Inquiry Driven Systems

	I put down the cup and turn to my mind. It is up to my mind to find the truth. But how? What grave uncertainty, whenever the mind feels overtaken by itself; when it, the seeker, is also the obscure country where it must seek and where all its baggage will be nothing to it. Seek? Not only that: create. It is face to face with something that does not yet exist and that only it can accomplish, and bring into its light.
	— Marcel Proust, In Search of Lost Time, [Pro, 1.48]

1. Introduction

1.1. Outline of the Project : Inquiry Into Inquiry

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.

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.

This is the most concise and comprehensive answer I know, but it is likely to sound facetious at this point. On the other hand, if I did not actually use the method of inquiry that I describe as inquiry, how could the results possibly be taken seriously? Correspondingly, the questions of methodological self-application and self-referential consistency will be found at the center of this research.

In truth, it is fully possible that every means at inquiry's disposal will ultimately find application in resolving the problem of inquiry's nature. Other than a restraint to valid methods of inquiry — what those are is part of the question — there is no reason to expect a prior limitation on the range of methods that might be required.

This only leads up to the question of priorities: Which methods do I think it wise to apply first? In this project I will give preference to two kinds of technique, one analytic and one synthetic.

The prevailing method of research I will exercise throughout this work involves representing problematic phenomena in a variety of formal systems and then implementing these representations in a computational medium as a way of clarifying the more complex descriptions that evolve.

Aside from its theoretical core, this research is partly empirical and partly heuristic. Therefore, I expect that the various components of methodology will need to be applied in an iterative or even opportunistic fashion, working on any edge of research that appears to be ready at a given time. If forced to anticipate the likely developments, I would sketch the possibilities roughly as follows.

The methodology that underlies this approach has two components: The analytic component involves describing the performance and competence of intelligent agents in the medium of various formal systems. The synthetic component involves implementing these formal systems and the descriptions they express in the form of computational interpreters or language processors.

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.

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.

Paradigm. Generic example that reflects significant properties of a target class of phenomena, often derived from a tradition of study.

Analysis. Effective analysis of concepts, capacities, structures, and functions in terms of fundamental operations and computable functions.

Work in this phase typically proceeds according to the following recipe.

1. Focus on a problematic phenomenon. This is a generic property or process that attracts one's interest, like intelligence or inquiry.
2. Gather under consideration significant examples of concrete systems or agents that exhibit the property or process in question.
3. Reflect on their common properties in a search for less obvious traits that might explain their more surprising features.
4. Check these accounts of the phenomenon in one of several ways. For example, one might (a) search out other systems or situations in nature that manifest the critical traits, or (b) implement the putative traits in computer simulations. If these hypothesized traits generate (give rise to, provide a basis for) the phenomenon of interest, either in nature or on the computer, then one has reason to consider them further as possible explanations.

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.

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.

Paraphrasis. "A method of accounting for fictions by explaining various purported terms away" (Quine, in Van Heijenoort, 216). See also (Whitehead and Russell, in Van Heijenoort, 217–223).

Synthesis. Regard computer programs as implementations of hypothetical or postulated faculties. Within the framework of experimental research, programs can serve as descriptive, modal, or normative hypotheses, that is, conjectures about how a process is actually accomplished in nature, speculations as to how it might be done in principle, or explorations of how it might be done better in the medium of technological extensions.

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.

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.

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.

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?

Given that the problem is inquiry and the method is inquiry, the test of progress and eventual success is just the measure of any inquiry's performance. According to my current understanding of inquiry, and the tentative model of inquiry that will guide this project, the criterion of an inquiry's competence is how well it succeeds in reducing the uncertainty of its agent about its object.

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.

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.

If it is desired to articulate the loop of self-application a bit further, computer models of inquiry can be seen as building a two-way bridge between experimental science and software engineering, allowing the results of each to be applied in the furtherance of the other.

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).

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.

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.

In the remainder of this section I will try to indicate as briefly as possible the nature of the problem that must be faced in this particular approach to inquiry, and to explain what a large share of the ensuing fuss will be directed toward clearing up.

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.

1.2.2. A Fugitive Canon

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

First, there is the issue of computational mediation. In using the sorts of sequences that computers go through to mediate discussion of the sorts of sequences that people go through, it becomes necessary to re-examine all of the facilitating assumptions that are commonly taken for granted in relating one human experience to another, that is, in describing and building structural relationships among the experiences of human agents.

Second, there is the problem of representing the general in the particular. How is it possible for the most particular imaginable things, namely, the transient experiential states of agents, to represent the most general imaginable things, namely, the agents' own conceptions of the abstract categories of experience?

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?

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.

The initial set of concepts I need to get discussion started are few. Assuming that a working set of ideas can be understood on informal grounds at the outset, I anticipate being able to formalize them to a greater degree as the project gets under way. Inquiry in general will be described as encompassing particular inquiries. Particular forms of inquiry, regarded as phenomenal processes, will be analyzed in terms of simpler kinds of phenomenal processes.

As a phenomenon, a particular way of doing inquiry is regarded as embodied in a faculty of inquiry, as possessed by an agent of inquiry. As a process, a particular example of inquiry is regarded as extended in time through a sequence of states, as experienced by its ongoing agent. It is envisioned that an agent or faculty of any generically described phenomenal process, inquiry included, could be started off from different initial states and would follow different trajectories of subsequent states, and yet there would be a recognizable quality or abstractable property that justifies invoking the name of the genus.

The steps of this analysis will be annotated below by making use of the following conventions. Lower case letters denote phenomena, processes, or faculties under investigation. Upper case letters denote classes of the same sorts of entities. Special use is made of the following symbols:

Y = genus of inquiry,

y = generic inquiry,

y0 = present inquiry.

Compositions of faculties are indicated by concatenating their names, posed in the sense that the right-indicated faculty applies to the left-indicated faculty, in the following form:

f \(\cdot\) g

A notation of the form

f >= g

indicates that f is greater than or equal to g in a decompositional series, in other words, f possesses g as a component.

The coset notation

F \(\cdot\) G

indicates a class of faculties of the form

f \(\cdot\) g,

with f in F and g in G.

Notations like

{?}, {?, ?}, {?, ?, ?}, …

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

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.

y = {?}

In giving the current inquiry a reflexive cast, as inquiry into inquiry, I have brought inquiry face to face with itself, inditing it to apply its action in pursuing a knowledge of its passion.

y0 = y \(\cdot\) y = {?}{?}

If this juxtaposition of characters is to have a meaningful issue, then the fullness of its instrumental and objective aspects must have recourse to easier actions and simpler objects.

y >= {?, ?}

Looking for an edge on each face of inquiry, as a plausible option for beginning to apply one to the other, I find what seems a likely pair. I begin with an aspect of instrumental inquiry that is easy to do, namely discussion, along with an aspect of objective inquiry that is unavoidable to discuss, namely formalization.

y >= {disc, form}

In accord with this plan, the body of this section is devoted to a discussion of formalization.

y0 = y \(\cdot\) y >= {d, f}{d, f} >= {f}{d}

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.

y0 = y \(\cdot\) y >= {d, f}{d, f} >= {d}{d}

Discussion is easy in general because its termination criterion is relaxed to the point of becoming otiose. A discussion of things in general can be pursued as an end in itself, with no consideration of any purpose but persevering in its current form, and this accounts for the virtually perpetual continuation of many a familiar and perennial discussion.

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.

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.

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:

1. Examine every notion of the casual intuition that enters into the informal discussion and inquire into its qualifications as a potential candidate for formalization.
2. Pay special attention to the nominal operations that are invoked to substantiate each tentative explanation of a critically important process. Often, but not infallibly, these can be detected appearing in the guise of "-ionized" terms, words ending in "-ion" that typically connote both a process and its result.
3. 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.

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.

y0 = y \(\cdot\) y >= {d, f}{d, f} >= {f}{f}

To illustrate the formal charge by taking the present matter to task, the word formalization is itself exemplary of the -ionized terms falling under the charge, and so it can be lionized as the nominal head of a prospectively formal discussion. The reader has a right to object at this point that I have not described what particular action I intend to convey under the heading of formalization, by no means enough to begin applying it to any term, much less itself. However, anyone can recognize on syntactic grounds that the word is an instance of the formal rule, purely from the character of its terminal -ion, and this can be done aside from all clues about the particular meaning that I intend it to have at the end of formalization.

Unlike a mechanical interpreter meeting with the declaration of an undefined term for the very first time, the human reader of this text has the advantage of a prior acquaintance with almost every term that might conceivably enter into informal discussion. And formalization is a stock term widely traded in the forums of ordinary and technical discussion, so the reader is bound to have met with it in the context of practical experience and to have attached a personal concept to it. Therefore, this inquiry into formalization begins with a writer and a reader in a state of limited uncertainty, each attaching a distribution of meanings in practice to the word formalization, but uncertain whether their diverse spectra of associations can presently constitute or eventually converge to compatible arrays of effective meaning.

To review: The concept of formalization itself is an item of informal discussion that might be investigated as a candidate for formalization. For each aspect or component of the formalization process that I plan to transport across the semi-permeable threshold from informal to formal discussion, the reader has permission to challenge it, plus an open invitation to question every further process that I mention as a part of its constitution, and to ask with regard to each item whether its registration has cleared up the account in any measure or merely rung up a higher charge on the running bill of fare.

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.

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.

F ⊆ D

In this section, I step back from the example of formalization and consider the general task of clarifying and communicating concepts by means of a properly directed discussion. Let this kind of motivated or measured discussion be referred to as a meditation, that is, "a discourse intended to express its author's reflections or to guide others in contemplation" (Webster's). The motive of a meditation is to mediate a certain object or intention, namely, the system of concepts intended for clarification or communication. The measure of a meditation is a system of values that permits its participants to tell how close they are to achieving its object. The letter "M" will be used to annotate this form of meditation.

F ⊆ M ⊆ D

This brings the discussion around to considering the intentional objects of measured discussions and the qualifications of a writer so motivated. Just what is involved in achieving the object of a motivated discussion? Can these intentions be formalized?

y0 = y \(\cdot\) y >= {d, f}{d, f} >= {d}{f}

• The writer's task is not to create meaning from nothing, but to construct a relation from the typical meanings that are available in ordinary discourse to the particular meanings that are intended to be the effects of a particular discussion.

In case there is difficulty with the meaning of the word meaning, I replace its use with references to a system of interpretation (SOI), a technical concept that will be increasingly formalized as this project proceeds. Thus, the writer's job description is reformulated as follows.

• The writer's task is not to create a system of interpretation (SOI) from nothing, but to construct a relation from the typical SOI's that are available in ordinary discourse to the particular SOI's that are intended to be the effects of a particular discussion.

This assignment begins with an informal system of interpretation (SOI₁), and builds a relation from it to another system of interpretation (SOI₂). The first is an informal SOI that amounts to a shared resource of writer and reader. The latter is a system of meanings in practice that is the current object of the writer's intention to recommend for the reader's consideration and, hopefully, edification. In order to have a compact term for highlighting the effects of a discussion that builds a relation between SOI's, I will call this aspect of the process narration.

It is the writer's ethical responsibility to ensure that a discourse is potentially edifying with respect to the reader's current SOI, and the reader's self-interest to evaluate whether a discourse is actually edifying from the perspective of the reader's present SOI.

Formally, the relation that the writer builds from SOI to SOI can always be cast or recast as a three-place relation, one whose staple element of structure is an ordered or indexed triple. One component of each triple is anchored in the interpreter of the moment, and the other two form a connection with the source and target SOI's of the current assignment.

Once this relation is built, a shift in the attention of any interpreter or a change in the present focus of discourse can leave the impression of a transformation taking place from SOI1 to SOI2, but this is more illusory (or allusory) than real. To be more precise, this style of transformation takes place on a virtual basis, and need not have the substantive impact (or import) that a substantial replacement of one SOI by another would imply. For a writer to affect a reader in this way would simply not be polite. A moment's consideration of the kinds of SOI-building worth having leads me to enumerate a few characteristics of polite discourse or considerate discussion.

If this form of SOI-building narrative is truly intended to edify and educate, whether pursued in monologue or dialogue fashion, then its action cannot be forcibly to replace the meanings in practice a sign already has with others of an arbitrary nature, but freely to augment the options for meaning and powers for choice in the resulting SOI.

As conditions for the possibility of considerate but significant narration, there are a couple of requirements placed on the writer and the reader. Considerate narration, constructing a relation from SOI to SOI in a politic fashion, cannot operate in an infectious or addictive manner, invading a SOI like a virus or a trojan horse, but must transfer its communication into the control of the receiving SOI. Significant communication, in which the receiving SOI is augmented by options for meaning and powers for choice that it did not have before, requires a SOI on the reader's part that is extensible in non-trivial ways.

At this point, the discussion has touched on a topic, in one of its manifold aspects, that it will encounter repeatedly, under a variety of aspects, throughout this work. In recognition of this circumstance, and to prepare the way for future discussion, it seems like a good idea to note a few of the aliases that this protean topic can be found lurking under, and to notice the logical relationships that exist among its several different appearances.

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?

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.

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.

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.

Many times, it is convenient to personify the interpretive organization as if it were embodied in the actions of a typical user of the framework or a substantive agent of the system. I will call this agent the interpreter of the moment. At other times, it may be necessary to analyze the action of interpretation more carefully. At these times, it is important to remember that this form of personification is itself a figure of speech, one that has no meaning outside a fairly flexible interpretive framework. Thus, the term interpreter can be a cipher analogous to the terms X, unknown, or to whom it may concern appearing in a system of potentially recursive constraints. As such, it serves in the role of an indeterminate symbol, in the end to be solved for a fitting value, but in the mean time conveying an appearance of knowledge in a place where very little is known about the subject itself.

A meta-language corresponds to what I call an interpretive framework. Besides a set of descriptions and conceptions, it embodies the whole collective activity of unexamined structures and automatic processes that are trusted by agents at a given moment to make its employment meaningful in practice. An interpretive framework is best understood as a form of conduct, that is, a comprehensive organization of related activities. In use, an interpretive framework operates to contain activity and constrain the engagement of agents to certain forms of active involvement and dynamic participation, and manifests itself only incidentally in the manipulation of compact symbols and isolated instruments. In short, though a framework may have pointer dials and portable tools attached to it, it is usually too incumbent and cumbersome to be easily moved on its own grounds, at least, it rests beyond the scope of any local effort to do so.

An interpretive framework (IF) is set to work when an agent or agency becomes involved in its organization and participates in the forms of activity that make it up. Often, an IF is founded and persists in operation long before any participant is able to reflect on its structure or to post a note of its character to the constituting members of the framework. In some cases, the rules of the IF in question forbid the act of reflecting on its form. In practice, to the extent that agents are actively involved in filling out the requisite forms and taking part in the step by step routines of the IF they may have little surplus memory capacity to memorandize the big picture even when it is permitted in principle.

An object language is a special case of the kind of formal system that is so completely formalized that it can be regarded as combinatorial object, an inactive image of a form of activity that is meant for the moment to be studied rather than joined. The supposition that there is a meaningful and well-defined distinction between object language and meta-language ordinarily goes unexamined. This means that the assumption of a distinction between them is de facto a part of the meta-language and not even an object of discussion in the object language. A slippery slope begins here. A failure to build reflective capacities into an interpretive framework can let go unchallenged the spurious opinion that presumes there can be only one way to draw a distinction between object language and meta-language.

The next natural development is to iterate the supposed distinction. This represents an attempt to formalize and thereby objectify parts of the meta-language, precipitating it like a new layer of pearl or crystal from the resident medium or mother liquor, and thereby preparing the decantation of a still more pervasive and ethereal meta-meta-language. The successive results of this process can have a positivistically intoxicating effect on the human intellect. But a not so happy side-effect leads the not quite mindful cerebration up and down a blind alley, chasing the specious impression that just beyond the realm of objective nature there lies a unique fractionation of permeabilities and a permanent hierarchy of effabilities in language.

The grounds of discussion I am raking over here constellate a rather striking scene, especially for something intended as a neutral backdrop. Unlike other concerns, the points I am making seem obvious to all reasonable people at the outset of discussion, and yet the difficulties that follow as inquiry develops get muddier and more grating the more one probes and stirs them up. A large measure of the blame, I think, can be charged to a misleading directive that people derive from the epithet meta, leading them to search for higher and higher levels of meaning and truth, on beyond language, on beyond any conceivable system of signs, and on beyond sense. Prolonged use of the prefix meta leads people to act as if a meta-language were step outside of ordinary language, or an artificial platform constructed above and beyond natural language, and then they forget that formal models are developments internal to the informal context. For this reason among others, I suggest replacing talk about rigidly stratified object languages and meta-languages with talk about contingent interpretive frameworks.

To avoid the types of cul-de-sac (cultist act) encountered above, I am taking some pains to ensure a reflective capacity for the interpretive frameworks I develop in this project. This is a capacity that natural languages always assume for themselves, instituting specialized discourses as developments that take place within their frame and not as constructs that lie beyond their scope. Any time the levels of recursive discussion become too involved to manage successfully, one needs to keep available the resource of instant wisdom, the modest but indispensable quantum of ready understanding, that restores itself on each return to the ordinary universe (OU).

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.

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.

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.

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.

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).

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.

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.

The POI's that define the intents and the purposes of this project are the closely related processes of inquiry and interpretation, so the MOI's that must be formulated are models of inquiry and interpretation, species of formal systems that are even more intimately bound up than usual with the IF's employed and the SOI's deployed in their ongoing development as models.

Since all of the interpretive systems and all of the process models that are being mentioned here come from the same broad family of mathematical objects, the different roles that they play in this investigation are mainly distinguished by variations in their manner and degree of formalization:

1. The typical POI comes from natural sources and casual conduct. It is not formalized in itself but only in the form of its image or model, and just to the extent that aspects of its structure and function are captured by a formal MOI. But the richness of any natural phenomenon or realistic process seldom falls within the metes and bounds of any final or finite formula.
2. Beyond the initial stages of investigation, the MOI is postulated as a completely formalized object, or is quickly on its way to becoming one. As such, it serves as a pivotal fulcrum and a point of application poised between the undefined reaches of phenomena and noumena, respectively, terms that serve more as directions of pointing than as denotations of entities. What enables the MOI to grasp these directions is the quite felicitous mathematical circumsatnce that there can be well-defined and finite relations between entities that are infinite and even indefinite in themselves. Indeed, exploiting this handle on infinity is the main trick of all computational models and effective procedures. It is how a finitely informed creature (FIC) can "make infinite use of finite means". Thus, my reason for calling the MOI cardinal or pivotal is that it forms a model in two senses, loosely analogical and more strictly logical, integrating twin roles of the model concept in a single focus.
3. Finally, the IF's and the SOI's 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.

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 three-place relations called sign relations. 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.

There is a close relationship between the pragmatic theory of signs and the pragmatic theory of inquiry. In fact, the correspondence between the two studies exhibits so many parallels and coincidences that it is often best to treat them as integral parts of one and the same subject. In a very real sense, inquiry is the process by which sign relations come to be established and continue to evolve. In other words, inquiry, "thinking" in its best sense, "is a term denoting the various ways in which things acquire significance" (Dewey). Thus, there is an active and intricate form of cooperation that needs to be appreciated and maintained between these converging modes of investigation. Its proper character is best understood by realizing that the theory of inquiry is adapted to study the developmental aspects of sign relations, a subject which the theory of signs is specialized to treat from structural and comparative points of view.

Because the examples in this section have been artificially constructed to be as simple as possible, their detailed elaboration can run the risk of trivializing the whole theory of sign relations. Still, these examples have subtleties of their own, and their careful treatment will serve to illustrate important issues in the general theory of signs.

Imagine a discussion between two people, Ann and Bob, and attend only to that aspect of their interpretive practice that involves the use of the following nouns and pronouns: "Ann", "Bob", "I", "you".

• The object domain of this discussion fragment is the set of two people {Ann, Bob}.

• The syntactic domain or the sign system of their discussion is limited to the set of four signs {"Ann", "Bob", "I", "You"}.

In their discussion, Ann and Bob are not only the passive objects of nominative and accusative references but also the active interpreters of the language that they use. The system of interpretation (SOI) associated with each language user can be represented in the form of an individual three-place relation called the sign relation of that interpreter.

Understood in terms of its set-theoretic extension, a sign relation L is a subset of a cartesian product O × S × I. Here, O, S, I are three sets that are known as the object domain, the sign domain, and the interpretant domain, respectively, of the sign relation L ⊆ O × S × I.

In general, the three domains of a sign relation can be any sets whatsoever, but the kinds of sign relations that are contemplated in a computational framework are usually constrained to having I ⊆ S. In this case, interpretants are just a special variety of signs, and this makes it convenient to lump signs and interpretants together into a single class called the syntactic domain. In the forthcoming examples, S and I are identical as sets, so the very same elements manifest themselves in two different roles of the sign relations in question. When it is necessary to refer to the whole set of objects and signs in the union of the domains O, S, I for a given sign relation L, one may refer to this set as the World of L and write W = W_L = O ∪ S ∪ I.

To facilitate an interest in the abstract structures of sign relations, and to keep the notations as brief as possible as the examples become more complicated, it serves to introduce the following general notations:

O	=	Object Domain
S	=	Sign Domain
I	=	Interpretant Domain

Introducing a few abbreviations for use in considering the present Example, we have the following data:

O	=	{Ann, Bob}	=	{A, B}
S	=	{"Ann", "Bob", "I", "You"}	=	{"A", "B", "i", "u"}
I	=	{"Ann", "Bob", "I", "You"}	=	{"A", "B", "i", "u"}

In the present example, S = I = syntactic domain.

The sign relation associated with a given interpreter J is denoted L_J  or L(J). Tables 1 and 2 give the sign relations associated with the interpreters A and B, respectively, putting them in the form of relational databases. Thus, the rows of each Table list the ordered triples of the form ‹o, s, i› that make up the corresponding sign relations, L_A , L_B  ⊆ O × S × I. It is often tempting to use the same names for objects and for relations involving these objects, but it is best to avoid this in a first approach, taking up the issues that this practice raises after the less problematic features of these relations have been treated.

Table 1. Sign Relation of Interpreter A

Object	Sign	Interpretant
A	"A"	"A"
A	"A"	"i"
A	"i"	"A"
A	"i"	"i"
B	"B"	"B"
B	"B"	"u"
B	"u"	"B"
B	"u"	"u"

Table 2. Sign Relation of Interpreter B

Object	Sign	Interpretant
A	"A"	"A"
A	"A"	"u"
A	"u"	"A"
A	"u"	"u"
B	"B"	"B"
B	"B"	"i"
B	"i"	"B"
B	"i"	"i"

These Tables codify a rudimentary level of interpretive practice for the agents A and B, and provide a basis for formalizing the initial semantics that is appropriate to their common syntactic domain. Each row of a Table names an object and two co-referent signs, making up an ordered triple of the form ‹o, s, i› that is called an elementary relation, that is, one element of the relation's set-theoretic extension.

Already in this elementary context, there are several different meanings that might attach to the project of a formal semiotics, or a formal theory of meaning for signs. In the process of discussing these alternatives, it is useful to introduce a few terms that are occasionally used in the philosophy of language to point out the needed distinctions.

One aspect of semantics is concerned with the reference that a sign has to its object, which is called its denotation. For signs in general, neither the existence nor the uniqueness of a denotation is guaranteed. Thus, the denotation of a sign can refer to a plural, a singular, or a vacuous number of objects. In the pragmatic theory of signs, these references are formalized as certain types of dyadic relations that are obtained by projection from the triadic sign relations.

The dyadic relation that constitutes the denotative component of a sign relation L is denoted Den(L). Information about the denotative component of semantics can be derived from L by taking its dyadic projection on the plane that is generated by the object domain and the sign domain, indicated by any one of the equivalent forms, Proj_OS L, L_OS , or L₁₂ , and defined as follows:

Den(L) = Proj_OS L = L_OS = {‹o, s› ∈ O × S : ‹o, s, i› ∈ L for some i ∈ I}.

Looking to the denotative aspects of the present example, various rows of the Tables specify that A uses "i" to denote A and "u" to denote B, whereas B uses "i" to denote B and "u" to denote A. It is utterly amazing that even these impoverished remnants of natural language use have properties that quickly bring the usual prospects of formal semantics to a screeching halt.

The other dyadic aspects of semantics that might be considered concern the reference that a sign has to its interpretant and the reference that an interpretant has to its object. As before, either type of reference can be multiple, unique, or empty in its collection of terminal points, and both can be formalized as different types of dyadic relations that are obtained as planar projections of the triadic sign relations.

The connection that a sign makes to an interpretant is called its connotation. In the general theory of sign relations, this aspect of semantics includes the references that a sign has to affects, concepts, impressions, intentions, mental ideas, and to the whole realm of an agent's mental states and allied activities, broadly encompassing intellectual associations, emotional impressions, motivational impulses, and real conduct. This complex ecosystem of references is unlikely ever to be mapped in much detail, much less completely formalized, but the tangible warp of its accumulated mass is commonly alluded to as the connotative import of language. Given a particular sign relation L, the dyadic relation that constitutes the connotative component of L is denoted Con(L).

The bearing that an interpretant has toward a common object of its sign and itself has no standard name. If an interpretant is considered to be a sign in its own right, then its independent reference to an object can be taken as belonging to another moment of denotation, but this omits the mediational character of the whole transaction.

Given the service that interpretants supply in furnishing a locus for critical, reflective, and explanatory glosses on objective scenes and their descriptive texts, it is easy to regard them as annotations both of objects and of signs, but this function points in the opposite direction to what is needed in this connection. What does one call the inverse of the annotation function? More generally asked, what is the converse of the annotation relation?

In light of these considerations, I find myself still experimenting with terms to suit this last-mentioned dimension of semantics. On a trial basis, I refer to it as the ideational, the intentional, or the canonical component of the sign relation, and I provisionally refer to the reference of an interpretant sign to its object as its ideation, its intention, or its conation. Given a particular sign relation L, the dyadic relation that constitutes the intentional component of L is denoted Int(L).

A full consideration of the connotative and intentional aspects of semantics would force a return to difficult questions about the true nature of the interpretant sign in the general theory of sign relations. It is best to defer these issues to a later discussion. Fortunately, omission of this material does not interfere with understanding the purely formal aspects of the present example.

Formally, these new aspects of semantics present no additional problem:

The connotative component of a sign relation L can be formalized as its dyadic projection on the plane generated by the sign domain and the interpretant domain, defined as follows:

Con(L) = Proj_SI L = L_SI = {‹s, i› ∈ S × I : ‹o, s, i› ∈ L for some o ∈ O}.

The intentional component of semantics for a sign relation L, or its second moment of denotation, is adequately captured by its dyadic projection on the plane generated by the object domain and interpretant domain, defined as follows:

Int(L) = Proj_OI L = L_OI = {‹o, i› ∈ O × I : ‹o, s, i› ∈ L for some s ∈ S}.

As it happens, the sign relations L_A and L_B in the present example are fully symmetric with respect to exchanging signs and interpretants, so all of the structure of (L_A)_OS  and (L_B)_OS  is merely echoed in (L_A)_OI  and (L_B)_OI , respectively.

Note on notation. When there is only one sign relation L_J  = L(J) associated with a given interpreter J, it is convenient to use the following forms of abbreviation:

JOS	=	Den(LJ )	=	ProjOS LJ	=	(LJ )OS	=	L(J)OS
JSI	=	Con(LJ )	=	ProjSI LJ	=	(LJ )SI	=	L(J)SI
JOI	=	Int(LJ )	=	ProjOI LJ	=	(LJ )OI	=	L(J)OI

The principal concern of this project is not with every conceivable sign relation but chiefly with those that are capable of supporting inquiry processes. In these, the relationship between the connotational and the denotational aspects of meaning is not wholly arbitrary. Instead, this relationship must be naturally constrained or deliberately designed in such a way that it:

1. Represents the embodiment of significant properties that have objective reality in the agent's domain.
2. Supports the achievement of particular purposes that have intentional value for the agent.

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.

1.3.4.3. Semiotic Equivalence Relations

If one examines the sign relations L_A and L_B that are associated with the interpreters A and B, 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 L_A and L_B is that their connotative components A_SI  and B_SI  constitute a pair of equivalence relations on their common syntactic domain S = I. It is convenient to refer to such structures as semiotic equivalence relations (SER's) since they equate signs that mean the same thing to somebody. Each of the SER's, A_SI , B_SI  ⊆ S × I = S × S partitions the whole collection of signs into semiotic equivalence classes (SEC's). 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 (SEP's) of the syntactic domain.

The main trouble with this notion of semantics in the present situation is that the two semiotic partitions for A and B 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 (POV).

Information about the different forms of semiotic equivalence induced by the interpreters A and B is summarized in Tables 3 and 4. The form of these Tables should suffice to explain what is meant by saying that the SEP's for A and B are orthogonal to each other.

Table 3. Semiotic Partition of Interpreter A

"A" "i"

"u" "B"

Table 4. Semiotic Partition of Interpreter B

"A" "u"

"i" "B"

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

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

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 ‹o, s, i› ∈ L, it is permissible to use "[o]_L" to mean the same thing as "[s]_L". 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 these terms, the SER for interpreter A yields the semiotic equations:

	["A"]A	=	["i"]A	,	["B"]A	=	["u"]A	,
or	"A"	=A	"i"	,	"B"	=A	"u"	,

and the semiotic partition: {{"A", "i"}, {"B", "u"}}.

In contrast, the SER for interpreter B yields the semiotic equations:

	["A"]B	=	["u"]B	,	["B"]B	=	["i"]B	,
or	"A"	=B	"u"	,	"B"	=B	"i"	,

and the semiotic partition: {{"A", "u"}, {"B", "i"}}.

1.3.4.4. Graphical Representations

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.

By way of terminology, a directed edge ‹x, y› is called an arc from point x to point y, and a self-loop ‹x, x› is called a sling at x.

The denotative components Den(A) and Den(B) can be represented as digraphs on the six points of their common world set W = O ∪ S ∪ I = {A, B, "A", "B", "i", "u"}. The arcs are given as follows:

1. Den(A) has an arc from each point of {"A", "i"} to A and from each point of {"B", "u"} to B.
2. Den(B) has an arc from each point of {"A", "u"} to A and from each point of {"B", "i"} to B.

Den(A) and Den(B) can be interpreted as transition digraphs that chart the succession of steps or the connection of states in a computational process. If the graph is read this way, the denotational arcs summarize the upshots of the computations that are involved when the interpreters A and B evaluate the signs in S according to their own frames of reference.

The connotative components Con(A) and Con(B) can be represented as digraphs on the four points of their common syntactic domain S = I = {"A", "B", "i", "u"}. Since Con(A) and Con(B) are SER's, 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:

1. Con(A) has the structure of a SER on S, with a sling at each of the points in S, two-way arcs between the points of {"A", "i"}, and two-way arcs between the points of {"B", "u"}.
2. Con(B) has the structure of a SER on S, with a sling at each of the points in S, two-way arcs between the points of {"A", "u"}, and two-way arcs between the points of {"B", "i"}.

Taken as transition digraphs, Con(A) and Con(B) highlight the associations that are permitted between equivalent signs, as this equivalence is judged by the interpreters A and B, 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.

1.3.4.5. Taking Stock

So far, my discussion of the discussion between A and B, 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 A and B, 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.

It seems like a good idea to pause at this point and reflect on the state of understanding that has been reached. In order to motivate further developments it will be useful to inventory two types of shortfall in the present state of discussion, the first having to do with the defects of my present discussion in revealing the relevant attributes of even so simple an example as the one I used to begin, the second having to do with the defects that this species of example exhibits within the genus of sign relations it is intended to illustrate.

As a general schema, I describe these respective limitations as the rhetorical and the objective defects that a discussion can have in addressing its intended object. The immediate concern is to remedy the insufficiencies of analysis that affect the treatment of the current case. The overarching task is to address the atypically simplistic features of this example as it falls within the class of sign relations that are relevant to actual inquiry.

The next few subsections will be concerned with the most problematic features of the A and B 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.

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 A and B, 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. Numerous bars to insight appear to interlock here. Each one is forged with a good aim in mind, if a bit single-minded in its coverage of the scene, and the whole gang is set to work innocently enough in the unavoidable circumstances of informal discussion. But a failure to absorb their amalgamated impact on the figurative representations and the analytic intentions of sign relations can lead to several types of false impression, both about the true characters of the tables presented here and about the proper utilities of their graphical equivalents to be implemented as data structures in the computer. The next few remarks are put forth in hopes of averting these brands of misreading.

The general character of this question can be expressed in the schematic terms that were used earlier to give a rough sketch of the modeling activity as a whole. How do the isolated SOI's of A and B relate to the interpretive framework that I am using to present them, and how does this IF operate, not only to objectify A and B as models of interpretation (MOI's), but simultaneously to embrace the present and the prospective SOI's of the current narrative, the implicit systems of interpretation that embody in turn the initial conditions and the final intentions of this whole discussion?

One way to see how this issue arises in the discussion of A and B is to recognize that each table of a sign relation is a complex sign in itself, each of whose syntactic constituents plays the role of a simpler sign. In other words, there is nothing but text to be seen on the page. In comparison to what it represents, the table is like a sign relation that has undergone a step of semantic ascent. It is as if the entire contents of the original sign relation have been transposed up a notch on the scale that registers levels of indirectness in reference, each item passing from a more objective to a more symbolic mode of presentation.

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.

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.

Taking this unsuspecting introduction of iconic signs as a serendipitous lesson, an important principle can be lifted from their style of success. They bring the search for models of intellectual processes to look for classes of representation that do not lean too heavily on local idioms for devising labels but rather suspend their abstract formal structures in qualities of media that can best be preserved through a wide variety of global transformations. In time these ventures will lead this project to contemplate various forms of graphical abstraction as supplying possibly the most solid sites for pouring the foundations of formal expression.

What does appear in one of these Tables? It is not the objects that appear under the Object heading, but only the signs of these objects. It is not even the signs and interpretants themselves that appear under the Sign and Interpretant headings, but only the remoter signs of them that are formed by quotation. The unformalized sign relation in which these signs of objects, signs of signs, and signs of interpretants have their role as such is not the one Tabled, but another one that operates behind the scenes to bring its image and intent to the reader.

To understand what the Table is meant to convey the reader has to participate in the informal and more accessory sign relation in order to follow its indications to the intended and more accessible sign relation. As logical or mathematical objects, the sign relations A and B do not exist in the medium of their Tables but are represented there by dint of the relevant structural properties that they share with these Tables. As fictional characters, the interpretive agents A and B do not exist in a uniquely literal sense but serve as typical literary figures to convey the intended formal account, standing in for concrete experiences with language use the likes of which are familiar to writer and reader alike.

The successful formalization of a focal sign relation cannot get around its reliance on prior forms of understanding, like the raw ability to follow indications whose components of competence are embodied in the vaster and largely unarticulated context of a peripheral sign relation. But the extent to which the analysis of a formal sign relation depends on a particular context or a particular interpreter is the extent to which an opportunity for understanding is undermined by a prior petition of the very principles to be explained. Thus, there is little satisfaction in special pleadings or ad hoc accounts of interpretive practice that cannot be transported across a multitude of contexts, media, and interpreters.

What does all this mean, in concrete form, for the proper appreciation of the present example? And looking beyond that, what does it mean in terms of concrete activities that need to be tackled by this work?

One task is to eliminate several types of formal confound that currently affect this investigation. Even though there is an essential tension to be maintained down the lines between casual and formal discussion, the traffic across these realms needs to be monitored carefully. There are identifiable sources of confusion that devolve from the context of informal discussion and invade the arena of formal study, subverting its necessary powers of reflection and undermining its overall effectiveness.

One serious form of contamination can be traced to the accidental circumstance that A and B and I all use the same proper names for A and B. 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.

1.3.4.8. The Conflict of Interpretations

One discrepancy that needs to be documented can be observed in the conflict of interpretations between A and B, 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.

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.

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.

Pronouns qualify as indices because their objective references cannot be traced without recovering further information about their actual context, not just their objective and syntactic contexts but the pragmatic context involved in their actualizing situation of use (SOU) or their realizing instance of use (IOU). To fulfill their duty to sense the reading of indices demands to be supplemented by a more determinate indication of their interpreter of reference, the agent that is responsible for putting them into active use at the moment in question.

Typical examples of indexical signs in programming languages are: (1) variables, signs that need to be bound to a syntactic context or an instantiation frame in order to have a determinate meaning, and (2) pointers, signs that serve particular interpreters operating relative to locally active environments as accessory addresses of modifiable memory contents. In any case something extra — some further information about the objective, syntactic, or interpretive context — must be added to the index in order to tell what it denotes.

If a real object can be regarded as a generic and permanent property that is shared by all of its specific and momentary instantiations, then it is possible to re-characterize indexical signs in the following terms: An index of an object is a property of an actual instance of that object. It is in this sense that indices are said to have actual but not essential connections to what they denote.

Saying that an index is a property of an instance of an object almost makes it sound as though the relation of an index to what it denotes could be defined in purely objective terms, as a product of the two dyadic relations, property of and instance of, and independently of any particular interpreter. But jumping to this conclusion would only produce an approximation to the truth, or a likely story, one that provokes the rejoinders: In whose approach? or Likely to whom?

Taking up these challenges provides a clue as to how a sign relation can appear to be nearly objective, moderately independent, or relatively composite, all within the medium of a particular framework for analysis and interpretation. Careful inspection of the context of definition reveals that it is not really the supposedly frame-free relations of properties and instances that suffice to compose the indexical connection. It is not enough that the separate links exist in principle to make something a property of an instance of something. In order to constitute a genuine sign relation, indexical or otherwise, each link must be recognized to exist by one and the same interpreter.

From this point of view, the object is considered to be something in the external world and the index is considered to be something that touches on the interpreter's experience, both of which subsume, though perhaps in different senses, the object instance (OI) that mediates their actual connection. Although the respective subsumptions, of OI to object and of OI to index, can appear to fall at first glance only within the reach of divergent senses, both must appeal for their eventual realization to a common sense, one that rests within the grasp of a single interpreter. Apparently then, the object instance is the sort of entity that can contribute to generating both the object and the experience, in this way connecting the diverse abstractions called objects and indices.

If a suitable framework of object instances can be found to rationalize an interpreter's experience with objects, then the actual connection that subsists between an object and its index becomes in this framework precisely the connection that exists between two properties of the same object instance, or between two sets intersecting in a common element. Relative to the appropriate framework, the actual connections needed to explain a global indexing operation can be identified, point for point, with the collective function of those joint instances or common elements.

At this stage of analysis, what were originally regarded as real objects have become hypostatic abstractions, extended as generic entities over classes of more transient objects, their instantiating actualizations. In this setting, a real object is now analogous to an extended property or a generative predicate, whose extension generates the trajectory of its momentary instances or the locus of its points in actual existence.

Persisting in this form of analysis appears to lead discussion to levels of existence that are in one way or another more real, more determinate, in a word, more objective than its original objects. If only a particular way of pursuing this form of analysis could be established as reaching a truly fundamental level of existence, then reason could not object to speaking of objects of objects, and even invoking the ultimate objects of objects, meaning the unique atoms at the base of the hierarchy that is formed by the descent of objects.

However, experience leads me to believe that forms of analysis are too peculiar to persons and communities, too dependent on their particular experiences and traditions, and overall too much bound to interpretive constitutions of learning and culture to ever be justly established as invariants of nature. In the end, or rather, by way of appeal to the many courts of final opinion, to invoke any particular form of analysis, no matter whether it is baseless or well-founded, is just another way of referring judgment to a particular interpreter, a contingent IF or a self-serving SOI. Consequently, every form of arbitration retains an irreducibly arbitrary element, and the best policy remains what it has always been, to maintain an honest index of that fact. 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.

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.

One such moment was reached in the preceding subsection, where the topics opened up by indexical signs invited the discussion to begin addressing much wider areas of concern. Eventually, to account for the effective operation of indexical signs I will have to invoke the concept of a real object and pursue the analysis of ostensible objects in terms of still more objective things. These are the extended multitudes of increasingly determinate objects that I will variously refer to as the actualizations, instantiations, realizations, etc. of objects, and on occasion (and not without reason) the objects of objects (OOO's).

Another such moment will arrive when I turn to developing suitable embodiments of sign relations within dynamically realistic systems. In order to implement interpreters as state transition systems, I will have to justify the idea that dynamic states are the real signs and proceed to reconstitute the customary types of signs as abstractions from still more significant tokens. These are the immediate occasions of sign-using transactions that I will tender as situations of use (SOU's) or instances of use (IOU's), plus the states and motions of dynamic systems that solely are able to realize these uses and discharge the obligations they incur to reality.

In every case, working within the framework of systems theory will lead this discussion toward systems and conditions of systems as the ultimate objects of investigation, implicated as the ends of both synthetic and analytic proceedings. Sign relations, initially formulated as relations among three arbitrary sets, will gradually have their original substrates replaced with three systems, the object, sign, and interpretant systems.

Since the roles of a sign relation are formally and pragmatically defined, they do not depend on the material aspects or the essential attributes of elements or domains. Therefore, it is conceivable that the very same system could appear in all three roles, and from this possibility arises much of the ensuing complications of the subject.

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: (1) 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.

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 A and B, and then, by taking up the lessons of this circumscribed arena as a paradigm for future investigation.

In order to explain the operation of sign relations corresponding to iconic and indexical signs in the A and B example, it becomes necessary to refer to potential objects of thought that are located, if they exist at all, outside the realm of the initial object set, that is, lying beyond the objects of thought present at the outset of discussion that one initially recognizes as objects of formally identified signs. In particular, it is incumbent on a satisfying explanation to invoke the abstract properties of objects and the actual instances of objects, where these properties and instances are normally assumed to be new objects of thought that are distinct from the objects to which they refer.

In the pragmatic account of things, thoughts are just signs in the mind of their thinker, so every object of a thought is the object of a sign, though perhaps in a sign relation that has not been fully formalized. Considered on these grounds, the search for a satisfactory context in which to explain the actions and effects of signs turns into a recursive process that potentially calls on ever higher levels of properties and ever deeper levels of instances that are found to stem from whatever objects instigated the search.

To make it serve as a paradigm for future developments, I repeat the basic pattern that has been observed with a slightly different emphasis. In order to explain the operation of icons and indices in a particular discussion, it is necessary to invoke the abstract properties of objects and the actual instances of objects, where by objects one initially comprehends a limited collection of objects of thought under discussion. If these properties and instances are themselves regarded as potential objects of thought, and if they are conceived to be definitively other than the objects whose properties and instances they happen to be, then every initial collection of objects is forced to expand on further consideration, in this way pointing to a world of objects of thought that extends in two directions beyond the originating frame of discussion.

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.

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.

This subsection presents the objective project (OP) that I plan to take up for investigating the forms of sign relations, and it outlines three objective levels (OL's) of formulation that guide the analytic and synthetic study of interpretive structure and regulate the prospective stages of implementing this plan in particular cases. The main purpose of these schematic conceptions is organizational, to provide a conceptual architecture for the burgeoning hierarchies of objects that arise in the generative processes of inquiry.

In the immediate context the objective project and the three levels of objective description are presented in broad terms. In the process of surveying a variety of problems that serve to instigate efforts in this general direction, I explore the prospects of a particular organon, or instrumental scheme for the analysis and synthesis of objects, that is intended to address these issues, and I give an overview of its design. In interpreting the sense of the word objective as it is used in this application, it may help to regard this objective project in the light of a telescopic analogy, with an objective being "a lens or system of lenses that forms an image of an object" (Webster's).

In the next three subsections after this one the focus returns to the separate levels of object structure, starting with the highest level of specification and treating the supporting levels in order of increasing detail. At each stage, the developing tools are applied to the analysis of concrete problems that arise in trying to clarify the structure and function of sign relations. For the present task, elaborations of this perspective are kept within the bounds of what is essential to deal with the example of A and B.

My use of the word object derives from the stock of the Greek root pragma, which captures all the senses needed to suggest the stability of concern and the dedication to purpose that are forever bound up in the constitution of objects and the institution of objectives. What it implies is that every object, objective, or objectivity is always somebody's object, objective, or objectivity.

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 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.

In this project I would like to treat the difference between construction and deconstruction as being more or less synonymous with the contrast between synthesis and analysis, but doing this without the introduction of too much distortion requires the intervention of a further distinction. Therefore, let it be recognized that all orientations to the constitutions of objects can be pursued in both regimented and radical fashions. In the weaker senses of the terms, analysis and synthesis work within a preset and limited regime of objects, construing each object as being composed from a fixed inventory of stock constituents. In the stronger senses, contracting for the application of these terms places a more strenuous demand on the would-be construer.

A radical form of analysis, in order to discern the contrasting intentions in everything construed as an object, requires interpreters to leave or at least re-place objects within the contexts of their live acquaintance, to reflect on their own motives and motifs for construing and employing objects in the ways they do, and to deconstruct how their own aims and biases enter into the form and use of objects.

A radical form of synthesis, in order to integrate ideas and information devolving from entirely different frameworks of interpretation (FOI's), requires interpreters to reconstruct isolated concepts and descriptions on a mutually compatible basis and to use means of composition that can constitute a medium for common sensibilities.

Thus, the radical project in all of these directions demands forms of interpretation, analysis, synthesis that can reflect a measure of light on the initially unstated assumptions of their prospective agents.

The foregoing considerations lead up to the organizing conception of an objective framework (OF), in which objects can be analyzed into sets of constituent objects, perhaps proceeding recursively to some limiting level where the fundamental objects of thought are thought to rest. If an OF is felt to be completely unique and uniquely complete, then people tend to regard it as constituting a veritable ontology, but I will not be able to go that far. The recognition of plural and fallible perspectives that goes with pragmatic forms of thinking does not see itself falling into line any time soon with any one or only one ontology.

On the opposite score, there is no reason to deny the possibility that a unique and complete OF exists. Indeed, the hope that such a standpoint does exist often provides inquiry with a beneficial regulative principle or a heuristic hypothesis to work on. It merely happens, for the run of finitely informed creatures (FIC's) at any rate, that the existence of an ideal framework is something to be established after the fact, at least nearer toward the end of inquiry than the present time marks.

In this project, an OF embodies one or more objective genres (OG's), also called forms of analysis (FOA's) or forms of synthesis (FOS's), each of which delivers its own rendition of a great chain of being for all the objects under its purview. In effect, each OG develops its own version of an ontological hierarchy (OH), designed independently of the others to capture an aspect of structure in its objective domain.

For now, the level of an OF operates as a catch-all, giving the projected discussion the elbow room it needs to range over an unspecified variety of different OG's and to place the particular OG's of active interest in a running context of comparative evaluations and developmental options.

Any given OG can appear under the alias of a form of analysis (FOA) or a form of synthesis (FOS), depending on the direction of prevailing interest. A notion frequently invoked for the same purpose is that of an ontological hierarchy (OH), but I will use this only provisionally, and only so long as it is clear that alternative ontologies can always be proposed for the same space of objects.

An OG embodies many objective motives or objective motifs (OM's). If an OG constitutes a genus, or generic pattern of object structure, then the OM's amount to its specific and individual exemplars. Thus, an OM can appear in the guise of a particular instance, trial, or "run" of the general form of analytic or synthetic procedure that accords with the protocols of a given OG.

In order to provide a way of talking about objective points of view in general without having to specify a particular level, I will use the term objective concern (OC) to cover any individual OF, OG, or OM.

An OG, in its general way, or an OM, in its individual way, begins by relating each object in its purview to a unique set of further objects, called the components, constituents, effects, ingredients, or instances of that object with respect to that objective concern (OC). As long as discussion remains fixed to what is visible within the scope of a particular OC, the collected effects of each object in view constitute its active ingredients, supplying it with a unique decomposition that defines it to a degree sufficient for all purposes conceivable within that discussion.

Contemplated from an outside perspective, however, the status of these effects as the defining unique determinants (DUD's) of each object under examination is something to be questioned. The supposed constituents of an object that are obvious with respect to one OC can be regarded with suspicion from the points of view of alternative OC's, and their apparent status as rock-bottom substantives can find itself reconstituted in the guise of provisional placeholders (placebos or excipients) that precipitately index the potential operation of more subtly active ingredients.

If a single OG could be unique and the realization of every object in it could be complete, then there might be some basis for saying that the elements of objects and the extensions of objects are known, and thus that the very objects of objects (OOO's) are determined by its plan. In practice, however, it takes a diversity of overlapping and not entirely systematic OG's to make up a moderately useful OF.

What gives an OG a definite constitution is the naming of a space of objects that falls under its purview and the setting down of a system of axioms that affects its generating relations. 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.

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.

1. An objective framework (OF) consists of one or more objective genres (OG's), also called forms of analysis (FOA's), forms of synthesis (FOS's), or ontological hierarchies (OH's). Typically, these span a diverse spectrum of formal characteristics and intended interpretations.
2. An OG is made up of one or more objective motives or objective motifs (OM's), sometimes regarded as particular instances of analysis (IOA's) or instances of synthesis (IOS's). All of the OM's governed by a particular OG exhibit a kinship of structures and intentions, and each OM roughly fits the pattern or follows in the footsteps of its guiding OG.
3. An OM can be identified with a certain moment of interpretation, one in which a particular dyadic relation appears to govern all the objects in its purview. Initially presented as an abstraction, an individual OM is commonly fleshed out by identifying it with its interpretive agent. As this practice amounts to a very loose form of personification, it is subject to all the dangers of its type and is bound eventually to engender a multitude of misunderstandings. In contexts where more precision is needed it is best to recognize that the application of an OM is restricted to special instants and limited intervals of time. This means that an individual OM must look to the interpretive moment (IM) of its immediate activity to find the materials available for both its concrete instantiation and its real implementation. Finally, having come round to the picture of an objective motive realized in an interpretive moment, this discussion has made a discrete advance toward the desired forms of dynamically realistic models, providing itself with what begins to look like the elemental states and dispositions needed to build fully actualized systems of interpretation.

A major theoretical task that remains outstanding for this project is to discover a minimally adequate basis for defining the state of uncertainty that an interpretive system has with respect to the questions it is able to formulate about the state of an object system. Achieving this would permit a measure of definiteness to be brought to the question of inquiry's nature, since it can be grasped intuitively that the gist of inquiry is to reduce an agent's level of uncertainty about its object, objective, or objectivity through appropriate changes of state.

Accordingly, one of the roles intended for this OF is to provide a set of standard formulations for describing the moment to moment uncertainty of interpretive systems. The formally definable concepts of the MOI (the objective case of a SOI) and the IM (the momentary state of a SOI) are intended to formalize the intuitive notions of a generic mental constitution and a specific mental disposition that usually serve in discussing states and directions of mind.

The structures present at each objective level are formulated by means of converse pairs of staging relations, prototypically symbolized by the signs \(\lessdot\) and \(\gtrdot\). At the more generic levels of OF's and OG's the staging operations associated with the generators \(\lessdot\) and \(\gtrdot\) involve the application of dyadic relations analogous to class membership \(\in\!\) and its converse \(\ni\!\), but the increasing amounts of parametric information that are needed to determine specific motives and detailed motifs give OM's the full power of triadic relations. Using the same pair of symbols to denote staging relations at all objective levels helps to prevent an excessive proliferation of symbols, but it means that the meaning of these symbols is always heavily dependent on context. In particular, even fundamental properties like the effective arity of the relations signified can vary from level to level.

The staging relations divide into two orientations, \(\lessdot\) versus \(\gtrdot\), indicating opposing senses of direction with respect to the distinction between analytic and synthetic projects:

The standing relations, indicated by \(\lessdot\), are analogous to the element of or membership relation \(\in\!\). Another interpretation of \(\lessdot\) is the instance of relation. At least with respect to the more generic levels of analysis, any distinction between these readings is immaterial to the formal interests and structural objectives of this discussion.

The propping relations, indicated by \(\gtrdot\), are analogous to the class of relation or converse of the membership relation. An alternate meaning for \(\gtrdot\) is the property of relation. Although it is possible to maintain a distinction here, this discussion is mainly interested in a level of formal structure to which this difference is irrelevant.

Although it may be logically redundant, it is useful in practice to introduce efficient symbolic devices for both directions of relation, \(\lessdot\) and \(\gtrdot\), and to maintain a formal calculus that treats analogous pairs of relations on an equal footing. Extra measures of convenience come into play when the relations are used as assignment operations to create titles, define terms, and establish offices of objects in the active contexts of given relations. Thus, I regard these dual relationships as symmetric primitives and use them as the generating relations of all three objective levels.

Next, I present several different ways of formalizing objective genres and motives. The reason for employing multiple descriptions is to capture the various ways that these patterns of organization appear in practice.

One way to approach the formalization of an objective genre \(G\!\) is through an indexed collection of dyadic relations:

\(G = \{ G_j \} = \{ G_j : j \in J \}\ \text{with}\ G_j \subseteq P_j \times Q_j\ (\forall j \in J).\)

Here, \(J\!\) is a set of actual (not formal) parameters used to index the OG, while \(P_j\!\) and \(Q_j\!\) are domains of objects (initially in the informal sense) that enter into the dyadic relations \(G_j\!\).

Aside from their indices, many of the \(G_j\!\) in \(G\!\) can be abstractly identical to each other. This would earn \(G\!\) the designation of a multi-family or a multi-set, but I prefer to treat the index \(j\!\) as a concrete part of the indexed relation \(G_j\!\), in this way distinguishing it from all other members of the indexed family \(G\!\).

Ordinarily, it is desirable to avoid making individual mention of the separately indexed domains, \(P_j\!\) and \(Q_j\!\) for all \(j\!\) in \(J\!\). Common strategies for getting around this trouble involve the introduction of additional domains, designed to encompass all the objects needed in given contexts. Toward this end, an adequate supply of intermediate domains, called the rudiments of universal mediation, can be defined as follows:

\(X_j = P_j \cup Q_j,\)

\(P = \textstyle \bigcup_j P_j,\)

\(Q = \textstyle \bigcup_j Q_j.\)

Ultimately, all of these totalitarian strategies end the same way, at first, by envisioning a domain \(X\!\) that is big enough to encompass all the objects of thought that might demand entry into a given discussion, and then, by invoking one of the following conventions:

Rubric of Universal Inclusion\[X = \textstyle \bigcup_j (P_j \cup Q_j).\]

Rubric of Universal Equality\[X = P_j = Q_j\ (\forall j \in J).\]

Working under either of these assumptions, \(G\!\) can be provided with a simplified form of presentation:

\(G = \{ G_j \} = \{ G_j : j \in J \}\ \text{with}\ G_j \subseteq X \times X\ (\forall j \in J).\)

However, it serves a purpose of this project to preserve the individual indexing of relational domains for while longer, or at least to keep this usage available as an alternative formulation. Generally speaking, it is always possible in principle to form the union required by the universal inclusion convention, or without loss of generality to assume the equality imposed by the universal equality convention. The problem is that the unions and equalities invoked by these rubrics may not be effectively definable or testable in a computational context. Further, even when these sets or tests can be constructed or certified by some computational agent or another, the pertinent question at any interpretive moment is whether each collection or constraint is actively being apprehended or warranted by the particular interpreter charged with responsibility for it by the indicated assignment of domains.

But an overall purpose of this formalism is to represent the objects and constituencies known to specific interpreters at definite moments of their interpretive proceedings, in other words, to depict the information about objective existence and constituent structure that is possessed, recognized, responded to, acted on, and followed up by concrete agents as they move through their immediate contexts of activity. Accordingly, keeping individual tabs on the relational domains \(P_j\!\) and \(Q_j\!\), though it does not solve this array of problems, does serve to mark the concern with particularity and to keep before the mind the issues of individual attention and responsibility that are appropriate to interpretive agents. In short, whether or not domains appear with explicit subscripts, one should always be ready to answer Who subscribes to these domains?

It is important to emphasize that the index set \(J\!\) and the particular attachments of indices to dyadic relations are part and parcel to \(G\!\), befitting the concrete character intended for the concept of an objective genre, which is expected to realistically embody in the character of each \(G_j\!\) both a local habitation and a name. For this reason, among others, the \(G_j\!\) can safely be referred to as individual dyadic relations. Since the classical notion of an individual as a perfectly determinate entity has no application in finite information contexts, it is safe to recycle this term to distinguish the terminally informative particulars that a concrete index \(j\!\) adds to its thematic object \(G_j\!\).

Depending on the prevailing direction of interest in the genre \(G\!\), \(\lessdot\) or \(\gtrdot\), the same symbol is used equivocally for all the relations \(G_j\!\). The \(G_j\!\) can be regarded as formalizing the objective motives that make up the genre \(G\!\), provided it is understood that the information corresponding to the parameter \(j\!\) constitutes an integral part of the motive or motif of \(G_j\!\).

In this formulation, \(G\!\) constitutes ontological hierarchy of a plenary type, one that determines the complete array of objects and relationships that are conceivable and describable within a given discussion. Operating with reference to the global field of possibilities presented by \(G\!\), each \(G_j\!\) corresponds to the specialized competence of a particular agent, selecting out the objects and links of the generic hierarchy that are known to, owing to, or owned by a given interpreter.

Another way to formalize the defining structure of an objective genre can be posed in terms of a relative membership relation or a notion of relative elementhood. The constitutional structure of a particular genre can be set up in a flexible manner by taking it in two stages, starting from the level of finer detail and working up to the big picture:

1. Each OM is constituted by what it means to be an object within it. What constitutes an object in a given OM can be fixed as follows:
1. In absolute terms, by specifying the domain of objects that fall under its purview. For the present, I assume that each OM inherits the same object domain \(X\!\) from its governing OG.
2. In relative terms, by specifying a converse pair of dyadic relations that (redundantly) determine two sets of facts:
1. What is an instance, example, member, or element of what, relative to the OM in question.
2. What is a property, quality, class, or set of what, relative to the OM in question.
2. The various OM's of a particular OG can be unified under its aegis by means of a single triadic relation, one that names an OM and a pair of objects and that holds when one object belongs to the other in the sense identified by the relevant OM. If it becomes absolutely essential to emphasize the relativity of elements, one may resort to calling them relements, in this way jostling the mind to ask: Relement to what?

The last and perhaps the best way to form an objective genre \(G\!\) is to present it as a triadic relation:

\(G = \{ (j, p, q) \} \subseteq J \times P \times Q ,\)

or:

\(G = \{ (j, x, y) \} \subseteq J \times X \times X .\)

Given an objective genre \(G\!\) whose motives are indexed by a set \(J\!\) and whose objects form a set \(X\!\), there is a triadic relation among a motive and a pair of objects that exists when the first object belongs to the second object according to that motive. This is called the standing relation of the genre, and it can be taken as one way of defining and establishing the genre. In the way that triadic relations usually give rise to dyadic operations, the associated standing operation of the genre can be thought of as a brand of assignment operation that makes one object belong to another in a certain sense, namely, in the sense indicated by the designated motive.

There is a partial converse of the standing relation that transposes the order in which the two object domains are mentioned. This is called the propping relation of the genre, and it can be taken as an alternate way of defining the genre.

\(G\!\uparrow \ = \ \{(j, q, p) \in J \times Q \times P : (j, p, q) \in G \},\)

or:

\(G\!\uparrow \ = \ \{(j, y, x) \in J \times X \times X : (j, x, y) \in G \}.\)

The following conventions are useful for discussing the set-theoretic extensions of the staging relations and staging operations of an objective genre:

The standing relation of a genre is denoted by the symbol \(:\!\lessdot\), pronounced set-in, with either of the following two type-markings:

\(:\!\lessdot\ \subseteq\ J \times P \times Q,\)

\(:\!\lessdot\ \subseteq\ J \times X \times X.\)

The propping relation of a genre is denoted by the symbol \(:\!\gtrdot\), pronounced set-on, with either of the following two type-markings:

\(:\!\gtrdot\ \subseteq\ J \times Q \times P,\)

\(:\!\gtrdot\ \subseteq\ J \times X \times X.\)

Often one's level of interest in a genre is purely generic. When the relevant genre is regarded as an indexed family of dyadic relations, \(G = \{ G_j \}\!\), then this generic interest is tantamount to having one's concern rest with the union of all the dyadic relations in the genre.

\(\textstyle \bigcup_J G = \textstyle \bigcup_j G_j = \{ (x, y) \in X \times X : (x, y) \in G_j\ (\exists j \in J) \}.\)

When the relevant genre is contemplated as a triadic relation, \(G \subseteq J \times X \times X\), then one is dealing with the projection of \(G\!\) on the object dyad \(X \times X\).

\(G_{XX} = \operatorname{proj}_{XX}(G) = \{ (x, y) \in X \times X : (j, x, y) \in G\ (\exists j \in J) \}.\)

On these occasions, the assertion that \((x, y)\!\) is in \(\cup_J G = G_{XX}\) can be indicated by any one of the following equivalent expressions:

\(G : x \lessdot y,\)	\(x \lessdot_G y,\)	\(x \lessdot y : G,\)
\(G : y \gtrdot x,\)	\(y \gtrdot_G x,\)	\(y \gtrdot x : G.\)

At other times explicit mention needs to be made of the interpretive perspective or individual dyadic relation that links two objects. To indicate that a triple consisting of a motive \(j\!\) and two objects \(x\!\) and \(y\!\) belongs to the standing relation of the genre, in symbols, \((j, x, y) \in\ :\!\lessdot\), or equally, to indicate that a triple consisting of a motive \(j\!\) and two objects \(y\!\) and \(x\!\) belongs to the propping relation of the genre, in symbols, \((j, y, x) \in\ :\!\gtrdot\), all of the following notations are equivalent:

\(j : x \lessdot y,\)	\(x \lessdot_j y,\)	\(x \lessdot y : j,\)
\(j : y \gtrdot x,\)	\(y \gtrdot_j x,\)	\(y \gtrdot x : j.\)

Assertions of these relations can be read in various ways, for example:

\(j : x \lessdot y\) \(j : y \gtrdot x\) \(x \lessdot_j y\) \(y \gtrdot_j x\) \(x \lessdot y : j\) \(y \gtrdot x : j\)

\(j\ \text{sets}\ x\ \text{in}\ y.\) \(j\ \text{sets}\ y\ \text{on}\ x.\) \(j\ \text{makes}\ x\ \text{an instance of}\ y.\) \(j\ \text{makes}\ y\ \text{a property of}\ x.\) \(j\ \text{thinks}\ x\ \text{an instance of}\ y.\) \(j\ \text{thinks}\ y\ \text{a property of}\ x.\) \(j\ \text{attests}\ x\ \text{an instance of}\ y.\) \(j\ \text{attests}\ y\ \text{a property of}\ x.\) \(j\ \text{appoints}\ x\ \text{an instance of}\ y.\) \(j\ \text{appoints}\ y\ \text{a property of}\ x.\) \(j\ \text{witnesses}\ x\ \text{an instance of}\ y.\) \(j\ \text{witnesses}\ y\ \text{a property of}\ x.\) \(j\ \text{interprets}\ x\ \text{an instance of}\ y.\) \(j\ \text{interprets}\ y\ \text{a property of}\ x.\) \(j\ \text{contributes}\ x\ \text{to}\ y.\) \(j\ \text{attributes}\ y\ \text{to}\ x.\) \(j\ \text{determines}\ x\ \text{an example of}\ y.\) \(j\ \text{determines}\ y\ \text{a quality of}\ x.\) \(j\ \text{evaluates}\ x\ \text{an example of}\ y.\) \(j\ \text{evaluates}\ y\ \text{a quality of}\ x.\) \(j\ \text{proposes}\ x\ \text{an example of}\ y.\) \(j\ \text{proposes}\ y\ \text{a quality of}\ x.\) \(j\ \text{musters}\ x\ \text{under}\ y.\) \(j\ \text{marshals}\ y\ \text{over}\ x.\) \(j\ \text{indites}\ x\ \text{among}\ y.\) \(j\ \text{ascribes}\ y\ \text{about}\ x.\) \(j\ \text{imputes}\ x\ \text{among}\ y.\) \(j\ \text{imputes}\ y\ \text{about}\ x.\) \(j\ \text{judges}\ x\ \text{beneath}\ y.\) \(j\ \text{judges}\ y\ \text{beyond}\ x.\) \(j\ \text{finds}\ x\ \text{preceding}\ y.\) \(j\ \text{finds}\ y\ \text{succeeding}\ x.\) \(j\ \text{poses}\ x\ \text{before}\ y.\) \(j\ \text{poses}\ y\ \text{after}\ x.\) \(j\ \text{forms}\ x\ \text{below}\ y.\) \(j\ \text{forms}\ y\ \text{above}\ x.\)

In making these free interpretations of genres and motifs, one needs to read them in a logical rather than a cognitive sense. A statement like "\(j\!\) thinks \(x\!\) an instance of \(y\!\)" should be understood as saying that "\(j\!\) is a thought with the logical import that \(x\!\) is an instance of \(y\!\)", and a statement like "\(j\!\) proposes \(y\!\) a property of \(x\!\)" should be taken to mean that "\(j\!\) is a proposition to the effect that \(y\!\) is a property of \(x\!\)".

These cautions are necessary to forestall the problems of intentional attitudes and contexts, something I intend to clarify later on in this project. At present, I regard the well-known opacities of this subject as arising from the circumstance that cognitive glosses tend to impute an unspecified order of extra reflection to each construal of the basic predicates. The way I plan to approach this issue is through a detailed analysis of the cognitive capacity for reflective thought, to be developed to the extent possible in formal terms by using sign relational models.

By way of anticipating the nature of the problem, consider the following examples to illustrate the contrast between logical and cognitive senses:

• In a cognitive context, if \(j\!\) is a considered opinion that \(S\!\) is true, and \(j\!\) is a considered opinion that \(T\!\) is true, then it does not have to automatically follow that \(j\!\) is a considered opinion that the conjunction \(S\ \operatorname{and}\ T\) is true, since an extra measure of consideration might conceivably be involved in cognizing the conjunction of \(S\!\) and \(T\!\).

• In a logical context, if \(j\!\) is a piece of evidence that \(S\!\) is true, and \(j\!\) is a piece of evidence that \(T\!\) is true, then it follows by these very facts alone that \(j\!\) is a piece of evidence that the conjunction \(S\ \operatorname{and}\ T\) is true. This is analogous to a situation where, if a person \(j\!\) draws a set of three lines, \(AB,\!\) \(BC,\!\) and \(AC,\!\) then \(j\!\) has drawn a triangle \(ABC,\!\) whether \(j\!\) recognizes the fact on reflection and further consideration or not.

Some readings of the staging relations are tantamount to statements of (a possibly higher order) model theory. For example, consider the predicate \(P : J \to \mathbb{B}\) defined by the following equivalence:

\(P(j) \quad \Leftrightarrow \quad j\ \text{proposes}\ x\ \text{an instance of}\ y.\)

Then \(P\!\) 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 \(J\!\). And yet all of these expressions are just elaborate ways of stating the underlying assertion which says that there exists a triple \((j, x, y)\!\) in the genre \(G (:\!\lessdot)\).

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.

A principal aim of using sign relations as formal models is to be capable of analyzing complex activities that arise in nature and human domains. Proceeding by the opportunistic mode of analysis by synthesis, one generates likely constructions from a stock of favored, familiar, and well-understood sign relations, the supply of which hopefully grows with time, constantly matching their formal properties against the structures encountered in the "wilds" of natural phenomena and human conduct. When salient traits of both the freely generated products and the widely gathered phenomena coincide in enough points, then the details of the constructs one has built for oneself can help to articulate a plausible hypothesis as to how the observable appearances might be explained.

A principal difficulty of using sign relations for this purpose arises from the very power of productivity they bring to bear in the process, the capacity of triadic relations to generate a welter of what are bound to be mostly arbitrary structures, with only a scattered few hoping to show any promise, but the massive profusion of which exceeds from the outset any reason's ability to sort them out and test them in practice. And yet, as the phenomena of interest become more complex, the chances grow slimmer that adequate explanations will be found in any of the thinner haystacks. In this respect, sign relations inherit the basic proclivities of set theory, which can be so successful and succinct in presenting and clarifying the properties of already found materials and hard won formal insights, and yet so overwhelming to use as a tool of random exploration and discovery.

The sign relations of \(A\!\) and \(B\!\), though natural in themselves as far as they go, were nevertheless introduced in an artificial fashion and presented by means of arbitrary stipulations. Sign relations that arise in more natural settings usually have a rationale, a reason for being as they are, and therefore become amenable to classification on the basis of the distinctive characters that make them what they are.

Consequently, naturally occurring sign relations can be expected to fall into species or natural kinds, and to have special properties that make them keep on occurring in nature. Moreover, cultivated varieties of sign relations, the kinds that have been converted to social purposes and found to be viable in actual practice, will have identifiable and especially effective properties by virtue of which their signs are rendered significant.

In the pragmatic theory of sign relations, three natural kinds of signs are recognized, under the names of icons, indices, and symbols. Examples of indexical or accessional signs figured significantly in the discussion of \(A\!\) and \(B\!\), as illustrated by the pronouns "i" and "u" in \(S.\!\) Examples of iconic or analogical signs were also present, though keeping to the background, in the very form of the sign relation Tables that were used to schematize the whole activity of each interpreter. Examples of symbolic or conventional signs, of course, abide even more deeply in the background, pervading the whole context and making up the very fabric of this discussion.

In order to deal with the array of issues presented so far in this subsection, all of which have to do with controlling the generative power of sign relations to serve the specific purposes of understanding, I apply the previously introduced concept of an objective genre (OG). This is intended to be a determinate purpose or a deliberate pattern of analysis and synthesis that one can identify as being active at given moments in a discussion and that affects what one regards as the relevant structural properties of its objects.

In the remainder of this subsection the concept of an OG is used informally, and only to the extent needed for a pressing application, namely, to rationalize the natural kinds that are claimed for signs and to clarify an important contrast that exists between icons and indices.

The OG I apply here is called the genre of properties and instances. One moves through its space, higher and lower in a particular ontology, by means of two dyadic relations, upward by taking a property of and downward by taking an instance of whatever object initially enters one's focus of attention. Each object of this OG is reckoned to be the unique common property of the set of objects that lie one step below it, objects that are in turn reckoned to be instances of the given object.

Pretty much the same relational structures could be found in the genre or paradigm of qualities and examples, but the use of examples here is polymorphous enough to include experiential, exegetic, and executable examples. This points the way to a series of related genres, for example, the OG's of principles and illustrations, laws and existents, precedents and exercises, and on to lessons and experiences. All in all, in their turn, these modulations of the basic OG show a way to shift the foundations of ontological hierarchies toward bases in individual and systematic experience, and thus to put existentially dynamic rollers under the blocks of what seem to be essentially invariant pyramids.

Any object of these OG's can be contemplated in the light of two potential relationships, namely, with respect to its chances of being an object quality or an object example of something else. In future references, abbreviated notations like \(\operatorname{OG} (\operatorname{Prop}, \operatorname{Inst})\) or \(\operatorname{OG} = (\operatorname{Prop}, \operatorname{Inst})\) will be used to specify particular genres, giving the intended interpretations of their generating relations \(\{ \lessdot,\gtrdot \}.\)

With respect to this OG, I can now characterize icons and indices. Icons are signs by virtue of being instances of properties of objects. Indices are signs by virtue of being properties of instances of objects.

Because the initial discussion seems to flow more smoothly if I apply dyadic relations on the left, I formulate these definitions as follows:

\(\begin{array}{llll} \text{For Icons:} & \operatorname{Sign} (\operatorname{Obj}) & = & \operatorname{Inst} (\operatorname{Prop} (\operatorname{Obj})). \\ \text{For Indices:} & \operatorname{Sign} (\operatorname{Obj}) & = & \operatorname{Prop} (\operatorname{Inst} (\operatorname{Obj})). \\ \end{array}\)

Imagine starting from the sign and retracing steps to reach the object, in this way finding the converses of these relations to be as follows:

\(\begin{array}{llll} \text{For Icons:} & \operatorname{Obj} (\operatorname{Sign}) & = & \operatorname{Inst} (\operatorname{Prop} (\operatorname{Sign})). \\ \text{For Indices:} & \operatorname{Obj} (\operatorname{Sign}) & = & \operatorname{Prop} (\operatorname{Inst} (\operatorname{Sign})). \\ \end{array}\)

In spite of the apparent duality between these patterns of composition, there is a significant asymmetry to be observed in the way that the insistent theme of realism interrupts the underlying genre. In order to understand this, it is necessary to note that the strain of pragmatic thinking I am using here takes its definition of reality from the word's original Scholastic sources, where the adjective real means having properties. Taken in this sense, reality is necessary but not sufficient to actuality, where actual means "existing in act and not merely potentially" (Webster's). To reiterate, actuality is sufficient but not necessary to reality. The distinction between the ideas is further pointed up by the fact that a potential can be real, and that its reality can be independent of any particular moment in which the power acts.

These abstract considerations would probably remain distant from the present concern, were it not for two points of connection:

1. Relative to the present genre, the distinction of reality, that can be granted to certain objects of thought and not to others, fulfills an analogous role to the distinction that singles out sets among classes in modern versions of set theory. Taking the membership relation \(\in\!\) as a predecessor relation in a pre-designated hierarchy of classes, a class attains the status of a set, and by dint of this becomes an object of more determinate discussion, simply if it has successors. Pragmatic reality is distinguished from both the medieval and the modern versions, however, by the fact that its reality is always a reality to somebody. This is due to the circumstance that it takes both an abstract property and a concrete interpreter to establish the practical reality of an object.
2. This project seeks articulations and implementations of intelligent activity within dynamically realistic systems. The individual stresses placed on articulation, implementation, actuality, dynamics, and reality collectively reinforce the importance of several issues:

• Systems theory, consistently pursued, eventually demands for its rationalization a distinct ontology, in which states of being and modes of action form the principal objects of thought, out of which the ordinary sorts of stably extended objects must be constructed. In the "grammar" of process philosophy, verbs and pronouns are more basic than nouns. In its influence on the course of this discussion, the emphasis on systematic action is tantamount to an objective genre that makes dynamic systems, their momentary states and their passing actions, become the ultimate objects of synthesis and analysis. Consequently, the drift of this inquiry will be turned toward conceiving actions, as traced out in the trajectories of systems, to be the primitive elements of construction, more fundamental in this objective genre than stationary objects extended in space. As a corollary, it expects to find that physical objects of the static variety have a derivative status in relation to the activities that orient agents, both organisms and organizations, toward purposeful objectives.

• At root, the notion of dynamics is concerned with power in the sense of potential. The brand of pragmatic thinking that I use in this work permits potential entities to be analyzed as real objects and conceptual objects to be constituted by the conception of their actual effects in practical instances. In the attempt to unify symbolic and dynamic approaches to intelligent systems (Upper and Lower Kingdoms?), there remains an insistent need to build conceptual bridges. A facility for relating objects to their actualizing instances and their instantiating actions lends many useful tools to an effort of this nature, in which the search for understanding cannot rest until each object and phenomenon has been reconstructed in terms of active occurrences and ways of being.

• In prospect of form, it does not matter whether one takes this project as a task of analyzing and articulating the actualizations of intelligence that already exist in nature, or whether one views it as a goal of synthesizing and artificing the potentials for intelligence that have yet to be conceived in practice. From a formal perspective, the analysis and the synthesis are just reciprocal ways of tracing or retracing the same generic patterns of potential structure that determine actual form.

Returning to the examination of icons and indices, and keeping the criterion of reality in mind, notice the radical difference that comes into play in recursive settings between the two types of contemplated moves that are needed to trace the respective signs back to their objects, that is, to discover their denotations:

1. Icon → Object. Taking the iconic sign as an initial instance, try to go up to a property and then down to a different or perhaps the same instance. This form of ascent does not require a distinct object, since reality of the sign is sufficient to itself. In other words, if the sign has any properties at all, then it is an icon of a real object, even if that object is only itself.
2. Index → Object. Taking the indexical sign as an initial property, try to go down to an instance and then up to a different or perhaps the same property. This form of descent requires a real instance to substantiate it, but not necessarily a distinct object. Consequently, the index always has a real connection to its object, even if that object is only itself.

In sum: For icons a separate reality is optional, for indices a separate reality is obligatory. As often happens with a form of analysis, each term under the indicated sum appears to verge on indefinite expansion:

1. For icons, the existence of a separate reality is optional. This means that the question of reality in the sign relation can depend on nothing more than the reality of each sign itself, on whether it has any property with respect to the OG in question. In effect, icons can rely on their own reality to faithfully provide a real object.
2. For indices, the existence of a separate reality is obligatory. And yet this reality need not affect the object of the sign. In essence, indices are satisfied with a basis in reality that need only reside in an actual object instance, one that establishes a real connection between the object and its index with regard to the OG in question.

Finally, suppose that \(M\!\) and \(N\!\) are hypothetical sign relations intended to capture all the iconic and indexical relationships, respectively, that a typical object \(x\!\) enjoys within its genre \(G.\!\) A sign relation in which every sign has the same kind of relation to its object under an assumed form of analysis is appropriately called a homogeneous sign relation. In particular, if \(H\!\) is a homogeneous sign relation in which every sign has either an iconic or an indexical relation to its object, then it is convenient to apply the corresponding adjective to the whole of \(H\!.\)

Typical sign relations of the iconic or indexical kind generate especially simple and remarkably stable sorts of interpretive processes. In arity, they could almost be classified as approximately dyadic, since most of their interesting structure is wrapped up in their denotative aspects, while their connotative functions are relegated to the tangential role of preserving the directions of their denotative axes. In a metaphorical but true sense, iconic and indexical sign relations equip objective frameworks with "gyroscopes", helping them maintain their interpretive perspectives in a persistent orientation toward their objective world.

Of course, every form of sign relation still depends on the agency of a proper interpreter to bring it to life, and every species of sign process stays forever relative to the interpreters that actually bring it to term. But it is a rather special circumstance by means of which the actions of icons and indices are able to turn on the existence of independently meaningful properties and instances, as recognized within an objective framework, and this means that the interpretive associations of these signs are not always as idiosyncratic as they might otherwise be.

The dispensation of consensual bonds in a common medium leaves room for many individual interpreters to inhabit a shared frame of reference, and for a diversity of transient interpretive moments to take up and consolidate a continuing perspective on a world of mutual interests. This further increases the likelihood that differing and developing interpreters will become able to participate in compatible views and coherent values in relation to the aggregate of things, to collate information from a variety of sources, and to bring concerted action to bear on an appreciable distribution of extended realities and intended objectives. Instead of the disparities due to parallax leading to disorder and paralysis, accounting for the distinctive points of view behind the discrepancies can give rise to stereoscopic perspectives. In a community of interpretation and inquiry that has all these virtues, each individual try at objectivity is a venture that all the interpreters are nonetheless able to call their own.

Is this prospect a utopian vision? Perhaps it is exactly that. But it is the hope that inquiry discovers resting first and last within itself, quietly guiding every other aim and motive of inquiry.

Turning to the language of objective concerns, what can now be said about the compositional structures of the iconic sign relation \(M\!\) and the indexical sign relation \(N\!\)? In preparation for this topic, a few additional steps must be taken to continue formalizing the concept of an objective genre and to begin developing a calculus for composing objective motifs.

I recall the objective genre of properties and instances and re-introduce the symbols \(\lessdot\) and \(\gtrdot\) for the converse pair of dyadic relations that generate it. Reverting to the convention I employ in formal discussions of applying relational operators on the right, it is convenient to express the relative terms "property of \(x\!\)" and "instance of \(x\!\)" by means of a case inflection on \(x\!,\) that is, as "\(x\!\)’s property" and "\(x\!\)’s instance", respectively. Described in this way, \(\operatorname{OG} (\operatorname{Prop}, \operatorname{Inst}) = \langle \lessdot, \gtrdot \rangle,\) where:

\(\begin{array}{lllllll} x \lessdot & = & x \operatorname{'s~Property} & = & \operatorname{Property~of}\ x & = & \operatorname{Object~above}\ x. \\ x \gtrdot & = & x \operatorname{'s~Instance} & = & \operatorname{Instance~of}\ x & = & \operatorname{Object~below}\ x. \\ \end{array}\)

A symbol like \(^{\backprime\backprime} x \lessdot ^{\prime\prime}\) or \(^{\backprime\backprime} x \gtrdot ^{\prime\prime}\) is called a catenation, where \(^{\backprime\backprime} x ^{\prime\prime}\) is the catenand and \(^{\backprime\backprime} \lessdot ^{\prime\prime}\) or \(^{\backprime\backprime} \gtrdot ^{\prime\prime}\) is the catenator. Due to the fact that \(^{\backprime\backprime} \lessdot ^{\prime\prime}\) and \(^{\backprime\backprime} \gtrdot ^{\prime\prime}\) indicate dyadic relations, the significance of these so-called unsaturated catenations can be rationalized as follows:

\(\begin{array}{lllll} x \lessdot & = & x\ \operatorname{is~the~Instance~of~what?} & = & x \operatorname{'s~Property}. \\ x \gtrdot & = & x\ \operatorname{is~the~Property~of~what?} & = & x \operatorname{'s~Instance}. \\ \end{array}\)

In this fashion, the definitions of icons and indices can be reformulated:

\(\begin{array}{lllll} x \operatorname{'s~Icon} & = & x \operatorname{'s~Property's~Instance} & = & x \lessdot \gtrdot \\ x \operatorname{'s~Index} & = & x \operatorname{'s~Instance's~Property} & = & x \gtrdot \lessdot \\ \end{array}\)

According to the definitions of the homogeneous sign relations \(M\!\) and \(N,\!\) we have:

\(\begin{array}{lllll} x \operatorname{'s~Icon} & = & x \cdot M_{OS} \\ x \operatorname{'s~Index} & = & x \cdot N_{OS} \\ \end{array}\)

Equating the results of these equations yields the analysis of \(M\!\) and \(N\!\) as forms of composition within the genre of properties and instances:

\(\begin{array}{lllll} x \operatorname{'s~Icon} & = & x \cdot M_{OS} & = & x \lessdot \gtrdot \\ x \operatorname{'s~Index} & = & x \cdot N_{OS} & = & x \gtrdot \lessdot \\ \end{array}\)

On the assumption (to be examined more closely later) that any object \(x\!\) can be taken as a sign, the converse relations appear to be manifestly identical to the originals:

\(\begin{array}{llllll} \text{For Icons:} & x \operatorname{'s~Object} & = & x \cdot M_{SO} & = & x \lessdot \gtrdot \\ \text{For Indices:} & x \operatorname{'s~Object} & = & x \cdot N_{SO} & = & x \gtrdot \lessdot \\ \end{array}\)

Abstracting from the applications to an otiose \(x\!\) delivers the results:

\(\begin{array}{llllll} \text{For Icons:} & M_{OS} & = & M_{SO} & = & \lessdot \gtrdot \\ \text{For Indices:} & N_{OS} & = & N_{SO} & = & \gtrdot \lessdot \\ \end{array}\)

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.

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 \(\operatorname{OG} (\operatorname{Prop}, \operatorname{Inst})\) have served to rough out the basic shapes of the more refined analytic instruments to be developed in this subsection.

The notion of an objective motive or objective motif (OM) is intended to specialize or personalize the application of objective genres to take particular interpreters into account. For example, pursuing the pattern of \(\operatorname{OG} (\operatorname{Prop}, \operatorname{Inst}),\) a prospective OM of this genre does not merely tell about the properties and instances that objects can have in general, it recognizes a particular arrangement of objects and supplies them with its own ontology, giving "a local habitation and a name" to the bunch. What matters to an OM is a particular collection of objects (of thought) and a personal selection of links that go from each object (of thought) to higher and lower objects (of thought), all things being relative to a subjective ontology or a live hierarchy of thought, one that is currently known to and actively pursued by a designated interpreter of those thoughts.

The cautionary details interspersed at critical points in the preceding paragraph are intended to keep this inquiry vigilant against a constant danger of using ontological language, namely, the illusion that one can analyze the being of any real object merely by articulating the grammar of one's own thoughts, that is, simply by parsing signs in the mind. As always, it is best to regard OG's and OM's as filters and reticles, as transparent templates that are used to view a space, constituting the structures of objects only in one respect at a time, but never with any assurance of totality.

With these refinements, the use of dyadic projections to investigate sign relations can be combined with the perspective of objective motives to factor the facets or decompose the components of sign relations in a more systematic fashion. Given a homogeneous sign relation \(H\!\) of iconic or indexical type, the dyadic projections \(H_{OS}\!\) and \(H_{OI}\!\) can be analyzed as compound relations over the basis supplied by the \(G_j\!\) in \(G.\!\) As an application that is sufficiently important in its own right, the investigation of icons and indices continues to provide a useful testing ground for breaking in likely proposals of concepts and notation.

To pursue the analysis of icons and indices at the next stage of formalization, fix the OG of this discussion to have the type \(\langle \lessdot, \gtrdot \rangle\) and let each sign relation under discussion be articulated in terms of an objective motif that tells what objects and signs, plus what mediating linkages through properties and instances, are assumed to be recognized by its interpreter.

Let \(X\!\) collect the objects of thought that fall within a particular OM, and let \(X\!\) include the whole world of a sign relation plus everything needed to support and contain it. That is, \(X\!\) collects all the types of things that go into a sign relation, \(O \cup S \cup I = W \subseteq X,\) plus whatever else in the way of distinct object qualities and object exemplars is discovered or established to be generated out of this basis by the relations of the OM.

In order to keep this \(X\!\) simple enough to contemplate on a single pass but still make it deep enough to cover the issues of interest at present, I limit \(X\!\) to having just three disjoint layers of things to worry about:

The top layer is the relevant class of object qualities:

\(Q = X_0 \lessdot = W \lessdot\)

The middle layer is the initial collection of objects and signs:

\(X_0 = W\!\)

The bottom layer is a suitable set of object exemplars:

\(E = X_0 \gtrdot = W \gtrdot\)

Recall the reading of the staging relations:

\(h : x \lessdot m\)	\(\Leftrightarrow\)	\(h\ \operatorname{regards}\ x\ \operatorname{as~an~instance~of}\ m.\)
\(h : m \gtrdot y\)	\(\Leftrightarrow\)	\(h\ \operatorname{regards}\ m\ \operatorname{as~a~property~of}\ y.\)
\(h : x \gtrdot n\)	\(\Leftrightarrow\)	\(h\ \operatorname{regards}\ x\ \operatorname{as~a~property~of}\ n.\)
\(h : n \lessdot y\)	\(\Leftrightarrow\)	\(h\ \operatorname{regards}\ n\ \operatorname{as~an~instance~of}\ y.\)

Express the analysis of icons and indices as follows:

\(\text{For Icons:}\!\)   \(M_{OS}\!\) \(\colon\!\) \(x \lessdot \gtrdot x \operatorname{'s~Sign}.\) \(\text{For Indices:}\!\)   \(N_{OS}\!\) \(\colon\!\) \(x \gtrdot \lessdot x \operatorname{'s~Sign}.\)

Let \(j\!\) and \(k\!\) be hypothetical interpreters that do the jobs of \(M\!\) and \(N,\!\) respectively:

\(\begin{array}{llllll} \text{For Icons:} & x \operatorname{'s~Sign} & = & x \cdot M_{OS} & = & x \lessdot_j \gtrdot_j \\ \text{For Indices:} & x \operatorname{'s~Sign} & = & x \cdot N_{OS} & = & x \gtrdot_k \lessdot_k \\ \end{array}\)

Factor out the names of the interpreters \(j\!\) and \(k\!\) to serve as identifiers of objective motifs:

\(\text{For Icons:}\!\)   \(j\!\) \(\colon\!\) \(x \lessdot \gtrdot x \operatorname{'s~Sign}.\) \(\text{For Indices:}\!\)   \(k\!\) \(\colon\!\) \(x \gtrdot \lessdot x \operatorname{'s~Sign}.\)

Finally, the constant motif names \(j\!\) and \(k\!\) can be collected to one side of a composition or distributed to its individual links:

\(\begin{array}{llllll} j : x \lessdot \gtrdot y & \Leftrightarrow & j : x \lessdot m & \operatorname{and} & j : m \gtrdot y & (\exists m \in Q). \\ k : x \gtrdot \lessdot y & \Leftrightarrow & k : x \gtrdot n & \operatorname{and} & k : n \lessdot y & (\exists n \in E). \\ \end{array}\)

These statements can be read to say:

• \(j\!\) thinks \(x\!\) an icon of \(y\!\) if and only if there is an \(m\!\) such that \(j\!\) thinks \(x\!\) an instance of \(m\!\) and \(j\!\) thinks \(m\!\) a property of \(y.\!\).

• \(k\!\) thinks \(x\!\) an index of \(y\!\) if and only if there is an \(n\!\) such that \(k\!\) thinks \(x\!\) a property of \(n\!\) and \(k\!\) thinks \(n\!\) an instance of \(y.\!\).

Readers who object to the anthropomorphism or the approximation of these statements can replace every occurrence of the verb thinks with the phrase interprets … as, or even the circumlocution acts in every formally relevant way as if, changing what must be changed elsewhere. For the moment, I am not concerned with the exact order of reflective sensitivity that goes into these interpretive linkages, but only with a rough outline of the pragmatic equivalence classes that are afforded by the potential conduct of their agents.

In the discussion of the dialogue between \(A\!\) and \(B\!\) it was allowed that the same signs \(^{\backprime\backprime} A ^{\prime\prime}\) and \(^{\backprime\backprime} B ^{\prime\prime}\) 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 \(^{\backprime\backprime} j ^{\prime\prime}\) and \(^{\backprime\backprime} k ^{\prime\prime}\) 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.

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.

Integrating divergent IF's and reconciling their objectifications is, generally speaking, a very difficult maneuver to carry out successfully. Two factors that contribute to the near intractability of this task can be described and addressed as follows.

1. The trouble is partly due to the ossified taxonomies and obligatory tactics that come through time and training to inhabit the conceptual landscapes of agents, especially if they have spent the majority of their time operating according to a single IF. The IF informs their activity in ways they no longer have to think about, and thus rarely find a reason to modify. But it also inhibits their interpretive and practical conduct to the customary ways of seeing and doing things that are granted by that framework, and it restricts them to the forms of intuition that are suggested and sanctioned by the operative IF. Without critical reflection, or a mechanism to make amendments to its own constitution, an IF tends to operate behind the scenes of observation in such a way as to obliterate any inkling of flexibility in thought or practice and to obstruct every hint or threat (so perceived) of conceptual revision.
2. Apparently it is so much easier to devise techniques for taking things apart than it is to find ways of putting them back together that there seem to be only a few heuristic strategies of general application that are available to guide the work of integration. A few of the tools and materials needed for these constructions have been illustrated in concrete form throughout the presentation of examples in this section. An overall survey of their principles can be summed up as follows.

• One integration heuristic is the lattice metaphor, also called the partial order or common denominator paradigm. When IF's can be objectified as OF's that are organized according to the principles of suitable orderings, then it is often possible to lift or extend these order properties to the space of frameworks themselves, and thereby to enable construction of the desired kinds of integrative frameworks as upper and lower bounds in the appropriate ordering.

• Another integration heuristic is the mosaic metaphor, also called the stereoscopic or inverse projection paradigm. This technique has been illustrated especially well by the methods used throughout this section to analyze the three-dimensional structures of sign relations. In fact, the picture of any sign relation offers a paradigm in microcosm for the macroscopic work of integration, showing how reductive aspects of structure can be projected from a shared but irreducible reality. The extent to which the full-bodied structure of a triadic sign relation can be reconstructed from its dyadic projections, although a limited extent in general, presents a near perfect epitome of the larger task in this situation, namely, to find an integrated framework that embodies the diverse facets of reality severally observed from inside the individual frameworks. Acting as gnomonic recipes for the higher order processes they limn and delimit, sign relations keep before the mind the ways in which a higher dimensional structure determines its fragmentary aspects but is not in general determined by them.

To express the nature of this integration task in logical terms, it combines elements of both proof theory and model theory, interweaving: (1) A phase that develops theories about the symbolic competence or knowledge of intelligent agents, using abstract formal systems to represent the theories and phenomenological data to constrain them; (2) A phase that seeks concrete models of these theories, looking to the kinds of mathematical structure that have a dynamic or system-theoretic interpretation, and compiling the constraints that a recursive conceptual analysis imposes on the ultimate elements of their construction.

The set of sign relations {A, B} is an example of an extremely simple formal system, encapsulating aspects of the symbolic competence and the pragmatic performance that might be exhibited by potentially intelligent interpretive agents, however abstractly and partially given at this stage of description. The symbols of a formal system like {A, B} can be held subject to abstract constraints, having their meanings in relation to each other determined by definitions and axioms (for example, the laws defining an equivalence relation), making it possible to manipulate the resulting information by means of the inference rules in a proof system. This illustrates the proof-theoretic aspect of a symbol system.

Suppose that a formal system like {A, B} is initially approached from a theoretical direction, in other words, by listing the abstract properties one thinks it ought to have. Then the existence of an extensional model that satisfies these constraints, as exhibited by the sign relation tables, demonstrates that one's theoretical description is logically consistent, even if the models that first come to mind are still a bit too abstractly symbolic and do not have all the dynamic concreteness that is demanded of system-theoretic interpretations. This amounts to the other side of the ledger, the model-theoretic aspect of a symbol system, at least insofar as the present account has dealt with it.

More is required of the modeler, however, in order to find the desired kinds of system-theoretic models (for example, state transition systems), and this brings the search for realizations of formal systems down to the toughest part of the exercise. Some of the problems that emerge were highlighted in the example of A and B. Although it is ordinarily possible to construct state transition systems in which the states of interpreters correspond relatively directly to the acceptations of the primitive signs given, the conflict of interpretations that develops between different interpreters from these prima facie implementations is a sign that there is something superficial about this approach.

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.

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?"

Pursuant to this particular way of setting out on the long-term quest, a more immediate goal of the current project is to understand the action of full-fledged symbols, insofar as they conduct themselves through the media of minds and quasi-minds. At this very point the quest is joined by the pragmatic investigations of signs and inquiry, which share this interest in chasing down symbols to their precursive lairs.

In the pragmatic theory of signs a symbol is a strangely insistent yet curiously indirect type of sign, one whose accordance with its object depends sheerly on the real possibility that it will be so interpreted. Taking on the nature of a bet, a symbol's prospective value trades on nothing more than the chance of acquiring the desired interpretant, and thus it can capitalize on the simple fact that what it proposes is not impossible. In this way it is possible to see that a formal principle is involved in the success of symbols. The elementary conceivability of a particular sign relation, the pure circumstance that renders it logically or mathematically possible, means that the formal constraint it places on its domains is always really and potentially there, awaiting its discovery and exploitation for the purposes of representation and communication.

In this question about the symbol's capacity for meaning, then, is found another contact between the theory of signs and the logic of inquiry. As C.S. Peirce expressed it:

Now, I ask, how is it that anything can be done with a symbol, without reflecting upon the conception, much less imagining the object that belongs to it? It is simply because the symbol has acquired a nature, which may be described thus, that when it is brought before the mind certain principles of its use — whether reflected on or not — by association immediately regulate the action of the mind; and these may be regarded as laws of the symbol itself which it cannot as a symbol transgress.

(Peirce, CE 1, 173).

Inference in general obviously supposes symbolization; and all symbolization is inference. For every symbol as we have seen contains information. And … all kinds of information involve inference. Inference, then, is symbolization. They are the same notions. Now we have already analyzed the notion of a symbol, and we have found that it depends upon the possibility of representations acquiring a nature, that is to say an immediate representative power. This principle is therefore the ground of inference in general.

(Peirce, CE 1, 280).

A symbol which has connotation and denotation contains information. Whatever symbol contains information contains more connotation than is necessary to limit its possible denotation to those things which it may denote. That is, every symbol contains more than is sufficient for a principle of selection.

(Peirce, CE 1, 282).

The information of a term is the measure of its superfluous comprehension. That is to say that the proper office of the comprehension is to determine the extension of the term. …           Every addition to the comprehension of a term, lessens its extension up to a certain point, after that further additions increase the information instead. …           And therefore as every term must have information, every term has superfluous comprehension. And, hence, whenever we make a symbol to express any thing or any attribute we cannot make it so empty that it shall have no superfluous comprehension.           I am going, next, to show that inference is symbolization and that the puzzle of the validity of scientific inference lies merely in this superfluous comprehension and is therefore entirely removed by a consideration of the laws of information.

(Peirce, CE 1, 467).

A full explanation of these statements, linking scientific inference, symbolization, and information together in such an integral fashion, would require an excursion into the pragmatic theory of information that Peirce was already presenting in lectures at Harvard as early as 1865. For now, let it suffice to say that this anticipation of the information concept, fully recognizing the reality of its dimension, would not sound too remote from the varieties of law abiding constraint exploitation that have become increasingly familiar since the dawn of cybernetics.

But more than this, Peirce's notion of information supplies an array of missing links that joins together in one scheme the logical roles of terms, propositions, and arguments, the semantic functions of denotation and connotation, and the practical methodology needed to address and measure the quantitative dimensions of information. This is precisely the kind of linkage that I need in this project to integrate the dynamic and symbolic aspects of inquiry.

Not by sheer coincidence, the task of understanding symbolic action, working up through icons and indices to the point of tackling symbols, is also one of the ultimate aims that the interpretive and objective frameworks being proposed here are intended to subserve.

An OF is a convenient stage for those works that have progressed far enough to make use of it, but in times of flux it must be remembered that an OF is only a hypostatic projection, that is, the virtual image, reified concept, or phantom limb of the IF that tentatively extends it.

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.

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.

The interpreter is an agency depicted over and above the sign relation, but in a very real sense it is simply identical with the whole of it. And so one is led to examine the relationship between the interpreter and the interpretant, the element falling within the sign relation to which the sign in actuality tends. The catch is that the whole of the intended sign relation is seldom known from the beginning of inquiry, and so the aimed for interpretant is often just as unknown as the rest.

These eventualities call for the elaboration of interpretive and objective frameworks in which not just the specious but the speculative purpose of a sign can be contemplated, permitting extensions of the initial data, through error and retrial, to satisfy emergent and recurring questions.

At last, even with the needed frameworks only partly shored up, I can finally ravel up and tighten one thread of this rambling investigation. All this time, steadily rising to answer the challenge about the identity of the interpreter, Who's there?, and the role of the interpretant, Stand and unfold yourself, has been the ready and abiding state of a certain system of interpretation, developing its character and gradually evolving its meaning through a series of imputations and extensions. Namely, the MOI (the SOI experienced as an object) can answer for the interpreter, to whatever extent that conduct can be formalized, and the IM (the SOI experienced in action, in statu nascendi) can serve as a proxy for the momentary thrust of interpretive dynamics, to whatever degree that process can be explicated.

To put a finer point on this result I can do no better at this stage of discussion than to recount the "metaphorical argument" that Peirce often used to illustrate the same conclusion.

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. 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? 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. … 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.

(Peirce, CE 1, 466–467).

It will take a while to develop the wealth of information that a suitably perspicacious and persistent IF would find implicit in this unassuming homily. The main innovations that this project can hope to add to the story are as follows:

1. To prescribe a context of effective systems theory (C'EST), one that can provide for the computational formalization of each intuitively given interpreter as a determinate model of interpretation (MOI). An appropriate set of concepts and methods would deal with the generic constitutions of interpreters, converting paraphrastic and periphrastic descriptions of their interpretive practice into relatively complete and concrete specifications of sign relations.
2. 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.

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.

It is always possible that the dual aspects of transformation and information are conceptually complementary and even non-orientable. That is, there may be no way to arrange our mental apparatus to grasp both sides at the same time, and the whole appearance that there are two sides may be an illusion of overly local and myopic perspectives. However, none of this should be taken for granted without proof.

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.

1.3.5. Discussion of Formalization : Specific Objects

"Knowledge" is a referring back: in its essence a regressus in infinitum. That which comes to a standstill (at a supposed causa prima, at something unconditioned, etc.) is laziness, weariness —

— Nietzsche, The Will to Power, [Nie, S575, 309]

With this preamble, I return to develop my own account of formalization, with special attention to the kind of step that leads from the inchoate chaos of casual discourse to a well-founded discussion of formal models. A formalization step, of the incipient kind being considered here, has the peculiar property that one can say with some definiteness where it ends, since it leads precisely to a well-defined formal model, but not with any definiteness where it begins. Any attempt to trace the steps of formalization backward toward their ultimate beginnings can lead to an interminable multiplicity of open-ended explorations. In view of these circumstances, let me limit my attention to the frame of the present inquiry and try to sum up what brings me to this point.

I ask whether it is possible to reason about inquiry in a way that leads to a productive end. I pose this question as an inquiry into inquiry, and I use the formula \(y_0 = y \cdot y\) to express the relationship between the present inquiry, \(y_0,\!\) and a generic inquiry, \(y.\!\) Then I propose a couple of components of inquiry, expressed in the form \(y \succ \{ d, f \},\) that appear to be worth investigating. Applying these components to each other, as must be done in the present inquiry, I am led to the current discussion of formalization, \(y_0 = y \cdot y \succ f \cdot d.\)

There is already much to question here. At least, so many repetitions of the same mysterious formula are bound to lead the reader to question its meaning.

1. The notion of a "generic inquiry" is ambiguous. Its meaning in practice depends on whether this descriptive term is interpreted literally or merely as a figure of speech. In the literal case, the name \(^{\backprime\backprime} y ^{\prime\prime}\) denotes a particular inquiry, \(y \in Y,\!\) one that is assumed to be equipotential or prototypical in a yet to be specified way. In the figurative case, the name \(^{\backprime\backprime} y ^{\prime\prime}\) is simply a variable that ranges over a collection \(Y\!\) of nominally conceivable inquiries.
2. On first reading, the recipe \(y_0 = y \cdot y\) 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.
3. Given the formula \(y_0 = y \cdot y,\) the subordination \(y \succ \{ d, f \},\) and the successive containments \(F \subseteq M \subseteq D,\) the \(y\!\) that looks into \(y\!\) is not restricted to examining \(y \operatorname{'s}\) immediate subordinates, \(d\!\) and \(f,\!\) but it can investigate any feature of \(y \operatorname{'s}\) overall context, whether objective, syntactic, interpretive, whether definitive or incidental, and finally it can question any supporting claim of the discussion. Moreover, the question \(y\!\) is not limited to the particular claims that are being made here, but applies to the abstract relations and the general notions that are invoked in making them. Among the many kinds of inquiry that suggest themselves, there are the following possibilities:
1. Inquiry into propositions about application and equality.
   Start with the formula \(y_0 = y \cdot y\) itself.
2. Inquiry into application ( \(\cdot\) ).
3. Inquiry into equality (\(=\!\)).
4. Inquiry into indices (for example, the \(0\) in \(y_0\!\)).
5. Inquiry into terms, namely, constants and variables.
   What are the functions of \(^{\backprime\backprime} y ^{\prime\prime}\) and \(^{\backprime\backprime} y_0 ^{\prime\prime}\) in this respect?
6. Inquiry into decomposition or subordination (\(\succ\)).
7. Inquiry into containment or inclusion. In particular, examine the claim that \(F \subseteq M \subseteq D\) which conditions 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.

If inquiry begins in doubt, then inquiry into inquiry begins in doubt about doubt. All things considered, the formula \(y_0 = y \cdot y\) has to be taken as the first attempt at a description of the problem, a hypothesis about the nature of inquiry, or an image that is tossed out by way of getting an initial fix on the object in question. Everything in this account so far, and everything else that I am likely to add, can only be reckoned as hypothesis, whose accuracy, pertinence, and usefulness can be tested, judged, and redeemed only after the fact of proposing it and after the facts to which it refers have themselves been gathered up.

A number of problems present themselves due to the context in which the present inquiry is aimed to present itself. The hypothesis that suggests itself to one person, as worth exploring at a particular time, does not always present itself to another person as worth exploring at the same time, or even necessarily to the same person at another time. In a community of inquiry that extends beyond an isolated person and in a process of inquiry that extends beyond a singular moment in time, it is therefore necessary to consider the nature of the communication process that the discussion of inquiry in general and the discussion of formalization in particular need to invoke for their ultimate utility.

Solitude and solipsism are no solution to the problems of community and communication, since even an isolated individual, if ever there was, is, or comes to be such a thing, has to maintain the lines of communication that are required to integrate past, present, and prospective selves — in other words, translating everything into present terms, the parts of one's actually present self that involve actual experiences and present observations, present expectations as reflective of actual memories, and present intentions as reflective of actual hopes. So the dialogue that one holds with oneself is every bit as problematic as the dialogue that one enters with others. Others only surprise one in other ways than one ordinarily surprises oneself.

I recognize inquiry as beginning with a surprising phenomenon or a problematic situation, more briefly described as a surprise or a problem, respectively. These are the types of moments that try our souls, the instances of events that instigate inquiry as an effort to achieve their own resolution. Surprises and problems are experienced as afflicted with an irritating uncertainty or a compelling difficulty, one that calls for a response on the part of the agent in question:

1. 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.
2. 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", a compact term that admits of expansion as a debt, a difference, a difficulty, a discrepancy, a dispersion, a distribution, a doubt, a duplicity, or a 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 deed, or in fact.

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.

1.3.5.1. The Will to Form

The power of form, the will to give form to oneself. "Happiness" admitted as a goal. Much strength and energy behind the emphasis on forms. The delight in looking at a life that seems so easy. — To the French, the Greeks looked like children.

— Nietzsche, The Will to Power, [Nie, S94, 58]

Let me see if can summarize as quickly as possible the problem that I see before me. Each time that I try to express my experience, to lend it a form that others can recognize, to put it in a shape that I myself can later recall, or to store it in a state that allows me the chance of its re-experience, I generate an image of the way things are, or at least a description of how things seem to me. I call this process "reflection", since it fabricates an image in a medium of signs that reflects an aspect of experience. Often this experience can be said to be "of" — what? — something that exists or persists at least partially outside the immediate experience, some action, event, or object that is imagined to inform the present experience, or perhaps some conduct of one's own that obtrudes for a moment into the world of others and meets with a reaction there. In all of these cases, where the experience is everted to refer to an object and becomes the attribute of something with an external aspect, something that is thus supposed to be a prior cause of the experience, the reflection on experience doubles as a reflection on that conduct, performance, or transaction that the experience is an experience "of". In short, if the experience has an eversion that makes it of an object, then its reflection is again a reflection that is also of this object.

Just at the point where one threatens to become 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. One still has the problem: 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.

1.3.5.2. The Forms of Reasoning

The most valuable insights are arrived at last; but the most valuable insights are methods.

— Nietzsche, The Will to Power, [Nie, S469, 261]

A certain arbitrariness has to be faced in the terms that one uses to talk about reasoning, to split it up into different parts and to sort it out into different types. It is like the arbitrary choice that one makes in assigning the midpoint of an interval to the subintervals on its sides. In setting out the forms of a nomenclature, in fitting the schemes of my terminology to the territory that it disturbs in the process of mapping, I cannot avoid making arbitrary choices, but I can aim for a strategy that is flexible enough to recognize its own alternatives and to accommodate the other options that lie within their scope. If I make the mark of deduction the fact that it reduces the number of terms, as it moves from the grounds to the end of an argument, then I am due to devise a name for the process that augments the number of terms, and thus prepares the grounds for any account of experience.

What name hints at the many ways that signs arise in regard to things? What name covers the manifest ways that a map takes over its territory? What name fits this naming of names, these proceedings that inaugurate a sign in the first place, that duly install it on the office of a term? What name suits all the actions of addition, annexation, incursion, and invention that instigate the initial bearing of signs on an object domain? In the interests of a "maximal analytic precision", it is fitting that I should try to sharpen this notion to the point where it applies purely to a simple act, that of entering a new term on the lists, in effect, of enlisting a new term to the ongoing account of experience. Thus, let me style this process as "adduction" or "production", in spite of the fact that the aim of precision is partially blunted by the circumstance that these words have well-worn uses in other contexts. In this way, I can isolate to some degree the singular step of adding a term, leaving it to a later point to distinguish the role that it plays in an argument.

As it stands, the words "adduction" and "production" could apply to the arbitrary addition of terms to a discussion, whether or not these terms participate in valid forms of argument or contribute to their mediation. Although there are a number of auxiliary terms, like "factorization", "mediation", or "resolution", that can help to pin down these meanings, it is also useful to have a word that can convey the exact sense meant. Therefore, I coin the term "obduction" to suggest the type of reasoning process that is opposite or converse to deduction and that introduces a middle term "in the way" as it passes from a subject to a predicate. Consider the adjunction to one's vocabulary that is comprised of these three words: "adduction", "production", "obduction". In particular, how do they appear in the light of their mutual applications to each other and especially with respect to their own reflexivities? Notice that the terms "adduction" and "production" apply to the ways that all three terms enter this general discussion, but that "obduction" applies only to their introduction only in specific contexts of argument.

Another dimension of variation that needs to be noted among these different types of processes is their status with regard to determimism. 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.

1.3.5.3. A Fork in the Road

On "logical semblance" — The concepts "individual" and "species" equally false and merely apparent. "Species" expresses only the fact that an abundance of similar creatures appear at the same time and that the tempo of their further growth and change is for a long time slowed down, so actual small continuations and increases are not very much noticed (— a phase of evolution in which the evolution is not visible, so an equilibrium seems to have been attained, making possible the false notion that a goal has been attained — and that evolution has a goal —).

— Nietzsche, The Will to Power, [Nie, S521, 282]

It is worth trying to discover, as I currently am, how many properties of inquiry can be derived from the simple fact that it needs to be able to apply to itself. I find three main ways to approach this issue, the problem of inquiry's self-application, or the question of its reflexivity:

1. One way attempts to continue the derivation in the manner of a necessary deduction, perhaps by reasoning in the following vein: If self-application is a property of inquiry, then it is sensible to inquire into the concept of application that makes this conceivable, and not just conceivable, but potentially fruitful.
2. Another way breaks off the attempt at a deductive development and puts forth a full-scale model of inquiry, one that has enough plausibility to be probated in the court of experience and enough specificity to be tested in the context of self-application.
3. The last way is a bit ambivalent in its indications, seeking as it does both the original unity and the ultimate synthesis at one and the same time. Perhaps it goes toward reversing the steps that lead up to this juncture, marking it down as an impasse, chalking it up as a learning experience, or admitting the failure of the imagined distinction to make a difference in reality. Whether this form of egress is interpreted as a backtracking correction or as a leaping forward to the next level of integration, it serves to erase the distinction between demonstration and exploration.

Without a clear sense of how many properties of inquiry are necessary consequences of its self-application and how many are merely accessory to it, or even whether some contradiction still lies lurking within the notion of reflexivity, I have no choice but to follow all three lines of inquiry wherever they lead, keeping an eye out for the synchronicities, the constructive collusions and the destructive collisions that may happen to occur among them.

The fictions that one introduces to shore up a shaky account of experience can often be discharged at a later stage of development, gradually replacing them with primitive elements of less and less dubious characters. Hypostases and hypotheses, the creative terms and the inventive propositions that one invokes 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 come to be cancelled 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.

1.3.5.4. A Forged Bond

The form counts as something enduring and therefore more valuable; but the form has merely been invented by us; and however often "the same form is attained", it does not mean that it is the same form — what appears is always something new, and it is only we, who are always comparing, who include the new, to the extent that it is similar to the old, in the unity of the "form". As if a type should be attained and, as it were, was intended by and inherent in the process of formation.

— Nietzsche, The Will to Power, [Nie, S521, 282]

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.

1.3.5.5. A Formal Account

Form, species, law, idea, purpose — in all these cases the same error is made of giving a false reality to a fiction, as if events were in some way obedient to something — an artificial distinction is made in respect of events between that which acts and that toward which the act is directed (but this "which" and this "toward" are only posited in obedience to our metaphysical-logical dogmatism: they are not "facts").

— Nietzsche, The Will to Power, [Nie, S521, 282]

In this section I consider the step of formalization that takes discussion from a large scale informal inquiry to a well-defined formal inquiry, establishing a relation between the implicit context and the explicit text.

In this project, formalization is used to produce formal models that represent relevant features of a phenomenon or process of interest. Thus, the formal model is what constitutes the image of formalization.

The role of formalization splits into two different cases depending on the intended use of the formal model. When the phenomenon of interest is external to the agent that is carrying out the formalization, then the model of that phenomenon can be developed without doing significant reflection on the formalization process itself. This is usually a more straightforward operation, since it avails itself of automatic competencies that are not themselves in question. However, …

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 to ever be fruitful and the scope of whose questions may not be focused well enough to ever 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.

1.3.5.6. Analogs, Icons, Models, Surrogates

One should not understand this compulsion to construct concepts, species, forms, purposes, laws ("a world of identical cases") as if they enabled us to fix the real world; but as a compulsion to arrange a world for ourselves in which our existence is made possible: — we thereby create a world which is calculable, simplified, comprehensible, etc., for us.

— Nietzsche, The Will to Power. [Nie, S521, 282]

This project makes pivotal use of certain formal models to represent the conceived structure in a phenomenon of interest. For my purposes, the phenomenon of interest is typically a process of interpretation (POI) or a process of inquiry (POI), two nominal species of process that will turn out to evolve from different points of view on the same form of conduct.

Commonly, a process of interest presents itself as the trajectory that an agent describes through an extended space of configurations. The work of conceptualization and formalization is to represent this process as a conceptual object in terms of a formal model. Depending on the point of view that is taken from moment to moment in this work, the formal model of interest may be cast either as a model of interpretation (MOI) or as a model of inquiry (MOI). As might be guessed, it will turn out that both descriptions refer essentially to the same subject, but this will take some development to become clear.

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.

1.3.5.7. Steps and Tests of Formalization

This same compulsion exists in the sense activities that support reason — by simplification, coarsening, emphasizing, and elaborating, upon which all "recognition", all ability to make oneself intelligible rests. Our needs have made our senses so precise that the "same apparent world" always reappears and has thus acquired the semblance of reality.

— Nietzsche, The Will to Power, [Nie, S521, 282]

A step of formalization moves the active focus of discussion from the presentational object or source domain to the representational object or target domain that constitutes the relevant MOI. If the structure in the source context is already formalized then the step of formalization can itself be formalized in an especially elegant and satisfying way as a structure-preserving map, homomorphism, or arrow of category theory.

The test of a formalization being complete is that a computer could in principle carry out the steps of the process exactly as represented in the formal model or image. It needs to be appreciated that this is a criterion of sufficiency to formal understanding and not of necessity relevant to material re-creation. The ordinary agents of informal discussion who address the task of formalization do not disappear in the process of completing it, since it is precisely for their understanding that the step is undertaken. Only if the phenomenon at issue were by its very nature solely a matter of form could its formal analogue constitute an authentic reproduction. But this potential consideration is far from the ordinary case I need to discuss at present.

In ordinary discussion, agents depend on the likely interpretations of others to give their common notions and 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.

1.3.5.8. The Referee

Our subjective compulsion to believe in logic only reveals that, long before logic itself entered our consciousness, we did nothing but introduce its postulates into events: now we discover them in events — we can no longer do otherwise — and imagine that this compulsion guarantees something connected with "truth".

— Nietzsche, The Will to Power, [Nie, S521, 282–283]

In a formal inquiry of the sort projected here, the less the discussants need to depend on the compliance of understanding interpreters the more they will necessarily understand at the end of the formalization. It might be thought that the ultimate zero of understanding expected on the part of the interpreter would correspond to the ultimate height of understanding demanded on the part of the formalizer, but this neglects the negative potential of misunderstanding, the sheer perversity of interpretation that true human creativity can bring to bear on any text. 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, something that is known to begin from a relatively neutral initial condition.

1.3.5.9. Partial Formalizations

It is we who created the "thing", the "identical thing", subject, attribute, activity, object, substance, form, after we had long pursued the process of making identical, coarse and simple. The world seems logical to us because we have made it logical.

— Nietzsche, The Will to Power, [Nie, S521, 283]

In many discussions the source context remains unformalized in itself, taking form only according to the image it receives in this or that individual MOI. In this case, the step of formalization is not a total function but 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, by ignoring elements of the source domain and collapsing material distinctions in an irregular fashion.

1.3.5.10. A Formal Utility

Ultimate solution. — We believe in reason: this, however, is the philosophy of gray concepts. Language depends on the most naive prejudices.

— Nietzsche, The Will to Power, [Nie, S522, 283]

The usefulness of the MOI is that it provides discussion with a compact image of the whole source domain.

The use of formalization as a pretermination criterion. One of the primary benefits of the requirement of formalization is to serve as a pretermination criterion.

A benefit of adopting the objective of formalization is that it equips discussion with a pretermination criterion.

The purpose of formalization is to identify a simpler version or to fashion a simpler image of a difficult inquiry, one that is well-defined and simple enough to assure its termination in a finite interval of space-time.

In formalization one tries to extract a simpler image of the larger inquiry, a context of participatory action that one is too embroiled in carrying out step by step to see as a whole. In the context of the recursive inquiry I have outlined, the step of formalization is intended to bring discussion appreciably closer to a solid base for the operational definition of inquiry.

1.3.5.11. A Formal Aesthetic

Now we read disharmonies and problems into things because we think only in the form of language — and thus believe in the "eternal truth" of "reason" (e.g., subject, attribute, etc.)

— Nietzsche, The Will to Power, [Nie, S522, 283]

Recognizing that the Latin word forma means not just form but also beauty supplies a clue that not all formal models are equally valuable for a purpose of interest. There is a certain quality of formal elegance, or select character, that is essential to the practical utility of the model.

The virtue of a good formal model is to provide discussion with a fitting image of the whole phenomenon of interest. The aim of formalization is to extract from an informal discussion or locate within a broader inquiry a clearer and simpler image of the whole activity. If the formalized precis or image is unusually apt it might be prized as a recapitulation or gnomon and said to capture the essence, the gist, of the nub of the whole affair.

A pragmatic qualification of this virtue requires that the image be formed quickly enough to take decisive action on. So the quality of being a result often takes precedence over the quality of the result. A definite result, however partial, is frequently reckoned to be better than having to wait for a complete picture that may never develop.

But an overly narrow or premature formalization, where the quality of the original phenomenon is too severely reduced in the formalized image, may result in destroying all interest in the result that does result.

1.3.5.12. A Formal Apology

We cease to think when we refuse to do so under the constraint of language; we barely reach the doubt that sees this limitation as a limitation.

— Nietzsche, The Will to Power, [Nie, S522, 283]

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 knowledge of their form alone, without having to know everything about their material content down to the depths that matter can go.

1.3.5.13. A Formal Suspicion

Rational thought is interpretation according to a scheme that we cannot throw off.

— Nietzsche, The Will to Power, [Nie, S522, 283]

I hope that the reader has arrived by now at an independent suspicion that the process of formalization is a microcosm nearly as complex as the whole subject of inquiry itself. Indeed, the initial formulation of a problem is tantamount to a mode of "representational inquiry". In many ways this first effort, that stirs from the torpor of ineffable unease to seek any sort of unity in the manifold of fragmented impressions, is the most difficult, subtle, and crucial kind of inquiry. It begins in doubt about even so much as a fair way to represent the problematic situation, but its result can predestine whether subsequent inquiry has any hope of success. There is very little in this brand of formal engagement and participatory representation that resembles the simple and disinterested act of holding a mirror, flat and featureless, up to nature.

If formalization really is a form of inquiry in itself, then its formulations have deductive consequences that can be tested. In other words, formal models have logical effects that reflect on their fitness to qualify as representations, and these effects can cause them to be rejected merely on the grounds of being a defective picture or a misleading conception of the source phenomenon. Therefore, it should be appreciated that software tailored to this task will probably need to spend more time in the alterations of backtracking than it will have occasion to trot out parades of ready-to-wear models.

Impelled by the mass of assembled clues from restarts and refits to the gathering form of a coherent direction, the inkling may have gradually accumulated in the reader that something of the same description has been treated in the pragmatic theory of inquiry under the heading of abductive reasoning. This is distinguished from inductive reasoning, that goes from the particular to the general, in that abductive reasoning must work from a mixed collection of generals and particulars toward a middle term, a formal intermediary that is more specific than the vague allusions gathered about its subject and more generic than the elusive instances fashioned to illustrate its prospective predicates.

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.

1.3.5.14. The Double Aspect of Concepts

Nothing is more erroneous than to make of psychical and physical phenomena the two faces, the two revelations of one and the same substance. Nothing is explained thereby: the concept "substance" is perfectly useless as an explanation. Consciousness in a subsidiary role, almost indifferent, superfluous, perhaps destined to vanish and give way to a perfect automatism —

— Nietzsche, The Will to Power, [Nie, S523, 283]

This work is a particular inquiry into the nature of inquiry in general. As a consequence, every conceptual construct that appears in it will take on a double aspect.

To illustrate, let take the concept of a "sign relation" as an example and let me use it to speak about my own agency in this inquiry. All I need to say about a sign relation at this point is that it is a three-place relation, and therefore can be imagined as a relational data-base with three columns, in this case naming the "object", the "sign", and the "interpretant" of the relation at each moment in time of the corresponding "sign process". At any given moment of this inquiry I will be participating in a certain sign relation that constitutes the informal context of my activity, the full nature of which I can barely hope to conceptualize in explicitly formal terms. At times, the object of this informal sign relation will itself be a sign relation, typically one that is already formalized or one that I have a better hope of formalizing, but it could conceivably be the original sign relation with which I began.

In such cases, when the object of a sign relation is also a sign relation, the general concept of a sign relation takes on a double duty:

1. The less formalized sign relation is used to mediate the inquiry. As a conceptual construct, it is not yet fully conceived or constructed at the moments of inquiry being considered. Perhaps it is better to regard it as a "concept under construction". Employed as a contextual apparatus, this sign relation serves an instrumental role in the study or construal of its objective sign relation.
2. The more formalized sign relation is mentioned as a substantive object to be contemplated and manipulated by the inquiry. As a conceptual construct, it exemplifies the role intended for it best if it is already as completely formalized as possible. It is being engaged as a substantive object of inquiry.

I have given this project a reflective or a recursive cast, describing it as inquiry into inquiry, and one of the consequences of this is that every concept employed in the work will take on a double aspect, divided role, or dual purpose. At any moment, the object inquiry of the moment is aimed to take on a formal definition, whereas the active inquiry …

1.3.5.15. A Formal Permission

If there are to be synthetic a priori judgments, then reason must be in a position to make connections: connection is a form. Reason must possess the capacity of giving form.

— Nietzsche, The Will to Power, [Nie, S530, 288]

1.3.5.16. A Formal Invention

Before there is "thought" (gedacht) there must have been "invention" (gedichtet); the construction of identical cases, of the appearance of sameness, is more primitive than the knowledge of sameness.

Nietzsche, The Will to Power, [Nie, S544, 293]

1.3.6. Recursion in Perpetuity

Will to truth is a making firm, a making true and durable, an abolition of the false character of things, a reinterpretation of it into beings. "Truth" is therefore not something there, that might be found or discovered — but something that must be created and that gives a name to a process, or rather to a will to overcome that has in itself no end — introducing truth, as a processus in infinitum, an active determining — not a becoming-conscious of something that is in itself firm and determined. It is a word for the "will to power".

— Nietzsche, The Will to Power, [Nie, S552, 298]

\(\cdots\)

Life is founded upon the premise of a belief in enduring and regularly recurring things; the more powerful life is, the wider must be the knowable world to which we, as it were, attribute being. Logicizing, rationalizing, systematizing as expedients of life.

— Nietzsche, The Will to Power, [Nie, S552, 298–299]

\(\cdots\)

Man projects his drive to truth, his "goal" in a certain sense, outside himself as a world that has being, as a metaphysical world, as a "thing-in-itself", as a world already in existence. His needs as creator invent the world upon which he works, anticipate it; this anticipation (this "belief" in truth) is his support.

— Nietzsche, The Will to Power, [Nie, S552, 299]

\(\cdots\)

1.3.7. Processus, Regressus, Progressus

From time immemorial we have ascribed the value of an action, a character, an existence, to the intention, the purpose for the sake of which one has acted or lived: this age-old idiosyncrasy finally takes a dangerous turn — provided, that is, that the absence of intention and purpose in events comes more and more to the forefront of consciousness.

— Nietzsche, The Will to Power, [Nie, S666, 351]

\(\cdots\)

Thus there seems to be in preparation a universal disvaluation: "Nothing has any meaning" — this melancholy sentence means "All meaning lies in intention, and if intention is altogether lacking, then meaning is altogether lacking, too".

— Nietzsche, The Will to Power, [Nie, S666, 351]

\(\cdots\)

In accordance with this valuation, one was constrained to transfer the value of life to a "life after death", or to the progressive development of ideas or of mankind or of the people or beyond mankind; but with that one had arrived at a progressus in infinitum of purposes: one was at last constrained to make a place for oneself in the "world process" (perhaps with the dysdaemonistic perspective that it was a process into nothingness).

— Nietzsche, The Will to Power, [Nie, S666, 351]

\(\cdots\)

1.3.8. Rondeau : Tempo di Menuetto

And do you know what "the world" is to me? Shall I show it to you in my mirror? This world: a monster of energy, without beginning, without end; a firm, iron magnitude of force that does not grow bigger or smaller, that does not expend itself but only transforms itself; as a whole, of unalterable size, a household without expenses or losses, but likewise without increase or income; enclosed by "nothingness" as by a boundary; not something blurry or wasted, not something endlessly extended, but set in a definite space as a definite force, and not a space that might be "empty" here or there, but rather as force throughout, as a play of forces and waves of forces, at the same time one and many, increasing here and at the same time decreasing there; a sea of forces flowing and rushing together, eternally changing, eternally flooding back, with tremendous years of recurrence, with an ebb and a flood of its forms; out of the simplest forms striving toward the most complex, out of the stillest, most rigid, coldest forms toward the hottest, most turbulent, most self-contradictory, and then again returning home to the simple out of this abundance, out of the play of contradictions back to the joy of concord, still affirming itself in this uniformity of its courses and its years, blessing itself as that which must return eternally, as a becoming that knows no satiety, no disgust, no weariness: this, my Dionysian world of the eternally self-creating, the eternally self-destroying, this mystery world of the twofold voluptuous delight, my "beyond good and evil", without goal, unless the joy of the circle is itself a goal; without will, unless a ring feels good will toward itself — do you want a name for this world? A solution for all its riddles? A light for you, too, you best-concealed, strongest, most intrepid, most midnightly men? — This world is the will to power — and nothing besides! And you yourselves are also this will to power — and nothing besides!

— Nietzsche, The Will to Power, [Nie, S1067, 549–550]

I have attempted in a narrative form to present an accurate picture of the formalization process as it develops in practice. Of course, accuracy must be distinguished from precision, for there are times when accuracy is better served by a vague outline that captures the manner of the subject than it is by a minute account that misses the mark entirely or catches each detail at the expense of losing the central point. Conveying the traffic between chaos and form under the restraint of an overbearing and excisive taxonomy would have sheared away half the picture and robbed the whole exchange of the lion's share of the duty.

At moments I could do no better than to break into metaphor, but I believe that a certain tolerance for metaphor, especially in the initial stages of formalization, is a necessary capacity for reaching beyond the secure boundaries of what is already comfortable to reason. Plus, a controlled transport of metaphor allows one to draw on the boundless store of ready analogies and germinal morphisms that every natural language provides for free.

Finally, it would leave an unfair impression to delete the characters of narrative and metaphor from the text of the story, and especially after they have had such a hand in creating it.

Even the most precise of established formulations cannot be protected from being reused in ways that initially appear as an abuse of language.

One of the most difficult questions about the development of intelligent systems is how the power of abstraction can arise, beginning from the kinds of formal systems where each symbol has one meaning at most. I think that the natural pathway of this evolution has to go through the obscure territory of ambiguity and metaphor.

A critical phase and a crucial step in the development of intelligent systems, biological or technological, is concerned with achieving a certain power of abstraction, but the real trick is for the budding intelligence to accomplish this without losing a grip on the material contents of the abstract categories, the labels and levels of which this power interposes and intercalates between essence and existence.

If one looks to the surface material of natural languages for signs of how this power of abstraction might arise, one finds a suggestive set of potential precursors in the phenomena of ambiguity, anaphora, and metaphor. Keeping this in mind throughout the project, I will pay close attention to the places where the power of abstraction seems to develop, especially in the guises of systematic ambiguity and controlled metaphor.

Paradoxically, and a bit ironically, if one's initial attempt to formalize semantics begins with the aim of stamping out ambiguity, metaphor, and all forms of figurative language use, then one may have precluded all hope of developing a capacity for abstraction at any later stage.

1.3.9. Reconnaissance

In every sort of project there are two things to consider: first, the absolute goodness of the project; in the second place, the facility of execution.           In the first respect it suffices that the project be acceptable and practicable in itself, that what is good in it be in the nature of the thing; here, for example, that the proposed education be suitable for man and well adapted to the human heart.           The second consideration depends on relations given in certain situations — relations accidental to the thing, which consequently are not necessary and admit of infinite variety.

Rousseau, Emile, or On Education, [Rou1, 34–35]

This section provides a glancing introduction to many subjects that cannot be treated in depth until much later in this work, but that need to be touched on at this point, if only in order to "prime the canvass" or to "set the tone" for the rest of this work, that is, to suggest the general philosophy, the implicit assumptions, and the basic conceptions that guide, limit, and underlie this approach to the subject of inquiry. In the process of achieving the aims of this preliminary survey, it is apparently necessary for me, on this occasion, to pick my way through a densely interwoven web, to wit, a pressing but by no means a clear context of informal discussion, and to work my way across and around a nearly invisible warp, a whit less wittingly, a network of not yet fully formalized thought that nevertheless informs discussion in its own way.

At every stage my work is bound by dint of the necessities that appear, to me, to occasion it, and thus my initial overture to a more developed inquiry is bound to continue in an indirect style. As this venture and each of its tentative subventures is compelled to try their supervening and intervening subjects in an array of oblique and incidental manners, I am continually forced to detect my likeliest directions of progress by gently teasing out only the most readily exposed clues from the context of tangent discourse, and I am consequently obliged to clarify my local chances of success by provisionally tugging loose only the most roughly isolated threads from this gradually explicated and formulated network. Accordingly, a reconnaissance of the immediate surroundings affords but a minimal opportunity to exercise options for creativity and imagination, and there is little choice but to pick up each subordinate subject in the midst of its action and to let go of it again while it is still in progress.

In the process of carrying out the present reconnaissance it is useful to illustrate the pragmatic theory of signs as it bears on a series of slightly less impoverished and somewhat more interesting materials, to demonstrate a few of the ways that the theory of signs can be applied to a selection of genuinely complex and problematic texts, specifically, poetic and lyrical texts that are elicited from natural language sources through the considerable art of creative authors. In keeping with the nonchalant provenance of these texts, I let them make their appearance on the scene of the present discussion in what may seem like a purely incidental way, and only gradually to acquire an explicit recognition.

1.3.9.1. The Informal Context

On either side the river lie
Long fields of barley and of rye,
That clothe the wold and meet the sky;
And thro' the field the road runs by
	To many-tower'd Camelot;
And up and down the people go,
Gazing where the lilies blow
Round an island there below,
	The island of Shalott.
		Tennyson, The Lady of Shalott, [Ten, 17]

One of the continuing difficulties of this work is the tension between the formal contexts of representation, where clarity and certainty are easiest to achieve, and the informal context of applications, where any degree of insight into the nature of the problems and the structure of the entanglements affecting it is eagerly awaited and earnestly desired. This tension is due to the distances that stretch across the expanses of these contexts, especially if one considers their more extreme poles, since there is no release given of the necessity to build connections, conduct negotiations, establish a continuum of reciprocal transactions, and maintain a community of working relationships that is capable of uniting their diversity into a coherent whole. Consequently, it is at the wide end of the hopper that the real problems of formalization can be seen to occur, where taking in too resistant and tangled a material can play havoc with the fragile mechanisms of the formalization process that the mind has scarcely been able to develop in its time to date.

It may be useful at this point of the discussion to insert a reminder of why it is apposite to delve into the difficulties of the informal context. The task of programming is to identify intellectual activities that are initially carried on in the informal context, especially those that have obscure aspects in need of clarification or onerous features in need of facilitation, to analyze the ends and the means of these activities until formal analogues can be found for some of their parts, thereby devising suitable surrogates for these components within the formal arena or the effective sphere, and finally to implement these formalizations within the efficient arena or the practical sphere.

Inquiry is an activity that still takes place largely in the informal context. Accordingly, much of what people instinctively and intuitively do in carrying out an inquiry is done without a fully explicit idea of why they proceed that way, or even a thorough reflection on what they hope to gain by their efforts. It may come as a shock to realize this, since most people regard their scientific inquiries, at least, as rational procedures that are founded on explicit knowledge and follow a host of established models. But the standard of rigor that I have in mind here refers to the kind of fully thorough formalization that it would take to create autonomous computer programs for inquiry, ones that are capable of carrying out significant aspects of complete inquiries on their own. The remoteness of that goal quickly becomes evident to any programmer who sets out in the general direction of trying to achieve it.

Willows whiten, aspens quiver,
Little breezes dusk and shiver
Thro' the wave that runs forever
By the island in the river
	Flowing down to Camelot.
Four gray walls, and four gray towers,
Overlook a space of flowers,
And the silent isle imbowers
	The Lady of Shalott.
		Tennyson, The Lady of Shalott, [Ten, 17]

Nothing says that everything can be formalized. Nothing says even that every intellectual process has a formal analogue, at least, nothing yet. Indeed, one is obliged to formulate the question whether every inquiry can be formalized, and one has to be prepared for the possibility that an informal inquiry may lead one to the ultimate conclusion that not every inquiry has a formalization. But how can these questions be any clearer than the terms inquiry and formalization that they invoke? At this point it does not appear that further clarity can be achieved until specific notions of inquiry and formalization are set forth.

Although it can be said that a few components of inquiry are partially formalized in current practice, even this much reference to the parts of inquiry involves the choice of particular models of inquiry and specific notions of formalization. Starting from a sign-theoretic setting, and with the aim of working toward a system-theoretic framework, I am led to ask the following questions:

1. What is a question, for instance, this one?
2. How do questions arise, for instance, this one?
3. How can the formulation of a question, for example, as this one is, catalyze the formulation of an answer, for example, as this is not?

These questions are concerned with the nature, origin, and development, in turn, of a class of entities called questions. One of the first questions that arises about these questions is whether a question can sensibly refer to a class of entities of which the question is itself imagined or intended to be a member. Putting this aside for a while, I can try to get a handle on the above three questions by placing them in different lights, that is, by interpreting them in different contexts:

1. To ask these questions in a sign-theoretic context is to ask about the nature, the origin, and the development of the entities called questions as a class of signs, in brief but sufficiently general terms, to inquire into the life of a question as a sign.
2. To re-pose these questions in a system-theoretic context is to inquire into the notion of a state of question, asking:
1. What sort of system is involved in its conception?
2. How does it arise within such a system?
3. How does it evolve over time?

By the margin, willow-veil'd,
Slide the heavy barges trail'd
By slow horses; and unhail'd
The shallop flitteth silken-sail'd
	Skimming down to Camelot:
But who hath seen her wave her hand?
Or at the casement seen her stand?
Or is she known in all the land,
	The Lady of Shalott?
		Tennyson, The Lady of Shalott, [Ten, 17]

I begin with the idea that a question is an unclear sign. The question can express a problematic situation or a surprising phenomenon, but of course it expresses it only obscurely, or else the inquiry is at an end. Answering the question is, generally speaking, a task of converting or replacing the initial sign with a clearer but logically equivalent sign, usually proceeding until a maximally clear sign or a sufficiently clear sign is achieved, or else until some convincing indication is developed that the initial sign has no meaning at all, or no sense worth pursuing.

What gives a person a sense that a sign has meaning, well before its meaning is clearly known? What makes one think that a sign leads to the objects and the ideas that give it meaning, while only a sign is before the mind? Are there good and proper ways to test the probable utility of a sign, short of following its indications out to the end? And how can one tell if one's sense of meaning is deluded, saving the resort that suffers the total consequences of belief, faith, or trust in the sign, namely, of acting on the ostensible meaning of the sign?

An inquiry begins, in general, with an unclear sign that appears to be indicating an obscure object to an unknown interpreter, that is, to an interpreter whose own nature is likely to be every bit as mysterious as the sign that is observed and the object that is indicated put together.

An inquiry viewed as a recursive procedure seeks to compute, to find, or to generate a satisfactory answer to a hard question by working its way back to related but easier questions, component questions on which the whole original question appears to depend, until a set of questions are found that are so basic and whose answers are so easy, so evident, or so obvious that the agent of inquiry already knows their answers or is quickly able to obtain them, whence the agent of the procedure can continue by building up an adequate answer to the instigating question in terms of its answers to these fundamental questions. The couple of phases that can be distinguished on logical grounds to be taking place within this process, whether in point of actual practice they proceed in exclusively serial, interactively dialectic, or independently parallel fashions, are usually described as the "analytic descent" (AD) and the "synthetic ascent" (SA) of the recursion in question.

Only reapers, reaping early
In among the bearded barley,
Hear a song that echoes cheerly
From the river winding clearly,
	Down to tower'd Camelot:
And by the moon the reaper weary,
Piling sheaves in uplands airy,
Listening, whispers, "'T is the fairy
	Lady of Shalott."
		Tennyson, The Lady of Shalott, [Ten, 17]

One of the continuing claims of this work is that the formal structures of sign relations are not only adequate to address the needs of building a basic commerce among objects, signs, and ideas but are ideally suited to the task of linking vastly different realms of objective realities and widely disparate realms of interpretive contexts. What accounts for the utility that sign relations enjoy as a staple element for this job, not only for establishing the connectivity and maintaining the integrity of the mind in the world, but for holding the world and the mind together?

This utility is largely due to the augmented arity of sign relations as triadic relations. This endows them with an ability to extend in several dimensions at once, to span the distances between the objective and the interpretive domains that the duties of denotation are likely to demand, while concurrently expanding the volumes of contextual dispersion that the courts of connotation are liable to exact in the process of waging their syntax. The use of sign relations represents a significant advance over the more restrictive employments of dyadic relations, which do not allow of extension in more than one dimension at a time, permitting no area to be swept out nor any volume to be enclosed. For these reasons, sign relations constitute an admirable way to distribute the tensions of the task of inquiry over a space that is adequate to carry their loads.

Incidentally, it needs to be noted that this inquiry into the utility of sign relations in inquiry is not so much a question of whether the mind makes use of sign relations, or something that is isomorphic to them by any other name, since an acquaintance with the comparative strengths of various arities of relations is enough to make it obvious that no other way is available for the mind to do the things it does, but it is more a matter of how aware the mind can be made of its use of sign relations, and of how explicitly it can learn to express itself in regard to the structures and the functions of the sign relations in which it works.

In view of this distinction, the issue for this inquiry is not so much a question about the bare facts of sign relation use themselves as it is a question about the abilities of sign-using agents to accomplish anything amounting to, analogous to, or approaching an awareness of these facts. This is a question about an additional aptitude of sign-bearing agents, an extra capacity for the articulation and the expression of the facts and the factors that affect their very bearing as agents, and it amounts to an aptness for "reflection" on the facilities, the facticities, and the faculties that factor into making up their own sign use. If nothing else, these reflections serve to settle the question of a name, permitting this ability to be called "reflection", however little else is known about it.

There she weaves by night and day
A magic web with colors gay.
She has heard a whisper say,
A curse is on her if she stay
	To look down to Camelot.
She knows not what the curse may be,
And so she weaveth steadily,
And little other care hath she,
	The Lady of Shalott.
		Tennyson, The Lady of Shalott, [Ten, 17]

The purpose of a sign, for instance, a name, an expression, a program, or a text, is to denote and possibly to describe an object, for instance, a thing, a situation, or an activity in the world. When the reality to be described is infinitely more complex than the typically finite resources that one has to describe it, then strategic uses of these resources are bound to occur. For example, elliptic, multiple, and repeated uses of signs are almost bound to be called for, involving the strategies of approximation, abstraction, and recursion, respectively.

The agent of a system of interpretation that is driven to the point of distraction by the task of describing an inexhaustibly complex reality has several strategies, aside from dropping the task altogether, that are available to it for recovering from a lapse of attention to its object:

1. The agent can resort to approximation. This involves accepting the limitations of attention and restricting one's intention to capturing, describing, or representing merely the most salient aspect, facet, fraction, or fragment of the objective reality.
2. The agent can resort to abstraction. …
3. The agent can resort to recursion. This tactic can in fact be considered as a special type of abstraction. …

A common feature of these techniques is the creation of a formal domain, a context that contains the conceptually manageable images of objective reality, a circumscribed arena for thought, one that the mind can range over without an intolerable fear of being overwhelmed by its complexity. In short, a formal arena, for all the strife that remains to it and for all the tension that it maintains with its informal surroundings, still affords a space for thought in which various forms of complete analysis and full comprehension are at least conceivable in principle. For all their illusory character, these meager comforts are not to be despised.

And moving thro' a mirror clear
That hangs before her all the year,
Shadows of the world appear.
There she sees the highway near
	Winding down to Camelot:
There the river eddy whirls,
And there the surly village-churls,
And the red cloaks of market girls,
	Pass onward from Shalott.
		Tennyson, The Lady of Shalott, [Ten, 17–18]

The formal plane stands like a mirror in relation to the informal scene. If it did not reflect the interests and represent the objects that endure within the informal context, no matter how dimly and slightly it is able to portray them, then what goes on in a formal domain would lose all its fascination. At least, it would have little hold on a healthy mentality. The various formal domains that an individual agent is able to grasp are set within the informal sphere like so many myriads of mirrored facets that are available to be cut on a complex gemstone. Each formal domain affords a medium for reflection and transmission, a momentary sliver of selective clarity that allows an agent who realizes it to reflect and to represent, if always a bit obscurely and partially, a miniscule share of the wealth of formal possibilities that is there to be apportioned out.

Each portion of this uncut stone provides a space, and thus supplies a "formal material", that can be used to embody a few of those aspects of action that are discerned, designed, desired, or destined to transpire in the grander setting that is incident on it, in a numinous context that appears to surround its brief flashes of insight from every side at once. Each selection of an optional cut precludes a wealth of others possible, forcing an agent with limited resources to make an existential choice. To put it succinctly, the original impulses and the ultimate objects of human activity are all wrapped up in the informal context, and a formal domain can maintain its peculiar motive and its particular rationale for existing only as a parasite on this larger host of instinctive reasons.

In other images, aside from a mirror, a formal domain can be compared to a circus arena, a theatrical stage, a motion picture, television, or other sort of projective screen, a congressional forum, indeed, to that greatest of all three-ring circuses, the government of certain republics that we all know and love. If the clonish characters, clownish figures, and other colonial representatives that carry on in the formal arena did not mimic in variously diverting and enlightening ways the concerns of their spectators in the stands, then there would hardly be much reason for attending to their antics. Even when the action in a formal arena appears to be designed as a contrast, more diverting than enlightening, or a recreation, more a comic relief from their momentary intensity than a serious resolution of the troubles that prevail in the ordinary realm, it still amounts to a strategic way of dealing with a problematic tension in the informal context.

Sometimes a troop of damsels glad,
An abbot on an ambling pad,
Sometimes a curly shepherd-lad,
Or long-hair'd page in crimson clad,
	Goes by to tower'd Camelot;
And sometimes thro' the mirror blue
The knights come riding two and two:
She hath no loyal knight and true,
	The Lady of Shalott.
		Tennyson, The Lady of Shalott, [Ten, 18]

Before I can continue any further, it is necessary to discuss a question of terminology that continues to bedevil this discussion with ambiguities: Is a "context" still a "text", and thus composed of signs throughout, or is it something else again, an object among objects of another order, or the incidental setting of an interpreter's referent and significant acts?

The reason I have to raise this question is to make its ambiguities, up til now remaining implicit, at least more explicit in future encounters. The reason I cannot settle this question is that the array of its answers is already too fixed in established usage, and so it seems unavoidable to rely on intelligent interpreters and context-sensitive interpretation to pick up the option that makes the most sense in and of a given context. Keeping this degree of flexibility in mind, that allows one to flip back and forth between the text and the context, and that leaves one all the while free to cycle through the objective, syntactic, and interpretive readings of the word "context", it is now possible to make the following observations about the relation of the formal to the informal context.

All human interests arise in and return to the informal context, an openly vague region of indefinite duration and ever-expanding scope. That is to say, all of the objectives that people instinctively value and all of the phenomena that people genuinely wish to understand are things that arise in informal conduct, are carried on in pursuit of it, develop in connection with it, and ultimately have their bearing on it. Indeed, the wellsprings that nourish a human interest in abstract forms are never in danger of escaping the watersheds of the informal sphere, and they promise by dint of their very nature never to totally inundate nor to wholly overflow the landscape that renders itself visible there. This fact is apparent from the circumstance that every formal domain is originally instituted as a flawed inclusion within the informal context, continues to develop its constitution as a wholly-dependent subsidiary of it, and sustains itself as worthy of attention only so long as it remains a sustaining contributor to it.

But in her web she still delights
To weave the mirror's magic sights,
For often thro' the silent nights
A funeral, with plumes and lights,
	And music, went to Camelot:
Or when the moon was overhead,
Came two young lovers lately wed;
"I am half-sick of shadows," said
	The Lady of Shalott.
		Tennyson, The Lady of Shalott, [Ten, 18]

To describe the question that instigates an inquiry in the language of the pragmatic theory of signs, the original situation of the inquirer is constituted by an "elementary sign relation", taking the form <o, s, i>. In other words, the initial state of an inquiry is constellated by an ordered triple of the form <o, s, i>, a triadic element that is known in this case to exist as a member of an otherwise unknown sign relation, if the truth were told, a sign relation that defines the whole conceivable world of the interpreter along with the nature of the interpreter itself. Given that the initial situation of an inquiry has this structure, there are just three different "directions of recursion" (DOR's) that the agent of the inquiry can take out of it.

On occasion, it is useful to consider a DOR as outlined by two factors: (1) There is the "line of recursion" (LOR) that extends more generally in a couple of directions, conventionally referred to as "up" and "down". (2) There is the "arrow of recursion" (AOR), a binary feature that is frequently but quite arbitrarily depicted as "positive" or "negative", and that picks out one of the two possible directions, "up" or "down", respectively. Since one is usually more concerned with the devolution of a complex power, that is, with the direction of analytic descent, the downward development, or the reductive progress of the recursion, it is common practice to point to DOR's and to advert to LOR's in a welter of loosely ambivalent ways, letting context determine the appropriate sense.

A bow-shot from her bower-eaves,
He rode between the barley sheaves,
The sun came dazzling thro' the leaves,
And flamed upon the brazen greaves
	Of bold Sir Lancelot.
A redcross knight forever kneel'd
To a lady in his shield,
That sparkled on the yellow field,
	Beside remote Shalott.
		Tennyson, The Lady of Shalott, [Ten, 18]

A process of interpretation can appear to be working solely and steadily on the signs that occupy a formal context — to emblaze it as an emblem: on an island, in a mirror, and all through the texture of a tapestry — at least, it can appear this way to an insufficiently attentive onlooker. But an agent of interpretation is obliged to keep a private counsel, to maintain a frame that adumbrates the limits of a personal scope, and so an interpreter recurs in addition to a boundary on, a connection to, or an interface with the informal context — returning to the figure blazed: every interloper on the scene silently resorts to the facile musings and the potentially delusive inspirations of looking down the road toward the secret aims of the finished text: its ideal reader, its eventual critique, its imagined interest, its hidden intention, and its ultimate importance. An interpreter keeps at this work within this confine and keeps at this station within this horizon only so long as the counsel that is kept in the depths of the self keeps on appearing as a consistent entity in and of itself and just so long as it comports with continuing to do so.

A recursive quest can lead in many different directions as it develops. It can lead agents to resources that they set out without knowing that they bring to the task, to abilities that they start out unaware even of having or stay oblivious to ever having, and to skills that they possess, whether they exercise them or not, but do not really know themselves to be in possession of, at least at first but perhaps forever, though they automatically, instinctively, and intuitively employ all the appropriate aptitudes whenever the occasion calls for them. This happens especially when learning is first occurring and agents are developing a particular type of skill, picking it up almost in passing, in conjunction with the actions that they are learning to exercise on special types of objects. In a related pattern of development, a recursive quest can lead agents to resources that they already think they have in their power but that they are hard pressed to account for when they ask themselves exactly how they accomplish the corresponding performances.

A recursion can "lead to" a resource in two senses: (1) It can have recourse to a resource as power that is meant to be used in carrying out another action, and merely in the pursuit of a more remote object, that is, as an ancillary, assumed, implicit, incidental, instrumental, mediate, or subservient power. (2) It can be brought face to face with the fact or the question of this power, as an entity that is explicitly mentioned or recognized as a problem, and thus be forced to reflect on the nature of this putative resource in and of itself.

The gemmy bridle glitter'd free,
Like to some branch of stars we see
Hung in the golden Galaxy.
The bridle bells rang merrily
	As he rode down to Camelot:
And from his blazon'd baldric slung
A mighty silver bugle hung,
And as he rode his armor rung,
	Beside remote Shalott.
		Tennyson, The Lady of Shalott, [Ten, 18]

Any attempt to present the informal context in anything approaching its full detail is likely to lead to so much conflict and confusion that it begins to appear more akin to a chaotic context or a formless void than it chances to resemble a merely casual or a purely incidental environ. For all intents and purposes, the informal context is a coalescence of many forces and influences and a loose coalition of disparate ambitions. These forces impact on the individual thinker in what can appear like a random fashion, especially at the beginnings of individual development. Broadly speaking, if one considers the "ways of thinking" (WOT's) that are made available to a thinker, then these factors can be divvied up according to their bearing on two wide divisons of their full array:

1. There are the WOT's that are prevalent in various communities of cultural, literary, practical, scientific, and technical discourse.
2. There are the WOT's that are peculiar to the individual thinker.

But this division in abstract terms, claiming to separate WOT's communal from WOT's personal, does not disentangle the synthetic unities that are fused and woven together in practice, especially in view of the fact that collective ways of thinking are actualized only by particular individuals. Indeed, for each established way of thinking there is a further parting of the ways, collectively speaking, between the ways that it purports to conduct itself and the ways that it actually conducts itself in practice. In order to tell the difference, individual thinkers have to participate in the corresponding forms of practical conduct.

The informal context enfolds a multitude of formal arenas, to selections of which the particular interpreters usually prefer to attach themselves. It transforms a space into a medium of reflection, a respite, a retreat, or a final resort that affords the agent of interpretation a stance from which to review the action and to reflect on its many possible meanings. The informal context is so much broader in scope than the formal arenas of discourse that are located within it that it does not matter if one styles it with the definite article "the" or the indefinite article "an", since no one imagines that a unique definition could ever be implied by the vagueness of its sweeping intension or imposed on the vastness of its continuing extension. It is in the informal context that a problem arising spontaneously is most likely to meet with its first expression, and if a writer is looking for a common stock of images and signs that can permit communication with the randomly encountered reader, then it is here that the author has the best chance of finding such a resource.

All in the blue unclouded weather
Thick-jewell'd shone the saddle-leather,
The helmet and the helmet-feather
Burn'd like one burning flame together,
	As he rode down to Camelot.
As often thro' the purple night,
Below the starry clusters bright,
Some bearded meteor, trailing light,
	Moves over still Shalott.
		Tennyson, The Lady of Shalott, [Ten, 18]

There is a "form of recursion" (FOR) that is a FOR for itself, that seeks above all to perpetuate itself, that never quite terminates by design and never quite reaches its end on purpose, but merely seizes the occasional diaeresis to pause for a while while a state of dynamic equilibrium or a moment of dialectical equipoise is achieved between its formal focus and the informal context. The FOR for itself recurs not to an absolute state or a static absolute but to a relationship between the ego and the entire world, between the fictional character or the hypostatic personality that is hypothesized to explain the occurrence of specific localized phenomena and something else again, a whole that is larger, more global, and better integrated, however elusive and undifferentiated it is in its integrity.

This "inclusive other" can be referred to as "nature", so long as this nature is understood as a form of being that is not alien to the ego and not wholly external to the agent, and it can be identified as the "self", so long as this identity is understood as a relation that is not alone a property of the ego and not wholly internal to the mind of the agent.

His broad clear brow in sunlight glow'd;
On burnish'd hooves his war-horse trode;
From underneath his helmet flow'd
His coal-black curls as on he rode,
	As he rode down to Camelot.
From the bank and from the river
He flash'd into the crystal mirror,
"Tirra lirra," by the river
	Sang Sir Lancelot.
		Tennyson, The Lady of Shalott, [Ten, 18]

There is a FOR for another whose nature is never to quit in its quest until its aim is within its clasp, though it knows how much chance there is for success, and it knows the reason why its reach exceeds its grasp. This FOR, too, never rests in and of itself, but unlike the FOR for itself it can be satisfied by achieving a particular alternative state that is distinct from its initial condition, by reaching another besides itself. This FOR, too, short of reaching its specific end, never quite terminates in its own right, not of its essence, nor by its intent, nor does it relent through any deliberate purpose of its own, but only by accident of an unforeseen circumstance or by dint of an incidental misfortune.

It needs to be examined whether this state of dynamic equilibrium, this condition of balance, equanimity, harmony, and peace can be described as an aim, an end, a goal, or a good that even the FOR for itself can take for itself.

She left the web, she left the loom,
She made three paces thro' the room,
She saw the water-lily bloom,
She saw the helmet and the plume,
	She look'd down to Camelot.
Out flew the web and floated wide;
The mirror crack'd from side to side;
"The curse is come upon me," cried
	The Lady of Shalott.
		Tennyson, The Lady of Shalott, [Ten, 18]

In stepping back from a "formally engaged existence" (FEE) to reflect on the activities that normally take place within its formal arena, in stepping away from the peculiar concerns that normally take precedence within its jurisdiction to those that prevail in more ordinary contexts — and unless one is empowered by some miracle of discursive transport to jump from one charmed circle of discussion to another without entailing the usual repercussions: of causing a considerable loss of continuity, or of suffering a significant shock of dissociation — then one commonly enters on, as an intervening stage of discourse, and passes through, as a transitional phase of discussion, a context that is convenient to call a "higher order level of discourse" (HOLOD). This new level of discussion allows for a fresh supply of signs and ideas that can serve to reinforce an agent's inherent but transient capacity for reflection, qualifying an observant agent as a deliberate interpreter of the events under survey.

Opening up a HOLOD affords an agent an almost blank book, constituted within the boundless contents of the informal context, for noting what appears in the formal arena that formally incited its initial formation. This actuates a barely biased count and a basically broader context for keeping track of what goes on in a target domain. In other words that can be used to hint at its potential, it provides an uncarved block and an ungraven image, an unsullied field and an untrod plain, an unfilled frame and an unsigned space, a grander sphere and a greater unity, a higher and a wider plateau, all in all, just the kind of global staging ground that is needed for reflection on the initial arena of discourse. It comes already equipped with a "higher order level of syntax" (HOLOS) that is needed for referring to the objects and the procedures of many different formal arenas, at least, it presents a generative promise of creating enough signs and articulating enough expressions to denote the more important aspects of the formal businesses that it is responsible for reflecting on, and it generally has all the other accoutrements that are appropriate to an expanded context of interpretation or an elevated level of discourse.

In forming a HOLOD one reaches into the informal context for the images and the methods to do so. As long as one is restricted by availability or habit to dyadic relations one tends to pay attention to either one of two complementary features of the situation at the expense of the other. One can attend to either (1) the transitions that occur between entities at a single level of discourse, or (2) the distinctions that exist between entities at different levels of discourse.

In the stormy east-wind straining,
The pale yellow woods were waning,
The broad stream in his banks complaining,
Heavily the low sky raining
	Over tower'd Camelot;
Down she came and found a boat
Beneath a willow left afloat,
And round about the prow she wrote
	The Lady of Shalott.
		Tennyson, The Lady of Shalott, [Ten, 18]

An "ostensibly recursive text" (ORT) is a text that cites itself by title at some site within its body. A "wholly ostensibly recursive literature" (WORL) is a litany, a liturgy, or any other body of texts that names its entire collective corpus at some locus of citation within its interior. I am using the words "cite" and "site" to emphasize the superficially syntactic character of these definitions, where the title of a text is conventionally indicated by capitals, by italics, by quotation, or by underscoring. If a text has a definite subject or an explicit theme, for instance, an object or a state of affairs to which it makes a denotative reference, then it is not unusual for this reference to be reused as the title of the text, but this is only the rudimentary beginnings of a true self-reference in the text. Although a genuine self-reference can take its inspiration from a text being named after something that it denotes, the reference in the text to the text itself becomes complete only when the name of the subject or the title of the theme is stretched to serve as the explicit denoter of the entire text.

The sort of ostentation that is made conspicuous in these definitions is neither necessary nor sufficient for an actual recursion to take place, since the actuality of the recursive circumstance depends on the action of the interpreter, one who is always free in principle to ignore or to subvert the suggestions of the text, who has the power to override the ostensible instructions that go with the territory of any ORT, and who is potentially invited to invent whatever innovations of interpretation are conceivably able to come to mind.

In reading the signs of ostensible recursion that appear within a text of this sort an interpreter is empowered, if not always explicitly entitled, to pick out a personal way of refining their implications from among the plenitude of possible options: to gloss them over or to read them anew, to reform the masses of their solid associations into a manifold body of interpenetrating interpretations or to refuse the resplendence of their canonical suggestions in the fires of freshly refulgent convictions and by dint of the impressions that redound from a host of novel directions, to regard their indications in the light of wholly familiar conventions or to regale their invitations in the hopes of a rather more sumptuous symposium, to reinforce their established denominations with a ruthless redundancy or to riddle their resorts to the rarefied reaches of rhyme and reason with repeated petitions for their reconciliation and restless researches to reconstruct the rationales of their resources until they are honeycombed with an array of rich connotations, to subtilize or to subvert, in short, to choose between thoroughly undermining or more thoroughly understanding the suggestions of its WORL.

And down the river's dim expanse —
Like some bold seer in a trance,
Seeing all his own mischance —
With a glassy countenance
	Did she look to Camelot.
And at the closing of the day
She loosed the chain, and down she lay;
The broad stream bore her far away,
	The Lady of Shalott.
		Tennyson, The Lady of Shalott, [Ten, 18]

Given the benefit of hindsight, or with some measure of due reflection, it is perhaps fair to say that no one should ever have expected that a property which is delimited solely on syntactic grounds would turn out to be anything more than ultimately shallow. But this recognition only leaves the true nature of recursion yet to be described. This is a task that can be duly inaugurated here but that has to be left unfinished in its present shape, as it occupies the greater body of the current work.

Unless a text calls for some sort of action on the part of the interpreter then the appearance of an ostensible recursion or a syntactic repetition also has little import for action, with the possible exception of making the reading a bit redundant or imparting a rhyme to its reverberations. Taken fully in the light that a general freedom of interpretation sheds on the subject of recursion, a syntactic resonance could just as easily be read to announce the occasion of a break from an automatic routine, to afford a rest from rote repetition, rather than heralding the advent of yet another ritual compulsion to repeat. This is the form of recall, the kind of recognition or recollection of the self, that is always patent amid the potential confusion of the reflected image, that is always open to the intelligent interpreter.

If one can establish the suggestion that an intelligent interpreter does not have to follow the suggestions of a text — establish it in the sense that most people recognize this principle of freedom in their own action, however stinting they are in granting it to their fellow interpreters and however skeptical they remain in extending the scope of its application to machines — then one is likely to feel more free to pursue the signs that a text spells out and to explore the actions that they suggest.

Now there is a form of conduct or a pattern of activity that naturally accompanies a text, no matter how inert its images may be, and this is the action of reading. If the act of reading can be led to induce work on a larger scale, then reading becomes akin to heeding. In the medium of an active interpretation a reading can inspire a form of performance, and legislative declarations acquire the executive force that is needed to constitute commands, injunctions, instructions, prescriptions, recipes, and programs. Under these conditions an ostensible recursion, the mere repetition of a sign in a context subordinate to its initial appearance, as in a title role, can serve to codify a perpetual process, a potential infinitude of action, all in a finite text, where only the details of a determinate application and the discretion of an individual interpreter can bring the perennating roots of life to bear fruit in a finite time.

Lying, robed in snowy white
That loosely flew to left and right —
The leaves upon her falling light —
Thro' the noises of the night
	She floated down to Camelot:
And as the boat-head wound along
The willowy hills and fields among,
They heard her singing her last song,
	The Lady of Shalott.
		Tennyson, The Lady of Shalott, [Ten, 18–19]

It is time to discuss a text of a type that bears a kinship to the ORT, whose cut as a whole is likened to the reclusive cousins of this caste, each one lying just within reach of a related ORT but keeping itself a pace away, staying at a discreet remove, reserving the full implications of its potential recursion against the day of a suitable interpretation, and all in all residing in similar manors of meaning to the ORT, though not so ostentatiously. Even if the manifold ways of reading the senses of such a text are not as conspicuous as those of an ORT, and if it is a fair complaint to say that the deliberate design that keeps it from being obvious can also keep it from ever becoming clear, there is in principle a key to unlocking its meaning, and the ulterior purpose of the text is simply to pass on this key.

For the lack of a better name, let the type of text that devolves in evidence here be called a "pseud-ORT" (PORT) or a "quasi-ORT" (QORT). These acronyms inherit the hedge word "ostensibly" from the ORT's that their individual namesakes beget, once they are interpreted as doing so. It is the main qualification of the indicated PORT's or QORT's, and the one that continues to be borne by them as the sole inherent property of their bearing. As before, this qualification is intended to serve as a caution to the reader that the properties ordinarily imputed to the text do not actually belong to the matter of the text, but that they properly belong to the agent and the process of the active interpretation, namely, the one that is actually carried out on the material supplied by the text. The adjoined pair of weasel words "pseudo" and "quasi" are intended to remind the reader that a PORT or a QORT falls short of even the order of specious recursion that is afforded by an ORT, but has to be nudged in the general direction of this development or this evolution through the intercession of artificial distortions or specialized modulations of the semantics that is applied to the text. Whether these extra epithets exacerbate the spurious character of the putative recursion or whether they take the edge off the order of ostentation that already occurs in an ORT is a question that can be deferred to a future time.

Heard a carol, mournful, holy,
Chanted loudly, chanted lowly,
Till her blood was frozen slowly,
And her eyes were darken'd wholly,
	Turn'd to tower'd Camelot;
For ere she reach'd upon the tide
The first house by the water-side,
Singing in her song she died,
	The Lady of Shalott.
		Tennyson, The Lady of Shalott, [Ten, 19]

If its ways are kept in the way intended, lacking only a fitting key to be unlocked, then the PORT or the QORT in question leads an interloper into a recursion only whenever the significance of certain analogies, comparisons, metaphors, or similes is recognized by that interpreter. Generally speaking, this happens only when the interpreter discovers that a set of "semiotic equations" (SEQ's), applying to signs that can be picked out from the text in specific senses, is conceivably in force. Expressed another way, the recursive or self-referent interpretation is actualized when the interpreter hypothesizes that the text in question bears up under a certain kind of additional intention, namely, that a system of "qualified identifications" (QUI's) ought to be applied to selected signs in the text.

These analogies and equations have the effect of creating novel forms of "semiotic equivalence relations" (SER's) that overlay the ostensible text. These relations generate further layers of "semiotic partitions" (SEP's), or families of "semiotic equivalence classes" (SEC's), that are typically restricted in their application to a specially selected sample of symbols in the text. Since these classes are generally of an abstract sort and frequently of a recondite kind, and since they are usually intended for the purposes of a specialized interpretation, their collective import on the sense of a text is conveniently summarized under the designation of an "abstract", "abstruse", "arcane", or "analogical recursion key" (ARK).

By way of summary, a PORT or a QORT is a type of text that approaches a definite ORT subject to the recognition of an ARK, and thus affords the opportunity of leading its reader to a recursive interpretation.

The writer borrows a vehicle from the informal context, adapts its forms to the current conditions, adopts the guises appurtenant to it, and aims to appropriate to a private advantage what appears as if it is asking to assist or is long ago abandoned along a public way. The writer instills this open form with a living significance, invests it with a new lease of meaning, inscribes it perhaps with a personal title or a suitable envoi, and sends it on its way, through whatever medium avails itself and to whatever party awaits it, without knowing how the sense of the message is destined to be appreciated when life in the ordinary sense is passed from its limbs and long after the flashes of its creation are frozen in the shapes of its reception. All in all, the writer has no choice but to assume the good graces of eventually finding a charitable interpretation.

Under tower and balcony,
By garden-wall and gallery,
A gleaming shape she floated by,
A corse between the houses high,
	Silent into Camelot.
Out upon the wharfs they came,
Knight and burgher, lord and dame,
And round the prow they read her name,
	The Lady of Shalott.
		Tennyson, The Lady of Shalott, [Ten, 19]

I assume that the reader has gleaned the existence of something beyond a purely accidental relation that runs between the text and the epitext, between the prose discussion and the succession of epigraphs, that are interwoven with each other throughout the course of this presentation. In general, it is best to let these incidental counterpoints develop in a loosely parallel but rough independence from each other, and to let them run through their corresponding paces not too strenuously interlocked. The rule is thus to lay out the principal lines of their generic motives, their arguments, plans, plots, and themes, without incurring the fear of inadvertent intersections looming near, and thus to string the beads of their selective articulations along the strands of their envisioned text without invoking the undue force of a final collusion among their mass. In spite of all that, I take the chance of bringing the various threads together at this point, in order to sound out their accords and discords, and to make a bolder exegesis of the relationships that they display.

Tennyson's poem The Lady of Shalott is akin to an ORT, but a bit more remote, since the name styled as "The Lady of Shalott", that the author invokes over the course of the text, is not at first sight the title of a poem, but a title that its character adopts and afterwards adapts as the name of a boat. It is only on a deeper reading that this text can be related to or transformed into a proper ORT. Operating on a general principle of interpretation, the reader is entitled to suspect that the author is trying to say something about himself, his life, and his work, and that he is likely to be exploiting for this purpose the figure of his ostensible character and the vehicle of his manifest text. If this is an aspect of the author's intention, whether conscious or unconscious, then the reader has a right to expect that several forms of analogy are key to understanding the full intention of the text.

Given the complexity and the subtlety of the epitext in this subsection, it makes sense to begin the detailed analysis of ORT's and their ilk with a much simpler example, and one that exemplifies a straightforward ORT. These preparations are undertaken at the beginning of the next section, after which it is feasible to return to the present example, to consider the formal analysis of PORT's and QORT's, to explain how the effects of meaning that are achieved in this general type of text are supported by its sign-theoretic structure, and to discuss how these semantic intents are facilitated by the infrastructure of the language that is employed.

Who is this? and what is here?
And in the lighted palace near
Died the sound of royal cheer;
And they cross'd themselves for fear,
	All the knights at Camelot:
But Lancelot mused a little space;
He said, "She has a lovely face;
God in his mercy lend her grace,
	The Lady of Shalott."
		Tennyson, The Lady of Shalott, [Ten, 19]

As it happens, many a text in literature or science that concerns itself with hypothetical creatures, mythical entities, or speculative figures, that contents itself with idealized models of actual situations, indulges itself with idle idylls that barely allude to the serious threats against human peace and social well-being that they betray, or satisfies itself with romantic images of real enough but unknown perils of the soul — none of these would hold the level of interest that it actually has if it did not make itself available to many different levels of interpretation, readings that go far beyond the levels of discourse where it ostensibly presents itself at first sight.

Although it is easy to pick out examples of sign relations that are already completely formalized, and thus to study them as combinatorial objects of a more or less independent interest, this tactic makes it all the more difficult to see what ties these impoverished examples to the kinds of sign relations that freely develop in the unformed environment and that inform all the more natural problems that one might encounter. Thus, in this section I make an effort to catch the formalization process in its very first steps, as it begins to dehisce the very seeds of its future development from the security of their enveloping integuments.

The form of initiatory task that a certain turn of mind arrives at only toward the end of its quest is not so much to describe the tensions that exist among contexts — those between the formal arenas, bowers, courts and the informal context that surrounds them all — as it is to exhibit these forces in action and to bear up under their influences on inquiry. The task is not so much to talk about the informal context, to the point of trying to exhaust it with words, as it is to anchor one's activity in the infinitudes of its unclaimed resources, to the depth that it allows this importunity, and to buoy the significant points of one's discussion, its channels, shallows, shoals, and shores, for the time that the tide permits this opportunity.

1.3.9.2. The Epitext

It is time to render more explicit a feature of the text in the previous subsection, to abstract the form that it realizes from the materials that it appropriates to fill out its pattern, to extract the generic structure of its devices as a style of presentation or a standard technique, and to make this formal resource available for use as future occasions warrant. To this end, let a succession of epigraphs, incidental to a main text but having a consistent purpose all their own, and illustrating the points of the main text in an exemplary, poignant, or succinct way, be referred to as an "epitext".

What is the point of this poem, or what kind of example do I make of it? It seems designed to touch on a point that is very near the heart of the inquiry into inquiry: This is the question of self-referential integrity, indeed, the very possibility of referential self-consistency. The point is whether a writer can produce a text that says something significant about the process that produces it. What "significant" means is open for discussion. Its scope is usually taken to encompass the general properties and the generic powers of the process in question. And from there the inquiry, if its double focus allows the drawing of a hasty inference, is thrown back into its elliptical orbit. It is not for long that the agent of inquiry remains in the possession of the inquiry itself, since the very purpose of inquiry is to escape from the throes of the uncertainty that threw it into action. And the writer does not expect to find a reader in the transits of the very same flux. So when the inquiry is done, all that one has to remember it by, and all that another has to reconstruct it from, is the text of inquiry that came to be produced in the process. The text is only an imago, an inactive image of a living process that does not wholly live in any of its works. The text is only a parable, a likely story about an action that ended, for all intents and purposes, a long time before or a short while ago. And the text is particular, finite, and discrete. So the problem is not insignificant, for the text of inquiry to say something of consequence, not just about its own small self, but about the process of inquiry that is capable of generating a modest array of texts of its kind. Nothing says that a text has to be constituted solely at a single level of discourse, that signs of novel, mysterious, and wholly altered characters have to be adduced in order to give it multiple levels of interpretation, or that an interpretive agent has to remain forever chained in the first tower of syntax that is needed to establish a provisional point of view. This signifies something weirder than the simple circumstance that texts intended at different levels of discourse can be laced, mixed, spliced, and woven together in an indiscriminate style. It means that each piece of text and each bit of subtext, in short, each sign that participates in the whole of a text, is potentially subject to multiple interpretations, coherent or not with the modes of interpretation that are applied to the contexts surrounding the sign.

Of course, there are difficulties to be faced in leaving a single-minded perspective, as there are troubles that arise in first rising above the flat lack of any perspective at all. If the perversity of polymorphism, that allows terms to be interpreted under many types, and the curse of recursion, that permits texts to have recourse to signifying themselves, could in fact be avoided in practice, then perhaps it would be better to disallow their mention and use altogether. Alas, these complexities are not so quickly dismissed, not if computers are intended to help people make use of their formal calculi and their symbolic languages in all of the ways that they are actually accustomed to use them.

There is an interaction that occurs between the issues of polymorphism and recursion that needs to be noted at this point. It is not always the text that hits its interpreter over the head with the glaring conceits of its subject and the obvious vanities of its self-reference that contains the subtlest forms of recursion. As long as its signs are subject to allegorical and metaphorical interpretations it is always possible that some of the readings of a text can refer to the process of writing itself, to the nature of the relationship that is craft or draft from the writer to the reader, and to all the adventitious uncertainties that affect any attempt at achieving a measure of understanding. In order for a text to refer to itself it need not take on any name for itself nor call itself by any given title. In order for a text to make reference to the interpreter who writes it, the interpreter who reads it, the means, the ends, or any other medium or party to its interpretation, it need not characterize any of these roles, scenes, or stages in a literal fashion within the measure of its lines, nor refer to any portion of their number under the assumptions of aliases, disguises, secret identities, or cryptic titles, whether put off or put on. Indeed, all of the signs that are chained together within the body of the text — the kind of a body, by the way, that appears to be able to absorb all of the signs that are applied to it — are constrained by the very nature of signs. They can do little more than ease the way toward a potential meaning, facilitate a desired understanding, or hint at a given interpretation of their senses.

There is no property of the text itself that is capable of constraining the freedom of interpretation. There is nothing at all that constrains the freedom of interpretation, nothing but the nature of the interpreter. Of course, I am referring to absolutes here, and disclaiming the force of absolute constraints. If it is in the nature of a particular interpreter, as all of the sensible ones are, to let the interpretation be constrained, moderately and relatively speaking, by the character of the signs within a well delimited text, then so be it. I am merely pointing out that the degrees of potential freedom are usually much greater than one is likely initially to think.

When it comes to recursion the freedom of interpretation is a two-edged sword, or perhaps a two-headed axe. It allows an interpreter to ignore the signs of ostensible recursion, and thus to escape the confines of a labyrinth whose blueprint develops from a compulsion to repeat. But it also makes it possible to see reflections of the self where none appear to be obvious, and thus to encounter a host of recursions where none is dictated by the text.

It is useful to sum up in the following way the nature of the potentially explosive interaction that falls out between polymorphism and recursion: In order for writers by means of their texts to refer to themselves, and in order for readers in terms of these texts to recognize themselves, it need only occur to an interpreter that a self-referent interpretation is conceivable, whether or not this is the obvious, original, or ostensible interpretation of the text.

It is due to this "freedom of interpretation" (FOI), that individualizes itself in identification with a particular "form of interpretation" (FOI), that every "liberty of interpretation" (LOI) is practically equivalent to its very own "law of interpretation" (LOI). In the end, it is the middle terms, form and liberty, that give the only grounds for making sense. When all is said and done, it is the middle grounds that leave the only room for practical action, since absolute freedom and absolute law are indiscernible from the absolute constraint of absolute chaos. Let me emphasize what this means by developing its implications for the use of certain phrases in common use and by detecting the bearing that it has on reforming the fashions of their understanding. References to "reflexive signs" and "recursive texts" are misnomers, useful as a way of pointing out obvious forms of potential self-reference, but neither sufficient nor necessary to determine whether a self-reference of signs or their users actually occurs. Like other properties that one is often tempted to make the mistake of attributing to signs in fashions that are absolutely exclusive rather than relatively independent of their users, reflexivity and recursivity are not properly properties that these signs possess all by themselves but features that they manifest in a particular exercise of their active senses and their live interpretation. To the extent that the course of interpretation and the direction of reference are under the control of a particular interpreter, the words "recursive", "reflexive", and "self-referent" do not describe any properties that are essential to signs or texts, codes or programs, but refer to the manner of their regard, in other words, to a feature of their interpreter.

This means that a recursive interpretation of a sign or a text can recur just so long as its interpreter has an interest in pursuing it. It can terminate, not just with the absolute extremes of an ideal object or an objective limit, that is, with states of perfect certainty or tokens of ultimate clarity, but also in the interpretive direction, that is, with forms of self-recognition and a conduct that arises from self-knowledge. In the meantime, between these points of final termination, a recursive interpretation can also pause on a temporary basis at any time that the degree of involvement of the interpreter is pushed beyond the limits of moderation, or any time that the level of interest for the interpreter drifts beyond or is driven outside the band of personal toleration.

1.3.9.3. The Formative Tension

The incidental arena or informal context is presently described in casual, derivative, or negative terms, simply as the not yet formal, and so this admittedly unruly region is currently depicted in ways that suggest a purely unformed and a wholly formless chaos, which it is not. Increasing experience with the formalization process can help one to develop a better appreciation of the informal context, and in time one can argue for a more positive characterization of this realm as a truly formative context. The formal domain is where risks are contemplated, but the formative context is where risks are taken. In this view, the informal context is more clearly seen as the off-stage staging ground where everything that appears on the formal scene is first assembled for a formal presentation. In taking this view, one is stepping back a bit in one's imagination from the scene that presses on one's attention, getting a sense of its frame and its stage, and becoming accustomed to see what appears in ever dimmer lights, in short, one is learning to reflect on the more obvious actions, to read their pretexts, and to detect the motives that end in them.

It is fair to assume that an agent of inquiry possesses a faculty of inquiry that is available for exercise in an informal context, that is, without being required to formalize its properties prior to their use. If this faculty of inquiry is a unity, then it appears as a whole on both sides of the "glass", that is, on both sides of the imaginary line that one pretends to draw between a formal arena and its informal context.

Recognizing the positive value of an informal context as an open forum or a formative space, it is possible to form the alignments of capacities that are indicated in Table 5.

Table 5.  Alignments of Capacities
o-------------------o-----------------------------o
|      Formal       |          Formative          |
o-------------------o-----------------------------o
|     Objective     |        Instrumental         |
|      Passive      |           Active            |
o-------------------o--------------o--------------o
|     Afforded      |  Possessed   |  Exercised   |
o-------------------o--------------o--------------o

The style of this discussion, based on the distinction between possession and exercise that arises so naturally in this context, stems from a root that is old indeed. In this connection, it is fruitful to compare the current alignments with those given in Aristotle's treatise On the Soul, a germinal textbook of psychology that ventures to analyze the concept of the mind, psyche, or soul to the point of arriving at a definition. The alignments of capacites, analogous correspondences, and illustrative materials outlined by Aristotle are summarized in Table 6.

Table 6.  Alignments of Capacities in Aristotle
o-------------------o-----------------------------o
|      Matter       |            Form             |
o-------------------o-----------------------------o
|   Potentiality    |          Actuality          |
|    Receptivity    |  Possession  |   Exercise   |
|       Life        |    Sleep     |    Waking    |
|        Wax        |         Impression          |
|        Axe        |    Edge      |   Cutting    |
|        Eye        |   Vision     |    Seeing    |
|       Body        |            Soul             |
o-------------------o-----------------------------o
|       Ship?       |           Sailor?           |
o-------------------o-----------------------------o

An attempt to synthesize the materials and the schemes that are given in Tables 5 and 6 leads to the alignments of capacities that are shown in Table 7. I do not pretend that the resulting alignments are perfect, since there is clearly some sort of twist taking place between the top and the bottom of this synthetic arrangement. Perhaps this is due to the alterations of case, tense, and grammatical category that occur throughout the paradigm, or perhaps it has something to do with the fact that the relationships through the middle of the Table are more analogical than categorical. For the moment I am content to leave all the paradoxes intact, taking the pattern of tensions and torsions as a puzzle for future study.

Table 7.  Synthesis of Alignments
o-------------------o-----------------------------o
|      Formal       |          Formative          |
o-------------------o-----------------------------o
|     Objective     |        Instrumental         |
|      Passive      |           Active            |
|     Afforded      |  Possessed   |  Exercised   |
|      To Hold      |   To Have    |    To Use    |
|    Receptivity    |  Possession  |   Exercise   |
|   Potentiality    |          Actuality          |
|      Matter       |            Form             |
o-------------------o-----------------------------o

Due to the importance of Aristotle's account for every discussion that follows it, not to mention for the many that follow it without knowing it, and because the issues it raises arise repeatedly throughout this work, I am going to cite an extended extract from the relevant text (Aristotle, On the Soul, 2.1), breaking up the argument into a number of individual premisses, stages, and examples.

a.	The theories of the soul (psyche) handed down by our predecessors have been sufficiently discussed; now let us start afresh, as it were, and try to determine (diorisai) what the soul is, and what definition (logos) of it will be most comprehensive (koinotatos).
b.	We describe one class of existing things as substance (ousia), and this we subdivide into three: (1) matter (hyle), which in itself is not an individual thing, (2) shape (morphe) or form (eidos), in virtue of which individuality is directly attributed, and (3) the compound of the two.
c.	Matter is potentiality (dynamis), while form is realization or actuality (entelecheia), and the word actuality is used in two senses, illustrated by the possession of knowledge (episteme) and the exercise of it (theorein).
d.	Bodies (somata) seem to be pre-eminently substances, and most particularly those which are of natural origin (physica), for these are the sources (archai) from which the rest are derived.
e.	But of natural bodies some have life (zoe) and some have not; by life we mean the capacity for self-sustenance, growth, and decay.
f.	Every natural body (soma physikon), then, which possesses life must be substance, and substance of the compound type (synthete).
g.	But since it is a body of a definite kind, viz., having life, the body (soma) cannot be soul (psyche), for the body is not something predicated of a subject, but rather is itself to be regarded as a subject, i.e., as matter.
h.	So the soul must be substance in the sense of being the form of a natural body, which potentially has life. And substance in this sense is actuality.
i.	The soul, then, is the actuality of the kind of body we have described. But actuality has two senses, analogous to the possession of knowledge and the exercise of it.
j.	Clearly (phaneron) actuality in our present sense is analogous to the possession of knowledge; for both sleep (hypnos) and waking (egregorsis) depend upon the presence of the soul, and waking is analogous to the exercise of knowledge, sleep to its possession (echein) but not its exercise (energein).
k.	Now in one and the same person the possession of knowledge comes first.
l.	The soul may therefore be defined as the first actuality of a natural body potentially possessing life; and such will be any body which possesses organs (organikon).
m.	(The parts of plants are organs too, though very simple ones: e.g., the leaf protects the pericarp, and the pericarp protects the seed; the roots are analogous to the mouth, for both these absorb food.)
n.	If then one is to find a definition which will apply to every soul, it will be "the first actuality of a natural body possessed of organs".
o.	So one need no more ask (zetein) whether body and soul are one than whether the wax (keros) and the impression (schema) it receives are one, or in general whether the matter of each thing is the same as that of which it is the matter; for admitting that the terms unity and being are used in many senses, the paramount (kyrios) sense is that of actuality.
p.	We have, then, given a general definition of what the soul is: it is substance in the sense of formula (logos), i.e., the essence of such-and-such a body.
q.	Suppose that an implement (organon), e.g. an axe, were a natural body; the substance of the axe would be that which makes it an axe, and this would be its soul; suppose this removed, and it would no longer be an axe, except equivocally. As it is, it remains an axe, because it is not of this kind of body that the soul is the essence or formula, but only of a certain kind of natural body which has in itself a principle of movement and rest.
r.	We must, however, investigate our definition in relation to the parts of the body.
s.	If the eye were a living creature, its soul would be its vision; for this is the substance in the sense of formula of the eye. But the eye is the matter of vision, and if vision fails there is no eye, except in an equivocal sense, as for instance a stone or painted eye.
t.	Now we must apply what we have found true of the part to the whole living body. For the same relation must hold good of the whole of sensation to the whole sentient body qua sentient as obtains between their respective parts.
u.	That which has the capacity to live is not the body which has lost its soul, but that which possesses its soul; so seed and fruit are potentially bodies of this kind.
v.	The waking state is actuality in the same sense as the cutting of the axe or the seeing of the eye, while the soul is actuality in the same sense as the faculty of the eye for seeing, or of the implement for doing its work.
w.	The body is that which exists potentially; but just as the pupil and the faculty of seeing make an eye, so in the other case the soul and body make a living creature.
x.	It is quite clear, then, that neither the soul nor certain parts of it, if it has parts, can be separated from the body; for in some cases the actuality belongs to the parts themselves. Not but what there is nothing to prevent some parts being separated, because they are not actualities of any body.
y.	It is also uncertain (adelon) whether the soul as an actuality bears the same relation to the body as the sailor (ploter) to the ship (ploion).
z.	This must suffice as an attempt to determine in rough outline the nature of the soul.

1.3.10. Recurring Themes

The overall purpose of the next several Sections is threefold:

1. To continue to illustrate the salient properties of sign relations in the medium of selected examples.
2. To demonstrate the use of sign relations to analyze and clarify a particular order of difficult symbols and complex texts, namely, those that involve recursive, reflective, or reflexive features.
3. To begin to suggest the implausibility of understanding this order of phenomena without using sign relations or something like them, namely, concepts with the power of triadic relations.

The prospective lines of an inquiry into inquiry cannot help but meet at various points, where a certain entanglement of the subjects of interest repeatedly has to be faced. The present discussion of sign relations is currently approaching one of these points. As the work progresses, the formal tools of logic and set theory become more and more indispensable to say anything significant or to produce any meaningful results in the study of sign relations. And yet it appears, at least from the vantage of the pragmatic perspective, that the best way to formalize, to justify, and to sharpen the use of these tools is by means of the sign relations that they involve. And so the investigation shuffles forward on two or more feet, shifting from a stance that fixes on a certain level of logic and set theory, using it to advance the understanding of sign relations, and then exploits the leverage of this new pivot to consider variations, and hopefully improvements, in the very language of concepts and terms that one uses to express questions about logic and sets, in all of its aspects, from syntax, to semantics, to the pragmatics of both human and computational interpreters.

The main goals of the present section are as follows:

1. To introduce a basic logical notation and a naive theory of sets, just enough to treat sign relations as the set-theoretic extensions of logically expressible concepts.
2. To use this modicum of formalism to define a number of conceptual constructs, useful in the analysis of more general sign relations.
3. To develop a proof format that is suitable for deriving facts about these constructs in a careful and potentially computational manner.
4. More incidentally, but increasingly effectively, to get a sense of how sign relations can be used to clarify the very languages that are used to talk about them.

1.3.10.1. Preliminary Notions

The present phase of discussion proceeds by recalling a series of basic definitions, refining them to deal with more specialized situations, and refitting them as necessary to cover larger families of sign relations.

In this discussion the word semantic is being used as a generic adjective to describe anything concerned with or related to meaning, whether denotative, connotative, or pragmatic, and without regard to how these different aspects of meaning are correlated with each other. The word semiotic is being used, more specifically, to indicate the connotative relationships that exist between signs, in particular, to stress the aspects of process and of potential for progress that are involved in the transitions between signs and their interpretants. Whenever the focus fails to be clear from the context of discussion, the modifiers denotative and referential are available to pinpoint the relationships that exist between signs and their objects. Finally, there is a common usage of the term pragmatic to highlight aspects of meaning that have to do with the context of use and the language user, but I reserve the use of this term to refer to the interpreter as an agent with a purpose, and thus to imply that all three aspects of sign relations are involved in the subject under discussion.

Recall the definitions of semiotic equivalence classes (SECs), semiotic partitions (SEPs), semiotic equations (SEQs), and semiotic equivalence relations (SERs), as in Segment 1.3.4.3.

The discussion up to this point is partial to examples of sign relations that enjoy especially nice properties, in particular, whose connotative components form equivalence relations and whose denotative components conform to these equivalences, in the sense that all of the signs in a single equivalence class always denote one and the same object. By way of liberalizing this discussion to more general cases of sign relations, this subsection develops a number of additional concepts for describing the internal relations of sign relations and makes a set of definitions that do not take the aforementioned features for granted.

The complete sign relation involved in a situation encompasses all the things that one thinks about and all the thoughts that one thinks about them while engaged in that situation, in other words, all the signs and ideas that flit through one's mind in relation to a domain of objects. Only a rarefied sample of this complete sign relation is bound to avail itself to reflective awareness, still less of it is likely to inspire a common interest in the community of inquiry at large, and only bits and pieces of it can be expected to suit themselves to a formal analysis. In view of these considerations, it is useful to have a general idea of the sampling relation that an investigator, oneself in particular, is likely to form between two sign relations: (1) the whole sign relation that one intends to study, and (2) the selective portion of it that one is able to pin down for a formal examination.

It is important to realize that a sampling relation, to express it roughly, is a special case of a sign relation. Aside from acting on sign relations and creating an association between sign relations, a sampling relation is also involved in a larger sign relation, at least, it can be subsumed within a general order of sign relations that allows sign relations themselves to be taken as the objects, the signs, and the interpretants of what can be called a higher order sign relation. Considered with respect to its full potential, its use, and its purpose, a sampling relation does not fall outside the closure of sign relations. To be precise, a sampling relation falls within the denotative component of a higher order sign relation, since the sign relation sampled is the object of study and the sample is taken as a sign of it.

With respect to the general variety of sampling relations there are a number of specific conceptions that are likely to be useful in this study, a few of which can now be discussed.

A bit of a sign relation is defined to be any subset of its extension, that is, an arbitrary selection from its set of ordered triples.

Described in relation to sampling relations, a bit of a sign relation is just the most arbitrary possible sample of it, and thus its occurring to mind implies the most general form of sampling relation to be in effect. In essence, it is just as if a bit of a sign relation, by virtue of its appearing in evidence, can always be interpreted as a bit of evidence that some sort of sampling relation is being applied.

1.3.10.2. Intermediary Notions

A number of additional definitions are relevant to sign relations whose connotative components constitute equivalence relations, if only in part.

A dyadic relation on a single set (DROSS) is a non-empty set of points plus a set of ordered pairs on these points. Until further notice, any reference to a dyadic relation is intended to be taken in this sense, in other words, as a reference to a DROSS. In a typical notation, the dyadic relation \(\underline{G} = (X, G) = (G^{(1)}, G^{(2)})\) is specified by giving the set of points \(X = G^{(1)}\!\) and the set of ordered pairs \(G = G^{(2)} \subseteq X \times X\) that go together to define the relation. In contexts where the set of points is understood, it is customary to call the whole relation \(\underline{G}\) by the name of the set \(G.\!\)

A subrelation of a dyadic relation \(\underline{G} = (X, G) = (G^{(1)}, G^{(2)})\) is a dyadic relation \(\underline{H} = (Y, H) = (H^{(1)}, H^{(2)})\) that has all of its points and pairs in \(\underline{G}\) more precisely, that has all of its points \(Y \subseteq X\) and all of its pairs \(H \subseteq G.\)

The induced subrelation on a subset (ISOS), taken with respect to the dyadic relation \(G \subseteq X \times X\) and the subset \(Y \subseteq X,\) is the maximal subrelation of \(G\!\) whose points belong to \(Y.\!\) In other words, it is the dyadic relation on \(Y\!\) whose extension contains all of the pairs of \(Y \times Y\) that appear in \(G.\!\) Since the construction of an ISOS is uniquely determined by the data of \(G\!\) and \(Y,\!\) it can be represented as a function of these arguments, as in the notation \(\operatorname{ISOS} (G, Y),\) which can be denoted more briefly as \(\underline{G}_Y.\!\). Using the symbol \(\bigcap\) to indicate the intersection of a pair of sets, the construction of \(\underline{G}_Y = \operatorname{ISOS} (G, Y)\) can be defined as follows:

\(\begin{array}{lll} \underline{G}_Y & = & (Y, \ G_Y) \\ \\ & = & (G_Y^{(1)}, \ G_Y^{(2)}) \\ \\ & = & (Y, \ \{ (x, y) \in Y\!\times\!Y : (x, y) \in G^{(2)} \}) \\ \\ & = & (Y, \ Y\!\times\!Y \, \bigcap \, G^{(2)}). \\ \end{array}\)

These definitions for dyadic relations can now be applied in a context where each bit of a sign relation that is being considered satisfies a special set of conditions, namely, if \(R\!\) is the relational bit under consideration:

1. Syntactic domain \(X\!\) = Sign domain \(S\!\) = Interpretant domain \(I.\!\)
2. Connotative component = \(R_{XX}\!\) = \(R_{SI}\!\) = Equivalence relation \(E.\!\)

Under these assumptions, and with regard to bits of sign relations that satisfy these conditions, it is useful to consider further selections of a specialized sort, namely, those that keep equivalent signs synonymous.

An arbit of a sign relation is a slightly more judicious bit of it, preserving a semblance of whatever SEP happens to rule over its signs, and respecting the semiotic parts of the sampled sign relation, when it has such parts. In other words, an arbit suggests an act of selection that represents the parts of the original SEP by means of the parts of the resulting SEP, that extracts an ISOS of each clique in the SER that it bothers to select any points at all from, and that manages to portray in at least this partial fashion all or none of every SEC that appears in the original sign relation.

1.3.10.3. Propositions and Sentences

The concept of a sign relation is typically extended as a set \(\mathcal{L} \subseteq \mathcal{O} \times \mathcal{S} \times \mathcal{I}.\) Because this extensional representation of a sign relation is one of the most natural forms that it can take up, along with being one of the most important forms that it is likely to be encountered in, a good amount of set-theoretic machinery is necessary to carry out a reasonably detailed analysis of sign relations in general.

For the purposes of this discussion, let it be supposed that each set \(Q,\!\) that comprises a subject of interest in a particular discussion or that constitutes a topic of interest in a particular moment of discussion, is a subset of a set \(X,\!\) one that is sufficiently universal relative to that discussion or big enough to cover everything that is being talked about in that moment. In a setting like this it is possible to make a number of useful definitions, to which we now turn.

The negation of a sentence \(s,\!\) written as \(^{\backprime\backprime} \, \underline{(} s \underline{)} \, ^{\prime\prime}\) and read as \(^{\backprime\backprime} \, \operatorname{not}\ s \, ^{\prime\prime},\) is a sentence that is true when \(s\!\) is false and false when \(s\!\) is true.

The complement of a set \(Q\!\) with respect to the universe \(X\!\) is denoted by \(^{\backprime\backprime} \, X\!-\!Q \, ^{\prime\prime},\) or simply by \(^{\backprime\backprime} \, {}^{_\sim} Q \, ^{\prime\prime}\) when the universe \(X\!\) is determinate, and is defined as the set of elements in \(X\!\) that do not belong to \(Q,\!\) that is:

\(\begin{array}{lllll} {}^{_\sim} Q & = & X\!-\!Q & = & \{ \, x \in X : \underline{(} x \in Q \underline{)} \, \}. \\ \end{array}\)

The relative complement of \(P\!\) in \(Q,\!\) for two sets \(P, Q \subseteq X,\) is denoted by \(^{\backprime\backprime} \, Q\!-\!P \, ^{\prime\prime}\) and defined as the set of elements in \(Q\!\) that do not belong to \(P,\!\) that is:

\(\begin{array}{lll} Q\!-\!P & = & \{ \, x \in X : x \in Q ~\operatorname{and}~ \underline{(} x \in P \underline{)} \, \}. \\ \end{array}\)

The intersection of \(P\!\) and \(Q,\!\) for two sets \(P, Q \subseteq X,\) is denoted by \(^{\backprime\backprime} \, P \cap Q \, ^{\prime\prime}\) and defined as the set of elements in \(X\!\) that belong to both \(P\!\) and \(Q.\!\)

\(\begin{array}{lll} P \cap Q & = & \{ \, x \in X : x \in P ~\operatorname{and}~ x \in Q \, \}. \\ \end{array}\)

The union of \(P\!\) and \(Q,\!\) for two sets \(P, Q \subseteq X,\) is denoted by \(^{\backprime\backprime} \, P \cup Q \, ^{\prime\prime}\) and defined as the set of elements in \(X\!\) that belong to at least one of \(P\!\) or \(Q.\!\)

\(\begin{array}{lll} P \cup Q & = & \{ \, x \in X : x \in P ~\operatorname{or}~ x \in Q \, \}. \\ \end{array}\)

The symmetric difference of \(P\!\) and \(Q,\!\) for two sets \(P, Q \subseteq X,\) is denoted by \(^{\backprime\backprime} \, P ~\hat{+}~ Q \, ^{\prime\prime}\) and is defined as the set of elements in \(X\!\) that belong to just one of \(P\!\) or \(Q.\!\)

\(\begin{array}{lll} P ~\hat{+}~ Q & = & \{ \, x \in X : x \in P\!-\!Q ~\operatorname{or}~ x \in Q\!-\!P \, \}. \\ \end{array}\)

The foregoing "definitions" are the bare essentials that are needed to get the rest of this discussion going, but they have to be regarded as almost purely informal in character, at least, at this stage of the game. In particular, these definitions all invoke the undefined notion of what a sentence is, they all rely on the reader's native intuition of what a set is, and they all derive their coherence and their meaning from the common understanding, but the equally casual use and unreflective acquaintance that just about everybody has of the logical connectives not, and, or, as these are expressed in natural language terms.

As formative definitions, these initial postulations neither acquire the privileged status of untouchable axioms and infallible intuitions nor do they deserve any special suspicion, at least, nothing over and above the reflective critique that one ought to apply to all important definitions. These dim beginnings of anything approaching genuine definitions also serve to accustom the mind's eye to a particular style of observation, that of seeing informal concepts presented in a formal frame, in a way that demands their increasing clarification. In this style of examination, the frame of the set-builder expression \(\{ x \in X : \underline{~~~} \}\) functions like the eye of the needle through which one is trying to transport a suitably rich import of mathematics.

Part the task of the remaining discussion is gradually to formalize the promissory notes that are represented by these terms and stipulations and to see whether their casual comprehension can be converted into an explicit subject matter, one that depends on grasping the corresponding collection of almost wholly, if still partially formalized conceptions. To this we now turn.

The binary domain is the set \(\mathbb{B} = \{ 0, 1 \}\) of two algebraic values, whose arithmetic operations obey the rules of \(\operatorname{GF}(2),\) the galois field of exactly two elements, whose addition and multiplication tables are tantamount to addition and multiplication of integers modulo 2.

The boolean domain is the set \(\underline\mathbb{B} = \{ \underline{0}, \underline{1} \}\) of two logical values, whose elements are read as false and true, or as falsity and truth, respectively.

At this point, I cannot tell whether the distinction between these two domains is slight or significant, and so this question must evolve its own answer, while I pursue a larger inquiry by means of its hypothesis. The weight of the matter appears to increase as the investigation moves from abstract, algebraic, and formal settings to contexts where logical semantics, natural language syntax, and concrete categories of grammar are compelling considerations. Speaking abstractly and roughly enough, it is often acceptable to identify these two domains, and up until this point there has rarely appeared to be a sufficient reason to keep their concepts separately in mind. The boolean domain \(\underline\mathbb{B}\) comes with at least two operations, though often under different names and always included in a number of others, that are analogous to the field operations of the binary domain \(\mathbb{B},\) and operations that are isomorphic to the rest of the boolean operations in \(\underline\mathbb{B}\) can always be built on the binary basis of \(\mathbb{B}.\)

Of course, as sets of the same cardinality, the domains \(\mathbb{B}\) and \(\underline\mathbb{B}\) and all of the structures that can be built on them become isomorphic at a high enough level of abstraction. Consequently, the main reason for making this distinction in the present context appears to be a matter more of grammar than an issue of logical and mathematical substance, namely, so that the signs \(^{\backprime\backprime} \underline{0} ^{\prime\prime}\) and \(^{\backprime\backprime} \underline{1} ^{\prime\prime}\) can appear with some semblance of syntactic legitimacy in linguistic contexts that call for a grammatical sentence or a sentence surrogate to represent the classes of sentences that are always false and always true, respectively. The signs \(^{\backprime\backprime} 0 ^{\prime\prime}\) and \(^{\backprime\backprime} 1 ^{\prime\prime},\) customarily read as nouns but not as sentences, fail to be suitable for this purpose. Whether these scruples, that are needed to conform to a particular choice of natural language context, are ultimately important, is another thing that remains to be determined.

The negation of a value \(x\!\) in \(\underline\mathbb{B},\) written \(^{\backprime\backprime} \underline{(} x \underline{)} ^{\prime\prime}\) or \(^{\backprime\backprime} \lnot x ^{\prime\prime}\) and read as \(^{\backprime\backprime} \operatorname{not}\ x ^{\prime\prime},\) is the boolean value \(\underline{(} x \underline{)} \in \underline\mathbb{B}\) that is \(\underline{1}\) when \(x\!\) is \(\underline{0}\) and \(\underline{0}\) when \(x\!\) is \(\underline{1}.\) Negation is a monadic operation on boolean values, that is, a function of the form \(f : \underline\mathbb{B} \to \underline\mathbb{B},\) as shown in Table 8.

Table 8. Negation Operation for the Boolean Domain

\(x\!\)	\(\underline{(} x \underline{)}\)
\(\underline{0}\)	\(\underline{1}\)
\(\underline{1}\)	\(\underline{0}\)

It is convenient to transport the product and the sum operations of \(\mathbb{B}\) into the logical setting of \(\underline\mathbb{B},\) where they can be symbolized by signs of the same character. This yields the following definitions of a product and a sum in \(\underline\mathbb{B}\) and leads to the following forms of multiplication and addition tables.

The product \(x \cdot y\) of two values \(x\!\) and \(y\!\) in \(\underline\mathbb{B}\) is given by Table 9. As a dyadic operation on boolean values, that is, a function of the form \(f : \underline\mathbb{B} \times \underline\mathbb{B} \to \underline\mathbb{B},\) the product corresponds to the logical operation of conjunction, written \(^{\backprime\backprime} x \land y ^{\prime\prime}\) or \(^{\backprime\backprime} x\!\And\!y ^{\prime\prime}\) and read as \(^{\backprime\backprime} x ~\operatorname{and}~ y ^{\prime\prime}.\) In accord with common practice, the multiplication sign is frequently omitted from written expressions of the product.

Table 9. Product Operation for the Boolean Domain

\(\cdot\!\)	\(\underline{0}\)	\(\underline{1}\)
\(\underline{0}\)	\(\underline{0}\)	\(\underline{0}\)
\(\underline{1}\)	\(\underline{0}\)	\(\underline{1}\)

The sum \(x + y\!\) of two values \(x\!\) and \(y\!\) in \(\underline\mathbb{B}\) is given in Table 10. As a dyadic operation on boolean values, that is, a function of the form \(f : \underline\mathbb{B} \times \underline\mathbb{B} \to \underline\mathbb{B},\) the sum corresponds to the logical operation of exclusive disjunction, usually read as \(^{\backprime\backprime} x ~\text{or}~ y ~\text{but not both} ^{\prime\prime}.\) Depending on the context, other signs and readings that invoke this operation are\[^{\backprime\backprime} x \ne y ^{\prime\prime}\] or \(^{\backprime\backprime} x \not\Leftrightarrow y ^{\prime\prime},\) read as \(^{\backprime\backprime} x ~\text{is not equal to}~ y ^{\prime\prime},\) \(^{\backprime\backprime} x ~\text{is not equivalent to}~ y ^{\prime\prime},\) or \(^{\backprime\backprime} \text{exactly one of}~ x, y ~\text{is true} ^{\prime\prime}.\)

Table 10. Sum Operation for the Boolean Domain

\(+\!\)	\(\underline{0}\)	\(\underline{1}\)
\(\underline{0}\)	\(\underline{0}\)	\(\underline{1}\)
\(\underline{1}\)	\(\underline{1}\)	\(\underline{0}\)

For sentences, the signs of equality (\(=\!\)) and inequality (\(\ne\!\)) are reserved to signify the syntactic identity and non-identity, respectively, of the literal strings of characters that make up the sentences in question, while the signs of equivalence (\(\Leftrightarrow\)) and inequivalence (\(\not\Leftrightarrow\)) refer to the logical values, if any, of these strings, and thus they signify the equality and inequality, respectively, of their conceivable boolean values. For the logical values themselves, the two pairs of symbols collapse in their senses to a single pair, signifying a single form of coincidence or a single form of distinction, respectively, between the boolean values of the entities involved.

In logical studies, one tends to be interested in all of the operations or all of the functions of a given type, at least, to the extent that their totalities and their individualities can be comprehended, and not just the specialized collections that define particular algebraic structures. Although the remainder of the dyadic operations on boolean values, in other words, the rest of the sixteen functions of the form \(f : \underline\mathbb{B} \times \underline\mathbb{B} \to \underline\mathbb{B},\) could be presented in the same way as the multiplication and addition tables, it is better to look for a more efficient style of representation, one that is able to express all of the boolean functions on the same number of variables on a roughly equal basis, and with a bit of luck, affords us with a calculus for computing with these functions.

The utility of a suitable calculus would involve, among other things:

1. Finding the values of given functions for given arguments.
2. Inverting boolean functions, that is, finding the fibers of boolean functions, or solving logical equations that are expressed in terms of boolean functions.
3. Facilitating the recognition of invariant forms that take boolean functions as their functional components.

The whole point of formal logic, the reason for doing logic formally and the measure that determines how far it is possible to reason abstractly, is to discover functions that do not vary as much as their variables do, in other words, to identify forms of logical functions that, though they express a dependence on the values of their constituent arguments, do not vary as much as possible, but approach the way of being a function that constant functions enjoy. Thus, the recognition of a logical law amounts to identifying a logical function, that, though it ostensibly depends on the values of its putative arguments, is not as variable in its values as the values of its variables are allowed to be.

The indicator function or the characteristic function of the set \(Q \subseteq X,\) written \(f_Q,\!\) is the map from the universe \(X\!\) to the boolean domain \(\underline\mathbb{B} = \{ \underline{0}, \underline{1} \}\) that is defined in the following ways:

1. Considered in extensional form, \(f_Q\!\) is the subset of \(X \times \underline\mathbb{B}\) that is given by the following formula:

   \(f_Q ~=~ \{ (x, y) \in X \times \underline\mathbb{B} ~:~ y = \underline{1} ~\Leftrightarrow~ x \in Q \}.\)

2. Considered in functional form, \(f_Q\!\) is the map from \(X\!\) to \(\underline\mathbb{B}\) that is given by the following condition:

   \(f_Q (x) ~=~ \underline{1} ~\Leftrightarrow~ x \in Q.\)

A proposition about things in the universe, for short, a proposition, is the same thing as an indicator function, that is, a function of the form \(f : X \to \underline\mathbb{B}.\) The convenience of this seemingly redundant usage is that it allows one to refer to an indicator function without having to specify right away, as a part of its designated subscript, exactly what set it indicates, even though a proposition always indicates some subset of its designated universe, and even though one will probably or eventually want to know exactly what subset that is.

According to the stated understandings, a proposition is a function that indicates a set, in the sense that a function associates values with the elements of a domain, some of which values can be interpreted to mark out for special consideration a subset of that domain. The way in which an indicator function is imagined to "indicate" a set can be expressed in terms of the following concepts.

The fiber of a codomain element \(y \in Y\!\) under a function \(f : X \to Y\) is the subset of the domain \(X\!\) that is mapped onto \(y,\!\) in short, it is \(f^{-1} (y) \subseteq X.\) In other language that is often used, the fiber of \(y\!\) under \(f\!\) is called the antecedent set, the inverse image, the level set, or the pre-image of \(y\!\) under \(f.\!\) All of these equivalent concepts are defined as follows:

\(\operatorname{Fiber~of}~ y ~\operatorname{under}~ f ~=~ f^{-1} (y) ~=~ \{ x \in X : f(x) = y \}.\)

In the special case where \(f\!\) is the indicator function \(f_Q\!\) of a set \(Q \subseteq X,\) the fiber of \(\underline{1}\) under \(f_Q\!\) is just the set \(Q\!\) back again:

\(\operatorname{Fiber~of}~ \underline{1} ~\operatorname{under}~ f_Q ~=~ f_Q ^{-1} (\underline{1}) ~=~ \{ x \in X : f_Q (x) = \underline{1} \} ~=~ Q.\)

In this specifically boolean setting, as in the more generally logical context, where truth under any name is especially valued, it is worth devoting a specialized notation to the fiber of truth in a proposition, to mark with particular ease and explicitness the set that it indicates. For this purpose, I introduce the use of fiber bars or ground signs, written as a frame of the form \([| \, \ldots \, |]\) around a sentence or the sign of a proposition, and whose application is defined as follows:

\(\operatorname{If}~ f : X \to \underline\mathbb{B},\)

\(\operatorname{then}~ [| f |] ~=~ f^{-1} (\underline{1}) ~=~ \{ x \in X : f(x) = \underline{1} \}.\)

Some may recognize here fledgling efforts to reinforce flights of Fregean semantics with impish pitches of Peircean semiotics. Some may deem it Icarean, all too Icarean.

The definition of a fiber, in either the general or the boolean case, is a purely nominal convenience for referring to the antecedent subset, the inverse image under a function, or the pre-image of a functional value. The definition of an operator on propositions, signified by framing the signs of propositions with fiber bars or ground signs, remains a purely notational device, and yet the notion of a fiber in a logical context serves to raise an interesting point. By way of illustration, it is legitimate to rewrite the above definition in the following form:

\(\operatorname{If}~ f : X \to \underline\mathbb{B},\)

\(\operatorname{then}~ [| f |] ~=~ f^{-1} (\underline{1}) ~=~ \{ x \in X : f(x) \}.\)

The set-builder frame \(\{ x \in X : \underline{~~~} \}\) requires a grammatical sentence or a sentential clause to fill in the blank, as with the sentence \(^{\backprime\backprime} f(x) = \underline{1} ^{\prime\prime}\) that serves to fill the frame in the initial definition of a logical fiber. And what is a sentence but the expression of a proposition, in other words, the name of an indicator function? As it happens, the sign \(^{\backprime\backprime} f(x) ^{\prime\prime}\) and the sentence \(^{\backprime\backprime} f(x) = \underline{1} ^{\prime\prime}\) represent the very same value to this context, for all \(x\!\) in \(X,\!\) that is, they will appear equal in their truth or falsity to any reasonable interpreter of signs or sentences in this context, and so either one of them can be tendered for the other, in effect, exchanged for the other, within this context, frame, and reception.

The sign \(^{\backprime\backprime} f(x) ^{\prime\prime}\) manifestly names the value \(f(x).\!\) This is a value that can be seen in many lights. It is, at turns:

1. The value that the proposition \(f\!\) has at the point \(x,\!\) in other words, the value that \(f\!\) bears at the point \(x\!\) where \(f\!\) is being evaluated, the value that \(f\!\) takes on with respect to the argument or the object \(x\!\) that the whole proposition is taken to be about.
2. The value that the proposition \(f\!\) not only takes up at the point \(x,\!\) but that it carries, conveys, transfers, or transports into the setting \(^{\backprime\backprime} \{ x \in X : \underline{~~~} \} ^{\prime\prime}\) or into any other context of discourse where \(f\!\) is meant to be evaluated.
3. The value that the sign \(^{\backprime\backprime} f(x) ^{\prime\prime}\) has in the context where it is placed, that it stands for in the context where it stands, and that it continues to stand for in this context just so long as the same proposition \(f\!\) and the same object \(x\!\) are borne in mind.
4. The value that the sign \(^{\backprime\backprime} f(x) ^{\prime\prime}\) represents to its full interpretive context as being its own logical interpretant, namely, the value that it signifies as its canonical connotation to any interpreter of the sign that is cognizant of the context in which it appears.

The sentence \(^{\backprime\backprime} f(x) = \underline{1} ^{\prime\prime}\) indirectly names what the sign \(^{\backprime\backprime} f(x) ^{\prime\prime}\) more directly names, that is, the value \(f(x).\!\) In other words, the sentence \(^{\backprime\backprime} f(x) = \underline{1} ^{\prime\prime}\) has the same value to its interpretive context that the sign \(^{\backprime\backprime} f(x) ^{\prime\prime}\) imparts to any comparable context, each by way of its respective evaluation for the same \(x \in X.\)

What is the relation among connoting, denoting, and evaluing, where the last term is coined to describe all the ways of bearing, conveying, developing, or evolving a value in, to, or into an interpretive context? In other words, when a sign is evaluated to a particular value, one can say that the sign evalues that value, using the verb in a way that is categorically analogous or grammatically conjugate to the times when one says that a sign connotes an idea or that a sign denotes an object. This does little more than provide the discussion with a weasel word, a term that is designed to avoid the main issue, to put off deciding the exact relation between formal signs and formal values, and ultimately to finesse the question about the nature of formal values, the question whether they are more akin to conceptual signs and figurative ideas or to the kinds of literal objects and platonic ideas that are independent of the mind.

These questions are confounded by the presence of certain peculiarities in formal discussions, especially by the fact that an equivalence class of signs is tantamount to a formal object. This has the effect of allowing an abstract connotation to work as a formal denotation. In other words, if the purpose of a sign is merely to lead its interpreter up to a sign in an equivalence class of signs, then it follows that this equivalence class is the object of the sign, that connotation can achieve denotation, at least, to some degree, and that the interpretant domain collapses with the object domain, at least, in some respect, all things being relative to the sign relation that embeds the discussion.

Introducing the realm of values is a stopgap measure that temporarily permits the discussion to avoid certain singularities in the embedding sign relation, and allowing the process of evaluation as a compromise mode of signification between connotation and denotation only manages to steer around a topic that eventually has to be mapped in full, but these strategies do allow the discussion to proceed a little further without having to answer questions that are too difficult to be settled fully or even tackled directly at this point. As far as the relations among connoting, denoting, and evaluing are concerned, it is possible that all of these constitute independent dimensions of significance that a sign might be able to enjoy, but since the notion of connotation is already generic enough to contain multitudes of subspecies, I am going to subsume, on a tentative basis, all of the conceivable modes of evaluing within the broader concept of connotation.

With this degree of flexibility in mind, one can say that the sentence \(^{\backprime\backprime} f(x) = \underline{1} ^{\prime\prime}\) latently connotes what the sign \(^{\backprime\backprime} f(x) ^{\prime\prime}\) patently connotes. Taken in abstraction, both syntactic entities fall into an equivalence class of signs that constitutes an abstract object, a thing of value that is identified by the sign \(^{\backprime\backprime} f(x) ^{\prime\prime},\) and thus an object that might as well be identified with the value \(f(x).\!\)

The upshot of this whole discussion of evaluation is that it allows us to rewrite the definitions of indicator functions and their fibers as follows:

The indicator function or the characteristic function of a set \(Q \subseteq X,\) written \(f_Q,\!\) is the map from \(X\!\) to the boolean domain \(\underline\mathbb{B} = \{ \underline{0}, \underline{1} \}\) that is defined in the following ways:

1. Considered in extensional form, \(f_Q\!\) is the subset of \(X \times \underline\mathbb{B}\) that is given by the following formula:

   \(f_Q ~=~ \{ (x, y) \in X \times \underline\mathbb{B} ~:~ y ~\Leftrightarrow~ x \in Q \}.\)

2. Considered in functional form, \(f_Q\!\) is the map from \(X\!\) to \(\underline\mathbb{B}\) that is given by the following condition:

   \(f_Q ~\Leftrightarrow~ x \in Q.\)

The fibers of truth and falsity under a proposition \(f : X \to \underline\mathbb{B}\) are subsets of \(X\!\) that are variously described as follows:

\(\begin{array}{lll} \text{The fiber of}~ \underline{1} ~\text{under}~ f & = & [| f |] \\ & = & f^{-1} (\underline{1}) \\ & = & \{ x \in X ~:~ f(x) = \underline{1} \} \\ & = & \{ x \in X ~:~ f(x) \}. \\ \\ \text{The fiber of}~ \underline{0} ~\text{under}~ f & = & {}^{_\sim} [| f |] \\ & = & f^{-1} (\underline{0}) \\ & = & \{ x \in X ~:~ f(x) = \underline{0} \} \\ & = & \{ x \in X ~:~ \underline{(} f(x) \underline{)} \, \}. \end{array}\)

Perhaps this looks like a lot of work for the sake of what seems to be such a trivial form of syntactic transformation, but it is an important step in loosening up the syntactic privileges that are held by the sign of logical equivalence \({}^{\backprime\backprime} \Leftrightarrow {}^{\prime\prime},\) as written between logical sentences, and the sign of equality \({}^{\backprime\backprime} = {}^{\prime\prime},\) as written between their logical values, or else between propositions and their boolean values, respectively. Doing this removes a longstanding but wholly unnecessary conceptual confound between the idea of an assertion and the notion of an equation, and it allows one to treat logical equality on a par with the other logical operations.

As a purely informal aid to interpretation, I frequently use the letters \(^{\backprime\backprime} p ^{\prime\prime}, ^{\backprime\backprime} q ^{\prime\prime}\) to denote propositions. This can serve to tip off the reader that a function is intended as the indicator function of a set, and thus it saves us the trouble of declaring the type \(f : X \to \underline\mathbb{B}\) each time that a function is introduced as a proposition.

Another convention of use in this context is to let underscored letters stand for \(k\!\)-tuples, lists, or sequences of objects. Typically, the elements of the \(k\!\)-tuple, list, or sequence are all of one type, and the underscored letter is typically the same basic character as the letters that are indexed or subscripted to denote the individual components of the \(k\!\)-tuple, list, or sequence. When the dimension of the \(k\!\)-tuple, list, or sequence is clear from context, the underscoring may be omitted. For example, the following patterns of construction are very often seen:

\(\begin{array}{lllclllcl} 1. & \text{If} & x_1, \dots, x_k & \in & X & \text{then} & \underline{x} = (x_1, \ldots, x_k) & \in & X^k. \\ 2. & \text{If} & x_1, \dots, x_k & : & X & \text{then} & \underline{x} = (x_1, \ldots, x_k) & : & X^k. \\ 3. & \text{If} & f_1, \dots, f_k & : & X \to Y & \text{then} & \underline{f} = (f_1, \ldots, f_k) & : & (X \to Y)^k. \\ \end{array}\)

There is usually felt to be a slight but significant distinction between a membership statement of the form \(^{\backprime\backprime} x \in X \, ^{\prime\prime}\) and a type indication of the form \(^{\backprime\backprime} x : X \, ^{\prime\prime},\) for instance, as they are used in the examples above. The difference that appears to be perceived in categorical statements, when those of the form \(^{\backprime\backprime} x \in X \, ^{\prime\prime}\) and those of the form \(^{\backprime\backprime} x : X \, ^{\prime\prime}\) are set in side by side comparisons with each other, is that a multitude of objects can be said to have the same type without having to posit the existence of a set to which they all belong. Without trying to decide whether I share this feeling or even fully understand the distinction in question, I can only try to maintain a style of notation that respects it to some degree. It is conceivable that the question of belonging to a set is rightly regarded as the more serious matter, one that concerns the reality of an object and the substance of a predicate, than the question of falling under a type, that may depend only on the way that a sign is interpreted and the way that information about an object is organized. When it comes to the kinds of hypothetical statements that appear in the present instance, those of the forms \(^{\backprime\backprime} x \in X ~\Leftrightarrow~ \underline{x} \in \underline{X} \, ^{\prime\prime}\) and \(^{\backprime\backprime} x : X ~\Leftrightarrow~ \underline{x} : \underline{X} \, ^{\prime\prime},\) these are usually read as implying some order of synthetic construction, one whose contingent consequences involve the constitution of a new space to contain the elements being compounded and the recognition of a new type to characterize the elements being moulded, respectively. In these applications, the statement about types is again taken to be less presumptive than the corresponding statement about sets, since the apodosis is intended to do nothing more than abbreviate and summarize what is already stated in the protasis.

A boolean connection of degree \(k,\!\) also known as a boolean function on \(k\!\) variables, is a map of the form \(F : \underline\mathbb{B}^k \to \underline\mathbb{B}.\) In other words, a boolean connection of degree \(k\!\) is a proposition about things in the universe \(X = \underline\mathbb{B}^k.\)

An imagination of degree \(k\!\) on \(X\!\) is a \(k\!\)-tuple of propositions about things in the universe \(X.\!\) By way of displaying the kinds of notation that are used to express this idea, the imagination \(\underline{f} = (f_1, \ldots, f_k)\) is given as a sequence of indicator functions \(f_j : X \to \underline\mathbb{B},\) for \(j = {}_1^k.\) All of these features of the typical imagination \(\underline{f}\) can be summed up in either one of two ways: either in the form of a membership statement, to the effect that \(\underline{f} \in (X \to \underline\mathbb{B})^k,\) or in the form of a type statement, to the effect that \(\underline{f} : (X \to \underline\mathbb{B})^k,\) though perhaps the latter form is slightly more precise than the former.

The play of images determined by \(\underline{f}\) and \(x,\!\) more specifically, the play of the imagination \(\underline{f} = (f_1, \ldots, f_k)\) that has to do with the element \(x \in X,\) is the \(k\!\)-tuple \(\underline{y} = (y_1, \ldots, y_k)\) of values in \(\underline\mathbb{B}\) that satisfies the equations \(y_j = f_j (x),\!\) for \(j = 1 ~\text{to}~ k.\)

A projection of \(\underline\mathbb{B}^k,\) written \(\pi_j\!\) or \(\operatorname{pr}_j,\!\) is one of the maps \(\pi_j : \underline\mathbb{B}^k \to \underline\mathbb{B},\) for \(j = 1 ~\text{to}~ k,\) that is defined as follows:

\(\begin{array}{cccccc} \text{If} & \underline{y} & = & (y_1, \ldots, y_k) & \in & \underline\mathbb{B}^k, \\ \\ \text{then} & \pi_j (\underline{y}) & = & \pi_j (y_1, \ldots, y_k) & = & y_j. \\ \end{array}\)

The projective imagination of \(\underline\mathbb{B}^k\) is the imagination \((\pi_1, \ldots, \pi_k).\)

A sentence about things in the universe, for short, a sentence, is a sign that denotes a proposition. In other words, a sentence is any sign that denotes an indicator function, any sign whose object is a function of the form \(f : X \to \underline\mathbb{B}.\)

To emphasize the empirical contingency of this definition, one can say that a sentence is any sign that is interpreted as naming a proposition, any sign that is taken to denote an indicator function, or any sign whose object happens to be a function of the form \(f : X \to \underline\mathbb{B}.\)

An expression is a type of sign, for instance, a term or a sentence, that has a value. In this conception of an expression, I am deliberately leaving a number of options open, like whether it amounts to a term or to a sentence and whether it ought to be accounted as denoting a value or as connoting a value. Perhaps the expression has different values under different lights, and perhaps it relates to them differently in different respects. In the end, what one calls an expression matters less than where its value lies. Of course, no matter whether one calls an expression a term or a sentence, if the value is an element in \(\underline\mathbb{B},\) then the expression affords the option of being treated as a sentence, meaning that it is subject to assertion and composition in the same way that any sentence is, having its value figure into the values of larger expressions through the linkages of sentential connectives, and allowing the consideration of what things in what universe the corresponding proposition indicates.

Expressions with this degree of flexibility in the types under which they can be interpreted are difficult to translate from their formal settings into more natural contexts. Indeed, the whole issue can be difficult to talk about, or even to think about, since the grammatical categories of sentences and noun phrases are not so fluid in natural language settings are they can made in artificial arenas.

To finesse the issue of whether an expression denotes or connotes its value, or else to create general term that covers what both possibilities have in common, one can say that an expression evalues its value.

An assertion is just a sentence that is being used in a certain way, namely, to indicate the indication of the indicator function that the sentence is usually used to denote. In other words, an assertion is a sentence that is being converted to a certain use or being interpreted in a certain role, and one whose immediate denotation is being pursued to its substantive indication, specifically, the fiber of truth of the proposition that the sentence potentially denotes. Thus, an assertion is a sentence that is held to denote the set of things in the universe of which the sentence is true.

Taken in a context of communication, an assertion is basically a request that the interpreter consider the things for which the sentence is true, in other words, to find the fiber of truth in the associated proposition, or to invert the indicator function that is denoted by the sentence with respect to its possible value of truth.

A denial of a sentence \(s\!\) is an assertion of its negation \(^{\backprime\backprime} \, \underline{(} s \underline{)} \, ^{\prime\prime}.\) It acts as a request to think about the things for which the sentence is false, in other words, to find the fiber of falsity in the indicted proposition, or to invert the indicator function that is denoted by the sentence with respect to its possible value of falsity.

According to this manner of definition, any sign that happens to denote a proposition, any sign that is taken as denoting an indicator function, by that very fact alone successfully qualifies as a sentence. That is, a sentence is any sign that actually succeeds in denoting a proposition, any sign that one way or another brings to mind, as its actual object, a function of the form \(f : X \to \underline\mathbb{B}.\)

There are several features of this definition that need be understood. Indeed, there are problems involved in this whole style of definition that need to be discussed, and this requires a slight digression.

1.3.10.3. Propositions and Sentences

The "binary domain" is the set !B! = {!0!, !1!} of two algebraic values,
whose arithmetic operations obey the rules of GF(2), the "galois field"
of exactly two elements, whose addition and multiplication tables are
tantamount to addition and multiplication of integers "modulo 2".

The "boolean domain" is the set %B% = {%0%, %1%} of two logical values,
whose elements are read as "false" and "true", or as "falsity" and "truth",
respectively.

At this point, I cannot tell whether the distinction between these two
domains is slight or significant, and so this question must evolve its
own answer, while I pursue a larger inquiry by means of its hypothesis.
The weight of the matter appears to increase as the investigation moves
from abstract, algebraic, and formal settings to contexts where logical
semantics, natural language syntax, and concrete categories of grammar
are compelling considerations.  Speaking abstractly and roughly enough,
it is often acceptable to identify these two domains, and up until this
point there has rarely appeared to be a sufficient reason to keep their
concepts separately in mind.  The boolean domain %B% comes with at least
two operations, though often under different names and always included
in a number of others, that are analogous to the field operations of the
binary domain !B!, and operations that are isomorphic to the rest of the
boolean operations in %B% can always be built on the binary basis of !B!.

Of course, as sets of the same cardinality, the domains !B! and %B%
and all of the structures that can be built on them become isomorphic
at a high enough level of abstraction.  Consequently, the main reason
for making this distinction in the immediate context appears to be more
a matter of grammar than an issue of logical and mathematical substance,
namely, so that the signs "%0%" and "%1%" can appear with a semblance of
syntactic legitimacy in linguistic contexts that call for a grammatical
sentence or a sentence surrogate to represent the classes of sentences
that are "always false" and "always true", respectively.  The signs
"0" and "1", customarily read as nouns but not as sentences, fail
to be suitable for this purpose.  Whether these scruples, that are
needed to conform to a particular choice of natural language context,
are ultimately important, is another thing I do not know at this point.

The "negation" of x, for x in %B%, written as "(x)"
and read as "not x", is the boolean value (x) in %B%
that is %1% when x is %0%, and %0% when x is %1%.

Thus, negation is a monadic operation on boolean
values, a function of the form (_) : %B% -> %B%.

It is convenient to transport the product and the sum operations of !B!
into the logical setting of %B%, where they can be symbolized by signs
of the same character, doubly underlined as necessary to avoid confusion.
This yields the following definitions of a "product" and a "sum" in %B%
and leads to the following forms of multiplication and addition tables.

The "product" of x and y, for values x, y in %B%, is given by Table 8.

Table 8.  Product Operation for the Boolean Domain
o---------o---------o---------o
|   %.%   #   %0%   |   %1%   |
o=========o=========o=========o
|   %0%   #   %0%   |   %0%   |
o---------o---------o---------o
|   %1%   #   %0%   |   %1%   |
o---------o---------o---------o

Viewed as a function on logical values, %.% : %B% x %B% -> %B%, the product corresponds to the logical operation that is commonly called "conjunction" and that is otherwise expressed as "x and y".  In accord with common practice, the raised dot ".", doubly underlined or otherwise, is frequently omitted from written expressions of the product.
The "sum" of x and y, for values x, y in %B%, is given by Table 9.

Table 9.  Sum Operation for the Boolean Domain
o---------o---------o---------o
|   %+%   #   %0%   |   %1%   |
o=========o=========o=========o
|   %0%   #   %0%   |   %1%   |
o---------o---------o---------o
|   %1%   #   %1%   |   %0%   |
o---------o---------o---------o

Viewed as a function on logical values, %+% : %B% x %B% -> %B%, the sum corresponds to the logical operation that is generally called "exclusive disjunction" and that is otherwise expressed as "x or y, but not both".  Depending on the context, a couple of other signs and readings that can invoke this operation are:

1.  "x =/= y", read "x is not equal to y", or "exactly one of x and y".
2.  "x <=/=> y", read "x is not equivalent to y", or "x opposes y".

For sentences, the signs of equality ("=") and inequality ("=/=") are reserved to signify the syntactic identity and non-identity, respectively, of the literal strings of characters that make up the sentences in question, while the signs of equivalence ("<=>") and inequivalence ("<=/=>") refer to the logical values, if any, of these strings, and serve to signify the equality and inequality, respectively, of their conceivable boolean values.  For the logical values themselves, the two pairs of symbols collapse in their senses to a single pair, signifying a single form of coincidence or a single form of distinction, respectively, between the boolean values of the entities involved.

In logical studies, one tends to be interested in all of the operations or all of the functions of a given type, at least, to the extent that their totalities and their individualities can be comprehended, and not just the specialized collections that define particular algebraic structures.

Although the rest of the conceivably possible dyadic operations on boolean values, in other words, the remainder of the sixteen functions f : %B% x %B% -> %B%, could be presented in the same way as the multiplication and addition tables, it is better to look for a more efficient style of representation, one that is able to express all of the boolean functions on the same number of variables on a roughly equal basis, and with a bit of luck, affords us with a calculus for computing with these functions.

The utility of a suitable calculus would involve, among other things:

1.  Finding the values of given functions for given arguments.
2.  Inverting boolean functions, that is, "finding the fibers" of boolean functions, or solving logical equations that are expressed in terms of boolean functions.
3.  Facilitating the recognition of invariant forms that take boolean functions as their functional components.

The whole point of formal logic, the reason for doing logic formally and the measure that determines how far it is possible to reason abstractly, is to discover functions that do not vary as much as their variables do, in other words, to identify forms of logical functions that, though they express a dependence on the values of their constituent arguments, do not vary as much as possible, but approach the way of being a function that constant functions enjoy.  Thus, the recognition of a logical law amounts to identifying a logical function, that, though it ostensibly depends on the values of its putative arguments, is not as variable in its values as the values of its variables are allowed to be.

The "indicator function" or the "characteristic function" of a set Q c X, written "f_Q", is the map from X to the boolean domain %B% = {%0%, %1%} that is defined in the following ways:

1.  Considered in extensional form, f_Q is the subset of X x %B% that is given by the following formula:

    f_Q  =  {<x, b> in X x %B%  :  b = %1%  <=>  x in Q}.

2.  Considered in functional form, f_Q is the map from X to %B% that is given by the following condition:

    f_Q (x) = %1%  <=>  x in Q.

A "proposition about things in the universe", for short, a "proposition", is the same thing as an indicator function, that is, a function of the form f : X -> %B%.  The convenience of this seemingly redundant usage is that it allows one to refer to an indicator function without having to specify right away, as a part of its designated subscript, exactly what set it indicates, even though a proposition always indicates some subset of its designated universe, and even though one will probably or eventually want to know exactly what subset that is.

According to the stated understandings, a proposition is a function that indicates a set, in the sense that a function associates values with the elements of a domain, some of which values can be interpreted to mark out for special consideration a subset of that domain.  The way in which an indicator function is imagined to "indicate" a set can be expressed in terms of the following concepts.

The "fiber" of a codomain element y in Y under a function f : X -> Y is the subset of the domain X that is mapped onto y, in short, it is f^(-1)(y) c X.  In other language that is often used, the fiber of y under f is called the "antecedent set", the "inverse image", the "level set", or the "pre-image" of y under f.  All of these equivalent concepts are defined as follows:
Fiber of y under f  =  f^(-1)(y)  =  {x in X  :  f(x) = y}.

In the special case where f is the indicator function f_Q of the set Q c X, the fiber of 1 under fQ is just the set Q back again:

Fiber of 1 under fQ  =  fQ-1(1)  =  {x in X  :  fQ(x) = 1}  =  Q.

In this specifically boolean setting, as in the more generally logical context, where "truth" under any name is especially valued, it is worth devoting a specialized notation to the "fiber of truth" in a proposition, to mark the set that it indicates with a particular ease and explicitness.

For this purpose, I introduce the use of "fiber bars" or "ground signs", written as "[| ... |]" around a sentence or the sign of a proposition, and whose application is defined as follows:

If  f : X -> %B%,

then  [| f |]  =  f^(-1)(%1%)  =  {x in X  :  f(x) = %1%}.

----

Some may recognize here fledgling efforts
to reinforce flights of Fregean semantics
with impish pitches of Peircean semiotics.
Some may deem it Icarean, all too Icarean.

1.3.10.3  Propositions & Sentences (cont.)

The definition of a fiber, in either the general or the boolean case,
is a purely nominal convenience for referring to the antecedent subset,
the inverse image under a function, or the pre-image of a functional value.
The definition of an operator on propositions, signified by framing the signs
of propositions with fiber bars or ground signs, remains a purely notational
device, and yet the notion of a fiber in a logical context serves to raise
an interesting point.  By way of illustration, it is legitimate to rewrite
the above definition in the following form:

If  f : X -> %B%,

then  [| f |]  =  f^(-1)(%1%)  =  {x in X  :  f(x)}.

The set-builder frame "{x in X  :  ... }" requires a grammatical sentence or
a sentential clause to fill in the blank, as with the sentence "f(x) = %1%"
that serves to fill the frame in the initial definition of a logical fiber.
And what is a sentence but the expression of a proposition, in other words,
the name of an indicator function?  As it happens, the sign "f(x)" and the
sentence "f(x) = %1%" represent the very same value to this context, for
all x in X, that is, they will appear equal in their truth or falsity
to any reasonable interpreter of signs or sentences in this context,
and so either one of them can be tendered for the other, in effect,
exchanged for the other, within this context, frame, and reception.

The sign "f(x)" manifestly names the value f(x).
This is a value that can be seen in many lights.
It is, at turns:

1.  The value that the proposition f has at the point x,
    in other words, the value that f bears at the point x
    where f is being evaluated, the value that f takes on
    with respect to the argument or the object x that the
    whole proposition is taken to be about.

2.  The value that the proposition f not only takes up at
    the point x, but that it carries, conveys, transfers,
    or transports into the setting "{x in X  :  ... }" or
    into any other context of discourse where f is meant
    to be evaluated.

3.  The value that the sign "f(x)" has in the context where it is placed,
    that it stands for in the context where it stands, and that it continues
    to stand for in this context just so long as the same proposition f and the
    same object x are borne in mind.

4.  The value that the sign "f(x)" represents to its full interpretive context
    as being its own logical interpretant, namely, the value that it signifies
    as its canonical connotation to any interpreter of the sign that is cognizant
    of the context in which it appears.

The sentence "f(x) = %1%" indirectly names what the sign "f(x)"
more directly names, that is, the value f(x).  In other words,
the sentence "f(x) = %1%" has the same value to its interpretive
context that the sign "f(x)" imparts to any comparable context,
each by way of its respective evaluation for the same x in X.

What is the relation among connoting, denoting, and "evaluing", where
the last term is coined to describe all the ways of bearing, conveying,
developing, or evolving a value in, to, or into an interpretive context?
In other words, when a sign is evaluated to a particular value, one can
say that the sign "evalues" that value, using the verb in a way that is
categorically analogous or grammatically conjugate to the times when one
says that a sign "connotes" an idea or that a sign "denotes" an object.
This does little more than provide the discussion with a "weasel word",
a term that is designed to avoid the main issue, to put off deciding the
exact relation between formal signs and formal values, and ultimately to
finesse the question about the nature of formal values, whether they are
more akin to conceptual signs and figurative ideas or to the kinds of
literal objects and platonic ideas that are independent of the mind.

These questions are confounded by the presence of certain peculiarities in
formal discussions, especially by the fact that an equivalence class of signs
is tantamount to a formal object.  This has the effect of allowing an abstract
connotation to work as a formal denotation.  In other words, if the purpose of
a sign is merely to lead its interpreter up to a sign in an equivalence class
of signs, then it follows that this equivalence class is the object of the
sign, that connotation can achieve denotation, at least, to some degree,
and that the interpretant domain collapses with the object domain,
at least, in some respect, all things being relative to the
sign relation that embeds the discussion.

Introducing the realm of "values" is a stopgap measure that temporarily
permits the discussion to avoid certain singularities in the embedding
sign relation, and allowing the process of "evaluation" as a compromise
mode of signification between connotation and denotation only manages to
steer around a topic that eventually has to be mapped in full, but these
strategies do allow the discussion to proceed a little further without
having to answer questions that are too difficult to be settled fully
or even tackled directly at this point.  As far as the relations among
connoting, denoting, and evaluing are concerned, it is possible that
all of these constitute independent dimensions of significance that
a sign might be able to enjoy, but since the notion of connotation
is already generic enough to contain multitudes of subspecies, I am
going to subsume, on a tentative basis, all of the conceivable modes
of "evaluing" within the broader concept of connotation.

With this degree of flexibility in mind, one can say that the sentence
"f(x) = %1%" latently connotes what the sign "f(x)" patently connotes.
Taken in abstraction, both syntactic entities fall into an equivalence
class of signs that constitutes an abstract object, a thing of value
that is "identified by" the sign "f(x)", and thus an object that might
as well be "identified with" the value f(x).

The upshot of this whole discussion of evaluation is that it allows one to
rewrite the definitions of indicator functions and their fibers as follows:

The "indicator function" or the "characteristic function" of a set Q c X,
written "f_Q", is the map from X to the boolean domain %B% = {%0%, %1%}
that is defined in the following ways:

1.  Considered in its extensional form, f_Q is the subset of X x %B%
    that is given by the following formula:

    f_Q  =  {<x, b> in X x %B%  :  b  <=>  x in Q}.

2.  Considered in its functional form, f_Q is the map from X to %B%
    that is given by the following condition:

    f_Q (x)  <=>  x in Q.

The "fibers" of truth and falsity under a proposition f : X -> %B%
are subsets of X that are variously described as follows:

1.  The fiber of %1% under f  =  [| f |]  =  f^(-1)(%1%)

                              =  {x in X  :  f(x) = %1%}

                              =  {x in X  :  f(x) }.

2.  The fiber of %0% under f  =  ~[| f |]  =  f^(-1)(%0%)

                              =  {x in X  :  f(x) = %0%}

                              =  {x in X  :  (f(x)) }.

Perhaps this looks like a lot of work for the sake of what seems to be
such a trivial form of syntactic transformation, but it is an important
step in loosening up the syntactic privileges that are held by the sign
of logical equivalence "<=>", as written between logical sentences, and
by the sign of equality "=", as written between their logical values, or
else between propositions and their boolean values.  Doing this removes
a longstanding but wholly unnecessary conceptual confound between the
idea of an "assertion" and notion of an "equation", and it allows one
to treat logical equality on a par with the other logical operations.

----

Where are we?  We just defined the concept of a functional fiber in several
of the most excruciating ways possible, but that's just because this method
of refining functional fibers is intended partly for machine consumputation,
so its schemata must be rendered free of all admixture of animate intuition.
However, just between us, a single picture may suffice to sum up the notion:

|   X-[| f |] ,  [| f |]   c   X
|   o       o   o   o   o      |
|    \     /     \  |  /       |
|     \   /       \ | /        | f
|      \ /         \|/         |
|       o           o          v
|   {  %0%    ,    %1%  }  =  %B%

For the sake of current reference:

| The "fibers" of truth and falsity in a proposition f : X -> %B%
| are the subsets [| f |] and X - [| f |] of X that are variously
| described as follows:
|
| The fiber of %1% under f
|
| =  [| f |]  =  f^(-1)(%1%)
|
| =  {x in X  :  f(x) = %1%}
|
| =  {x in X  :  f(x) }.
|
| The fiber of %0% under f
|
| =  ~[| f |]  =  f^(-1)(%0%)
|
| =   {x in X  :  f(x) = %0%}
|
| =   {x in X  :  (f(x)) }.

Oh, by the way, the outer parentheses in "(f(g))" signify negation.
I did not have here the "stricken parentheses" that I normally use.

Why are we doing this?  The immediate reason -- whose critique I defer --
has to do with finding a modus vivendi, whether a working compromise or
a genuine integration, between the assertive-declarative languages and
the functional-procedural languages that we have available for the sake
of conceptual-logical-ontological analysis, clarification, description,
inference, problem-solving, programming, representation, or whatever.

In the next few installments, I will be working toward the definition
of an operation called the "stretch".  This is related to the concept
from category theory that is called a "pullback".  As a few will know
the uses of that already, maybe there's hope of stretching the number.

----

In this episode, I compile a collection of definitions,
leading up to the particular conception of a "sentence"
that I'll be using throughout the rest of this inquiry.

1.3.10.3  Propositions & Sentences (cont.)

As a purely informal aid to interpretation, I frequently use the letters
"p", "q" to denote propositions.  This can serve to tip off the reader
that a function is intended as the indicator function of a set, and
it saves us the trouble of declaring the type f : X -> %B% each
time that a function is introduced as a proposition.

Another convention of use in this context is to let boldface letters
stand for k-tuples, lists, or sequences of objects.  Typically, the
elements of the k-tuple, list, or sequence are all of one type, and
typically the boldface letter is of the same basic character as the
indexed or subscripted letters that are used denote the components
of the k-tuple, list, or sequence.  When the dimension of elements
and functions is clear from the context, we may elect to drop the
bolding of characters that name k-tuples, lists, and sequences.

For example:

1.  If x_1, ..., x_k in X,       then #x# = <x_1, ..., x_k> in X' = X^k.

2.  If x_1, ..., x_k  : X,       then #x# = <x_1, ..., x_k>  : X' = X^k.

3.  If f_1, ..., f_k  : X -> Y,  then #f# = <f_1, ..., f_k>  : (X -> Y)^k.

There is usually felt to be a slight but significant distinction between
the "membership statement" that uses the sign "in" as in Example (1) and
the "type statement" that uses the sign ":" as in examples (2) and (3).
The difference that appears to be perceived in categorical statements,
when those of the form "x in X" and those of the form "x : X" are set
in side by side comparisons with each other, is that a multitude of
objects can be said to have the same type without having to posit
the existence of a set to which they all belong.  Without trying
to decide whether I share this feeling or even fully understand
the distinction in question, I can only try to maintain a style
of notation that respects it to some degree.  It is conceivable
that the question of belonging to a set is rightly sensed to be
the more serious matter, one that has to do with the reality of
an object and the substance of a predicate, than the question of
falling under a type, that may have more to do with the way that
a sign is interpreted and the way that information about an object
is organized.  When it comes to the kinds of hypothetical statements
that appear in these Examples, those of the form "x in X => #x# in X'"
and "x : X => #x# : X'", these are usually read as implying some order
of synthetic construction, one whose contingent consequences involve the
constitution of a new space to contain the elements being compounded and
the recognition of a new type to characterize the elements being moulded,
respectively.  In these applications, the statement about types is again
taken to be less presumptive than the corresponding statement about sets,
since the apodosis is intended to do nothing more than to abbreviate and
to summarize what is already stated in the protasis.

A "boolean connection" of degree k, also known as a "boolean function"
on k variables, is a map of the form F : %B%^k -> %B%.  In other words,
a boolean connection of degree k is a proposition about things in the
universe X = %B%^k.

An "imagination" of degree k on X is a k-tuple of propositions about things
in the universe X.  By way of displaying the various kinds of notation that
are used to express this idea, the imagination #f# = <f_1, ..., f_k> is given
as a sequence of indicator functions f_j : X -> %B%, for j = 1 to k.  All of
these features of the typical imagination #f# can be summed up in either one
of two ways:  either in the form of a membership statement, to the effect that
#f# is in (X -> %B%)^k, or in the form of a type statement, to the effect that
#f# : (X -> %B%)^k, though perhaps the latter form is slightly more precise than
the former.

The "play of images" that is determined by #f# and x, more specifically,
the play of the imagination #f# = <f_1, ..., f_k> that has to with the
element x in X, is the k-tuple #b# = <b_1, ..., b_k> of values in %B%
that satisfies the equations b_j = f_j (x), for all j = 1 to k.

A "projection" of %B%^k, typically denoted by "p_j" or "pr_j",
is one of the maps p_j : %B%^k -> %B%, for j = 1 to k, that is
defined as follows:

If         #b#   =       <b_1, ..., b_k>           in  %B%^k,

then  p_j (#b#)  =  p_j (<b_1, ..., b_k>)  =  b_j  in  %B%.

The "projective imagination" of %B%^k is the imagination <p_1, ..., p_k>.

A "sentence about things in the universe", for short, a "sentence",
is a sign that denotes a proposition.  In other words, a sentence is
any sign that denotes an indicator function, any sign whose object is
a function of the form f : X �> B.

To emphasize the empirical contingency of this definition, one can say
that a sentence is any sign that is interpreted as naming a proposition,
any sign that is taken to denote an indicator function, or any sign whose
object happens to be a function of the form f : X �> B.

----

I finish out the Subsection on "Propositions & Sentences" with
an account of how I use concepts like "assertion" and "denial".

1.3.10.3  Propositions & Sentences (cont.)

An "expression" is a type of sign, for instance, a term or a sentence,
that has a value.  In forming this conception of an expression, I am
deliberately leaving a number of options open, for example, whether
the expression amounts to a term or to a sentence and whether it
ought to be accounted as denoting a value or as connoting a value.
Perhaps the expression has different values under different lights,
and perhaps it relates to them differently in different respects.
In the end, what one calls an expression matters less than where
its value lies.  Of course, no matter whether one chooses to call
an expression a "term" or a "sentence", if the value is an element
of %B%, then the expression affords the option of being treated as
a sentence, meaning that it is subject to assertion and composition
in the same way that any sentence is, having its value figure into
the values of larger expressions through the linkages of sentential
connectives, and affording us the consideration of what things in
what universe the corresponding proposition happens to indicate.

Expressions with this degree of flexibility in the types under
which they can be interpreted are difficult to translate from
their formal settings into more natural contexts.  Indeed,
the whole issue can be difficult to talk about, or even
to think about, since the grammatical categories of
sentential clauses and noun phrases are rarely so
fluid in natural language settings are they can
be rendered in artificially formal arenas.

To finesse the issue of whether an expression denotes or connotes its value,
or else to create a general term that covers what both possibilities have
in common, one can say that an expression "evalues" its value.

An "assertion" is just a sentence that is being used in a certain way,
namely, to indicate the indication of the indicator function that the
sentence is usually used to denote.  In other words, an assertion is
a sentence that is being converted to a certain use or that is being
interpreted in a certain role, and one whose immediate denotation is
being pursued to its substantive indication, specifically, the fiber
of truth of the proposition that the sentence potentially denotes.
Thus, an assertion is a sentence that is held to denote the set of
things in the universe for which the sentence is held to be true.

Taken in a context of communication, an assertion is basically a request
that the interpreter consider the things for which the sentence is true,
in other words, to find the fiber of truth in the associated proposition,
or to invert the indicator function that is denoted by the sentence with
respect to its possible value of truth.

A "denial" of a sentence z is an assertion of its negation -(z)-.
The denial acts as a request to think about the things for which the
sentence is false, in other words, to find the fiber of falsity in the
indicted proposition, or to invert the indicator function that is being
denoted by the sentence with respect to its possible value of falsity.

According to this manner of definition, any sign that happens to denote
a proposition, any sign that is taken as denoting an indicator function,
by that very fact alone successfully qualifies as a sentence.  That is,
a sentence is any sign that actually succeeds in denoting a proposition,
any sign that one way or another brings to mind, as its actual object,
a function of the form f : X �> B.

There are many features of this definition that need to be understood.
Indeed, there are problems involved in this whole style of definition
that need to be discussed, and doing this requires a slight excursion.

1.3.10.4. Empirical Types and Rational Types

In this subsection, I want to examine the style of definition that I used to define a sentence as a type of sign, to adapt its application to other problems of defining types, and to draw a lesson of general significance.

Notice that I am defining a sentence in terms of what it denotes, and not in terms of its structure as a sign.  In this way of reckoning, a sign is not a sentence on account of any property that it has in itself, but only due to the sign relation that actually happens to interpret it.  This makes the property of being a sentence a question of actualities and contingent relations, not merely a question of potentialities and absolute categories.  This does nothing to alter the level of interest that one is bound to have in the structures of signs, it merely shifts the import of the question from the logical plane of definition to the pragmatic plane of effective action.  As a practical matter, of course, some signs are better for a given purpose than others, more conducive to a particular result than others, and more effective in achieving an assigned objective than others, and the reasons for this are at least partly explained by the relationships that can be found to exist among a sign's structure, its object, and the sign relation that fits them.

Notice the general character of this development.  I start by defining a type of sign according to the type of object that it happens to denote, ignoring at first the structural potential that it brings to the task.  According to this mode of definition, a type of sign is singled out from other signs in terms of the type of object that it actually denotes and not according to the type of object that it is designed or destined to denote, nor in terms of the type of structure that it possesses in itself.  This puts the empirical categories, the classes based on actualities, at odds with the rational categories, the classes based on intentionalities.
In hopes that this much explanation is enough to rationalize the account of types that I am using, I break off the digression at this point and return to the main discussion.

1.3.10.5. Articulate Sentences

A sentence is "articulate" (1) if it has a significant form, a compound constitution, or a non�trivial structure as a sign, and (2) if there is an informative relationship that exists between its structure as a sign and the proposition that it happens to denote.  A sentence of this kind is typically given in the form of a "description", an "expression", or a "formula", in other words, as an articulated sign or a well�structured element of a formal language.  As a general rule, the class of sentences that one is willing to contemplate is compiled from a particular brand of complex signs and syntactic strings, those that are put together from the basic building blocks of a formal language and held in a special esteem for the roles that they play within its grammar.  However, even if a typical sentence is a sign that is generated by a formal regimen, having its form, its meaning, and its use governed by the principles of a comprehensive grammar, the class of sentences that one has a mind to contemplate can also include among its number many other signs of an arbitrary nature.

Frequently this "formula" has a "variable" in it that "ranges over" the universe U.  A "variable" is an ambiguous or equivocal sign that can be interpreted as denoting any element of the set that it "ranges over".

If a sentence denotes a proposition f : U �> B, then the "value" of the sentence with regard to u C U is the value f(u) of the proposition at u, where "0" is interpreted as "false" and "1" is interpreted as "true".

Since the value of a sentence or a proposition depends on the universe of discourse to which it is "referred", and since it also depends on the element of the universe with regard to which it is evaluated, it is usual to say that a sentence or a proposition "refers" to a universe and to its elements, though perhaps in a variety of different senses.  Furthermore, a proposition, acting in the role of as an indicator function, "refers" to the elements that it "indicates", namely, the elements on which it takes a positive value.  In order to sort out the possible confusions that are capable of arising here, I need to examine how these various notions of reference are related to the notion of denotation that is used in the pragmatic theory of sign relations.

One way to resolve the various senses of "reference" that arise in this setting is to make the following sorts of distinctions among them.  Let the reference of a sentence or a proposition to a universe of discourse, the one that it acquires by way of taking on any interpretation at all, be taken as its "general reference", the kind of reference that one can safely ignore as irrelevant, at least, so long as one stays immersed in only one context of discourse or only one moment of discussion.  Let the references that an indicator function f has to the elements on which it evaluates to 0 be called its "negative references".  Let the references that an indicator function f has to the elements on which it evaluates to 1 be called its "positive references" or its "indications".  Finally, unspecified references to the "references" of a sentence, a proposition, or an indicator function can be taken by default as references to their specific, positive references.

The universe of discourse for a sentence, the set whose elements the sentence is interpreted to be about, is not a property of the sentence by itself, but of the sentence in the presence of its interpretation.  Independently of how many explicit variables a sentence contains, its value can always be interpreted as depending on any number of implicit variables.  For instance, even a sentence with no explicit variable, a constant expression like "0" or "1", can be taken to denote a constant proposition of the form c : U �> B.  Whether or not it has an explicit variable, I always take a sentence as referring to a proposition, one whose values refer to elements of a universe U.

Notice that the letters "P" and "Q", interpreted as signs that denote indicator functions P, Q : U �> B, have the character of sentences in relation to propositions, at least, they have the same status in this abstract discussion as genuine sentences have in concrete discussions.  This illustrates the relation between sentences and propositions as a special case of the relation between signs and objects.

To assist the reading of informal examples, I frequently use the letters "s", "t", and "S", "T" to denote sentences.  Thus, it is conceivable to have a situation where S = "P" and where P : U �> B.  Altogether, this means that the sign "S" denotes the sentence S, that the sentence S is the sentence "P", and that the sentence "P" denotes the proposition or the indicator function P : U �> B.  In settings where it is necessary to keep track of a large number of sentences, I use subscripted letters like "e1", ..., "en" to refer to the various expressions.

A "sentential connective" is a sign, a coordinated sequence of signs, a significant pattern of arrangement, or any other syntactic device that can be used to connect a number of sentences together in order to form a single sentence.  If k is the number of sentences that are connected, then the connective is said to be of order k.  If the sentences acquire a logical relationship by this means, and are not just strung together by this mechanism, then the connective is called a "logical connective".  If the value of the constructed sentence depends on the values of the component sentences in such a way that the value of the whole is a boolean function of the values of the parts, then the connective is called a "propositional connective".

1.3.10.6. Stretching Principles

There is a principle, of constant use in this work, that needs to be made explicit.  In order to give it a name, I refer to this idea as the "stretching principle".  Expressed in different ways, it says that:

1.	Any relation of values extends to a relation of what is valued.

2.	Any statement about values says something about the things that are given these values.

3.	Any association among a range of values establishes an association among the domains of things that these values are the values of.

4.	Any connection between two values can be stretched to create a connection, of analogous form, between the objects, persons, qualities, or relationships that are valued in these connections.

5.	For every operation on values, there is a corresponding operation on the actions, conducts, functions, procedures, or processes that lead to these values, as well as there being analogous operations on the objects that instigate all of these various proceedings.
Nothing about the application of the stretching principle guarantees that the analogues it generates will be as useful as the material it works on.  It is another question entirely whether the links that are forged in this fashion are equal in their strength and apposite in their bearing to the tried and true utilities of the original ties, but in principle they are always there.

In particular, a connection F : Bk �> B can be understood to indicate a relation among boolean values, namely, the k�ary relation F�1(1) c Bk.  If these k values are values of things in a universe U, that is, if one imagines each value in a k�tuple of values to be the functional image that results from evaluating an element of U under one of its possible aspects of value, then one has in mind the k propositions fj : U �> B, for j = 1 to k, in sum, one embodies the imagination f = <f1, ..., fk>.  Together, the imagination f ? (U �> B)k and the connection F : Bk �> B stretch each other to cover the universe U, yielding a new proposition P : U �> B.
To encapsulate the form of this general result, I define a composition that takes an imagination f = <f1, ..., fk> C (U �> B)k and a boolean connection F : Bk �> B and gives a proposition P : U �> B.  Depending on the situation, specifically, according to whether many F and many f, a single F and many f, or many F and a single f are being considered, respectively, I refer to this P under one of three descriptions:

1.	In a general setting, where the connection F and the imagination f are both permitted to take up a variety of concrete possibilities, call P the "stretch of F and f from U to B", and write it in the style of a composition as "F $ f".  This is meant to suggest that the symbol "$", here read as "stretch", denotes an operator of the form 

$ : (Bk �> B) x (U �> B)k �> (U �> B).

2.	In a setting where the connection F is fixed but the imagination f is allowed to vary over a wide range of possibilities, call P the "stretch of F to f on U", and write it in the style "F$f", exactly as if "F$" denotes an operator F$ : (U �> B)k �> (U �> B) that is derived from F and applied to f, ultimately yielding a proposition F$f : U �> B.

3.	In a setting where the imagination f is fixed but the connection F is allowed to range over wide variety of possibilities, call P the "stretch of f by F to B", and write it in the style "f$F", exactly as if "f$" denotes an operator f$ : (Bk �> B) �> (U �> B) that is derived from f and applied to F, ultimately yielding a proposition f$F : U �> B.
Because this notation is only used in settings where the imagination f : (U �> B)k and the connection F : Bk �> B are distinguished by their types, it does not really matter whether one writes "F $ f" or "f $ F" for the initial composition.

Just as a sentence is a sign that denotes a proposition, which thereby serves to indicate a set, a propositional connective is a provision of syntax whose mediate effect is to denote an operation on propositions, which thereby manages to indicate the result of an operation on sets.  In order to see how these compound forms of indication can be defined, it is useful to go through the steps that are needed to construct them.  In general terms, the ingredients of the construction are as follows:

1.	An imagination of degree k on U, in other words, a k�tuple of propositions fj : U �> B, for j = 1 to k, or an object of the form f = <f1, ..., fk> : (U �> B)k.

2.	A connection of degree k, in other words, a proposition about things in Bk, or a boolean function of the form F : Bk �> B.
From these materials, it is required to construct a proposition P : U �> B such that P(u) = F(f1(u), ..., fk(u)), for all u C U.  The desired construction is determined as follows:

The cartesian power Bk, as a cartesian product, is characterized by the possession of a "projective imagination" p = <p1, ..., pk> of degree k on Bk, along with the property that any imagination f = <f1, ..., fk> of degree k on an arbitrary set W determines a unique map f! : W �> Bk, the play of whose projective images <p1(f!(w), ..., pk(f!(w)) on the functional image f!(w) matches the play of images <f1(w), ..., fk(w)> under f, term for term and at every element w in W.

Just to be on the safe side, I state this again in more standard terms.  The cartesian power Bk, as a cartesian product, is characterized by the possession of k projection maps pj : Bk �> B, for j = 1 to k, along with the property that any k maps fj : W �> B, from an arbitrary set W to B, determine a unique map f! : W �> Bk such that pj(f!(w)) = fj(w), for all j = 1 to k, and for all w C W.

Now suppose that the arbitrary set W in this construction is just the relevant universe U.  Given that the function f! : U �> Bk is uniquely determined by the imagination f : (U �> B)k, that is, by the k�tuple of propositions f = <f1, ..., fk>, it is safe to identify f! and f as being a single function, and this makes it convenient on many occasions to refer to the identified function by means of its explicitly descriptive name "<f1, ..., fk>".  This facility of address is especially appropriate whenever a concrete term or a constructive precision is demanded by the context of discussion.

1.3.10.7. Stretching Operations

The preceding discussion of stretch operations is slightly more general than is called for in the present context, and so it is probably a good idea to draw out the particular implications that are needed right away.

If F : Bk �> B is a boolean function on k variables, then it is possible to define a mapping F$ : (U �> B)k �> (U �> B), in effect, an operation that takes k propositions into a single proposition, where F$ satisfies the following conditions:

F$(f1, ..., fk)	:	U �> B
:
F$(f1, ..., fk)(u)	=	F(f(u))

	=	F(<f1, ..., fk>(u))

	=	F(f1(u), ..., fk(u)).

Thus, F$ is what a propositional connective denotes, a particular way of connecting the propositions that are denoted by a number of sentences into a proposition that is denoted by a single sentence.

Now "fX" is sign that denotes the proposition fX, and it certainly seems like a sufficient sign for it.  Why is there is a need to recognize any other signs of it?
If one takes a sentence as a type of sign that denotes a proposition and a proposition as a type of function whose values serve to indicate a set, then one needs a way to grasp the overall relation between the sentence and the set as taking place within a "higher order" (HO) sign relation.

Roughly sketched, the relations of denotation and indication that exist among sets, propositions, sentences, and values can be diagrammed as in Table 10.

Table 10.  Levels of Indication
Object
Sign
Higher Order Sign
Set
Proposition
Sentence
f-1(v)
f
“F”
X
1
“1”
~X
0
“0” 		Sign	Higher Order Sign
	Set	Proposition	Sentence
	f�1(v)	f	"f"
	X	1	"1"

	~X	0	"0"

Strictly speaking, a proposition is too abstract to be a sign, and so the contents of Table 10 have to be taken with the indicated grains of salt.  Propositions, as indicator functions, are abstract mathematical objects, not any kinds of syntactic elements, and so propositions cannot literally constitute the orders of concrete signs that remain of ultimate interest in the pragmatic theory of signs, or in any theory of effective meaning.  Therefore, it needs to be understood that a proposition f can be said to "indicate" a set X only insofar as the values of 1 and 0 that it assigns to the elements of the universe U are positive and negative indications, respectively, of the elements in X, and thus indications of the set X and of its complement ~X = U � X, respectively.  It is actually these values, when rendered by a concrete implementation of the indicator function f, that are the actual signs of the objects that are inside the set X and the objects that are outside the set X, respectively.

In order to deal with the HO sign relations that are involved in this situation, I introduce a couple of new notations:

1.	To mark the relation of denotation between a sentence S and the proposition that it denotes, let the "spiny bracket" notation "[S]" be used for "the indicator function denoted by the sentence S".

2.	To mark the relation of denotation between a proposition P and the set that it indicates, let the "spiny brace" notation "{X}" be used for "the indicator function of the set X".

Notice that the spiny bracket operator "[ ]" takes one "downstream", in accord with the usual direction of denotation, from a sign to its object, while the spiny brace operator "{ }" takes one "upstream", against the usual direction of denotation, and thus from an object to its sign.
In order to make these notations useful in practice, it is necessary to note of a couple of their finer points, points that might otherwise seem too fine to take much trouble over.  For this reason, I express their usage a bit more carefully as follows:

1.	Let "spiny brackets", like "[ ]", be placed around a name of a sentence S, as in the expression "[S]", or else around a token appearance of the sentence itself, to serve as a name for the proposition that S denotes.

2.	Let "spiny braces", like "{ }", be placed around a name of a set X, as in the expression "{X}", to serve as a name for the indicator function fX.

Table 11 illustrates the use of this notation, listing in each column several different but equivalent ways of referring to the same entity.

Table 11.  Illustrations of Notation
	Object	Sign	Higher Order Sign
	Set	Proposition	Sentence
	X	P	S

	|[S]|	[S]	S

	|P|	P	"P"

	|fX|	fX	"fX"

	X	{X}	"{X}"

In particular, one can observe the following relations and formulas, all of a purely notational character:

1.	If the sentence S denotes the proposition P : U �> B, then [S] = P.

2.	If the sentence S denotes the proposition P : U �> B
such that |P| = P�1(1) = X c U, then [S] = P = fX = {X}.

3.	X	=	{u C U : u C X}

		=	|{X}|	=	{X}�1(1)

		=	|fX|	=	fX�1(1).
4.	{X}	=	{ {u C U : u C X} }

		=	[u C X]

		=	fX.

Now if a sentence S really denotes a proposition P, and if the notation "[S]" is merely meant to supply another name for the proposition that S already denotes, then why is there any need for the additional notation?  It is because the interpretive mind habitually races from the sentence S, through the proposition P that it denotes, and on to the set X = P�1(1) that the proposition P indicates, often jumping to the conclusion that the set X is the only thing that the sentence S is intended to denote.  This HO sign situation and the mind's inclination when placed within its setting calls for a linguistic mechanism or a notational device that is capable of analyzing the compound action and controlling its articulate performance, and this requires a way to interrupt the flow of assertion that typically takes place from S to P to X.

1.3.10.8. The Cactus Patch

Thus, what looks to us like a sphere of scientific knowledge more accurately should be represented as the inside of a highly irregular and spiky object, like a pincushion or porcupine, with very sharp extensions in certain directions, and virtually no knowledge in immediately adjacent areas.  If our intellectual gaze could shift slightly, it would alter each quill's direction, and suddenly our entire reality would change.
		(Herbert Bernstein, NWOK, 38).
In this and the four subsections that follow, I describe a calculus for representing propositions as sentences, in other words, as syntactically defined sequences of signs, and for manipulating these sentences chiefly in the light of their semantically defined contents, in other words, with respect to their logical values as propositions.  In their computational representation, the expressions of this calculus parse into a class of tree�like data structures called "painted cacti".  This is a family of graph�theoretic data structures that can be observed to have especially nice properties, turning out to be not only useful from a computational standpoint but also quite interesting from a theoretical point of view.  The rest of this subsection serves to motivate the development of this calculus and treats a number of general issues that surround the topic.
In order to facilitate the use of propositions as indicator functions it helps to acquire a flexible notation for referring to propositions in that light, for interpreting sentences in a corresponding role, and for negotiating the requirements of mutual sense between these two domains.  If none of the formalisms that are readily available or in common use are able to meet the design requirements that come to mind, then it is necessary to contemplate the design of a new language that is especially tailored to the purpose.  In the present application, there is a pressing need to devise a general calculus for composing propositions, computing their values on particular arguments, and inverting their indications to arrive at the sets of things in the universe that are indicated by them.
For computational purposes, it is convenient to have a middle ground or an intermediate language for negotiating between the koine of sentences regarded as strings of literal characters and the realm of propositions regarded as objects of logical value, even if this renders it necessary to introduce an artificial medium of exchange between these two domains.  If one envisions these computations to be carried out in any organized fashion, and ultimately or partially by means of the familiar sorts of machines, then the strings that represent these logical propositions are likely to find themselves parsed into tree�like data structures at some stage of the game.  With regard to their abstract structures as graphs, there are several species of graph�theoretic data structures that can be used to accomplish this job in a reasonably effective and efficient way.
Over the course of this project, I plan to use two species of graphs:
1.  "painted and rooted cacti" (PARCA).
2.  "painted and rooted conifers" (PARCO).
For now, it is enough to discuss the former class of data structures, leaving the consideration of the latter class to a part of the project where their distinctive features are key to developments at that stage.  Accordingly, within the context of the current patch of discussion, or until it becomes necessary to attach further notice to the conceivable varieties of parse graphs, the acronym "PARC" is sufficient to indicate the pertinent genus of abstract graph that is under consideration.
By way of making these tasks feasible to carry out on a regular basis, a prospective language designer is required not only to supply a fluent medium for the expression of propositions, but further to accompany the assertions of their sentences with a canonical mechanism for teasing out the fibers of their indicator functions.  Accordingly, with regard to a body of conceivable propositions, one needs to furnish a standard array of techniques for following the threads of their indications from their objective universe to their values for the mind and back again, that is, for tracing the clues that sentences provide from the universe of their objects to the signs of their values, and, in turn, from signs to objects.  Ultimately, one seeks to render propositions so functional as indicators of sets and so essential for examining the equality of sets that they can constitute a veritable criterion for the practical conceivability of sets.  Tackling this task requires me to introduce a number of new definitions and a collection of additional notational devices, to which I now turn.
Depending on whether a formal language is called by the type of sign that makes it up or whether it is named after the type of object that its signs are intended to denote, one may refer to this cactus language as a "sentential calculus" or as a "propositional calculus", respectively.
When the syntactic definition of the language is well enough understood, then the language can begin to acquire a semantic function.  In natural circumstances, the syntax and the semantics are likely to be engaged in a process of co�evolution, whether in ontogeny or in phylogeny, that is, the two developments probably form parallel sides of a single bootstrap.  But this is not always the easiest way, at least, at first, to formally comprehend the nature of their action or the power of their interaction.
According to the customary mode of formal reconstruction, the language is first presented in terms of its syntax, in other words, as a formal language of strings called "sentences", amounting to a particular subset of the possible strings that can be formed on a finite alphabet of signs.  A syntactic definition of the "cactus language", one that proceeds along purely formal lines, is carried out in the next subsection.  After that, the development of the language's more concrete aspects can be seen as a matter of defining two functions:  The first is a function that takes each sentence of the language into a computational data structure, to be exact, a tree�like parse graph called a "painted cactus".  The second is a function that takes each sentence of the language, or its interpolated parse graph, into a logical proposition, in effect, ending up with an indicator function as the object denoted by the sentence.
The discussion of syntax brings up a number of associated issues that have to be clarified before going on.  These are questions of "style", that is, the sort of description, "grammar", or theory that one finds available or chooses as preferable for a given language.  These issues are discussed in the subsection after next (Subsection 10).
There is an aspect of syntax that is so schematic in its basic character that it can be conveyed by computational data structures, so algorithmic in its uses that it can be automated by routine mechanisms, and so fixed in its nature that its practical exploitation can be served by the usual devices of computation.  Because it involves the transformation of signs, it can be recognized as an aspect of semiotics.  Since it can be carried out in abstraction from meaning, it is not up to the level of semantics, much less a complete pragmatics, though it does incline to the pragmatic aspects of computation that are auxiliary to and incidental to the human use of language.  Therefore, I refer to this aspect of formal language use as the "algorithmics" or the "mechanics" of language processing.  A mechanical conversion of the "cactus language" into its associated data structures is discussed in Subsection 11.
In the usual way of proceeding on formal grounds, meaning is added by giving each "grammatical sentence", or each syntactically distinguished string, an interpretation as a logically meaningful sentence, in effect, providing each abstractly well�formed sentence with a proposition for it to denote.  A semantic interpretation of the "cactus language" is carried out in Subsection 12.

References

• Aristotle, "On The Soul", in Aristotle, Volume 8, W.S. Hett (trans.), William Heinemann, London, UK, 1936, 1986.

• Charniak, E., and McDermott, D.V. (1985), Introduction to Artificial Intelligence, Addison-Wesley, Reading, MA.

• Charniak, E., Riesbeck, C.K., and McDermott, D.V. (1980), Artificial Intelligence Programming, Lawrence Erlbaum Associates, Hillsdale, NJ.

• Dewey, John (1910/1991), How We Think, Prometheus Books, Buffalo, NY. Originally published 1910.

• Holland, J.H., Holyoak, K.J., Nisbett, R.E., and Thagard, P.R. (1986), Induction : Processes of Inference, Learning, and Discovery, MIT Press, Cambridge, MA.

• O'Rorke, P. (1990), "Review of AAAI 1990 Spring Symposium on Automated Abduction", SIGART Bulletin, Vol. 1, No. 3, ACM Press, October 1990, pp. 12–17.

• Pearl, J. (1991), Probabilistic Reasoning in Intelligent Systems : Networks of Plausible Inference, Revised 2nd printing, Morgan Kaufmann, San Mateo, CA.

• Peng, Y., and Reggia, J.A. (1990), Abductive Inference Models for Diagnostic Problem-Solving, Springer-Verlag, New York, NY.

• Shakespeare, William (1988), William Shakespeare : The Complete Works, Compact Edition, S. Wells and G. Taylor (eds.), Oxford University Press, Oxford, UK.

• Sowa, J.F. (1984), Conceptual Structures : Information Processing in Mind and Machine, Addison-Wesley, Reading, MA.

• Sowa, J.F. (ed., 1991), Principles of Semantic Networks : Explorations in the Representation of Knowledge, Morgan Kaufmann, San Mateo, CA.

Document History

Title:	Inquiry Driven Systems : An Inquiry Into Inquiry
Author:	Jon Awbrey
Revised:	14 Apr 2004, Draft 10.04
Revised:	07 Apr 2003, Draft 10.01
Revised:	02 Mar 2003, Draft 10.00
Revised:	23 Jun 2002, Draft 8.76
Revised:	10 Jun 2002, Draft 8.75
Revised:	06 Jan 2002, Draft 8.70
Revised:	08 Jan 2001, Draft 8b
Revised:	30 Jun 2000, Draft 8.2
Revised:	01 Oct 1999, Draft 7e
Revised:	23 Jun 1999, Draft 7b
Created:	23 Jun 1996
Advisor:	M.A. Zohdy
Setting:	Oakland University, Rochester, Michigan, USA

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