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.

Focus on a problematic phenomenon. This is a generic property or process that attracts one's interest, like intelligence or inquiry.
Gather under consideration significant examples of concrete systems or agents that exhibit the property or process in question.
Reflect on their common properties in a search for less obvious traits that might explain their more surprising features.
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:

Examine every notion of the casual intuition that enters into the informal discussion and inquire into its qualifications as a potential candidate for formalization.
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.
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:

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.
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.
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:

J_OS	=	Den(L_J)	=	Proj_OSL_J	=	(L_J)_OS	=	L(J)_OS
J_SI	=	Con(L_J)	=	Proj_SIL_J	=	(L_J)_SI	=	L(J)_SI
J_OI	=	Int(L_J)	=	Proj_OIL_J	=	(L_J)_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:

Represents the embodiment of significant properties that have objective reality in the agent's domain.
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:

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

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"}.
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.

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.
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.
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:

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:

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.
In relative terms, by specifying a converse pair of dyadic relations that (redundantly) determine two sets of facts:

What is an instance, example, member, or element of what, relative to the OM in question.
What is a property, quality, class, or set of what, relative to the OM in question.

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:

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.
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:

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.
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:

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.
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.

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.
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:

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.
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.

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.
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.
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:

Inquiry into propositions about application and equality.
Start with the formula $y_0 = y \cdot y$ itself.
Inquiry into application ( $\cdot$ ).
Inquiry into equality ($=\!$).
Inquiry into indices (for example, the $0$ in $y_0\!$).
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?
Inquiry into decomposition or subordination ($\succ$).
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:

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.
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:

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.
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.
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:

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.
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:

What is a question, for instance, this one?
How do questions arise, for instance, this one?
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:

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.
To re-pose these questions in a system-theoretic context is to inquire into the notion of a state of question, asking:

What sort of system is involved in its conception?
How does it arise within such a system?
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:

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.
The agent can resort to abstraction. …
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:

There are the WOT's that are prevalent in various communities of cultural, literary, practical, scientific, and technical discourse.
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:

To continue to illustrate the salient properties of sign relations in the medium of selected examples.
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.
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:

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.
To use this modicum of formalism to define a number of conceptual constructs, useful in the analysis of more general sign relations.
To develop a proof format that is suitable for deriving facts about these constructs in a careful and potentially computational manner.
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:

Syntactic domain $X\!$ = Sign domain $S\!$ = Interpretant domain $I.\!$
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 $^{\backprime\backprime} \, X\!-\!Q \, ^{\prime\prime}$ and is defined as the set of elements in $X\!$ that do not belong to $Q\!$. When the universe $X\!$ is fixed throughout a given discussion, the complement $X\!-\!Q$ may be denoted either by $^{\backprime\backprime} \thicksim \! Q \, ^{\prime\prime}$ or by $^{\backprime\backprime} \, \tilde{Q} \, ^{\prime\prime}$. Thus we have the following series of equivalences:

$\begin{array}{lllllll} \tilde{Q} & = & \thicksim \! Q & = & X\!-\!Q & = & \{ \, x \in X : \underline{(} x \in Q \underline{)} \, \}. \\ \end{array}$

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:

Finding the values of given functions for given arguments.
Inverting boolean functions, that is, finding the fibers of boolean functions, or solving logical equations that are expressed in terms of boolean functions.
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:

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

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

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.
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.
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.
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:

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

$f_Q ~=~ \{ (x, y) \in X \times \underline\mathbb{B} ~:~ y ~\Leftrightarrow~ x \in Q \}.$
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 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 $\underline\mathbb{B},$ 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 affording 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 contrived 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 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 discourse for which the sentence is held to be true.

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

A denial of a sentence $s\!$ is an assertion of its negation $^{\backprime\backprime} \, \underline{(} s \underline{)} \, ^{\prime\prime}.$ 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 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 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 Segment, 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.

I defined 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 the sign itself 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 $X.\!$ 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 $f : X \to \underline\mathbb{B},$ then the value of the sentence with regard to $x \in X$ is the value $f(x)\!$ of the proposition at $x,\!$ where $^{\backprime\backprime} \underline{0} ^{\prime\prime}$ is interpreted as false and $^{\backprime\backprime} \underline{1} ^{\prime\prime}$ 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 $\underline{0}$ be called its negative references. Let the references that an indicator function $f\!$ has to the elements on which it evaluates to $\underline{1}$ 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 $^{\backprime\backprime} \underline{0} ^{\prime\prime}$ or $^{\backprime\backprime} \underline{1} ^{\prime\prime},$ can be taken to denote a constant proposition of the form $c : X \to \underline\mathbb{B}.$ 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 $X.\!$

Notice that the letters $^{\backprime\backprime} p ^{\prime\prime}$ and $^{\backprime\backprime} q ^{\prime\prime},$ interpreted as signs that denote indicator functions $p, q : X \to \underline\mathbb{B},$ 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 $^{\backprime\backprime} s ^{\prime\prime}$ and $^{\backprime\backprime} t ^{\prime\prime},$ to denote sentences. Thus, it is conceivable to have a situation where $s ~=~ ^{\backprime\backprime} p ^{\prime\prime}$ and where $p : X \to \underline\mathbb{B}.$ Altogether, this means that the sign $^{\backprime\backprime} s ^{\prime\prime}$ denotes the sentence $s,\!$ that the sentence $s\!$ is the sentence $^{\backprime\backprime} p ^{\prime\prime},$ and that the sentence $^{\backprime\backprime} p ^{\prime\prime}$ denotes the proposition or the indicator function $p : X \to \underline\mathbb{B}.$ In settings where it is necessary to keep track of a large number of sentences, I use subscripted letters like ${}^{\backprime\backprime} e_1 {}^{\prime\prime}, \, \ldots, \, {}^{\backprime\backprime} e_n {}^{\prime\prime}$ 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:

Any relation of values extends to a relation of what is valued.
Any statement about values says something about the things that are given these values.
Any association among a range of values establishes an association among the domains of things that these values are the values of.
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.
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 $F : \underline\mathbb{B}^k \to \underline\mathbb{B}$ can be understood to indicate a relation among boolean values, namely, the $k\!$-ary relation $F^{-1} (\underline{1}) \subseteq \underline\mathbb{B}^k.$ If these $k\!$ values are values of things in a universe $X,\!$ 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 $X\!$ under one of its possible aspects of value, then one has in mind the $k\!$ propositions $f_j : X \to \underline\mathbb{B},$ for $j = 1 ~\text{to}~ k,$ in sum, one embodies the imagination $\underline{f} = (f_1, \ldots, f_k).$ Together, the imagination $\underline{f} \in (X \to \underline\mathbb{B})^k$ and the connection $F : \underline\mathbb{B}^k \to \underline\mathbb{B}$ stretch each other to cover the universe $X,\!$ yielding a new proposition $p : X \to \underline\mathbb{B}.$

To encapsulate the form of this general result, I define a composition that takes an imagination $\underline{f} = (f_1, \ldots, f_k) \in (X \to \underline\mathbb{B})^k$ and a boolean connection $F : \underline\mathbb{B}^k \to \underline\mathbb{B}$ and gives a proposition $p : X \to \underline\mathbb{B}.$ Depending on the situation, specifically, according to whether many $F\!$ and many $\underline{f},$ a single $F\!$ and many $\underline{f},$ or many $F\!$ and a single $\underline{f}$ are being considered, respectively, the proposition $p\!$ thus constructed may be referred to under one of three descriptions:

In a general setting, where the connection $F\!$ and the imagination $\underline{f}$ are both permitted to take up a variety of concrete possibilities, call $p\!$ the stretch of $F\!$ and $\underline{f}$ from $X\!$ to $\underline\mathbb{B},$, and write it in the style of a composition as $F ~\$~ \underline{f}.$ This is meant to suggest that the symbol $^{\backprime\backprime} $ ^{\prime\prime},$ here read as stretch, denotes an operator of the form:

$\$ : (\underline\mathbb{B}^k \to \underline\mathbb{B}) \times (X \to \underline\mathbb{B})^k \to (X \to \underline\mathbb{B}).$
In a setting where the connection $F\!$ is fixed but the imagination $\underline{f}$ is allowed to vary over a wide range of possibilities, call $p\!$ the stretch of $F\!$ to $\underline{f}$ on $X,\!$ and write it in the style $F^\$ \underline{f},$ exactly as if $^{\backprime\backprime} F^\$ \, ^{\prime\prime}$ denotes an operator $F^\$ : (X \to \underline\mathbb{B})^k \to (X \to \underline\mathbb{B})$ that is derived from $F\!$ and applied to $\underline{f},$ ultimately yielding a proposition $F^\$ \underline{f} : X \to \underline\mathbb{B}.$
In a setting where the imagination$\underline{f}$ is fixed but the connection $F\!$ is allowed to range over wide variety of possibilities, call $p\!$ the stretch of $\underline{f}$ by $F\!$ to $\underline\mathbb{B},$ and write it in the style $\underline{f}^\$ F,$ exactly as if $^{\backprime\backprime} \underline{f}^\$ \, ^{\prime\prime}$ denotes an operator $\underline{f}^\$ : (\underline\mathbb{B}^k \to \underline\mathbb{B}) \to (X \to \underline\mathbb{B}$ that is derived from $\underline{f}$ and applied to $F,\!$ ultimately yielding a proposition $\underline{f}^\$ F : X \to \underline\mathbb{B}.$

Because this notation is only used in settings where the imagination $\underline{f} : (X \to \underline\mathbb{B})^k$ and the connection $F : \underline\mathbb{B}^k \to \underline\mathbb{B}$ are distinguished by their types, it does not really matter whether one writes $^{\backprime\backprime} F ~\$~ \underline{f} \, ^{\prime\prime}$ or $^{\backprime\backprime} \underline{f} ~\$~ F \, ^{\prime\prime}$ 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:

An imagination of degree $k\!$ on $X,\!$ in other words, a $k\!$-tuple of propositions $f_j : X \to \underline\mathbb{B},$ for $j = 1 ~\text{to}~ k,$ or an object of the form $\underline{f} = (f_1, \ldots, f_k) : (X \to \underline\mathbb{B})^k.$
A connection of degree $k,\!$ in other words, a proposition about things in $\underline\mathbb{B}^k,$ or a boolean function of the form $F : \underline\mathbb{B}^k \to \underline\mathbb{B}.$

From these materials, it is required to construct a proposition $p : X \to \underline\mathbb{B}$ such that $p(x) = F(f_1 (x), \ldots, f_k (x)),$ for all $x \in X.$ The desired construction is determined as follows:

The cartesian power $\underline\mathbb{B}^k,$ as a cartesian product, is characterized by the possession of a projective imagination $\pi = (\pi_1, \ldots, \pi_k)$ of degree $k\!$ on $\underline\mathbb{B}^k,$ along with the property that any imagination $\underline{f} = (f_1, \ldots, f_k)$ of degree $k\!$ on an arbitrary set $W\!$ determines a unique map $f! : W \to \underline\mathbb{B}^k,$ the play of whose projective images $(\pi_1 (f!(w), \ldots, \pi_k (f!(w))$ on the functional image $f!(w)\!$ matches the play of images $(f_1 (w), \ldots, f_k (w))$ under $\underline{f},$ 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 $\underline\mathbb{B}^k,$ as a cartesian product, is characterized by the possession of $k\!$ projection maps $\pi_j : \underline\mathbb{B}^k \to \underline\mathbb{B},$ for $j = 1 ~\text{to}~ k,$ along with the property that any $k\!$ maps $f_j : W \to \underline\mathbb{B},$ from an arbitrary set $W\!$ to $\underline\mathbb{B},$ determine a unique map $f! : W \to \underline\mathbb{B}^k$ such that $\pi_j (f!(w)) = f_j (w),\!$ for all $j = 1 ~\text{to}~ k,$ and for all $w \in W.$

Now suppose that the arbitrary set $W\!$ in this construction is just the relevant universe $X.\!$ Given that the function $f! : X \to \underline\mathbb{B}^k$ is uniquely determined by the imagination $\underline{f} : (X \to \underline\mathbb{B})^k,$ that is, by the $k\!$-tuple of propositions $\underline{f} = (f_1, \ldots, f_k),$ it is safe to identify $f!\!$ and $\underline{f}$ 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 $^{\backprime\backprime} (f_1, \ldots, f_k) \, ^{\prime\prime}.$ 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 $F : \underline\mathbb{B}^k \to \underline\mathbb{B}$ is a boolean function on $k\!$ variables, then it is possible to define a mapping $F^\$ : (X \to \underline\mathbb{B})^k \to (X \to \underline\mathbb{B}),$ in effect, an operation that takes $k\!$ propositions into a single proposition, where $F^\$$ satisfies the following conditions:

$\begin{array}{lcl} F^\$ (f_1, \ldots, f_k) & : & X \to \underline\mathbb{B} \\ \\ F^\$ (f_1, \ldots, f_k) (x) & = & F(\underline{f} (x)) \\ & = & F((f_1, \ldots, f_k) (x)) \\ & = & F(f_1 (x), \ldots, f_k (x)). \\ \end{array}$

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 $^{\backprime\backprime} f_Q \, ^{\prime\prime}$ is sign that denotes the proposition $f_Q,\!$ 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 sign relation.

Roughly sketched, the relations of denotation and indication that exist among sets, propositions, sentences, and values can be diagrammed as in Table 11.

**Table 11. Levels of Indication**
$\operatorname{Object}$	$\operatorname{Sign}$	$\operatorname{Higher~Order~Sign}$
$\operatorname{Set}$	$\operatorname{Proposition}$	$\operatorname{Sentence}$
$f^{-1} (y)\!$	$f\!$	$^{\backprime\backprime} f \, ^{\prime\prime}$
$Q\!$	$\underline{1}$	$^{\backprime\backprime} \underline{1} ^{\prime\prime}$
${}^{_\sim} Q$	$\underline{0}$	$^{\backprime\backprime} \underline{0} ^{\prime\prime}$

Strictly speaking, a proposition is too abstract to be a sign, and so the contents of Table 11 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 $Q\!$ only insofar as the values of $\underline{1}$ and $\underline{0}$ that it assigns to the elements of the universe $X\!$ are positive and negative indications, respectively, of the elements in $Q,\!$ and thus indications of the set $Q\!$ and of its complement ${}^{_\sim} Q = X\!-\!Q,$ 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 $Q\!$ and the objects that are outside the set $Q,\!$ respectively.

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

To mark the relation of denotation between a sentence $s\!$ and the proposition that it denotes, let the drop notation $\downharpoonleft s \downharpoonright$ be used for the indicator function denoted by the sentence $s.\!$
To mark the relation of denotation between a proposition $q\!$ and the set that it indicates, let the lift notation $\upharpoonleft Q \upharpoonright$ be used for the indicator function of the set $Q.\!$

Notice that the drop operator $\downharpoonleft \cdots \downharpoonright$ takes one "downstream", in accord with the direction of denotation, from a sign to its object, while the lift operator $\upharpoonleft \cdots \upharpoonright$ takes one "upstream", against the 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:

Let the down hooks $\downharpoonleft \cdots \downharpoonright$ be placed around the name of a sentence $s,\!$ as in the expression $^{\backprime\backprime} \downharpoonleft s \downharpoonright \, ^{\prime\prime},$ or else around a token appearance of the sentence itself, to serve as a name for the proposition that $s\!$ denotes.
Let the up hooks $\upharpoonleft \cdots \upharpoonright$ be placed around a name of a set $Q,\!$ as in the expression $^{\backprime\backprime} \upharpoonleft Q \upharpoonright \, ^{\prime\prime},$ to serve as a name for the indicator function $f_Q.\!$

Table 12 illustrates the use of this notation, listing in each column several different but equivalent ways of referring to the same entity.

**Table 12. Ilustrations of Notation**
$\operatorname{Object}$	$\operatorname{Sign}$	$\operatorname{Higher~Order~Sign}$
$\operatorname{Set}$	$\operatorname{Proposition}$	$\operatorname{Sentence}$
$Q\!$	$q\!$	$s\!$
$[\| \downharpoonleft s \downharpoonright \|]$	$\downharpoonleft s \downharpoonright$	$s\!$
$[\| q \|]\!$	$q\!$	$^{\backprime\backprime} q \, ^{\prime\prime}$
$[\| f_Q \|]\!$	$f_Q\!$	$^{\backprime\backprime} f_Q \, ^{\prime\prime}$
$Q\!$	$\upharpoonleft Q \upharpoonright$	$^{\backprime\backprime} \upharpoonleft Q \upharpoonright \, ^{\prime\prime}$

In particular, one observes the following relations and formulas:

1.	Let the sentence $s\!$ denote the proposition $q,\!$
		where	$q : X \to \underline\mathbb{B}.$
	Then we have the notational equivalence:
		$\downharpoonleft s \downharpoonright ~=~ q.$
2.	Let the sentence $s\!$ denote the proposition $q,\!$
		where	$q : X \to \underline\mathbb{B}$
		and	$[\| q \|] ~=~ q^{-1} (\underline{1}) ~=~ Q \subseteq X.$
	Then we have the notational equivalences:
		$\downharpoonleft s \downharpoonright ~=~ q ~=~ f_Q ~=~ \upharpoonleft Q \upharpoonright.$
3.	$Q\!$	$=\!$	$\{ x \in X : x \in Q \}$
		$=\!$	$[\| \upharpoonleft X \upharpoonright \|] ~=~ \upharpoonleft X \upharpoonright^{-1} (\underline{1})$
		$=\!$	$[\| f_Q \|] ~=~ f_Q^{-1} (\underline{1}).$
4.	$\upharpoonleft Q \upharpoonright$	$=\!$	$\upharpoonleft \{ x \in X : x \in Q \} \upharpoonright$
		$=\!$	$\downharpoonleft x \in Q \downharpoonright$
		$=\!$	$f_Q.\!$

Now if a sentence $s\!$ really denotes a proposition $q,\!$ and if the notation $^{\backprime\backprime} \downharpoonleft s \downharpoonright \, ^{\prime\prime}$ 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 $q\!$ that it denotes, and on to the set $Q = q^{-1} (\underline{1})$ that the proposition $q\!$ indicates, often jumping to the conclusion that the set $Q\!$ is the only thing that the sentence $s\!$ is intended to denote. This higher order sign situation and the mind's inclination when placed in 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 $q\!$ to $Q.\!$

1.3.11. 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 J. Bernstein, "Idols of Modern Science", [HJB, 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 the two domains. If none of the formalisms that are readily available or in common use are able to meet all of 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 express 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:

Painted And Rooted Cacti (PARCAI).
Painted And Rooted Conifers (PARCOI).

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 graphs that are 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 1.3.10.10).

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 1.3.10.11.

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, equipping or providing each abstractly well-formed sentence with a logical proposition for it to denote. A semantic interpretation of the cactus language is carried out in Subsection 1.3.10.12.

1.3.11.1. The Cactus Language : Syntax

Picture two different configurations of such an irregular shape, superimposed on each other in space, like a double exposure photograph. Of the two images, the only part which coincides is the body. The two different sets of quills stick out into very different regions of space. The objective reality we see from within the first position, seemingly so full and spherical, actually agrees with the shifted reality only in the body of common knowledge. In every direction in which we look at all deeply, the realm of discovered scientific truth could be quite different. Yet in each of those two different situations, we would have thought the world complete, firmly known, and rather round in its penetration of the space of possible knowledge.

— Herbert J. Bernstein, "Idols of Modern Science", [HJB, 38]

In this Subsection, I describe the syntax of a family of formal languages that I intend to use as a sentential calculus, and thus to interpret for the purpose of reasoning about propositions and their logical relations. In order to carry out the discussion, I need a way of referring to signs as if they were objects like any others, in other words, as the sorts of things that are subject to being named, indicated, described, discussed, and renamed if necessary, that can be placed, arranged, and rearranged within a suitable medium of expression, or else manipulated in the mind, that can be articulated and decomposed into their elementary signs, and that can be strung together in sequences to form complex signs. Signs that have signs as their objects are called higher order signs, and this is a topic that demands an apt formalization, but in due time. The present discussion requires a quicker way to get into this subject, even if it takes informal means that cannot be made absolutely precise.

As a temporary notation, let the relationship between a particular sign $s\!$ and a particular object $o\!$, namely, the fact that $s\!$ denotes $o\!$ or the fact that $o\!$ is denoted by $s\!$, be symbolized in one of the following two ways:

$\begin{array}{lccc} 1. & s & \rightarrow & o \\ \\ 2. & o & \leftarrow & s \\ \end{array}$

Now consider the following paradigm:

$\begin{array}{llccc} 1. & \operatorname{If} & ^{\backprime\backprime}\operatorname{A}^{\prime\prime} & \rightarrow & \operatorname{Ann}, \\ & \operatorname{that~is}, & ^{\backprime\backprime}\operatorname{A}^{\prime\prime} & \operatorname{denotes} & \operatorname{Ann}, \\ & \operatorname{then} & \operatorname{A} & = & \operatorname{Ann} \\ & \operatorname{and} & \operatorname{Ann} & = & \operatorname{A}. \\ & \operatorname{Thus} & ^{\backprime\backprime}\operatorname{Ann}^{\prime\prime} & \rightarrow & \operatorname{A}, \\ & \operatorname{that~is}, & ^{\backprime\backprime}\operatorname{Ann}^{\prime\prime} & \operatorname{denotes} & \operatorname{A}. \\ \end{array}$

$\begin{array}{llccc} 2. & \operatorname{If} & \operatorname{Bob} & \leftarrow & ^{\backprime\backprime}\operatorname{B}^{\prime\prime}, \\ & \operatorname{that~is}, & \operatorname{Bob} & \operatorname{is~denoted~by} & ^{\backprime\backprime}\operatorname{B}^{\prime\prime}, \\ & \operatorname{then} & \operatorname{Bob} & = & \operatorname{B} \\ & \operatorname{and} & \operatorname{B} & = & \operatorname{Bob}. \\ & \operatorname{Thus} & \operatorname{B} & \leftarrow & ^{\backprime\backprime}\operatorname{Bob}^{\prime\prime}, \\ & \operatorname{that~is}, & \operatorname{B} & \operatorname{is~denoted~by} & ^{\backprime\backprime}\operatorname{Bob}^{\prime\prime}. \\ \end{array}$

When I say that the sign "blank" denotes the sign " ", it means that the string of characters inside the first pair of quotation marks can be used as another name for the string of characters inside the second pair of quotes. In other words, "blank" is a higher order sign whose object is " ", and the string of five characters inside the first pair of quotation marks is a sign at a higher level of signification than the string of one character inside the second pair of quotation marks. This relationship can be abbreviated in either one of the following ways:

$\begin{array}{lll} ^{\backprime\backprime}\operatorname{~}^{\prime\prime} & \leftarrow & ^{\backprime\backprime}\operatorname{blank}^{\prime\prime} \\ \\ ^{\backprime\backprime}\operatorname{blank}^{\prime\prime} & \rightarrow & ^{\backprime\backprime}\operatorname{~}^{\prime\prime} \\ \end{array}$

Using the raised dot "$\cdot$" as a sign to mark the articulation of a quoted string into a sequence of possibly shorter quoted strings, and thus to mark the concatenation of a sequence of quoted strings into a possibly larger quoted string, one can write:

$\begin{array}{lllll} ^{\backprime\backprime}\operatorname{~}^{\prime\prime} & \leftarrow & ^{\backprime\backprime}\operatorname{blank}^{\prime\prime} & = & ^{\backprime\backprime}\operatorname{b}^{\prime\prime} \, \cdot \, ^{\backprime\backprime}\operatorname{l}^{\prime\prime} \, \cdot \, ^{\backprime\backprime}\operatorname{a}^{\prime\prime} \, \cdot \, ^{\backprime\backprime}\operatorname{n}^{\prime\prime} \, \cdot \, ^{\backprime\backprime}\operatorname{k}^{\prime\prime} \\ \end{array}$

This usage allows us to refer to the blank as a type of character, and also to refer any blank we choose as a token of this type, referring to either of them in a marked way, but without the use of quotation marks, as I just did. Now, since a blank is just what the name "blank" names, it is possible to represent the denotation of the sign " " by the name "blank" in the form of an identity between the named objects, thus:

$\begin{array}{lll} ^{\backprime\backprime}\operatorname{~}^{\prime\prime} & = & \operatorname{blank} \\ \end{array}$

With these kinds of identity in mind, it is possible to extend the use of the "$\cdot$" sign to mark the articulation of either named or quoted strings into both named and quoted strings. For example:

$\begin{array}{lclcl} ^{\backprime\backprime}\operatorname{~~}^{\prime\prime} & = & ^{\backprime\backprime}\operatorname{~}^{\prime\prime} \, \cdot \, ^{\backprime\backprime}\operatorname{~}^{\prime\prime} & = & \operatorname{blank} \, \cdot \, \operatorname{blank} \\ \\ ^{\backprime\backprime}\operatorname{~blank}^{\prime\prime} & = & ^{\backprime\backprime}\operatorname{~}^{\prime\prime} \, \cdot \, ^{\backprime\backprime}\operatorname{blank}^{\prime\prime} & = & \operatorname{blank} \, \cdot \, ^{\backprime\backprime}\operatorname{blank}^{\prime\prime} \\ \\ ^{\backprime\backprime}\operatorname{blank~}^{\prime\prime} & = & ^{\backprime\backprime}\operatorname{blank}^{\prime\prime} \, \cdot \, ^{\backprime\backprime}\operatorname{~}^{\prime\prime} & = & ^{\backprime\backprime}\operatorname{blank}^{\prime\prime} \, \cdot \, \operatorname{blank} \end{array}$

A few definitions from formal language theory are required at this point.

An alphabet is a finite set of signs, typically, $\mathfrak{A} = \{ \mathfrak{a}_1, \ldots, \mathfrak{a}_n \}.$

A string over an alphabet $\mathfrak{A}$ is a finite sequence of signs from $\mathfrak{A}.$

The length of a string is just its length as a sequence of signs.

The empty string is the unique sequence of length 0. It is sometimes denoted by an empty pair of quotation marks, $^{\backprime\backprime\prime\prime},$ but more often by the Greek symbols epsilon or lambda.

A sequence of length $k > 0\!$ is typically presented in the concatenated forms:

$s_1 s_2 \ldots s_{k-1} s_k\!$

or

$s_1 \cdot s_2 \cdot \ldots \cdot s_{k-1} \cdot s_k$

with $s_j \in \mathfrak{A}$ for all $j = 1 \ldots k.$

Two alternative notations are often useful:

$\varepsilon$	=	$^{\backprime\backprime\prime\prime}$	=	the empty string.
$\underline\varepsilon$	=	$\{ \varepsilon \}$	=	the language consisting of a single empty string.

The kleene star $\mathfrak{A}^*$ of alphabet $\mathfrak{A}$ is the set of all strings over $\mathfrak{A}.$ In particular, $\mathfrak{A}^*$ includes among its elements the empty string $\varepsilon.$

The kleene plus $\mathfrak{A}^+$ of an alphabet $\mathfrak{A}$ is the set of all positive length strings over $\mathfrak{A},$ in other words, everything in $\mathfrak{A}^*$ but the empty string.

A formal language $\mathfrak{L}$ over an alphabet $\mathfrak{A}$ is a subset of $\mathfrak{A}^*.$ In brief, $\mathfrak{L} \subseteq \mathfrak{A}^*.$ If $s\!$ is a string over $\mathfrak{A}$ and if $s\!$ is an element of $\mathfrak{L},$ then it is customary to call $s\!$ a sentence of $\mathfrak{L}.$ Thus, a formal language $\mathfrak{L}$ is defined by specifying its elements, which amounts to saying what it means to be a sentence of $\mathfrak{L}.$

One last device turns out to be useful in this connection. If $s\!$ is a string that ends with a sign $t,\!$ then $s \cdot t^{-1}$ is the string that results by deleting from $s\!$ the terminal $t.\!$

In this context, I make the following distinction:

To delete an appearance of a sign is to replace it with an appearance of the empty string "".
To erase an appearance of a sign is to replace it with an appearance of the blank symbol " ".

A token is a particular appearance of a sign.

The informal mechanisms that have been illustrated in the immediately preceding discussion are enough to equip the rest of this discussion with a moderately exact description of the so-called cactus language that I intend to use in both my conceptual and my computational representations of the minimal formal logical system that is variously known to sundry communities of interpretation as propositional logic, sentential calculus, or more inclusively, zeroth order logic (ZOL).

The painted cactus language $\mathfrak{C}$ is actually a parameterized family of languages, consisting of one language $\mathfrak{C}(\mathfrak{P})$ for each set $\mathfrak{P}$ of paints.

The alphabet $\mathfrak{A} = \mathfrak{M} \cup \mathfrak{P}$ is the disjoint union of two sets of symbols:

$\mathfrak{M}$ is the alphabet of measures, the set of punctuation marks, or the collection of syntactic constants that is common to all of the languages $\mathfrak{C}(\mathfrak{P}).$ This set of signs is given as follows:

$\begin{array}{lccccccccccc} \mathfrak{M} & = & \{ & \mathfrak{m}_1 & , & \mathfrak{m}_2 & , & \mathfrak{m}_3 & , & \mathfrak{m}_4 & \} \\ & = & \{ & ^{\backprime\backprime} \, \operatorname{~} \, ^{\prime\prime} & , & ^{\backprime\backprime} \, \operatorname{(} \, ^{\prime\prime} & , & ^{\backprime\backprime} \, \operatorname{,} \, ^{\prime\prime} & , & ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime} & \} \\ & = & \{ & \operatorname{blank} & , & \operatorname{links} & , & \operatorname{comma} & , & \operatorname{right} & \} \\ \end{array}$
$\mathfrak{P}$ is the palette, the alphabet of paints, or the collection of syntactic variables that is peculiar to the language $\mathfrak{C}(\mathfrak{P}).$ This set of signs is given as follows:

$\mathfrak{P} = \{ \mathfrak{p}_j : j \in J \}.$

The easiest way to define the language $\mathfrak{C}(\mathfrak{P})$ is to indicate the general sorts of operations that suffice to construct the greater share of its sentences from the specified few of its sentences that require a special election. In accord with this manner of proceeding, I introduce a family of operations on strings of $\mathfrak{A}^*$ that are called syntactic connectives. If the strings on which they operate are exclusively sentences of $\mathfrak{C}(\mathfrak{P}),$ then these operations are tantamount to sentential connectives, and if the syntactic sentences, considered as abstract strings of meaningless signs, are given a semantics in which they denote propositions, considered as indicator functions over some universe, then these operations amount to propositional connectives.

Rather than presenting the most concise description of these languages right from the beginning, it serves comprehension to develop a picture of their forms in gradual stages, starting from the most natural ways of viewing their elements, if somewhat at a distance, and working through the most easily grasped impressions of their structures, if not always the sharpest acquaintances with their details.

The first step is to define two sets of basic operations on strings of $\mathfrak{A}^*.$

The concatenation of one string $s_1\!$ is just the string $s_1.\!$

The concatenation of two strings $s_1, s_2\!$ is the string $s_1 \cdot s_2.\!$

The concatenation of the $k\!$ strings $(s_j)_{j = 1}^k$ is the string of the form $s_1 \cdot \ldots \cdot s_k.\!$
The surcatenation of one string $s_1\!$ is the string $^{\backprime\backprime} \, \operatorname{(} \, ^{\prime\prime} \, \cdot \, s_1 \, \cdot \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime}.$

The surcatenation of two strings $s_1, s_2\!$ is $^{\backprime\backprime} \, \operatorname{(} \, ^{\prime\prime} \, \cdot \, s_1 \, \cdot \, ^{\backprime\backprime} \, \operatorname{,} \, ^{\prime\prime} \, \cdot \, s_2 \, \cdot \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime}.$

The surcatenation of the $k\!$ strings $(s_j)_{j = 1}^k$ is the string of the form $^{\backprime\backprime} \, \operatorname{(} \, ^{\prime\prime} \, \cdot \, s_1 \, \cdot \, ^{\backprime\backprime} \, \operatorname{,} \, ^{\prime\prime} \, \cdot \, \ldots \, \cdot \, ^{\backprime\backprime} \, \operatorname{,} \, ^{\prime\prime} \, \cdot \, s_k \, \cdot \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime}.$

These definitions can be made a little more succinct by defining the following sorts of generic operators on strings:

The concatenation $\operatorname{Conc}_{j=1}^k$ of the sequence of $k\!$ strings $(s_j)_{j=1}^k$ is defined recursively as follows:

$\operatorname{Conc}_{j=1}^1 s_j \ = \ s_1.$
For $\ell > 1,\!$

$\operatorname{Conc}_{j=1}^\ell s_j \ = \ \operatorname{Conc}_{j=1}^{\ell - 1} s_j \, \cdot \, s_\ell.$

The surcatenation $\operatorname{Surc}_{j=1}^k$ of the sequence of $k\!$ strings $(s_j)_{j=1}^k$ is defined recursively as follows:

$\operatorname{Surc}_{j=1}^1 s_j \ = \ ^{\backprime\backprime} \, \operatorname{(} \, ^{\prime\prime} \, \cdot \, s_1 \, \cdot \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime}.$
For $\ell > 1,\!$

$\operatorname{Surc}_{j=1}^\ell s_j \ = \ \operatorname{Surc}_{j=1}^{\ell - 1} s_j \, \cdot \, ( \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime} \, )^{-1} \, \cdot \, ^{\backprime\backprime} \, \operatorname{,} \, ^{\prime\prime} \, \cdot \, s_\ell \, \cdot \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime}.$

The definitions of these syntactic operations can now be organized in a slightly better fashion by making a few additional conventions and auxiliary definitions.

The conception of the $k\!$-place concatenation operation can be extended to include its natural prequel:

$\operatorname{Conc}^0 \ = \ ^{\backprime\backprime\prime\prime}$ = the empty string.

Next, the construction of the $k\!$-place concatenation can be broken into stages by means of the following conceptions:

The precatenation $\operatorname{Prec} (s_1, s_2)$ of the two strings $s_1, s_2\!$ is the string that is defined as follows:

$\operatorname{Prec} (s_1, s_2) \ = \ s_1 \cdot s_2.$
The concatenation of the sequence of $k\!$ strings $s_1, \ldots, s_k\!$ can now be defined as an iterated precatenation over the sequence of $k+1\!$ strings that begins with the string $s_0 = \operatorname{Conc}^0 \, = \, ^{\backprime\backprime\prime\prime}$ and then continues on through the other $k\!$ strings:

$\operatorname{Conc}_{j=0}^0 s_j \ = \ \operatorname{Conc}^0 \ = \ ^{\backprime\backprime\prime\prime}.$
For $\ell > 0,\!$

$\operatorname{Conc}_{j=1}^\ell s_j \ = \ \operatorname{Prec}(\operatorname{Conc}_{j=0}^{\ell - 1} s_j, s_\ell).$

The conception of the $k\!$-place surcatenation operation can be extended to include its natural "prequel":

$\operatorname{Surc}^0 \ = \ ^{\backprime\backprime} \, \operatorname{()} \, ^{\prime\prime}.$

Finally, the construction of the $k\!$-place surcatenation can be broken into stages by means of the following conceptions:
1. A subclause in $\mathfrak{A}^*$ is a string that ends with a $^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime}.$
2. The subcatenation $\operatorname{Subc} (s_1, s_2)$ of a subclause $s_1\!$ by a string $s_2\!$ is the string that is defined as follows:
  
  $\operatorname{Subc} (s_1, s_2) \ = \ s_1 \, \cdot \, ( \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime} \, )^{-1} \, \cdot \, ^{\backprime\backprime} \, \operatorname{,} \, ^{\prime\prime} \, \cdot \, s_2 \, \cdot \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime}.$
3. The surcatenation of the $k\!$ strings $s_1, \ldots, s_k\!$ can now be defined as an iterated subcatenation over the sequence of $k+1\!$ strings that starts with the string $s_0 \ = \ \operatorname{Surc}^0 \ = \ ^{\backprime\backprime} \, \operatorname{()} \, ^{\prime\prime}$ and then continues on through the other $k\!$ strings:

Notice that the expressions $\operatorname{Conc}_{j=0}^0 s_j$ and $\operatorname{Surc}_{j=0}^0 s_j$ are defined in such a way that the respective operators $\operatorname{Conc}^0$ and $\operatorname{Surc}^0$ simply ignore, in the manner of constants, whatever sequences of strings $s_j\!$ may be listed as their ostensible arguments.

Having defined the basic operations of concatenation and surcatenation on arbitrary strings, in effect, giving them operational meaning for the all-inclusive language $\mathfrak{L} = \mathfrak{A}^*,$ it is time to adjoin the notion of a more discriminating grammaticality, in other words, a more properly restrictive concept of a sentence.

If $\mathfrak{L}$ is an arbitrary formal language over an alphabet of the sort that we are talking about, that is, an alphabet of the form $\mathfrak{A} = \mathfrak{M} \cup \mathfrak{P},$ then there are a number of basic structural relations that can be defined on the strings of $\mathfrak{L}.$

1.	$s\!$ is the concatenation of $s_1\!$ and $s_2\!$ in $\mathfrak{L}$ if and only if
	$s_1\!$ is a sentence of $\mathfrak{L},$ $s_2\!$ is a sentence of $\mathfrak{L},$ and
	$s = s_1 \cdot s_2.$
2.	$s\!$ is the concatenation of the $k\!$ strings $s_1, \ldots, s_k\!$ in $\mathfrak{L},$
	if and only if $s_j\!$ is a sentence of $\mathfrak{L},$ for all $j = 1 \ldots k,$ and
	$s = \operatorname{Conc}_{j=1}^k s_j = s_1 \cdot \ldots \cdot s_k.$
3.	$s\!$ is the discatenation of $s_1\!$ by $t\!$ if and only if
	$s_1\!$ is a sentence of $\mathfrak{L},$ $t\!$ is an element of $\mathfrak{A},$ and
	$s_1 = s \cdot t.$
	When this is the case, one more commonly writes:
	$s = s_1 \cdot t^{-1}.$
4.	$s\!$ is a subclause of $\mathfrak{L}$ if and only if
	$s\!$ is a sentence of $\mathfrak{L}$ and $s\!$ ends with a $^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime}.$
5.	$s\!$ is the subcatenation of $s_1\!$ by $s_2\!$ if and only if
	$s_1\!$ is a subclause of $\mathfrak{L},$ $s_2\!$ is a sentence of $\mathfrak{L},$ and
	$s = s_1 \, \cdot \, ( \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime} \, )^{-1} \, \cdot \, ^{\backprime\backprime} \, \operatorname{,} \, ^{\prime\prime} \, \cdot \, s_2 \, \cdot \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime}.$
6.	$s\!$ is the surcatenation of the $k\!$ strings $s_1, \ldots, s_k\!$ in $\mathfrak{L},$
	if and only if $s_j\!$ is a sentence of $\mathfrak{L},$ for all $j = 1 \ldots k,\!$ and
	$s \ = \ \operatorname{Surc}_{j=1}^k s_j \ = \ ^{\backprime\backprime} \, \operatorname{(} \, ^{\prime\prime} \, \cdot \, s_1 \, \cdot \, ^{\backprime\backprime} \, \operatorname{,} \, ^{\prime\prime} \, \cdot \, \ldots \, \cdot \, ^{\backprime\backprime} \, \operatorname{,} \, ^{\prime\prime} \, \cdot \, s_k \, \cdot \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime}.$

The converses of these decomposition relations are tantamount to the corresponding forms of composition operations, making it possible for these complementary forms of analysis and synthesis to articulate the structures of strings and sentences in two directions.

The painted cactus language with paints in the set $\mathfrak{P} = \{ p_j : j \in J \}$ is the formal language $\mathfrak{L} = \mathfrak{C} (\mathfrak{P}) \subseteq \mathfrak{A}^* = (\mathfrak{M} \cup \mathfrak{P})^*$ that is defined as follows:

PC 1.	The blank symbol $m_1\!$ is a sentence.
PC 2.	The paint $p_j\!$ is a sentence, for each $j\!$ in $J.\!$
PC 3.	$\operatorname{Conc}^0$ and $\operatorname{Surc}^0$ are sentences.
PC 4.	For each positive integer $k,\!$
	if $s_1, \ldots, s_k\!$ are sentences,
	then $\operatorname{Conc}_{j=1}^k s_j$ is a sentence,
	and $\operatorname{Surc}_{j=1}^k s_j$ is a sentence.

As usual, saying that $s\!$ is a sentence is just a conventional way of stating that the string $s\!$ belongs to the relevant formal language $\mathfrak{L}.$ An individual sentence of $\mathfrak{C} (\mathfrak{P}),$ for any palette $\mathfrak{P},$ is referred to as a painted and rooted cactus expression (PARCE) on the palette $\mathfrak{P},$ or a cactus expression, for short. Anticipating the forms that the parse graphs of these PARCE's will take, to be described in the next Subsection, the language $\mathfrak{L} = \mathfrak{C} (\mathfrak{P})$ is also described as the set $\operatorname{PARCE} (\mathfrak{P})$ of PARCE's on the palette $\mathfrak{P},$ more generically, as the PARCE's that constitute the language $\operatorname{PARCE}.$

A bare PARCE, a bit loosely referred to as a bare cactus expression, is a PARCE on the empty palette $\mathfrak{P} = \varnothing.$ A bare PARCE is a sentence in the bare cactus language, $\mathfrak{C}^0 = \mathfrak{C} (\varnothing) = \operatorname{PARCE}^0 = \operatorname{PARCE} (\varnothing).$ This set of strings, regarded as a formal language in its own right, is a sublanguage of every cactus language $\mathfrak{C} (\mathfrak{P}).$ A bare cactus expression is commonly encountered in practice when one has occasion to start with an arbitrary PARCE and then finds a reason to delete or to erase all of its paints.

Only one thing remains to cast this description of the cactus language into a form that is commonly found acceptable. As presently formulated, the principle PC 4 appears to be attempting to define an infinite number of new concepts all in a single step, at least, it appears to invoke the indefinitely long sequences of operators, $\operatorname{Conc}^k$ and $\operatorname{Surc}^k,$ for all $k > 0.\!$ As a general rule, one prefers to have an effectively finite description of conceptual objects, and this means restricting the description to a finite number of schematic principles, each of which involves a finite number of schematic effects, that is, a finite number of schemata that explicitly relate conditions to results.

A start in this direction, taking steps toward an effective description of the cactus language, a finitary conception of its membership conditions, and a bounded characterization of a typical sentence in the language, can be made by recasting the present description of these expressions into the pattern of what is called, more or less roughly, a formal grammar.

A notation in the style of $S :> T\!$ is now introduced, to be read among many others in this manifold of ways:

$S\ \operatorname{covers}\ T$

$S\ \operatorname{governs}\ T$

$S\ \operatorname{rules}\ T$

$S\ \operatorname{subsumes}\ T$

$S\ \operatorname{types~over}\ T$

The form $S :> T\!$ is here recruited for polymorphic employment in at least the following types of roles:

To signify that an individually named or quoted string $T\!$ is being typed as a sentence $S\!$ of the language of interest $\mathfrak{L}.$
To express the fact or to make the assertion that each member of a specified set of strings $T \subseteq \mathfrak{A}^*$ also belongs to the syntactic category $S,\!$ the one that qualifies a string as being a sentence in the relevant formal language $\mathfrak{L}.$
To specify the intension or to signify the intention that every string that fits the conditions of the abstract type $T\!$ must also fall under the grammatical heading of a sentence, as indicated by the type $S,\!$ all within the target language $\mathfrak{L}.$

In these types of situation the letter $^{\backprime\backprime} S \, ^{\prime\prime}$ that signifies the type of a sentence in the language of interest, is called the initial symbol or the sentence symbol of a candidate formal grammar for the language, while any number of letters like $^{\backprime\backprime} T \, ^{\prime\prime}$ signifying other types of strings that are necessary to a reasonable account or a rational reconstruction of the sentences that belong to the language, are collectively referred to as intermediate symbols.

Combining the singleton set $\{ ^{\backprime\backprime} S \, ^{\prime\prime} \}$ whose sole member is the initial symbol with the set $\mathfrak{Q}$ that assembles together all of the intermediate symbols results in the set $\{ ^{\backprime\backprime} S \, ^{\prime\prime} \} \cup \mathfrak{Q}$ of non-terminal symbols. Completing the package, the alphabet $\mathfrak{A}$ of the language is also known as the set of terminal symbols. In this discussion, I will adopt the convention that $\mathfrak{Q}$ is the set of intermediate symbols, but I will often use $q\!$ as a typical variable that ranges over all of the non-terminal symbols, $q \in \{ ^{\backprime\backprime} S \, ^{\prime\prime} \} \cup \mathfrak{Q}.$ Finally, it is convenient to refer to all of the symbols in $\{ ^{\backprime\backprime} S \, ^{\prime\prime} \} \cup \mathfrak{Q} \cup \mathfrak{A}$ as the augmented alphabet of the prospective grammar for the language, and accordingly to describe the strings in $( \{ ^{\backprime\backprime} S \, ^{\prime\prime} \} \cup \mathfrak{Q} \cup \mathfrak{A} )^*$ as the augmented strings, in effect, expressing the forms that are superimposed on a language by one of its conceivable grammars. In certain settings it becomes desirable to separate the augmented strings that contain the symbol $^{\backprime\backprime} S \, ^{\prime\prime}$ from all other sorts of augmented strings. In these situations the strings in the disjoint union $\{ ^{\backprime\backprime} S \, ^{\prime\prime} \} \cup (\mathfrak{Q} \cup \mathfrak{A} )^*$ are known as the sentential forms of the associated grammar.

In forming a grammar for a language statements of the form $W :> W',\!$ where $W\!$ and $W'\!$ are augmented strings or sentential forms of specified types that depend on the style of the grammar that is being sought, are variously known as characterizations, covering rules, productions, rewrite rules, subsumptions, transformations, or typing rules. These are collected together into a set $\mathfrak{K}$ that serves to complete the definition of the formal grammar in question.

Correlative with the use of this notation, an expression of the form $T <: S,\!$ read to say that $T\!$ is covered by $S,\!$ can be interpreted to say that $T\!$ is of the type $S.\!$ Depending on the context, this can be taken in either one of two ways:

Treating $T\!$ as a string variable, it means that the individual string $T\!$ is typed as $S.\!$
Treating $T\!$ as a type name, it means that any instance of the type $T\!$ also falls under the type $S.\!$

In accordance with these interpretations, an expression of the form $t <: T\!$ can be read in all of the ways that one typically reads an expression of the form $t : T.\!$

There are several abuses of notation that commonly tolerated in the use of covering relations. The worst offense is that of allowing symbols to stand equivocally either for individual strings or else for their types. There is a measure of consistency to this practice, considering the fact that perfectly individual entities are rarely if ever grasped by means of signs and finite expressions, which entails that every appearance of an apparent token is only a type of more particular tokens, and meaning in the end that there is never any recourse but to the sort of discerning interpretation that can decide just how each sign is intended. In view of all this, I continue to permit expressions like $t <: T\!$ and $T <: S,\!$ where any of the symbols $t, T, S\!$ can be taken to signify either the tokens or the subtypes of their covering types.

Note. For some time to come in the discussion that follows, although I will continue to focus on the cactus language as my principal object example, my more general purpose will be to develop the subject matter of the formal languages and grammars. I will do this by taking up a particular method of stepwise refinement and using it to extract a rigorous formal grammar for the cactus language, starting with little more than a rough description of the target language and applying a systematic analysis to develop a sequence of increasingly more effective and more exact approximations to the desired grammar.

Employing the notion of a covering relation it becomes possible to redescribe the cactus language $\mathfrak{L} = \mathfrak{C} (\mathfrak{P})$ in the following ways.

Grammar 1

Grammar 1 is something of a misnomer. It is nowhere near exemplifying any kind of a standard form and it is only intended as a starting point for the initiation of more respectable grammars. Such as it is, it uses the terminal alphabet $\mathfrak{A} = \mathfrak{M} \cup \mathfrak{P}$ that comes with the territory of the cactus language $\mathfrak{C} (\mathfrak{P}),$ it specifies $\mathfrak{Q} = \varnothing,$ in other words, it employs no intermediate symbols, and it embodies the covering set $\mathfrak{K}$ as listed in the following display.

$\mathfrak{C} (\mathfrak{P}) : \text{Grammar 1}\!$

$\mathfrak{Q} = \varnothing$

$\begin{array}{rcll} 1. & S & :> & m_1 \ = \ ^{\backprime\backprime} \operatorname{~} ^{\prime\prime} \\ 2. & S & :> & p_j, \, \text{for each} \, j \in J \\ 3. & S & :> & \operatorname{Conc}^0 \ = \ ^{\backprime\backprime\prime\prime} \\ 4. & S & :> & \operatorname{Surc}^0 \ = \ ^{\backprime\backprime} \, \operatorname{()} \, ^{\prime\prime} \\ 5. & S & :> & S^* \\ 6. & S & :> & ^{\backprime\backprime} \, \operatorname{(} \, ^{\prime\prime} \, \cdot \, S \, \cdot \, ( \, ^{\backprime\backprime} \operatorname{,} ^{\prime\prime} \, \cdot \, S \, )^* \, \cdot \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime} \\ \end{array}$

In this formulation, the last two lines specify that:

The concept of a sentence in $\mathfrak{L}$ covers any concatenation of sentences in $\mathfrak{L},$ in effect, any number of freely chosen sentences that are available to be concatenated one after another.
The concept of a sentence in $\mathfrak{L}$ covers any surcatenation of sentences in $\mathfrak{L},$ in effect, any string that opens with a $^{\backprime\backprime} \, \operatorname{(} \, ^{\prime\prime},$ continues with a sentence, possibly empty, follows with a finite number of phrases of the form $^{\backprime\backprime} \operatorname{,} ^{\prime\prime} \, \cdot \, S,$ and closes with a $^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime}.$

This appears to be just about the most concise description of the cactus language $\mathfrak{C} (\mathfrak{P})$ that one can imagine, but there are a couple of problems that are commonly felt to afflict this style of presentation and to make it less than completely acceptable. Briefly stated, these problems turn on the following properties of the presentation:

The invocation of the kleene star operation is not reduced to a manifestly finitary form.
The type $S\!$ that indicates a sentence is allowed to cover not only itself but also the empty string.

I will discuss these issues at first in general, and especially in regard to how the two features interact with one another, and then I return to address in further detail the questions that they engender on their individual bases.

In the process of developing a grammar for a language, it is possible to notice a number of organizational, pragmatic, and stylistic questions, whose moment to moment answers appear to decide the ongoing direction of the grammar that develops and the impact of whose considerations work in tandem to determine, or at least to influence, the sort of grammar that turns out. The issues that I can see arising at this point I can give the following prospective names, putting off the discussion of their natures and the treatment of their details to the points in the development of the present example where they evolve their full import.

The degree of intermediate organization in a grammar.
The distinction between empty and significant strings, and thus the distinction between empty and significant types of strings.
The principle of intermediate significance. This is a constraint on the grammar that arises from considering the interaction of the first two issues.

In responding to these issues, it is advisable at first to proceed in a stepwise fashion, all the better to accommodate the chances of pursuing a series of parallel developments in the grammar, to allow for the possibility of reversing many steps in its development, indeed, to take into account the near certain necessity of having to revisit, to revise, and to reverse many decisions about how to proceed toward an optimal description or a satisfactory grammar for the language. Doing all this means exploring the effects of various alterations and innovations as independently from each other as possible.

The degree of intermediate organization in a grammar is measured by how many intermediate symbols it has and by how they interact with each other by means of its productions. With respect to this issue, Grammar 1 has no intermediate symbols at all, $\mathfrak{Q} = \varnothing,$ and therefore remains at an ostensibly trivial degree of intermediate organization. Some additions to the list of intermediate symbols are practically obligatory in order to arrive at any reasonable grammar at all, other inclusions appear to have a more optional character, though obviously useful from the standpoints of clarity and ease of comprehension.

One of the troubles that is perceived to affect Grammar 1 is that it wastes so much of the available potential for efficient description in recounting over and over again the simple fact that the empty string is present in the language. This arises in part from the statement that $S :> S^*,\!$ which implies that:

$\begin{array}{lcccccccccccc} S & :> & S^* & = & \underline\varepsilon & \cup & S & \cup & S \cdot S & \cup & S \cdot S \cdot S & \cup & \ldots \\ \end{array}$

There is nothing wrong with the more expansive pan of the covered equation, since it follows straightforwardly from the definition of the kleene star operation, but the covering statement to the effect that $S :> S^*\!$ is not a very productive piece of information, in the sense of telling very much about the language that falls under the type of a sentence $S.\!$ In particular, since it implies that $S :> \underline\varepsilon,$ and since $\underline\varepsilon \cdot \mathfrak{L} \, = \, \mathfrak{L} \cdot \underline\varepsilon \, = \, \mathfrak{L},$ for any formal language $\mathfrak{L},$ the empty string $\varepsilon$ is counted over and over in every term of the union, and every non-empty sentence under $S\!$ appears again and again in every term of the union that follows the initial appearance of $S.\!$ As a result, this style of characterization has to be classified as true but not very informative. If at all possible, one prefers to partition the language of interest into a disjoint union of subsets, thereby accounting for each sentence under its proper term, and one whose place under the sum serves as a useful parameter of its character or its complexity. In general, this form of description is not always possible to achieve, but it is usually worth the trouble to actualize it whenever it is.

Suppose that one tries to deal with this problem by eliminating each use of the kleene star operation, by reducing it to a purely finitary set of steps, or by finding an alternative way to cover the sublanguage that it is used to generate. This amounts, in effect, to recognizing a type, a complex process that involves the following steps:

Noticing a category of strings that is generated by iteration or recursion.
Acknowledging the fact that it needs to be covered by a non-terminal symbol.
Making a note of it by instituting an explicitly-named grammatical category.

In sum, one introduces a non-terminal symbol for each type of sentence and each part of speech or sentential component that is generated by means of iteration or recursion under the ruling constraints of the grammar. In order to do this one needs to analyze the iteration of each grammatical operation in a way that is analogous to a mathematically inductive definition, but further in a way that is not forced explicitly to recognize a distinct and separate type of expression merely to account for and to recount every increment in the parameter of iteration.

Returning to the case of the cactus language, the process of recognizing an iterative type or a recursive type can be illustrated in the following way. The operative phrases in the simplest sort of recursive definition are its initial part and its generic part. For the cactus language $\mathfrak{C} (\mathfrak{P}),$ one has the following definitions of concatenation as iterated precatenation and of surcatenation as iterated subcatenation, respectively:

$\begin{array}{llll} 1. & \operatorname{Conc}_{j=1}^0 & = & ^{\backprime\backprime\prime\prime} \\ \\ & \operatorname{Conc}_{j=1}^k S_j & = & \operatorname{Prec} (\operatorname{Conc}_{j=1}^{k-1} S_j, S_k) \\ \\ 2. & \operatorname{Surc}_{j=1}^0 & = & ^{\backprime\backprime} \, \operatorname{()} \, ^{\prime\prime} \\ \\ & \operatorname{Surc}_{j=1}^k S_j & = & \operatorname{Subc} (\operatorname{Surc}_{j=1}^{k-1} S_j, S_k) \\ \\ \end{array}$

In order to transform these recursive definitions into grammar rules, one introduces a new pair of intermediate symbols, $\operatorname{Conc}$ and $\operatorname{Surc},$ corresponding to the operations that share the same names but ignoring the inflexions of their individual parameters $j\!$ and $k.\!$ Recognizing the type of a sentence by means of the initial symbol $S\!$ and interpreting $\operatorname{Conc}$ and $\operatorname{Surc}$ as names for the types of strings that are generated by concatenation and by surcatenation, respectively, one arrives at the following transformation of the ruling operator definitions into the form of covering grammar rules:

$\begin{array}{llll} 1. & \operatorname{Conc} & :> & ^{\backprime\backprime\prime\prime} \\ \\ & \operatorname{Conc} & :> & \operatorname{Conc} \cdot S \\ \\ 2. & \operatorname{Surc} & :> & ^{\backprime\backprime} \, \operatorname{()} \, ^{\prime\prime} \\ \\ & \operatorname{Surc} & :> & ^{\backprime\backprime} \, \operatorname{(} \, ^{\prime\prime} \, \cdot \, S \, \cdot \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime} \\ \\ & \operatorname{Surc} & :> & \operatorname{Surc} \, \cdot \, ( \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime} \, )^{-1} \, \cdot \, ^{\backprime\backprime} \operatorname{,} ^{\prime\prime} \, \cdot \, S \, \cdot \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime} \end{array}$

As given, this particular fragment of the intended grammar contains a couple of features that are desirable to amend.

Given the covering $S :> \operatorname{Conc},$ the covering rule $\operatorname{Conc} :> \operatorname{Conc} \cdot S$ says no more than the covering rule $\operatorname{Conc} :> S \cdot S.$ Consequently, all of the information contained in these two covering rules is already covered by the statement that $S :> S \cdot S.$
A grammar rule that invokes a notion of decatenation, deletion, erasure, or any other sort of retrograde production, is frequently considered to be lacking in elegance, and a there is a style of critique for grammars that holds it preferable to avoid these types of operations if it is at all possible to do so. Accordingly, contingent on the prescriptions of the informal rule in question, and pursuing the stylistic dictates that are writ in the realm of its aesthetic regime, it becomes necessary for us to backtrack a little bit, to temporarily withdraw the suggestion of employing these elliptical types of operations, but without, of course, eliding the record of doing so.

Grammar 2

One way to analyze the surcatenation of any number of sentences is to introduce an auxiliary type of string, not in general a sentence, but a proper component of any sentence that is formed by surcatenation. Doing this brings one to the following definition:

A tract is a concatenation of a finite sequence of sentences, with a literal comma $^{\backprime\backprime} \operatorname{,} ^{\prime\prime}$ interpolated between each pair of adjacent sentences. Thus, a typical tract $T\!$ takes the form:

$\begin{array}{lllllllllll} T & = & S_1 & \cdot & ^{\backprime\backprime} \operatorname{,} ^{\prime\prime} & \cdot & \ldots & \cdot & ^{\backprime\backprime} \operatorname{,} ^{\prime\prime} & \cdot & S_k \\ \end{array}$

A tract must be distinguished from the abstract sequence of sentences, $S_1, \ldots, S_k,\!$ where the commas that appear to come to mind, as if being called up to separate the successive sentences of the sequence, remain as partially abstract conceptions, or as signs that retain a disengaged status on the borderline between the text and the mind. In effect, the types of commas that appear to follow in the abstract sequence continue to exist as conceptual abstractions and fail to be cognized in a wholly explicit fashion, whether as concrete tokens in the object language, or as marks in the strings of signs that are able to engage one's parsing attention.

Returning to the case of the painted cactus language $\mathfrak{L} = \mathfrak{C} (\mathfrak{P}),$ it is possible to put the currently assembled pieces of a grammar together in the light of the presently adopted canons of style, to arrive a more refined analysis of the fact that the concept of a sentence covers any concatenation of sentences and any surcatenation of sentences, and so to obtain the following form of a grammar:

$\mathfrak{C} (\mathfrak{P}) : \text{Grammar 2}\!$

$\mathfrak{Q} = \{ \, ^{\backprime\backprime} T \, ^{\prime\prime} \, \}$

$\begin{array}{rcll} 1. & S & :> & \varepsilon \\ 2. & S & :> & m_1 \\ 3. & S & :> & p_j, \, \text{for each} \, j \in J \\ 4. & S & :> & S \, \cdot \, S \\ 5. & S & :> & ^{\backprime\backprime} \, \operatorname{(} \, ^{\prime\prime} \, \cdot \, T \, \cdot \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime} \\ 6. & T & :> & S \\ 7. & T & :> & T \, \cdot \, ^{\backprime\backprime} \operatorname{,} ^{\prime\prime} \, \cdot \, S \\ \end{array}$

In this rendition, a string of type $T\!$ is not in general a sentence itself but a proper part of speech, that is, a strictly lesser component of a sentence in any suitable ordering of sentences and their components. In order to see how the grammatical category $T\!$ gets off the ground, that is, to detect its minimal strings and to discover how its ensuing generations get started from these, it is useful to observe that the covering rule $T :> S\!$ means that $T\!$ inherits all of the initial conditions of $S,\!$ namely, $T \, :> \, \varepsilon, m_1, p_j.$ In accord with these simple beginnings it comes to parse that the rule $T \, :> \, T \, \cdot \, ^{\backprime\backprime} \operatorname{,} ^{\prime\prime} \, \cdot \, S,$ with the substitutions $T = \varepsilon$ and $S = \varepsilon$ on the covered side of the rule, bears the germinal implication that $T \, :> \, ^{\backprime\backprime} \operatorname{,} ^{\prime\prime}.$

Grammar 2 achieves a portion of its success through a higher degree of intermediate organization. Roughly speaking, the level of organization can be seen as reflected in the cardinality of the intermediate alphabet $\mathfrak{Q} = \{ \, ^{\backprime\backprime} T \, ^{\prime\prime} \, \}$ but it is clearly not explained by this simple circumstance alone, since it is taken for granted that the intermediate symbols serve a purpose, a purpose that is easily recognizable but that may not be so easy to pin down and to specify exactly. Nevertheless, it is worth the trouble of exploring this aspect of organization and this direction of development a little further.

Grammar 3

Although it is not strictly necessary to do so, it is possible to organize the materials of our developing grammar in a slightly better fashion by recognizing two recurrent types of strings that appear in the typical cactus expression. In doing this, one arrives at the following two definitions:

A rune is a string of blanks and paints concatenated together. Thus, a typical rune $R\!$ is a string over $\{ m_1 \} \cup \mathfrak{P},$ possibly the empty string:

$R\ \in\ ( \{ m_1 \} \cup \mathfrak{P} )^*$

When there is no possibility of confusion, the letter $^{\backprime\backprime} R \, ^{\prime\prime}$ can be used either as a string variable that ranges over the set of runes or else as a type name for the class of runes. The latter reading amounts to the enlistment of a fresh intermediate symbol, $^{\backprime\backprime} R \, ^{\prime\prime} \in \mathfrak{Q},$ as a part of a new grammar for $\mathfrak{C} (\mathfrak{P}).$ In effect, $^{\backprime\backprime} R \, ^{\prime\prime}$ affords a grammatical recognition for any rune that forms a part of a sentence in $\mathfrak{C} (\mathfrak{P}).$ In situations where these variant usages are likely to be confused, the types of strings can be indicated by means of expressions like $r <: R\!$ and $W <: R.\!$

A foil is a string of the form $^{\backprime\backprime} \, \operatorname{(} \, ^{\prime\prime} \, \cdot \, T \, \cdot \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime},$ where $T\!$ is a tract. Thus, a typical foil $F\!$ has the form:

$\begin{array}{lllllllllllllll} F & = & ^{\backprime\backprime} \, \operatorname{(} \, ^{\prime\prime} & \cdot & S_1 & \cdot & ^{\backprime\backprime} \operatorname{,} ^{\prime\prime} & \cdot & \ldots & \cdot & ^{\backprime\backprime} \operatorname{,} ^{\prime\prime} & \cdot & S_k & \cdot & ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime} \\ \end{array}$

This is just the surcatenation of the sentences $S_1, \ldots, S_k.\!$ Given the possibility that this sequence of sentences is empty, and thus that the tract $T\!$ is the empty string, the minimum foil $F\!$ is the expression $^{\backprime\backprime} \, \operatorname{()} \, ^{\prime\prime}.$ Explicitly marking each foil $F\!$ that is embodied in a cactus expression is tantamount to recognizing another intermediate symbol, $^{\backprime\backprime} F \, ^{\prime\prime} \in \mathfrak{Q},$ further articulating the structures of sentences and expanding the grammar for the language $\mathfrak{C} (\mathfrak{P}).$ All of the same remarks about the versatile uses of the intermediate symbols, as string variables and as type names, apply again to the letter $^{\backprime\backprime} F \, ^{\prime\prime}.$

$\mathfrak{C} (\mathfrak{P}) : \text{Grammar 3}\!$

$\mathfrak{Q} = \{ \, ^{\backprime\backprime} F \, ^{\prime\prime}, \, ^{\backprime\backprime} R \, ^{\prime\prime}, \, ^{\backprime\backprime} T \, ^{\prime\prime} \, \}$

$\begin{array}{rcll} 1. & S & :> & R \\ 2. & S & :> & F \\ 3. & S & :> & S \, \cdot \, S \\ 4. & R & :> & \varepsilon \\ 5. & R & :> & m_1 \\ 6. & R & :> & p_j, \, \text{for each} \, j \in J \\ 7. & R & :> & R \, \cdot \, R \\ 8. & F & :> & ^{\backprime\backprime} \, \operatorname{(} \, ^{\prime\prime} \, \cdot \, T \, \cdot \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime} \\ 9. & T & :> & S \\ 10. & T & :> & T \, \cdot \, ^{\backprime\backprime} \operatorname{,} ^{\prime\prime} \, \cdot \, S \\ \end{array}$

In Grammar 3, the first three Rules say that a sentence (a string of type $S\!$), is a rune (a string of type $R\!$), a foil (a string of type $F\!$), or an arbitrary concatenation of strings of these two types. Rules 4 through 7 specify that a rune $R\!$ is an empty string $\varepsilon,$ a blank symbol $m_1,\!$ a paint $p_j,\!$ or any concatenation of strings of these three types. Rule 8 characterizes a foil $F\!$ as a string of the form $^{\backprime\backprime} \, \operatorname{(} \, ^{\prime\prime} \, \cdot \, T \, \cdot \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime},$ where $T\!$ is a tract. The last two Rules say that a tract $T\!$ is either a sentence $S\!$ or else the concatenation of a tract, a comma, and a sentence, in that order.

At this point in the succession of grammars for $\mathfrak{C} (\mathfrak{P}),$ the explicit uses of indefinite iterations, like the kleene star operator, are now completely reduced to finite forms of concatenation, but the problems that some styles of analysis have with allowing non-terminal symbols to cover both themselves and the empty string are still present.

Any degree of reflection on this difficulty raises the general question: What is a practical strategy for accounting for the empty string in the organization of any formal language that counts it among its sentences? One answer that presents itself is this: If the empty string belongs to a formal language, it suffices to count it once at the beginning of the formal account that enumerates its sentences and then to move on to more interesting materials.

Returning to the case of the cactus language $\mathfrak{C} (\mathfrak{P}),$ in other words, the formal language $\operatorname{PARCE}$ of painted and rooted cactus expressions, it serves the purpose of efficient accounting to partition the language into the following couple of sublanguages:

The emptily painted and rooted cactus expressions make up the language $\operatorname{EPARCE}$ that consists of a single empty string as its only sentence. In short:

$\operatorname{EPARCE} \ = \ \underline\varepsilon \ = \ \{ \varepsilon \}$
The significantly painted and rooted cactus expressions make up the language $\operatorname{SPARCE}$ that consists of everything else, namely, all of the non-empty strings in the language $\operatorname{PARCE}.$ In sum:

$\operatorname{SPARCE} \ = \ \operatorname{PARCE} \setminus \varepsilon$

As a result of marking the distinction between empty and significant sentences, that is, by categorizing each of these three classes of strings as an entity unto itself and by conceptualizing the whole of its membership as falling under a distinctive symbol, one obtains an equation of sets that connects the three languages being marked:

$\operatorname{SPARCE} \ = \ \operatorname{PARCE} \ - \ \operatorname{EPARCE}$

In sum, one has the disjoint union:

$\operatorname{PARCE} \ = \ \operatorname{EPARCE} \ \cup \ \operatorname{SPARCE}$

For brevity in the present case, and to serve as a generic device in any similar array of situations, let $S\!$ be the type of an arbitrary sentence, possibly empty, and let $S'\!$ be the type of a specifically non-empty sentence. In addition, let $\underline\varepsilon$ be the type of the empty sentence, in effect, the language $\underline\varepsilon = \{ \varepsilon \}$ that contains a single empty string, and let a plus sign $^{\backprime\backprime} + ^{\prime\prime}$ signify a disjoint union of types. In the most general type of situation, where the type $S\!$ is permitted to include the empty string, one notes the following relation among types:

$S \ = \ \underline\varepsilon \ + \ S'$

With the distinction between empty and significant expressions in mind, I return to the grasp of the cactus language $\mathfrak{L} = \mathfrak{C} (\mathfrak{P}) = \operatorname{PARCE} (\mathfrak{P})$ that is afforded by Grammar 2, and, taking that as a point of departure, explore other avenues of possible improvement in the comprehension of these expressions. In order to observe the effects of this alteration as clearly as possible, in isolation from any other potential factors, it is useful to strip away the higher levels intermediate organization that are present in Grammar 3, and start again with a single intermediate symbol, as used in Grammar 2. One way of carrying out this strategy leads on to a grammar of the variety that will be articulated next.

Grammar 4

If one imposes the distinction between empty and significant types on each non-terminal symbol in Grammar 2, then the non-terminal symbols $^{\backprime\backprime} S \, ^{\prime\prime}$ and $^{\backprime\backprime} T \, ^{\prime\prime}$ give rise to the expanded set of non-terminal symbols $^{\backprime\backprime} S \, ^{\prime\prime}, \, ^{\backprime\backprime} S' \, ^{\prime\prime}, \, ^{\backprime\backprime} T \, ^{\prime\prime}, \, ^{\backprime\backprime} T' \, ^{\prime\prime},$ leaving the last three of these to form the new intermediate alphabet. Grammar 4 has the intermediate alphabet $\mathfrak{Q} \, = \, \{ \, ^{\backprime\backprime} S' \, ^{\prime\prime}, \, ^{\backprime\backprime} T \, ^{\prime\prime}, \, ^{\backprime\backprime} T' \, ^{\prime\prime} \, \},$ with the set $\mathfrak{K}$ of covering rules as listed in the next display.

$\mathfrak{C} (\mathfrak{P}) : \text{Grammar 4}\!$

$\mathfrak{Q} = \{ \, ^{\backprime\backprime} S' \, ^{\prime\prime}, \, ^{\backprime\backprime} T \, ^{\prime\prime}, \, ^{\backprime\backprime} T' \, ^{\prime\prime} \, \}$

$\begin{array}{rcll} 1. & S & :> & \varepsilon \\ 2. & S & :> & S' \\ 3. & S' & :> & m_1 \\ 4. & S' & :> & p_j, \, \text{for each} \, j \in J \\ 5. & S' & :> & ^{\backprime\backprime} \, \operatorname{(} \, ^{\prime\prime} \, \cdot \, T \, \cdot \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime} \\ 6. & S' & :> & S' \, \cdot \, S' \\ 7. & T & :> & \varepsilon \\ 8. & T & :> & T' \\ 9. & T' & :> & T \, \cdot \, ^{\backprime\backprime} \operatorname{,} ^{\prime\prime} \, \cdot \, S \\ \end{array}$

In this version of a grammar for $\mathfrak{L} = \mathfrak{C} (\mathfrak{P}),$ the intermediate type $T\!$ is partitioned as $T = \underline\varepsilon + T',$ thereby parsing the intermediate symbol $T\!$ in parallel fashion with the division of its overlying type as $S = \underline\varepsilon + S'.$ This is an option that I will choose to close off for now, but leave it open to consider at a later point. Thus, it suffices to give a brief discussion of what it involves, in the process of moving on to its chief alternative.

There does not appear to be anything radically wrong with trying this approach to types. It is reasonable and consistent in its underlying principle, and it provides a rational and a homogeneous strategy toward all parts of speech, but it does require an extra amount of conceptual overhead, in that every non-trivial type has to be split into two parts and comprehended in two stages. Consequently, in view of the largely practical difficulties of making the requisite distinctions for every intermediate symbol, it is a common convention, whenever possible, to restrict intermediate types to covering exclusively non-empty strings.

For the sake of future reference, it is convenient to refer to this restriction on intermediate symbols as the intermediate significance constraint. It can be stated in a compact form as a condition on the relations between non-terminal symbols $q \in \{ \, ^{\backprime\backprime} S \, ^{\prime\prime} \, \} \cup \mathfrak{Q}$ and sentential forms $W \in \{ \, ^{\backprime\backprime} S \, ^{\prime\prime} \, \} \cup (\mathfrak{Q} \cup \mathfrak{A})^*.$

$\text{Condition On Intermediate Significance}\!$

$\begin{array}{lccc} \text{If} & q & :> & W \\ \text{and} & W & = & \varepsilon \\ \text{then} & q & = & ^{\backprime\backprime} S \, ^{\prime\prime} \\ \end{array}$

If this is beginning to sound like a monotone condition, then it is not absurd to sharpen the resemblance and render the likeness more acute. This is done by declaring a couple of ordering relations, denoting them under variant interpretations by the same sign, $^{\backprime\backprime}\!< \, ^{\prime\prime}.$

The ordering $^{\backprime\backprime}\!< \, ^{\prime\prime}$ on the set of non-terminal symbols, $q \in \{ \, ^{\backprime\backprime} S \, ^{\prime\prime} \, \} \cup \mathfrak{Q},$ ordains the initial symbol $^{\backprime\backprime} S \, ^{\prime\prime}$ to be strictly prior to every intermediate symbol. This is tantamount to the axiom that $^{\backprime\backprime} S \, ^{\prime\prime} < q,$ for all $q \in \mathfrak{Q}.$
The ordering $^{\backprime\backprime}\!< \, ^{\prime\prime}$ on the collection of sentential forms, $W \in \{ \, ^{\backprime\backprime} S \, ^{\prime\prime} \, \} \cup (\mathfrak{Q} \cup \mathfrak{A})^*,$ ordains the empty string to be strictly minor to every other sentential form. This is stipulated in the axiom that $\varepsilon < W,$ for every non-empty sentential form $W.\!$

Given these two orderings, the constraint in question on intermediate significance can be stated as follows:

$\text{Condition On Intermediate Significance}\!$

$\begin{array}{lccc} \text{If} & q & :> & W \\ \text{and} & q & > & ^{\backprime\backprime} S \, ^{\prime\prime} \\ \text{then} & W & > & \varepsilon \\ \end{array}$

Achieving a grammar that respects this convention typically requires a more detailed account of the initial setting of a type, both with regard to the type of context that incites its appearance and also with respect to the minimal strings that arise under the type in question. In order to find covering productions that satisfy the intermediate significance condition, one must be prepared to consider a wider variety of calling contexts or inciting situations that can be noted to surround each recognized type, and also to enumerate a larger number of the smallest cases that can be observed to fall under each significant type.

Grammar 5

With the foregoing array of considerations in mind, one is gradually led to a grammar for $\mathfrak{L} = \mathfrak{C} (\mathfrak{P})$ in which all of the covering productions have either one of the following two forms:

$\begin{array}{ccll} S & :> & \varepsilon & \\ q & :> & W, & \text{with} \ q \in \{ \, ^{\backprime\backprime} S \, ^{\prime\prime} \, \} \cup \mathfrak{Q} \ \text{and} \ W \in (\mathfrak{Q} \cup \mathfrak{A})^+ \\ \end{array}$

A grammar that fits into this mold is called a context-free grammar. The first type of rewrite rule is referred to as a special production, while the second type of rewrite rule is called an ordinary production. An ordinary derivation is one that employs only ordinary productions. In ordinary productions, those that have the form $q :> W,\!$ the replacement string $W\!$ is never the empty string, and so the lengths of the augmented strings or the sentential forms that follow one another in an ordinary derivation, on account of using the ordinary types of rewrite rules, never decrease at any stage of the process, up to and including the terminal string that is finally generated by the grammar. This type of feature is known as the non-contracting property of productions, derivations, and grammars. A grammar is said to have the property if all of its covering productions, with the possible exception of $S :> \varepsilon,$ are non-contracting. In particular, context-free grammars are special cases of non-contracting grammars. The presence of the non-contracting property within a formal grammar makes the length of the augmented string available as a parameter that can figure into mathematical inductions and motivate recursive proofs, and this handle on the generative process makes it possible to establish the kinds of results about the generated language that are not easy to achieve in more general cases, nor by any other means even in these brands of special cases.

Grammar 5 is a context-free grammar for the painted cactus language that uses $\mathfrak{Q} = \{ \, ^{\backprime\backprime} S' \, ^{\prime\prime}, \, ^{\backprime\backprime} T \, ^{\prime\prime} \, \},$ with $\mathfrak{K}$ as listed in the next display.

$\mathfrak{C} (\mathfrak{P}) : \text{Grammar 5}\!$

$\mathfrak{Q} = \{ \, ^{\backprime\backprime} S' \, ^{\prime\prime}, \, ^{\backprime\backprime} T \, ^{\prime\prime} \, \}$

$\begin{array}{rcll} 1. & S & :> & \varepsilon \\ 2. & S & :> & S' \\ 3. & S' & :> & m_1 \\ 4. & S' & :> & p_j, \, \text{for each} \, j \in J \\ 5. & S' & :> & S' \, \cdot \, S' \\ 6. & S' & :> & ^{\backprime\backprime} \, \operatorname{()} \, ^{\prime\prime} \\ 7. & S' & :> & ^{\backprime\backprime} \, \operatorname{(} \, ^{\prime\prime} \, \cdot \, T \, \cdot \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime} \\ 8. & T & :> & ^{\backprime\backprime} \, \operatorname{,} \, ^{\prime\prime} \\ 9. & T & :> & S' \\ 10. & T & :> & T \, \cdot \, ^{\backprime\backprime} \, \operatorname{,} \, ^{\prime\prime} \\ 11. & T & :> & T \, \cdot \, ^{\backprime\backprime} \, \operatorname{,} \, ^{\prime\prime} \, \cdot \, S' \\ \end{array}$

Finally, it is worth trying to bring together the advantages of these diverse styles of grammar, to whatever extent that they are compatible. To do this, a prospective grammar must be capable of maintaining a high level of intermediate organization, like that arrived at in Grammar 2, while respecting the principle of intermediate significance, and thus accumulating all the benefits of the context-free format in Grammar 5. A plausible synthesis of most of these features is given in Grammar 6.

Grammar 6

Grammar 6 has the intermediate alphabet $\mathfrak{Q} = \{ \, ^{\backprime\backprime} S' \, ^{\prime\prime}, \, ^{\backprime\backprime} F \, ^{\prime\prime}, \, ^{\backprime\backprime} R \, ^{\prime\prime}, \, ^{\backprime\backprime} T \, ^{\prime\prime} \, \},$ with the production set $\mathfrak{K}$ as listed in the next display.

$\mathfrak{C} (\mathfrak{P}) : \text{Grammar 6}\!$

$\mathfrak{Q} = \{ \, ^{\backprime\backprime} S' \, ^{\prime\prime}, \, ^{\backprime\backprime} F \, ^{\prime\prime}, \, ^{\backprime\backprime} R \, ^{\prime\prime}, \, ^{\backprime\backprime} T \, ^{\prime\prime} \, \}$

$\begin{array}{rcll} 1. & S & :> & \varepsilon \\ 2. & S & :> & S' \\ 3. & S' & :> & R \\ 4. & S' & :> & F \\ 5. & S' & :> & S' \, \cdot \, S' \\ 6. & R & :> & m_1 \\ 7. & R & :> & p_j, \, \text{for each} \, j \in J \\ 8. & R & :> & R \, \cdot \, R \\ 9. & F & :> & ^{\backprime\backprime} \, \operatorname{()} \, ^{\prime\prime} \\ 10. & F & :> & ^{\backprime\backprime} \, \operatorname{(} \, ^{\prime\prime} \, \cdot \, T \, \cdot \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime} \\ 11. & T & :> & ^{\backprime\backprime} \, \operatorname{,} \, ^{\prime\prime} \\ 12. & T & :> & S' \\ 13. & T & :> & T \, \cdot \, ^{\backprime\backprime} \, \operatorname{,} \, ^{\prime\prime} \\ 14. & T & :> & T \, \cdot \, ^{\backprime\backprime} \, \operatorname{,} \, ^{\prime\prime} \, \cdot \, S' \\ \end{array}$

The preceding development provides a typical example of how an initially effective and conceptually succinct description of a formal language, but one that is terse to the point of allowing its prospective interpreter to waste exorbitant amounts of energy in trying to unravel its implications, can be converted into a form that is more efficient from the operational point of view, even if slightly more ungainly in regard to its elegance.

The basic idea behind all of this machinery remains the same: Besides the select body of formulas that are introduced as boundary conditions, it merely institutes the following general rule:

$\operatorname{If}$	the strings $S_1, \ldots, S_k\!$ are sentences,
$\operatorname{Then}$	their concatenation in the form
	$\operatorname{Conc}_{j=1}^k S_j \ = \ S_1 \, \cdot \, \ldots \, \cdot \, S_k$
	is a sentence,
$\operatorname{And}$	their surcatenation in the form
	$\operatorname{Surc}_{j=1}^k S_j \ = \ ^{\backprime\backprime} \, \operatorname{(} \, ^{\prime\prime} \, \cdot \, S_1 \, \cdot \, ^{\backprime\backprime} \, \operatorname{,} \, ^{\prime\prime} \, \cdot \, \ldots \, \cdot \, ^{\backprime\backprime} \, \operatorname{,} \, ^{\prime\prime} \, \cdot \, S_k \, \cdot \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime}$
	is a sentence.

1.3.11.2. Generalities About Formal Grammars

It is fitting to wrap up the foregoing developments by summarizing the notion of a formal grammar that appeared to evolve in the present case. For the sake of future reference and the chance of a wider application, it is also useful to try to extract the scheme of a formalization that potentially holds for any formal language. The following presentation of the notion of a formal grammar is adapted, with minor modifications, from the treatment in (DDQ, 60–61).

A formal grammar $\mathfrak{G}$ is given by a four-tuple $\mathfrak{G} = ( \, ^{\backprime\backprime} S \, ^{\prime\prime}, \, \mathfrak{Q}, \, \mathfrak{A}, \, \mathfrak{K} \, )$ that takes the following form of description:

$^{\backprime\backprime} S \, ^{\prime\prime}$ is the initial, special, start, or sentence symbol. Since the letter $^{\backprime\backprime} S \, ^{\prime\prime}$ serves this function only in a special setting, its employment in this role need not create any confusion with its other typical uses as a string variable or as a sentence variable.
$\mathfrak{Q} = \{ q_1, \ldots, q_m \}$ is a finite set of intermediate symbols, all distinct from $^{\backprime\backprime} S \, ^{\prime\prime}.$
$\mathfrak{A} = \{ a_1, \dots, a_n \}$ is a finite set of terminal symbols, also known as the alphabet of $\mathfrak{G},$ all distinct from $^{\backprime\backprime} S \, ^{\prime\prime}$ and disjoint from $\mathfrak{Q}.$ Depending on the particular conception of the language $\mathfrak{L}$ that is covered, generated, governed, or ruled by the grammar $\mathfrak{G},$ that is, whether $\mathfrak{L}$ is conceived to be a set of words, sentences, paragraphs, or more extended structures of discourse, it is usual to describe $\mathfrak{A}$ as the alphabet, lexicon, vocabulary, liturgy, or phrase book of both the grammar $\mathfrak{G}$ and the language $\mathfrak{L}$ that it regulates.
$\mathfrak{K}$ is a finite set of characterizations. Depending on how they come into play, these are variously described as covering rules, formations, productions, rewrite rules, subsumptions, transformations, or typing rules.

To describe the elements of $\mathfrak{K}$ it helps to define some additional terms:

The symbols in $\{ \, ^{\backprime\backprime} S \, ^{\prime\prime} \, \} \cup \mathfrak{Q} \cup \mathfrak{A}$ form the augmented alphabet of $\mathfrak{G}.$
The symbols in $\{ \, ^{\backprime\backprime} S \, ^{\prime\prime} \, \} \cup \mathfrak{Q}$ are the non-terminal symbols of $\mathfrak{G}.$
The symbols in $\mathfrak{Q} \cup \mathfrak{A}$ are the non-initial symbols of $\mathfrak{G}.$
The strings in $( \{ \, ^{\backprime\backprime} S \, ^{\prime\prime} \, \} \cup \mathfrak{Q} \cup \mathfrak{A} )^*$ are the augmented strings for $\mathfrak{G}.$
The strings in $\{ \, ^{\backprime\backprime} S \, ^{\prime\prime} \, \} \cup (\mathfrak{Q} \cup \mathfrak{A})^*$ are the sentential forms for $\mathfrak{G}.$

Each characterization in $\mathfrak{K}$ is an ordered pair of strings $(S_1, S_2)\!$ that takes the following form:

$S_1 \ = \ Q_1 \cdot q \cdot Q_2,$

$S_2 \ = \ Q_1 \cdot W \cdot Q_2.$

In this scheme, $S_1\!$ and $S_2\!$ are members of the augmented strings for $\mathfrak{G},$ more precisely, $S_1\!$ is a non-empty string and a sentential form over $\mathfrak{G},$ while $S_2\!$ is a possibly empty string and also a sentential form over $\mathfrak{G}.$

Here also, $q\!$ is a non-terminal symbol, that is, $q \in \{ \, ^{\backprime\backprime} S \, ^{\prime\prime} \, \} \cup \mathfrak{Q},$ while $Q_1, Q_2,\!$ and $W\!$ are possibly empty strings of non-initial symbols, a fact that can be expressed in the form, $Q_1, Q_2, W \in (\mathfrak{Q} \cup \mathfrak{A})^*.$

In practice, the couplets in $\mathfrak{K}$ are used to derive, to generate, or to produce sentences of the corresponding language $\mathfrak{L} = \mathfrak{L} (\mathfrak{G}).$ The language $\mathfrak{L}$ is then said to be governed, licensed, or regulated by the grammar $\mathfrak{G},$ a circumstance that is expressed in the form $\mathfrak{L} = \langle \mathfrak{G} \rangle.$ In order to facilitate this active employment of the grammar, it is conventional to write the abstract characterization $(S_1, S_2)\!$ and the specific characterization $(Q_1 \cdot q \cdot Q_2, \ Q_1 \cdot W \cdot Q_2)$ in the following forms, respectively:

$\begin{array}{lll} S_1 & :> & S_2 \\ Q_1 \cdot q \cdot Q_2 & :> & Q_1 \cdot W \cdot Q_2 \\ \end{array}$

In this usage, the characterization $S_1 :> S_2\!$ is tantamount to a grammatical license to transform a string of the form $Q_1 \cdot q \cdot Q_2$ into a string of the form $Q1 \cdot W \cdot Q2,$ in effect, replacing the non-terminal symbol $q\!$ with the non-initial string $W\!$ in any selected, preserved, and closely adjoining context of the form $Q1 \cdot \underline{~~~} \cdot Q2.$ In this application the notation $S_1 :> S_2\!$ can be read to say that $S_1\!$ produces $S_2\!$ or that $S_1\!$ transforms into $S_2.\!$

An immediate derivation in $\mathfrak{G}$ is an ordered pair $(W, W')\!$ of sentential forms in $\mathfrak{G}$ such that:

$\begin{array}{llll} W = Q_1 \cdot X \cdot Q_2, & W' = Q_1 \cdot Y \cdot Q_2, & \text{and} & (X, Y) \in \mathfrak{K}. \end{array}$

As noted above, it is usual to express the condition $(X, Y) \in \mathfrak{K}$ by writing $X :> Y \, \text{in} \, \mathfrak{G}.$

The immediate derivation relation is indicated by saying that $W\!$ immediately derives $W',\!$ by saying that $W'\!$ is immediately derived from $W\!$ in $\mathfrak{G},$ and also by writing:

$W ::> W'.\!$

A derivation in $\mathfrak{G}$ is a finite sequence $(W_1, \ldots, W_k)\!$ of sentential forms over $\mathfrak{G}$ such that each adjacent pair $(W_j, W_{j+1})\!$ of sentential forms in the sequence is an immediate derivation in $\mathfrak{G},$ in other words, such that:

$W_j ::> W_{j+1},\ \text{for all}\ j = 1\ \text{to}\ k - 1.$

If there exists a derivation $(W_1, \ldots, W_k)\!$ in $\mathfrak{G},$ one says that $W_1\!$ derives $W_k\!$ in $\mathfrak{G}$ or that $W_k\!$ is derivable from $W_1\!$ in $\mathfrak{G},$ and one typically summarizes the derivation by writing:

$W_1 :\!*\!:> W_k.\!$

The language $\mathfrak{L} = \mathfrak{L} (\mathfrak{G}) = \langle \mathfrak{G} \rangle$ that is generated by the formal grammar $\mathfrak{G} = ( \, ^{\backprime\backprime} S \, ^{\prime\prime}, \, \mathfrak{Q}, \, \mathfrak{A}, \, \mathfrak{K} \, )$ is the set of strings over the terminal alphabet $\mathfrak{A}$ that are derivable from the initial symbol $^{\backprime\backprime} S \, ^{\prime\prime}$ by way of the intermediate symbols in $\mathfrak{Q}$ according to the characterizations in $\mathfrak{K}.$ In sum:

$\mathfrak{L} (\mathfrak{G}) \ = \ \langle \mathfrak{G} \rangle \ = \ \{ \, W \in \mathfrak{A}^* \, : \, ^{\backprime\backprime} S \, ^{\prime\prime} \, :\!*\!:> \, W \, \}.$

Finally, a string $W\!$ is called a word, a sentence, or so on, of the language generated by $\mathfrak{G}$ if and only if $W\!$ is in $\mathfrak{L} (\mathfrak{G}).$

1.3.11.3. The Cactus Language : Stylistics

As a result, we can hardly conceive of how many possibilities there are for what we call objective reality. Our sharp quills of knowledge are so narrow and so concentrated in particular directions that with science there are myriads of totally different real worlds, each one accessible from the next simply by slight alterations — shifts of gaze — of every particular discipline and subspecialty.

— Herbert J. Bernstein, "Idols of Modern Science", [HJB, 38]

This Subsection highlights an issue of style that arises in describing a formal language. In broad terms, I use the word style to refer to a loosely specified class of formal systems, typically ones that have a set of distinctive features in common. For instance, a style of proof system usually dictates one or more rules of inference that are acknowledged as conforming to that style. In the present context, the word style is a natural choice to characterize the varieties of formal grammars, or any other sorts of formal systems that can be contemplated for deriving the sentences of a formal language.

In looking at what seems like an incidental issue, the discussion arrives at a critical point. The question is: What decides the issue of style? Taking a given language as the object of discussion, what factors enter into and determine the choice of a style for its presentation, that is, a particular way of arranging and selecting the materials that come to be involved in a description, a grammar, or a theory of the language? To what degree is the determination accidental, empirical, pragmatic, rhetorical, or stylistic, and to what extent is the choice essential, logical, and necessary? For that matter, what determines the order of signs in a word, a sentence, a text, or a discussion? All of the corresponding parallel questions about the character of this choice can be posed with regard to the constituent part as well as with regard to the main constitution of the formal language.

In order to answer this sort of question, at any level of articulation, one has to inquire into the type of distinction that it invokes, between arrangements and orders that are essential, logical, and necessary and orders and arrangements that are accidental, rhetorical, and stylistic. As a rough guide to its comprehension, a logical order, if it resides in the subject at all, can be approached by considering all of the ways of saying the same things, in all of the languages that are capable of saying roughly the same things about that subject. Of course, the all that appears in this rule of thumb has to be interpreted as a fittingly qualified sort of universal. For all practical purposes, it simply means all of the ways that a person can think of and all of the languages that a person can conceive of, with all things being relative to the particular moment of investigation. For all of these reasons, the rule must stand as little more than a rough idea of how to approach its object.

If it is demonstrated that a given formal language can be presented in any one of several styles of formal grammar, then the choice of a format is accidental, optional, and stylistic to the very extent that it is free. But if it can be shown that a particular language cannot be successfully presented in a particular style of grammar, then the issue of style is no longer free and rhetorical, but becomes to that very degree essential, necessary, and obligatory, in other words, a question of the objective logical order that can be found to reside in the object language.

As a rough illustration of the difference between logical and rhetorical orders, consider the kinds of order that are expressed and exhibited in the following conjunction of implications:

$X \Rightarrow Y\ \operatorname{and}\ Y \Rightarrow Z.$

Here, there is a happy conformity between the logical content and the rhetorical form, indeed, to such a degree that one hardly notices the difference between them. The rhetorical form is given by the order of sentences in the two implications and the order of implications in the conjunction. The logical content is given by the order of propositions in the extended implicational sequence:

$X\ \le\ Y\ \le\ Z.$

To see the difference between form and content, or manner and matter, it is enough to observe a few of the ways that the expression can be varied without changing its meaning, for example:

$Z \Leftarrow Y\ \operatorname{and}\ Y \Leftarrow X.$

Any style of declarative programming, also called logic programming, depends on a capacity, as embodied in a programming language or other formal system, to describe the relation between problems and solutions in logical terms. A recurring problem in building this capacity is in bridging the gap between ostensibly non-logical orders and the logical orders that are used to describe and to represent them. For instance, to mention just a couple of the most pressing cases, and the ones that are currently proving to be the most resistant to a complete analysis, one has the orders of dynamic evolution and rhetorical transition that manifest themselves in the process of inquiry and in the communication of its results.

This patch of the ongoing discussion is concerned with describing a particular variety of formal languages, whose typical representative is the painted cactus language $\mathfrak{L} = \mathfrak{C} (\mathfrak{P}).$ It is the intention of this work to interpret this language for propositional logic, and thus to use it as a sentential calculus, an order of reasoning that forms an active ingredient and a significant component of all logical reasoning. To describe this language, the standard devices of formal grammars and formal language theory are more than adequate, but this only raises the next question: What sorts of devices are exactly adequate, and fit the task to a "T"? The ultimate desire is to turn the tables on the order of description, and so begins a process of eversion that evolves to the point of asking: To what extent can the language capture the essential features and laws of its own grammar and describe the active principles of its own generation? In other words: How well can the language be described by using the language itself to do so?

In order to speak to these questions, I have to express what a grammar says about a language in terms of what a language can say on its own. In effect, it is necessary to analyze the kinds of meaningful statements that grammars are capable of making about languages in general and to relate them to the kinds of meaningful statements that the syntactic sentences of the cactus language might be interpreted as making about the very same topics. So far in the present discussion, the sentences of the cactus language do not make any meaningful statements at all, much less any meaningful statements about languages and their constitutions. As of yet, these sentences subsist in the form of purely abstract, formal, and uninterpreted combinatorial constructions.

Before the capacity of a language to describe itself can be evaluated, the missing link to meaning has to be supplied for each of its strings. This calls for a dimension of semantics and a notion of interpretation, topics that are taken up for the case of the cactus language $\mathfrak{C} (\mathfrak{P})$ in Subsection 1.3.10.12. Once a plausible semantics is prescribed for this language it will be possible to return to these questions and to address them in a meaningful way.

The prominent issue at this point is the distinct placements of formal languages and formal grammars with respect to the question of meaning. The sentences of a formal language are merely the abstract strings of abstract signs that happen to belong to a certain set. They do not by themselves make any meaningful statements at all, not without mounting a separate effort of interpretation, but the rules of a formal grammar make meaningful statements about a formal language, to the extent that they say what strings belong to it and what strings do not. Thus, the formal grammar, a formalism that appears to be even more skeletal than the formal language, still has bits and pieces of meaning attached to it. In a sense, the question of meaning is factored into two parts, structure and value, leaving the aspect of value reduced in complexity and subtlety to the simple question of belonging. Whether this single bit of meaningful value is enough to encompass all of the dimensions of meaning that we require, and whether it can be compounded to cover the complexity that actually exists in the realm of meaning — these are questions for an extended future inquiry.

Perhaps I ought to comment on the differences between the present and the standard definition of a formal grammar, since I am attempting to strike a compromise with several alternative conventions of usage, and thus to leave certain options open for future exploration. All of the changes are minor, in the sense that they are not intended to alter the classes of languages that are able to be generated, but only to clear up various ambiguities and sundry obscurities that affect their conception.

Primarily, the conventional scope of non-terminal symbols was expanded to encompass the sentence symbol, mainly on account of all the contexts where the initial and the intermediate symbols are naturally invoked in the same breath. By way of compensating for the usual exclusion of the sentence symbol from the non-terminal class, an equivalent distinction was introduced in the fashion of a distinction between the initial and the intermediate symbols, and this serves its purpose in all of those contexts where the two kind of symbols need to be treated separately.

At the present point, I remain a bit worried about the motivations and the justifications for introducing this distinction, under any name, in the first place. It is purportedly designed to guarantee that the process of derivation at least gets started in a definite direction, while the real questions have to do with how it all ends. The excuses of efficiency and expediency that I offered as plausible and sufficient reasons for distinguishing between empty and significant sentences are likely to be ephemeral, if not entirely illusory, since intermediate symbols are still permitted to characterize or to cover themselves, not to mention being allowed to cover the empty string, and so the very types of traps that one exerts oneself to avoid at the outset are always there to afflict the process at all of the intervening times.

If one reflects on the form of grammar that is being prescribed here, it looks as if one sought, rather futilely, to avoid the problems of recursion by proscribing the main program from calling itself, while allowing any subprogram to do so. But any trouble that is avoidable in the part is also avoidable in the main, while any trouble that is inevitable in the part is also inevitable in the main. Consequently, I am reserving the right to change my mind at a later stage, perhaps to permit the initial symbol to characterize, to cover, to regenerate, or to produce itself, if that turns out to be the best way in the end.

Before I leave this Subsection, I need to say a few things about the manner in which the abstract theory of formal languages and the pragmatic theory of sign relations interact with each other.

Formal language theory can seem like an awfully picky subject at times, treating every symbol as a thing in itself the way it does, sorting out the nominal types of symbols as objects in themselves, and singling out the passing tokens of symbols as distinct entities in their own rights. It has to continue doing this, if not for any better reason than to aid in clarifying the kinds of languages that people are accustomed to use, to assist in writing computer programs that are capable of parsing real sentences, and to serve in designing programming languages that people would like to become accustomed to use. As a matter of fact, the only time that formal language theory becomes too picky, or a bit too myopic in its focus, is when it leads one to think that one is dealing with the thing itself and not just with the sign of it, in other words, when the people who use the tools of formal language theory forget that they are dealing with the mere signs of more interesting objects and not with the objects of ultimate interest in and of themselves.

As a result, there a number of deleterious effects that can arise from the extreme pickiness of formal language theory, arising, as is often the case, when formal theorists forget the practical context of theorization. It frequently happens that the exacting task of defining the membership of a formal language leads one to think that this object and this object alone is the justifiable end of the whole exercise. The distractions of this mediate objective render one liable to forget that one's penultimate interest lies always with various kinds of equivalence classes of signs, not entirely or exclusively with their more meticulous representatives.

When this happens, one typically goes on working oblivious to the fact that many details about what transpires in the meantime do not matter at all in the end, and one is likely to remain in blissful ignorance of the circumstance that many special details of language membership are bound, destined, and pre-determined to be glossed over with some measure of indifference, especially when it comes down to the final constitution of those equivalence classes of signs that are able to answer for the genuine objects of the whole enterprise of language. When any form of theory, against its initial and its best intentions, leads to this kind of absence of mind that is no longer beneficial in all of its main effects, the situation calls for an antidotal form of theory, one that can restore the presence of mind that all forms of theory are meant to augment.

The pragmatic theory of sign relations is called for in settings where everything that can be named has many other names, that is to say, in the usual case. Of course, one would like to replace this superfluous multiplicity of signs with an organized system of canonical signs, one for each object that needs to be denoted, but reducing the redundancy too far, beyond what is necessary to eliminate the factor of "noise" in the language, that is, to clear up its effectively useless distractions, can destroy the very utility of a typical language, which is intended to provide a ready means to express a present situation, clear or not, and to describe an ongoing condition of experience in just the way that it seems to present itself. Within this fleshed out framework of language, moreover, the process of transforming the manifestations of a sign from its ordinary appearance to its canonical aspect is the whole problem of computation in a nutshell.

It is a well-known truth, but an often forgotten fact, that nobody computes with numbers, but solely with numerals in respect of numbers, and numerals themselves are symbols. Among other things, this renders all discussion of numeric versus symbolic computation a bit beside the point, since it is only a question of what kinds of symbols are best for one's immediate application or for one's selection of ongoing objectives. The numerals that everybody knows best are just the canonical symbols, the standard signs or the normal terms for numbers, and the process of computation is a matter of getting from the arbitrarily obscure signs that the data of a situation are capable of throwing one's way to the indications of its character that are clear enough to motivate action.

Having broached the distinction between propositions and sentences, one can see its similarity to the distinction between numbers and numerals. What are the implications of the foregoing considerations for reasoning about propositions and for the realm of reckonings in sentential logic? If the purpose of a sentence is just to denote a proposition, then the proposition is just the object of whatever sign is taken for a sentence. This means that the computational manifestation of a piece of reasoning about propositions amounts to a process that takes place entirely within a language of sentences, a procedure that can rationalize its account by referring to the denominations of these sentences among propositions.

The application of these considerations in the immediate setting is this: Do not worry too much about what roles the empty string $\varepsilon \, = \, ^{\backprime\backprime\prime\prime}$ and the blank symbol $m_1 \, = \, ^{\backprime\backprime} \operatorname{~} ^{\prime\prime}$ are supposed to play in a given species of formal languages. As it happens, it is far less important to wonder whether these types of formal tokens actually constitute genuine sentences than it is to decide what equivalence classes it makes sense to form over all of the sentences in the resulting language, and only then to bother about what equivalence classes these limiting cases of sentences are most conveniently taken to represent.

These concerns about boundary conditions betray a more general issue. Already by this point in discussion the limits of the purely syntactic approach to a language are beginning to be visible. It is not that one cannot go a whole lot further by this road in the analysis of a particular language and in the study of languages in general, but when it comes to the questions of understanding the purpose of a language, of extending its usage in a chosen direction, or of designing a language for a particular set of uses, what matters above all else are the pragmatic equivalence classes of signs that are demanded by the application and intended by the designer, and not so much the peculiar characters of the signs that represent these classes of practical meaning.

Any description of a language is bound to have alternative descriptions. More precisely, a circumscribed description of a formal language, as any effectively finite description is bound to be, is certain to suggest the equally likely existence and the possible utility of other descriptions. A single formal grammar describes but a single formal language, but any formal language is described by many different formal grammars, not all of which afford the same grasp of its structure, provide an equivalent comprehension of its character, or yield an interchangeable view of its aspects. Consequently, even with respect to the same formal language, different formal grammars are typically better for different purposes.

With the distinctions that evolve among the different styles of grammar, and with the preferences that different observers display toward them, there naturally comes the question: What is the root of this evolution?

One dimension of variation in the styles of formal grammars can be seen by treating the union of languages, and especially the disjoint union of languages, as a sum, by treating the concatenation of languages as a product, and then by distinguishing the styles of analysis that favor sums of products from those that favor products of sums as their canonical forms of description. If one examines the relation between languages and grammars carefully enough to see the presence and the influence of these different styles, and when one comes to appreciate the ways that different styles of grammars can be used with different degrees of success for different purposes, then one begins to see the possibility that alternative styles of description can be based on altogether different linguistic and logical operations.

It possible to trace this divergence of styles to an even more primitive division, one that distinguishes the additive or the parallel styles from the multiplicative or the serial styles. The issue is somewhat confused by the fact that an additive analysis is typically expressed in the form of a series, in other words, a disjoint union of sets or a linear sum of their independent effects. But it is easy enough to sort this out if one observes the more telling connection between parallel and independent. Another way to keep the right associations straight is to employ the term sequential in preference to the more misleading term serial. Whatever one calls this broad division of styles, the scope and sweep of their dimensions of variation can be delineated in the following way:

The additive or parallel styles favor sums of products $(\textstyle\sum\prod)$ as canonical forms of expression, pulling sums, unions, co-products, and logical disjunctions to the outermost layers of analysis and synthesis, while pushing products, intersections, concatenations, and logical conjunctions to the innermost levels of articulation and generation. In propositional logic, this style leads to the disjunctive normal form (DNF).
The multiplicative or serial styles favor products of sums $(\textstyle\prod\sum)$ as canonical forms of expression, pulling products, intersections, concatenations, and logical conjunctions to the outermost layers of analysis and synthesis, while pushing sums, unions, co-products, and logical disjunctions to the innermost levels of articulation and generation. In propositional logic, this style leads to the conjunctive normal form (CNF).

There is a curious sort of diagnostic clue that often serves to reveal the dominance of one mode or the other within an individual thinker's cognitive style. Examined on the question of what constitutes the natural numbers, an additive thinker tends to start the sequence at 0, while a multiplicative thinker tends to regard it as beginning at 1.

In any style of description, grammar, or theory of a language, it is usually possible to tease out the influence of these contrasting traits, namely, the additive attitude versus the mutiplicative tendency that go to make up the particular style in question, and even to determine the dominant inclination or point of view that establishes its perspective on the target domain.

In each style of formal grammar, the multiplicative aspect is present in the sequential concatenation of signs, both in the augmented strings and in the terminal strings. In settings where the non-terminal symbols classify types of strings, the concatenation of the non-terminal symbols signifies the cartesian product over the corresponding sets of strings.

In the context-free style of formal grammar, the additive aspect is easy enough to spot. It is signaled by the parallel covering of many augmented strings or sentential forms by the same non-terminal symbol. Expressed in active terms, this calls for the independent rewriting of that non-terminal symbol by a number of different successors, as in the following scheme:

$\begin{matrix} q & :> & W_1 \\ \\ \cdots & \cdots & \cdots \\ \\ q & :> & W_k \\ \end{matrix}$

It is useful to examine the relationship between the grammatical covering or production relation $(:>\!)$ and the logical relation of implication $(\Rightarrow),$ with one eye to what they have in common and one eye to how they differ. The production $q :> W\!$ says that the appearance of the symbol $q\!$ in a sentential form implies the possibility of exchanging it for $W.\!$ Although this sounds like a possible implication, to the extent that $q\!$ implies a possible $W\!$ or that $q\!$ possibly implies $W,\!$ the qualifiers possible and possibly are the critical elements in these statements, and they are crucial to the meaning of what is actually being implied. In effect, these qualifications reverse the direction of implication, yielding $^{\backprime\backprime} \, q \Leftarrow W \, ^{\prime\prime}$ as the best analogue for the sense of the production.

One way to sum this up is to say that non-terminal symbols have the significance of hypotheses. The terminal strings form the empirical matter of a language, while the non-terminal symbols mark the patterns or the types of substrings that can be noticed in the profusion of data. If one observes a portion of a terminal string that falls into the pattern of the sentential form $W,\!$ then it is an admissible hypothesis, according to the theory of the language that is constituted by the formal grammar, that this piece not only fits the type $q\!$ but even comes to be generated under the auspices of the non-terminal symbol $^{\backprime\backprime} q ^{\prime\prime}.$

A moment's reflection on the issue of style, giving due consideration to the received array of stylistic choices, ought to inspire at least the question: "Are these the only choices there are?" In the present setting, there are abundant indications that other options, more differentiated varieties of description and more integrated ways of approaching individual languages, are likely to be conceivable, feasible, and even more ultimately viable. If a suitably generic style, one that incorporates the full scope of logical combinations and operations, is broadly available, then it would no longer be necessary, or even apt, to argue in universal terms about which style is best, but more useful to investigate how we might adapt the local styles to the local requirements. The medium of a generic style would yield a viable compromise between additive and multiplicative canons, and render the choice between parallel and serial a false alternative, at least, when expressed in the globally exclusive terms that are currently most commonly adopted to pose it.

One set of indications comes from the study of machines, languages, and computation, especially the theories of their structures and relations. The forms of composition and decomposition that are generally known as parallel and serial are merely the extreme special cases, in variant directions of specialization, of a more generic form, usually called the cascade form of combination. This is a well-known fact in the theories that deal with automata and their associated formal languages, but its implications do not seem to be widely appreciated outside these fields. In particular, it dispells the need to choose one extreme or the other, since most of the natural cases are likely to exist somewhere in between.

Another set of indications appears in algebra and category theory, where forms of composition and decomposition related to the cascade combination, namely, the semi-direct product and its special case, the wreath product, are encountered at higher levels of generality than the cartesian products of sets or the direct products of spaces.

In these domains of operation, one finds it necessary to consider also the co-product of sets and spaces, a construction that artificially creates a disjoint union of sets, that is, a union of spaces that are being treated as independent. It does this, in effect, by indexing, coloring, or preparing the otherwise possibly overlapping domains that are being combined. What renders this a chimera or a hybrid form of combination is the fact that this indexing is tantamount to a cartesian product of a singleton set, namely, the conventional index, color, or affix in question, with the individual domain that is entering as a factor, a term, or a participant in the final result.

One of the insights that arises out of Peirce's logical work is that the set operations of complementation, intersection, and union, along with the logical operations of negation, conjunction, and disjunction that operate in isomorphic tandem with them, are not as fundamental as they first appear. This is because all of them can be constructed from or derived from a smaller set of operations, in fact, taking the logical side of things, from either one of two sole sufficient operators, called amphecks by Peirce, strokes by those who re-discovered them later, and known in computer science as the NAND and the NNOR operators. For this reason, that is, by virtue of their precedence in the orders of construction and derivation, these operations have to be regarded as the simplest and the most primitive in principle, even if they are scarcely recognized as lying among the more familiar elements of logic.

I am throwing together a wide variety of different operations into each of the bins labeled additive and multiplicative, but it is easy to observe a natural organization and even some relations approaching isomorphisms among and between the members of each class.

The relation between logical disjunction and set-theoretic union and the relation between logical conjunction and set-theoretic intersection ought to be clear enough for the purposes of the immediately present context. In any case, all of these relations are scheduled to receive a thorough examination in a subsequent discussion (Subsection 1.3.10.13). But the relation of a set-theoretic union to a category-theoretic co-product and the relation of a set-theoretic intersection to a syntactic concatenation deserve a closer look at this point.

The effect of a co-product as a disjointed union, in other words, that creates an object tantamount to a disjoint union of sets in the resulting co-product even if some of these sets intersect non-trivially and even if some of them are identical in reality, can be achieved in several ways. The most usual conception is that of making a separate copy, for each part of the intended co-product, of the set that is intended to go there. Often one thinks of the set that is assigned to a particular part of the co-product as being distinguished by a particular color, in other words, by the attachment of a distinct index, label, or tag, being a marker that is inherited by and passed on to every element of the set in that part. A concrete image of this construction can be achieved by imagining that each set and each element of each set is placed in an ordered pair with the sign of its color, index, label, or tag. One describes this as the injection of each set into the corresponding part of the co-product.

For example, given the sets $P\!$ and $Q,\!$ overlapping or not, one can define the indexed or marked sets $P_{[1]}\!$ and $Q_{[2]},\!$ amounting to the copy of $P\!$ into the first part of the co-product and the copy of $Q\!$ into the second part of the co-product, in the following manner:

$\begin{array}{lllll} P_{[1]} & = & (P, 1) & = & \{ (x, 1) : x \in P \}, \\ Q_{[2]} & = & (Q, 2) & = & \{ (x, 2) : x \in Q \}. \\ \end{array}$

Using the coproduct operator ($\textstyle\coprod$) for this construction, the sum, the coproduct, or the disjointed union of $P\!$ and $Q\!$ in that order can be represented as the ordinary union of $P_{[1]}\!$ and $Q_{[2]}.\!$

$\begin{array}{lll} P \coprod Q & = & P_{[1]} \cup Q_{[2]}. \\ \end{array}$

The concatenation $\mathfrak{L}_1 \cdot \mathfrak{L}_2$ of the formal languages $\mathfrak{L}_1$ and $\mathfrak{L}_2$ is just the cartesian product of sets $\mathfrak{L}_1 \times \mathfrak{L}_2$ without the extra $\times$'s, but the relation of cartesian products to set-theoretic intersections and thus to logical conjunctions is far from being clear. One way of seeing a type of relation is to focus on the information that is needed to specify each construction, and thus to reflect on the signs that are used to carry this information. As a first approach to the topic of information, according to a strategy that seeks to be as elementary and as informal as possible, I introduce the following set of ideas, intended to be taken in a very provisional way.

A stricture is a specification of a certain set in a certain place, relative to a number of other sets, yet to be specified. It is assumed that one knows enough to tell if two strictures are equivalent as pieces of information, but any more determinate indications, like names for the places that are mentioned in the stricture, or bounds on the number of places that are involved, are regarded as being extraneous impositions, outside the proper concern of the definition, no matter how convenient they are found to be for a particular discussion. As a schematic form of illustration, a stricture can be pictured in the following shape:

$^{\backprime\backprime}$

$\ldots \times X \times Q \times X \times \ldots$

$^{\prime\prime}$

A strait is the object that is specified by a stricture, in effect, a certain set in a certain place of an otherwise yet to be specified relation. Somewhat sketchily, the strait that corresponds to the stricture just given can be pictured in the following shape:

$\ldots \times X \times Q \times X \times \ldots$

In this picture $Q\!$ is a certain set and $X\!$ is the universe of discourse that is relevant to a given discussion. Since a stricture does not, by itself, contain a sufficient amount of information to specify the number of sets that it intends to set in place, or even to specify the absolute location of the set that its does set in place, it appears to place an unspecified number of unspecified sets in a vague and uncertain strait. Taken out of its interpretive context, the residual information that a stricture can convey makes all of the following potentially equivalent as strictures:

$\begin{array}{ccccccc} ^{\backprime\backprime} Q ^{\prime\prime} & , & ^{\backprime\backprime} X \times Q \times X ^{\prime\prime} & , & ^{\backprime\backprime} X \times X \times Q \times X \times X ^{\prime\prime} & , & \ldots \\ \end{array}$

With respect to what these strictures specify, this leaves all of the following equivalent as straits:

$\begin{array}{ccccccc} Q & = & X \times Q \times X & = & X \times X \times Q \times X \times X & = & \ldots \\ \end{array}$

Within the framework of a particular discussion, it is customary to set a bound on the number of places and to limit the variety of sets that are regarded as being under active consideration, and it is also convenient to index the places of the indicated relations, and of their encompassing cartesian products, in some fixed way. But the whole idea of a stricture is to specify a strait that is capable of extending through and beyond any fixed frame of discussion. In other words, a stricture is conceived to constrain a strait at a certain point, and then to leave it literally embedded, if tacitly expressed, in a yet to be fully specified relation, one that involves an unspecified number of unspecified domains.

A quantity of information is a measure of constraint. In this respect, a set of comparable strictures is ordered on account of the information that each one conveys, and a system of comparable straits is ordered in accord with the amount of information that it takes to pin each one of them down. Strictures that are more constraining and straits that are more constrained are placed at higher levels of information than those that are less so, and entities that involve more information are said to have a greater complexity in comparison with those entities that involve less information, that are said to have a greater simplicity.

In order to create a concrete example, let me now institute a frame of discussion where the number of places in a relation is bounded at two, and where the variety of sets under active consideration is limited to the typical subsets $P\!$ and $Q\!$ of a universe $X.\!$ Under these conditions, one can use the following sorts of expression as schematic strictures:

$\begin{matrix} ^{\backprime\backprime} X ^{\prime\prime} & ^{\backprime\backprime} P ^{\prime\prime} & ^{\backprime\backprime} Q ^{\prime\prime} \\ \\ ^{\backprime\backprime} X \times X ^{\prime\prime} & ^{\backprime\backprime} X \times P ^{\prime\prime} & ^{\backprime\backprime} X \times Q ^{\prime\prime} \\ \\ ^{\backprime\backprime} P \times X ^{\prime\prime} & ^{\backprime\backprime} P \times P ^{\prime\prime} & ^{\backprime\backprime} P \times Q ^{\prime\prime} \\ \\ ^{\backprime\backprime} Q \times X ^{\prime\prime} & ^{\backprime\backprime} Q \times P ^{\prime\prime} & ^{\backprime\backprime} Q \times Q ^{\prime\prime} \\ \end{matrix}$

These strictures and their corresponding straits are stratified according to their amounts of information, or their levels of constraint, as follows:

$\begin{array}{lcccc} \text{High:} & ^{\backprime\backprime} P \times P ^{\prime\prime} & ^{\backprime\backprime} P \times Q ^{\prime\prime} & ^{\backprime\backprime} Q \times P ^{\prime\prime} & ^{\backprime\backprime} Q \times Q ^{\prime\prime} \\ \\ \text{Med:} & ^{\backprime\backprime} P ^{\prime\prime} & ^{\backprime\backprime} X \times P ^{\prime\prime} & ^{\backprime\backprime} P \times X ^{\prime\prime} \\ \\ \text{Med:} & ^{\backprime\backprime} Q ^{\prime\prime} & ^{\backprime\backprime} X \times Q ^{\prime\prime} & ^{\backprime\backprime} Q \times X ^{\prime\prime} \\ \\ \text{Low:} & ^{\backprime\backprime} X ^{\prime\prime} & ^{\backprime\backprime} X \times X ^{\prime\prime} \\ \end{array}$

Within this framework, the more complex strait $P \times Q$ can be expressed in terms of the simpler straits, $P \times X$ and $X \times Q.$ More specifically, it lends itself to being analyzed as their intersection, in the following way:

$\begin{array}{lllll} P \times Q & = & P \times X & \cap & X \times Q. \\ \end{array}$

From here it is easy to see the relation of concatenation, by virtue of these types of intersection, to the logical conjunction of propositions. The cartesian product $P \times Q$ is described by a conjunction of propositions, namely, $P_{[1]} \land Q_{[2]},$ subject to the following interpretation:

$P_{[1]}\!$ asserts that there is an element from the set $P\!$ in the first place of the product.
$Q_{[2]}\!$ asserts that there is an element from the set $Q\!$ in the second place of the product.

The integration of these two pieces of information can be taken in that measure to specify a yet to be fully determined relation.

In a corresponding fashion at the level of the elements, the ordered pair $(p, q)\!$ is described by a conjunction of propositions, namely, $p_{[1]} \land q_{[2]},$ subject to the following interpretation:

$p_{[1]}\!$ says that $p\!$ is in the first place of the product element under construction.
$q_{[2]}\!$ says that $q\!$ is in the second place of the product element under construction.

Notice that, in construing the cartesian product of the sets $P\!$ and $Q\!$ or the concatenation of the languages $\mathfrak{L}_1$ and $\mathfrak{L}_2$ in this way, one shifts the level of the active construction from the tupling of the elements in $P\!$ and $Q\!$ or the concatenation of the strings that are internal to the languages $\mathfrak{L}_1$ and $\mathfrak{L}_2$ to the concatenation of the external signs that it takes to indicate these sets or these languages, in other words, passing to a conjunction of indexed propositions, $P_{[1]}\!$ and $Q_{[2]},\!$ or to a conjunction of assertions, $(\mathfrak{L}_1)_{[1]}$ and $(\mathfrak{L}_2)_{[2]},$ that marks the sets or the languages in question for insertion in the indicated places of a product set or a product language, respectively. In effect, the subscripting by the indices $^{\backprime\backprime} [1] ^{\prime\prime}$ and $^{\backprime\backprime} [2] ^{\prime\prime}$ can be recognized as a special case of concatenation, albeit through the posting of editorial remarks from an external mark-up language.

In order to systematize the relations that strictures and straits placed at higher levels of complexity, constraint, information, and organization have with those that are placed at the associated lower levels, I introduce the following pair of definitions:

The $j^\text{th}\!$ excerpt of a stricture of the form $^{\backprime\backprime} \, S_1 \times \ldots \times S_k \, ^{\prime\prime},$ regarded within a frame of discussion where the number of places is limited to $k,\!$ is the stricture of the form $^{\backprime\backprime} \, X \times \ldots \times S_j \times \ldots \times X \, ^{\prime\prime}.$ In the proper context, this can be written more succinctly as the stricture $^{\backprime\backprime} \, (S_j)_{[j]} \, ^{\prime\prime},$ an assertion that places the $j^\text{th}\!$ set in the $j^\text{th}\!$ place of the product.

The $j^\text{th}\!$ extract of a strait of the form $S_1 \times \ldots \times S_k,\!$ constrained to a frame of discussion where the number of places is restricted to $k,\!$ is the strait of the form $X \times \ldots \times S_j \times \ldots \times X.$ In the appropriate context, this can be denoted more succinctly by the stricture $^{\backprime\backprime} \, (S_j)_{[j]} \, ^{\prime\prime},$ an assertion that places the $j^\text{th}\!$ set in the $j^\text{th}\!$ place of the product.

In these terms, a stricture of the form $^{\backprime\backprime} \, S_1 \times \ldots \times S_k \, ^{\prime\prime}$ can be expressed in terms of simpler strictures, to wit, as a conjunction of its $k\!$ excerpts:

$\begin{array}{lll} ^{\backprime\backprime} \, S_1 \times \ldots \times S_k \, ^{\prime\prime} & = & ^{\backprime\backprime} \, (S_1)_{[1]} \, ^{\prime\prime} \, \land \, \ldots \, \land \, ^{\backprime\backprime} \, (S_k)_{[k]} \, ^{\prime\prime}. \end{array}$

In a similar vein, a strait of the form $S_1 \times \ldots \times S_k\!$ can be expressed in terms of simpler straits, namely, as an intersection of its $k\!$ extracts:

$\begin{array}{lll} S_1 \times \ldots \times S_k & = & (S_1)_{[1]} \, \cap \, \ldots \, \cap \, (S_k)_{[k]}. \end{array}$

There is a measure of ambiguity that remains in this formulation, but it is the best that I can do in the present informal context.

1.3.11.4. The Cactus Language : Mechanics

We are only now beginning to see how this works. Clearly one of the mechanisms for picking a reality is the sociohistorical sense of what is important — which research program, with all its particularity of knowledge, seems most fundamental, most productive, most penetrating. The very judgments which make us push narrowly forward simultaneously make us forget how little we know. And when we look back at history, where the lesson is plain to find, we often fail to imagine ourselves in a parallel situation. We ascribe the differences in world view to error, rather than to unexamined but consistent and internally justified choice.

— Herbert J. Bernstein, "Idols of Modern Science", [HJB, 38]

In this Subsection, I discuss the mechanics of parsing the cactus language into the corresponding class of computational data structures. This provides each sentence of the language with a translation into a computational form that articulates its syntactic structure and prepares it for automated modes of processing and evaluation. For this purpose, it is necessary to describe the target data structures at a fairly high level of abstraction only, ignoring the details of address pointers and record structures and leaving the more operational aspects of implementation to the imagination of prospective programmers. In this way, I can put off to another stage of elaboration and refinement the description of the program that constructs these pointers and operates on these graph-theoretic data structures.

The structure of a painted cactus, insofar as it presents itself to the visual imagination, can be described as follows. The overall structure, as given by its underlying graph, falls within the species of graph that is commonly known as a rooted cactus, and the only novel feature that it adds to this is that each of its nodes can be painted with a finite sequence of paints, chosen from a palette that is given by the parametric set $\{ \, ^{\backprime\backprime} \operatorname{~} ^{\prime\prime} \, \} \cup \mathfrak{P} = \{ m_1 \} \cup \{ p_1, \ldots, p_k \}.$

It is conceivable, from a purely graph-theoretical point of view, to have a class of cacti that are painted but not rooted, and so it is frequently necessary, for the sake of precision, to more exactly pinpoint the target species of graphical structure as a painted and rooted cactus (PARC).

A painted cactus, as a rooted graph, has a distinguished node that is called its root. By starting from the root and working recursively, the rest of its structure can be described in the following fashion.

Each node of a PARC consists of a graphical point or vertex plus a finite sequence of attachments, described in relative terms as the attachments at or to that node. An empty sequence of attachments defines the empty node. Otherwise, each attachment is one of three kinds: a blank, a paint, or a type of PARC that is called a lobe.

Each lobe of a PARC consists of a directed graphical cycle plus a finite sequence of accoutrements, described in relative terms as the accoutrements of or on that lobe. Recalling the circumstance that every lobe that comes under consideration comes already attached to a particular node, exactly one vertex of the corresponding cycle is the vertex that comes from that very node. The remaining vertices of the cycle have their definitions filled out according to the accoutrements of the lobe in question. An empty sequence of accoutrements is taken to be tantamount to a sequence that contains a single empty node as its unique accoutrement, and either one of these ways of approaching it can be regarded as defining a graphical structure that is called a needle or a terminal edge. Otherwise, each accoutrement of a lobe is itself an arbitrary PARC.

Although this definition of a lobe in terms of its intrinsic structural components is logically sufficient, it is also useful to characterize the structure of a lobe in comparative terms, that is, to view the structure that typifies a lobe in relation to the structures of other PARC's and to mark the inclusion of this special type within the general run of PARC's. This approach to the question of types results in a form of description that appears to be a bit more analytic, at least, in mnemonic or prima facie terms, if not ultimately more revealing. Working in this vein, a lobe can be characterized as a special type of PARC that is called an unpainted root plant (UR-plant).

An UR-plant is a PARC of a simpler sort, at least, with respect to the recursive ordering of structures that is being followed here. As a type, it is defined by the presence of two properties, that of being planted and that of having an unpainted root. These are defined as follows:

A PARC is planted if its list of attachments has just one PARC.
A PARC is UR if its list of attachments has no blanks or paints.

In short, an UR-planted PARC has a single PARC as its only attachment, and since this attachment is prevented from being a blank or a paint, the single attachment at its root has to be another sort of structure, that which we call a lobe.

To express the description of a PARC in terms of its nodes, each node can be specified in the fashion of a functional expression, letting a citation of the generic function name "$\operatorname{Node}$" be followed by a list of arguments that enumerates the attachments of the node in question, and letting a citation of the generic function name "$\operatorname{Lobe}$" be followed by a list of arguments that details the accoutrements of the lobe in question. Thus, one can write expressions of the following forms:

$1.\!$	$\operatorname{Node}^0$	$=\!$	$\operatorname{Node}()$
		$=\!$	a node with no attachments.
	$\operatorname{Node}_{j=1}^k C_j$	$=\!$	$\operatorname{Node} (C_1, \ldots, C_k)$
		$=\!$	a node with the attachments $C_1, \ldots, C_k.$
$2.\!$	$\operatorname{Lobe}^0$	$=\!$	$\operatorname{Lobe}()$
		$=\!$	a lobe with no accoutrements.
	$\operatorname{Lobe}_{j=1}^k C_j$	$=\!$	$\operatorname{Lobe} (C_1, \ldots, C_k)$
		$=\!$	a lobe with the accoutrements $C_1, \ldots, C_k.$

Working from a structural description of the cactus language, or any suitable formal grammar for $\mathfrak{C} (\mathfrak{P}),$ it is possible to give a recursive definition of the function called $\operatorname{Parse}$ that maps each sentence in $\operatorname{PARCE} (\mathfrak{P})$ to the corresponding graph in $\operatorname{PARC} (\mathfrak{P}).$ One way to do this proceeds as follows:

The parse of the concatenation $\operatorname{Conc}_{j=1}^k$ of the $k\!$ sentences $(s_j)_{j=1}^k$ is defined recursively as follows:

$\operatorname{Parse} (\operatorname{Conc}^0) ~=~ \operatorname{Node}^0.$
For $k > 0,\!$

$\operatorname{Parse} (\operatorname{Conc}_{j=1}^k s_j) ~=~ \operatorname{Node}_{j=1}^k \operatorname{Parse} (s_j).$

The parse of the surcatenation $\operatorname{Surc}_{j=1}^k$ of the $k\!$ sentences $(s_j)_{j=1}^k$ is defined recursively as follows:

$\operatorname{Parse} (\operatorname{Surc}^0) ~=~ \operatorname{Lobe}^0.$
For $k > 0,\!$

$\operatorname{Parse} (\operatorname{Surc}_{j=1}^k s_j) ~=~ \operatorname{Lobe}_{j=1}^k \operatorname{Parse} (s_j).$

For ease of reference, Table 13 summarizes the mechanics of these parsing rules.

Table 13. Algorithmic Translation Rules

$\text{Sentence in PARCE}\!$

$\xrightarrow{\operatorname{Parse}}$

$\text{Graph in PARC}\!$

$\operatorname{Conc}^0$	$\xrightarrow{\operatorname{Parse}}$	$\operatorname{Node}^0$
$\operatorname{Conc}_{j=1}^k s_j$	$\xrightarrow{\operatorname{Parse}}$	$\operatorname{Node}_{j=1}^k \operatorname{Parse} (s_j)$

$\operatorname{Surc}^0$	$\xrightarrow{\operatorname{Parse}}$	$\operatorname{Lobe}^0$
$\operatorname{Surc}_{j=1}^k s_j$	$\xrightarrow{\operatorname{Parse}}$	$\operatorname{Lobe}_{j=1}^k \operatorname{Parse} (s_j)$

A substructure of a PARC is defined recursively as follows. Starting at the root node of the cactus $C,\!$ any attachment is a substructure of $C.\!$ If a substructure is a blank or a paint, then it constitutes a minimal substructure, meaning that no further substructures of $C\!$ arise from it. If a substructure is a lobe, then each one of its accoutrements is also a substructure of $C,\!$ and has to be examined for further substructures.

The concept of substructure can be used to define varieties of deletion and erasure operations that respect the structure of the abstract graph. For the purposes of this depiction, a blank symbol $^{\backprime\backprime} ~ ^{\prime\prime}$ is treated as a primer, in other words, as a clear paint or a neutral tint. In effect, one is letting $m_1 = p_0.\!$ In this frame of discussion, it is useful to make the following distinction:

To delete a substructure is to replace it with an empty node, in effect, to reduce the whole structure to a trivial point.
To erase a substructure is to replace it with a blank symbol, in effect, to paint it out of the picture or to overwrite it.

A bare PARC, loosely referred to as a bare cactus, is a PARC on the empty palette $\mathfrak{P} = \varnothing.$ In other veins, a bare cactus can be described in several different ways, depending on how the form arises in practice.

Leaning on the definition of a bare PARCE, a bare PARC can be described as the kind of a parse graph that results from parsing a bare cactus expression, in other words, as the kind of a graph that issues from the requirements of processing a sentence of the bare cactus language $\mathfrak{C}^0 = \operatorname{PARCE}^0.$
To express it more in its own terms, a bare PARC can be defined by tracing the recursive definition of a generic PARC, but then by detaching an independent form of description from the source of that analogy. The method is sufficiently sketched as follows:

A bare PARC is a PARC whose attachments are limited to blanks and bare lobes.
A bare lobe is a lobe whose accoutrements are limited to bare PARC's.

In practice, a bare cactus is usually encountered in the process of analyzing or handling an arbitrary PARC, the circumstances of which frequently call for deleting or erasing all of its paints. In particular, this generally makes it easier to observe the various properties of its underlying graphical structure.

1.3.11.5. The Cactus Language : Semantics

Alas, and yet what are you, my written and painted thoughts! It is not long ago that you were still so many-coloured, young and malicious, so full of thorns and hidden spices you made me sneeze and laugh — and now? You have already taken off your novelty and some of you, I fear, are on the point of becoming truths: they already look so immortal, so pathetically righteous, so boring!

— Nietzsche, Beyond Good and Evil, [Nie-2, ¶ 296]

In this Subsection, I describe a particular semantics for the painted cactus language, telling what meanings I aim to attach to its bare syntactic forms. This supplies an interpretation for this parametric family of formal languages, but it is good to remember that it forms just one of many such interpretations that are conceivable and even viable. In deed, the distinction between the object domain and the sign domain can be observed in the fact that many languages can be deployed to depict the same set of objects and that any language worth its salt is bound to to give rise to many different forms of interpretive saliency.

In formal settings, it is common to speak of interpretation as if it created a direct connection between the signs of a formal language and the objects of the intended domain, in other words, as if it determined the denotative component of a sign relation. But a closer attention to what goes on reveals that the process of interpretation is more indirect, that what it does is to provide each sign of a prospectively meaningful source language with a translation into an already established target language, where already established means that its relationship to pragmatic objects is taken for granted at the moment in question.

With this in mind, it is clear that interpretation is an affair of signs that at best respects the objects of all of the signs that enter into it, and so it is the connotative aspect of semiotics that is at stake here. There is nothing wrong with my saying that I interpret a sentence of a formal language as a sign that refers to a function or to a proposition, so long as you understand that this reference is likely to be achieved by way of more familiar and perhaps less formal signs that you already take to denote those objects.

On entering a context where a logical interpretation is intended for the sentences of a formal language there are a few conventions that make it easier to make the translation from abstract syntactic forms to their intended semantic senses. Although these conventions are expressed in unnecessarily colorful terms, from a purely abstract point of view, they do provide a useful array of connotations that help to negotiate what is otherwise a difficult transition. This terminology is introduced as the need for it arises in the process of interpreting the cactus language.

The task of this Subsection is to specify a semantic function for the sentences of the cactus language $\mathfrak{L} = \mathfrak{C}(\mathfrak{P}),$ in other words, to define a mapping that "interprets" each sentence of $\mathfrak{C}(\mathfrak{P})$ as a sentence that says something, as a sentence that bears a meaning, in short, as a sentence that denotes a proposition, and thus as a sign of an indicator function. When the syntactic sentences of a formal language are given a referent significance in logical terms, for example, as denoting propositions or indicator functions, then each form of syntactic combination takes on a corresponding form of logical significance.

By way of providing a logical interpretation for the cactus language, I introduce a family of operators on indicator functions that are called propositional connectives, and I distinguish these from the associated family of syntactic combinations that are called sentential connectives, where the relationship between these two realms of connection is exactly that between objects and their signs. A propositional connective, as an entity of a well-defined functional and operational type, can be treated in every way as a logical or a mathematical object, and thus as the type of object that can be denoted by the corresponding form of syntactic entity, namely, the sentential connective that is appropriate to the case in question.

There are two basic types of connectives, called the blank connectives and the bound connectives, respectively, with one connective of each type for each natural number $k = 0, 1, 2, 3, \ldots.$

The blank connective of $k\!$ places is signified by the concatenation of the $k\!$ sentences that fill those places.

For the special case of $k = 0,\!$ the blank connective is taken to be an empty string or a blank symbol — it does not matter which, since both are assigned the same denotation among propositions.

For the generic case of $k > 0,\!$ the blank connective takes the form $s_1 \cdot \ldots \cdot s_k.$ In the type of data that is called a text, the use of the center dot $(\cdot)$ is generally supplanted by whatever number of spaces and line breaks serve to improve the readability of the resulting text.
The bound connective of $k\!$ places is signified by the surcatenation of the $k\!$ sentences that fill those places.

For the special case of $k = 0,\!$ the bound connective is taken to be an empty closure — an expression enjoying one of the forms $\underline{(} \underline{)}, \, \underline{(} ~ \underline{)}, \, \underline{(} ~~ \underline{)}, \, \ldots$ with any number of blank symbols between the parentheses — all of which are assigned the same logical denotation among propositions.

For the generic case of $k > 0,\!$ the bound connective takes the form $\underline{(} s_1, \ldots, s_k \underline{)}.$

At this point, there are actually two different dialects, scripts, or modes of presentation for the cactus language that need to be interpreted, in other words, that need to have a semantic function defined on their domains.

There is the literal formal language of strings in $\operatorname{PARCE} (\mathfrak{P}),$ the painted and rooted cactus expressions that constitute the language $\mathfrak{L} = \mathfrak{C} (\mathfrak{P}) \subseteq \mathfrak{A}^* = (\mathfrak{M} \cup \mathfrak{P})^*.$
There is the figurative formal language of graphs in $\operatorname{PARC} (\mathfrak{P}),$ the painted and rooted cacti themselves, a parametric family of graphs or a species of computational data structures that is graphically analogous to the language of literal strings.

Of course, these two modalities of formal language, like written and spoken natural languages, are meant to have compatible interpretations, and so it is usually sufficient to give just the meanings of either one. All that remains is to provide a codomain or a target space for the intended semantic function, in other words, to supply a suitable range of logical meanings for the memberships of these languages to map into. Out of the many interpretations that are formally possible to arrange, one way of doing this proceeds by making the following definitions:

The conjunction $\operatorname{Conj}_j^J q_j$ of a set of propositions, $\{ q_j : j \in J \},$ is a proposition that is true if and only if every one of the $q_j\!$ is true.

$\operatorname{Conj}_j^J q_j$ is true $\Leftrightarrow$ $q_j\!$ is true for every $j \in J.$
The surjunction $\operatorname{Surj}_j^J q_j$ of a set of propositions, $\{ q_j : j \in J \},$ is a proposition that is true if and only if exactly one of the $q_j\!$ is untrue.

$\operatorname{Surj}_j^J q_j$ is true $\Leftrightarrow$ $q_j\!$ is untrue for unique $j \in J.$

If the number of propositions that are being joined together is finite, then the conjunction and the surjunction can be represented by means of sentential connectives, incorporating the sentences that represent these propositions into finite strings of symbols.

If $J\!$ is finite, for instance, if $J\!$ consists of the integers in the interval $j = 1 ~\text{to}~ k,$ and if each proposition $q_j\!$ is represented by a sentence $s_j,\!$ then the following strategies of expression are open:

The conjunction $\operatorname{Conj}_j^J q_j$ can be represented by a sentence that is constructed by concatenating the $s_j\!$ in the following fashion:

$\operatorname{Conj}_j^J q_j ~\leftrightsquigarrow~ s_1 s_2 \ldots s_k.$
The surjunction $\operatorname{Surj}_j^J q_j$ can be represented by a sentence that is constructed by surcatenating the $s_j\!$ in the following fashion:

$\operatorname{Surj}_j^J q_j ~\leftrightsquigarrow~ \underline{(} s_1, s_2, \ldots, s_k \underline{)}.$

If one opts for a mode of interpretation that moves more directly from the parse graph of a sentence to the potential logical meaning of both the PARC and the PARCE, then the following specifications are in order:

A cactus rooted at a particular node is taken to represent what that node denotes, its logical denotation or its logical interpretation.

The logical denotation of a node is the logical conjunction of that node's arguments, which are defined as the logical denotations of that node's attachments. The logical denotation of either a blank symbol or an empty node is the boolean value $\underline{1} = \operatorname{true}.$ The logical denotation of the paint $\mathfrak{p}_j\!$ is the proposition $p_j,\!$ a proposition that is regarded as primitive, at least, with respect to the level of analysis that is represented in the current instance of $\mathfrak{C} (\mathfrak{P}).$
The logical denotation of a lobe is the logical surjunction of that lobe's arguments, which are defined as the logical denotations of that lobe's accoutrements. As a corollary, the logical denotation of the parse graph of $\underline{(} \underline{)},$ otherwise called a needle, is the boolean value $\underline{0} = \operatorname{false}.$

If one takes the point of view that PARCs and PARCEs amount to a pair of intertranslatable languages for the same domain of objects, then denotation brackets of the form $\downharpoonleft \ldots \downharpoonright$ can be used to indicate the logical denotation $\downharpoonleft C_j \downharpoonright$ of a cactus $C_j\!$ or the logical denotation $\downharpoonleft s_j \downharpoonright$ of a sentence $s_j.\!$

Tables 14 and 15 summarize the relations that serve to connect the formal language of sentences with the logical language of propositions. Between these two realms of expression there is a family of graphical data structures that arise in parsing the sentences and that serve to facilitate the performance of computations on the indicator functions. The graphical language supplies an intermediate form of representation between the formal sentences and the indicator functions, and the form of mediation that it provides is very useful in rendering the possible connections between the other two languages conceivable in fact, not to mention in carrying out the necessary translations on a practical basis. These Tables include this intermediate domain in their Central Columns. Between their First and Middle Columns they illustrate the mechanics of parsing the abstract sentences of the cactus language into the graphical data structures of the corresponding species. Between their Middle and Final Columns they summarize the semantics of interpreting the graphical forms of representation for the purposes of reasoning with propositions.

Table 14. Semantic Translation : Functional Form

$\operatorname{Sentence}$

$\xrightarrow[\operatorname{~~~~~~~~~~}]{\operatorname{Parse}}$

$\operatorname{Graph}$

$\xrightarrow[\operatorname{~~~~~~~~~~}]{\operatorname{Denotation}}$

$\operatorname{Proposition}$

$s_j\!$

$\xrightarrow{\operatorname{~~~~~~~~~~}}$

$C_j\!$

$\xrightarrow{\operatorname{~~~~~~~~~~}}$

$q_j\!$

$\operatorname{Conc}^0$	$\xrightarrow{\operatorname{~~~~~~~~~~}}$	$\operatorname{Node}^0$	$\xrightarrow{\operatorname{~~~~~~~~~~}}$	$\underline{1}$
$\operatorname{Conc}^k_j s_j$	$\xrightarrow{\operatorname{~~~~~~~~~~}}$	$\operatorname{Node}^k_j C_j$	$\xrightarrow{\operatorname{~~~~~~~~~~}}$	$\operatorname{Conj}^k_j q_j$

$\operatorname{Surc}^0$	$\xrightarrow{\operatorname{~~~~~~~~~~}}$	$\operatorname{Lobe}^0$	$\xrightarrow{\operatorname{~~~~~~~~~~}}$	$\underline{0}$
$\operatorname{Surc}^k_j s_j$	$\xrightarrow{\operatorname{~~~~~~~~~~}}$	$\operatorname{Lobe}^k_j C_j$	$\xrightarrow{\operatorname{~~~~~~~~~~}}$	$\operatorname{Surj}^k_j q_j$

Table 15. Semantic Translation : Equational Form

$\downharpoonleft \operatorname{Sentence} \downharpoonright$

$\stackrel{\operatorname{Parse}}{=}$

$\downharpoonleft \operatorname{Graph} \downharpoonright$

$\stackrel{\operatorname{Denotation}}{=}$

$\operatorname{Proposition}$

$\downharpoonleft s_j \downharpoonright$

$=\!$

$\downharpoonleft C_j \downharpoonright$

$=\!$

$q_j\!$

$\downharpoonleft \operatorname{Conc}^0 \downharpoonright$	$=\!$	$\downharpoonleft \operatorname{Node}^0 \downharpoonright$	$=\!$	$\underline{1}$
$\downharpoonleft \operatorname{Conc}^k_j s_j \downharpoonright$	$=\!$	$\downharpoonleft \operatorname{Node}^k_j C_j \downharpoonright$	$=\!$	$\operatorname{Conj}^k_j q_j$

$\downharpoonleft \operatorname{Surc}^0 \downharpoonright$	$=\!$	$\downharpoonleft \operatorname{Lobe}^0 \downharpoonright$	$=\!$	$\underline{0}$
$\downharpoonleft \operatorname{Surc}^k_j s_j \downharpoonright$	$=\!$	$\downharpoonleft \operatorname{Lobe}^k_j C_j \downharpoonright$	$=\!$	$\operatorname{Surj}^k_j q_j$

Aside from their common topic, the two Tables present slightly different ways of conceptualizing the operations that go to establish their maps. Table 14 records the functional associations that connect each domain with the next, taking the triplings of a sentence $s_j,\!$ a cactus $C_j,\!$ and a proposition $q_j\!$ as basic data, and fixing the rest by recursion on these. Table 15 records these associations in the form of equations, treating sentences and graphs as alternative kinds of signs, and generalizing the denotation bracket operator to indicate the proposition that either denotes. It should be clear at this point that either scheme of translation puts the sentences, the graphs, and the propositions that it associates with each other roughly in the roles of the signs, the interpretants, and the objects, respectively, whose triples define an appropriate sign relation. Indeed, the "roughly" can be made "exactly" as soon as the domains of a suitable sign relation are specified precisely.

A good way to illustrate the action of the conjunction and surjunction operators is to demonstrate how they can be used to construct the boolean functions on any finite number of variables. Let us begin by doing this for the first three cases, $k = 0, 1, 2.\!$

A boolean function $F^{(0)}\!$ on $0\!$ variables is just an element of the boolean domain $\underline\mathbb{B} = \{ \underline{0}, \underline{1} \}.$ Table 16 shows several different ways of referring to these elements, just for the sake of consistency using the same format that will be used in subsequent Tables, no matter how degenerate it tends to appear in the initial case.

**Table 16. Boolean Functions on Zero Variables**
$F\!$	$F\!$	$F()\!$	$F\!$
$\underline{0}$	$F_0^{(0)}\!$	$\underline{0}$	$(~)$
$\underline{1}$	$F_1^{(0)}\!$	$\underline{1}$	$((~))$

Column 1 lists each boolean element or boolean function under its ordinary constant name or under a succinct nickname, respectively.

Column 2 lists each boolean function in a style of function name $F_j^{(k)}\!$ that is constructed as follows: The superscript $(k)\!$ gives the dimension of the functional domain, that is, the number of its functional variables, and the subscript $j\!$ is a binary string that recapitulates the functional values, using the obvious translation of boolean values into binary values.

Column 3 lists the functional values for each boolean function, or possibly a boolean element appearing in the guise of a function, for each combination of its domain values.

Column 4 shows the usual expressions of these elements in the cactus language, conforming to the practice of omitting the underlines in display formats. Here I illustrate also the convention of using the expression $^{\backprime\backprime} ((~)) ^{\prime\prime}$ as a visible stand-in for the expression of the logical value $\operatorname{true},$ a value that is minimally represented by a blank expression that tends to elude our giving it much notice in the context of more demonstrative texts.

Table 17 presents the boolean functions on one variable, $F^{(1)} : \underline\mathbb{B} \to \underline\mathbb{B},$ of which there are precisely four.

**Table 17. Boolean Functions on One Variable**
$F\!$	$F\!$	$F(x)\!$		$F\!$
		$F(\underline{1})$	$F(\underline{0})$
$F_0^{(1)}\!$	$F_{00}^{(1)}\!$	$\underline{0}$	$\underline{0}$	$(~)$
$F_1^{(1)}\!$	$F_{01}^{(1)}\!$	$\underline{0}$	$\underline{1}$	$(x)\!$
$F_2^{(1)}\!$	$F_{10}^{(1)}\!$	$\underline{1}$	$\underline{0}$	$x\!$
$F_3^{(1)}\!$	$F_{11}^{(1)}\!$	$\underline{1}$	$\underline{1}$	$((~))$

Here, Column 1 codes the contents of Column 2 in a more concise form, compressing the lists of boolean values, recorded as bits in the subscript string, into their decimal equivalents. Naturally, the boolean constants reprise themselves in this new setting as constant functions on one variable. Thus, one has the synonymous expressions for constant functions that are expressed in the next two chains of equations:

$\begin{matrix} F_0^{(1)} & = & F_{00}^{(1)} & = & \underline{0} ~:~ \underline\mathbb{B} \to \underline\mathbb{B} \\ \\ F_3^{(1)} & = & F_{11}^{(1)} & = & \underline{1} ~:~ \underline\mathbb{B} \to \underline\mathbb{B} \end{matrix}$

As for the rest, the other two functions are easily recognized as corresponding to the one-place logical connectives, or the monadic operators on $\underline\mathbb{B}.$ Thus, the function $F_1^{(1)} = F_{01}^{(1)}$ is recognizable as the negation operation, and the function $F_2^{(1)} = F_{10}^{(1)}$ is obviously the identity operation.

Table 18 presents the boolean functions on two variables, $F^{(2)} : \underline\mathbb{B}^2 \to \underline\mathbb{B},$ of which there are precisely sixteen.

**Table 18. Boolean Functions on Two Variables**
$F\!$	$F\!$	$F(x, y)\!$				$F\!$
		$F(\underline{1}, \underline{1})$	$F(\underline{1}, \underline{0})$	$F(\underline{0}, \underline{1})$	$F(\underline{0}, \underline{0})$
$F_{0}^{(2)}\!$	$F_{0000}^{(2)}\!$	$\underline{0}$	$\underline{0}$	$\underline{0}$	$\underline{0}$	$(~)$
$F_{1}^{(2)}\!$	$F_{0001}^{(2)}\!$	$\underline{0}$	$\underline{0}$	$\underline{0}$	$\underline{1}$	$(x)(y)\!$
$F_{2}^{(2)}\!$	$F_{0010}^{(2)}\!$	$\underline{0}$	$\underline{0}$	$\underline{1}$	$\underline{0}$	$(x) y\!$
$F_{3}^{(2)}\!$	$F_{0011}^{(2)}\!$	$\underline{0}$	$\underline{0}$	$\underline{1}$	$\underline{1}$	$(x)\!$
$F_{4}^{(2)}\!$	$F_{0100}^{(2)}\!$	$\underline{0}$	$\underline{1}$	$\underline{0}$	$\underline{0}$	$x (y)\!$
$F_{5}^{(2)}\!$	$F_{0101}^{(2)}\!$	$\underline{0}$	$\underline{1}$	$\underline{0}$	$\underline{1}$	$(y)\!$
$F_{6}^{(2)}\!$	$F_{0110}^{(2)}\!$	$\underline{0}$	$\underline{1}$	$\underline{1}$	$\underline{0}$	$(x, y)\!$
$F_{7}^{(2)}\!$	$F_{0111}^{(2)}\!$	$\underline{0}$	$\underline{1}$	$\underline{1}$	$\underline{1}$	$(x y)\!$
$F_{8}^{(2)}\!$	$F_{1000}^{(2)}\!$	$\underline{1}$	$\underline{0}$	$\underline{0}$	$\underline{0}$	$x y\!$
$F_{9}^{(2)}\!$	$F_{1001}^{(2)}\!$	$\underline{1}$	$\underline{0}$	$\underline{0}$	$\underline{1}$	$((x, y))\!$
$F_{10}^{(2)}\!$	$F_{1010}^{(2)}\!$	$\underline{1}$	$\underline{0}$	$\underline{1}$	$\underline{0}$	$y\!$
$F_{11}^{(2)}\!$	$F_{1011}^{(2)}\!$	$\underline{1}$	$\underline{0}$	$\underline{1}$	$\underline{1}$	$(x (y))\!$
$F_{12}^{(2)}\!$	$F_{1100}^{(2)}\!$	$\underline{1}$	$\underline{1}$	$\underline{0}$	$\underline{0}$	$x\!$
$F_{13}^{(2)}\!$	$F_{1101}^{(2)}\!$	$\underline{1}$	$\underline{1}$	$\underline{0}$	$\underline{1}$	$((x)y)\!$
$F_{14}^{(2)}\!$	$F_{1110}^{(2)}\!$	$\underline{1}$	$\underline{1}$	$\underline{1}$	$\underline{0}$	$((x)(y))\!$
$F_{15}^{(2)}\!$	$F_{1111}^{(2)}\!$	$\underline{1}$	$\underline{1}$	$\underline{1}$	$\underline{1}$	$((~))$

As before, all of the boolean functions of fewer variables are subsumed in this Table, though under a set of alternative names and possibly different interpretations. Just to acknowledge a few of the more notable pseudonyms:

The constant function $\underline{0} ~:~ \underline\mathbb{B}^2 \to \underline\mathbb{B}$ appears under the name $F_{0}^{(2)}.$

The constant function $\underline{1} ~:~ \underline\mathbb{B}^2 \to \underline\mathbb{B}$ appears under the name $F_{15}^{(2)}.$

The negation and identity of the first variable are $F_{3}^{(2)}$ and $F_{12}^{(2)},$ respectively.

The negation and identity of the second variable are $F_{5}^{(2)}$ and $F_{10}^{(2)},$ respectively.

The logical conjunction is given by the function $F_{8}^{(2)} (x, y) = x \cdot y.$

The logical disjunction is given by the function $F_{14}^{(2)} (x, y) = \underline{((} ~x~ \underline{)(} ~y~ \underline{))}.$

Functions expressing the conditionals, implications, or if-then statements are given in the following ways:

\[[x \Rightarrow y] = F_{11}^{(2)} (x, y) = \underline{(} ~x~ \underline{(} ~y~ \underline{))} = [\operatorname{not}~ x ~\operatorname{without}~ y].\]

\[[x \Leftarrow y] = F_{13}^{(2)} (x, y) = \underline{((} ~x~ \underline{)} ~y~ \underline{)} = [\operatorname{not}~ y ~\operatorname{without}~ x].\]

The function that corresponds to the biconditional, the equivalence, or the if and only statement is exhibited in the following fashion:

\[[x \Leftrightarrow y] = [x = y] = F_{9}^{(2)} (x, y) = \underline{((} ~x~,~y~ \underline{))}.\]

Finally, there is a boolean function that is logically associated with the exclusive disjunction, inequivalence, or not equals statement, algebraically associated with the binary sum operation, and geometrically associated with the symmetric difference of sets. This function is given by:

\[[x \neq y] = [x + y] = F_{6}^{(2)} (x, y) = \underline{(} ~x~,~y~ \underline{)}.\]

Let me now address one last question that may have occurred to some. What has happened, in this suggested scheme of functional reasoning, to the distinction that is quite pointedly made by careful logicians between (1) the connectives called conditionals and symbolized by the signs $(\rightarrow)$ and $(\leftarrow),$ and (2) the assertions called implications and symbolized by the signs $(\Rightarrow)$ and $(\Leftarrow)$, and, in a related question: What has happened to the distinction that is equally insistently made between (3) the connective called the biconditional and signified by the sign $(\leftrightarrow)$ and (4) the assertion that is called an equivalence and signified by the sign $(\Leftrightarrow)$? My answer is this: For my part, I am deliberately avoiding making these distinctions at the level of syntax, preferring to treat them instead as distinctions in the use of boolean functions, turning on whether the function is mentioned directly and used to compute values on arguments, or whether its inverse is being invoked to indicate the fibers of truth or untruth under the propositional function in question.

1.3.11.6. Stretching Exercises

Taking up the preceding arrays of particular connections, namely, the boolean functions on up to two variables, $F^{(k)} : \underline\mathbb{B}^k \to \underline\mathbb{B},$ for $k\!$ in $\{ 0, 1, 2 \},\!$ it is possible to illustrate the use of the stretch operation in a variety of concrete cases.

For example, suppose that $F\!$ is a connection of the form $F : \underline\mathbb{B}^2 \to \underline\mathbb{B},$ that is, any one of the sixteen possibilities in Table 18, while $p\!$ and $q\!$ are propositions of the form $p, q : X \to \underline\mathbb{B},$ that is, propositions about things in the universe $X,\!$ or else the indicators of sets contained in $X.\!$

Then one has the imagination $\underline{f} = (f_1, f_2) = (p, q) : (X \to \underline\mathbb{B})^2,$ and the stretch of the connection $F\!$ to $\underline{f}$ on $X\!$ amounts to a proposition $F^\$ (p, q) : X \to \underline\mathbb{B}$ that may be read as the stretch of $F\!$ to $p\!$ and $q.\!$ If one is concerned with many different propositions about things in $X,\!$ or if one is abstractly indifferent to the particular choices for $p\!$ and $q,\!$ then one may detach the operator $F^\$ : (X \to \underline\mathbb{B}))^2 \to (X \to \underline\mathbb{B})),$ called the stretch of $F\!$ over $X,\!$ and consider it in isolation from any concrete application.

When the cactus notation is used to represent boolean functions, a single $\$$ sign at the end of the expression is enough to remind the reader that the connections are meant to be stretched to several propositions on a universe $X.\!$

For example, take the connection $F : \underline\mathbb{B}^2 \to \underline\mathbb{B}$ such that:

\[F(x, y) ~=~ F_{6}^{(2)} (x, y) ~=~ \underline{(}~x~,~y~\underline{)}\]

The connection in question is a boolean function on the variables $x, y\!$ that returns a value of $\underline{1}$ just when just one of the pair $x, y\!$ is not equal to $\underline{1},$ or what amounts to the same thing, just when just one of the pair $x, y\!$ is equal to $\underline{1}.$ There is clearly an isomorphism between this connection, viewed as an operation on the boolean domain $\underline\mathbb{B} = \{ \underline{0}, \underline{1} \},$ and the dyadic operation on binary values $x, y \in \mathbb{B} = \operatorname{GF}(2)$ that is otherwise known as $x + y\!.$

The same connection $F : \underline\mathbb{B}^2 \to \underline\mathbb{B}$ can also be read as a proposition about things in the universe $X = \underline\mathbb{B}^2.$ If $s\!$ is a sentence that denotes the proposition $F,\!$ then the corresponding assertion says exactly what one states in uttering the sentence $^{\backprime\backprime} \, x ~\operatorname{is~not~equal~to}~ y \, ^{\prime\prime}.$ In such a case, one has $\downharpoonleft s \downharpoonright \, = F,$ and all of the following expressions are ordinarily taken as equivalent descriptions of the same set:

\(\begin{array}{lll} [| \downharpoonleft s \downharpoonright |] & = & [| F |] \\[6pt] & = & F^{-1} (\underline{1}) \\[6pt] & = & \{~ (x, y) \in \underline\mathbb{B}^2 ~:~ s ~\} \\[6pt] & = & \{~ (x, y) \in \underline\mathbb{B}^2 ~:~ F(x, y) = \underline{1} ~\} \\[6pt] & = & \{~ (x, y) \in \underline\mathbb{B}^2 ~:~ F(x, y) ~\} \\[6pt] & = & \{~ (x, y) \in \underline\mathbb{B}^2 ~:~ \underline{(}~x~,~y~\underline{)} = \underline{1} ~\} \\[6pt] & = & \{~ (x, y) \in \underline\mathbb{B}^2 ~:~ \underline{(}~x~,~y~\underline{)} ~\} \\[6pt] & = & \{~ (x, y) \in \underline\mathbb{B}^2 ~:~ x ~\operatorname{exclusive~or}~ y ~\} \\[6pt] & = & \{~ (x, y) \in \underline\mathbb{B}^2 ~:~ \operatorname{just~one~true~of}~ x, y ~\} \\[6pt] & = & \{~ (x, y) \in \underline\mathbb{B}^2 ~:~ x ~\operatorname{not~equal~to}~ y ~\} \\[6pt] & = & \{~ (x, y) \in \underline\mathbb{B}^2 ~:~ x \nLeftrightarrow y ~\} \\[6pt] & = & \{~ (x, y) \in \underline\mathbb{B}^2 ~:~ x \neq y ~\} \\[6pt] & = & \{~ (x, y) \in \underline\mathbb{B}^2 ~:~ x + y ~\}. \end{array}\)

Notice the distinction, that I continue to maintain at this point, between the logical values $\{ \operatorname{falsehood}, \operatorname{truth} \}$ and the algebraic values $\{ 0, 1 \}.\!$ This makes it legitimate to write a sentence directly into the righthand side of a set-builder expression, for instance, weaving the sentence $s\!$ or the sentence $^{\backprime\backprime} \, x ~\operatorname{is~not~equal~to}~ y \, ^{\prime\prime}$ into the context $^{\backprime\backprime} \, \{ (x, y) \in \underline{B}^2 : \ldots \} \, ^{\prime\prime},$ thereby obtaining the corresponding expressions listed above. It also allows us to assert the proposition $F(x, y)\!$ in a more direct way, without detouring through the equation $F(x, y) = \underline{1},$ since it already has a value in $\{ \operatorname{falsehood}, \operatorname{true} \},$ and thus can be taken as tantamount to an actual sentence.

If the appropriate safeguards can be kept in mind, avoiding all danger of confusing propositions with sentences and sentences with assertions, then the marks of these distinctions need not be forced to clutter the account of the more substantive indications, that is, the ones that really matter. If this level of understanding can be achieved, then it may be possible to relax these restrictions, along with the absolute dichotomy between algebraic and logical values, which tends to inhibit the flexibility of interpretation.

This covers the properties of the connection $F(x, y) = \underline{(}~x~,~y~\underline{)},$ treated as a proposition about things in the universe $X = \underline\mathbb{B}^2.$ Staying with this same connection, it is time to demonstrate how it can be "stretched" to form an operator on arbitrary propositions.

To continue the exercise, let $p\!$ and $q\!$ be arbitrary propositions about things in the universe $X,\!$ that is, maps of the form $p, q : X \to \underline\mathbb{B},$ and suppose that $p, q\!$ are indicator functions of the sets $P, Q \subseteq X,$ respectively. In other words, we have the following data:

$\begin{matrix} p & = & \upharpoonleft P \upharpoonright & : & X \to \underline\mathbb{B} \\ \\ q & = & \upharpoonleft Q \upharpoonright & : & X \to \underline\mathbb{B} \\ \\ (p, q) & = & (\upharpoonleft P \upharpoonright, \upharpoonleft Q \upharpoonright) & : & (X \to \underline\mathbb{B})^2 \\ \end{matrix}$

Then one has an operator $F^\$,$ the stretch of the connection $F\!$ over $X,\!$ and a proposition $F^\$ (p, q),$ the stretch of $F\!$ to $(p, q)\!$ on $X,\!$ with the following properties:

$\begin{array}{ccccl} F^\$ & = & \underline{(} \ldots, \ldots \underline{)}^\$ & : & (X \to \underline\mathbb{B})^2 \to (X \to \underline\mathbb{B}) \\ \\ F^\$ (p, q) & = & \underline{(}~p~,~q~\underline{)}^\$ & : & X \to \underline\mathbb{B} \\ \end{array}$

As a result, the application of the proposition $F^\$ (p, q)$ to each $x \in X$ returns a logical value in $\underline\mathbb{B},$ all in accord with the following equations:

$\begin{matrix} F^\$ (p, q)(x) & = & \underline{(}~p~,~q~\underline{)}^\$ (x) & \in & \underline\mathbb{B} \\ \\ \Updownarrow & & \Updownarrow \\ \\ F(p(x), q(x)) & = & \underline{(}~p(x)~,~q(x)~\underline{)} & \in & \underline\mathbb{B} \\ \end{matrix}$

For each choice of propositions $p\!$ and $q\!$ about things in $X,\!$ the stretch of $F\!$ to $p\!$ and $q\!$ on $X\!$ is just another proposition about things in $X,\!$ a simple proposition in its own right, no matter how complex its current expression or its present construction as $F^\$ (p, q) = \underline{(}~p~,~q~\underline{)}^\$$ makes it appear in relation to $p\!$ and $q.\!$ Like any other proposition about things in $X,\!$ it indicates a subset of $X,\!$ namely, the fiber that is variously described in the following ways:

$\begin{array}{lll} [| F^\$ (p, q) |] & = & [| \underline{(}~p~,~q~\underline{)}^\$ |] \\[6pt] & = & (F^\$ (p, q))^{-1} (\underline{1}) \\[6pt] & = & \{~ x \in X ~:~ F^\$ (p, q)(x) ~\} \\[6pt] & = & \{~ x \in X ~:~ \underline{(}~p~,~q~\underline{)}^\$ (x) ~\} \\[6pt] & = & \{~ x \in X ~:~ \underline{(}~p(x)~,~q(x)~\underline{)} ~\} \\[6pt] & = & \{~ x \in X ~:~ p(x) + q(x) ~\} \\[6pt] & = & \{~ x \in X ~:~ p(x) \neq q(x) ~\} \\[6pt] & = & \{~ x \in X ~:~ \upharpoonleft P \upharpoonright (x) ~\neq~ \upharpoonleft Q \upharpoonright (x) ~\} \\[6pt] & = & \{~ x \in X ~:~ x \in P ~\nLeftrightarrow~ x \in Q ~\} \\[6pt] & = & \{~ x \in X ~:~ x \in P\!-\!Q ~\operatorname{or}~ x \in Q\!-\!P ~\} \\[6pt] & = & \{~ x \in X ~:~ x \in P\!-\!Q ~\cup~ Q\!-\!P ~\} \\[6pt] & = & \{~ x \in X ~:~ x \in P + Q ~\} \\[6pt] & = & P + Q ~\subseteq~ X \\[6pt] & = & [|p|] + [|q|] ~\subseteq~ X \end{array}$

1.3.12. Syntactic Transformations

To discuss the import of the above definitions in greater depth, it serves to establish a number of logical relations and set-theoretic identities that can be found to hold among this array of conceptions and constructions. Facilitating this task requires in turn a number of auxiliary concepts and notations.

The diverse notions of indication under discussion are expressed in a variety of different notations, in particular, the logical language of sentences, the functional language of propositions, and the geometric language of sets. Thus, one way to explain the relationships that exist among these concepts is to describe the translations that they induce among the allied families of notation.

1.3.12.1. Syntactic Transformation Rules

A good way to summarize these translations and to organize their use in practice is by means of the syntactic transformation rules (STRs) that partially formalize them. A rudimentary example of a STR is readily mined from the raw materials that are already available in this area of discussion. To begin, let the definition of an indicator function be recorded in the following form:

$\text{Definition 1}\!$

	$\text{If}\!$	$Q ~\subseteq~ X$
	$\text{then}\!$	$\upharpoonleft Q \upharpoonright ~:~ X \to \underline\mathbb{B}$
	$\text{such that:}\!$

$\operatorname{D1a.}$

$\upharpoonleft Q \upharpoonright (x) ~\Leftrightarrow~ x \in Q$

$\forall x \in X$

In practice, a definition like this is commonly used to substitute one logically equivalent expression or sentence for another in a context where the conditions of using the definition this way are satisfied and where the change is perceived to advance a proof. This employment of a definition can be expressed in the form of a STR that allows one to exchange two expressions of logically equivalent forms for one another in every context where their logical values are the only consideration. To be specific, the logical value of an expression is the value in the boolean domain $\underline\mathbb{B} = \{ \underline{0}, \underline{1} \} = \{ \operatorname{false}, \operatorname{true} \}$ that the expression stands for in its context or represents to its interpreter.

In the case of Definition 1, the corresponding STR permits one to exchange a sentence of the form $x \in Q$ with an expression of the form $\upharpoonleft Q \upharpoonright (x)$ in any context that satisfies the conditions of its use, namely, the conditions of the definition that lead up to the stated equivalence. The relevant STR is recorded in Rule 1. By way of convention, I list the items that fall under a rule roughly in order of their ascending conceptual subtlety or their increasing syntactic complexity, without regard to their normal or typical orders of exchange, since this can vary widely from case to case.

		$\text{Rule 1}\!$
$\text{If}\!$	$Q \subseteq X$
$\text{then}\!$	$\upharpoonleft Q \upharpoonright ~:~ X \to \underline\mathbb{B}$
$\text{and if}\!$	$x \in X$
$\text{then}\!$	$\text{the following are equivalent:}\!$
$\text{R1a.}\!$	$x \in Q$
$\text{R1b.}\!$	$\upharpoonleft Q \upharpoonright (x)$

Conversely, any rule of this sort, properly qualified by the conditions under which it applies, can be turned back into a summary statement of the logical equivalence that is involved in its application. This mode of conversion a static principle and a transformational rule, that is, between a statement of equivalence and an equivalence of statements, is so automatic that it is usually not necessary to make a separate note of the "horizontal" versus the "vertical" versions.

As another example of a STR, consider the following logical equivalence, that holds for any $Q \subseteq X$ and for all $x \in X.$

$\upharpoonleft Q \upharpoonright (x) ~\Leftrightarrow~ \upharpoonleft Q \upharpoonright (x) = \underline{1}.$

In practice, this logical equivalence is used to exchange an expression of the form $\upharpoonleft Q \upharpoonright (x)$ with a sentence of the form $\upharpoonleft Q \upharpoonright (x) = \underline{1}$ in any context where one has a relatively fixed $Q \subseteq X$ in mind and where one is conceiving $x \in X$ to vary over its whole domain, namely, the universe $X.\!$ This leads to the STR that is given in Rule 2.

		$\text{Rule 2}\!$
$\text{If}\!$	$f : X \to \underline\mathbb{B}$
$\text{and}\!$	$x \in X$
$\text{then}\!$	$\text{the following are equivalent:}\!$
$\text{R2a.}\!$	$f(x)\!$
$\text{R2b.}\!$	$f(x) = \underline{1}$

Rules like these can be chained together to establish extended rules, just so long as their antecedent conditions are compatible. For example, Rules 1 and 2 combine to give the equivalents that are listed in Rule 3. This follows from a recognition that the function $\upharpoonleft Q \upharpoonright ~:~ X \to \underline\mathbb{B}$ that is introduced in Rule 1 is an instance of the function $f : X \to \underline\mathbb{B}$ that is mentioned in Rule 2. By the time one arrives in the "consequence box" of either Rule, then, one has in mind a comparatively fixed $Q \subseteq X,$ a proposition $f\!$ or $\upharpoonleft Q \upharpoonright$ about things in $X,\!$ and a variable argument $x \in X.$

$\operatorname{Rule~3}$

	$\text{If}\!$	$Q ~\subseteq~ X$
	$\text{and}\!$	$x ~\in~ X$
	$\text{then}\!$	$\text{the following are equivalent:}\!$

$\operatorname{R3a.}$	$x ~\in~ Q$	$\operatorname{R3a~:~R1a}$
		$::\!$
$\operatorname{R3b.}$	$\upharpoonleft Q \upharpoonright (x)$	$\operatorname{R3b~:~R1b}$ $\operatorname{R3b~:~R2a}$
		$::\!$
$\operatorname{R3c.}$	$\upharpoonleft Q \upharpoonright (x) ~=~ \underline{1}$	$\operatorname{R3c~:~R2b}$

A large stock of rules can be derived in this way, by chaining together segments that are selected from a stock of previous rules, with perhaps the whole process of derivation leading back to an axial body or a core stock of rules, with all recurring to and relying on an axiomatic basis. In order to keep track of their derivations, as their pedigrees help to remember the reasons for trusting their use in the first place, derived rules can be annotated by citing the rules from which they are derived.

In the present discussion, I am using a particular style of annotation for rule derivations, one that is called proof by grammatical paradigm or proof by syntactic analogy. The annotations in the right hand margin of the Rule Box interweave the numerators and the denominators of the paradigm being employed, in other words, the alternating terms of comparison in a sequence of analogies. Taking the syntactic transformations marked in the Rule Box one at a time, each step is licensed by its formal analogy to a previously established rule.

For example, the annnotation $X_1 : A_1 :: X_2 : A_2\!$ may be read to say that $X_1\!$ is to $A_1\!$ as $X_2\!$ is to $A_2,\!$ where the step from $A_1\!$ to $A_2\!$ is permitted by a previously accepted rule.

This can be illustrated by considering the derivation of Rule 3 in the augmented form that follows:

$\begin{array}{lcclc} \text{R3a.} & x \in Q & \text{is to} & \text{R1a.} & x \in Q \\[6pt] & & \text{as} & & \\[6pt] \text{R3b.} & \upharpoonleft Q \upharpoonright (x) & \text{is to} & \text{R1b.} & \upharpoonleft Q \upharpoonright (x) \\[6pt] & & \text{and} & & \\[6pt] \text{R3b.} & \upharpoonleft Q \upharpoonright (x) & \text{is to} & \text{R2a.} & f(x) \\[6pt] & & \text{as} & & \\[6pt] \text{R3c.} & \upharpoonleft Q \upharpoonright (x) = \underline{1} & \text{is to} & \text{R2b.} & f(x) = \underline{1} \end{array}$

Notice how the sequence of analogies pivots on the term $\text{R3b},\!$ viewing it first under the aegis of $\text{R1b},\!$ as the second term of the first analogy, and then turning to view it again under the guise of $\text{R2a},\!$ as the first term of the second analogy.

By way of convention, rules that are tailored to a particular application, case, or subject, and rules that are adapted to a particular goal, object, or purpose, I frequently refer to as Facts.

Besides linking rules together into extended sequences of equivalents, there is one other way that is commonly used to get new rules from old. Novel starting points for rules can be obtained by extracting pairs of equivalent expressions from a sequence that falls under an established rule and then stating their equality in the appropriate form of equation.

For example, extracting the expressions $\text{R3a}\!$ and $\text{R3c}\!$ that are given as equivalents in Rule 3 and explicitly stating their equivalence produces the equation recorded in Corollary 1.

		$\text{Corollary 1}\!$
$\text{If}\!$	$Q \subseteq X$
$\text{and}\!$	$x \in X$
$\text{then}\!$	$\text{the following statement is true:}\!$
$\text{C1a.}\!$	$x \in Q ~\Leftrightarrow~ \upharpoonleft Q \upharpoonright (x) = \underline{1}$	$\text{R3a} \Leftrightarrow \text{R3c}$

There are a number of issues, that arise especially in establishing the proper use of STRs, that are appropriate to discuss at this juncture. The notation $\downharpoonleft s \downharpoonright$ is intended to represent the proposition denoted by the sentence $s.\!$ There is only one problem with the use of this form. There is, in general, no such thing as "the" proposition denoted by $s.\!$ Generally speaking, if a sentence is taken out of context and considered across a variety of different contexts, there is no unique proposition that it can be said to denote. But one is seldom ever speaking at the maximum level of generality, or even found to be thinking of it, and so this notation is usually meaningful and readily understandable whenever it is read in the proper frame of mind. Still, once the issue is raised, the question of how these meanings and understandings are possible has to be addressed, especially if one desires to express the regulations of their syntax in a partially computational form. This requires a closer examination of the very notion of context, and it involves engaging in enough reflection on the contextual evaluation of sentences that the relevant principles of its successful operation can be discerned and rationalized in explicit terms.

A sentence that is written in a context where it represents a value of $\underline{1}$ or $\underline{0}$ as a function of things in the universe $X,\!$ where it stands for a value of $\operatorname{truth}$ or $\operatorname{falsehood},$ depending on how the signs that constitute its proper syntactic arguments are interpreted as denoting objects in $X,\!$ in other words, where it is bound to lead its interpreter to view its own truth or falsity as determined by a choice of objects in $X,\!$ is a sentence that might as well be written in the context $\downharpoonleft \ldots \downharpoonright,$ whether this frame is explicitly marked around it or not.

More often than not, the context of interpretation fixes the denotations of most of the signs that make up a sentence, and so it is safe to adopt the convention that only those signs whose objects are not already fixed are free to vary in their denotations. Thus, only the signs that remain in default of prior specification are subject to treatment as variables, with a decree of functional abstraction hanging over all of their heads.

$\downharpoonleft x \in Q \downharpoonright ~=~ \lambda (x, \in, Q).(x \in Q).$

Going back to Rule 1, we see that it lists a pair of concrete sentences and authorizes exchanges in either direction between the syntactic structures that have these two forms. But a sentence is any sign that denotes a proposition, and so there are any number of less obvious sentences that can be added to this list, extending the number of items that are licensed to be exchanged. For example, a larger collection of equivalent sentences is recorded in Rule 4.

		$\text{Rule 4}\!$
$\text{If}\!$	$Q \subseteq X ~\text{is fixed}$
$\text{and}\!$	$x \in X ~\text{is varied}$
$\text{then}\!$	$\text{the following are equivalent:}\!$
$\text{R4a.}\!$	$x \in Q$
$\text{R4b.}\!$	$\downharpoonleft x \in Q \downharpoonright$
$\text{R4c.}\!$	$\downharpoonleft x \in Q \downharpoonright (x)$
$\text{R4d.}\!$	$\upharpoonleft Q \upharpoonright (x)$
$\text{R4e.}\!$	$\upharpoonleft Q \upharpoonright (x) = \underline{1}$

The first and last items on this list, namely, the sentence $\text{R4a}\!$ stating $x \in Q$ and the sentence $\text{R4e}\!$ stating $\upharpoonleft Q \upharpoonright (x) = \underline{1},$ are just the pair of sentences from Rule 3 whose equivalence for all $x \in X$ is usually taken to define the idea of an indicator function $\upharpoonleft Q \upharpoonright ~:~ X \to \underline\mathbb{B}.$ At first sight, the inclusion of the other items appears to involve a category confusion, in other words, to mix the modes of interpretation and to create an array of mismatches between their ostensible types and the ruling type of a sentence. On reflection, and taken in context, these problems are not as serious as they initially seem. For example, the expression $^{\backprime\backprime} \downharpoonleft x \in Q \downharpoonright \, ^{\prime\prime}$ ostensibly denotes a proposition, but if it does, then it evidently can be recognized, by virtue of this very fact, to be a genuine sentence. As a general rule, if one can see it on the page, then it cannot be a proposition but can at most be a sign of one.

The use of the basic logical connectives can be expressed in the form of a STR as follows:

$\text{Logical Translation Rule 0}\!$

	$\text{If}\!$	$s_j ~\text{is a sentence about things in the universe X}$
	$\text{and}\!$	$p_j ~\text{is a proposition about things in the universe X}$
	$\text{such that:}\!$
	$\text{L0a.}\!$	$\downharpoonleft s_j \downharpoonright ~=~ p_j, ~\text{for all}~ j \in J,$
	$\text{then}\!$	$\text{the following equations are true:}\!$

	$\text{L0b.}\!$	$\downharpoonleft \operatorname{Conc}_j^J s_j \downharpoonright$	$=\!$	$\operatorname{Conj}_j^J \downharpoonleft s_j \downharpoonright$	$=\!$	$\operatorname{Conj}_j^J p_j$
	$\text{L0c.}\!$	$\downharpoonleft \operatorname{Surc}_j^J s_j \downharpoonright$	$=\!$	$\operatorname{Surj}_j^J \downharpoonleft s_j \downharpoonright$	$=\!$	$\operatorname{Surj}_j^J p_j$

As a general rule, the application of a STR involves the recognition of an antecedent condition and the facilitation of a consequent condition. The antecedent condition is a state whose initial expression presents a match, in a formal sense, to one of the sentences that are listed in the STR, and the consequent condition is achieved by taking its suggestions seriously, in other words, by following its sequence of equivalents and implicants to some other link in its chain.

Generally speaking, the application of a rule involves the recognition of an antecedent condition as a case that falls under a clause of the rule. This means that the antecedent condition is able to be captured in the form, conceived in the guise, expressed in the manner, grasped in the pattern, or recognized in the shape of one of the sentences in a list of equivalents or a chain of implicants.

A condition is amenable to a rule if any of its conceivable expressions formally match any of the expressions that are enumerated by the rule. Further, it requires the relegation of the other expressions to the production of a result. Thus, there is the choice of an initial expression that needs to be checked on input for whether it fits the antecedent condition and there are several types of output that are generated as a consequence, only a few of which are usually needed at any given time.

Editing Note. Need a transition here. Give a brief description of the Tables of Translation Rules that have now been moved to the Appendices, and then move on to the rest of the Definitions and Proof Schemata.

A rule that allows one to turn equivalent sentences into identical propositions:

$(S \Leftrightarrow T) \quad \Leftrightarrow \quad (\downharpoonleft S \downharpoonright = \downharpoonleft T \downharpoonright)$

Compare:

$\downharpoonleft v = w \downharpoonright (v, w)$

$\downharpoonleft v(u) = w(u) \downharpoonright (u)$

Editing Note. The last draft I can find has 5 variants for the next box, "Value Rule 1", and I can't tell right off which I meant to use. Until I can get back to this, here's a link to the collection of variants:

Value Rule 1

$\operatorname{Evaluation~Rule~1}$

	$\text{If}\!$	$f, g ~:~ X \to \underline\mathbb{B}$
	$\text{and}\!$	$x ~\in~ X$
	$\text{then}\!$	$\text{the following are equivalent:}\!$


$\operatorname{E1a.}$	$f(x) ~=~ g(x)$	$\operatorname{E1a~:~V1a}$
		$::\!$
$\operatorname{E1b.}$	$f(x) ~\Leftrightarrow~ g(x)$	$\operatorname{E1b~:~V1b}$
		$::\!$
$\operatorname{E1c.}$	$\underline{((}~ f(x) ~,~ g(x) ~\underline{))}$	$\operatorname{E1c~:~V1c}$ $\operatorname{E1c~:~$1a}$
		$::\!$
$\operatorname{E1d.}$	$\underline{((}~ f ~,~ g ~\underline{))}^\$ (x)$	$\operatorname{E1d~:~$1b}$

$\operatorname{Definition~2}$

	$\text{If}\!$	$P, Q ~\subseteq~ X$
	$\text{then}\!$	$\text{the following are equivalent:}\!$

	$\operatorname{D2a.}$	$P ~=~ Q$
	$\operatorname{D2b.}$	$\overset{X}{\underset{x}{\forall}}~ (x \in P ~\Leftrightarrow~ x \in Q)$

$\operatorname{Definition~3}$

	$\text{If}\!$	$f, g ~:~ X \to Y$
	$\text{then}\!$	$\text{the following are equivalent:}\!$

	$\operatorname{D3a.}$	$f ~=~ g$
	$\operatorname{D3b.}$	$\overset{X}{\underset{x}{\forall}}~ (f(x) ~=~ g(x))$

$\operatorname{Definition~4}$

	$\text{If}\!$	$Q ~\subseteq~ X$
	$\text{then}\!$	$\text{the following are identical subsets of}~ X \times \underline\mathbb{B}:$

	$\operatorname{D4a.}$	$\upharpoonleft Q \upharpoonright$
	$\operatorname{D4b.}$	$\{ (x, y) \in X \times \underline\mathbb{B} ~:~ y ~=~ \downharpoonleft x \in Q \downharpoonright$

$\operatorname{Definition~5}$

	$\text{If}\!$	$Q ~\subseteq~ X$
	$\text{then}\!$	$\text{the following are identical propositions} ~:~ X \to \underline\mathbb{B}$

	$\operatorname{D5a.}$	$\upharpoonleft Q \upharpoonright$
	$\operatorname{D5b.}$	$\downharpoonleft x \in Q \downharpoonright$

Given an indexed set of sentences, $s_j\!$ for $j \in J,$ it is possible to consider the logical conjunction of the corresponding propositions. Various notations for this concept are be useful in various contexts, a sufficient sample of which are recorded in Definition 6.

$\operatorname{Definition~6}$

	$\text{If}\!$	$\text{each string}~ s_j, ~\text{as}~ j ~\text{ranges over the set}~ J,$
		$\text{is a sentence about things in the universe}~ X~$
	$\text{then}\!$	$\text{the following are equivalent:}\!$

	$\operatorname{D6a.}$	$\overset{J}{\underset{j}{\forall}}~ s_j$
	$\operatorname{D6b.}$	$\operatorname{Conj}_j^J s_j$

$\operatorname{Definition~7}$

	$\text{If}\!$	$s, t ~\text{are sentences about things in the universe}~ X$
	$\text{then}\!$	$\text{the following are equivalent:}\!$

	$\operatorname{D7a.}$	$s ~\Leftrightarrow~ t$
	$\operatorname{D7b.}$	$\downharpoonleft s \downharpoonright ~=~ \downharpoonleft t \downharpoonright$

$\operatorname{Rule~5}$

	$\text{If}\!$	$P, Q ~\subseteq~ X$
	$\text{then}\!$	$\text{the following are equivalent:}\!$

$\operatorname{R5a.}$	$P ~=~ Q$	$\operatorname{R5a~:~D2a}$
		$::\!$
$\operatorname{R5b.}$	$\overset{X}{\underset{x}{\forall}}~ (x \in P ~\Leftrightarrow~ x \in Q)$	$\operatorname{R5b~:~D2b}$ $\operatorname{R5b~:~D7a}$
		$::\!$
$\operatorname{R5c.}$	$\overset{X}{\underset{x}{\forall}}~ (\downharpoonleft x \in P \downharpoonright ~=~ \downharpoonleft x \in Q \downharpoonright)$	$\operatorname{R5c~:~D7b}$ $\operatorname{R5c~:~\_\_?\_\_}$
		$::\!$
$\operatorname{R5d.}$	$\begin{matrix} \{ (x, y) \in X \times \underline\mathbb{B} ~:~ y ~=~ \downharpoonleft x \in P \downharpoonright \\ = \\ \{ (x, y) \in X \times \underline\mathbb{B} ~:~ y ~=~ \downharpoonleft x \in Q \downharpoonright \end{matrix}$	$\operatorname{R5d~:~\_\_?\_\_}$ $\operatorname{R5d~:~D5b}$
		$::\!$
$\operatorname{R5e.}$	$\upharpoonleft P \upharpoonright ~=~ \upharpoonleft Q \upharpoonright$	$\operatorname{R5e~:~D5a}$

$\operatorname{Rule~6}$

	$\text{If}\!$	$f, g ~:~ X \to Y$
	$\text{then}\!$	$\text{the following are equivalent:}\!$

$\operatorname{R6a.}$	$f ~=~ g$	$\operatorname{R6a~:~D3a}$
		$::\!$
$\operatorname{R6b.}$	$\overset{X}{\underset{x}{\forall}}~ (f(x) ~=~ g(x))$	$\operatorname{R6b~:~D3b}$ $\operatorname{R6b~:~D6a}$
		$::\!$
$\operatorname{R6c.}$	$\operatorname{Conj_x^X}~ (f(x) ~=~ g(x))$	$\operatorname{R6c~:~D6b}$

$\operatorname{Rule~7}$

	$\text{If}\!$	$p, q ~:~ X \to \underline\mathbb{B}$
	$\text{then}\!$	$\text{the following are equivalent:}\!$

$\operatorname{R7a.}$	$p ~=~ q$	$\operatorname{R7a~:~R6a}$
		$::\!$
$\operatorname{R7b.}$	$\overset{X}{\underset{x}{\forall}}~ (p(x) ~=~ q(x))$	$\operatorname{R7b~:~R6b}$
		$::\!$
$\operatorname{R7c.}$	$\operatorname{Conj_x^X}~ (p(x) ~=~ q(x))$	$\operatorname{R7c~:~R6c}$ $\operatorname{R7c~:~P1a}$
		$::\!$
$\operatorname{R7d.}$	$\operatorname{Conj_x^X}~ (p(x) ~\Leftrightarrow~ q(x))$	$\operatorname{R7d~:~P1b}$
		$::\!$
$\operatorname{R7e.}$	$\operatorname{Conj_x^X}~ \underline{((}~ p(x) ~,~ q(x) ~\underline{))}$	$\operatorname{R7e~:~P1c}$ $\operatorname{R7e~:~$1a}$
		$::\!$
$\operatorname{R7f.}$	$\operatorname{Conj_x^X}~ \underline{((}~ p ~,~ q ~\underline{))}^\$ (x)$	$\operatorname{R7f~:~$1b}$

Editing Note. Check earlier and later drafts to see where $\text{P1a, P1b, P1c}~$ came from. Are these just placeholders for the Value or Evaluation Rules?

$\operatorname{Rule~8}$

	$\text{If}\!$	$s, t ~\text{are sentences about things in}~ X$
	$\text{then}\!$	$\text{the following are equivalent:}\!$

$\operatorname{R8a.}$	$s ~\Leftrightarrow~ t$	$\operatorname{R8a~:~D7a}$
		$::\!$
$\operatorname{R8b.}$	$\downharpoonleft s \downharpoonright ~=~ \downharpoonleft t \downharpoonright$	$\operatorname{R8b~:~D7b}$ $\operatorname{R8b~:~R7a}$
		$::\!$
$\operatorname{R8c.}$	$\overset{X}{\underset{x}{\forall}}~ (\downharpoonleft s \downharpoonright (x) ~=~ \downharpoonleft t \downharpoonright (x))$	$\operatorname{R8c~:~R7b}$
		$::\!$
$\operatorname{R8d.}$	$\operatorname{Conj_x^X}~ (\downharpoonleft s \downharpoonright (x) ~=~ \downharpoonleft t \downharpoonright (x))$	$\operatorname{R8d~:~R7c}$
		$::\!$
$\operatorname{R8e.}$	$\operatorname{Conj_x^X}~ (\downharpoonleft s \downharpoonright (x) ~\Leftrightarrow~ \downharpoonleft t \downharpoonright (x))$	$\operatorname{R8e~:~R7d}$
		$::\!$
$\operatorname{R8f.}$	$\operatorname{Conj_x^X}~ \underline{((}~ \downharpoonleft s \downharpoonright (x) ~,~ \downharpoonleft t \downharpoonright (x) ~\underline{))}$	$\operatorname{R8f~:~R7e}$
		$::\!$
$\operatorname{R8g.}$	$\operatorname{Conj_x^X}~ \underline{((}~ \downharpoonleft s \downharpoonright ~,~ \downharpoonleft t \downharpoonright ~\underline{))}^\$ (x)$	$\operatorname{R8g~:~R7f}$

For instance, the observation that expresses the equality of sets in terms of their indicator functions can be formalized according to the pattern in Rule 9, namely, at lines R9a, R9b, and R9c, and these components of Rule 9 can be cited in future uses by their indices in this list. Using Rule 7, annotated as R7, to adduce a few properties of indicator functions to the account, it is possible to extend Rule 9 by another few steps, referenced as R9d, R9e, R9f, and R9g.

$\operatorname{Rule~9}$

	$\text{If}\!$	$P, Q ~\subseteq~ X$
	$\text{then}\!$	$\text{the following are equivalent:}\!$

$\operatorname{R9a.}$	$P ~=~ Q$	$\operatorname{R9a~:~R5a}$
		$::\!$
$\operatorname{R9b.}$	$\upharpoonleft P \upharpoonright ~=~ \upharpoonleft Q \upharpoonright$	$\operatorname{R9b~:~R5e}$ $\operatorname{R9b~:~R7a}$
		$::\!$
$\operatorname{R9c.}$	$\overset{X}{\underset{x}{\forall}}~ (\upharpoonleft P \upharpoonright (x) ~=~ \upharpoonleft Q \upharpoonright (x))$	$\operatorname{R9c~:~R7b}$
		$::\!$
$\operatorname{R9d.}$	$\operatorname{Conj_x^X}~ (\upharpoonleft P \upharpoonright (x) ~=~ \upharpoonleft Q \upharpoonright (x))$	$\operatorname{R9d~:~R7c}$
		$::\!$
$\operatorname{R9e.}$	$\operatorname{Conj_x^X}~ (\upharpoonleft P \upharpoonright (x) ~\Leftrightarrow~ \upharpoonleft Q \upharpoonright (x))$	$\operatorname{R9e~:~R7d}$
		$::\!$
$\operatorname{R9f.}$	$\operatorname{Conj_x^X}~ \underline{((}~ \upharpoonleft P \upharpoonright (x) ~,~ \upharpoonleft Q \upharpoonright (x) ~\underline{))}$	$\operatorname{R9f~:~R7e}$
		$::\!$
$\operatorname{R9g.}$	$\operatorname{Conj_x^X}~ \underline{((}~ \upharpoonleft P \upharpoonright ~,~ \upharpoonleft Q \upharpoonright ~\underline{))}^\$ (x)$	$\operatorname{R9g~:~R7f}$

$\operatorname{Rule~10}$

	$\text{If}\!$	$P, Q ~\subseteq~ X$
	$\text{then}\!$	$\text{the following are equivalent:}\!$

$\operatorname{R10a.}$	$P ~=~ Q$	$\operatorname{R10a~:~D2a}$
		$::\!$
$\operatorname{R10b.}$	$\overset{X}{\underset{x}{\forall}}~ (x \in P ~\Leftrightarrow~ x \in Q)$	$\operatorname{R10b~:~D2b}$ $\operatorname{R10b~:~R8a}$
		$::\!$
$\operatorname{R10c.}$	$\downharpoonleft x \in P \downharpoonright ~=~ \downharpoonleft x \in Q \downharpoonright$	$\operatorname{R10c~:~R8b}$
		$::\!$
$\operatorname{R10d.}$	$\overset{X}{\underset{x}{\forall}}~ \downharpoonleft x \in P \downharpoonright (x) ~=~ \downharpoonleft x \in Q \downharpoonright (x)$	$\operatorname{R10d~:~R8c}$
		$::\!$
$\operatorname{R10e.}$	$\operatorname{Conj_x^X}~ (\downharpoonleft x \in P \downharpoonright (x) ~=~ \downharpoonleft x \in Q \downharpoonright (x))$	$\operatorname{R10e~:~R8d}$
		$::\!$
$\operatorname{R10f.}$	$\operatorname{Conj_x^X}~ (\downharpoonleft x \in P \downharpoonright (x) ~\Leftrightarrow~ \downharpoonleft x \in Q \downharpoonright (x))$	$\operatorname{R10f~:~R8e}$
		$::\!$
$\operatorname{R10g.}$	$\operatorname{Conj_x^X}~ \underline{((}~ \downharpoonleft x \in P \downharpoonright (x) ~,~ \downharpoonleft x \in Q \downharpoonright (x) ~\underline{))}$	$\operatorname{R10g~:~R8f}$
		$::\!$
$\operatorname{R10h.}$	$\operatorname{Conj_x^X}~ \underline{((}~ \downharpoonleft x \in P \downharpoonright ~,~ \downharpoonleft x \in Q \downharpoonright ~\underline{))}^\$ (x)$	$\operatorname{R10h~:~R8g}$

$\operatorname{Rule~11}$

	$\text{If}\!$	$Q ~\subseteq~ X$
	$\text{then}\!$	$\text{the following are equivalent:}\!$

$\operatorname{R11a.}$	$Q ~=~ \{ x \in X ~:~ s \}$	$\operatorname{R11a~:~R5a}$
		$::\!$
$\operatorname{R11b.}$	$\upharpoonleft Q \upharpoonright ~=~ \upharpoonleft \{ x \in X ~:~ s \} \upharpoonright$	$\operatorname{R11b~:~R5e}$
		$::\!$
$\operatorname{R11c.}$	$\begin{array}{lcl} \upharpoonleft Q \upharpoonright & \subseteq & X \times \underline\mathbb{B} \\ \upharpoonleft Q \upharpoonright & = & \{ (x, y) \in X \times \underline\mathbb{B} ~:~ y = \, \downharpoonleft s \downharpoonright (x) \} \end{array}$	$\operatorname{R11c~:~\_\_?\_\_}$ $\operatorname{R11c~:~\_\_?\_\_}$
		$::\!$
$\operatorname{R11d.}$	$\begin{array}{ccccl} \upharpoonleft Q \upharpoonright & : & X & \to & \underline\mathbb{B} \\ \upharpoonleft Q \upharpoonright & : & x & \mapsto & \downharpoonleft s \downharpoonright (x) \end{array}$	$\operatorname{R11d~:~\_\_?\_\_}$ $\operatorname{R11d~:~\_\_?\_\_}$
		$::\!$
$\operatorname{R11e.}$	$\overset{X}{\underset{x}{\forall}}~ \upharpoonleft Q \upharpoonright (x) ~=~ \downharpoonleft s \downharpoonright (x)$	$\operatorname{R11e~:~\_\_?\_\_}$ $\operatorname{R11e~:~\_\_?\_\_}$
		$::\!$
$\operatorname{R11f.}$	$\upharpoonleft Q \upharpoonright ~=~ \downharpoonleft s \downharpoonright$	$\operatorname{R11f~:~\_\_?\_\_}$

An application of Rule 11 involves the recognition of an antecedent condition as a case under the Rule, that is, as a condition that matches one of the sentences in the Rule's chain of equivalents, and it requires the relegation of the other expressions to the production of a result. Thus, there is the choice of an initial expression that has to be checked on input for whether it fits the antecedent condition, and there is the choice of three types of output that are generated as a consequence, only one of which is generally needed at any given time. More often than not, though, a rule is applied in only a few of its possible ways. The usual antecedent and the usual consequents for Rule 11 can be distinguished in form and specialized in practice as follows:

$\operatorname{R11a}$ marks the usual starting place for an application of the Rule, that is, the standard form of antecedent condition that is likely to lead to an invocation of the Rule.

$\operatorname{R11b}$ records the trivial consequence of applying the up-spar operator $\upharpoonleft \cdots \upharpoonright$ to both sides of the initial equation.

$\operatorname{R11c}$ gives a version of the indicator function with $\upharpoonleft X \upharpoonright ~\subseteq~ X \times \underline\mathbb{B},$ called the extensional or relational form of the indicator function.

$\operatorname{R11d}$ gives a version of the indicator function with $\upharpoonleft X \upharpoonright ~:~ X \to \underline\mathbb{B},$ called its functional form.

Applying Rule 9, Rule 8, and the Logical Rules to the special case where $s \Leftrightarrow (X = Y),$ one obtains the following general Fact:

$\operatorname{Fact~1}$

	$\text{If}\!$	$P, Q ~\subseteq~ X$
	$\text{then}\!$	$\text{the following are equivalent:}\!$

$\operatorname{F1a.}$	$s \quad \Leftrightarrow \quad (P ~=~ Q)$	$\operatorname{F1a~:~R9a}$
		$::\!$
$\operatorname{F1b.}$	$s \quad \Leftrightarrow \quad (\upharpoonleft P \upharpoonright ~=~ \upharpoonleft Q \upharpoonright)$	$\operatorname{F1b~:~R9b}$
		$::\!$
$\operatorname{F1c.}$	$s \quad \Leftrightarrow \quad \overset{X}{\underset{x}{\forall}}~ (\upharpoonleft P \upharpoonright (x) ~=~ \upharpoonleft Q \upharpoonright (x))$	$\operatorname{F1c~:~R9c}$
		$::\!$
$\operatorname{F1d.}$	$s \quad \Leftrightarrow \quad \operatorname{Conj_x^X}~ (\upharpoonleft P \upharpoonright (x) ~=~ \upharpoonleft Q \upharpoonright (x))$	$\operatorname{F1d~:~R9d}$ $\operatorname{F1d~:~R8a}$
		$::\!$
$\operatorname{F1e.}$	$\downharpoonleft s \downharpoonright \quad = \quad \downharpoonleft \operatorname{Conj_x^X}~ (\upharpoonleft P \upharpoonright (x) ~=~ \upharpoonleft Q \upharpoonright (x)) \downharpoonright$	$\operatorname{F1e~:~R8b}$ $\operatorname{F1e~:~\_\_?\_\_}$
		$::\!$
$\operatorname{F1f.}$	$\downharpoonleft s \downharpoonright \quad = \quad \operatorname{Conj_x^X}~ \downharpoonleft (\upharpoonleft P \upharpoonright (x) ~=~ \upharpoonleft Q \upharpoonright (x)) \downharpoonright$	$\operatorname{F1f~:~\_\_?\_\_}$ $\operatorname{F1f~:~\_\_?\_\_}$
		$::\!$
$\operatorname{F1g.}$	$\downharpoonleft s \downharpoonright \quad = \quad \operatorname{Conj_x^X}~ \underline{((}~ \upharpoonleft P \upharpoonright (x) ~,~ \upharpoonleft Q \upharpoonright (x) ~\underline{))}$	$\operatorname{F1g~:~\_\_?\_\_}$ $\operatorname{F1g~:~\_\_?\_\_}$
		$::\!$
$\operatorname{F1h.}$	$\downharpoonleft s \downharpoonright \quad = \quad \operatorname{Conj_x^X}~ \underline{((}~ \upharpoonleft P \upharpoonright ~,~ \upharpoonleft Q \upharpoonright ~\underline{))}^\$ (x)$	$\operatorname{F1h~:~~\_\_?\_\_}$

1.3.12.2. Derived Equivalence Relations

One seeks a method of general application for approaching the individual sign relation, a way to select an aspect of its form, to analyze it with regard to its intrinsic structure, and to classify it in comparison with other sign relations. With respect to a particular sign relation, one approach that presents itself is to examine the relation between signs and interpretants that is given directly by its connotative component and to compare it with the various forms of derived, indirect, mediate, or peripheral relationships that can be found to exist among signs and interpretants by way of secondary considerations or subsequent studies. Of especial interest are the relationships among signs and interpretants that can be obtained by working through the collections of objects that they commonly or severally denote.

A classic way of showing that two sets are equal is to show that every element of the first belongs to the second and that every element of the second belongs to the first. The problem with this strategy is that one can exhaust a considerable amount of time trying to prove that two sets are equal before it occurs to one to look for a counterexample, that is, an element of the first that does not belong to the second or an element of the second that does not belong to the first, in cases where that is precisely what one ought to be seeking. It would be nice if there were a more balanced, impartial, or neutral way to go about this task, one that did not require such an undue commitment to either side, a technique that helps to pinpoint the counterexamples when they exist, and a method that keeps in mind the original relation of proving that and showing that to probing, testing, and seeing whether.

A different way of seeing that two sets are equal, or of seeing whether two sets are equal, is based on the following observation:

Two sets are equal as sets

$\iff$

The indicator functions of the two sets are equal as functions

$\iff$

The values of the two indicator functions are equal to each other on all domain elements.

It is important to notice the hidden quantifier, of a universal kind, that lurks in all three equivalent statements but is only revealed in the last.

In making the next set of definitions and in using the corresponding terminology it is taken for granted that all of the references of signs are relative to a particular sign relation $L \subseteq O \times S \times I$ that either remains to be specified or is already understood. Further, I continue to assume that $S = I,\!$ in which case this set is called the syntactic domain of $L.\!$

In the following definitions, let $L \subseteq O \times S \times I,$ let $S = I,\!$ and let $x, y \in S.\!$

Recall the definition of $\operatorname{Con} (L),$ the connotative component of a sign relation $L,\!$ in the following form:

$\operatorname{Con} (L) ~=~ L_{SI} ~=~ \{ (s, i) \in S \times I ~:~ (o, s, i) \in L ~\text{for some}~ o \in O \}.$

Equivalent expressions for this concept are recorded in Definition 8.

$\operatorname{Definition~8}$

	$\text{If}\!$	$L ~\subseteq~ O \times S \times I$
	$\text{then}\!$	$\text{the following are identical subsets of}~ S \times I \, :$

	$\operatorname{D8a.}$	$L_{SI}\!$
	$\operatorname{D8b.}$	$\operatorname{Con}^L$
	$\operatorname{D8c.}$	$\operatorname{Con}(L)$
	$\operatorname{D8d.}$	$\operatorname{proj}_{SI}(L)$
	$\operatorname{D8e.}$	$\{ (s, i) \in S \times I ~:~ (o, s, i) \in L ~\operatorname{for~some}~ o \in O \}$

Editing Note. Need a discussion of converse relations here. Perhaps it would work to introduce the operators that Peirce used for the converse of a dyadic relative $\ell,$ namely, $K\ell ~=~ k\!\cdot\!\ell ~=~ \breve\ell.$

The dyadic relation $L_{IS}\!$ that is the converse of the connotative relation $L_{SI}\!$ can be defined directly in the following fashion:

$\overset{\smile}{\operatorname{Con}(L)} ~=~ L_{IS} ~=~ \{ (i, s) \in I \times S ~:~ (o, s, i) \in L ~\text{for some}~ o \in O \}.$

A few of the many different expressions for this concept are recorded in Definition 9.

$\operatorname{Definition~9}$

	$\text{If}\!$	$L ~\subseteq~ O \times S \times I$
	$\text{then}\!$	$\text{the following are identical subsets of}~ I \times S \, :$

	$\operatorname{D9a.}$	$L_{IS}\!$
	$\operatorname{D9b.}$	$\overset{\smile}{L_{SI}}$
	$\operatorname{D9c.}$	$\overset{\smile}{\operatorname{Con}^L}$
	$\operatorname{D9d.}$	$\overset{\smile}{\operatorname{Con}(L)}$
	$\operatorname{D9e.}$	$\operatorname{proj}_{IS}(L)$
	$\operatorname{D9f.}$	$\operatorname{Conv}(\operatorname{Con}(L))$
	$\operatorname{D9g.}$	$\{ (i, s) \in I \times S ~:~ (o, s, i) \in L ~\operatorname{for~some}~ o \in O \}$

Recall the definition of $\operatorname{Den} (L),$ the denotative component of $L,\!$ in the following form:

$\operatorname{Den} (L) ~=~ L_{OS} ~=~ \{ (o, s) \in O \times S ~:~ (o, s, i) \in L ~\text{for some}~ i \in I \}.$

Equivalent expressions for this concept are recorded in Definition 10.

$\operatorname{Definition~10}$

	$\text{If}\!$	$L ~\subseteq~ O \times S \times I$
	$\text{then}\!$	$\text{the following are identical subsets of}~ O \times S \, :$

	$\operatorname{D10a.}$	$L_{OS}\!$
	$\operatorname{D10b.}$	$\operatorname{Den}^L$
	$\operatorname{D10c.}$	$\operatorname{Den}(L)$
	$\operatorname{D10d.}$	$\operatorname{proj}_{OS}(L)$
	$\operatorname{D10e.}$	$\{ (o, s) \in O \times S ~:~ (o, s, i) \in L ~\operatorname{for~some}~ i \in I \}$

The dyadic relation $L_{SO}\!$ that is the converse of the denotative relation $L_{OS}\!$ can be defined directly in the following fashion:

$\overset{\smile}{\operatorname{Den}(L)} ~=~ L_{SO} ~=~ \{ (s, o) \in S \times O ~:~ (o, s, i) \in L ~\text{for some}~ i \in I \}.$

A few of the many different expressions for this concept are recorded in Definition 11.

$\operatorname{Definition~11}$

	$\text{If}\!$	$L ~\subseteq~ O \times S \times I$
	$\text{then}\!$	$\text{the following are identical subsets of}~ S \times O \, :$

	$\operatorname{D11a.}$	$L_{SO}\!$
	$\operatorname{D11b.}$	$\overset{\smile}{L_{OS}}$
	$\operatorname{D11c.}$	$\overset{\smile}{\operatorname{Den}^L}$
	$\operatorname{D11d.}$	$\overset{\smile}{\operatorname{Den}(L)}$
	$\operatorname{D11e.}$	$\operatorname{proj}_{SO}(L)$
	$\operatorname{D11f.}$	$\operatorname{Conv}(\operatorname{Den}(L))$
	$\operatorname{D11g.}$	$\{ (s, o) \in S \times O ~:~ (o, s, i) \in L ~\operatorname{for~some}~ i \in I \}$

The denotation of $x\!$ in $L,\!$ written $\operatorname{Den}(L, x),$ is defined as follows:

$\operatorname{Den}(L, x) ~=~ \{ o \in O ~:~ (o, x) \in \operatorname{Den}(L) \}.$

In other words:

$\operatorname{Den}(L, x) ~=~ \{ o \in O ~:~ (o, x, i) \in L ~\text{for some}~ i \in I \}.$

Equivalent expressions for this concept are recorded in Definition 12.

$\operatorname{Definition~12}$

	$\text{If}\!$	$L ~\subseteq~ O \times S \times I$
	$\text{and}\!$	$x ~\in~ S$
	$\text{then}\!$	$\text{the following are identical subsets of}~ O \, :$

	$\operatorname{D12a.}$	$L_{OS} \cdot x$
	$\operatorname{D12b.}$	$\operatorname{Den}^L \cdot x$
	$\operatorname{D12c.}$	$\operatorname{Den}^L \|_x$
	$\operatorname{D12d.}$	$\operatorname{Den}^L (-, x)$
	$\operatorname{D12e.}$	$\operatorname{Den}(L, x)$
	$\operatorname{D12f.}$	$\operatorname{Den}(L) \cdot x$
	$\operatorname{D12g.}$	$\{ o \in O ~:~ (o, x) \in \operatorname{Den}(L) \}$
	$\operatorname{D12h.}$	$\{ o \in O ~:~ (o, x, i) \in L ~\operatorname{for~some}~ i \in I \}$

Signs are equiferent if they refer to all and only the same objects, that is, if they have exactly the same denotations. In other language for the same relation, signs are said to be denotatively equivalent or referentially equivalent, but it is probably best to check whether the extension of this concept over the syntactic domain is really a genuine equivalence relation before jumping to the conclusions that are implied by these latter terms.

To define the equiference of signs in terms of their denotations, one says that $x\!$ is equiferent to $y\!$ under $L,\!$ and writes $x ~\overset{L}{=}~ y,\!$ to mean that $\operatorname{Den}(L, x) = \operatorname{Den}(L, y).$ Taken in extension, this notion of a relation between signs induces an equiference relation on the syntactic domain.

For each sign relation $L,\!$ this yields a binary relation $\operatorname{Der}(L) \subseteq S \times I$ that is defined as follows:

$\operatorname{Der}(L) ~=~ Der^L ~=~ \{ (x, y) \in S \times I ~:~ \operatorname{Den}(L, x) = \operatorname{Den}(L, y) \}.$

These definitions and notations are recorded in the following display.

$\operatorname{Definition~13}$

	$\text{If}\!$	$L ~\subseteq~ O \times S \times I$
	$\text{then}\!$	$\text{the following are identical subsets of}~ S \times I \, :$

	$\operatorname{D13a.}$	$\operatorname{Der}^L$
	$\operatorname{D13b.}$	$\operatorname{Der}(L)$
	$\operatorname{D13c.}$	$\{ (x, y) \in S \times I ~:~ \operatorname{Den}^L\|_x = \operatorname{Den}^L\|_y \}$
	$\operatorname{D13d.}$	$\{ (x, y) \in S \times I ~:~ \operatorname{Den}(L, x) = \operatorname{Den}(L, y) \}$

The relation $\operatorname{Der}(L)$ is defined and the notation $x ~\overset{L}{=}~ y$ is meaningful in every situation where the corresponding denotation operator $\operatorname{Den}(-,-)$ makes sense, but it remains to check whether this relation enjoys the properties of an equivalence relation.

Reflexive property.

Is it true that $x ~\overset{L}{=}~ x$ for every $x \in S = I$?

By definition, $x ~\overset{L}{=}~ x$ if and only if $\operatorname{Den}(L, x) = \operatorname{Den}(L, x).$

Thus, the reflexive property holds in any setting where the denotations $\operatorname{Den}(L, x)$ are defined for all signs $x\!$ in the syntactic domain of $R.\!$
Symmetric property.

Does $x ~\overset{L}{=}~ y$ imply $y ~\overset{L}{=}~ x$ for all $x, y \in S$?

In effect, does $\operatorname{Den}(L, x) = \operatorname{Den}(L, y)$ imply $\operatorname{Den}(L, y) = \operatorname{Den}(L, x)$ for all signs $x\!$ and $y\!$ in the syntactic domain $S\!$?

Yes, so long as the sets $\operatorname{Den}(L, x)$ and $\operatorname{Den}(L, y)$ are well-defined, a fact which is already being assumed.
Transitive property.

Does $x ~\overset{L}{=}~ y$ and $y ~\overset{L}{=}~ z$ imply $x ~\overset{L}{=}~ z$ for all $x, y, z \in S$?

To belabor the point, does $\operatorname{Den}(L, x) = \operatorname{Den}(L, y)$ and $\operatorname{Den}(L, y) = \operatorname{Den}(L, z)$ imply $\operatorname{Den}(L, x) = \operatorname{Den}(L, z)$ for all $x, y, z \in S$?

Yes, once again, under the stated conditions.

It should be clear at this point that any question about the equiference of signs reduces to a question about the equality of sets, specifically, the sets that are indexed by these signs. As a result, so long as these sets are well-defined, the issue of whether equiference relations induce equivalence relations on their syntactic domains is almost as trivial as it initially appears.

Taken in its set-theoretic extension, a relation of equiference induces a denotative equivalence relation (DER) on its syntactic domain $S = I.\!$ This leads to the formation of denotative equivalence classes (DECs), denotative partitions (DEPs), and denotative equations (DEQs) on the syntactic domain. But what does it mean for signs to be equiferent?

Notice that this is not the same thing as being semiotically equivalent, in the sense of belonging to a single semiotic equivalence class (SEC), falling into the same part of a semiotic partition (SEP), or having a semiotic equation (SEQ) between them. It is only when very felicitous conditions obtain, establishing a concord between the denotative and the connotative components of a sign relation, that these two ideas coalesce.

In general, there is no necessity that the equiference of signs, that is, their denotational equivalence or their referential equivalence, induces the same equivalence relation on the syntactic domain as that defined by their semiotic equivalence, even though this state of accord seems like an especially desirable situation. This makes it necessary to find a distinctive nomenclature for these structures, for which I adopt the term denotative equivalence relations (DERs). In their train they bring the allied structures of denotative equivalence classes (DECs) and denotative partitions (DEPs), while the corresponding statements of denotative equations (DEQs) are expressible in the form $x ~\overset{L}{=}~ y.$

The uses of the equal sign for denoting equations or equivalences are recalled and extended in the following ways:

If $E\!$ is an arbitrary equivalence relation, then the equation $x =_E y\!$ means that $(x, y) \in E.$
If $L\!$ is a sign relation such that $L_{SI}\!$ is a SER on $S = I,\!$ then the semiotic equation $x =_L y\!$ means that $(x, y) \in L_{SI}.$
If $L\!$ is a sign relation such that $F\!$ is its DER on $S = I,\!$ then the denotative equation $x ~\overset{L}{=}~ y$ means that $(x, y) \in F,$ in other words, that $\operatorname{Den}(L, x) = \operatorname{Den}(L, y).$

The use of square brackets for denoting equivalence classes is recalled and extended in the following ways:

If $E\!$ is an arbitrary equivalence relation, then $[x]_E\!$ is the equivalence class of $x\!$ under $E.\!$
If $L\!$ is a sign relation such that $\operatorname{Con}(L)$ is a SER on $S = I,\!$ then $[x]_L\!$ is the SEC of $x\!$ under $\operatorname{Con}(L).$
If $L\!$ is a sign relation such that $\operatorname{Der}(L)$ is a DER on $S = I,\!$ then $[x]^L\!$ is the DEC of $x\!$ under $\operatorname{Der}(L).$

By applying the form of Fact 1 to the special case where $X = \operatorname{Den}(L, x)$ and $Y = \operatorname{Den}(L, y),$ one obtains the following facts.

$\operatorname{Fact~2.1}$

	$\text{If}\!$	$L ~\subseteq~ O \times S \times I$
	$\text{then}\!$	$\text{the following are identical subsets of}~ S \times I :$

$\operatorname{F2.1a.}$	$\operatorname{Der}^L$	$\operatorname{F2.1a~:~D13a}$
		$::\!$
$\operatorname{F2.1b.}$	$\operatorname{Der}(L)$	$\operatorname{F2.1b~:~D13b}$
		$::\!$
$\operatorname{F2.1c.}$	$\begin{array}{ll} \{ & (x, y) \in S \times I ~: \\ & \operatorname{Den}(L, x) ~=~ \operatorname{Den}(L, y) \\ \} & \\ \end{array}$	$\operatorname{F2.1c~:~D13c}$ $\operatorname{F2.1c~:~R9a}$
		$::\!$
$\operatorname{F2.1d.}$	$\begin{array}{ll} \{ & (x, y) \in S \times I ~: \\ & \upharpoonleft \operatorname{Den}(L, x) \upharpoonright ~=~ \upharpoonleft \operatorname{Den}(L, y) \upharpoonright \\ \} & \\ \end{array}$	$\operatorname{F2.1d~:~R9b}$
		$::\!$
$\operatorname{F2.1e.}$	$\begin{array}{ll} \{ & (x, y) \in S \times I ~: \\ & \overset{O}{\underset{o}{\forall}}~ \upharpoonleft \operatorname{Den}(L, x) \upharpoonright (o) ~=~ \upharpoonleft \operatorname{Den}(L, y) \upharpoonright (o) \\ \} & \\ \end{array}$	$\operatorname{F2.1e~:~R9c}$
		$::\!$
$\operatorname{F2.1f.}$	$\begin{array}{ll} \{ & (x, y) \in S \times I ~: \\ & \underset{o \in O}{\operatorname{Conj}}~ \upharpoonleft \operatorname{Den}(L, x) \upharpoonright (o) ~=~ \upharpoonleft \operatorname{Den}(L, y) \upharpoonright (o) \\ \} & \\ \end{array}$	$\operatorname{F2.1f~:~R9d}$
		$::\!$
$\operatorname{F2.1g.}$	$\begin{array}{ll} \{ & (x, y) \in S \times I ~: \\ & \underset{o \in O}{\operatorname{Conj}}~ \underline{((}~ \upharpoonleft \operatorname{Den}(L, x) \upharpoonright (o) ~,~ \upharpoonleft \operatorname{Den}(L, y) \upharpoonright (o) ~\underline{))} \\ \} & \\ \end{array}$	$\operatorname{F2.1g~:~R9e}$
		$::\!$
$\operatorname{F2.1h.}$	$\begin{array}{ll} \{ & (x, y) \in S \times I ~: \\ & \underset{o \in O}{\operatorname{Conj}}~ \underline{((}~ \upharpoonleft \operatorname{Den}(L, x) \upharpoonright ~,~ \upharpoonleft \operatorname{Den}(L, y) \upharpoonright ~\underline{))}^\$ (o) \\ \} & \\ \end{array}$	$\operatorname{F2.1h~:~R9f}$ $\operatorname{F2.1h~:~D12e}$
		$::\!$
$\operatorname{F2.1i.}$	$\begin{array}{ll} \{ & (x, y) \in S \times I ~: \\ & \underset{o \in O}{\operatorname{Conj}}~ \underline{((}~ \upharpoonleft L_{OS} \cdot x \upharpoonright ~,~ \upharpoonleft L_{OS} \cdot y \upharpoonright ~\underline{))}^\$ (o) \\ \} & \\ \end{array}$	$\operatorname{F2.1i~:~D12a}$

$\operatorname{Fact~2.2}$

	$\text{If}\!$	$L ~\subseteq~ O \times S \times I$
	$\text{then}\!$	$\text{the following are equivalent:}\!$


$\operatorname{F2.2a.}$	$\begin{array}{cccl} \operatorname{Der}^L & = & \{ & (x, y) \in S \times I ~: \\ & & & \begin{array}{ccl} \underset{o \in O}{\operatorname{Conj}} \\ & ( & \upharpoonleft \operatorname{Den}^L x \upharpoonright (o) \\ & = & \upharpoonleft \operatorname{Den}^L y \upharpoonright (o) \\ & ) & \\ \end{array} \\ & & \} & \\ \end{array}$	$\operatorname{F2.2a~:~R11a}$
		$::\!$
$\operatorname{F2.2b.}$	$\begin{array}{ccccl} \upharpoonleft \operatorname{Der}^L \upharpoonright & = & \upharpoonleft & \{ & (x, y) \in S \times I ~: \\ & & & & \begin{array}{ccl} \underset{o \in O}{\operatorname{Conj}} \\ & ( & \upharpoonleft \operatorname{Den}^L x \upharpoonright (o) \\ & = & \upharpoonleft \operatorname{Den}^L y \upharpoonright (o) \\ & ) & \\ \end{array} \\ & & & \} & \\ & & \upharpoonright & & \\ \end{array}$	$\operatorname{F2.2b~:~R11b}$
		$::\!$
$\operatorname{F2.2c.}$	$\begin{array}{cccl} \upharpoonleft \operatorname{Der}^L \upharpoonright & = & \{ & (x, y, z) \in S \times I \times \underline\mathbb{B} ~:~ z = \\ & & & \begin{array}{cccl} \downharpoonleft & \underset{o \in O}{\operatorname{Conj}} \\ & & ( & \upharpoonleft \operatorname{Den}^L x \upharpoonright (o) \\ & & = & \upharpoonleft \operatorname{Den}^L y \upharpoonright (o) \\ & & ) & \\ \downharpoonright & & \\ \end{array} \\ & & \} & \\ \end{array}$	$\operatorname{F2.2c~:~R11c}$
		$::\!$
$\operatorname{F2.2d.}$	$\begin{array}{cccl} \upharpoonleft \operatorname{Der}^L \upharpoonright & = & \{ & (x, y, z) \in S \times I \times \underline\mathbb{B} ~:~ z = \\ & & & \begin{array}{cccl} \underset{o \in O}{\operatorname{Conj}} \\ & \downharpoonleft & ( & \upharpoonleft \operatorname{Den}^L x \upharpoonright (o) \\ & & = & \upharpoonleft \operatorname{Den}^L y \upharpoonright (o) \\ & & ) & \\ & \downharpoonright & & \\ \end{array} \\ & & \} & \\ \end{array}$	$\operatorname{F2.2d~:~Log}$
		$::\!$
$\operatorname{F2.2e.}$	$\begin{array}{cccl} \upharpoonleft \operatorname{Der}^L \upharpoonright & = & \{ & (x, y, z) \in S \times I \times \underline\mathbb{B} ~:~ z = \\ & & & \begin{array}{ccl} \underset{o \in O}{\operatorname{Conj}} \\ & \underline{((} & \upharpoonleft \operatorname{Den}^L x \upharpoonright (o) \\ & , & \upharpoonleft \operatorname{Den}^L y \upharpoonright (o) \\ & \underline{))} & \\ \end{array} \\ & & \} & \\ \end{array}$	$\operatorname{F2.2e~:~Log}$
		$::\!$
$\operatorname{F2.2f.}$	$\begin{array}{cccl} \upharpoonleft \operatorname{Der}^L \upharpoonright & = & \{ & (x, y, z) \in S \times I \times \underline\mathbb{B} ~:~ z = \\ & & & \begin{array}{cll} \underset{o \in O}{\operatorname{Conj}} \\ & \underline{((} & \upharpoonleft \operatorname{Den}^L x \upharpoonright \\ & , & \upharpoonleft \operatorname{Den}^L y \upharpoonright \\ & \underline{))}^\$ & (o) \\ \end{array} \\ & & \} & \\ \end{array}$	$\operatorname{F2.2f~:~$~}$

$\operatorname{Fact~2.3}$

	$\text{If}\!$	$L ~\subseteq~ O \times S \times I$
	$\text{then}\!$	$\text{the following are equivalent:}\!$


$\operatorname{F2.3a.}$	$\begin{array}{cccl} \operatorname{Der}^L & = & \{ & (x, y) \in S \times I ~: \\ & & & \begin{array}{ccl} \underset{o \in O}{\operatorname{Conj}} \\ & ( & \upharpoonleft \operatorname{Den}^L x \upharpoonright (o) \\ & = & \upharpoonleft \operatorname{Den}^L y \upharpoonright (o) \\ & ) & \\ \end{array} \\ & & \} & \\ \end{array}$	$\operatorname{F2.3a~:~R11a}$
		$::\!$
$\operatorname{F2.3b.}$	$\begin{array}{ccccl} \upharpoonleft \operatorname{Der}^L \upharpoonright (x, y) & = & \downharpoonleft & \underset{o \in O}{\operatorname{Conj}} \\ & & & & \begin{array}{cl} ( & \upharpoonleft \operatorname{Den}^L x \upharpoonright (o) \\ = & \upharpoonleft \operatorname{Den}^L y \upharpoonright (o) \\ ) & \\ \end{array} \\ & & \downharpoonright & & \\ \end{array}$	$\operatorname{F2.3b~:~R11d}$
		$::\!$
$\operatorname{F2.3c.}$	$\begin{array}{ccccl} \upharpoonleft \operatorname{Der}^L \upharpoonright (x, y) & = & \underset{o \in O}{\operatorname{Conj}} \\ & & & \begin{array}{ccl} \downharpoonleft & ( & \upharpoonleft \operatorname{Den}^L x \upharpoonright (o) \\ & = & \upharpoonleft \operatorname{Den}^L y \upharpoonright (o) \\ & ) & \\ \downharpoonright & & \\ \end{array} \\ & & & \\ \end{array}$	$\operatorname{F2.3c~:~Log}$
		$::\!$
$\operatorname{F2.3d.}$	$\begin{array}{ccccl} \upharpoonleft \operatorname{Der}^L \upharpoonright (x, y) & = & \underset{o \in O}{\operatorname{Conj}} \\ & & & \begin{array}{ccl} \downharpoonleft & ( & \upharpoonleft \operatorname{Den}^L \upharpoonright (o, x) \\ & = & \upharpoonleft \operatorname{Den}^L \upharpoonright (o, y) \\ & ) & \\ \downharpoonright & & \\ \end{array} \\ & & & \\ \end{array}$	$\operatorname{F2.3d~:~Def}$
		$::\!$
$\operatorname{F2.3e.}$	$\begin{array}{ccccl} \upharpoonleft \operatorname{Der}^L \upharpoonright (x, y) & = & \underset{o \in O}{\operatorname{Conj}} \\ & & & \begin{array}{cl} \underline{((} & \upharpoonleft \operatorname{Den}^L \upharpoonright (o, x) \\ , & \upharpoonleft \operatorname{Den}^L \upharpoonright (o, y) \\ \underline{))} & \\ \end{array} \\ & & & \\ \end{array}$	$\operatorname{F2.3e~:~Log}$ $\operatorname{F2.3e~:~D10b}$
		$::\!$
$\operatorname{F2.3f.}$	$\begin{array}{ccccl} \upharpoonleft \operatorname{Der}^L \upharpoonright (x, y) & = & \underset{o \in O}{\operatorname{Conj}} \\ & & & \begin{array}{cl} \underline{((} & \upharpoonleft L_{OS} \upharpoonright (o, x) \\ , & \upharpoonleft L_{OS} \upharpoonright (o, y) \\ \underline{))} & \\ \end{array} \\ & & & \\ \end{array}$	$\operatorname{F2.3f~:~D10a}$

1.3.12.3. Digression on Derived Relations

A better understanding of derived equivalence relations (DERs) can be achieved by placing their constructions within a more general context and thus comparing the associated type of derivation operation, namely, the one that takes a triadic relation $L\!$ into a dyadic relation $\operatorname{Der}(L),$ with other types of operations on triadic relations. The proper setting would permit a comparative study of all their constructions from a basic set of projections and a full array of compositions on dyadic relations.

To that end, let the derivation $\operatorname{Der}(L)$ be expressed in the following way:

$\upharpoonleft \operatorname{Der}(L) \upharpoonright (x, y) \quad = \quad \underset{o \in O}{\operatorname{Conj}} ~\underline{((}~ \upharpoonleft L_{SO} \upharpoonright (x, o) ~,~ \upharpoonleft L_{OS} \upharpoonright (o, y) ~\underline{))}~.$

From this may be abstracted a way of composing two dyadic relations that have a domain in common. For example, let $P \subseteq X \times M$ and $Q \subseteq M \times Y$ be dyadic relations that have the middle domain $M\!$ in common. Then we may define a form of composition, notated $P \circeq Q,$ where $P \circeq Q ~\subseteq~ X \times Y$ is defined as follows:

$\upharpoonleft P \circeq Q \upharpoonright (x, y) \quad = \quad \underset{m \in M}{\operatorname{Conj}} ~\underline{((}~ \upharpoonleft P \upharpoonright (x, m) ~,~ \upharpoonleft Q \upharpoonright (m, y) ~\underline{))}~.$

Compare this with the usual form of composition, typically notated $P \circ Q$ and defined as follows:

$\upharpoonleft P \circ Q \upharpoonright (x, y) \quad = \quad \underset{m \in M}{\operatorname{Disj}} ~\upharpoonleft P \upharpoonright (x, m) ~\cdot~ \upharpoonleft Q \upharpoonright (m, y)~.$

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.

Proust, Marcel (1913–1927), In Search of Lost Time, Christopher Prendergast (general editor), Penguin Books, London, UK, 2002, 6 volumes:

The Way by Swann's (1913), Lydia Davis (trans.)
In the Shadow of Young Girls in Flower (1919), James Grieve (trans.)
The Guermantes Way (1920–1921), Mark Treharne (trans.)
Sodom and Gomorrah (1921–1922), John Sturrock (trans.)
The Prisoner (1923), Carol Clark (trans.)
The Fugitive (1925), Peter Collier (trans.)
Finding Time Again (1927), Ian Patterson (trans.)

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:	May-Jun 2004	Draft 11.00	Inquiry List: 1 2 3
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:	26 Nov 2001	Draft 8.63	Arisbe List, SUO List
Revised:	08 Jan 2001	Draft 8b
Revised:	30 Jun 2000	Draft 8.2	Arisbe Site: 1
Revised:	11 Feb 2000	Draft 8a
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

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

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

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

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

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

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

\(\operatorname{Object}\)	\(\operatorname{Sign}\)	\(\operatorname{Higher~Order~Sign}\)
\(\operatorname{Set}\)	\(\operatorname{Proposition}\)	\(\operatorname{Sentence}\)
\(f^{-1} (y)\!\)	\(f\!\)	\(^{\backprime\backprime} f \, ^{\prime\prime}\)
\(Q\!\)	\(\underline{1}\)	\(^{\backprime\backprime} \underline{1} ^{\prime\prime}\)
\({}^{_\sim} Q\)	\(\underline{0}\)	\(^{\backprime\backprime} \underline{0} ^{\prime\prime}\)

1.	Let the sentence \(s\!\) denote the proposition \(q,\!\)
		where	\(q : X \to \underline\mathbb{B}.\)
	Then we have the notational equivalence:
		\(\downharpoonleft s \downharpoonright ~=~ q.\)
2.	Let the sentence \(s\!\) denote the proposition \(q,\!\)
		where	\(q : X \to \underline\mathbb{B}\)
		and	\([\| q \|] ~=~ q^{-1} (\underline{1}) ~=~ Q \subseteq X.\)
	Then we have the notational equivalences:
		\(\downharpoonleft s \downharpoonright ~=~ q ~=~ f_Q ~=~ \upharpoonleft Q \upharpoonright.\)
3.	\(Q\!\)	\(=\!\)	\(\{ x \in X : x \in Q \}\)
		\(=\!\)	\([\| \upharpoonleft X \upharpoonright \|] ~=~ \upharpoonleft X \upharpoonright^{-1} (\underline{1})\)
		\(=\!\)	\([\| f_Q \|] ~=~ f_Q^{-1} (\underline{1}).\)
4.	\(\upharpoonleft Q \upharpoonright\)	\(=\!\)	\(\upharpoonleft \{ x \in X : x \in Q \} \upharpoonright\)
		\(=\!\)	\(\downharpoonleft x \in Q \downharpoonright\)
		\(=\!\)	\(f_Q.\!\)

\(\varepsilon\)	=	\(^{\backprime\backprime\prime\prime}\)	=	the empty string.
\(\underline\varepsilon\)	=	\(\{ \varepsilon \}\)	=	the language consisting of a single empty string.

1.	\(s\!\) is the concatenation of \(s_1\!\) and \(s_2\!\) in \(\mathfrak{L}\) if and only if
	\(s_1\!\) is a sentence of \(\mathfrak{L},\) \(s_2\!\) is a sentence of \(\mathfrak{L},\) and
	\(s = s_1 \cdot s_2.\)
2.	\(s\!\) is the concatenation of the \(k\!\) strings \(s_1, \ldots, s_k\!\) in \(\mathfrak{L},\)
	if and only if \(s_j\!\) is a sentence of \(\mathfrak{L},\) for all \(j = 1 \ldots k,\) and
	\(s = \operatorname{Conc}_{j=1}^k s_j = s_1 \cdot \ldots \cdot s_k.\)
3.	\(s\!\) is the discatenation of \(s_1\!\) by \(t\!\) if and only if
	\(s_1\!\) is a sentence of \(\mathfrak{L},\) \(t\!\) is an element of \(\mathfrak{A},\) and
	\(s_1 = s \cdot t.\)
	When this is the case, one more commonly writes:
	\(s = s_1 \cdot t^{-1}.\)
4.	\(s\!\) is a subclause of \(\mathfrak{L}\) if and only if
	\(s\!\) is a sentence of \(\mathfrak{L}\) and \(s\!\) ends with a \(^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime}.\)
5.	\(s\!\) is the subcatenation of \(s_1\!\) by \(s_2\!\) if and only if
	\(s_1\!\) is a subclause of \(\mathfrak{L},\) \(s_2\!\) is a sentence of \(\mathfrak{L},\) and
	\(s = s_1 \, \cdot \, ( \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime} \, )^{-1} \, \cdot \, ^{\backprime\backprime} \, \operatorname{,} \, ^{\prime\prime} \, \cdot \, s_2 \, \cdot \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime}.\)
6.	\(s\!\) is the surcatenation of the \(k\!\) strings \(s_1, \ldots, s_k\!\) in \(\mathfrak{L},\)
	if and only if \(s_j\!\) is a sentence of \(\mathfrak{L},\) for all \(j = 1 \ldots k,\!\) and
	\(s \ = \ \operatorname{Surc}_{j=1}^k s_j \ = \ ^{\backprime\backprime} \, \operatorname{(} \, ^{\prime\prime} \, \cdot \, s_1 \, \cdot \, ^{\backprime\backprime} \, \operatorname{,} \, ^{\prime\prime} \, \cdot \, \ldots \, \cdot \, ^{\backprime\backprime} \, \operatorname{,} \, ^{\prime\prime} \, \cdot \, s_k \, \cdot \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime}.\)

PC 1.	The blank symbol \(m_1\!\) is a sentence.
PC 2.	The paint \(p_j\!\) is a sentence, for each \(j\!\) in \(J.\!\)
PC 3.	\(\operatorname{Conc}^0\) and \(\operatorname{Surc}^0\) are sentences.
PC 4.	For each positive integer \(k,\!\)
	if \(s_1, \ldots, s_k\!\) are sentences,
	then \(\operatorname{Conc}_{j=1}^k s_j\) is a sentence,
	and \(\operatorname{Surc}_{j=1}^k s_j\) is a sentence.

\(\operatorname{If}\)	the strings \(S_1, \ldots, S_k\!\) are sentences,
\(\operatorname{Then}\)	their concatenation in the form
	\(\operatorname{Conc}_{j=1}^k S_j \ = \ S_1 \, \cdot \, \ldots \, \cdot \, S_k\)
	is a sentence,
\(\operatorname{And}\)	their surcatenation in the form
	\(\operatorname{Surc}_{j=1}^k S_j \ = \ ^{\backprime\backprime} \, \operatorname{(} \, ^{\prime\prime} \, \cdot \, S_1 \, \cdot \, ^{\backprime\backprime} \, \operatorname{,} \, ^{\prime\prime} \, \cdot \, \ldots \, \cdot \, ^{\backprime\backprime} \, \operatorname{,} \, ^{\prime\prime} \, \cdot \, S_k \, \cdot \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime}\)
	is a sentence.

\(1.\!\)	\(\operatorname{Node}^0\)	\(=\!\)	\(\operatorname{Node}()\)
		\(=\!\)	a node with no attachments.
	\(\operatorname{Node}_{j=1}^k C_j\)	\(=\!\)	\(\operatorname{Node} (C_1, \ldots, C_k)\)
		\(=\!\)	a node with the attachments \(C_1, \ldots, C_k.\)
\(2.\!\)	\(\operatorname{Lobe}^0\)	\(=\!\)	\(\operatorname{Lobe}()\)
		\(=\!\)	a lobe with no accoutrements.
	\(\operatorname{Lobe}_{j=1}^k C_j\)	\(=\!\)	\(\operatorname{Lobe} (C_1, \ldots, C_k)\)
		\(=\!\)	a lobe with the accoutrements \(C_1, \ldots, C_k.\)

\(F\!\)	\(F\!\)	\(F()\!\)	\(F\!\)
\(\underline{0}\)	\(F_0^{(0)}\!\)	\(\underline{0}\)	\((~)\)
\(\underline{1}\)	\(F_1^{(0)}\!\)	\(\underline{1}\)	\(((~))\)

\(F\!\)	\(F\!\)	\(F(x)\!\)		\(F\!\)
		\(F(\underline{1})\)	\(F(\underline{0})\)
\(F_0^{(1)}\!\)	\(F_{00}^{(1)}\!\)	\(\underline{0}\)	\(\underline{0}\)	\((~)\)
\(F_1^{(1)}\!\)	\(F_{01}^{(1)}\!\)	\(\underline{0}\)	\(\underline{1}\)	\((x)\!\)
\(F_2^{(1)}\!\)	\(F_{10}^{(1)}\!\)	\(\underline{1}\)	\(\underline{0}\)	\(x\!\)
\(F_3^{(1)}\!\)	\(F_{11}^{(1)}\!\)	\(\underline{1}\)	\(\underline{1}\)	\(((~))\)

\(F\!\)	\(F\!\)	\(F(x, y)\!\)				\(F\!\)
		\(F(\underline{1}, \underline{1})\)	\(F(\underline{1}, \underline{0})\)	\(F(\underline{0}, \underline{1})\)	\(F(\underline{0}, \underline{0})\)
\(F_{0}^{(2)}\!\)	\(F_{0000}^{(2)}\!\)	\(\underline{0}\)	\(\underline{0}\)	\(\underline{0}\)	\(\underline{0}\)	\((~)\)
\(F_{1}^{(2)}\!\)	\(F_{0001}^{(2)}\!\)	\(\underline{0}\)	\(\underline{0}\)	\(\underline{0}\)	\(\underline{1}\)	\((x)(y)\!\)
\(F_{2}^{(2)}\!\)	\(F_{0010}^{(2)}\!\)	\(\underline{0}\)	\(\underline{0}\)	\(\underline{1}\)	\(\underline{0}\)	\((x) y\!\)
\(F_{3}^{(2)}\!\)	\(F_{0011}^{(2)}\!\)	\(\underline{0}\)	\(\underline{0}\)	\(\underline{1}\)	\(\underline{1}\)	\((x)\!\)
\(F_{4}^{(2)}\!\)	\(F_{0100}^{(2)}\!\)	\(\underline{0}\)	\(\underline{1}\)	\(\underline{0}\)	\(\underline{0}\)	\(x (y)\!\)
\(F_{5}^{(2)}\!\)	\(F_{0101}^{(2)}\!\)	\(\underline{0}\)	\(\underline{1}\)	\(\underline{0}\)	\(\underline{1}\)	\((y)\!\)
\(F_{6}^{(2)}\!\)	\(F_{0110}^{(2)}\!\)	\(\underline{0}\)	\(\underline{1}\)	\(\underline{1}\)	\(\underline{0}\)	\((x, y)\!\)
\(F_{7}^{(2)}\!\)	\(F_{0111}^{(2)}\!\)	\(\underline{0}\)	\(\underline{1}\)	\(\underline{1}\)	\(\underline{1}\)	\((x y)\!\)
\(F_{8}^{(2)}\!\)	\(F_{1000}^{(2)}\!\)	\(\underline{1}\)	\(\underline{0}\)	\(\underline{0}\)	\(\underline{0}\)	\(x y\!\)
\(F_{9}^{(2)}\!\)	\(F_{1001}^{(2)}\!\)	\(\underline{1}\)	\(\underline{0}\)	\(\underline{0}\)	\(\underline{1}\)	\(((x, y))\!\)
\(F_{10}^{(2)}\!\)	\(F_{1010}^{(2)}\!\)	\(\underline{1}\)	\(\underline{0}\)	\(\underline{1}\)	\(\underline{0}\)	\(y\!\)
\(F_{11}^{(2)}\!\)	\(F_{1011}^{(2)}\!\)	\(\underline{1}\)	\(\underline{0}\)	\(\underline{1}\)	\(\underline{1}\)	\((x (y))\!\)
\(F_{12}^{(2)}\!\)	\(F_{1100}^{(2)}\!\)	\(\underline{1}\)	\(\underline{1}\)	\(\underline{0}\)	\(\underline{0}\)	\(x\!\)
\(F_{13}^{(2)}\!\)	\(F_{1101}^{(2)}\!\)	\(\underline{1}\)	\(\underline{1}\)	\(\underline{0}\)	\(\underline{1}\)	\(((x)y)\!\)
\(F_{14}^{(2)}\!\)	\(F_{1110}^{(2)}\!\)	\(\underline{1}\)	\(\underline{1}\)	\(\underline{1}\)	\(\underline{0}\)	\(((x)(y))\!\)
\(F_{15}^{(2)}\!\)	\(F_{1111}^{(2)}\!\)	\(\underline{1}\)	\(\underline{1}\)	\(\underline{1}\)	\(\underline{1}\)	\(((~))\)

		\(\text{Rule 1}\!\)
\(\text{If}\!\)	\(Q \subseteq X\)
\(\text{then}\!\)	\(\upharpoonleft Q \upharpoonright ~:~ X \to \underline\mathbb{B}\)
\(\text{and if}\!\)	\(x \in X\)
\(\text{then}\!\)	\(\text{the following are equivalent:}\!\)
\(\text{R1a.}\!\)	\(x \in Q\)
\(\text{R1b.}\!\)	\(\upharpoonleft Q \upharpoonright (x)\)

		\(\text{Rule 2}\!\)
\(\text{If}\!\)	\(f : X \to \underline\mathbb{B}\)
\(\text{and}\!\)	\(x \in X\)
\(\text{then}\!\)	\(\text{the following are equivalent:}\!\)
\(\text{R2a.}\!\)	\(f(x)\!\)
\(\text{R2b.}\!\)	\(f(x) = \underline{1}\)

		\(\text{Corollary 1}\!\)
\(\text{If}\!\)	\(Q \subseteq X\)
\(\text{and}\!\)	\(x \in X\)
\(\text{then}\!\)	\(\text{the following statement is true:}\!\)
\(\text{C1a.}\!\)	\(x \in Q ~\Leftrightarrow~ \upharpoonleft Q \upharpoonright (x) = \underline{1}\)	\(\text{R3a} \Leftrightarrow \text{R3c}\)

		\(\text{Rule 4}\!\)
\(\text{If}\!\)	\(Q \subseteq X ~\text{is fixed}\)
\(\text{and}\!\)	\(x \in X ~\text{is varied}\)
\(\text{then}\!\)	\(\text{the following are equivalent:}\!\)
\(\text{R4a.}\!\)	\(x \in Q\)
\(\text{R4b.}\!\)	\(\downharpoonleft x \in Q \downharpoonright\)
\(\text{R4c.}\!\)	\(\downharpoonleft x \in Q \downharpoonright (x)\)
\(\text{R4d.}\!\)	\(\upharpoonleft Q \upharpoonright (x)\)
\(\text{R4e.}\!\)	\(\upharpoonleft Q \upharpoonright (x) = \underline{1}\)