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~~'''Author: [[User:Jon Awbrey|Jon Awbrey]]'''~~

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• [[Inquiry Driven Systems|Contents]]

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• [[Directory:Jon Awbrey/Papers/Inquiry Driven Systems|Contents]]

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• [[Inquiry Driven Systems : Part 1|Part 1]]

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• [[Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Part 1|Part 1]]

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• [[Inquiry Driven Systems : Part 2|Part 2]]

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• [[Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Part 2|Part 2]]

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• [[Inquiry Driven Systems : Part 3|Part 3]]

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• [[Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Part 3|Part 3]]

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• [[Inquiry Driven Systems : Part 4|Part 4]]

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• [[Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Part 4|Part 4]]

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• [[Inquiry Driven Systems : Part 5|Part 5]]

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• [[Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Part 5|Part 5]]

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• [[Inquiry Driven Systems : Part 6|Part 6]]

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• [[Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Part 6|Part 6]]

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• [[Inquiry Driven Systems : Part 7|Part 7]]

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• [[Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Part 7|Part 7]]

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• [[Inquiry Driven Systems : Part 8|Part 8]]

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• [[Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Part 8|Part 8]]

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• [[Inquiry Driven Systems : Appendices|Appendices]]

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• [[Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Appendices|Appendices]]

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• [[Inquiry Driven Systems : References|References]]

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• [[Directory:Jon Awbrey/Papers/Inquiry Driven Systems : References|References]]

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• [[Inquiry Driven Systems : Document History|Document History]]

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• [[Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Document History|Document History]]

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</div>

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=~~=Introduction==~~

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==~~=Outline of the Project : Inquiry Into Inquiry=~~==

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==1. Introduction==

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====Problem====

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===1.1. Outline of the Project : Inquiry Into Inquiry===

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====1.1.1. Problem====

This research is oriented toward a single problem: What is the nature of inquiry? I intend to address crucial questions about the operation, organization, and computational facilitation of inquiry, taking inquiry to encompass the general trend of all forms of reasoning that lead to the features of scientific investigation as their ultimate development.

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====Method====

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====1.1.2. Method====

How will I approach this problem about the nature of inquiry? The simplest answer is this: I will apply the method of inquiry to the problem of inquiry's nature.

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If everything goes according to the pattern I have observed in previous work, the principal facets of analytic and synthetic procedure will each be prefaced by its own distinctive phase of preparatory activity, where the basic materials needed for further investigation are brought together for comparative study. Taking these initial stages into consideration, I can describe the main modalities of this research in greater detail.

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=====The Paradigmatic and Process-Analytic Phase=====

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=====1.1.2.1. The Paradigmatic and Process-Analytic Phase=====

In this phase I describe the performance and competence of intelligent agents in terms of various formal systems. For aspects of an inquiry process that affect its dynamic or temporal performance I will typically use representations modeled on finite automata and differential systems. For aspects of an inquiry faculty that reflect its formal or symbolic competence I will commonly use representations like formal grammars, logical calculi, constraint-based axiom systems, and rule-based theories in association with different proof styles.

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The last option of the last step already overlaps with the synthetic phase of work. Viewing this procedure within the frame of experimental research, it is important to recognize that computer programs can fill the role of hypotheses, testable (defeasible or falsifiable) construals of how a process is actually, might be possibly, or ought to be optimally carried out.

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=====The Paraphrastic and Faculty-Synthetic Phase=====

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=====1.1.2.2. The Paraphrastic and Faculty-Synthetic Phase=====

The closely allied techniques of task analysis and software development that are known as ''step-wise refinement'' and ''top-down programming'' in computer science (Wirth 1976, 49, 303) have a long ancestry in logic and philosophy, going back to a strategy for establishing or discharging contextual definitions known as ''paraphrasis''. All of these methods are founded on the idea of providing meaning for operational specifications, ''definitions in use'', ''alleged descriptions'', or ''incomplete symbols''. No excessive generosity with the resources of meaning is intended, though. In practice, a larger share of the routine is spent detecting meaningless fictions rather than discovering meaningful concepts.

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For the purposes of this project, I will take ''paraphrastic definition'' to denote the analysis of formal specifications and contextual constraints to derive effective implementations of a process or its faculty. This is carried out by considering what the faculty in question is required to do in the many contexts it is expected to serve, and then by analyzing these formal specifications in order to design computer programs that fulfill them.

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=====Reprise of Methods=====

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=====1.1.2.3. Reprise of Methods=====

In summary, the whole array of methods will be typical of the top-down strategies used in artificial intelligence research (AIR), involving the conceptual and operational analysis of higher-order cognitive capacities with an eye toward the modeling, grounding, and support of these faculties in the form of effective computer programs. The toughest part of this discipline is in making sure that one does “come down”, that is, in finding guarantees that the analytic reagents and synthetic apparatus that one applies are actually effective, reducing the fat of speculation into something that will wash.

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Finally, I ought to observe a hedge against betting too much on this or any neat arrangement of research stages. It should not be forgotten that the flourishing of inquiry evolves its own forms of organic integrity. No matter how one tries to tease them apart, the various tendrils of research tend to interleave and intertwine as they will.

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====Criterion====

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====1.1.3. Criterion====

When is enough enough? What measure can I use to tell if my effort is working? What information is critical in deciding whether my exercise of the method is advancing my state of knowledge toward a solution of the problem?

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What are the practical tests of whether the results of inquiry succeed in reducing uncertainty? Two gains are often cited: Successful results of inquiry provide the agent with increased powers of prediction and control as to how the object system will behave in given circumstances. If a common theme is desired, at the price of a finely equivocal thread, it can be said that the agent has gained in its power of determination. Hence, more certainty is exhibited by less hesitation, more determination is manifested by less vacillation.

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====Application====

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====1.1.4. Application====

Where can the results be used? Knowledge about the nature of inquiry can be applied. It can be used to improve our personal competence at inquiry. It can be used to build software support for the tasks involved in inquiry.

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In yet another development, computer models of learning and reasoning form a linkage among cognitive psychology (the descriptive study of how we think), artificial intelligence (the prospective study of how we might think), and the logic of operations research (the normative study of how we ought to think in order to achieve the goals of reasoning).

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===Onus of the Project : No Way But Inquiry===

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===1.2. Onus of the Project : No Way But Inquiry===

At the beginning of inquiry there is nothing for me to work with but the actual constellation of doubts and beliefs that I have at the moment. Beliefs that operate at the deepest levels can be so taken for granted that they rarely if ever obtrude on awareness. Doubts that oppress in the most obvious ways are still known only as debits and droughts, as the absence of something, one knows not what, and a desire that obliges one only to try. Obscure forms of oversight provide an impulse to replenish the condition of privation but never out of necessity afford a sense of direction. One senses there ought to be a way out at once, or ordered ways to overcome obstruction, or organized or otherwise ways to obviate one's opacity of omission and rescue a secure motivation from the array of conflicting possibilities. In the roughest sense of the word, any action that does in fact lead out of this onerous state can be regarded as a form of “inquiry”. Only later, in moments of more leisurely inquiry, when it comes down to classifying and comparing the manner of escapes that can be recounted, does it become possible to recognize the ways in which certain general patterns of strategy are routinely more successful in the long run than others.

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====A Modulating Prelude====

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====1.2.1. A Modulating Prelude====

If I aim to devise the kind of computational support that can give the greatest assistance to inquiry, then it must be able to come in at the very beginning, to be of service in the kinds of formless and negative conditions that I just described, and to help people navigate a way through the constellations of contingent, incomplete, and contradictory indications that they actually find themselves sailing under at present.

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Toward the end of this discussion I will be using highly concrete mathematical models, or very specific families of combinatorial objects, to represent the abstract structures of experiential sequences that agents pass through. If these primitive and simplified models are to be regarded as something more than mere toys, and if the relations of particular experiences to particular models, along with the structural relationships that exist within the field of experiences and again within the collection of models, are not to be dismissed as category confusions, then I will need to develop a toolbox of logical techniques that can be used to justify these constructions. The required technology of categorical and relational notions will be developed in the process of addressing its basic task: To show how the same conceptual categories can be applied to materials and models of experience that are radically diverse in their specific contents and peculiar to the states of the particular agents to which they attach.

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====A Fugitive Canon====

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====1.2.2. A Fugitive Canon====

The principal difficulties associated with this task appear to spring from two roots.

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Finally, not altogether as an afterthought, there is a question that binds these issues together. How does it make sense to apply one's individual conceptions of the abstract categories of experience, not only to the experiences of oneself and others, but in points of form to compare them with the structures present in mathematical models?

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===Option of the Project : A Way Up To Inquiry===

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===1.3. Option of the Project : A Way Up To Inquiry===

I begin with an informal examination of the concept of inquiry. This section takes as its subjects the supposed faculty of inquiry in general and the present inquiry into inquiry in particular, and attempts to analyze them in relation to each other on formal principles alone.

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serve as proxies for unknown components and indicate tentative analyses of faculties in question.

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====Initial Analysis of Inquiry : Allegro Aperto====

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====1.3.1. Initial Analysis of Inquiry : Allegro Aperto====

If the faculty of inquiry is a coherent power, then it has an active or instrumental face, a passive or objective face, and a substantial body of connections between them.

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:: <math>y_0 = y \cdot y >\!\!= \{ d , f \} \{ d , f \} >\!\!= \{ f \} \{ d \}\!</math>

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====Discussion of Discussion====

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====1.3.2. Discussion of Discussion====

But first, I nearly skipped a step. Though it might present itself as an interruption, a topic so easy that I almost omitted it altogether deserves at least a passing notice.

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There's a catch here that applies to all living creatures: In order to keep talking one has to keep living. This brings discussion back to its role in inquiry, considered as an adaptation of living creatures designed to help them deal with their not so virtual environments. If discussion is constrained to the envelope of life and required to contribute to the trend of inquiry, instead of representing a kind of internal opposition, then it must be possible to tighten up the loose account and elevate the digressionary narrative into a properly directed inquiry. This brings an end to my initial discussion of ''discussion''.

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====Discussion of Formalization : General Topics====

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====1.3.3. Discussion of Formalization : General Topics====

Because this project makes constant use of formal models of phenomenal processes, it is appropriate at this point to introduce the understanding of formalization that I will use throughout this work and to preview a concrete example of its application.

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=====A Formal Charge=====

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=====1.3.3.1. A Formal Charge=====

An introduction to the topic of formalization, if proper, is obliged to begin informally. But it will be my constant practice to keep a formal eye on the whole proceedings. What this form of observation reveals must be kept silent for the most part at first, but I see no rule against sharing with the reader the general order of this watch:

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# Ask yourself, with regard to each postulant faculty in the current account, explicitly charged or otherwise, whether you can imagine any recipe, any program, any rule of procedure for carrying out the form, if not the substance, of what it does, or an aspect thereof.

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=====A Formalization of Formalization?=====

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=====1.3.3.2. A Formalization of Formalization?=====

An immediate application of the above rules is presented here, in hopes of giving the reader a concrete illustration of their use in a ready example, but the issues raised can quickly diverge into yet another distracting digression, one not so easily brought under control as the discussion of discussion, but whose complexity probably approaches that of the entire task. Therefore, a partial adumbration of its character will have to suffice for the present.

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The reader can follow this example with every concept that I mention in the explanation of formalization, and again in the larger investigation of inquiry, and be assured that it is has not often slipped my attention to at least venture the same, though a delimitation of each exploration in its present state of completion would be far too tedious and tenuous to escape expurgation.

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=====A Formalization of Discussion?=====

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=====1.3.3.3. A Formalization of Discussion?=====

The previous section took the concept of ''formalization'' as an example of a topic that a writer might try to translate from informal to formal discussion, perhaps as a way of clarifying the general concept to an optimal degree, or perhaps as a way of communicating a particular concept of it to a reader. In either case the formalization process, that aims to translate a concept from informal to formal discussion, is itself mediated by a form of discussion: (1) that interpreters conduct as a part of their ongoing monologue with themselves, or (2) that a writer (speaker) conducts in real or imagined dialogue with a reader (hearer). In view of this, I see no harm in letting the concept of discussion be stretched to cover all attempted processes of formalization.

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On several occasions, this discussion of inquiry will arrive at a form of ''aesthetic deduction'', in general terms, a piece of reasoning that ends with a design recommendation, in this case, where an analysis of the general purposes and interests of inquiry leads to the conclusion that a certain property of discussion is an admirable one, and that the quality in question forms an essential part of the implicit value system that is required to guide inquiry and make it what it is meant to be, a method for advancing toward desired forms of knowledge. After a collection of admirable qualities has been recognized as cohering together into a unity, it becomes natural to ask: What is the underlying reality that inheres in these qualities, and what are the logical relations that bind them together into the qualifications of inquiry and a definition of exactly what is desired for knowledge?

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=====A Concept of Formalization=====

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=====1.3.3.4. A Concept of Formalization=====

The concept of formalization is intended to cover the whole collection of activities that serve to build a relation between casual discussions, those that take place in the ordinary context of informal discourse, and formal discussions, those that make use of completely formalized models. To make a long story short, formalization is the narrative operation or active relation that construes the situational context in the form of a definite text. The end product that results from the formalization process is analogous to a snapshot or a candid picture, a relational or functional image that captures an aspect of the casual circumstances.

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Relations between casual and formal discussion are often treated in terms of a distinction between two languages, the ''meta-language'' and the ''object language'', linguistic systems that take complementary roles in filling out the discussion of interest. In the usual approach, issues of formalization are addressed by postulating a distinction between the meta-language, the descriptions and conceptions from ordinary language and technical discourse that can be used without being formalized, and the object language, the domain of structures and processes that can be studied as a completely formalized object.

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=====A Formal Approach=====

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=====1.3.3.5. A Formal Approach=====

I plan to approach the issue of formalization from a slightly different angle, proceeding through an analysis of the medium of interpretation and developing an effective conception of ''interpretive frameworks'' or ''interpretive systems''. This concept refers to any organized system of interpretive practice, ranging from those used in everyday speech, to the ones that inform technical discourse, to the kinds of completely formalized symbol systems that one can safely regard as mathematical objects. Depending on the degree of objectification that it possesses from one's point of view, the same system of conduct can be variously described as an interpretive framework (IF), interpretive system (IS), interpretive object (IO), or object system (OS). These terms are merely suggestive — no rigid form of classification is intended.

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From this angle of approach, let us try to view afresh the manner of drawing distinctions between various levels of formalization in language. Once again, I begin in the context of ordinary discussion, and if there is any distinction to be drawn between objective and instrumental languages then it must be possible to describe it within the frame of this informally discursive universe.

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=====A Formal Development=====

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=====1.3.3.6. A Formal Development=====

The point of view I take on the origin and development of formal models is that they arise with agents retracing structures that already exist in the context of informal activity, until gradually the most relevant and frequently reinforced patterns become emphasized and emboldened enough to continue their development as nearly autonomous styles, in brief, as ''genres'' growing out of a particular ''paradigm''.

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Taking the position that formal models develop within the framework of informal discussion, the questions that become important to ask of a prospective formal model are (1) whether it highlights the structure of its supporting context in a transparent form of emphasis and a relevant reinforcement of salient features, and (2) whether it reveals the active ingredients of its source materials in a critically reflective recapitulation or an analytically representative recipe, or (3) whether it insistently obscures what little fraction of its domain it manages to cover.

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=====A Formal Persuasion=====

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=====1.3.3.7. A Formal Persuasion=====

An interpretive system can be taken up with very little fanfare, since it does not enjoin one to declare undying allegiance to a particular point of view or to assign each piece of text in view to a sovereign territory, but only to entertain different points of view on the use of symbols. The chief design consideration for an interpretive system is that it must never function as a virus or addiction. Its suggestions must always be, initially and finally, purely optional adjunctions to whatever interpretive framework was already in place before it installed itself on the scene. Interpretive systems are not constituted in the faith that anything nameable will always be dependable, nor articulated in fixed principles that determine what must be doubted and what must not, but rest only in a form of self-knowledge that recognizes the doubts and beliefs that one actually has at each given moment.

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Before this project is done I will need to have developed an analytic and computational theory of interpreters and interpretive frameworks. In the aspects of this theory that I can anticipate at this point, an interpreter or interpretive framework is exemplified by a collective activity of symbol-using practices like those that might be found embodied in a person, a community, or a culture. Each one forms a moderately free and independent perspective, with no objective rankings of supremacy in practice that all interpretive frameworks are likely to support at any foreseeable moment in their fields of view. Of course, each interpreter initially enters discussion operating as if its own perspective were ''meta'' in comparison to all the others, but a well-developed interpretive framework is likely to have acquired the notion and taken notice of the fact that this is not likely to be a universally shared opinion (USO).

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====Discussion of Formalization : Concrete Examples====

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====1.3.4. Discussion of Formalization : Concrete Examples====

The previous section outlined a variety of general issues surrounding the concept of formalization. The following section will plot the specific objectives of this project in constructing formal models of intellectual processes. In this section I wish to take a breather between these abstract discussions in order to give their main ideas a few points of contact with terra firma. To do this, I examine a selection of concrete examples, artificially constructed to approach the minimum levels of non-trivial complexity, that are intended to illustrate the kinds of mathematical objects I have in mind using as formal models.

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=====Formal Models : A Sketch=====

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=====1.3.4.1. Formal Models : A Sketch=====

To sketch the features of the modeling activity that are relevant to the immediate purpose: The modeler begins with a ''phenomenon of interest'' or a ''process of interest'' (POI) and relates it to a formal ''model of interest'' (MOI), the whole while working within a particular ''interpretive framework'' (IF) and relating the results from one ''system of interpretation'' (SOI) to another, or to a subsequent development of the same SOI.

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# Finally, the IFs and the SOIs always remain partly out of sight, caught up in various stages of explicit notice between casual informality and partial formalization, with no guarantee or even much likelihood of a completely articulate formulation being forthcoming or even possible. Still, it is usually worth the effort to try lifting one edge or another of these frameworks and backdrops into the light, at least for a time.

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=====Sign Relations : A Primer=====

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=====1.3.4.2. Sign Relations : A Primer=====

To the extent that their structures and functions can be discussed at all, it is likely that all of the formal entities that are destined to develop in this approach to inquiry will be instances of a class of [[triadic relation|three-place relation]]s called ''[[sign relation]]s''. At any rate, all of the formal structures that I have examined so far in this area have turned out to be easily converted to or ultimately grounded in sign relations. This class of triadic relations constitutes the main study of the ''pragmatic theory of signs'', a branch of logical philosophy devoted to understanding all types of symbolic representation and communication.

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Therefore, my attention is directed mainly toward understanding the forms of correlation, coordination, and cooperation among the various components of sign relations that form the necessary conditions for carrying out coherent inquiries.

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=====Semiotic Equivalence Relations=====

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=====1.3.4.3. Semiotic Equivalence Relations=====

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If one examines the sign relations <math>L_\~~mathrm~~{A}\!</math> and <math>L_\~~mathrm~~{B}\!</math> that are associated with the interpreters <math>\~~mathrm~~{A}\!</math> and <math>\~~mathrm~~{B},\!</math> respectively, one observes that they have many contingent properties that are not possessed by sign relations in general. One nice property possessed by the sign relations <math>L_\~~mathrm~~{A}\!</math> and <math>L_\~~mathrm~~{B}\!</math> is that their connotative components <math>\~~mathrm~~{A}_{SI}\!</math> and <math>\~~mathrm~~{B}_{SI}\!</math> constitute a pair of equivalence ~~relations~~ on their common syntactic domain <math>S = I.\!</math> It is convenient to refer to such structures as ''[[semiotic equivalence ~~relations~~]]'' (SERs) since they equate signs that mean the same thing to somebody. Each of the SERs, <math>\~~mathrm~~{A}_{SI}, \~~mathrm~~{B}_{SI} \subseteq S \times I = S \times S,\!</math> partitions the whole collection of signs into ''semiotic equivalence ~~classes~~'' (SECs). This makes for a strong form of representation in that the structure of the participants' common object domain is reflected or reconstructed, part for part, in the structure of each of their ''semiotic ~~partitions~~'' (SEPs) of the syntactic domain.

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If one examines the sign relations <math>L_\text{A}\!</math> and <math>L_\text{B}\!</math> that are associated with the interpreters <math>\text{A}\!</math> and <math>\text{B}\!</math>, respectively, one observes that they have many contingent properties that are not possessed by sign relations in general. One nice property possessed by the sign relations <math>L_\text{A}\!</math> and <math>L_\text{B}\!</math> is that their connotative components <math>\text{A}_{SI}\!</math> and <math>\text{B}_{SI}\!</math> constitute a pair of [[equivalence relation]]s on their common syntactic domain <math>S = I\!</math>. It is convenient to refer to such structures as ''[[semiotic equivalence relation]]s'' (SERs) since they equate signs that mean the same thing to somebody. Each of the SERs, <math>\text{A}_{SI}, \text{B}_{SI} \subseteq S \times I = S \times S\!</math>, partitions the whole collection of signs into ''[[semiotic equivalence class]]es'' (SECs). This makes for a strong form of representation in that the structure of the participants' common object domain is reflected or reconstructed, part for part, in the structure of each of their ''[[semiotic partition]]s'' (SEPs) of the syntactic domain.

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The main trouble with this notion of semantics in the present situation is that the two semiotic partitions for <math>\~~mathrm~~{A}\!</math> and <math>\~~mathrm~~{B}\!</math> are not the same, indeed, they are orthogonal to each other. This makes it difficult to interpret either one of the partitions or equivalence relations on the syntactic domain as corresponding to any sort of objective structure or invariant reality, independent of the individual interpreter's point of view.

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The main trouble with this notion of semantics in the present situation is that the two semiotic partitions for <math>\text{A}\!</math> and <math>\text{B}\!</math> are not the same, indeed, they are orthogonal to each other. This makes it difficult to interpret either one of the partitions or equivalence relations on the syntactic domain as corresponding to any sort of objective structure or invariant reality, independent of the individual interpreter's point of view.

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Information about the different forms of semiotic equivalence induced by the interpreters <math>\~~mathrm~~{A}\!</math> and <math>\~~mathrm~~{B}\!</math> is summarized in Tables 3 and 4. The form of these Tables should suffice to explain what is meant by saying that the SEPs for <math>\~~mathrm~~{A}\!</math> and <math>\~~mathrm~~{B}\!</math> are orthogonal to each other.

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Information about the different forms of semiotic equivalence induced by the interpreters <math>\text{A}\!</math> and <math>\text{B}\!</math> is summarized in Tables 3 and 4. The form of these Tables should suffice to explain what is meant by saying that the SEPs for <math>\text{A}\!</math> and <math>\text{B}\!</math> are orthogonal to each other.

<br>

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{| align="center" border="0" cellpadding="4" cellspacing="0" style="text-align:center; width:100%"

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| width="50%" | <math>{}^{\backprime\backprime} \~~mathrm~~{A} {}^{\prime\prime}\!</math>

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| width="50%" | <math>{}^{\backprime\backprime} \text{A} {}^{\prime\prime}\!</math>

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| width="50%" | <math>{}^{\backprime\backprime} \~~mathrm~~{i} {}^{\prime\prime}\!</math>

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| width="50%" | <math>{}^{\backprime\backprime} \text{i} {}^{\prime\prime}\!</math>

|}

|-

|

{| align="center" border="0" cellpadding="4" cellspacing="0" style="text-align:center; width:100%"

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| width="50%" | <math>{}^{\backprime\backprime} \~~mathrm~~{u} {}^{\prime\prime}\!</math>

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| width="50%" | <math>{}^{\backprime\backprime} \text{u} {}^{\prime\prime}\!</math>

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| width="50%" | <math>{}^{\backprime\backprime} \~~mathrm~~{B} {}^{\prime\prime}\!</math>

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| width="50%" | <math>{}^{\backprime\backprime} \text{B} {}^{\prime\prime}\!</math>

|}

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{| align="center" border="0" cellpadding="12" cellspacing="0" style="text-align:center; width:50%"

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| <math>{}^{\backprime\backprime} \~~mathrm~~{A} {}^{\prime\prime}\!</math>

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| <math>{}^{\backprime\backprime} \text{A} {}^{\prime\prime}\!</math>

|-

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| <math>{}^{\backprime\backprime} \~~mathrm~~{u} {}^{\prime\prime}\!</math>

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| <math>{}^{\backprime\backprime} \text{u} {}^{\prime\prime}\!</math>

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{| align="center" border="0" cellpadding="12" cellspacing="0" style="text-align:center; width:50%"

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| <math>{}^{\backprime\backprime} \~~mathrm~~{i} {}^{\prime\prime}\!</math>

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| <math>{}^{\backprime\backprime} \text{i} {}^{\prime\prime}\!</math>

|-

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| <math>{}^{\backprime\backprime} \~~mathrm~~{B} {}^{\prime\prime}\!</math>

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| <math>{}^{\backprime\backprime} \text{B} {}^{\prime\prime}\!</math>

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<br>

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To discuss these types of situations further, I introduce the square bracket notation <math>[x]_E\!</math> for ''the equivalence class of the element <math>x\!</math> under the equivalence relation <math>E.\!</math>'' A statement that the elements <math>x\!</math> and <math>y\!</math> are equivalent under <math>E\!</math> is called an ''equation'', and can be written in either one of two ways, as <math>[x]_E = [y]_E\!</math> or as <math>x =_E y.\!</math>

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To discuss these types of situations further, I introduce the square bracket notation <math>[x]_E\!</math> for ''the equivalence class of the element <math>x\!</math> under the equivalence relation <math>E\!</math>''. A statement that the elements <math>x\!</math> and <math>y\!</math> are equivalent under <math>E\!</math> is called an ''equation'', and can be written in either one of two ways, as <math>[x]_E = [y]_E\!</math> or as <math>x =_E y\!</math>.

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In the application to sign relations I extend this notation in the following ways. When <math>L\!</math> is a sign relation whose ''syntactic projection'' or connotative component <math>L_{SI}\!</math> is an equivalence relation on <math>S,\!</math> I write <math>[s]_L\!</math> for ''the equivalence class of <math>s\!</math> under <math>L_{SI}.\!</math>'' A statement that the signs <math>x\!</math> and <math>y\!</math> are synonymous under a semiotic equivalence relation <math>L_{SI}\!</math> is called a ''semiotic equation'' (SEQ) and can be written in either of the forms: <math>[x]_L = [y]_L\!</math> or <math>x =_L y.\!</math>

+

In the application to sign relations I extend this notation in the following ways. When <math>L\!</math> is a sign relation whose ''syntactic projection'' or connotative component <math>L_{SI}\!</math> is an equivalence relation on <math>S\!</math>, I write <math>[s]_L\!</math> for ''the equivalence class of <math>s\!</math> under <math>L_{SI}\!</math>''. A statement that the signs <math>x\!</math> and <math>y\!</math> are synonymous under a semiotic equivalence relation <math>L_{SI}\!</math> is called a ''semiotic equation'' (SEQ), and can be written in either of the forms: <math>[x]_L = [y]_L\!</math> or <math>x =_L y\!</math>.

−

In many situations there is one further adaptation of the square bracket notation that can be useful. Namely, when there is known to exist a particular triple <math>{(o, s, i) \in L},\!</math> it is permissible to use <math>[o]_L\!</math> to mean the same thing as <math>[s]_L\!</math>. These modifications are designed to make the notation for semiotic equivalence classes harmonize as well as possible with the frequent use of similar devices for the denotations of signs and expressions.

+

In many situations there is one further adaptation of the square bracket notation that can be useful. Namely, when there is known to exist a particular triple <math>(o, s, i) \in L\!</math>, it is permissible to use <math>[o]_L\!</math> to mean the same thing as <math>[s]_L\!</math>. These modifications are designed to make the notation for semiotic equivalence classes harmonize as well as possible with the frequent use of similar devices for the denotations of signs and expressions.

−

~~The semiotic equivalence relation~~ for interpreter <math>\~~mathrm~~{A}\!</math> yields the ~~following~~ semiotic equations:

+

In these terms, the SER for interpreter <math>\text{A}\!</math> yields the semiotic equations:

−

{| ~~align~~="~~center~~" ~~style~~="~~text-align:center; width:100~~%"

+

{| cellpadding="10"

−

|

+

| width="10%" |  

−

<math>\~~begin~~{~~matrix~~}

+

| <math>[{}^{\backprime\backprime} \text{A} {}^{\prime\prime}]_\text{A}\!</math>

−

[ {}^{\backprime\backprime} \~~mathrm~~{A} {}^{\prime\prime} ]_~~{L_~~\~~mathrm~~{A}}

+

| <math>=\!</math>

−

& = &

+

| <math>[{}^{\backprime\backprime} \text{i} {}^{\prime\prime}]_\text{A}\!</math>

−

[ {}^{\backprime\backprime} \~~mathrm~~{i} {}^{\prime\prime} ]_~~{L_~~\~~mathrm~~{A}}

+

| width="20%" |  

−

\~~\[6pt]~~

+

| <math>[{}^{\backprime\backprime} \text{B} {}^{\prime\prime}]_\text{A}\!</math>

−

[ {}^{\backprime\backprime} \~~mathrm~~{B} {}^{\prime\prime} ]_{L_\~~mathrm~~{A}}

+

| <math>=\!</math>

−

& = &

+

| <math>[{}^{\backprime\backprime} \text{u} {}^{\prime\prime}]_\text{A}\!</math>

−

[ {}^{\backprime\backprime} \~~mathrm~~{u} {}^{\prime\prime} ]_{L_\~~mathrm~~{A}}

+

|-

−

\~~end~~{~~matrix~~}</math>

+

| width="10%" | or

+

|  <math>{}^{\backprime\backprime} \text{A} {}^{\prime\prime}\!</math>

+

| <math>=_\text{A}\!</math>

+

|  <math>{}^{\backprime\backprime} \text{i} {}^{\prime\prime}\!</math>

+

| width="20%" |  

+

|  <math>{}^{\backprime\backprime} \text{B} {}^{\prime\prime}\!</math>

+

| <math>=_\text{A}\!</math>

+

|  <math>{}^{\backprime\backprime} \text{u} {}^{\prime\prime}\!</math>

|}

−

or

+

and the semiotic partition: <math>\{ \{ {}^{\backprime\backprime} \text{A} {}^{\prime\prime}, {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \} , \{ {}^{\backprime\backprime} \text{B} {}^{\prime\prime}, {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \} \}\!</math>.

−

~~{| align="center" style="text-align:center; width:100%"~~

+

In contrast, the SER for interpreter <math>\text{B}\!</math> yields the semiotic equations:

−

|

−

<math>\~~begin{matrix}~~

−

~~{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}~~

−

~~& =_{L_\mathrm{A}} &~~

−

~~{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}~~

−

~~\\[6pt]~~

−

~~{}^{\backprime\backprime} \mathrm~~{B} ~~{}^{\prime\prime}~~

−

~~& =_{L_\mathrm{A}} &~~

−

~~{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}~~

−

\~~end{matrix}~~</math>

−

|}

−

~~Thus it induces the semiotic partition:~~

+

{| cellpadding="10"

−

+

| width="10%" |  

−

{| ~~align="center"~~ cellpadding="12" ~~style~~="~~text-align:center; width:100~~%"

+

| <math>[{}^{\backprime\backprime} \text{A} {}^{\prime\prime}]_\text{B}\!</math>

−

| <math>~~\{ \{~~ {}^{\backprime\backprime} \~~mathrm~~{A} {}^{\prime\prime}, {}^{\backprime\backprime} \~~mathrm~~{i} {}^{\prime\prime} \}, \{ {}^{\backprime\backprime} \~~mathrm~~{B} {}^{\prime\prime}, {}^{\backprime\backprime} \~~mathrm~~{u} {}^{\prime\prime} \~~} \}.\!</math>~~

+

| <math>=\!</math>

−

|}

+

| <math>[{}^{\backprime\backprime} \text{u} {}^{\prime\prime}]_\text{B}\!</math>

−

+

| width="20%" |  

−

~~The semiotic equivalence relation for interpreter <math>\mathrm~~{B}\!</math> ~~yields the following semiotic equations:~~

+

| <math>[{}^{\backprime\backprime} \text{B} {}^{\prime\prime}]_\text{B}\!</math>

−

+

| <math>=\!</math>

−

{| ~~align~~="~~center" style="text-align:center; width:100~~%"

+

| <math>[{}^{\backprime\backprime} \text{i} {}^{\prime\prime}]_\text{B}\!</math>

−

|

+

|-

−

<math>~~\begin{matrix}~~

+

| width="10%" | or

−

[ {}^{\backprime\backprime} \~~mathrm~~{A} {}^{\prime\prime} ]_~~{L_~~\~~mathrm~~{B}}

+

|  <math>{}^{\backprime\backprime} \text{A} {}^{\prime\prime}\!</math>

−

& ~~= &~~

+

| <math>=_\text{B}\!</math>

−

[ {}^{\backprime\backprime} \~~mathrm~~{u} {}^{\prime\prime} ~~]_{L_~~\~~mathrm{B}}~~

+

|  <math>{}^{\backprime\backprime} \text{u} {}^{\prime\prime}\!</math>

−

~~\\[6pt]~~

+

| width="20%" |  

−

[ {}^{\backprime\backprime} \~~mathrm~~{B} {}^{\prime\prime} ]_~~{L_~~\~~mathrm~~{B}}

+

|  <math>{}^{\backprime\backprime} \text{B} {}^{\prime\prime}\!</math>

−

& ~~= &~~

+

| <math>=_\text{B}\!</math>

−

[ {}^{\backprime\backprime} \~~mathrm~~{i} {}^{\prime\prime} ~~]_{L_~~\~~mathrm{B}}~~

+

|  <math>{}^{\backprime\backprime} \text{i} {}^{\prime\prime}\!</math>

−

~~\end{matrix}~~</math>

|}

−

or

+

and the semiotic partition: <math>\{ \{ {}^{\backprime\backprime} \text{A} {}^{\prime\prime}, {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \} , \{ {}^{\backprime\backprime} \text{B} {}^{\prime\prime}, {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \} \}\!</math>.

−

~~{| align~~=~~"center" style~~=~~"text-align:center; width:100%"~~

+

=====1.3.4.4. Graphical Representations=====

−

|

−

~~<math>\begin{matrix}~~

−

~~{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}~~

−

& =~~_{L_\mathrm{B}} &~~

−

~~{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}~~

−

~~\\[6pt]~~

−

~~{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}~~

−

& =~~_{L_\mathrm{B}} &~~

−

~~{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}~~

−

~~\end{matrix}</math>~~

−

|}

−

~~Thus it induces the semiotic partition:~~

+

The dyadic components of sign relations can be given graph-theoretic representations, as ''digraphs'' (or ''directed graphs''), that provide concise pictures of their structural and potential dynamic properties.

−

~~{| align="center" cellpadding="12" style="text-align:center; width:100%"~~

+

By way of terminology, a directed edge <math>(x, y)\!</math> is called an ''arc'' from point <math>x\!</math> to point <math>y\!</math>, and a self-loop <math>(x, x)\!</math> is called a ''sling'' at <math>x\!</math>.

−

| <math>~~\{ \{ {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}~~, ~~{}^{~~\~~backprime~~\~~backprime}~~ \~~mathrm{u} {}^{\prime\prime} \}~~, ~~\{ {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}~~, ~~{}^{~~\~~backprime\backprime} \mathrm{i} {}^{\prime\prime} \} \}.~~\!</math>

−

|}

−

==~~===Graphical Representations=====~~

+

The denotative components <math>\operatorname{Den}(\text{A})\!</math> and <math>\operatorname{Den}(\text{B})\!</math> can be represented as digraphs on the six points of their common world set <math>W = O \cup S \cup I = \{ \text{A}, \text{B}, {}^{\backprime\backprime} \text{A} {}^{\prime\prime}, {}^{\backprime\backprime} \text{B} {}^{\prime\prime}, {}^{\backprime\backprime} \text{i} {}^{\prime\prime}, {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \}\!</math>. The arcs are given as follows:

−

~~The dyadic components of sign relations can be given graph-theoretic representations, as ''digraphs''~~ (~~or ''directed graphs''~~), ~~that provide concise pictures~~ of ~~their structural and potential dynamic properties~~.

+

:: <math>\operatorname{Den}(\text{A})\!</math> has an arc from each point of <math>\{ {}^{\backprime\backprime} \text{A} {}^{\prime\prime}, {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \}\!</math> to <math>\text{A}\!</math> and from each point of <math>\{ {}^{\backprime\backprime} \text{B} {}^{\prime\prime}, {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \}\!</math> to <math>\text{B}\!</math>.

−

~~By way of terminology, a directed edge~~ <math>(~~x, y~~)\!</math> ~~is called~~ an ''arc'' from point <math>x\!</math> to ~~point~~ <math>y,\!</math> and ~~a self-loop~~ <math>(x, x)\!</math> ~~is called a ''sling'' at~~ <math>x.\!</math>

+

:: <math>\operatorname{Den}(\text{B})\!</math> has an arc from each point of <math>\{ {}^{\backprime\backprime} \text{A} {}^{\prime\prime}, {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \}\!</math> to <math>\text{A}\!</math> and from each point of <math>\{ {}^{\backprime\backprime} \text{B} {}^{\prime\prime}, {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \}\!</math> to <math>\text{B}\!</math>.

−

~~The denotative components~~ <math>\~~mathrm~~{Den}(L_\~~mathrm~~{A})\!</math> and <math>\~~mathrm~~{Den}(L_\~~mathrm~~{B})\!</math> can be ~~represented~~ as digraphs on the ~~six points~~ of ~~their common world set~~ <math>~~W = O~~ \~~cup S \cup I =~~\!</math> <math>\~~{ \mathrm{A}, \mathrm~~{B}~~, {}^{~~\backprime\backprime} \mathrm{A} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \}.\!</math> ~~The arcs are given as follows:~~

+

<math>\operatorname{Den}(\text{A})\!</math> and <math>\operatorname{Den}(\text{B})\!</math> can be interpreted as ''transition digraphs'' that chart the succession of steps or the connection of states in a computational process. If the graphs are read this way, the denotational arcs summarize the ''upshots'' of the computations that are involved when the interpreters <math>\text{A}\!</math> and <math>\text{B}\!</math> evaluate the signs in <math>S\!</math> according to their own frames of reference.

−

~~{| align="center" cellspacing="6" width="90%"~~

+

The connotative components <math>\operatorname{Con}(\text{A})\!</math> and <math>\operatorname{Con}(\text{B})\!</math> can be represented as digraphs on the four points of their common syntactic domain <math>S = I = \{ {}^{\backprime\backprime} \text{A} {}^{\prime\prime}, {}^{\backprime\backprime} \text{B} {}^{\prime\prime}, {}^{\backprime\backprime} \text{i} {}^{\prime\prime}, {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \}\!</math>. Since <math>\operatorname{Con}(\text{A})\!</math> and <math>\operatorname{Con}(\text{B})\!</math> are SERs, their digraphs conform to the pattern that is manifested by all digraphs of equivalence relations. In general, a digraph of an equivalence relation falls into connected components that correspond to the parts of the associated partition, with a complete digraph on the points of each part, and no other arcs. In the present case, the arcs are given as follows:

−

|

−

~~<p>~~<math>\~~mathrm~~{~~Den~~}(L_\~~mathrm~~{A})\!</math> ~~has an arc from each point of~~ <math>\{ {}~~^{\backprime~~\~~backprime} \mathrm~~{~~A} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime} \~~}\!</math> ~~to <math>\mathrm{A}\!</math> and an arc from each point~~ of <math>\{ {}^{\backprime\backprime} \~~mathrm~~{B} {}^{\prime\prime}, {}^{\backprime\backprime} \~~mathrm~~{u} {}^{\prime\prime} ~~\}\!</math> to <math>\mathrm{B}.\!</math></p>~~

−

|-

−

|

−

~~<p><math>\mathrm{Den}(L_\mathrm{B})\!</math> has an arc from each point of <math>\{~~ {}^{\backprime\backprime} \~~mathrm~~{A} {}^{\prime\prime}, {}^{\backprime\backprime} \~~mathrm~~{u} {}^{\prime\prime} \}\!</math> to <math>\~~mathrm~~{A}\!</math> and ~~an arc from each point of~~ <math>\{ {}^{\~~backprime\backprime} \mathrm~~{B} ~~{}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime} \}~~\!</math> to ~~<math>\mathrm{B}~~.~~\!</math></p>~~

−

|}

−

<math>\~~mathrm~~{~~Den~~}(L_\~~mathrm~~{A})\!</math> ~~and~~ <math>~~\mathrm{Den}(L_\mathrm{B})~~\!</math> ~~can be interpreted as ''transition digraphs'' that chart the succession~~ of ~~steps or~~ the ~~connection of states~~ in ~~a computational process. If the graphs are read this~~ way~~, the denotational~~ arcs ~~summarize~~ the ~~''upshots''~~ of ~~the computations that are involved when the interpreters~~ <math>\~~mathrm~~{A}\!</math> and <math>\~~mathrm~~{B}\~~!</math> evaluate the signs in <math>S~~\!</math> ~~according to their own frames of reference~~.

+

:: <math>\operatorname{Con}(\text{A})\!</math> has the structure of a SER on <math>S\!</math>, with a sling at each of the points in <math>S\!</math>, two-way arcs between the points of <math>\{ {}^{\backprime\backprime} \text{A} {}^{\prime\prime}, {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \}\!</math>, and two-way arcs between the points of <math>\{ {}^{\backprime\backprime} \text{B} {}^{\prime\prime}, {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \}\!</math>.

−

~~The connotative components~~ <math>\~~mathrm~~{Con}(L_\~~mathrm~~{A})\!</math> ~~and~~ <math>~~\mathrm{Con}(L_\mathrm{B})~~\!</math> ~~can be represented as digraphs on~~ the ~~four~~ points ~~of their common syntactic domain~~ <math>S ~~= I =~~\!</math> <math>\{ {}^{\backprime\backprime} \~~mathrm~~{A} {}^{\prime\prime}, {}^{\backprime\backprime} \~~mathrm~~{B} {}^{\prime\prime}, {}^{\backprime\backprime} \~~mathrm~~{i} {}^{\prime\prime}, {}^{\backprime\backprime} \~~mathrm~~{u} {}^{\prime\prime} \}~~.\!</math> Since <math>\mathrm{Con}(L_\mathrm{A})~~\!</math> and <math>\mathrm{Con}(L_\mathrm{B})\!</math> are semiotic equivalence relations, their digraphs conform to the pattern that is manifested by all digraphs of equivalence relations. In general, a digraph of an equivalence relation falls into connected components that correspond to the parts of the associated partition, with a complete digraph on the points of each part, and no other arcs. ~~In the present case, the arcs are given as follows:~~

+

:: <math>\operatorname{Con}(\text{B})\!</math> has the structure of a SER on <math>S\!</math>, with a sling at each of the points in <math>S\!</math>, two-way arcs between the points of <math>\{ {}^{\backprime\backprime} \text{A} {}^{\prime\prime}, {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \}\!</math>, and two-way arcs between the points of <math>\{ {}^{\backprime\backprime} \text{B} {}^{\prime\prime}, {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \}\!</math>.

−

~~{| align="center" cellspacing="6" width="90%"~~

+

Taken as transition digraphs, <math>\operatorname{Con}(\text{A})\!</math> and <math>\operatorname{Con}(\text{B})\!</math> highlight the associations that are permitted between equivalent signs, as this equivalence is judged by the interpreters <math>\text{A}\!</math> and <math>\text{B}\!</math>, respectively.

−

|

−

<p><math>\mathrm{Con}(L_\mathrm{A})\!</math> has the structure of a semiotic equivalence relation on <math>S,\!</math> with a sling at each point of <math>S,\!</math> arcs in both directions between the points of <math>\{ {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime} \},\!</math> and arcs in both directions between the points of <math>\{ {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \}.\!</math></p>

−

|-

−

|

−

<p><math>\mathrm{Con}(L_\mathrm{B})\!</math> has the structure of a semiotic equivalence relation on <math>S,\!</math> with a sling at each point of <math>S,\!</math> arcs in both directions between the points of <math>\{ {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \},\!</math> and arcs in both directions between the points of <math>\{ {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime} \}.\!</math></p>

−

|}

−

Taken as transition digraphs, <math>\~~mathrm~~{Con}(L_\~~mathrm~~{A})\!</math> and <math>\~~mathrm~~{Con}(L_\~~mathrm~~{B})\!</math> highlight the associations that are permitted between equivalent signs, as this equivalence is judged by the interpreters <math>\~~mathrm~~{A}\!</math> and <math>\~~mathrm~~{B},\!</math> respectively.

The theme running through the last three subsections, that associates different interpreters and different aspects of interpretation with different sorts of relational structures on the same set of points, heralds a topic that will be developed extensively in the sequel.

−

=====Taking Stock=====

+

=====1.3.4.5. Taking Stock=====

So far, my discussion of the discussion between <math>\text{A}\!</math> and <math>\text{B}\!</math>, in the picture that it gives of sign relations and their connection to the imagined processes of interpretation and inquiry, can best be described as fragmentary. In the story of <math>\text{A}\!</math> and <math>\text{B}\!</math>, a sample of typical language use has been drawn from the context of informal discussion and partially formalized in the guise of two independent sign relations, but no unified conception of the commonly understood interpretive practices in such a situation has yet been drafted.

Line 759: Line 727:

The next few subsections will be concerned with the most problematic features of the <math>\text{A}\!</math> and <math>\text{B}\!</math> dialogue, especially with the sorts of difficulties that are clues to significant deficits in theory and technique, and that point out directions for future improvements.

−

=====The “Meta” Question=====

+

=====1.3.4.6. The “Meta” Question=====

There is one point of common contention that I finessed from play in my handling of the discussion between <math>\text{A}\!</math> and <math>\text{B}\!</math>, even though it lies in plain view on both their Tables. This is the troubling business, recalcitrant to analysis precisely because its operations race on so heedlessly ahead of thought and grind on so routinely beneath its notice, that concerns the placement of object languages within the frame of a meta-language.

Line 771: Line 739:

Sign relations themselves, like any real objects of discussion, are either too abstract or too concrete to reside in the medium of communication, but can only find themselves represented there. The tables and graphs that are used to represent sign relations are themselves complex signs, involving a step of denotation to reach the sign relation intended. The intricacies of this step demand interpretive agents who are able, over and above executing all the rudimentary steps of denotation, to orchestrate the requisite kinds of concerted steps. This performance in turn requires a whole array of techniques to match the connotations of complex signs and to test their alternative styles of representation for semiotic equivalence. Analogous to the ways that matrices represent linear transformations and that multiplication tables represent group operations, a large part of the usefulness of these complex signs comes from the fact that they are not just conventional symbols for their objects but iconic representations of their structure.

−

=====Iconic Signs=====

+

=====1.3.4.7. Iconic Signs=====

In the pragmatic theory of signs, an ''icon'' is a sign that accomplishes its representation, including the projects of denotation and connotation, by virtue of properties that it shares with its object. In the case of relational tables and graphs, interpreted as iconic representations or analogous expressions of logical and mathematical objects, the pivotal properties are formal and abstract in character. Since a uniform translation through any dimension (of sight, of sound, or imagination) does not affect the structural properties of a configuration of signs in relation to each other, this may help to explain how tables and graphs, in spite of their semantic shiftiness, can succeed in representing sign relations without essential distortion.

Line 789: Line 757:

One serious form of contamination can be traced to the accidental circumstance that <math>\text{A}\!</math> and <math>\text{B}\!</math> and I all use the same proper names for <math>\text{A}\!</math> and <math>\text{B}\!</math>. This renders it is impossible to tell, purely from the tokens that are being tendered, whether it is a formal or a casual transaction that forms the issue of the moment. It also means that a formalization of the writer's and the reader's accessory sign relations would have several portions that look identical to pieces of those Tables under formal review.

−

=====The Conflict of Interpretations=====

+

=====1.3.4.8. The Conflict of Interpretations=====

One discrepancy that needs to be documented can be observed in the conflict of interpretations between <math>\text{A}\!</math> and <math>\text{B}\!</math>, as reflected in the lack of congruity between their semiotic partitions of the syntactic domain. This is a problematic but realistic feature of the present example. That is, it represents a type of problem with the interpretation of pronouns (indexical signs or bound variables) that actually arises in practice when attempting to formalize the semantics of natural, logical, and programming languages. On this account, the deficiency resides with the present analysis, and the burden remains to clarify exactly what is going on here.

Line 795: Line 763:

Notice, however, that I have deliberately avoided dealing with indexical tokens in the usual ways, namely, by seeking to eliminate all semantic ambiguities from the initial formalization. Instead, I have preserved this aspect of interpretive discrepancy as one of the essential phenomena or inescapable facts in the realm of pragmatic semantics, tantamount to the irreducible nature of perspective diversity. I believe that the desired competence at this faculty of language will come, not from any strategy of substitution that constantly replenishes bound variables with their objective referents on every fixed occasion, but from a pattern of recognition that keeps indexical signs persistently attached to their interpreters of reference.

−

=====Indexical Signs=====

+

=====1.3.4.9. Indexical Signs=====

In the pragmatic theory of signs, an ''index'' is a sign that achieves its representation of an object by virtue of an actual connection with it. Though real and objective, however, the indexical connection can be purely incidental and even a bit accidental. Its effectiveness depends only on the fact that an object in actual existence has many properties, definitive and derivative, any number of which can serve as its signs. Indices of an object reside among its more tangential sorts of attributes, its accidental or accessory features, which are really the properties of some but not all points in the locus of its existential actualization.

Line 821: Line 789:

Therefore, I consider any supposed form of ''ontological descent'' to be, more likely, just one among many possible forms of ''semantic descent'', each one of which details a particular way to reformulate objects as signs of more determinate objects, and every one of which operates with respect to its assumed form of analysis or its tacit analytic framework.

−

=====Sundry Problems=====

+

=====1.3.4.10. Sundry Problems=====

There are moments in the development of an analytic discussion when a thing initially described as a single object under a single sign needs to be reformulated as a congeries extending over more determinate objects. If the usage of the original singular sign is preserved, as it often is, then the multitude of new instances that one comes to fathom beneath the old object's superficial appearance gradually serve to reconstitute the singular sign's denotation in the fashion of a plural reference.

Line 835: Line 803:

A related source of conceptual turbulence stems from the circumstance that, even though a certain aesthetic dynamics attracts the mind toward sign relational systems that are capable of reflecting on, commenting on, and thus ''counter-rolling'' their own behavior, it is still important to distinguish in every active instance the part of the system that is doing the discussing from the part of the system that is being discussed. To do this, interpreters need two things: the senses to discern the essential tensions that typically prevail between the formal pole and the informal arena, and (2) the language to articulate, aside from their potential roles, the moment by moment placement of dynamic elements and systematic components with respect to this field of polarities.

−

=====Review and Prospect=====

+

=====1.3.4.11. Review and Prospect=====

What has been learned from the foregoing study of icons and indices? The import of this examination can be sized up in two stages, at first, by reflecting on the action of both the formal and the casual signs that were found to be operating in and around the discussion of <math>\text{A}\!</math> and <math>\text{B}\!</math>, and then, by taking up the lessons of this circumscribed arena as a paradigm for future investigation.

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Can this manner of recursively searching for explanation be established as well-founded? In order to organize the expanding circle of thoughts and the growing wealth of objects that are envisioned within its scheme, it helps to introduce a set of organizing conceptions. Doing this will be the business of the next four Subsections.

−

=====Objective Plans and Levels=====

+

=====1.3.4.12. Objective Plans and Levels=====

In accounting for the special characters of icons and indices that arose in previous discussions, it was necessary to open the domain of objects coming under formal consideration to include unspecified numbers of properties and instances of whatever objects were initially set down. This is a general phenomenon, affecting every motion toward explanation whether pursued by analytic or synthetic means. What it calls for in practice is a way of organizing growing domains of objects, without having to specify in advance all the objects there are.

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In other words, objectivity is always a matter of interpretation. It is concerned with and quantified by the magnitude of the consensus that a matter is bound to have at the end of inquiry, but in no way does this diminish or dismiss the fact that the fated determination is something on which any particular collection of current opinions are granted to differ. In principle, there begins to be a degree of objectivity as soon as something becomes an object to somebody, and the issue of whether this objective waxes or wanes in time is bound up with the number of observers that are destined to concur on it.

−

The critical question is not whether a thing is an object of thought and discussion, but what sort of thought and discussion it is an object of. How does one determine the character of this thought and discussion? And should this query be construed as a task of finding or ~~a task~~ of making? Whether it appeals to arts of acquisition, production, or discernment, and however one expects to decide or decode the conduct it requires, the character of the thought and discussion in view is sized up and riddled out in turn by looking at the whole domain of objects and the pattern of relations among them that it actively charts and encompasses. This makes what is usually called ''subjectivity'' a special case of what I must call ''objectivity'', since the interpretive and perspectival elements are ab initio operative and cannot be eliminated from any conceivable form of discernment, including their own.

+

The critical question is not whether a thing is an object of thought and discussion, but what sort of thought and discussion it is an object of. How does one determine the character of this thought and discussion? And should this query be construed as a task of finding or of making? Whether it appeals to arts of acquisition, production, or discernment, and however one expects to decide or decode the conduct it requires, the character of the thought and discussion in view is sized up and riddled out in turn by looking at the whole domain of objects and the pattern of relations among them that it actively charts and encompasses. This makes what is usually called ''subjectivity'' a special case of what I must call ''objectivity'', since the interpretive and perspectival elements are ab initio operative and cannot be eliminated from any conceivable form of discernment, including their own.

Consequently, analyses of objects and syntheses of objects are always analyses and syntheses to somebody. Both modes of approaching the constitutions of objects lead to the sorts of approximation that are appropriate to particular agents and able to be appropriated by whole communities of interpretation. By way of relief, on occasions when this motive of consistency hobbles discussion too severely, I will resort to using chimeras like ''object-analytic'' and ''object-synthetic'', paying the price of biasing the constitution of objects in one direction or another.

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What gives an OM a determinate character from moment to moment is the particular selection of objects and linkages from its governing OG that it can say it has appropriated, apprehended, or actualized, that is, the portion of its OG that it can say actually belongs to it, and whether they make up a lot or a little, the roles it can say it has made its own.

−

In setting out the preceding characterization, I have reiterated what is likely to seem like an anthropomorphism, prefacing each requirement of the candidate OM with the qualification ''it can say''. This is done in order to emphasize that an ~~OM's~~ command of a share of its OG is partly a function of the interpretive effability that it brings to bear on the object domain and partly a matter of the expressive power that it is able to dictate over its own development.

+

In setting out the preceding characterization, I have reiterated what is likely to seem like an anthropomorphism, prefacing each requirement of the candidate OM with the qualification ''it can say''. This is done in order to emphasize that an OMs command of a share of its OG is partly a function of the interpretive effability that it brings to bear on the object domain and partly a matter of the expressive power that it is able to dictate over its own development.

−

=====Formalization of OF : Objective Levels=====

+

=====1.3.4.13. Formalization of OF : Objective Levels=====

The three levels of objective detail to be discussed are referred to as the objective ''framework'', ''genre'', and ''motive'' that one finds actively involved in organizing, guiding, and regulating a particular inquiry.

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Then <math>P\!</math> is a proposition that applies to a domain of propositions, or elements with the evidentiary import of propositions, and its models are therefore conceived to be certain propositional entities in <math>J.\!</math> And yet all of these expressions are just elaborate ways of stating the underlying assertion which says that there exists a triple <math>(j, x, y)\!</math> in the genre <math>G (:\!\lessdot).</math>

−

=====Application of OF : Generic Level=====

+

=====1.3.4.14. Application of OF : Generic Level=====

Given an ontological framework that can provide multiple perspectives and moving platforms for dealing with object structure, in other words, that can organize diverse hierarchies and developing orders of objects, attention can now return to the discussion of sign relations as models of intellectual processes.

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This appears to suggest that icons and their objects are icons of each other, and that indices and their objects are indices of each other. Are the results of these symbolic manipulations really to be trusted? Given that there is no mention of the interpretive agent to whom these sign relations are supposed to appear, one might well suspect that these results can only amount to approximate truths or potential verities.

−

=====Application of OF : Motive Level=====

+

=====1.3.4.15. Application of OF : Motive Level=====

Now that an adequate variety of formal tools have been set in order and the workspace afforded by an objective framework has been rendered reasonably clear, the structural theory of sign relations can be pursued with greater precision. In support of this aim, the concept of an objective genre and the particular example provided by <math>\operatorname{OG} (\operatorname{Prop}, \operatorname{Inst})\!</math> have served to rough out the basic shapes of the more refined analytic instruments to be developed in this subsection.

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In the discussion of the dialogue between <math>\text{A}\!</math> and <math>\text{B}\!</math> it was allowed that the same signs <math>{}^{\backprime\backprime} \text{A} {}^{\prime\prime}\!</math> and <math>{}^{\backprime\backprime} \text{B} {}^{\prime\prime}\!</math> could reference the different categories of things they name with a deliberate duality and a systematic ambiguity. Used informally as a part of the peripheral discussion, they indicate the entirety of the sign relations themselves. Used formally within the focal dialogue, they denote the objects of two particular sign relations. In just this way, or an elaboration of it, the signs <math>{}^{\backprime\backprime} j {}^{\prime\prime}\!</math> and <math>{}^{\backprime\backprime} k {}^{\prime\prime}\!</math> can have their meanings extended to encompass both the objective motifs that inform and regulate experience and the object experiences that fill out and substantiate their forms.

−

=====The Integration of Frameworks=====

+

=====1.3.4.16. The Integration of Frameworks=====

A large number of the problems arising in this work have to do with the integration of different interpretive frameworks over a common objective basis, or the prospective bases provided by shared objectives. The main concern of this project continues to be the integration of dynamic and symbolic frameworks for understanding intelligent systems, concentrating on the kinds of interpretive agents that are capable of being involved in inquiry.

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The integration of model-theoretic and proof-theoretic aspects of ''physical symbol systems'', besides being closely analogous to the integration of denotative and connotative aspects of sign relations, is also relevant to the job of integrating dynamic and symbolic frameworks for intelligent systems. This is so because the search for dynamic realizations of symbol systems is only a more pointed exercise in model theory, where the mathematical materials made available for modeling are further constrained by system-theoretic principles, like being able to say what the states are and how the transitions are determined.

−

=====Recapitulation : A Brush with Symbols=====

+

=====1.3.4.17. Recapitulation : A Brush with Symbols=====

A common goal of work in artificial intelligence and cognitive simulation is to understand how is it possible for intelligent life to evolve from elements available in the primordial sea. Simply put, the question is: “What's in the brine that ink may character?”

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When the IF and the OF sketched here have been developed far enough, I hope to tell wherein and whereof a sign is able, by its very character, to address itself to a purpose, one determined by its objective nature and determining, in a measure, that of its intended interpreter, to the extent that it makes the other wiser than the other would otherwise be.

−

=====C'est Moi=====

+

=====1.3.4.18. C'est Moi=====

From the emblem unfurled on a tapestry to tease out the working of its loom and spindle, a charge to bind these frameworks together is drawn by necessity from a single request: ''To whom is the sign addressed?'' The easy, all too easy answer comes ''To whom it may concern'', but this works more to put off the question than it acts as a genuine response. To say that a sign relation is intended for the use of its interpreter, unless one has ready an independent account of that agent's conduct, only rephrases the initial question about the end of interpretation.

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<p>I think we need to reflect upon the circumstance that every word implies some proposition or, what is the same thing, every word, concept, symbol has an equivalent term — or one which has become identified with it, — in short, has an ''interpretant''.</p>

−

<p>Consider, what a word or symbol is; it is a sort of representation. Now a representation is something which stands for something. … A thing cannot stand for something without standing ''to'' something ''for'' that something. Now, what is this that a word stands ''to''? Is it a person?</p>

+

<p>Consider, what a word or symbol is; it is a sort of representation. Now a representation is something which stands for something. … A thing cannot stand for something without standing ''to'' something ''for'' that something. Now, what is this that a word stands ''to'' ? Is it a person?</p>

<p>We usually say that the word ''homme'' stands to a Frenchman for ''man''. It would be a little more precise to say that it stands to the Frenchman's mind — to his memory. It is still more accurate to say that it addresses a particular remembrance or image in that memory. And what ''image'', what remembrance? Plainly, the one which is the mental equivalent of the word ''homme'' — in short, its interpretant. Whatever a word addresses then or ''stands to'', is its interpretant or identified symbol. …</p>

−

<p>The interpretant of a term, then, and that which it stands to are identical. Hence, since it is of the very essence of a symbol that it should stand ''to'' something, every symbol — every word and every ''conception'' — must have an interpretant — or what is the same thing, must have information or implication.</p>

+

<p>The interpretant of a term, then, and that which it stands to are identical. Hence, since it is of the very essence of a symbol that it should stand to something, every symbol — every word and every ''conception'' — must have an interpretant — or what is the same thing, must have information or implication.</p>

|-

| align="right" | (Peirce, CE 1, 466–467).

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# To prepare a fully dynamic basis for actualizing interpretants. This means that an interpretant addressed by the interpretation of a sign would not be left in the form of a detached token or abstract memory image to be processed by a hypothetical but largely nondescript interpreter, but realized as a definite type of state configuration in a qualitative dynamic system. To fathom what should be the symbolic analogue of a ''state with momentum'' has presented this project with difficulties both conceptual and terminological. So far in this project, I have attempted to approach the character of an active sign-theoretic state in terms of an ''interpretive moment'' (IM), ''information state'' (IS), ''attended token'' (AT), ''situation of use'' (SOU), or ''instance of use'' (IOU). A successful concept would capture the transient dispositions that drive interpreters to engage in specific forms of inquiry, defining their ongoing state of uncertainty with regard to objects and questions of immediate concern.

−

=====Entr'acte=====

+

=====1.3.4.19. Entr'acte=====

Have I pointed at this problem from enough different directions to convey an idea of its location and extent? Here is one more variation on the theme. I believe that our theoretical empire is bare in spots. There does not exist yet in the field a suitably comprehensive concept of a dynamic system moving through a variable state of information. This conceptual gap apparently forces investigators to focus on one aspect or the other, on the dynamic bearing or the information borne, but leaves their studies unable to integrate the several perspectives into a full-dimensioned picture of the evolving knowledge system.

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Whatever the case, to constantly focus on the restricted aspects of dynamics adequately covered by currently available concepts leads one to ignore the growing body of symbolic knowledge that the states of systems potentially carry. Conversely, to leap from the relatively secure grounds of physically based dynamics into the briar patch of formally defined symbol systems often marks the last time that one has sufficient footing on the dynamic landscape to contemplate any form of overarching law, or any rule to prospectively govern the evolution of reflective knowledge. This is one of the reasons I continue to strive after the key ideas here. If straw is all that one has in reach, then ships and shelters will have to be built from straw.

−

====Discussion of Formalization : Specific Objects====

+

====1.3.5. Discussion of Formalization : Specific Objects====

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−

<li>The term ''generic inquiry'' is ambiguous. Its meaning in practice depends on whether the description of an inquiry as being generic is interpreted literally or merely as a figure of speech. In the literal case, the name <math>{}^{\backprime\backprime} y {}^{\prime\prime}\!</math> denotes a particular inquiry, <math>y \in Y,\!</math> one that is assumed to be prototypical in yet to be specified ways. In the figurative case, the name <math>{}^{\backprime\backprime} y {}^{\prime\prime}\!</math> is simply a variable that ranges over a collection <math>Y\!</math> of nominally conceivable inquiries.</li>

+

<li> The term ''generic inquiry'' is ambiguous. Its meaning in practice depends on whether the description of an inquiry as being generic is interpreted literally or merely as a figure of speech. In the literal case, the name <math>{}^{\backprime\backprime} y {}^{\prime\prime}\!</math> denotes a particular inquiry, <math>y \in Y,\!</math> one that is assumed to be prototypical in yet to be specified ways. In the figurative case, the name <math>{}^{\backprime\backprime} y {}^{\prime\prime}\!</math> is simply a variable that ranges over a collection <math>Y\!</math> of nominally conceivable inquiries.</li>

−

<li>First encountered, the recipe <math>y_0 = y \cdot y\!</math> appears to specify that the present inquiry is constituted by taking everything denoted by the most general concept of inquiry that the present inquirer can imagine and inquiring into it by means of the most general capacity for inquiry that this same inquirer can muster.</li>

+

<li> First encountered, the recipe <math>y_0 = y \cdot y\!</math> appears to specify that the present inquiry is constituted by taking everything denoted by the most general concept of inquiry that the present inquirer can imagine and inquiring into it by means of the most general capacity for inquiry that this same inquirer can muster.</li>

−

<li>Contemplating the formula <math>{}^{\backprime\backprime} y_0 = y \cdot y {}^{\prime\prime}\!</math> in the context of the subordination <math>y >\!\!= \{ d, f \}\!</math> and the successive containments <math>F \subseteq M \subseteq D,\!</math> the <math>y\!</math> that inquires into <math>y\!</math> is not restricted to examining <math>y \operatorname{'s}\!</math> immediate subordinates, <math>d\!</math> and <math>f,\!</math> but it can investigate any feature of <math>y \operatorname{'s}\!</math> overall context, whether objective, syntactic, interpretive, and whether definitive or incidental, and finally it can question any supporting claim of the discussion. Moreover, the question <math>y\!</math> is not limited to the particular claims that are being made here, but applies to the abstract relations and the general concepts that are invoked in making them. Among the many kinds of inquiry that suggest themselves, there are the following possibilities:</li>

+

<li> Contemplating the formula <math>{}^{\backprime\backprime} y_0 = y \cdot y {}^{\prime\prime}\!</math> in the context of the subordination <math>y >\!\!= \{ d, f \}\!</math> and the successive containments <math>F \subseteq M \subseteq D,\!</math> the <math>y\!</math> that inquires into <math>y\!</math> is not restricted to examining <math>y \operatorname{'s}\!</math> immediate subordinates, <math>d\!</math> and <math>f,\!</math> but it can investigate any feature of <math>y \operatorname{'s}\!</math> overall context, whether objective, syntactic, interpretive, and whether definitive or incidental, and finally it can question any supporting claim of the discussion. Moreover, the question <math>y\!</math> is not limited to the particular claims that are being made here, but applies to the abstract relations and the general concepts that are invoked in making them. Among the many kinds of inquiry that suggest themselves, there are the following possibilities:</li>

−

<li>Inquiry into propositions about application and equality. One may well begin with the forms of application and equality that are invoked in the formula <math>{}^{\backprime\backprime} y_0 = y \cdot y {}^{\prime\prime}\!</math> itself.</li>

+

<li> Inquiry into propositions about application and equality. One may well begin with the forms of application and equality that are invoked in the formula <math>{}^{\backprime\backprime} y_0 = y \cdot y {}^{\prime\prime}\!</math> itself.</li>

−

<li>Inquiry into application <math>(\cdot),\!</math> for example, the way that the term <math>{}^{\backprime\backprime} y \cdot y {}^{\prime\prime}\!</math> indicates the application of <math>y\!</math> to <math>y\!</math> in the formula <math>{}^{\backprime\backprime} y_0 = y \cdot y {}^{\prime\prime}.\!</math></li>

+

<li> Inquiry into application <math>(\cdot),\!</math> for example, the way that the term <math>{}^{\backprime\backprime} y \cdot y {}^{\prime\prime}\!</math> indicates the application of <math>y\!</math> to <math>y\!</math> in the formula <math>{}^{\backprime\backprime} y_0 = y \cdot y {}^{\prime\prime}.\!</math></li>

−

<li>Inquiry into equality <math>(=),\!</math> for example, the meaning of the equal sign in the formula <math>{}^{\backprime\backprime} y_0 = y \cdot y {}^{\prime\prime}.\!</math></li>

+

<li> Inquiry into equality <math>(=),\!</math> for example, the meaning of the equal sign in the formula <math>{}^{\backprime\backprime} y_0 = y \cdot y {}^{\prime\prime}.\!</math></li>

−

<li>Inquiry into indices, for example, the significance of <math>{}^{\backprime\backprime} 0 {}^{\prime\prime}\!</math> in <math>{}^{\backprime\backprime} y_0 {}^{\prime\prime}.\!</math></li>

+

<li> Inquiry into indices, for example, the significance of <math>{}^{\backprime\backprime} 0 {}^{\prime\prime}\!</math> in <math>{}^{\backprime\backprime} y_0 {}^{\prime\prime}.\!</math></li>

−

<li>Inquiry into terms, specifically, constants and variables. What are the functions of <math>{}^{\backprime\backprime} y {}^{\prime\prime}\!</math> and <math>{}^{\backprime\backprime} y_0 {}^{\prime\prime}\!</math> in this respect?</li>

+

<li> Inquiry into terms, specifically, constants and variables. What are the functions of <math>{}^{\backprime\backprime} y {}^{\prime\prime}\!</math> and <math>{}^{\backprime\backprime} y_0 {}^{\prime\prime}\!</math> in this respect?</li>

−

<li>Inquiry into decomposition or subordination, for example, as invoked by the sign <math>{}^{\backprime\backprime} >\!\!= {}^{\prime\prime}\!</math> in the formula <math>{}^{\backprime\backprime} y >\!\!= \{ d, f \} {}^{\prime\prime}.\!</math></li>

+

<li> Inquiry into decomposition or subordination, for example, as invoked by the sign <math>{}^{\backprime\backprime} >\!\!= {}^{\prime\prime}\!</math> in the formula <math>{}^{\backprime\backprime} y >\!\!= \{ d, f \} {}^{\prime\prime}.\!</math></li>

−

<li>Inquiry into containment or inclusion. In particular, examine the assumption that formalization <math>F\!</math>, mediation <math>M\!</math>, and discussion <math>D\!</math> are ordered as <math>F \subseteq M \subseteq D\!</math>, a claim that determines the chances that a formalization has an object, the degree to which a formalization can be carried out by means of a discussion, and the extent to which an object of formalization can be conveyed by a form of discussion.</li>

+

<li> Inquiry into containment or inclusion. In particular, examine the assumption that formalization <math>F\!</math>, mediation <math>M\!</math>, and discussion <math>D\!</math> are ordered as <math>F \subseteq M \subseteq D\!</math>, a claim that determines the chances that a formalization has an object, the degree to which a formalization can be carried out by means of a discussion, and the extent to which an object of formalization can be conveyed by a form of discussion.</li>

</ol></ol>

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−

<li>A ''surprise'' calls for an explanation to resolve the uncertainty that is present in it. This uncertainty is associated with a difference between observations and expectations.~~</li>~~

+

<li> A ''surprise'' calls for an explanation to resolve the uncertainty that is present in it. This uncertainty is associated with a difference between observations and expectations.

−

<li>A ''problem'' calls for a plan of action to resolve the difficulty that is present in it. This difficulty is associated with a difference between observations and intentions.</li>

+

<li> A ''problem'' calls for a plan of action to resolve the difficulty that is present in it. This difficulty is associated with a difference between observations and intentions.

+

To express this diversity in a unified formula, both types of inquiry begin with a ''delta'' <math>(\Delta),\!</math> a compact symbol that admits a spectrum of expansions: debt, difference, difficulty, discrepancy, dispersion, distribution, doubt, duplicity, or duty.</li>

</ol>

−

To express this diversity in a unified formula, both types of inquiry begin with a ''delta'' <math>(\Delta),\!</math> a compact symbol that admits a spectrum of expansions:  debt, difference, difficulty, discrepancy, dispersion, distribution, doubt, duplicity, or duty.

Expressed another way, inquiry begins with a doubt about one's object, whether this means what is true of a case, an object, or a world, what to do about reaching a goal, or whether the hoped-for goal is really good for oneself — with all that these questions lead to in essence, in action, or in fact.

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Perhaps there is an inexhaustible reality that issues in these apparent mysteries and recurrent crises, but, by the time I say this much, I am already indulging in a finite image, a hypothesis about what is going on. If nothing else, then, one finds again the familiar pattern, where the formative relation between the informal and the formal merely serves to remind one anew of the relation between the infinite and the finite.

−

=====The Will to Form=====

+

=====1.3.5.1. The Will to Form=====

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Just on the point of becoming lost in the morass of words for describing experience and the nuances of their interpretation, one can adopt a formal perspective, and realize that the relation among objects, experiences, and reflective images is formally analogous to the relation among objects, signs, and interpretant signs that is covered by the pragmatic theory of signs. The problem remains: ''How'' are the expressions of experience everted to form the exterior faces of extended objects and exploited to embed them in their external circumstances, and no matter whether this object with an outer face is oneself or another? Here, one needs to understand that expressions of experience include the original experiences themselves, at least, to the extent that they permit themselves to be recognized and reflected in ongoing experience. But now, from the formal point of view, ''How'' means only: To describe the formal conditions of a formal possibility.

−

=====The Forms of Reasoning=====

+

=====1.3.5.2. The Forms of Reasoning=====

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Another dimension of variation that needs to be noted among these different types of processes is their status with regard to determinism. Given the ordinary case of a well-formed syllogism, deduction is a fully deterministic process, since the middle term to be eliminated is clearly marked by its appearance in a pair of premisses. But if one is given nothing but the fact that forms this conclusion, or starts with a fact that is barely suspected to be the conclusion of a possible deduction, then there are many other middle terms and many other premisses that might be construed to result in this fact. Therefore, adduction and production, for all their uncontrolled generality, but even obduction, in spite of its specificity, cannot be treated as deterministic processes. Only in degenerate cases, where the number of terms in a discussion is extremely limited, or where the availability of middle terms is otherwise restricted, can it happen that these processes become deterministic.

−

=====A Fork in the Road=====

+

=====1.3.5.3. A Fork in the Road=====

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The fictions that one devises to shore up a shaky account of experience can often be discharged at a later stage of development, gradually coming to be replaced with primitive elements of less and less dubious characters. Hypostases and hypotheses, the creative terms and the inventive propositions that one coins to account for otherwise ineffable experiences, are tokens that are subject to a later account. Under recurring examination, many such tokens are found to be ciphers, marks that no one will miss if they are canceled out altogether. The symbolic currencies that tend to survive lend themselves to being exchanged for stronger and more settled constructions, in other words, for concrete definitions and explicit demonstrations, gradually leading to primitive elements of more and more durable utilities.

−

=====A Forged Bond=====

+

=====1.3.5.4. A Forged Bond=====

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A unity can be forged among the methods by noticing the following connections among them. All the while that one proceeds deductively, the primitive elements, the definitions and the axioms, must still be introduced hypothetically, notwithstanding the support they get from common sense and widespread assent. And the whole symbolic system that is constructed through hypothesis and deduction must still be tested in experience to see if it serves any purpose to maintain it.

−

=====A Formal Account=====

+

=====1.3.5.5. A Formal Account=====

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In a recursive context, a principal benefit of the formalization step is to find constituents of inquiry with reduced complexities, drawing attention from the context of informal inquiry, whose stock of questions may not be grasped well enough ever to be fruitful and the scope of whose questions may not be focused well enough ever to see an answer, and concentrating effort in an arena of formalized inquiry, where the questions are posed well enough to have some hope of bearing productive answers in a finite time.

−

=====Analogs, Icons, Models, Surrogates=====

+

=====1.3.5.6. Analogs, Icons, Models, Surrogates=====

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In this work, the basic structure of each MOI is adopted from the pragmatic theory of signs and the general account of its operation is derived from the pragmatic theory of inquiry. The indispensable utility of these formal models hinges on the circumstance that each MOI, whether playing its part in interpretation or in inquiry, is always a ''model'' in two important senses of the word. First, it is a model in the logical sense that its structure satisfies a formal theory or an abstract specification. Second, it is a model in the analogical sense that it represents an aspect of the structure that is present in another object or domain.

−

=====Steps and Tests of Formalization=====

+

=====1.3.5.7. Steps and Tests of Formalization=====

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In ordinary discussion, agents of inquiry and interpretation depend on the likely interpretations of others to give their common notions and their shared notations a meaning in practice. This means that a high level of implicit understanding is relied on to ground each informal inquiry in practice. The entire framework of logical assumptions and interpretive activities that is needed to shore up this platform will itself resist analysis, since it is precisely to save the effort of repeating routine analyses that the whole infrastructure is built.

−

=====A Puckish Referee=====

+

=====1.3.5.8. A Puckish Referee=====

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But computers are initially just as incapable of misunderstanding as they are of understanding. Therefore, it actually forms a moderate compromise to address the task of interpretation to a computational system, a thing that is known to begin from a moderately neutral initial condition.

−

=====Partial Formalizations=====

+

=====1.3.5.9. Partial Formalizations=====

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In many discussions the source context remains unformalized in itself, taking form only according to the image it receives in one or another individual MOI. In cases like these, the step of formalization does not amount to a total function but is limited to a partial mapping from the source to the target. Such a partial representation is analogous to a sampling operation. It is not defined on every point of the source domain but assigns values only to a proper selection of source elements. Thus, a partial formalization can be regarded as achieving its form of simplification in a loose way, ignoring elements of the source domain and collapsing material distinctions in irregular fashions.

−

=====A Formal Utility=====

+

=====1.3.5.10. A Formal Utility=====

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In the context of the recursive inquiry that I have outlined, the step of formalization is intended to bring discussion appreciably closer to a solid base for the operational definition of inquiry.

−

=====A Formal Aesthetic=====

+

=====1.3.5.11. A Formal Aesthetic=====

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But an overly narrow or premature formalization, where the nature of the phenomenon of interest is too much denatured in the formal image, may result in destroying all interest in the result that does result.

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=====A Formal Apology=====

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=====1.3.5.12. A Formal Apology=====

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Seizing the advantage of this formal flexibility makes it possible to take abstract leaps over a multitude of material obstacles, to reason about many properties of objects and processes from a knowledge of their form alone, without having to know everything about their material content down to the depths that matter can go.

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=====A Formal Suspicion=====

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=====1.3.5.13. A Formal Suspicion=====

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In a recursive context, the function of formalization is to relate a difficult problem to a simpler problem, breaking the original inquiry into two parts, the step of formalization and the rest of the inquiry, both of which branches it is hoped will be nearer to solid ground and easier to grasp than the original question.

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=====The Double Aspect of Concepts=====

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=====1.3.5.14. The Double Aspect of Concepts=====

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I have given this inquiry a reflective or recursive cast, portraying it as an inquiry into inquiry, and one of the consequences of this picture is that every concept employed in the work will take on a divided role, double aspect, or dual purpose. At any moment, the object inquiry of the moment is aimed to take on a formal definition, while the active inquiry need not acknowledge any image that it does not recognize as reflecting itself, nor is it bound by any horizon that does not capture its spirit.

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=====A Formal Permission=====

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=====1.3.5.15. A Formal Permission=====

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|}

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=====A Formal Invention=====

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=====1.3.5.16. A Formal Invention=====

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|}

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====Recursion in Perpetuity====

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====1.3.6. Recursion in Perpetuity====

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<math>\cdots\!</math>

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====Processus, Regressus, Progressus====

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====1.3.7. Processus, Regressus, Progressus====

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<math>\cdots\!</math>

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====Rondeau : Tempo di Menuetto====

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====1.3.8. Rondeau : Tempo di Menuetto====

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----

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• [[Inquiry Driven Systems|Contents]]

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• [[Directory:Jon Awbrey/Papers/Inquiry Driven Systems|Contents]]

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• [[Inquiry Driven Systems : Part 1|Part 1]]

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• [[Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Part 1|Part 1]]

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• [[Inquiry Driven Systems : Part 2|Part 2]]

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• [[Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Part 2|Part 2]]

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• [[Inquiry Driven Systems : Part 3|Part 3]]

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• [[Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Part 3|Part 3]]

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• [[Inquiry Driven Systems : Part 4|Part 4]]

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• [[Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Part 4|Part 4]]

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• [[Inquiry Driven Systems : Part 5|Part 5]]

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• [[Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Part 5|Part 5]]

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• [[Inquiry Driven Systems : Part 6|Part 6]]

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• [[Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Part 6|Part 6]]

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• [[Inquiry Driven Systems : Part 7|Part 7]]

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• [[Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Part 7|Part 7]]

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• [[Inquiry Driven Systems : Part 8|Part 8]]

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• [[Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Part 8|Part 8]]

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• [[Inquiry Driven Systems : Appendices|Appendices]]

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• [[Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Appendices|Appendices]]

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• [[Inquiry Driven Systems : References|References]]

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• [[Directory:Jon Awbrey/Papers/Inquiry Driven Systems : References|References]]

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• [[Inquiry Driven Systems : Document History|Document History]]

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• [[Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Document History|Document History]]

•

</div>

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[[Category:Artificial Intelligence]]

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~~[[Category:Automated Research Tools]]~~

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~~[[Category:Charles Sanders Peirce]]~~

[[Category:Critical Thinking]]

[[Category:Cybernetics]]

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~~[[Category:Differential Logic]]~~

[[Category:Education]]

[[Category:Hermeneutics]]

[[Category:Information Systems]]

[[Category:Inquiry]]

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~~[[Category:Inquiry Driven Systems]]~~

[[Category:Intelligence Amplification]]

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~~[[Category:Intelligence Augmentation]]~~

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~~[[Category:Intelligent Systems]]~~

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~~[[Category:Interpretation]]~~

[[Category:Learning Organizations]]

[[Category:Knowledge Representation]]

[[Category:Logic]]

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~~[[Category:Mathematics]]~~

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~~[[Category:Optimization]]~~

[[Category:Philosophy]]

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~~[[Category:Planning]]~~

[[Category:Pragmatics]]

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~~[[Category:Problem Solving]]~~

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~~[[Category:Scientific Explanation]]~~

[[Category:Semantics]]

[[Category:Semiotics]]

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~~[[Category:Software Engineering]]~~

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~~[[Category:Systems Engineering]]~~

[[Category:Systems Science]]

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~~[[Category:Systems Theory]]~~

Jon Awbrey

12,080

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Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Part 1 (view source)

Revision as of 15:14, 15 April 2017

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