This site is supported by donations to The OEIS Foundation.

# Inquiry Driven Systems • Part 2

Author: Jon Awbrey

## Background

### 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.

#### The Informal Context

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Tennyson's poem The Lady of Shalott is akin to an ORT, but a bit more remote, since the name styled as “The Lady of Shalott”, that the author invokes over the course of the text, is not at first sight the title of a poem, but a title 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 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 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.

#### 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.

#### 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.

 ${\displaystyle {\text{Formal}}}$ ${\displaystyle {\text{Formative}}}$ ${\displaystyle {\text{Objective}}}$ ${\displaystyle {\text{Instrumental}}}$ ${\displaystyle {\text{Passive}}}$ ${\displaystyle {\text{Active}}}$ ${\displaystyle {\text{Afforded}}}$ ${\displaystyle {\text{Possessed}}}$ ${\displaystyle {\text{Exercised}}}$

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.

 ${\displaystyle {\text{Matter}}}$ ${\displaystyle {\text{Form}}}$ ${\displaystyle {\text{Potentiality}}}$ ${\displaystyle {\text{Actuality}}}$ ${\displaystyle {\text{Receptivity}}}$ ${\displaystyle {\text{Possession}}}$ ${\displaystyle {\text{Exercise}}}$ ${\displaystyle {\text{Life}}}$ ${\displaystyle {\text{Sleep}}}$ ${\displaystyle {\text{Waking}}}$ ${\displaystyle {\text{Wax}}}$ ${\displaystyle {\text{Impression}}}$ ${\displaystyle {\text{Axe}}}$ ${\displaystyle {\text{Edge}}}$ ${\displaystyle {\text{Cutting}}}$ ${\displaystyle {\text{Eye}}}$ ${\displaystyle {\text{Vision}}}$ ${\displaystyle {\text{Seeing}}}$ ${\displaystyle {\text{Body}}}$ ${\displaystyle {\text{Soul}}}$ ${\displaystyle {\text{Ship?}}}$ ${\displaystyle {\text{Sailor?}}}$

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.

 ${\displaystyle {\text{Formal}}}$ ${\displaystyle {\text{Formative}}}$ ${\displaystyle {\text{Objective}}}$ ${\displaystyle {\text{Instrumental}}}$ ${\displaystyle {\text{Passive}}}$ ${\displaystyle {\text{Active}}}$ ${\displaystyle {\text{Afforded}}}$ ${\displaystyle {\text{Possessed}}}$ ${\displaystyle {\text{Exercised}}}$ ${\displaystyle {\text{To Hold}}}$ ${\displaystyle {\text{To Have}}}$ ${\displaystyle {\text{To Use}}}$ ${\displaystyle {\text{Receptivity}}}$ ${\displaystyle {\text{Possession}}}$ ${\displaystyle {\text{Exercise}}}$ ${\displaystyle {\text{Potentiality}}}$ ${\displaystyle {\text{Actuality}}}$ ${\displaystyle {\text{Matter}}}$ ${\displaystyle {\text{Form}}}$

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.

1. 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).
2. 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.
3. 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).
4. 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.
5. But of natural bodies some have life (zoe) and some have not; by life we mean the capacity for self-sustenance, growth, and decay.
6. Every natural body (soma physikon), then, which possesses life must be substance, and substance of the compound type (synthete).
7. 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.
8. 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.
9. 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.
10. 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).
11. Now in one and the same person the possession of knowledge comes first.
12. 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).
13. (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.)
14. 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”.
15. 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.
16. 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.
17. 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.
18. We must, however, investigate our definition in relation to the parts of the body.
19. 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.
20. 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.
21. 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.
22. 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.
23. 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.
24. 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.
25. 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).
26. This must suffice as an attempt to determine in rough outline the nature of the soul.

### Recurring Themes

The overall purpose of the next several Sections is threefold:

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

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

The main goals of the present section are as follows:

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

#### 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.

#### 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 ${\displaystyle {\underline {G}}=(X,G)=(G^{(1)},G^{(2)})}$ is specified by giving the set of points ${\displaystyle X=G^{(1)}}$ and the set of ordered pairs ${\displaystyle G=G^{(2)}\subseteq X\times X}$ that go together to define the relation. In contexts where the set of points is understood, it is customary to call the whole relation ${\displaystyle {\underline {G}}}$ by the name of the set ${\displaystyle G.}$

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

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

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

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 ${\displaystyle R}$ is the relational bit under consideration:

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

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.

#### Propositions and Sentences

The concept of a sign relation is typically extended as a set ${\displaystyle L\subseteq O\times S\times I.}$ 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 in which it is likely to be encountered, 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 ${\displaystyle Q,}$ 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 ${\displaystyle X,}$ 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 ${\displaystyle s}$, written as ${\displaystyle {}^{\backprime \backprime }{\texttt {(}}s{\texttt {)}}\,{}^{\prime \prime }}$ and read as ${\displaystyle {}^{\backprime \backprime }\,\mathrm {not} \ s\,{}^{\prime \prime },}$ is a sentence that is true when ${\displaystyle s}$ is false and false when ${\displaystyle s}$ is true.

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

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

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

 ${\displaystyle {\begin{array}{lll}Q\!-\!P&=&\{x\in X:x\in Q~\mathrm {and} ~{\texttt {(}}x\in P{\texttt {)}}\}.\end{array}}}$

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

 ${\displaystyle {\begin{array}{lll}P\cap Q&=&\{x\in X:x\in P~\mathrm {and} ~x\in Q\}.\end{array}}}$

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

 ${\displaystyle {\begin{array}{lll}P\cup Q&=&\{x\in X:x\in P~\mathrm {or} ~x\in Q\}.\end{array}}}$

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

 ${\displaystyle {\begin{array}{lll}P~{\hat {+}}~Q&=&\{x\in X:x\in P\!-\!Q~\mathrm {or} ~x\in Q\!-\!P\}.\end{array}}}$

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 ${\displaystyle \{x\in X:{\underline {~~~}}\}}$ 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 ${\displaystyle {\mathbb {B} =\{0,1\}}}$ of two algebraic values, whose arithmetic operations obey the rules of ${\displaystyle \mathrm {GF} (2),}$ 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 ${\displaystyle {\underline {\mathbb {B} }}=\{{\underline {0}},{\underline {1}}\}}$ 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 ${\displaystyle {\underline {\mathbb {B} }}}$ 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 ${\displaystyle \mathbb {B} ,}$ and operations that are isomorphic to the rest of the boolean operations in ${\displaystyle {\underline {\mathbb {B} }}}$ can always be built on the binary basis of ${\displaystyle \mathbb {B} .}$

Of course, as sets of the same cardinality, the domains ${\displaystyle \mathbb {B} }$ and ${\displaystyle {\underline {\mathbb {B} }}}$ 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 ${\displaystyle {}^{\backprime \backprime }{\underline {0}}{}^{\prime \prime }}$ and ${\displaystyle {}^{\backprime \backprime }{\underline {1}}{}^{\prime \prime }}$ 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 ${\displaystyle {}^{\backprime \backprime }0{}^{\prime \prime }}$ and ${\displaystyle {}^{\backprime \backprime }1{}^{\prime \prime },}$ 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 ${\displaystyle x}$ in ${\displaystyle {\underline {\mathbb {B} }},}$ written ${\displaystyle {}^{\backprime \backprime }{\texttt {(}}x{\texttt {)}}{}^{\prime \prime }}$ or ${\displaystyle {}^{\backprime \backprime }\lnot x{}^{\prime \prime }}$ and read as ${\displaystyle {}^{\backprime \backprime }\mathrm {not} \ x{}^{\prime \prime },}$ is the boolean value ${\displaystyle {\texttt {(}}x{\texttt {)}}\in {\underline {\mathbb {B} }}}$ that is ${\displaystyle {\underline {1}}}$ when ${\displaystyle x}$ is ${\displaystyle {\underline {0}}}$ and ${\displaystyle {\underline {0}}}$ when ${\displaystyle x}$ is ${\displaystyle {\underline {1}}.}$ Negation is a monadic operation on boolean values, that is, a function of the form ${\displaystyle f:{\underline {\mathbb {B} }}\to {\underline {\mathbb {B} }},}$ as shown in Table 8.

 ${\displaystyle x}$ ${\displaystyle {\texttt {(}}x{\texttt {)}}}$ ${\displaystyle {\underline {0}}}$ ${\displaystyle {\underline {1}}}$ ${\displaystyle {\underline {1}}}$ ${\displaystyle {\underline {0}}}$

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

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

 ${\displaystyle \cdot }$ ${\displaystyle {\underline {0}}}$ ${\displaystyle {\underline {1}}}$ ${\displaystyle {\underline {0}}}$ ${\displaystyle {\underline {0}}}$ ${\displaystyle {\underline {0}}}$ ${\displaystyle {\underline {1}}}$ ${\displaystyle {\underline {0}}}$ ${\displaystyle {\underline {1}}}$

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

 ${\displaystyle +}$ ${\displaystyle {\underline {0}}}$ ${\displaystyle {\underline {1}}}$ ${\displaystyle {\underline {0}}}$ ${\displaystyle {\underline {0}}}$ ${\displaystyle {\underline {1}}}$ ${\displaystyle {\underline {1}}}$ ${\displaystyle {\underline {1}}}$ ${\displaystyle {\underline {0}}}$

For sentences, the signs of equality ${\displaystyle (=)}$ and inequality ${\displaystyle (\neq )}$ 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 ${\displaystyle (\Leftrightarrow )}$ and inequivalence ${\displaystyle (\not \Leftrightarrow )}$ 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 ${\displaystyle f:{\underline {\mathbb {B} }}\times {\underline {\mathbb {B} }}\to {\underline {\mathbb {B} }},}$ could be presented in the same way as the multiplication and addition tables, it is better to look for a more efficient style of representation, one that is able to express all of the boolean functions on the same number of variables on a roughly equal basis, and with a bit of luck, affords us with a calculus for computing with these functions.

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

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

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

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

1. Considered in extensional form, ${\displaystyle f_{Q}}$ is the subset of ${\displaystyle X\times {\underline {\mathbb {B} }}}$ that is given by the following formula:

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

2. Considered in functional form, ${\displaystyle f_{Q}}$ is the map from ${\displaystyle X}$ to ${\displaystyle {\underline {\mathbb {B} }}}$ that is given by the following condition:

${\displaystyle f_{Q}(x)~=~{\underline {1}}~\Leftrightarrow ~x\in Q.}$

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

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

In the special case where ${\displaystyle f}$ is the indicator function ${\displaystyle f_{Q}}$ of a set ${\displaystyle Q\subseteq X,}$ the fiber of ${\displaystyle {\underline {1}}}$ under ${\displaystyle f_{Q}}$ is just the set ${\displaystyle Q}$ back again:

 ${\displaystyle \mathrm {Fiber~of} ~{\underline {1}}~\mathrm {under} ~f_{Q}~=~f_{Q}^{-1}({\underline {1}})~=~\{x\in X:f_{Q}(x)={\underline {1}}\}~=~Q.}$

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 ${\displaystyle [|\,\ldots \,|]}$ around a sentence or the sign of a proposition, and whose application is defined as follows:

 ${\displaystyle \mathrm {If} ~f:X\to {\underline {\mathbb {B} }},}$ ${\displaystyle \mathrm {then} ~[|f|]~=~f^{-1}({\underline {1}})~=~\{x\in X:f(x)={\underline {1}}\}.}$

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:

 ${\displaystyle \mathrm {If} ~f:X\to {\underline {\mathbb {B} }},}$ ${\displaystyle \mathrm {then} ~[|f|]~=~f^{-1}({\underline {1}})~=~\{x\in X:f(x)\}.}$

The set-builder frame ${\displaystyle \{x\in X:{\underline {~~~}}\}}$ requires a grammatical sentence or a sentential clause to fill in the blank, as with the sentence ${\displaystyle {}^{\backprime \backprime }f(x)={\underline {1}}{}^{\prime \prime }}$ 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 ${\displaystyle {}^{\backprime \backprime }f(x){}^{\prime \prime }}$ and the sentence ${\displaystyle {}^{\backprime \backprime }f(x)={\underline {1}}{}^{\prime \prime }}$ represent the very same value to this context, for all ${\displaystyle x}$ in ${\displaystyle X,}$ 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 ${\displaystyle {}^{\backprime \backprime }f(x){}^{\prime \prime }}$ manifestly names the value ${\displaystyle f(x).}$ This is a value that can be seen in many lights. It is, at turns:

1. The value that the proposition ${\displaystyle f}$ has at the point ${\displaystyle x,}$ in other words, the value that ${\displaystyle f}$ bears at the point ${\displaystyle x}$ where ${\displaystyle f}$ is being evaluated, the value that ${\displaystyle f}$ takes on with respect to the argument or the object ${\displaystyle x}$ that the whole proposition is taken to be about.
2. The value that the proposition ${\displaystyle f}$ not only takes up at the point ${\displaystyle x,}$ but that it carries, conveys, transfers, or transports into the setting ${\displaystyle {}^{\backprime \backprime }\{x\in X:{\underline {~~~}}\}{}^{\prime \prime }}$ or into any other context of discourse where ${\displaystyle f}$ is meant to be evaluated.
3. The value that the sign ${\displaystyle {}^{\backprime \backprime }f(x){}^{\prime \prime }}$ 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 ${\displaystyle f}$ and the same object ${\displaystyle x}$ are borne in mind.
4. The value that the sign ${\displaystyle {}^{\backprime \backprime }f(x){}^{\prime \prime }}$ 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 ${\displaystyle {}^{\backprime \backprime }f(x)={\underline {1}}{}^{\prime \prime }}$ indirectly names what the sign ${\displaystyle {}^{\backprime \backprime }f(x){}^{\prime \prime }}$ more directly names, that is, the value ${\displaystyle f(x).}$ In other words, the sentence ${\displaystyle {}^{\backprime \backprime }f(x)={\underline {1}}{}^{\prime \prime }}$ has the same value to its interpretive context that the sign ${\displaystyle {}^{\backprime \backprime }f(x){}^{\prime \prime }}$ imparts to any comparable context, each by way of its respective evaluation for the same ${\displaystyle x\in X.}$

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 ${\displaystyle {}^{\backprime \backprime }f(x)={\underline {1}}{}^{\prime \prime }}$ latently connotes what the sign ${\displaystyle {}^{\backprime \backprime }f(x){}^{\prime \prime }}$ 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 ${\displaystyle {}^{\backprime \backprime }f(x){}^{\prime \prime },}$ and thus an object that might as well be identified with the value ${\displaystyle f(x).}$

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 ${\displaystyle Q\subseteq X,}$ written ${\displaystyle f_{Q},}$ is the map from ${\displaystyle X}$ to the boolean domain ${\displaystyle {\underline {\mathbb {B} }}=\{{\underline {0}},{\underline {1}}\}}$ that is defined in the following ways:

1. Considered in extensional form, ${\displaystyle f_{Q}}$ is the subset of ${\displaystyle X\times {\underline {\mathbb {B} }}}$ that is given by the following formula:

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

2. Considered in functional form, ${\displaystyle f_{Q}}$ is the map from ${\displaystyle X}$ to ${\displaystyle {\underline {\mathbb {B} }}}$ that is given by the following condition:

${\displaystyle f_{Q}~\Leftrightarrow ~x\in Q.}$

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

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

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 ${\displaystyle {}^{\backprime \backprime }\Leftrightarrow {}^{\prime \prime },}$ as written between logical sentences, and the sign of equality ${\displaystyle {}^{\backprime \backprime }={}^{\prime \prime },}$ 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 ${\displaystyle {}^{\backprime \backprime }p{}^{\prime \prime },{}^{\backprime \backprime }q{}^{\prime \prime }}$ 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 ${\displaystyle f:X\to {\underline {\mathbb {B} }}}$ each time that a function is introduced as a proposition.

Another convention of use in this context is to let underscored letters stand for ${\displaystyle k}$-tuples, lists, or sequences of objects. Typically, the elements of the ${\displaystyle k}$-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 ${\displaystyle k}$-tuple, list, or sequence. When the dimension of the ${\displaystyle k}$-tuple, list, or sequence is clear from context, the underscoring may be omitted. For example, the following patterns of construction are very often seen:

 ${\displaystyle {\begin{array}{lllclllcl}1.&{\text{If}}&x_{1},\dots ,x_{k}&\in &X&{\text{then}}&{\underline {x}}=(x_{1},\ldots ,x_{k})&\in &X^{k}.\\2.&{\text{If}}&x_{1},\dots ,x_{k}&:&X&{\text{then}}&{\underline {x}}=(x_{1},\ldots ,x_{k})&:&X^{k}.\\3.&{\text{If}}&f_{1},\dots ,f_{k}&:&X\to Y&{\text{then}}&{\underline {f}}=(f_{1},\ldots ,f_{k})&:&(X\to Y)^{k}.\\\end{array}}}$

There is usually felt to be a slight but significant distinction between a membership statement of the form ${\displaystyle {}^{\backprime \backprime }x\in X\,{}^{\prime \prime }}$ and a type indication of the form ${\displaystyle {}^{\backprime \backprime }x:X\,{}^{\prime \prime },}$ 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 ${\displaystyle {}^{\backprime \backprime }x\in X\,{}^{\prime \prime }}$ and those of the form ${\displaystyle {}^{\backprime \backprime }x:X\,{}^{\prime \prime }}$ 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 ${\displaystyle {}^{\backprime \backprime }x\in X~\Leftrightarrow ~{\underline {x}}\in {\underline {X}}\,{}^{\prime \prime }}$ and ${\displaystyle {}^{\backprime \backprime }x:X~\Leftrightarrow ~{\underline {x}}:{\underline {X}}\,{}^{\prime \prime },}$ 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 ${\displaystyle k,}$ also known as a boolean function on ${\displaystyle k}$ variables, is a map of the form ${\displaystyle F:{\underline {\mathbb {B} }}^{k}\to {\underline {\mathbb {B} }}.}$ In other words, a boolean connection of degree ${\displaystyle k}$ is a proposition about things in the universe ${\displaystyle X={\underline {\mathbb {B} }}^{k}.}$

An imagination of degree ${\displaystyle k}$ on ${\displaystyle X}$ is a ${\displaystyle k}$-tuple of propositions about things in the universe ${\displaystyle X.}$ By way of displaying the kinds of notation that are used to express this idea, the imagination ${\displaystyle {\underline {f}}=(f_{1},\ldots ,f_{k})}$ is given as a sequence of indicator functions ${\displaystyle f_{j}:X\to {\underline {\mathbb {B} }},}$ for ${\displaystyle j={}_{1}^{k}.}$ All of these features of the typical imagination ${\displaystyle {\underline {f}}}$ can be summed up in either one of two ways: either in the form of a membership statement, to the effect that ${\displaystyle {\underline {f}}\in (X\to {\underline {\mathbb {B} }})^{k},}$ or in the form of a type statement, to the effect that ${\displaystyle {\underline {f}}:(X\to {\underline {\mathbb {B} }})^{k},}$ though perhaps the latter form is slightly more precise than the former.

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

A projection of ${\displaystyle {\underline {\mathbb {B} }}^{k},}$ written ${\displaystyle \pi _{j}}$ or ${\displaystyle \mathrm {pr} _{j},}$ is one of the maps ${\displaystyle \pi _{j}:{\underline {\mathbb {B} }}^{k}\to {\underline {\mathbb {B} }},}$ for ${\displaystyle j=1~{\text{to}}~k,}$ that is defined as follows:

 ${\displaystyle {\begin{array}{cccccc}{\text{If}}&{\underline {y}}&=&(y_{1},\ldots ,y_{k})&\in &{\underline {\mathbb {B} }}^{k},\\\\{\text{then}}&\pi _{j}({\underline {y}})&=&\pi _{j}(y_{1},\ldots ,y_{k})&=&y_{j}.\\\end{array}}}$

The projective imagination of ${\displaystyle {\underline {\mathbb {B} }}^{k}}$ is the imagination ${\displaystyle (\pi _{1},\ldots ,\pi _{k}).}$

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 ${\displaystyle f:X\to {\underline {\mathbb {B} }}.}$

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 ${\displaystyle f:X\to {\underline {\mathbb {B} }}.}$

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 ${\displaystyle {\underline {\mathbb {B} }},}$ 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 ${\displaystyle s}$ is an assertion of its negation ${\displaystyle {}^{\backprime \backprime }\,{\texttt {(}}s{\texttt {)}}\,{}^{\prime \prime }.}$ 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 ${\displaystyle f:X\to {\underline {\mathbb {B} }}.}$

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.

#### 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.

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

Notice that the letters ${\displaystyle {}^{\backprime \backprime }p{}^{\prime \prime }}$ and ${\displaystyle {}^{\backprime \backprime }q{}^{\prime \prime },}$ interpreted as signs that denote indicator functions ${\displaystyle p,q:X\to {\underline {\mathbb {B} }},}$ 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 ${\displaystyle {}^{\backprime \backprime }s{}^{\prime \prime }}$ and ${\displaystyle {}^{\backprime \backprime }t{}^{\prime \prime },}$ to denote sentences. Thus, it is conceivable to have a situation where ${\displaystyle s~=~{}^{\backprime \backprime }p{}^{\prime \prime }}$ and where ${\displaystyle p:X\to {\underline {\mathbb {B} }}.}$ Altogether, this means that the sign ${\displaystyle {}^{\backprime \backprime }s{}^{\prime \prime }}$ denotes the sentence ${\displaystyle s,}$ that the sentence ${\displaystyle s}$ is the sentence ${\displaystyle {}^{\backprime \backprime }p{}^{\prime \prime },}$ and that the sentence ${\displaystyle {}^{\backprime \backprime }p{}^{\prime \prime }}$ denotes the proposition or the indicator function ${\displaystyle p:X\to {\underline {\mathbb {B} }}.}$ In settings where it is necessary to keep track of a large number of sentences, I use subscripted letters like ${\displaystyle {}^{\backprime \backprime }e_{1}{}^{\prime \prime },\,\ldots ,\,{}^{\backprime \backprime }e_{n}{}^{\prime \prime }}$ 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 ${\displaystyle k}$ is the number of sentences that are connected, then the connective is said to be of order ${\displaystyle k.}$ 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.

#### Stretching Principles

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

1. Any relation of values extends to a relation of what is valued.
2. Any statement about values says something about the things that are given these values.
3. Any association among a range of values establishes an association among the domains of things that these values are the values of.
4. Any connection between two values can be stretched to create a connection, of analogous form, between the objects, persons, qualities, or relationships that are valued in these connections.
5. For every operation on values, there is a corresponding operation on the actions, conducts, functions, procedures, or processes that lead to these values, as well as there being analogous operations on the objects that instigate all of these various proceedings.

Nothing about the application of the stretching principle guarantees that the analogues it generates will be as useful as the material it works on. It is another question entirely whether the links that are forged in this fashion are equal in their strength and apposite in their bearing to the tried and true utilities of the original ties, but in principle they are always there.

In particular, a connection ${\displaystyle F:{\underline {\mathbb {B} }}^{k}\to {\underline {\mathbb {B} }}}$ can be understood to indicate a relation among boolean values, namely, the ${\displaystyle k}$-ary relation ${\displaystyle F^{-1}({\underline {1}})\subseteq {\underline {\mathbb {B} }}^{k}.}$ If these ${\displaystyle k}$ values are values of things in a universe ${\displaystyle X,}$ that is, if one imagines each value in a ${\displaystyle k}$-tuple of values to be the functional image that results from evaluating an element of ${\displaystyle X}$ under one of its possible aspects of value, then one has in mind the ${\displaystyle k}$ propositions ${\displaystyle f_{j}:X\to {\underline {\mathbb {B} }},}$ for ${\displaystyle j=1~{\text{to}}~k,}$ in sum, one embodies the imagination ${\displaystyle {\underline {f}}=(f_{1},\ldots ,f_{k}).}$ Together, the imagination ${\displaystyle {\underline {f}}\in (X\to {\underline {\mathbb {B} }})^{k}}$ and the connection ${\displaystyle F:{\underline {\mathbb {B} }}^{k}\to {\underline {\mathbb {B} }}}$ stretch each other to cover the universe ${\displaystyle X,}$ yielding a new proposition ${\displaystyle p:X\to {\underline {\mathbb {B} }}.}$

To encapsulate the form of this general result, I define a composition that takes an imagination ${\displaystyle {\underline {f}}=(f_{1},\ldots ,f_{k})\in (X\to {\underline {\mathbb {B} }})^{k}}$ and a boolean connection ${\displaystyle F:{\underline {\mathbb {B} }}^{k}\to {\underline {\mathbb {B} }}}$ and gives a proposition ${\displaystyle p:X\to {\underline {\mathbb {B} }}.}$ Depending on the situation, specifically, according to whether many ${\displaystyle F}$ and many ${\displaystyle {\underline {f}},}$ a single ${\displaystyle F}$ and many ${\displaystyle {\underline {f}},}$ or many ${\displaystyle F}$ and a single ${\displaystyle {\underline {f}}}$ are being considered, respectively, the proposition ${\displaystyle p}$ thus constructed may be referred to under one of three descriptions:

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

${\displaystyle \:({\underline {\mathbb {B} }}^{k}\to {\underline {\mathbb {B} }})\times (X\to {\underline {\mathbb {B} }})^{k}\to (X\to {\underline {\mathbb {B} }}).}$

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

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

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

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

From these materials, it is required to construct a proposition ${\displaystyle p:X\to {\underline {\mathbb {B} }}}$ such that ${\displaystyle p(x)=F(f_{1}(x),\ldots ,f_{k}(x)),}$ for all ${\displaystyle x\in X.}$ The desired construction is determined as follows:

The cartesian power ${\displaystyle {\underline {\mathbb {B} }}^{k},}$ as a cartesian product, is characterized by the possession of a projective imagination ${\displaystyle \pi =(\pi _{1},\ldots ,\pi _{k})}$ of degree ${\displaystyle k}$ on ${\displaystyle {\underline {\mathbb {B} }}^{k},}$ along with the property that any imagination ${\displaystyle {\underline {f}}=(f_{1},\ldots ,f_{k})}$ of degree ${\displaystyle k}$ on an arbitrary set ${\displaystyle W}$ determines a unique map ${\displaystyle f!:W\to {\underline {\mathbb {B} }}^{k},}$ the play of whose projective images ${\displaystyle (\pi _{1}(f!(w),\ldots ,\pi _{k}(f!(w))}$ on the functional image ${\displaystyle {f!(w)}}$ matches the play of images ${\displaystyle (f_{1}(w),\ldots ,f_{k}(w))}$ under ${\displaystyle {\underline {f}},}$ term for term and at every element ${\displaystyle w}$ in ${\displaystyle W.}$

Just to be on the safe side, I state this again in more standard terms. The cartesian power ${\displaystyle {\underline {\mathbb {B} }}^{k},}$ as a cartesian product, is characterized by the possession of ${\displaystyle k}$ projection maps ${\displaystyle \pi _{j}:{\underline {\mathbb {B} }}^{k}\to {\underline {\mathbb {B} }},}$ for ${\displaystyle j=1~{\text{to}}~k,}$ along with the property that any ${\displaystyle k}$ maps ${\displaystyle f_{j}:W\to {\underline {\mathbb {B} }},}$ from an arbitrary set ${\displaystyle W}$ to ${\displaystyle {\underline {\mathbb {B} }},}$ determine a unique map ${\displaystyle f!:W\to {\underline {\mathbb {B} }}^{k}}$ such that ${\displaystyle \pi _{j}(f!(w))=f_{j}(w),}$ for all ${\displaystyle j=1~{\text{to}}~k,}$ and for all ${\displaystyle w\in W.}$

Now suppose that the arbitrary set ${\displaystyle W}$ in this construction is just the relevant universe ${\displaystyle X.}$ Given that the function ${\displaystyle f!:X\to {\underline {\mathbb {B} }}^{k}}$ is uniquely determined by the imagination ${\displaystyle {\underline {f}}:(X\to {\underline {\mathbb {B} }})^{k},}$ that is, by the ${\displaystyle k}$-tuple of propositions ${\displaystyle {\underline {f}}=(f_{1},\ldots ,f_{k}),}$ it is safe to identify ${\displaystyle f!}$ and ${\displaystyle {\underline {f}}}$ 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 ${\displaystyle {}^{\backprime \backprime }(f_{1},\ldots ,f_{k})\,{}^{\prime \prime }.}$ This facility of address is especially appropriate whenever a concrete term or a constructive precision is demanded by the context of discussion.

#### 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 ${\displaystyle F:{\underline {\mathbb {B} }}^{k}\to {\underline {\mathbb {B} }}}$ is a boolean function on ${\displaystyle k}$ variables, then it is possible to define a mapping ${\displaystyle F^{\}:(X\to {\underline {\mathbb {B} }})^{k}\to (X\to {\underline {\mathbb {B} }}),}$ in effect, an operation that takes ${\displaystyle k}$ propositions into a single proposition, where ${\displaystyle F^{\}}$ satisfies the following conditions:

 ${\displaystyle {\begin{array}{lcl}F^{\}(f_{1},\ldots ,f_{k})&:&X\to {\underline {\mathbb {B} }}\\\\F^{\}(f_{1},\ldots ,f_{k})(x)&=&F({\underline {f}}(x))\\&=&F((f_{1},\ldots ,f_{k})(x))\\&=&F(f_{1}(x),\ldots ,f_{k}(x)).\\\end{array}}}$

Thus, ${\displaystyle F^{\}}$ 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 ${\displaystyle {}^{\backprime \backprime }f_{Q}\,{}^{\prime \prime }}$ is sign that denotes the proposition ${\displaystyle f_{Q},}$ 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.

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

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 ${\displaystyle f}$ can be said to "indicate" a set ${\displaystyle Q}$ only insofar as the values of ${\displaystyle {\underline {1}}}$ and ${\displaystyle {\underline {0}}}$ that it assigns to the elements of the universe ${\displaystyle X}$ are positive and negative indications, respectively, of the elements in ${\displaystyle Q,}$ and thus indications of the set ${\displaystyle Q}$ and of its complement ${\displaystyle {}^{_{\sim }}Q=X\!-\!Q,}$ respectively. It is actually these values, when rendered by a concrete implementation of the indicator function ${\displaystyle f,}$ that are the actual signs of the objects that are inside the set ${\displaystyle Q}$ and the objects that are outside the set ${\displaystyle Q,}$ respectively.

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

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

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

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

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

 ${\displaystyle {\text{Object}}}$ ${\displaystyle {\text{Sign}}}$ ${\displaystyle {\text{Higher Order Sign}}}$ ${\displaystyle {\text{Set}}}$ ${\displaystyle {\text{Proposition}}}$ ${\displaystyle {\text{Sentence}}}$ ${\displaystyle Q}$ ${\displaystyle q}$ ${\displaystyle s}$ ${\displaystyle [|\downharpoonleft s\downharpoonright |]}$ ${\displaystyle \downharpoonleft s\downharpoonright }$ ${\displaystyle s}$ ${\displaystyle [|q|]}$ ${\displaystyle q}$ ${\displaystyle {}^{\backprime \backprime }q\,{}^{\prime \prime }}$ ${\displaystyle [|f_{Q}|]}$ ${\displaystyle f_{Q}}$ ${\displaystyle {}^{\backprime \backprime }f_{Q}\,{}^{\prime \prime }}$ ${\displaystyle Q}$ ${\displaystyle \upharpoonleft Q\upharpoonright }$ ${\displaystyle {}^{\backprime \backprime }\upharpoonleft Q\upharpoonright \,{}^{\prime \prime }}$

In particular, one observes the following relations and formulas:

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

Now if a sentence ${\displaystyle s}$ really denotes a proposition ${\displaystyle q,}$ and if the notation ${\displaystyle {}^{\backprime \backprime }\downharpoonleft s\downharpoonright \,{}^{\prime \prime }}$ is merely meant to supply another name for the proposition that ${\displaystyle s}$ already denotes, then why is there any need for the additional notation? It is because the interpretive mind habitually races from the sentence ${\displaystyle s,}$ through the proposition ${\displaystyle q}$ that it denotes, and on to the set ${\displaystyle Q=q^{-1}({\underline {1}})}$ that the proposition ${\displaystyle q}$ indicates, often jumping to the conclusion that the set ${\displaystyle Q}$ is the only thing that the sentence ${\displaystyle s}$ 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 ${\displaystyle s}$ to ${\displaystyle q}$ to ${\displaystyle Q.}$