions", posed here to fall into "ex " and "in " kinds, the strategy just described affords a way of approaching the problem that is less like a bridge than a pole vault, taking its pivot on a fixed set of narrowly circumscribed sign relations to make a transit from extensional to intensional outlooks on their form. With time and reflection, the logical depth of the supposed distinction, the "pretension" of maintaining a couple of separate but equal tensions in isolation from each other, does not withstand a persistent probing. Accordingly, the gulf between the two realms can always be fathomed by a finitely informed creature, in fact, by the very form of interpreter that created the fault in the first place. Consequently, converting the form of a transient vault into the substance of a usable bridge requires in adjunction only that initially pliable and ultimately tensile sorts of connecting lines be conducted along the tracery of the vault until the work of castling the gap can begin.
As a way of resolving the discriminated "tensions", posed here to fall into "ex " and "in " kinds, the strategy just described affords a way of approaching the problem that is less like a bridge than a pole vault, taking its pivot on a fixed set of narrowly circumscribed sign relations to make a transit from extensional to intensional outlooks on their form. With time and reflection, the logical depth of the supposed distinction, the "pretension" of maintaining a couple of separate but equal tensions in isolation from each other, does not withstand a persistent probing. Accordingly, the gulf between the two realms can always be fathomed by a finitely informed creature, in fact, by the very form of interpreter that created the fault in the first place. Consequently, converting the form of a transient vault into the substance of a usable bridge requires in adjunction only that initially pliable and ultimately tensile sorts of connecting lines be conducted along the tracery of the vault until the work of castling the gap can begin.
In the pragmatic theory of signs, the word "representation" is a technical term that is synonymous with the word "sign", in other words, it applies to an entity in the most general category of things that can enter into sign relations in the roles of signs and interpretants. Thus, in this usage the scope of the term "representation" includes all sorts of syntactic, descriptive, and conceptual entities, a range of options I will frequently find it convenient to suggest by drawing on a pair of stock phrases: "terms and concepts" (TAC's) in a conjuctive context, versus "terms or concepts" (TOC's) in a disjunctive context.
In the pragmatic theory of signs, the word "representation" is a technical term that is synonymous with the word "sign", in other words, it applies to an entity in the most general category of things that can enter into sign relations in the roles of signs and interpretants. Thus, in this usage the scope of the term "representation" includes all sorts of syntactic, descriptive, and conceptual entities, a range of options I will frequently find it convenient to suggest by drawing on a pair of stock phrases: "terms and concepts" (TACs) in a conjuctive context, versus "terms or concepts" (TOCs) in a disjunctive context.
In mathematics, the word "representation" is commonly reserved for referring to a "homomorphism", that is, a linear transformation or a structure preserving mapping h: X >Y between mathematical "objects", that is, structure bearing spaces in a category of comparable domains.
In mathematics, the word "representation" is commonly reserved for referring to a "homomorphism", that is, a linear transformation or a structure preserving mapping h: X >Y between mathematical "objects", that is, structure bearing spaces in a category of comparable domains.
Generally speaking, one is free to interpret references to generalized objects either as indications of partially formed analogues or as partially informed descriptions of their corresponding species. I refer to these alternatives as the "object theoretic" and the "sign theoretic" options, respectively. The first interpretation assumes that vague and general references still have denotations, merely to vague and general objects. The second interpretation ascribes the partialities of information to the characters of the signs and expressions that are doing the denoting. In most cases that arise in casual discussion the choice between these conventions is purely stylistic. However, in many of the more intricate situations that arise in formal discussion the object choice often fails utterly, and whenever the utmost care is required it will usually be the attention to signs that saves the day.
Generally speaking, one is free to interpret references to generalized objects either as indications of partially formed analogues or as partially informed descriptions of their corresponding species. I refer to these alternatives as the "object theoretic" and the "sign theoretic" options, respectively. The first interpretation assumes that vague and general references still have denotations, merely to vague and general objects. The second interpretation ascribes the partialities of information to the characters of the signs and expressions that are doing the denoting. In most cases that arise in casual discussion the choice between these conventions is purely stylistic. However, in many of the more intricate situations that arise in formal discussion the object choice often fails utterly, and whenever the utmost care is required it will usually be the attention to signs that saves the day.
In order to speak of generalized orders of relations I need to outline the dimensions of variation along which I intend the characters of already familiar orders of relations to be broadened. Generally speaking, the taxonomic features of n place relations that I wish to liberalize can be read off from their "local incidence properties" (LIP's).
In order to speak of generalized orders of relations I need to outline the dimensions of variation along which I intend the characters of already familiar orders of relations to be broadened. Generally speaking, the taxonomic features of n place relations that I wish to liberalize can be read off from their "local incidence properties" (LIPs).
Definition. A "local incidence property" of an n place relation R is one that is based on the following sorts of data. Suppose R c X1x...xXn. Pick an element x in one of the domains Xi of R. Let "R&x@i" denote a subset of R called the "flag of R with x at i", or the "x@i flag of R". The "local flag" R&x@i c R is defined as follows:
Definition. A "local incidence property" of an n place relation R is one that is based on the following sorts of data. Suppose R c X1x...xXn. Pick an element x in one of the domains Xi of R. Let "R&x@i" denote a subset of R called the "flag of R with x at i", or the "x@i flag of R". The "local flag" R&x@i c R is defined as follows:
Definition. An n place relation R c X1x...xXn is called "P regular at i" iff every flag of R with x at i is P, letting x range over the domain Xi, in symbols, iff P(R&x@i) is true for all x C Xi.
Definition. An n place relation R c X1x...xXn is called "P regular at i" iff every flag of R with x at i is P, letting x range over the domain Xi, in symbols, iff P(R&x@i) is true for all x C Xi.
Of particular interest are the local incidence properties of relations that can be calculated from the cardinalities of their local flags, and these are naturally called "numerical incidence properties" (NIP's).
Of particular interest are the local incidence properties of relations that can be calculated from the cardinalities of their local flags, and these are naturally called "numerical incidence properties" (NIPs).
For example, R is said to be "k regular at i" or "k regular at Xi" if and only if the cardinality |R&x@i| = k for all x C Xi. In a similar fashion, one can define the NIP's "<k regular at i", ">k regular at i", and so on. For ease of reference, I record a few of these definitions here:
For example, R is said to be "k regular at i" or "k regular at Xi" if and only if the cardinality |R&x@i| = k for all x C Xi. In a similar fashion, one can define the NIPs "<k regular at i", ">k regular at i", and so on. For ease of reference, I record a few of these definitions here:
R is k regular at i iff |R&x@i| = k for all x C Xi.
R is k regular at i iff |R&x@i| = k for all x C Xi.
Returning to dyadic relations, it is useful to describe some familiar classes of objects in terms of their local and numerical incidence properties. Let R c SxT be an arbitrary dyadic relation. The following properties of R can then be defined:
Returning to dyadic relations, it is useful to describe some familiar classes of objects in terms of their local and numerical incidence properties. Let R c SxT be an arbitrary dyadic relation. The following properties of R can then be defined:
R is total at S iff R is 1 regular at S.
R is total at S iff R is ³1 regular at S.
R is total at T iff R is 1 regular at T.
R is total at T iff R is ³1 regular at T.
R is tubular at S iff R is 1 regular at S.
R is tubular at S iff R is £1 regular at S.
R is tubular at T iff R is 1 regular at T.
R is tubular at T iff R is £1 regular at T.
If R is tubular at S, then R is called a "partial function" or "prefunction" from S to T, often indicated by writing R = p : S ~> T.
If R is tubular at S, then R is called a "partial function" or "prefunction" from S to T, often indicated by writing R = p : S ~> T.
A "moderate equivalence relation" (MER) on the "modus" M c X is a relation on X whose restriction to M is an equivalence relation on M. In symbols, R c XxX such that R|M c MxM is an equivalence relation. Notice that the subset of restriction, or modus M, is a part of the definition, so the same relation R on X could be a MER or not depending on the choice of M. In spite of how it sounds, a moderate equivalence relation can have more ordered pairs in it than the ordinary sort of equivalence relation on the same set.
A "moderate equivalence relation" (MER) on the "modus" M c X is a relation on X whose restriction to M is an equivalence relation on M. In symbols, R c XxX such that R|M c MxM is an equivalence relation. Notice that the subset of restriction, or modus M, is a part of the definition, so the same relation R on X could be a MER or not depending on the choice of M. In spite of how it sounds, a moderate equivalence relation can have more ordered pairs in it than the ordinary sort of equivalence relation on the same set.
In applying the equivalence class notation to a sign relation R, the definitions and examples considered so far only cover the case where the connotative component RSI is a total equivalence relation on the whole syntactic domain S. The next job is to adapt this usage to PER's.
In applying the equivalence class notation to a sign relation R, the definitions and examples considered so far only cover the case where the connotative component RSI is a total equivalence relation on the whole syntactic domain S. The next job is to adapt this usage to PERs.
If R is a sign relation whose syntactic projection RSI is a PER on S, then I still write "[s]R" for the "equivalence class of s under RSI". But now, [s]R can be empty if s has no interpretant, that is, if s lies outside the "adequately meaningful" subset of the syntactic domain, where synonymy and equivalence of meaning are defined. Otherwise, if s has an i then it also has an o, by the definition of RSI. In this case, there is a triple <o, s, i> C R, and it is permissible to let [o]R = [s]R.
If R is a sign relation whose syntactic projection RSI is a PER on S, then I still write "[s]R" for the "equivalence class of s under RSI". But now, [s]R can be empty if s has no interpretant, that is, if s lies outside the "adequately meaningful" subset of the syntactic domain, where synonymy and equivalence of meaning are defined. Otherwise, if s has an i then it also has an o, by the definition of RSI. In this case, there is a triple <o, s, i> C R, and it is permissible to let [o]R = [s]R.
One of the most important design considerations that goes into building the requisite software system is how well it furthers certain lines of abstraction and generalization. One of these dimensions of abstraction or directions of generalization is discussed in this section, where I attempt to unify its many appearances under the theme of "partiality". This name is chosen to suggest the desired sense of abstract intention since the extensions of concepts that it favors and for which it leaves room are outgrowths of the limitation that finite signs and expressions can never provide more than partial information about the richness of individual detail that is always involved in any real object. All in all, this modicum of tolerance for uncertainty is the very play in the wheels of determinism that provides a significant chance for luck to play a part in the finer steps toward finishing every real objective.
One of the most important design considerations that goes into building the requisite software system is how well it furthers certain lines of abstraction and generalization. One of these dimensions of abstraction or directions of generalization is discussed in this section, where I attempt to unify its many appearances under the theme of "partiality". This name is chosen to suggest the desired sense of abstract intention since the extensions of concepts that it favors and for which it leaves room are outgrowths of the limitation that finite signs and expressions can never provide more than partial information about the richness of individual detail that is always involved in any real object. All in all, this modicum of tolerance for uncertainty is the very play in the wheels of determinism that provides a significant chance for luck to play a part in the finer steps toward finishing every real objective.
If one needs a slogan to entitle this form of propagation, it is only that "Necessity is the mother of invention". In other words, it is precisely this lack of perfect information that yields the opportunity for novel forms of speciation to develop among finitely informed creatures (FIC's), and just this need of perfect information that drives the evolving forms of independent determination and spontaneous creation in any area, no matter how well the arena is circumscribed by the restrictions of signs.
If one needs a slogan to entitle this form of propagation, it is only that "Necessity is the mother of invention". In other words, it is precisely this lack of perfect information that yields the opportunity for novel forms of speciation to develop among finitely informed creatures (FICs), and just this need of perfect information that drives the evolving forms of independent determination and spontaneous creation in any area, no matter how well the arena is circumscribed by the restrictions of signs.
In tracing the echoes of this theme, it is necessary to reflect on the circumstance that degenerate sign relations happen to be perfectly possible in practice, and it is desirable to provide a critical method that can address the facts of their flaws in theoretically insightful terms. Relative to particular environments of interpretation, nothing proscribes the occurrence of sign relations that are defective in any of their various facets, namely: (1) with signs that fail to denote or connote, (2) with interpretants that lack of being faithfully represented or reliably objectified, and (3) with objects that make no impression or remain ineffable in the preferred medium.
In tracing the echoes of this theme, it is necessary to reflect on the circumstance that degenerate sign relations happen to be perfectly possible in practice, and it is desirable to provide a critical method that can address the facts of their flaws in theoretically insightful terms. Relative to particular environments of interpretation, nothing proscribes the occurrence of sign relations that are defective in any of their various facets, namely: (1) with signs that fail to denote or connote, (2) with interpretants that lack of being faithfully represented or reliably objectified, and (3) with objects that make no impression or remain ineffable in the preferred medium.
The most familiar illustrations of information theoretic "partiality", "partial indication", or "signs bearing partial information about objects" occur every time one uses a general name, for example, the name of a genus, class, or set. Almost as commonly, the formula that expresses a logical proposition can be regarded as a partial specification of its logical models or satisfying interpretations. Just as the name of a genus or class can be taken as a "partially informed reference" or a "plural indefinite reference" (PIR) to one of its species or elements, so the name of an n place relation can be viewed as a PIR to one of its elementary relations or n tuples, and the formula or expression of a proposition can be understood as a PIR to one its models or satisfying interpretations. For brevity, this variety of referential indetermination can be called the "generic partiality" of signs as information bearers.
The most familiar illustrations of information theoretic "partiality", "partial indication", or "signs bearing partial information about objects" occur every time one uses a general name, for example, the name of a genus, class, or set. Almost as commonly, the formula that expresses a logical proposition can be regarded as a partial specification of its logical models or satisfying interpretations. Just as the name of a genus or class can be taken as a "partially informed reference" or a "plural indefinite reference" (PIR) to one of its species or elements, so the name of an n place relation can be viewed as a PIR to one of its elementary relations or n tuples, and the formula or expression of a proposition can be understood as a PIR to one its models or satisfying interpretations. For brevity, this variety of referential indetermination can be called the "generic partiality" of signs as information bearers.
Note. In this discussion I will not systematically distinguish between the logical entity typically called a "proposition" or "statement" and the syntactic entity usually called an "expression", "formula", or "sentence". Instead, I work on the assumption that both types of entity are always involved in everything one proposes and also on the hope that context will determine which aspect of proposing is most apt. For precision, the abstract category of propositions proper will have to be reconstituted as logical equivalence classes of syntactically diverse expressions. For the present, I will use the phrase "propositional expression" whenever it is necessary to call particular attention to the syntactic entity. Likewise, I will not always separate "higher order propositions" (HOP's), that is, propositions about propositions, from their corresponding formulations in the guise of "higher order propositional expressions" (HOPE's).
Note. In this discussion I will not systematically distinguish between the logical entity typically called a "proposition" or "statement" and the syntactic entity usually called an "expression", "formula", or "sentence". Instead, I work on the assumption that both types of entity are always involved in everything one proposes and also on the hope that context will determine which aspect of proposing is most apt. For precision, the abstract category of propositions proper will have to be reconstituted as logical equivalence classes of syntactically diverse expressions. For the present, I will use the phrase "propositional expression" whenever it is necessary to call particular attention to the syntactic entity. Likewise, I will not always separate "higher order propositions" (HOPs), that is, propositions about propositions, from their corresponding formulations in the guise of "higher order propositional expressions" (HOPEs).
Even though "partial information" is the usual case of information (as rendered by signs about objects) I will continue to use this phrase, for all its informative redundancy, to emphasize the issues of partial definition, specification, and determination that arise under the pervasive theme of "partiality".
Even though "partial information" is the usual case of information (as rendered by signs about objects) I will continue to use this phrase, for all its informative redundancy, to emphasize the issues of partial definition, specification, and determination that arise under the pervasive theme of "partiality".
When the subject matter of discussion is bounded by a universal set X, out of which all objects referred to must come, then every PIR to an object can be identified with the name or formula (sign or expression) of a subset S c X, or with that of its selector function S# : X > B. Conceptually, one imagines generating all the objects in X and then selecting out the ones that satisfy some test for membership in S.
When the subject matter of discussion is bounded by a universal set X, out of which all objects referred to must come, then every PIR to an object can be identified with the name or formula (sign or expression) of a subset S c X, or with that of its selector function S# : X > B. Conceptually, one imagines generating all the objects in X and then selecting out the ones that satisfy some test for membership in S.
In a realistic computational framework, however, when the domain of interest is given generatively in a genuine sense of the word, that is, defined solely in terms of the primitive elements and operations that are needed to generate it, and when the resource limitations in actual effect make it impractical to enumerate all the possibilities in advance of selecting the adumbrated subset, then the implementation of PIR's becomes a genuine computational problem.
In a realistic computational framework, however, when the domain of interest is given generatively in a genuine sense of the word, that is, defined solely in terms of the primitive elements and operations that are needed to generate it, and when the resource limitations in actual effect make it impractical to enumerate all the possibilities in advance of selecting the adumbrated subset, then the implementation of PIRs becomes a genuine computational problem.
Considered in its application to n place relations, the generic brand of partial specification constitutes a rather limited type of partiality, in that every element conceived as falling under the specified relation, no matter how indistinctly indicated, is still envisioned to maintain its full arity and to remain every bit a complete, though unknown, n tuple. Still, there is a simple way to extend the concept of generic partiality in a significant fashion, achieving a form of PIR's to relations by making use of "higher order propositions" (HOP's).
Considered in its application to n place relations, the generic brand of partial specification constitutes a rather limited type of partiality, in that every element conceived as falling under the specified relation, no matter how indistinctly indicated, is still envisioned to maintain its full arity and to remain every bit a complete, though unknown, n tuple. Still, there is a simple way to extend the concept of generic partiality in a significant fashion, achieving a form of PIRs to relations by making use of "higher order propositions" (HOPs).
Extending the concept of generic partiality, by iterating the principle on which it is based, leads to higher order propositions about elementary relations, or propositions about relations, as one way to achieve partial specifications of relations, or PIR's to relations.
Extending the concept of generic partiality, by iterating the principle on which it is based, leads to higher order propositions about elementary relations, or propositions about relations, as one way to achieve partial specifications of relations, or PIRs to relations.
This direction of generalization expands the scope of PIR's by means of an analogical extension, and can be charted in the following manner. If the sign or expression (name or formula) of an n place relation can be interpreted as a proposition about n tuples and thus as a PIR to an elementary relation, then a higher order proposition about n tuples is a proposition about n place relations that can be used to formulate a PIR to an n place relation.
This direction of generalization expands the scope of PIRs by means of an analogical extension, and can be charted in the following manner. If the sign or expression (name or formula) of an n place relation can be interpreted as a proposition about n tuples and thus as a PIR to an elementary relation, then a higher order proposition about n tuples is a proposition about n place relations that can be used to formulate a PIR to an n place relation.
In order to formalize these ideas, it is helpful to have notational devices for switching back and forth among different ways of exemplifying what is abstractly the same contents of information, in particular, for translating among sets, their logical expressions, and their functional indications.
In order to formalize these ideas, it is helpful to have notational devices for switching back and forth among different ways of exemplifying what is abstractly the same contents of information, in particular, for translating among sets, their logical expressions, and their functional indications.
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In a computational framework, indeed, in any constructively analytic and practically applied setting, the problem of working with insufficient information to fully determine one's object is a constant feature that goes with the territory of "finite information constructions" (FIC's). The fineness of detail that is able to be specified by formal symbols is continually bedeviled by the frustrating truncations of every signal to a finite code and by the resistive constrictions of every intention to the restrictive confines of what can actually be conducted. Of course, one tries to get around the more finessible limitations, but the figurative extensions that one hopes to achieve by recourse to quasi circular definitions and by reversion to parable and hyperbole — all of these tactics appeal to a pre established aptness of reception on the part of interpreters that begs the very question of a determinate understanding and that risks falling short of the exact attitude needed for success. At any rate, the indirect strategy of approach relies on such large reserves of enthymeme to fuel its course that the grasp of a period to set bounds on its argument and fix a term to its conclusion is often found diverging in ways that both underreach and overreach its object, and well founded or not the search for a generic method of definition typically ends so completely dumbfounded that it often trails off into the inescapable vacuity of a quasi terminal ellipsis ...
In a computational framework, indeed, in any constructively analytic and practically applied setting, the problem of working with insufficient information to fully determine one's object is a constant feature that goes with the territory of "finite information constructions" (FICs). The fineness of detail that is able to be specified by formal symbols is continually bedeviled by the frustrating truncations of every signal to a finite code and by the resistive constrictions of every intention to the restrictive confines of what can actually be conducted. Of course, one tries to get around the more finessible limitations, but the figurative extensions that one hopes to achieve by recourse to quasi circular definitions and by reversion to parable and hyperbole — all of these tactics appeal to a pre established aptness of reception on the part of interpreters that begs the very question of a determinate understanding and that risks falling short of the exact attitude needed for success. At any rate, the indirect strategy of approach relies on such large reserves of enthymeme to fuel its course that the grasp of a period to set bounds on its argument and fix a term to its conclusion is often found diverging in ways that both underreach and overreach its object, and well founded or not the search for a generic method of definition typically ends so completely dumbfounded that it often trails off into the inescapable vacuity of a quasi terminal ellipsis ...
This section treats the problems of insufficient information and indeterminate objects under the heading of "partializations", using this as a briefer term for the information theoretic generalizations of the relevant object domains that take the use of indeterminate denotations, or partial determinations of objects, explicitly into account.
This section treats the problems of insufficient information and indeterminate objects under the heading of "partializations", using this as a briefer term for the information theoretic generalizations of the relevant object domains that take the use of indeterminate denotations, or partial determinations of objects, explicitly into account.
Ultimately, one must be prepared to work with probability distributions that are defined on entire spaces O of the relevant objects or outcomes. But probability distributions are just a special class of functions f : O > [0, 1] c R, where R is the real line, and this means that the corresponding theory of partializations involves the dual aspect of the domain O, dealing with the "functionals" defined on it, or the functions that map it into "coefficient" spaces. And since it is unavoidable in a computational framework, one way or another every type of coefficient information, real or otherwise, must be approached bit by bit. That is, all information is defined in terms of the either or decisions that must be made to really and practically determine it. So, to make a long story short, one might as well approach this dual aspect by starting with the functions f : O > B = {0, 1}, in effect, with the logic of propositions.
Ultimately, one must be prepared to work with probability distributions that are defined on entire spaces O of the relevant objects or outcomes. But probability distributions are just a special class of functions f : O > [0, 1] c R, where R is the real line, and this means that the corresponding theory of partializations involves the dual aspect of the domain O, dealing with the "functionals" defined on it, or the functions that map it into "coefficient" spaces. And since it is unavoidable in a computational framework, one way or another every type of coefficient information, real or otherwise, must be approached bit by bit. That is, all information is defined in terms of the either or decisions that must be made to really and practically determine it. So, to make a long story short, one might as well approach this dual aspect by starting with the functions f : O > B = {0, 1}, in effect, with the logic of propositions.
I turn now to the question of "partially specified" (PS) relations, or “partially informed relations” (PIR's), in other words, to the explicit treatment of relations in terms of the information that is logically possessed or actually expressed about them. There seem to be several ways to approach the concept of an n place PIR and the supporting notion of a PS n tuple. Since the term "partial relation" is already implicitly in use for the general class of relations that are not necessarily total on any of their domains, I will coin the term "pro relation", on analogy with "pronoun" and "proposition", to denote an expression of information about a relation, a contingent indication that, if and when completed, conceivably points to a particular relation.
I turn now to the question of "partially specified" (PS) relations, or “partially informed relations” (PIRs), in other words, to the explicit treatment of relations in terms of the information that is logically possessed or actually expressed about them. There seem to be several ways to approach the concept of an n place PIR and the supporting notion of a PS n tuple. Since the term "partial relation" is already implicitly in use for the general class of relations that are not necessarily total on any of their domains, I will coin the term "pro relation", on analogy with "pronoun" and "proposition", to denote an expression of information about a relation, a contingent indication that, if and when completed, conceivably points to a particular relation.
One way to deal with "partially informed categories" (PIC's) of n place relations is to contemplate incomplete relational forms or schemata. Regarded over the years chiefly in logical and intensional terms, constructs of roughly this type have been variously referred to as "rhemes" or "rhemata" (Peirce), "unsaturated relations" (Frege), or "frames" (in current AI literature). Expressed in extensional terms, talking about PIC's of n place relations is tantamount to admitting elementary relations with missing elements. The question is not just syntactic — How to represent an n tuple with empty places? — but also semantic — How to make sense of an n tuple with less than n elements?
One way to deal with "partially informed categories" (PICs) of n place relations is to contemplate incomplete relational forms or schemata. Regarded over the years chiefly in logical and intensional terms, constructs of roughly this type have been variously referred to as "rhemes" or "rhemata" (Peirce), "unsaturated relations" (Frege), or "frames" (in current AI literature). Expressed in extensional terms, talking about PICs of n place relations is tantamount to admitting elementary relations with missing elements. The question is not just syntactic — How to represent an n tuple with empty places? — but also semantic — How to make sense of an n tuple with less than n elements?
In order to deal with PIR's in a thoroughly consistent fashion, it appears necessary to contemplate elementary relations that present themselves as being "unsaturated" (in Frege's usage of that term), in other words, to consider elements of a presumptive product space that in some sense "wanna be" n tuples or "would be" sequences of a certain length, but are currently missing components in some of their places.
In order to deal with PIRs in a thoroughly consistent fashion, it appears necessary to contemplate elementary relations that present themselves as being "unsaturated" (in Frege's usage of that term), in other words, to consider elements of a presumptive product space that in some sense "wanna be" n tuples or "would be" sequences of a certain length, but are currently missing components in some of their places.
To the extent that the issues of partialization become obvious at the level of symbols and can be dealt with by elementary syntactic means, they initially make their appearance in terms of the various ways that data can go missing.
To the extent that the issues of partialization become obvious at the level of symbols and can be dealt with by elementary syntactic means, they initially make their appearance in terms of the various ways that data can go missing.
The alternate notation "a^b" is provided for the ordered pair <a, b>. This choice of representation for ordered pairs is especially apt in the case of "concrete indices" (CI's) and "localized addresses" (LA's), where one wants the lead item to serve as a pointed reminder of the itemized content, as in i^Xi = <i, Xi>, and it helps to stress the individuality of each member in the indexed family, as in G = {Gj} = {j^Gj} = {<j, Gj>}.
The alternate notation "a^b" is provided for the ordered pair <a, b>. This choice of representation for ordered pairs is especially apt in the case of "concrete indices" (CIs) and "localized addresses" (LAs), where one wants the lead item to serve as a pointed reminder of the itemized content, as in i^Xi = <i, Xi>, and it helps to stress the individuality of each member in the indexed family, as in G = {Gj} = {j^Gj} = {<j, Gj>}.
The "caret" (^) device works well in any situation where one desires to accentuate the fact that a formal subscript is being reclaimed and elevated to the status of an actual parameter. By way of the operation indicated by the caret character the index bound to an object term can be rehabilitated as a full fledged component of an elementary relation, thereby schematically embedding the indicated object in the experiential space of a typical agent.
The "caret" (^) device works well in any situation where one desires to accentuate the fact that a formal subscript is being reclaimed and elevated to the status of an actual parameter. By way of the operation indicated by the caret character the index bound to an object term can be rehabilitated as a full fledged component of an elementary relation, thereby schematically embedding the indicated object in the experiential space of a typical agent.
These equivalences depend on the existence of natural isomorphisms between different ways of constructing n place product spaces, that is, on the associativity of pairwise products, a not altogether trivial result (MacLane, CatWorkMath, ch. 7).
These equivalences depend on the existence of natural isomorphisms between different ways of constructing n place product spaces, that is, on the associativity of pairwise products, a not altogether trivial result (MacLane, CatWorkMath, ch. 7).
2. Higher order indications (HOI's)?
2. Higher order indications (HOIs)?
^x = < , x> x^ = <x, > ?
^x = < , x> x^ = <x, > ?
In talking and thinking, often in just that order, about properties and classes of relations, one is always invoking, explicitly or implicitly, a particular background, a limited field of experience, actual or potential, against which each object of "discussion and thought" (DAT) figures. Expressing the matter in the idiom of logical inquiry, one brings to mind a preconceived universe of discourse U or a restricted domain of discussion X, and then contemplates ...
In talking and thinking, often in just that order, about properties and classes of relations, one is always invoking, explicitly or implicitly, a particular background, a limited field of experience, actual or potential, against which each object of "discussion and thought" (DAT) figures. Expressing the matter in the idiom of logical inquiry, one brings to mind a preconceived universe of discourse U or a restricted domain of discussion X, and then contemplates ...
This direction of generalization expands the scope of PIR's by means of an analogical extension, and can be charted in the following manner. If the name of a relation can be taken as a PIR to elementary relations, that is, if the formula of an n place relation can be interpreted as a proposition about n tuples, then a PIR to relations themselves can be formulated as a proposition about relations and thus as a HOPE about elementary relations or n tuples.
This direction of generalization expands the scope of PIRs by means of an analogical extension, and can be charted in the following manner. If the name of a relation can be taken as a PIR to elementary relations, that is, if the formula of an n place relation can be interpreted as a proposition about n tuples, then a PIR to relations themselves can be formulated as a proposition about relations and thus as a HOPE about elementary relations or n tuples.
One way to extend the generic brand of partiality among relations in a non trivial direction can be charted as follows. If the name or formula of a relation is a PIR to elementary relations, that is, if a sign or expression an n place relation is interpreted as a proposition about n tuples, then a PIR to relations ...
One way to extend the generic brand of partiality among relations in a non trivial direction can be charted as follows. If the name or formula of a relation is a PIR to elementary relations, that is, if a sign or expression an n place relation is interpreted as a proposition about n tuples, then a PIR to relations ...
The power set notation can be used to provide an alternative description of relations. In the case where S is a cartesian product, say S = X1x...xXn, then each n place relation described as a subset of S, say as R c X1x...xXn, is equally well described as an element of Pow (S), in other words, as R C Pow (X1x...xXn).
The power set notation can be used to provide an alternative description of relations. In the case where S is a cartesian product, say S = X1x...xXn, then each n place relation described as a subset of S, say as R c X1x...xXn, is equally well described as an element of Pow (S), in other words, as R C Pow (X1x...xXn).
2. The set of triples of dyadic relations, with pairwise cartesian products chosen in a pre arranged order from a triple of three sets <X, Y, Z>, is called the "dyadic explosion" of XxYxZ. This object is denoted by "Explo (X, Y, Z; 2)", read as the "explosion of XxYxZ by 2's", or more simply as "X, Y, Z, choose 2", and defined as follows:
2. The set of triples of dyadic relations, with pairwise cartesian products chosen in a pre arranged order from a triple of three sets <X, Y, Z>, is called the "dyadic explosion" of XxYxZ. This object is denoted by "Explo (X, Y, Z; 2)", read as the "explosion of XxYxZ by 2s", or more simply as "X, Y, Z, choose 2", and defined as follows:
Explo (X, Y, Z; 2) = Pow (XxY) x Pow (XxZ) x Pow (YxZ).
Explo (X, Y, Z; 2) = Pow (XxY) x Pow (XxZ) x Pow (YxZ).
Fragments
Fragments
Finally, the set of triples of dyadic relations, with pairwise cartesian products chosen in a pre arranged order from a collection of three sets {X, Y, Z}, is called the "dyadic explosion" of {X, Y, Z}. This object is denoted as "Explo (X, Y, Z; 2)", read as the "explosion of XxYxZ by 2's" or simply as "X, Y, Z, choose 2", and is defined as follows:
Finally, the set of triples of dyadic relations, with pairwise cartesian products chosen in a pre arranged order from a collection of three sets {X, Y, Z}, is called the "dyadic explosion" of {X, Y, Z}. This object is denoted as "Explo (X, Y, Z; 2)", read as the "explosion of XxYxZ by 2s" or simply as "X, Y, Z, choose 2", and is defined as follows:
Explo (X, Y, Z; 2) = Pow (XxY) x Pow (XxZ) x Pow (YxZ).
Explo (X, Y, Z; 2) = Pow (XxY) x Pow (XxZ) x Pow (YxZ).
1. One analytic method takes it as a maxim for the logic of context that: "Every sign or text is indexed by the context in which it occurs." This means that all signs, including indices, are themselves indexed, though initially only tacitly, by the objective situation, the syntactic context, and the actual interpreter that makes use of them.
1. One analytic method takes it as a maxim for the logic of context that: "Every sign or text is indexed by the context in which it occurs." This means that all signs, including indices, are themselves indexed, though initially only tacitly, by the objective situation, the syntactic context, and the actual interpreter that makes use of them.
To begin formalizing this brand of supplementation, it is necessary to mark salient aspects of the situational, contextual, and inclusively interpretive features of sign usage that were previously held tacit. In effect, signs once regarded as primitive objects need to be newly analyzed as categorical abstractions that cover multitudes of existential sign instances or "signs in use" (SIU's).
To begin formalizing this brand of supplementation, it is necessary to mark salient aspects of the situational, contextual, and inclusively interpretive features of sign usage that were previously held tacit. In effect, signs once regarded as primitive objects need to be newly analyzed as categorical abstractions that cover multitudes of existential sign instances or "signs in use" (SIUs).
To develop these dimensions of the A and B dialogue, I will attempt to articulate these interpretive parameters of signs by means of subscripts or superscripts attached to the signs or their quotations, in this way constituting a brand of "situated signs" or "attributed remarks".
To develop these dimensions of the A and B dialogue, I will attempt to articulate these interpretive parameters of signs by means of subscripts or superscripts attached to the signs or their quotations, in this way constituting a brand of "situated signs" or "attributed remarks".
or "A"A =B "A"B =B "i"A =B "u"B.
or "A"A =B "A"B =B "i"A =B "u"B.
Consequently, both SER's now induce the same semantic partition on S:
Consequently, both SERs now induce the same semantic partition on S:
{{ "A"A, "A"B, "i"A, "u"B }, { "B"A, "B"B, "i"B, "u"A }}.
{{ "A"A, "A"B, "i"A, "u"B }, { "B"A, "B"B, "i"B, "u"A }}.
1. Just as a sign represents its object and becomes associated with more or less equivalent signs in the minds of interpretive agents, the corpus of signs that embodies a SOI represents in a collective way its own proper object, intended objective, or "try at objectivity" (TAO).
1. Just as a sign represents its object and becomes associated with more or less equivalent signs in the minds of interpretive agents, the corpus of signs that embodies a SOI represents in a collective way its own proper object, intended objective, or "try at objectivity" (TAO).
2. Just as the relationship of a sign to its semantic objects and interpretive associates can be formalized within a single sign relation, the relation of a dynamically changing SOI to its reference environment, developmental goals, and desired characteristics of interpretive performance can be formalized by means of a higher order sign relation, one that further establishes a grounds of comparison for relating the growing SOI, not only to its former and future selves, but to a diverse company of other SOI's.
2. Just as the relationship of a sign to its semantic objects and interpretive associates can be formalized within a single sign relation, the relation of a dynamically changing SOI to its reference environment, developmental goals, and desired characteristics of interpretive performance can be formalized by means of a higher order sign relation, one that further establishes a grounds of comparison for relating the growing SOI, not only to its former and future selves, but to a diverse company of other SOIs.
From an outside perspective the distinction between a sign and its object is usually regarded as obvious, though agents operating in the thick of a SOI often act as though they cannot see the difference. Nevertheless, as a rule in practice, a sign is not a good thing to be confused with its object. Even in the rare and usually controversial cases where an identity of substance is contemplated, usually only for the sake of argument, there is still a distinction of roles to be maintained between the sign and its object. Just so, ...
From an outside perspective the distinction between a sign and its object is usually regarded as obvious, though agents operating in the thick of a SOI often act as though they cannot see the difference. Nevertheless, as a rule in practice, a sign is not a good thing to be confused with its object. Even in the rare and usually controversial cases where an identity of substance is contemplated, usually only for the sake of argument, there is still a distinction of roles to be maintained between the sign and its object. Just so, ...
Another form of realization lets each element of the object domain O correspond to the autonomous intention of the systematic agent to denote an object, achieve an objective, or broadly speaking to accomplish any other purpose with respect to an object in its domain. In this interpretation, the object X is a control parameter that brings the system Y into line with realizing a target set [X]Y.
Another form of realization lets each element of the object domain O correspond to the autonomous intention of the systematic agent to denote an object, achieve an objective, or broadly speaking to accomplish any other purpose with respect to an object in its domain. In this interpretation, the object X is a control parameter that brings the system Y into line with realizing a target set [X]Y.
Tables 75 and 76 show how the sign relations for A and B can be filled out as finite state processes in conformity with the interpretive principles just described. Rather than letting the actions go undefined for some combinations of inputs C O and states C S, transitions have been added that take the interpreters from whatever else they might have been thinking about to the SEC's of their objects. In either modality of realization, cognitive or control oriented, the abstract structure of the resulting sign process is exactly the same.
Tables 75 and 76 show how the sign relations for A and B can be filled out as finite state processes in conformity with the interpretive principles just described. Rather than letting the actions go undefined for some combinations of inputs C O and states C S, transitions have been added that take the interpreters from whatever else they might have been thinking about to the SECs of their objects. In either modality of realization, cognitive or control oriented, the abstract structure of the resulting sign process is exactly the same.
Table 75. Sign Process of Interpreter A
Table 75. Sign Process of Interpreter A
Y@X = "Y at X" = @[X]Y = [X]Y U { Arcs into [X]Y }.
Y@X = "Y at X" = @[X]Y = [X]Y U { Arcs into [X]Y }.
In effect, this discussion of dynamic realizations of sign relations has advanced from considering SEP's as partitioning the set of points in S to considering attractors as partitioning the set of arcs in SxI = SxS.
In effect, this discussion of dynamic realizations of sign relations has advanced from considering SEPs as partitioning the set of points in S to considering attractors as partitioning the set of arcs in SxI = SxS.
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As a flexible and fairly general strategy for describing reflective extensions it is convenient to take the following tack. Given a syntactic domain S, there is an independent formal language F = F(S) = S<<>>, to be called "the free quotational extension of S", that can be generated from S by embedding each of its signs to any depth of quotation marks. In F, the quoting operation can be regarded as a syntactic generator that is inherently free of constraining relations. In other words, for every s C S, the sequence s, <s>, <<s>>, ... contains nothing but pairwise distinct elements in F no matter how far it is produced. The set F(s) = s<<>> c F that collects the elements of this sequence is called "the subset of F generated from s by quotation".
As a flexible and fairly general strategy for describing reflective extensions it is convenient to take the following tack. Given a syntactic domain S, there is an independent formal language F = F(S) = S<<>>, to be called "the free quotational extension of S", that can be generated from S by embedding each of its signs to any depth of quotation marks. In F, the quoting operation can be regarded as a syntactic generator that is inherently free of constraining relations. In other words, for every s C S, the sequence s, <s>, <<s>>, ... contains nothing but pairwise distinct elements in F no matter how far it is produced. The set F(s) = s<<>> c F that collects the elements of this sequence is called "the subset of F generated from s by quotation".
Against this background, other varieties of reflective extension can be specified by means of semantic equations (SEQ's) that are considered to be imposed on the elements of F. Taking the reflective extensions Ref1 (A) and Ref1 (B) as the first orders of a "free" project toward reflective closure, variant extensions can be described by relating their entries with those of comparable members in the standard sequences Refn (A) and Refn (B).
Against this background, other varieties of reflective extension can be specified by means of semantic equations (SEQs) that are considered to be imposed on the elements of F. Taking the reflective extensions Ref1 (A) and Ref1 (B) as the first orders of a "free" project toward reflective closure, variant extensions can be described by relating their entries with those of comparable members in the standard sequences Refn (A) and Refn (B).
A variant pair of reflective extensions, Ref1(A|E1) and Ref1(B|E1), are presented in Tables 79 and 80, respectively. These are identical to the corresponding "free" variants, Ref1(A) and Ref1(B), with the exception of those entries that are constrained by the system of semantic equations:
A variant pair of reflective extensions, Ref1(A|E1) and Ref1(B|E1), are presented in Tables 79 and 80, respectively. These are identical to the corresponding "free" variants, Ref1(A) and Ref1(B), with the exception of those entries that are constrained by the system of semantic equations:
This has the effect of making all levels of quotation equivalent.
This has the effect of making all levels of quotation equivalent.
By calling attention to their intended status as "semantic" equations, meaning that signs are being set equal in the SEC's they inhabit or the objects they denote, I hope to emphasize that these equations are able to say something significant about objects.
By calling attention to their intended status as "semantic" equations, meaning that signs are being set equal in the SECs they inhabit or the objects they denote, I hope to emphasize that these equations are able to say something significant about objects.
??? Redo F(S) over W ??? Use WF = O U F ???
??? Redo F(S) over W ??? Use WF = O U F ???
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Before this complex of relationships can be formalized in much detail, I must introduce linguistic devices for generating "higher order signs", used to indicate other signs, and "situated signs", indexed by the names of their users, their contexts of use, and other types of information incidental to their usage in general. This leads to the consideration of SOI's that maintain recursive mechanisms for naming everything within their purview. This "nominal generosity" gives them a new order of generative capacity, producing a sufficient number of distinctive signs to name all the objects and then name the names that are needed in a given discussion.
Before this complex of relationships can be formalized in much detail, I must introduce linguistic devices for generating "higher order signs", used to indicate other signs, and "situated signs", indexed by the names of their users, their contexts of use, and other types of information incidental to their usage in general. This leads to the consideration of SOIs that maintain recursive mechanisms for naming everything within their purview. This "nominal generosity" gives them a new order of generative capacity, producing a sufficient number of distinctive signs to name all the objects and then name the names that are needed in a given discussion.
Symbolic systems for quoting inscriptions and ascribing quotations are associated in metamathematics with "godel numberings" of formal objects, enumerative functions that provide systematic but ostensibly arbitrary reference numbers for the signs and expressions in a formal language. Assuming these signs and expressions denote anything at all, their formal enumerations become the "codes" of formal objects, just as programs taken literally are code names for certain mathematical objects known as computable functions. Partial forms of specification not withstanding, these codes are the only complete modes of representation that formal objects can have in the medium of mechanical activity.
Symbolic systems for quoting inscriptions and ascribing quotations are associated in metamathematics with "godel numberings" of formal objects, enumerative functions that provide systematic but ostensibly arbitrary reference numbers for the signs and expressions in a formal language. Assuming these signs and expressions denote anything at all, their formal enumerations become the "codes" of formal objects, just as programs taken literally are code names for certain mathematical objects known as computable functions. Partial forms of specification not withstanding, these codes are the only complete modes of representation that formal objects can have in the medium of mechanical activity.
Ordinarily, the extra degree of attention to syntax that is needed for critical reflection on interpretive processes is called into play by means of syntactic operators and diacritical devices acting at the level of individual signs and elementary expressions. For example, quotation marks are used to force one type of "semantic ascent", causing signs to be treated as objects and marking points of interpretive shift as they occur in the syntactic medium. But these operators and devices must be symbolized, and these symbols must be interpreted. Consequently, there is no way to avoid the invocation of a cohering interpretive framework, one that needs to be specialized for analytic purposes.
Ordinarily, the extra degree of attention to syntax that is needed for critical reflection on interpretive processes is called into play by means of syntactic operators and diacritical devices acting at the level of individual signs and elementary expressions. For example, quotation marks are used to force one type of "semantic ascent", causing signs to be treated as objects and marking points of interpretive shift as they occur in the syntactic medium. But these operators and devices must be symbolized, and these symbols must be interpreted. Consequently, there is no way to avoid the invocation of a cohering interpretive framework, one that needs to be specialized for analytic purposes.
The best way to achieve the desired type of reflective capacity is by attaching a parameter to the IF used as an instrument of formal study, specifying certain choices or interpretive presumptions that affect the entire context of discussion. The aesthetic distance needed to arrive at a formal perspective on sign relations is maintained, not by jury rigging ordinary discussion with locally effective syntactic devices, but by asking the reader to consider certain dimensions of parametric variation in the global IF's used to comprehend the sign relations under study.
The best way to achieve the desired type of reflective capacity is by attaching a parameter to the IF used as an instrument of formal study, specifying certain choices or interpretive presumptions that affect the entire context of discussion. The aesthetic distance needed to arrive at a formal perspective on sign relations is maintained, not by jury rigging ordinary discussion with locally effective syntactic devices, but by asking the reader to consider certain dimensions of parametric variation in the global IFs used to comprehend the sign relations under study.
The interpretive parameter of paramount importance to this work is one that is critical to reflection. It can be presented as a choice between two alternative conventions, affecting the way one reflexively regards each sign in a text: (1) as a sign provoking interest only in passing, exchanged for the sake of a meaningful object it is always taken for granted to have, or (2) as a sign comprising an interest in and of itself, a state of a system or a modification of a medium that can signify an external value but does not necessarily denote anything else at all. I will name these options for responding to signs according to the aspects of character that are most appreciated in their net effects, whether signs for the sake of objects, or signs for their own sake, respectively.
The interpretive parameter of paramount importance to this work is one that is critical to reflection. It can be presented as a choice between two alternative conventions, affecting the way one reflexively regards each sign in a text: (1) as a sign provoking interest only in passing, exchanged for the sake of a meaningful object it is always taken for granted to have, or (2) as a sign comprising an interest in and of itself, a state of a system or a modification of a medium that can signify an external value but does not necessarily denote anything else at all. I will name these options for responding to signs according to the aspects of character that are most appreciated in their net effects, whether signs for the sake of objects, or signs for their own sake, respectively.
Whenever a greater degree of care is required, it becomes necessary to replace the object convention with the sign convention, which presumes to take for granted only what can be obvious to all observers, namely, the phenomenal appearances and temporal occurrences of objectified states of systems. To be sure, these modulations of media are still presented as signs, but only potentially as signs of other things. It goes with the territory of the formal language context to constantly check the inveterate impulses of the literate mind, to reflect on its automatic reflex toward meaning, to inhibit its uncontrolled operation, and to pause long enough in the rush to judgment to question whether its constant presumption of a motive is itself innocent.
Whenever a greater degree of care is required, it becomes necessary to replace the object convention with the sign convention, which presumes to take for granted only what can be obvious to all observers, namely, the phenomenal appearances and temporal occurrences of objectified states of systems. To be sure, these modulations of media are still presented as signs, but only potentially as signs of other things. It goes with the territory of the formal language context to constantly check the inveterate impulses of the literate mind, to reflect on its automatic reflex toward meaning, to inhibit its uncontrolled operation, and to pause long enough in the rush to judgment to question whether its constant presumption of a motive is itself innocent.
In order to deal with these issues of discourse analysis in an explicit way, it is necessary to have in place a technical notation for marking the very kinds of interpretive assumptions that normally go unmarked. Thus, I will describe a set of devices for annotating certain kinds of interpretive contingencies, called the "discourse analysis frames" (DAF's) or the "global interpretive frames" (GIF's), that can be operative at any given moment in a particular context of discussion.
In order to deal with these issues of discourse analysis in an explicit way, it is necessary to have in place a technical notation for marking the very kinds of interpretive assumptions that normally go unmarked. Thus, I will describe a set of devices for annotating certain kinds of interpretive contingencies, called the "discourse analysis frames" (DAFs) or the "global interpretive frames" (GIFs), that can be operative at any given moment in a particular context of discussion.
To mark a context of discussion where a particular set J of interpretive conventions is being maintained, I use labeled brackets of the following two forms: "unitary", as "{J| ... |J}, or "divided", as {J| ... | ... |J}. The unitary form encloses a context of discussion by delimiting a range of text whose reading is subject to the interpretive constraints J. The divided form specifies the objects, signs, and interpretive information in accord with which a species of discussion is generated. Labeled brackets enclosing contexts can be nested in their scopes, with interpretive data on each outer envelope applying to every inclusion. Labeled brackets arranging the "conversation pieces" or the "generators and relations" of a topic can lead to discussions that spill outside their frames, and thus are permitted to constitute overlapping contexts.
To mark a context of discussion where a particular set J of interpretive conventions is being maintained, I use labeled brackets of the following two forms: "unitary", as "{J| ... |J}, or "divided", as {J| ... | ... |J}. The unitary form encloses a context of discussion by delimiting a range of text whose reading is subject to the interpretive constraints J. The divided form specifies the objects, signs, and interpretive information in accord with which a species of discussion is generated. Labeled brackets enclosing contexts can be nested in their scopes, with interpretive data on each outer envelope applying to every inclusion. Labeled brackets arranging the "conversation pieces" or the "generators and relations" of a topic can lead to discussions that spill outside their frames, and thus are permitted to constitute overlapping contexts.