Inquiry Driven Systems : Part 2

• Contents • Part 1 • Part 2 • Part 3 • Part 4 • Part 5 • Part 6 • Appendices • References • Document History •

Part 2.

2.1. 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.

2.1.1. The Informal Context

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

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

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

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

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

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

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

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

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

To ask these questions in a sign-theoretic context is to ask about the nature, the origin, and the development of the entities called questions as a class of signs, in brief but sufficiently general terms, to inquire into the life of a question as a sign.
To re-pose these questions in a system-theoretic context is to inquire into the notion of a state of question, asking:

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

2.1.2. The Epitext

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

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

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

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

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

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

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

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

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

2.1.3. The Formative Tension

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

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

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

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

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

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

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

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

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

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

2.2. Recurring Themes

The overall purpose of the next several Sections is threefold:

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

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

The main goals of the present section are as follows:

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

2.2.1. Preliminary Notions

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

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

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

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

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

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

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

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

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

2.2.2. Intermediary Notions

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

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

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

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

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

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

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

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

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

2.2.3. Propositions and Sentences

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

**Table 8. Negation Operation for the Boolean Domain**
$x\!$	$\underline{(} x \underline{)}$
$\underline{0}$	$\underline{1}$
$\underline{1}$	$\underline{0}$

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

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

**Table 9. Product Operation for the Boolean Domain**
$\cdot\!$	$\underline{0}$	$\underline{1}$
$\underline{0}$	$\underline{0}$	$\underline{0}$
$\underline{1}$	$\underline{0}$	$\underline{1}$

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

**Table 10. Sum Operation for the Boolean Domain**
$+\!$	$\underline{0}$	$\underline{1}$
$\underline{0}$	$\underline{0}$	$\underline{1}$
$\underline{1}$	$\underline{1}$	$\underline{0}$

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

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

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

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

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

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

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

$f_Q ~=~ \{ (x, y) \in X \times \underline\mathbb{B} ~:~ y = \underline{1} ~\Leftrightarrow~ x \in Q \}.$
Considered in functional form, $f_Q\!$ is the map from $X\!$ to $\underline\mathbb{B}$ that is given by the following condition:

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

$f_Q ~=~ \{ (x, y) \in X \times \underline\mathbb{B} ~:~ y ~\Leftrightarrow~ x \in Q \}.$
Considered in functional form, $f_Q\!$ is the map from $X\!$ to $\underline\mathbb{B}$ that is given by the following condition:

$f_Q ~\Leftrightarrow~ x \in Q.$

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

2.2.4. Empirical Types and Rational Types

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

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

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

2.2.5. Articulate Sentences

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

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

If a sentence denotes a proposition $f : X \to \underline\mathbb{B},$ then the value of the sentence with regard to $x \in X$ is the value $f(x)\!$ of the proposition at $x,\!$ where $^{\backprime\backprime} \underline{0} ^{\prime\prime}$ is interpreted as false and $^{\backprime\backprime} \underline{1} ^{\prime\prime}$ is interpreted as true.

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

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

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

Notice that the letters $^{\backprime\backprime} p ^{\prime\prime}$ and $^{\backprime\backprime} q ^{\prime\prime},$ interpreted as signs that denote indicator functions $p, q : X \to \underline\mathbb{B},$ have the character of sentences in relation to propositions, at least, they have the same status in this abstract discussion as genuine sentences have in concrete discussions. This illustrates the relation between sentences and propositions as a special case of the relation between signs and objects.

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

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

2.2.6. Stretching Principles

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

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

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

In particular, a connection $F : \underline\mathbb{B}^k \to \underline\mathbb{B}$ can be understood to indicate a relation among boolean values, namely, the $k\!$-ary relation $F^{-1} (\underline{1}) \subseteq \underline\mathbb{B}^k.$ If these $k\!$ values are values of things in a universe $X,\!$ that is, if one imagines each value in a $k\!$-tuple of values to be the functional image that results from evaluating an element of $X\!$ under one of its possible aspects of value, then one has in mind the $k\!$ propositions $f_j : X \to \underline\mathbb{B},$ for $j = 1 ~\text{to}~ k,$ in sum, one embodies the imagination $\underline{f} = (f_1, \ldots, f_k).$ Together, the imagination $\underline{f} \in (X \to \underline\mathbb{B})^k$ and the connection $F : \underline\mathbb{B}^k \to \underline\mathbb{B}$ stretch each other to cover the universe $X,\!$ yielding a new proposition $p : X \to \underline\mathbb{B}.$

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

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

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

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

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

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

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

The cartesian power $\underline\mathbb{B}^k,$ as a cartesian product, is characterized by the possession of a projective imagination $\pi = (\pi_1, \ldots, \pi_k)$ of degree $k\!$ on $\underline\mathbb{B}^k,$ along with the property that any imagination $\underline{f} = (f_1, \ldots, f_k)$ of degree $k\!$ on an arbitrary set $W\!$ determines a unique map $f! : W \to \underline\mathbb{B}^k,$ the play of whose projective images $(\pi_1 (f!(w), \ldots, \pi_k (f!(w))$ on the functional image $f!(w)\!$ matches the play of images $(f_1 (w), \ldots, f_k (w))$ under $\underline{f},$ term for term and at every element $w\!$ in $W.\!$

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

Now suppose that the arbitrary set $W\!$ in this construction is just the relevant universe $X.\!$ Given that the function $f! : X \to \underline\mathbb{B}^k$ is uniquely determined by the imagination $\underline{f} : (X \to \underline\mathbb{B})^k,$ that is, by the $k\!$-tuple of propositions $\underline{f} = (f_1, \ldots, f_k),$ it is safe to identify $f!\!$ and $\underline{f}$ as being a single function, and this makes it convenient on many occasions to refer to the identified function by means of its explicitly descriptive name $^{\backprime\backprime} (f_1, \ldots, f_k) \, ^{\prime\prime}.$ This facility of address is especially appropriate whenever a concrete term or a constructive precision is demanded by the context of discussion.

2.2.7. Stretching Operations

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

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

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

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

Now $^{\backprime\backprime} f_Q \, ^{\prime\prime}$ is sign that denotes the proposition $f_Q,\!$ and it certainly seems like a sufficient sign for it. Why is there is a need to recognize any other signs of it?

If one takes a sentence as a type of sign that denotes a proposition and a proposition as a type of function whose values serve to indicate a set, then one needs a way to grasp the overall relation between the sentence and the set as taking place within a higher order sign relation.

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

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

Strictly speaking, a proposition is too abstract to be a sign, and so the contents of Table 11 have to be taken with the indicated grains of salt. Propositions, as indicator functions, are abstract mathematical objects, not any kinds of syntactic elements, and so propositions cannot literally constitute the orders of concrete signs that remain of ultimate interest in the pragmatic theory of signs, or in any theory of effective meaning. Therefore, it needs to be understood that a proposition $f\!$ can be said to "indicate" a set $Q\!$ only insofar as the values of $\underline{1}$ and $\underline{0}$ that it assigns to the elements of the universe $X\!$ are positive and negative indications, respectively, of the elements in $Q,\!$ and thus indications of the set $Q\!$ and of its complement ${}^{_\sim} Q = X\!-\!Q,$ respectively. It is actually these values, when rendered by a concrete implementation of the indicator function $f,\!$ that are the actual signs of the objects that are inside the set $Q\!$ and the objects that are outside the set $Q,\!$ respectively.

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

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

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

In order to make these notations useful in practice, it is necessary to note of a couple of their finer points, points that might otherwise seem too fine to take much trouble over. For this reason, I express their usage a bit more carefully as follows:

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

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

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

In particular, one observes the following relations and formulas:

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

Now if a sentence $s\!$ really denotes a proposition $q,\!$ and if the notation $^{\backprime\backprime} \downharpoonleft s \downharpoonright \, ^{\prime\prime}$ is merely meant to supply another name for the proposition that $s\!$ already denotes, then why is there any need for the additional notation? It is because the interpretive mind habitually races from the sentence $s,\!$ through the proposition $q\!$ that it denotes, and on to the set $Q = q^{-1} (\underline{1})$ that the proposition $q\!$ indicates, often jumping to the conclusion that the set $Q\!$ is the only thing that the sentence $s\!$ is intended to denote. This higher order sign situation and the mind's inclination when placed in its setting calls for a linguistic mechanism or a notational device that is capable of analyzing the compound action and controlling its articulate performance, and this requires a way to interrupt the flow of assertion that typically takes place from $s\!$ to $q\!$ to $Q.\!$

2.3. The Cactus Patch

Thus, what looks to us like a sphere of scientific knowledge more accurately should be represented as the inside of a highly irregular and spiky object, like a pincushion or porcupine, with very sharp extensions in certain directions, and virtually no knowledge in immediately adjacent areas. If our intellectual gaze could shift slightly, it would alter each quill's direction, and suddenly our entire reality would change.

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

In this and the four subsections that follow, I describe a calculus for representing propositions as sentences, in other words, as syntactically defined sequences of signs, and for manipulating these sentences chiefly in the light of their semantically defined contents, in other words, with respect to their logical values as propositions. In their computational representation, the expressions of this calculus parse into a class of tree-like data structures called painted cacti. This is a family of graph-theoretic data structures that can be observed to have especially nice properties, turning out to be not only useful from a computational standpoint but also quite interesting from a theoretical point of view. The rest of this subsection serves to motivate the development of this calculus and treats a number of general issues that surround the topic.

In order to facilitate the use of propositions as indicator functions it helps to acquire a flexible notation for referring to propositions in that light, for interpreting sentences in a corresponding role, and for negotiating the requirements of mutual sense between the two domains. If none of the formalisms that are readily available or in common use are able to meet all of the design requirements that come to mind, then it is necessary to contemplate the design of a new language that is especially tailored to the purpose. In the present application, there is a pressing need to devise a general calculus for composing propositions, computing their values on particular arguments, and inverting their indications to arrive at the sets of things in the universe that are indicated by them.

For computational purposes, it is convenient to have a middle ground or an intermediate language for negotiating between the koine of sentences regarded as strings of literal characters and the realm of propositions regarded as objects of logical value, even if this renders it necessary to introduce an artificial medium of exchange between these two domains. If one envisions these computations to be carried out in any organized fashion, and ultimately or partially by means of the familiar sorts of machines, then the strings that express these logical propositions are likely to find themselves parsed into tree-like data structures at some stage of the game. With regard to their abstract structures as graphs, there are several species of graph-theoretic data structures that can be used to accomplish this job in a reasonably effective and efficient way.

Over the course of this project, I plan to use two species of graphs:

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

For now, it is enough to discuss the former class of data structures, leaving the consideration of the latter class to a part of the project where their distinctive features are key to developments at that stage. Accordingly, within the context of the current patch of discussion, or until it becomes necessary to attach further notice to the conceivable varieties of parse graphs, the acronym "PARC" is sufficient to indicate the pertinent genus of abstract graphs that are under consideration.

By way of making these tasks feasible to carry out on a regular basis, a prospective language designer is required not only to supply a fluent medium for the expression of propositions, but further to accompany the assertions of their sentences with a canonical mechanism for teasing out the fibers of their indicator functions. Accordingly, with regard to a body of conceivable propositions, one needs to furnish a standard array of techniques for following the threads of their indications from their objective universe to their values for the mind and back again, that is, for tracing the clues that sentences provide from the universe of their objects to the signs of their values, and, in turn, from signs to objects. Ultimately, one seeks to render propositions so functional as indicators of sets and so essential for examining the equality of sets that they can constitute a veritable criterion for the practical conceivability of sets. Tackling this task requires me to introduce a number of new definitions and a collection of additional notational devices, to which I now turn.

Depending on whether a formal language is called by the type of sign that makes it up or whether it is named after the type of object that its signs are intended to denote, one may refer to this cactus language as a sentential calculus or as a propositional calculus, respectively.

When the syntactic definition of the language is well enough understood, then the language can begin to acquire a semantic function. In natural circumstances, the syntax and the semantics are likely to be engaged in a process of co-evolution, whether in ontogeny or in phylogeny, that is, the two developments probably form parallel sides of a single bootstrap. But this is not always the easiest way, at least, at first, to formally comprehend the nature of their action or the power of their interaction.

According to the customary mode of formal reconstruction, the language is first presented in terms of its syntax, in other words, as a formal language of strings called sentences, amounting to a particular subset of the possible strings that can be formed on a finite alphabet of signs. A syntactic definition of the cactus language, one that proceeds along purely formal lines, is carried out in the next Subsection. After that, the development of the language's more concrete aspects can be seen as a matter of defining two functions:

The first is a function that takes each sentence of the language into a computational data structure, to be exact, a tree-like parse graph called a painted cactus.
The second is a function that takes each sentence of the language, or its interpolated parse graph, into a logical proposition, in effect, ending up with an indicator function as the object denoted by the sentence.

The discussion of syntax brings up a number of associated issues that have to be clarified before going on. These are questions of style, that is, the sort of description, grammar, or theory that one finds available or chooses as preferable for a given language. These issues are discussed in the Subsection after next (Subsection 1.3.10.10).

There is an aspect of syntax that is so schematic in its basic character that it can be conveyed by computational data structures, so algorithmic in its uses that it can be automated by routine mechanisms, and so fixed in its nature that its practical exploitation can be served by the usual devices of computation. Because it involves the transformation of signs, it can be recognized as an aspect of semiotics. Since it can be carried out in abstraction from meaning, it is not up to the level of semantics, much less a complete pragmatics, though it does incline to the pragmatic aspects of computation that are auxiliary to and incidental to the human use of language. Therefore, I refer to this aspect of formal language use as the algorithmics or the mechanics of language processing. A mechanical conversion of the cactus language into its associated data structures is discussed in Subsection 1.3.10.11.

In the usual way of proceeding on formal grounds, meaning is added by giving each grammatical sentence, or each syntactically distinguished string, an interpretation as a logically meaningful sentence, in effect, equipping or providing each abstractly well-formed sentence with a logical proposition for it to denote. A semantic interpretation of the cactus language is carried out in Subsection 1.3.10.12.

2.3.1. The Cactus Language : Syntax

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

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

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

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

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

Now consider the following paradigm:

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

or

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

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

Two alternative notations are often useful:

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

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

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

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

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

In this context, I make the following distinction:

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

A token is a particular appearance of a sign.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

$S\ \operatorname{covers}\ T$

$S\ \operatorname{governs}\ T$

$S\ \operatorname{rules}\ T$

$S\ \operatorname{subsumes}\ T$

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

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

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

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

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

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

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

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

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

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

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

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

2.3.1.1. Grammar 1

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

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

$\mathfrak{Q} = \varnothing$

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

In this formulation, the last two lines specify that:

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

2.3.1.2. Grammar 2

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

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

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

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

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

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

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

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

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

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

2.3.1.3. Grammar 3

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

In sum, one has the disjoint union:

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

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

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

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

2.3.1.4. Grammar 4

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

2.3.1.5. Grammar 5

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

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

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

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

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

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

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

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

2.3.1.6. Grammar 6

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

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

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

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

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

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

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

2.3.2. Generalities About Formal Grammars

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

$W ::> W'.\!$

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

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

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

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

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

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

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

2.3.3. The Cactus Language : Stylistics

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

$^{\backprime\backprime}$

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

$^{\prime\prime}$

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

2.3.4. The Cactus Language : Mechanics

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Table 13. Algorithmic Translation Rules

$\text{Sentence in PARCE}\!$

$\xrightarrow{\operatorname{Parse}}$

$\text{Graph in PARC}\!$

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

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

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

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

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

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

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

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

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

2.3.5. The Cactus Language : Semantics

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Table 14. Semantic Translation : Functional Form

$\operatorname{Sentence}$

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

$\operatorname{Graph}$

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

$\operatorname{Proposition}$

$s_j\!$

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

$C_j\!$

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

$q_j\!$

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

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

Table 15. Semantic Translation : Equational Form

$\downharpoonleft \operatorname{Sentence} \downharpoonright$

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

$\downharpoonleft \operatorname{Graph} \downharpoonright$

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

$\operatorname{Proposition}$

$\downharpoonleft s_j \downharpoonright$

$=\!$

$\downharpoonleft C_j \downharpoonright$

$=\!$

$q_j\!$

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

2.3.6. Stretching Exercises

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

2.4. Syntactic Transformations

We have been examining several distinct but closely related notions of indication. To discuss the import of these ideas in greater depth, it serves to establish a number of logical relations and set-theoretic identities that can be found to hold among their roughly parallel arrays of conceptions and constructions. Facilitating this task requires in turn a number of auxiliary concepts and notations. The notions of indication in question are expressed in a variety of different notations, enumerated as follows:

The functional language of propositions
The logical language of sentences
The geometric language of sets

Thus, one way to explain the relationships that hold among these concepts is to describe the translations that are induced among their allied families of notation.

2.4.1. Syntactic Transformation Rules

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

$\text{Definition 1}\!$

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

$\operatorname{D1a.}$

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

$\forall x \in X$

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

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

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

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

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

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

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

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

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

$\operatorname{Rule~3}$

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Compare:

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

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

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

Value Rule 1

$\operatorname{Evaluation~Rule~1}$

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


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

$\operatorname{Definition~2}$

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

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

$\operatorname{Definition~3}$

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

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

$\operatorname{Definition~4}$

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

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

$\operatorname{Definition~5}$

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

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

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

$\operatorname{Definition~6}$

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

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

$\operatorname{Definition~7}$

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

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

$\operatorname{Rule~5}$

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

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

$\operatorname{Rule~6}$

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

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

$\operatorname{Rule~7}$

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

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

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

$\operatorname{Rule~8}$

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

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

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

$\operatorname{Rule~9}$

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

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

$\operatorname{Rule~10}$

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

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

$\operatorname{Rule~11}$

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

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

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

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

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

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

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

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

$\operatorname{Fact~1}$

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

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

2.4.2. Derived Equivalence Relations

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

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

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

Two sets are equal as sets

$\iff$

The indicator functions of the two sets are equal as functions

$\iff$

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

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

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

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

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

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

Equivalent expressions for this concept are recorded in Definition 8.

$\operatorname{Definition~8}$

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

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

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

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

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

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

$\operatorname{Definition~9}$

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

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

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

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

Equivalent expressions for this concept are recorded in Definition 10.

$\operatorname{Definition~10}$

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

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

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

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

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

$\operatorname{Definition~11}$

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

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

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

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

In other words:

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

Equivalent expressions for this concept are recorded in Definition 12.

$\operatorname{Definition~12}$

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

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

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

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

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

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

These definitions and notations are recorded in the following display.

$\operatorname{Definition~13}$

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

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

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

Reflexive property.

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

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

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

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

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

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

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

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

Yes, once again, under the stated conditions.

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

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

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

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

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

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

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

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

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

$\operatorname{Fact~2.1}$

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

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

$\operatorname{Fact~2.2}$

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


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

$\operatorname{Fact~2.3}$

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


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

2.4.3. Digression on Derived Relations

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

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

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

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

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

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

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

• Contents • Part 1 • Part 2 • Part 3 • Part 4 • Part 5 • Part 6 • Appendices • References • Document History •

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

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

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

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

\(\operatorname{Object}\)	\(\operatorname{Sign}\)	\(\operatorname{Higher~Order~Sign}\)
\(\operatorname{Set}\)	\(\operatorname{Proposition}\)	\(\operatorname{Sentence}\)
\(Q\!\)	\(q\!\)	\(s\!\)
\([\| \downharpoonleft s \downharpoonright \|]\)	\(\downharpoonleft s \downharpoonright\)	\(s\!\)
\([\| q \|]\!\)	\(q\!\)	\(^{\backprime\backprime} q \, ^{\prime\prime}\)
\([\| f_Q \|]\!\)	\(f_Q\!\)	\(^{\backprime\backprime} f_Q \, ^{\prime\prime}\)
\(Q\!\)	\(\upharpoonleft Q \upharpoonright\)	\(^{\backprime\backprime} \upharpoonleft Q \upharpoonright \, ^{\prime\prime}\)

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

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

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

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

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

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

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

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

\(F\!\)	\(F\!\)	\(F(x, y)\!\)				\(F\!\)
		\(F(\underline{1}, \underline{1})\)	\(F(\underline{1}, \underline{0})\)	\(F(\underline{0}, \underline{1})\)	\(F(\underline{0}, \underline{0})\)
\(F_{0}^{(2)}\!\)	\(F_{0000}^{(2)}\!\)	\(\underline{0}\)	\(\underline{0}\)	\(\underline{0}\)	\(\underline{0}\)	\((~)\)
\(F_{1}^{(2)}\!\)	\(F_{0001}^{(2)}\!\)	\(\underline{0}\)	\(\underline{0}\)	\(\underline{0}\)	\(\underline{1}\)	\((x)(y)\!\)
\(F_{2}^{(2)}\!\)	\(F_{0010}^{(2)}\!\)	\(\underline{0}\)	\(\underline{0}\)	\(\underline{1}\)	\(\underline{0}\)	\((x) y\!\)
\(F_{3}^{(2)}\!\)	\(F_{0011}^{(2)}\!\)	\(\underline{0}\)	\(\underline{0}\)	\(\underline{1}\)	\(\underline{1}\)	\((x)\!\)
\(F_{4}^{(2)}\!\)	\(F_{0100}^{(2)}\!\)	\(\underline{0}\)	\(\underline{1}\)	\(\underline{0}\)	\(\underline{0}\)	\(x (y)\!\)
\(F_{5}^{(2)}\!\)	\(F_{0101}^{(2)}\!\)	\(\underline{0}\)	\(\underline{1}\)	\(\underline{0}\)	\(\underline{1}\)	\((y)\!\)
\(F_{6}^{(2)}\!\)	\(F_{0110}^{(2)}\!\)	\(\underline{0}\)	\(\underline{1}\)	\(\underline{1}\)	\(\underline{0}\)	\((x, y)\!\)
\(F_{7}^{(2)}\!\)	\(F_{0111}^{(2)}\!\)	\(\underline{0}\)	\(\underline{1}\)	\(\underline{1}\)	\(\underline{1}\)	\((x y)\!\)
\(F_{8}^{(2)}\!\)	\(F_{1000}^{(2)}\!\)	\(\underline{1}\)	\(\underline{0}\)	\(\underline{0}\)	\(\underline{0}\)	\(x y\!\)
\(F_{9}^{(2)}\!\)	\(F_{1001}^{(2)}\!\)	\(\underline{1}\)	\(\underline{0}\)	\(\underline{0}\)	\(\underline{1}\)	\(((x, y))\!\)
\(F_{10}^{(2)}\!\)	\(F_{1010}^{(2)}\!\)	\(\underline{1}\)	\(\underline{0}\)	\(\underline{1}\)	\(\underline{0}\)	\(y\!\)
\(F_{11}^{(2)}\!\)	\(F_{1011}^{(2)}\!\)	\(\underline{1}\)	\(\underline{0}\)	\(\underline{1}\)	\(\underline{1}\)	\((x (y))\!\)
\(F_{12}^{(2)}\!\)	\(F_{1100}^{(2)}\!\)	\(\underline{1}\)	\(\underline{1}\)	\(\underline{0}\)	\(\underline{0}\)	\(x\!\)
\(F_{13}^{(2)}\!\)	\(F_{1101}^{(2)}\!\)	\(\underline{1}\)	\(\underline{1}\)	\(\underline{0}\)	\(\underline{1}\)	\(((x)y)\!\)
\(F_{14}^{(2)}\!\)	\(F_{1110}^{(2)}\!\)	\(\underline{1}\)	\(\underline{1}\)	\(\underline{1}\)	\(\underline{0}\)	\(((x)(y))\!\)
\(F_{15}^{(2)}\!\)	\(F_{1111}^{(2)}\!\)	\(\underline{1}\)	\(\underline{1}\)	\(\underline{1}\)	\(\underline{1}\)	\(((~))\)

		\(\text{Rule 1}\!\)
\(\text{If}\!\)	\(Q \subseteq X\)
\(\text{then}\!\)	\(\upharpoonleft Q \upharpoonright ~:~ X \to \underline\mathbb{B}\)
\(\text{and if}\!\)	\(x \in X\)
\(\text{then}\!\)	\(\text{the following are equivalent:}\!\)
\(\text{R1a.}\!\)	\(x \in Q\)
\(\text{R1b.}\!\)	\(\upharpoonleft Q \upharpoonright (x)\)

		\(\text{Rule 2}\!\)
\(\text{If}\!\)	\(f : X \to \underline\mathbb{B}\)
\(\text{and}\!\)	\(x \in X\)
\(\text{then}\!\)	\(\text{the following are equivalent:}\!\)
\(\text{R2a.}\!\)	\(f(x)\!\)
\(\text{R2b.}\!\)	\(f(x) = \underline{1}\)

		\(\text{Corollary 1}\!\)
\(\text{If}\!\)	\(Q \subseteq X\)
\(\text{and}\!\)	\(x \in X\)
\(\text{then}\!\)	\(\text{the following statement is true:}\!\)
\(\text{C1a.}\!\)	\(x \in Q ~\Leftrightarrow~ \upharpoonleft Q \upharpoonright (x) = \underline{1}\)	\(\text{R3a} \Leftrightarrow \text{R3c}\)

		\(\text{Rule 4}\!\)
\(\text{If}\!\)	\(Q \subseteq X ~\text{is fixed}\)
\(\text{and}\!\)	\(x \in X ~\text{is varied}\)
\(\text{then}\!\)	\(\text{the following are equivalent:}\!\)
\(\text{R4a.}\!\)	\(x \in Q\)
\(\text{R4b.}\!\)	\(\downharpoonleft x \in Q \downharpoonright\)
\(\text{R4c.}\!\)	\(\downharpoonleft x \in Q \downharpoonright (x)\)
\(\text{R4d.}\!\)	\(\upharpoonleft Q \upharpoonright (x)\)
\(\text{R4e.}\!\)	\(\upharpoonleft Q \upharpoonright (x) = \underline{1}\)