This site is supported by donations to The OEIS Foundation.

# Inquiry Driven Systems • Part 13

Author: Jon Awbrey

## Reflective Interpretive Frameworks (cont.)

### Recursive Aspects

Note. This section will most likely be rendered obsolete once all the planned changes in notation are worked through the text, as I will make a point of always distinguishing the interpretive agents ${\displaystyle {\text{A}}}$ and ${\displaystyle {\text{B}}}$ from the corresponding sign relations ${\displaystyle L({\text{A}})}$ and ${\displaystyle L({\text{B}}).}$

There is a one piece of unfinished business concerning the presentation of this example that deserves further comment.

Since the objects of reference ${\displaystyle {\text{A}}}$ and ${\displaystyle {\text{B}}}$ are imagined to be interpretive agents, it is convenient to use their names to denote the corresponding sign relations. Thus, the interpreters ${\displaystyle {\text{A}}}$ and ${\displaystyle {\text{B}}}$ are self-referent and mutually referent to the extent that they have names for themselves and each other. However, their discussion as a whole fails to contain any term for itself, and it even lacks a full set of grammatical cases for the objects in it. Whether these recursions and omissions cause any problems for my discussion will depend on the level of interpretive sophistication, not of ${\displaystyle {\text{A}}}$ and ${\displaystyle {\text{B}},}$ but of the external systems of interpretation that are brought to bear on it.

In defining the activities of interpreters ${\displaystyle {\text{A}}}$ and ${\displaystyle {\text{B}}}$ as sign relations, I have implicitly specified set-theoretic equations of the following form:

 ${\displaystyle {\begin{array}{lllllll}{\text{A}}&=&\{&({\text{A}},{}^{\backprime \backprime }{\text{A}}{}^{\prime \prime },{}^{\backprime \backprime }{\text{A}}{}^{\prime \prime }),&\ldots ,&({\text{A}},{}^{\backprime \backprime }{\text{i}}{}^{\prime \prime },{}^{\backprime \backprime }{\text{i}}{}^{\prime \prime }),&\\&&&({\text{B}},{}^{\backprime \backprime }{\text{B}}{}^{\prime \prime },{}^{\backprime \backprime }{\text{B}}{}^{\prime \prime }),&\ldots ,&({\text{B}},{}^{\backprime \backprime }{\text{u}}{}^{\prime \prime },{}^{\backprime \backprime }{\text{u}}{}^{\prime \prime })&\},\\[10pt]{\text{B}}&=&\{&({\text{A}},{}^{\backprime \backprime }{\text{A}}{}^{\prime \prime },{}^{\backprime \backprime }{\text{A}}{}^{\prime \prime }),&\ldots ,&({\text{A}},{}^{\backprime \backprime }{\text{u}}{}^{\prime \prime },{}^{\backprime \backprime }{\text{u}}{}^{\prime \prime }),&\\&&&({\text{B}},{}^{\backprime \backprime }{\text{B}}{}^{\prime \prime },{}^{\backprime \backprime }{\text{B}}{}^{\prime \prime }),&\ldots ,&({\text{B}},{}^{\backprime \backprime }{\text{i}}{}^{\prime \prime },{}^{\backprime \backprime }{\text{i}}{}^{\prime \prime })&\}.\end{array}}}$

The way I read these equations, they do not attempt to define the entities ${\displaystyle {\text{A}}}$ and ${\displaystyle {\text{B}}}$ in terms of themselves and each other. Instead, they define the whole instrumental activity of each interpreter, a highly complex duty, in terms of the interpreters' more perfunctory roles as objects of reference, and in terms of their associated actions on signs as mere tokens of each other's existence. In other words, the recursion does not have recourse to the full fledged faculties of ${\displaystyle {\text{A}}}$ and ${\displaystyle {\text{B}}}$ but only to their more inert excipients, their rote performances as inactive objects and passive images whose properly reduced complexities provide grounds for permitting the recursion to fly.

### Patterns of Self-Reference

In setting out the plan of a full scale RIF there is an unspoken promise to justify eventually the thematic motives that experimentally tolerate its indulgence in self-reference, and it seems that this implicit hope for a full atonement in time is a key to the tensions of the work being borne. This section, in order to inspire confidence in the prospects of a RIF being achievable, and by way of allaying widespread suspicions about all types of self-reference, examines several forms of circular referral and notes that not all contemplation of self-reference is incurably vicious.

In this section I consider signs, expressions, sign relations, and systems of interpretation (SOIs) that involve forms of self-reference. Because it is the abstract forms of self-reference that constitute the chief interest of this study, I collect this whole subject matter under the heading patterns of self-reference (POSRs). With respect to this domain I entertain the classification of POSRs in two different ways.

In this section I take notice of a broad family of formal structures that I refer to as patterns of self-reference (POSRs), because they seem to have in common the proposed description of a formal object by means of recursive or circular references. In their basic characters, POSRs range from the familiar to the strange, from the obvious to the problematic, and from the legitimate to the spurious. Often a POSR is best understood as a formal object in its own right, or as a formal sign that foreshadows a definite object, but occasionally a POSR can only be interpreted as something in the character of a syntactic pattern, one that goes into the making of a questionable specification and represents merely a dubious attempt to indicate or describe an object. All in all, POSRs range from the kinds of functions and objects, or programs and data structures, that are successfully defined by recursion to the sorts of vitiating circles that doom every attempt to define an unknown term in terms of itself.

Because POSRs span the spectrum from the moderately straightforward to the deliberately misleading, there is a need for ways to tell them apart, at least, before pursuing their consequences too far. Of course, if one cannot rest without having all computable functions at one's command, then no program can tell all the good and bad programs apart. But if one can be satisfied with a somewhat more modest domain, then there is hope for a way, an experimental, fallible, and incremental way, but a way nonetheless, that eventually leads one to know the good and ultimately keeps one away from the bad.

When it comes to their propriety, POSRs are found on empirical grounds to fall into two varieties: the exculpable and the indictable kinds. Thus, it is reasonable to attempt an empirical distinction, proposing to let experience mark each POSR as an excusable self-reference (ESR) or an improper self-reference (ISR), as the case may be. But empirical grounds can be a hard basis to fall back on, since a recourse to actual experience with POSRs can risk an agent's participation in pretended sign relations and promissory representations that amount in the end to nothing more than forms of interpretive futility. Therefore, one seeks an arrangement of methods in general or an ordering of options in these special cases that makes the empirical trial a court of last resort and that avoids resorting to the actual experience of interpretation as a routine matter of course.

First, I recognize an empirical distinction that seems to exist between the less problematic and the more problematic varieties of self-reference, allowing POSRs to be sorted according to the consequential features that they have in actual experience. There are the good sorts, those cleared up to the limits of accumulated experience as innocuous usages and even as probable utilities, and then there are the bad sorts, those marked by hard experience as definitely problematic.

Next, I search for an intuitive distinction that can be supposed to exist between the good and the bad sorts of POSRs, invoking a formal character or computable predicate of a POSR whose prior inspection can provide interpreters with a definitive indication or a decisive piece of information as to whether a POSR is good or bad, without forcing them to undergo the consequences of its actual use.

Before I can pin down what is involved in finding these intuitive characters and distinctions, it is necessary to discuss the concept of intuition that is relevant here. This issue requires a substantial digression and is taken up in the next section. After that, the concrete examples I take to be acceptable POSRs are presented.

### Practical Intuitions

[Variant] I use the word intuition in a pragmatic sense, at least, in a sense that is available to the word after the critique of pragmatism has purged it of certain vacuities that occasionally affect its use, especially the illusions of incorrigibility that place it beyond practical competence. In essence and etymology, and thus rehabilitated to its practical senses, an intuition is just an awareness, perhaps with a certain wariness, but certainly with no aura of infallibility. Thus, when I use the word intuition without further qualifications, it is intended to refer to a practical intuition of this kind, the only kind that has a recurring usefulness, and it ought to suggest the kinds of casual intuitions and fallible insights that intelligent agents ordinarily have, and that constitute their unformalized approximate knowledge of a given domain.

[Variant] The concept of intuition I am using here is a pragmatic one, referring to the kinds of casual intuitions and fallible insights that go to make up an agent's incompletely formalized and approximate knowledge of an object or domain. This sense of intuition differs from its technical meaning in various other philosophies, where it refers to a supposed modality of knowledge that involves an immediate cognition of an object, for instance, a direct perception of a fact about an object or an infallible apprehension of a fundamental truth about the world. Whatever the case, this makes an intuition a piece of knowledge about an object that is determined solely by something that exists outside the knower, and this can only be the object in itself, or what is called the transcendental object.

[Variant] This involves a particular notion of what constitutes an intuition, not any direct perception, immediate cognition, or otherwise infallible piece of knowledge that an agent might be supposed to have about an object or domain but only the modality of unformalized approximate knowledge that an agent actually has at the beginning of inquiry, with all the risks of casual intuition and fallible insight that go into it.

[Variant] The pragmatic concept of intuition is at odds with its technical meaning in certain other philosophies, where an intuition is supposed to be an immediate cognition of an object, perhaps a direct perception of a fact about an object or a state of affairs, perhaps an assured apprehension of a fundamental truth about the entire world of possible experience. The inference from this immediacy is supposed to be that an intuition is unmediated, therefore pure, therefore infallible.

[Variant] This candidate for an argument is a bit too quick, I think, so let me review its qualifications at a pettier pace. According to the way of thinking under examination, intuition is knowledge of an object that is not determined by previous knowledge of the same object. That is, an intuition is a piece of knowledge about an object that is determined solely by something that exists outside the knower, and this can only be the object in itself, or what is called the transcendental object. Accordingly, if the process that plants a bit of information in an agent's mind is not mediated by anything else under the agent's control, or by any previous step that the agent can help to determine, then the data in question is beyond correction, and thus acquires a status that is literally incorrigible.

There is a reason why the issue of immediacy has come up at this point. If there is a truth in the idea that “all thought takes place in signs”, as signs are understood in their pragmatic theory, then it means that thought in general, and inquiry in particular, is mediated by a process of interpretation that advances through the connotative components of one or many sign relations in the orbits of their denotative objects. Depending partly on the other assumptions that one makes about the nature of physical processes in the world, this constrains the models of embodied reasoning that one can entertain as being available to inquiry. One way of reading the implications of this “mediation” leads to the conclusion that thought is mediated by a potentially continuous process of interpretation, whose formal study requires the contemplation of potentially continuous sign relations.

Under common assumptions about the nature of causal processes, the possibility of continuity in sign relations becomes a logical necessity. This forces the distinction of immediacy to be recognized as a purely interpretive value, one that is attributed to a sign by a particular interpreter, and it renders the character of an intuition relative to the interpreter that is so impressed by it. The decision to interpret a datum of experience as an immediate sign is itself the result of a process of inference that says it is OK to do so, but it can be simply indicative of one interpreter's lack of interest or lack of capacity for pursuing the matter further.

The decision by an interpreter to treat a fact as immediate, often in spite of every indication to the contrary, can still be respected as such, but there need be nothing in the fact of the matter that makes it so. Nothing about their interpretive designation affects the logical status of axioms and primitives to be regarded as unproven truths and undefined meanings, respectively, but it does mark these entitlements as privileges that can be enjoyed uncontested only within a circumscribed system of reasoning, the whole of which system remains subject to being judged in competition with contending systems.

In contrast, uncommon assumptions about causality can lead to the consideration of discrete sign relations as complete entities in and of themselves. It is on these grounds, where the conceptual possibility of continuous sign relations meets the practical necessity of discrete sign relations, that the broader philosophy of pragmatism must come to terms with the narrower constraints of computing, indeed, where both this theory and this practice must begin to reckon with the forms of bounded rationality that are available to finite information creatures.

### Examples of Self-Reference

For ease of reference, I introduce the following terminology. With respect to the empirical dimension, a good POSR is described as an exculpable self-reference (ESR) while a bad POSR is described as an indictable self-reference (ISR). With respect to the intuitive dimension, a good POSR is depicted as an explicative self-reference (ESR) while a bad POSR is depicted as an implicative self-reference (ISR). Here, underscored acronyms are used to mark the provisionally settled, hypothetically tentative, or status quo condition of these casually intuitive categories.

These categories of POSRs can be discussed in greater detail as follows:

1. There is an empirical distinction that appears to impose itself on the varieties of self-reference, separating the forms that lead to trouble in thought and communication from the forms that do not. And there is a pragmatic reason for being interested in this distinction, the motive being to avoid the corresponding types of trouble in reflective thinking. Whether this apparent distinction can hold up under close examination is a good question to consider at a later point. But the real trouble to be faced at the moment is that an empirical distinction is a post hoc mark, a difference that makes itself obvious only after the possibly unpleasant facts to be addressed are already present in experience. Consequently, its certain recognition comes too late to avert the adverse portions of those circumstances that its very recognition is desired to avoid.

According to the form of this empirical distinction, a POSR can be classified either as an exculpable self-reference (ESR) or as an indictable self-reference (ISR). The distinction and the categories to either side of it are intended to sort out the POSRs that are safe and effective to use in thought and communication from the POSRs that can be hazardous to the health of inquiry.

More explicitly, the distinction between ESRs and ISRs is intended to capture the differences that exist between the following cases:

1. ESRs are POSRs that cause no apparent problems in thought or communication, often appearing as practiaclly useful in many contexts and even as logically necessary in some contexts.

2. ISRs are POSRs that lead to various sorts of trouble in the attempt to reason with them or to reason about them, that is, to use them consistently or even to decide for or against their use.

I refer to this as an empirical distinction in spite of the fact that the domain of experience in question is decidedly a formal one, because it rests on the kinds of concrete experiences and grows through the kinds of unforeseen developments that are ever the hallmark of experimental knowledge.

There is a pragmatic motive involved in this effort to classify the forms of self-reference, namely, to avoid certain types of trouble that seem to arise in reasoning by means of self referent forms. Accordingly, there is an obvious difference in the uses of self referent forms that is of focal interest here, but it presents itself as an empirical distinction, that is, an after the fact feature or post hoc mark. Namely, there are forms of self-reference that prove themselves useful in practice, being conducive to both thought and communication, and then there are forms that always seem to lead to trouble. The difference is evident enough after the impact of their effects has begun to set in, but it is not always easy to recognize these facts in advance of risking the very circumstances of confusion that one desires a classification to avoid.

In summary, one has the following problem. There is found an empirical distinction between different kinds of self-reference, one that becomes evident and is easy to judge after the onset of their effects has begun to set in, between the kinds of self-reference that lead to trouble and the kinds that do not. But what kinds of intuitive features, properties that one could recognize before the fact, would serve to distinguish the immanent and imminent empirical categories before one has gone through the trouble of suffering their effects?

Thus, one has the problem of translating between a given collection of empirical categories and a suitable collection of intuitive categories, the latter being of a kind that can be judged before the facts of experience have become inevitable, hoping thereby to correlate the two dimensions in such a way that the categories of intuition about POSRs can foretell the categories of experience with POSRs.

2. In a tentative approach to the subject of self-reference, I notice a principled distinction between two varieties of self-reference, that I call constitutional, implicative, or intrinsic self-reference (ISR) and extra-constitutional, explicative, or extrinsic self-reference (ESR), respectively.

1. ISR
2. ESR

In the rest of this section I put aside the question of defining a thing, symbol, or concept in terms of itself, which promises to be an exercise in futility, and consider only the possibility of explaining, explicating, or elaborating a thing, symbol, or concept in terms of itself. In this connection I attach special importance to a particular style of exposition, one that reformulates one's initial idea of an object in terms of the active implications or the effective consequences that its presence in a situation or its recognition and use in an application constitutes for the practical agent concerned. This style of pragmatic reconstruction can serve a useful purpose in clarifying the information one possesses about the object, sign, or idea of concern. Properly understood, it is marks the effective reformulation of ideas in ways that are akin to the more reductive sorts of operational definition, but overall is both more comprehensive and more pointedly related to the pragmatic agent, or the actual interpreter of the symbols and concepts in question.

The pending example of a POSR is, of course, the system composed of a pair of sign relations ${\displaystyle \{L({\text{A}}),L({\text{B}})\},}$ where the nouns and pronouns in each sign relation refer to the hypostatic agents ${\displaystyle {\text{A}}}$ and ${\displaystyle {\text{B}}}$ that are known solely as embodiments of the sign relations ${\displaystyle L({\text{A}})}$ and ${\displaystyle L({\text{B}}).}$ But this example, as reduced as it is, already involves an order of complexity that needs to be approached in more discrete stages than the ones enumerated in the current account. Therefore, it helps to take a step back from the full variety of sign relations and to consider related classes of POSRs that are typically simpler in principle.

1. The first class of POSRs I want to consider is diverse in form and content and has many names, but the feature that seems to unite all its instances is a self-commenting or self-documenting character. Typically, this means a partially self-documenting (PSD) character. As species of formal structures, PSD data structures are rife throughout computer science, and PSD developmental sequences turn up repeatedly in mathematics, logic, and proof theory. For the sake of euphony and ease of reference I collect this class of PSD POSRs under the name of auto-graphs (AGs).

The archetype of all auto-graphs is perhaps the familiar model of the natural numbers ${\displaystyle \mathbb {N} }$ as a sequence of sets, each of whose successive sets collects all and only the previous sets of the sequence:

 ${\displaystyle \{\},\quad \{\{\}\},\quad \{\{\},\{\{\}\}\},\quad \{\{\},\{\{\}\},\{\{\},\{\{\}\}\}\},\quad \ldots }$

This is the purest example of a PSD developmental sequence, where each member of the sequence documents the prior history of the development. This AG is akin to many kinds of PSD data structures that are found to be of constant use in computing. As a natural precursor to many kinds of intelligent data structures, it forms the inveterate backbone of a primitive capacity for intelligence. That is, this sequence has the sort of developing structure that can support the initial growth of learning in many species of creature constructions with adaptive constitutions, while it remains supple enough to supply an articulate skeleton for the evolving process of reflective inquiry. But this takes time to see.

For future reference, I refer to this model of natural numbers as “MON”. The very familiarity of this MON means that one reflexively proceeds from reading the signs of its set notation to thinking of its sets as mathematical objects, with little awareness of the sign relation that mediates the process, or even much reflection after the fact that is independent of the reflections recorded. Thus, even though this MON documents a process of reflective development, it need inspire no extra reflection on the acts of understanding needed to follow its directions.

In order to render this MON instructive for the development of a RIF, something intended to be a deliberately self-conscious construction, it is important to remedy the excessive lucidity of this MONs reflections, the confusing mix of opacity and transparency that comes in proportion to one's very familiarity with an object and that is compounded by one's very fluency in a language. To do this, it is incumbent on a proper analysis of the situation to slow the MON down, to interrupt one's own comprehension of its developing intent, and to articulate the details of the sign process that mediates it much more carefully than is customary.

These goals can be achieved by singling out the formal language that is used by this MON to denote its set theoretic objects. This involves separating the object domain ${\displaystyle {O=O_{\text{MON}}}}$ from the sign domain ${\displaystyle {S=S_{\text{MON}}},}$ paying closer attention to the naive level of set notation that is actually used by this MON, and treating its primitive set theoretic expressions as a formal language all its own.

Thus, I need to discuss a variety of formal languages on the following alphabet:

 ${\displaystyle {\underline {\underline {X}}}={\underline {\underline {X}}}_{\text{MON}}=\{~{}^{\backprime \backprime }~{}^{\prime \prime }~,~{}^{\backprime \backprime },{}^{\prime \prime }~,~{}^{\backprime \backprime }\{{}^{\prime \prime }~,~{}^{\backprime \backprime }\}{}^{\prime \prime }~\}.}$

Because references to an alphabet of punctuation marks can be difficult to process in the ordinary style of text, it helps to have alternative ways of naming these symbols.

First, I use raised angle brackets ${\displaystyle {}^{\langle }\ldots {}^{\rangle },}$ or supercilia, as alternate forms of quotation marks.

 ${\displaystyle {\underline {\underline {X}}}={\underline {\underline {X}}}_{\text{MON}}=\{~{}^{\langle }~{}^{\rangle }~,~{}^{\langle },{}^{\rangle }~,~{}^{\langle }\{{}^{\rangle }~,~{}^{\langle }\}{}^{\rangle }~\}.}$

Second, I use a collection of conventional names to refer to the symbols.

 ${\displaystyle {\underline {\underline {X}}}={\underline {\underline {X}}}_{\text{MON}}=\{{\text{blank}},{\text{comma}},{\text{lbrace}},{\text{rbrace}}\}.}$

Although it is possible to present this MON in a way that dispenses with blanks and commas, the more expansive language laid out here turns out to have capacities that are useful beyond this immediate context.

2. Reflection principles in propositional calculus. Many statements about the order are also statements in the order. Many statements in the order are already statements about the order.

3. Next, I consider a class of POSRs that turns up in group theory. [Variant] The next class of POSRs I want to discuss is one that arises in group theory.

4. Although it is seldom recognized, a similar form of self-reference appears in the study of group representations, and more generally, in the study of homomorphic representations of any mathematical structure. In particular, this type of ESR arises from the regular representation of a group in terms of its action on itself, that is, in the collection of effects that each element has on the all the individual elements of the group.

There are several ways to side-step the issue of self-reference in this situation. Typically, they are used in combination to avoid the problematic features of a self-referential procedure and thus to effectively rationalize the representation.

[Variant] As a preliminary study, it is useful to take up the slightly simpler brand of self-reference occurring in the topic of regular representations and to use it to make a first reconnaissance of the larger terrain.

[Variant] As a first foray into the area I use the topic of group representations to illustrate the theme of extra-constitutional self-reference. To provide the discussion with concrete material I examine a couple of small groups, picking examples that incidentally serve a double purpose and figure more substantially in a later stage of this project.

Each way of rationalizing the apparent self-reference begins by examining more carefully one of the features of the ostensibly circular formulation:

 ${\displaystyle x_{i}=\{(x_{1},~x_{1}\cdot x_{i}),\ldots ,(x_{n},~x_{n}\cdot x_{i})\}.}$
1. One approach examines the apparent equality of the expressions.
2. Another approach examines the nature of the objects that are invoked.

### Three Views of Systems

In this work I am using the word system in three different ways, in senses that refer to an object system (OS), a temporal system (TS), and a formal system (FS), respectively. This section describes these three ways of looking at a system, first in abstract isolation from each other, as though they reflected wholly separate species of systems, and then in concrete connection with each other, as the wholly apparent aspects of a single, underlying, systematic integrity. Finally, I close out the purely speculative parts of these considerations by showing how they come to bear on the present example, a collection of potentially meaningful actions pressed into the form of dialogue between ${\displaystyle {\text{A}}}$ and ${\displaystyle {\text{B}}.}$

1. An object system (OS) is an arbitrary collection of elements that present themselves to be of interest in a particular situation of inquiry. Formally, an OS is little more than a set. It represents a first attempt to unify a manifold of phenomena under a common concept, to aggregate the objects of discussion and thought that are relevant to the situation, and to include them in a general class. Typically, an OS begins as nothing more than a gathering together of actual or proposed objects. To serve its purpose, it need afford no more than an initial point of departure for staking out a tentative course of inquiry, and it can continue to be useful throughout inquiry, if only as a peg to hang new observations and contemplations on as the investigation proceeds.
2. A temporal system (TS) has states of being and the ability to move through sequences of states. Thus, it exists at a point in a space of states, undergoes transitions from state to state, and has the power, potential, or possibility of moving through various sequences of states. In doing this, the moment to moment existence of the typical TS sweeps out a characteristic succession of points in a space of states. When there is a definite constraint on the sequence of states that can occur, then one can begin to speak of a determinate, though not necessarily a deterministic dynamic process. In the sequel, the concept of a TS is used in an informal way, to refer to the most general kind of dynamic system conceivable, that is, an OS in which there is at least the barest notion of change or process that can serve to initiate discussion and that can continue to form the subject of further analysis.
3. A formal system (FS) contains the signs, expressions, and forms of argumentation that embody a particular way of talking and thinking about the objects in a designated OS. For the agent that uses a given FS, its design determines the way that these objects are perceived, described, and reasoned about, and the details of its constitution have consequences for all the processes of observation, contemplation, logical expression, articulate communication, and controlled action that it helps to mediate. Thus, the FS serves two main types of purposes: (a) As a formal language, it permits the articulation of an agent's observations with respect to the actual and proposed properties of an object system. (b) In addition, it embodies a system of practices, including techniques of argumentation, that are useful in representing reasoning about the properties and activities of the object system and that give the FS meaning and bearing with respect to the objective world.

There is a standard form of disclaimer that needs to be attached to this scheme of categories, qualifying any claim that it might be interpreted as making about the ontological status of the proposed distinctions. As often as not, the three categories of systems identified above do not correspond to materially different types of underlying entities so much as different stages in their development, or only in the development of discussions about them. As always, these distinctions do not reveal the essential categories and the substantial divergences of real systems so much as they reflect different ways of viewing them.

The need for a note of caution at this point is due to a persistent but unfortunate tendency of the symbol-using mentality, one that forms a potentially deleterious side effect to the necessary analytic capacity. Namely, having once discovered the many splendored facets of each real object worth looking into, the mind never ceases from trying to force its imagined categories of descriptive expressions (CODEs) down into the original categories of real entities (COREs). In spite of every contrary impression, the deeper-lying substrate of existence is solely responsible for funding the phenomenal appearances of the world.

Out of this tendency of the symbol-using mentality arises a constant difficulty with every theory of every reality. Namely, every use of a theoretical framework to view an underlying reality leads the user to forget, temporarily, that the reality is anything but its appearance, image, or representation in that framework. Logically speaking, there is an inalienable spectre of negation involved in every form of apparition, imagination, or representation. This abnegation would be complete if it were not for the possibility held out that some underlying realities may nevertheless be capable of representing themselves over time.

The relationship of objects in an underlying reality to their images in a theoretical framework is a topic that this discussion will return to repeatedly as the work progresses. In sum, for now, all of the following statements are approximations to the truth. At any given moment, the image is usually not the object. At times, it can almost be anything but the object. It is even entirely possible, oddly enough, that the image is nothing but the negation of the object, but as often as not it enjoys a more complex relationship than that of sheer opposition. Over time, in some instances, the image can become nearly indistinguishable from its object, but whether this is a good thing or not, in the long run, I cannot tell. The sense of the resulting identification, the bearing of the image on its object, depends on exactly how and how exactly this final coincidence comes about.

One of the goals of this work, indeed, of the whole pragmatic theory of sign relations, is an adequate understanding of the relationship between underlying reality objects and theoretical framework images. The purpose and also the criterion of an adequate understanding is this: It would prevent an interpretive agent, even while immersed in the context of a pertinent sign relation and deliberately taking part in a share of its conduct, from ever being confused again about the different roles of objects and images.

If one assumes that there is a unique and all-inclusive universe, and thus only one kind of system in essence that generates the phenomenon known as the whole objective world, then this integral form of universe is bound to enjoy all three aspects of systems phenomena in full measure. Then the task for a fully system-theoretic and reflective inquiry is to see how all of these aspects of systems can be integrated into a single mode of realization.

In many cases the three senses of the word system reflect distinctive orders of structure and function in the types of systems indicated, suggesting that there is something essential and substantive about the distinctions between objects, changes, and forms. With regard to the underlying reality, however, these differences can be as artificial as any that conventional language poses between nouns, verbs, and sentences. Of course, when the underlying system is degenerate, or not fully realized in all the relevant aspects, then it is fair to say that it falls under some categories more than others. In the general case, however, the three senses of the word system merely embody the spectrum of attitudes and intentions that observing and interpreting agents can take up with respect to the same underlying type of system.

An object system may seem little more than a set, the barest attempt to unify a manifold of interesting phenomena under a common concept, but no object system becomes an object of discussion and thought without invoking the informal precursors of formal systems, in other words, systems of practices, casually taken up, that reflection has the power to formalize in time. And any formal system, put to work in practice, has a temporal and dynamic aspect, especially in the transitions taking place from sign to interpretant sign that fill out its connotative component. Thus, a formal system implicitly involves a temporal system, even if its own object system is not itself temporal in nature but rests in a stable, a static, or an abstract state.

Formal systems and their systems of practice are subject to conversion into object systems, becoming the objects of higher order formal systems through the operation of a critical intellectual step usually called reflection.

Using the pragmatic theory of sign relations, I regard every object system in the context of a particular formal system. I take these two as one, for now, because a formal system and its object system are defined in relation to each other and are not really separable in practice. Later, I will discuss a form of independence that can exist between the two, but only in the derivative sense that many formal systems can be brought to bear on what turn out to be equivalent object systems.

Any physical system, subject to recognizably lawful constraints, can generally be turned to use as a channel of communication, contingent only on the limitations imposed by its inherent informational capacity. Therefore, any object system of sufficient capacity that resides under an agent's interpretive control can used as a medium for language and converted to convey the more specialized formal system.

In every situation the three kinds of system, or views of a system, are naturally related to each other through the concept of a sign relation. Applied in their turn, sign relations contain within themselves the germ of a particular idea, that no system can be called complete until it has the means to reflect on its own nature, at least in some measure. Thus, by integrating the three senses of the word system within the notion of a sign relation, I am trying to make it as easy as possible to move around in a space of apparently indispensable perspectives. To wit, regarding sign relations as formal objects in and of themselves, an intelligent agent needs the capacities: (1) to reflect on the objective forms of their phenomenal appearances, and (2) to participate in the active forms of their interpretive conduct. Further, an agent needs the flexibility to take up each of these stances toward sign relations at will, reflecting on them or joining in them as the situation demands.

I close this section by discussing the relationship among the three views of systems that are relevant to the example of ${\displaystyle {\text{A}}}$ and ${\displaystyle {\text{B}}.}$

[Variant] How do these three perspectives bear on the example of ${\displaystyle {\text{A}}}$ and ${\displaystyle {\text{B}}}$?

[Variant] In order to show how these three perspectives bear on the present inquiry, I will now discuss the relationship they exhibit in the example of ${\displaystyle {\text{A}}}$ and ${\displaystyle {\text{B}}.}$

In the present example, concerned with the form of communication that takes place between the interpreters ${\displaystyle {\text{A}}}$ and ${\displaystyle {\text{B}},}$ the topic of interest is not the type of dynamics that would change one of the original objects, ${\displaystyle {\text{A}}}$ or ${\displaystyle {\text{B}},}$ into the other. Thus, the object system is nothing more than the object domain ${\displaystyle O=\{{\text{A}},{\text{B}}\}}$ shared between the sign relations ${\displaystyle L({\text{A}})}$ and ${\displaystyle L({\text{B}}).}$ In this case, where the object system reduces to an abstract set, falling under the action of a trivial dynamics, one says that the object system is stable or static. In more developed examples, when the dynamics at the level of the object system becomes more interesting, the objects in the object system are usually referred to as objective configurations or object states. Later examples will take on object systems that enjoy significant variations in the sequences of their objective states.

### Building Bridges Between Representations

On the way to integrating dynamic and symbolic approaches to systems there is one important watershed that has to be crossed and recrossed, time and time again. This is a form of continental divide that decides between two alternative and exclusive modes of description (MODs) or categories of representation (CORs), and marks a writer's moment to moment selection of extensional representation (ER), on the one side, or intensional representation (IR), on the other. To apply the theme, in this section I address the task of building conceptual bridges between two different ways of describing or representing sign relations: (1) the ER that describes a sign relation in terms of its instances, and (2) the IR that describes a sign relation in terms of its properties.

It is best to begin the work of bridge-building on informal grounds, using concrete examples of ERs and IRs and taking advantage of basic ideas about their relationship that are readily available to every reader. After the overall scheme of construction is roughed out in this fashion, I plan to revisit the concept of representation in a more formal style, examining the balance of its in- and ex- “tensions” with a sharper eye to the relevant details and a greater chance of compassing the depths of form that arise between the two points of view.

The task of building this bridge is not trivial. In places, the basic elements of construction are yet to be forged from the available stocks, in others, the needed materials still lie in their ores, awaiting a suitable process to extract them, refine them, and bring them to a usable state. Due to the difficulties of this task and the length of time it will take to carry it out, I think it is advisable to establish two points of reference before setting to work.

1. As a way of providing sufficient motivation for the effort, I will indicate the importance of this bridge with respect to the aims of inquiry in general.
2. As a guard against a host of precipitous shortcuts that have been tried in the past, I will point out as clearly as possible a few of the obstacles that need to be surmounted. Once their structures are rightly understood, the obstructions that lie in the path of this bridge can be chalked up to experience with the reality of its construction, turned to use as stepping stones in the advance of its ultimate course, and given a fitting place in the progress of instruction.

Terms referring to properties of sign relations make it possible to formulate propositions about sign relations, either as occasioned by a clear and present example or in abstraction from any concrete instance. In turn, this makes it possible to carry on chains of reasoning about the properties of sign relations in detachment from the presence of actual cases that may or may not come to mind in the immediate present. This mode of abstraction, invoking the kind of IR that is involved in mediating every form of propositional reasoning, gives logic its wings and can lead to theories of great conceptual power, but it incurs the risk of leading reasoning astray into realms of irreferent pretension, eventually degenerating into spurious sounds that signify nothing.

It is only by means of an IR that logical reasoning, properly speaking, is able to begin. The stringency of this precept, if it is taken too strictly as a starting condition and applied solely in absolute terms, would be correctly perceived as demanding a provision that is jarring to every brand of good sense. But it was never meant to be taken this severely. In practice, the starkness of this tentative stipulation is moderated by the degree of fuzziness that still continues to reside in the interpretive distinction between ERs and IRs.

The alleged distinction between ERs and IRs, when it is projected to have a global application, remains arbitrary so long as it is taken at that level of abstraction, and it comes to take on the semblance of a definition only in relation to the interpretive conduct of a particular arbiter. No representation in actual practice is purely of one sort or the other, nor fails to have the characters of both types as a part of its mix. In other words, extensions and intensions are only abstractions from a profounder “tension” that is logically prior but functionally intermediate to them both, and every representation of any use will have its aspect of extensional particularity permeated by its aspect of intensional generality.

Toward the end of this construction I hope it will become clear that this bridge is a project intermediate in scale between the elementary linkage of signs to interpretants that is built into every sign relation and all the courses of conduct that go to span the gulf and build communication between vastly different systems of interpretation. In the meantime, there are strong analogies that make the architecture of this bridge parallel in form to the structures existing at both ends of the scale, shaping it in congruence with patterns of action that reside at both the micro and the macro levels. Observing these similarities and their lines of potential use as they arise will serve to guide the current work.

A sign relation is a complex object and its representations, insofar as they faithfully preserve its structure, are complex signs. Accordingly, the problems of translating between ERs and IRs of sign relations, of detecting when representations alleged to be of sign relations do indeed represent objects of the specified character, and of recognizing whether different representations do or do not represent the same sign relation as their common object — these are the familiar questions that would be asked of the signs and interpretants in a simple sign relation, but this time asked at a higher level, in regard to the complex signs and complex interpretants that are posed by the different stripes of representation. At the same time, it should be obvious that these are also the natural questions to be faced in building a bridge between representations.

How many different sorts of entities are conceivably involved in translating between ERs and IRs of sign relations? To address this question it helps to introduce a system of type notations that can be used to keep track of the various sorts of things, or the varieties of objects of thought, that are generated in the process of answering it. Table 47.1 summarizes the basic types of things that are needed in this pursuit, while the rest can be derived by constructions of the form ${\displaystyle X~\mathrm {of} ~Y,}$ notated ${\displaystyle X(Y)}$ or just ${\displaystyle XY,}$ for any basic types ${\displaystyle X}$ and ${\displaystyle Y.}$ The constructed types of things involved in the ERs and IRs of sign relations are listed in Tables 47.2 and 47.3, respectively.

 ${\displaystyle {\text{Type}}}$ ${\displaystyle {\text{Symbol}}}$ ${\displaystyle {\begin{array}{l}{\text{Property}}\\{\text{Sign}}\\{\text{Set}}\\{\text{Triple}}\\{\text{Underlying Element}}\end{array}}}$ ${\displaystyle {\begin{matrix}P\\{\underline {S}}\\S\\T\\U\end{matrix}}}$

 ${\displaystyle {\text{Type}}}$ ${\displaystyle {\text{Symbol}}}$ ${\displaystyle {\text{Construction}}}$ ${\displaystyle {\text{Relation}}}$ ${\displaystyle {R}}$ ${\displaystyle {S(T(U))}}$

 ${\displaystyle {\text{Type}}}$ ${\displaystyle {\text{Symbol}}}$ ${\displaystyle {\text{Construction}}}$ ${\displaystyle {\text{Relation}}}$ ${\displaystyle P(R)}$ ${\displaystyle P(S(T(U)))}$

Nothing as yet in this scheme of types says that all of the entities playing a part in the discussion are necessarily distinct, but only that there are this many roles to fill.

Let ${\displaystyle {\underline {S}}}$ be the type of signs, ${\displaystyle S}$ the type of sets, ${\displaystyle T}$ the type of triples, and ${\displaystyle U}$ the type of underlying objects. Now consider the various sorts of things, or the varieties of objects of thought, that are invoked on each side, annotating each type as it is mentioned:

ERs of sign relations describe them as sets ${\displaystyle (Ss)}$ of triples ${\displaystyle (Ts)}$ of underlying elements ${\displaystyle (Us).}$ This makes for three levels of objective structure that must be put in coordination with each other, a task that is projected to be carried out in the appropriate OF of sign relations. Corresponding to this aspect of structure in the OF, there is a parallel aspect of structure in the IF of sign relations. Namely, the accessory sign relations that are used to discuss a targeted sign relation need to have signs for sets ${\displaystyle {({\underline {S}}Ss)},}$ signs for triples ${\displaystyle {({\underline {S}}Ts)},}$ and signs for the underlying elements ${\displaystyle {({\underline {S}}Us)}.}$ This accounts for three levels of syntactic structure in the IF of sign relations that must be coordinated with each other and also with the targeted levels of objective structure.

[Variant] IRs of sign relations describe them in terms of properties ${\displaystyle (Ps)}$ that are taken as primitive entities in their own right. /// refer to properties ${\displaystyle (Ps)}$ of transactions ${\displaystyle (Ts)}$ of underlying elements ${\displaystyle (Us).}$

[Variant] IRs of sign relations refer to properties of sets ${\displaystyle (PSs),}$ properties of triples ${\displaystyle (PTs),}$ and properties of underlying elements ${\displaystyle (PUs).}$ This amounts to three more levels of objective structure in the OF of the IR that need to be coordinated with each other and interlaced with the OF of the ER if the two are to be brought into the same discussion, possibly for the purpose of translating either into the other. Accordingly, the accessory sign relations that are used to discuss an IR of a targeted sign relation need to have ${\displaystyle {\underline {S}}PSs,}$ ${\displaystyle {\underline {S}}PTs,}$ and ${\displaystyle {\underline {S}}PUs.}$

### Extensional Representations of Sign Relations

Up to this point, the concept of a sign relation has been discussed largely in terms of ERs. The sign relations ${\displaystyle L({\text{A}})}$ and ${\displaystyle L({\text{B}})}$ were initially described as collections of transactions among three participants and formalized as sets of triples of underlying elements.

Other examples of ERs are widely distributed throughout the foregoing discussion of ${\displaystyle {\text{A}}}$ and ${\displaystyle {\text{B}}.}$ The extensional mode of description is prevalent, not only in the presentation of sign relations by means of relational data tables, but also in the presentation of dyadic projections by means of digraphs. This manner of presentation follows the natural order of acquaintance with abstract relations, since the extensional mode of description is the category of representation that usually prevails whenever it is necessary to provide a detailed treatment of simple examples or an exhaustive account of individual instances.

Starting from a standpoint in concrete constructions, the easiest way to begin developing an explicit treatment of ERs is to gather the relevant materials in the forms already presented, to fill out the missing details and expand the abbreviated contents of these forms, and to review their full structures in a more formal light. Consequently, this section inaugurates the formal discussion of ERs by taking a second look at the interpreters ${\displaystyle {\text{A}}}$ and ${\displaystyle {\text{B}},}$ recollecting the Tables of their sign relations and finishing up the Tables of their dyadic components. Since the form of the sign relations ${\displaystyle L({\text{A}})}$ and ${\displaystyle L({\text{B}})}$ no longer presents any novelty, I can exploit their second presentation as a first opportunity to examine a selection of finer points, previously overlooked. Also, in the process of reviewing this material it is useful to anticipate a number of incidental issues that are reaching the point of becoming critical within this discussion and to begin introducing the generic types of technical devices that are needed to deal with them.

The next set of Tables summarizes the ERs of ${\displaystyle L({\text{A}})}$ and ${\displaystyle L({\text{B}}).}$ For ease of reference, Tables 48.1 and 49.1 repeat the contents of Tables 1 and 2, respectively, the only difference being that appearances of ordinary quotation marks ${\displaystyle ({}^{\backprime \backprime }\ldots {}^{\prime \prime })}$ are transcribed as invocations of the arch operator ${\displaystyle ({}^{\langle }\ldots {}^{\rangle }).}$ The reason for this slight change of notation will be explained shortly. The denotative components ${\displaystyle \mathrm {Den} ({\text{A}})}$ and ${\displaystyle \mathrm {Den} ({\text{B}})}$ are shown in the first two columns of Tables 48.2 and 49.2, respectively, while the third column gives the transition from sign to object as an ordered pair ${\displaystyle (s,o).}$ The connotative components ${\displaystyle \mathrm {Con} ({\text{A}})}$ and ${\displaystyle \mathrm {Con} ({\text{B}})}$ are shown in the first two columns of Tables 48.3 and 49.3, respectively, while the third column gives the transition from sign to interpretant as an ordered pair ${\displaystyle (s,i).}$

 ${\displaystyle {\text{Object}}}$ ${\displaystyle {\text{Sign}}}$ ${\displaystyle {\text{Interpretant}}}$ ${\displaystyle {\begin{matrix}{\text{A}}\\{\text{A}}\\{\text{A}}\\{\text{A}}\end{matrix}}}$ ${\displaystyle {\begin{matrix}{}^{\langle }{\text{A}}{}^{\rangle }\\{}^{\langle }{\text{A}}{}^{\rangle }\\{}^{\langle }{\text{i}}{}^{\rangle }\\{}^{\langle }{\text{i}}{}^{\rangle }\end{matrix}}}$ ${\displaystyle {\begin{matrix}{}^{\langle }{\text{A}}{}^{\rangle }\\{}^{\langle }{\text{i}}{}^{\rangle }\\{}^{\langle }{\text{A}}{}^{\rangle }\\{}^{\langle }{\text{i}}{}^{\rangle }\end{matrix}}}$ ${\displaystyle {\begin{matrix}{\text{B}}\\{\text{B}}\\{\text{B}}\\{\text{B}}\end{matrix}}}$ ${\displaystyle {\begin{matrix}{}^{\langle }{\text{B}}{}^{\rangle }\\{}^{\langle }{\text{B}}{}^{\rangle }\\{}^{\langle }{\text{u}}{}^{\rangle }\\{}^{\langle }{\text{u}}{}^{\rangle }\end{matrix}}}$ ${\displaystyle {\begin{matrix}{}^{\langle }{\text{B}}{}^{\rangle }\\{}^{\langle }{\text{u}}{}^{\rangle }\\{}^{\langle }{\text{B}}{}^{\rangle }\\{}^{\langle }{\text{u}}{}^{\rangle }\end{matrix}}}$

 ${\displaystyle {\text{Object}}}$ ${\displaystyle {\text{Sign}}}$ ${\displaystyle {\text{Transition}}}$ ${\displaystyle {\begin{matrix}{\text{A}}\\{\text{A}}\end{matrix}}}$ ${\displaystyle {\begin{matrix}{}^{\langle }{\text{A}}{}^{\rangle }\\{}^{\langle }{\text{i}}{}^{\rangle }\end{matrix}}}$ ${\displaystyle {\begin{matrix}({}^{\langle }{\text{A}}{}^{\rangle },{\text{A}})\\({}^{\langle }{\text{i}}{}^{\rangle },{\text{A}})\end{matrix}}}$ ${\displaystyle {\begin{matrix}{\text{B}}\\{\text{B}}\end{matrix}}}$ ${\displaystyle {\begin{matrix}{}^{\langle }{\text{B}}{}^{\rangle }\\{}^{\langle }{\text{u}}{}^{\rangle }\end{matrix}}}$ ${\displaystyle {\begin{matrix}({}^{\langle }{\text{B}}{}^{\rangle },{\text{B}})\\({}^{\langle }{\text{u}}{}^{\rangle },{\text{B}})\end{matrix}}}$

 ${\displaystyle {\text{Sign}}}$ ${\displaystyle {\text{Interpretant}}}$ ${\displaystyle {\text{Transition}}}$ ${\displaystyle {\begin{matrix}{}^{\langle }{\text{A}}{}^{\rangle }\\{}^{\langle }{\text{A}}{}^{\rangle }\\{}^{\langle }{\text{i}}{}^{\rangle }\\{}^{\langle }{\text{i}}{}^{\rangle }\end{matrix}}}$ ${\displaystyle {\begin{matrix}{}^{\langle }{\text{A}}{}^{\rangle }\\{}^{\langle }{\text{i}}{}^{\rangle }\\{}^{\langle }{\text{A}}{}^{\rangle }\\{}^{\langle }{\text{i}}{}^{\rangle }\end{matrix}}}$ ${\displaystyle {\begin{matrix}({}^{\langle }{\text{A}}{}^{\rangle },{}^{\langle }{\text{A}}{}^{\rangle })\\({}^{\langle }{\text{A}}{}^{\rangle },{}^{\langle }{\text{i}}{}^{\rangle })\\({}^{\langle }{\text{i}}{}^{\rangle },{}^{\langle }{\text{A}}{}^{\rangle })\\({}^{\langle }{\text{i}}{}^{\rangle },{}^{\langle }{\text{i}}{}^{\rangle })\end{matrix}}}$ ${\displaystyle {\begin{matrix}{}^{\langle }{\text{B}}{}^{\rangle }\\{}^{\langle }{\text{B}}{}^{\rangle }\\{}^{\langle }{\text{u}}{}^{\rangle }\\{}^{\langle }{\text{u}}{}^{\rangle }\end{matrix}}}$ ${\displaystyle {\begin{matrix}{}^{\langle }{\text{B}}{}^{\rangle }\\{}^{\langle }{\text{u}}{}^{\rangle }\\{}^{\langle }{\text{B}}{}^{\rangle }\\{}^{\langle }{\text{u}}{}^{\rangle }\end{matrix}}}$ ${\displaystyle {\begin{matrix}({}^{\langle }{\text{B}}{}^{\rangle },{}^{\langle }{\text{B}}{}^{\rangle })\\({}^{\langle }{\text{B}}{}^{\rangle },{}^{\langle }{\text{u}}{}^{\rangle })\\({}^{\langle }{\text{u}}{}^{\rangle },{}^{\langle }{\text{B}}{}^{\rangle })\\({}^{\langle }{\text{u}}{}^{\rangle },{}^{\langle }{\text{u}}{}^{\rangle })\end{matrix}}}$

 ${\displaystyle {\text{Object}}}$ ${\displaystyle {\text{Sign}}}$ ${\displaystyle {\text{Interpretant}}}$ ${\displaystyle {\begin{matrix}{\text{A}}\\{\text{A}}\\{\text{A}}\\{\text{A}}\end{matrix}}}$ ${\displaystyle {\begin{matrix}{}^{\langle }{\text{A}}{}^{\rangle }\\{}^{\langle }{\text{A}}{}^{\rangle }\\{}^{\langle }{\text{u}}{}^{\rangle }\\{}^{\langle }{\text{u}}{}^{\rangle }\end{matrix}}}$ ${\displaystyle {\begin{matrix}{}^{\langle }{\text{A}}{}^{\rangle }\\{}^{\langle }{\text{u}}{}^{\rangle }\\{}^{\langle }{\text{A}}{}^{\rangle }\\{}^{\langle }{\text{u}}{}^{\rangle }\end{matrix}}}$ ${\displaystyle {\begin{matrix}{\text{B}}\\{\text{B}}\\{\text{B}}\\{\text{B}}\end{matrix}}}$ ${\displaystyle {\begin{matrix}{}^{\langle }{\text{B}}{}^{\rangle }\\{}^{\langle }{\text{B}}{}^{\rangle }\\{}^{\langle }{\text{i}}{}^{\rangle }\\{}^{\langle }{\text{i}}{}^{\rangle }\end{matrix}}}$ ${\displaystyle {\begin{matrix}{}^{\langle }{\text{B}}{}^{\rangle }\\{}^{\langle }{\text{i}}{}^{\rangle }\\{}^{\langle }{\text{B}}{}^{\rangle }\\{}^{\langle }{\text{i}}{}^{\rangle }\end{matrix}}}$

 ${\displaystyle {\text{Object}}}$ ${\displaystyle {\text{Sign}}}$ ${\displaystyle {\text{Transition}}}$ ${\displaystyle {\begin{matrix}{\text{A}}\\{\text{A}}\end{matrix}}}$ ${\displaystyle {\begin{matrix}{}^{\langle }{\text{A}}{}^{\rangle }\\{}^{\langle }{\text{u}}{}^{\rangle }\end{matrix}}}$ ${\displaystyle {\begin{matrix}({}^{\langle }{\text{A}}{}^{\rangle },{\text{A}})\\({}^{\langle }{\text{u}}{}^{\rangle },{\text{A}})\end{matrix}}}$ ${\displaystyle {\begin{matrix}{\text{B}}\\{\text{B}}\end{matrix}}}$ ${\displaystyle {\begin{matrix}{}^{\langle }{\text{B}}{}^{\rangle }\\{}^{\langle }{\text{i}}{}^{\rangle }\end{matrix}}}$ ${\displaystyle {\begin{matrix}({}^{\langle }{\text{B}}{}^{\rangle },{\text{B}})\\({}^{\langle }{\text{i}}{}^{\rangle },{\text{B}})\end{matrix}}}$

 ${\displaystyle {\text{Sign}}}$ ${\displaystyle {\text{Interpretant}}}$ ${\displaystyle {\text{Transition}}}$ ${\displaystyle {\begin{matrix}{}^{\langle }{\text{A}}{}^{\rangle }\\{}^{\langle }{\text{A}}{}^{\rangle }\\{}^{\langle }{\text{u}}{}^{\rangle }\\{}^{\langle }{\text{u}}{}^{\rangle }\end{matrix}}}$ ${\displaystyle {\begin{matrix}{}^{\langle }{\text{A}}{}^{\rangle }\\{}^{\langle }{\text{u}}{}^{\rangle }\\{}^{\langle }{\text{A}}{}^{\rangle }\\{}^{\langle }{\text{u}}{}^{\rangle }\end{matrix}}}$ ${\displaystyle {\begin{matrix}({}^{\langle }{\text{A}}{}^{\rangle },{}^{\langle }{\text{A}}{}^{\rangle })\\({}^{\langle }{\text{A}}{}^{\rangle },{}^{\langle }{\text{u}}{}^{\rangle })\\({}^{\langle }{\text{u}}{}^{\rangle },{}^{\langle }{\text{A}}{}^{\rangle })\\({}^{\langle }{\text{u}}{}^{\rangle },{}^{\langle }{\text{u}}{}^{\rangle })\end{matrix}}}$ ${\displaystyle {\begin{matrix}{}^{\langle }{\text{B}}{}^{\rangle }\\{}^{\langle }{\text{B}}{}^{\rangle }\\{}^{\langle }{\text{i}}{}^{\rangle }\\{}^{\langle }{\text{i}}{}^{\rangle }\end{matrix}}}$ ${\displaystyle {\begin{matrix}{}^{\langle }{\text{B}}{}^{\rangle }\\{}^{\langle }{\text{i}}{}^{\rangle }\\{}^{\langle }{\text{B}}{}^{\rangle }\\{}^{\langle }{\text{i}}{}^{\rangle }\end{matrix}}}$ ${\displaystyle {\begin{matrix}({}^{\langle }{\text{B}}{}^{\rangle },{}^{\langle }{\text{B}}{}^{\rangle })\\({}^{\langle }{\text{B}}{}^{\rangle },{}^{\langle }{\text{i}}{}^{\rangle })\\({}^{\langle }{\text{i}}{}^{\rangle },{}^{\langle }{\text{B}}{}^{\rangle })\\({}^{\langle }{\text{i}}{}^{\rangle },{}^{\langle }{\text{i}}{}^{\rangle })\end{matrix}}}$

### Intensional Representations of Sign Relations

The next three sections consider how the ERs of ${\displaystyle L({\text{A}})}$ and ${\displaystyle L({\text{B}})}$ can be translated into a variety of different IRs. For the purposes of this introduction, only “faithful” translations between the different categories of representation are contemplated. This means that the conversion from ER to IR is intended to convey what is essentially the same information about ${\displaystyle L({\text{A}})}$ and ${\displaystyle L({\text{B}}),}$ to preserve all the relevant structural details implied by their various modes of description, but to do it in a way that brings selected aspects of their objective forms to light. General considerations surrounding the task of translation are taken up in this section, while the next two sections lay out different ways of carrying it through.

The larger purpose of this discussion is to serve as an introduction, not just to the special topic of devising IRs for sign relations, but to the general issue of producing, using, and comprehending IRs for any kind of relation or any domain of formal objects. It is hoped that a careful study of these simple IRs can inaugurate a degree of insight into the broader arenas of formalism of which they occupy an initial niche and into the wider landscapes of discourse of which they inhabit a natural corner, in time progressing up to the axiomatic presentation of formal theories about combinatorial domains and other mathematical objects.

For the sake of maximum clarity and reusability of results, I begin by articulating the abstract skeleton of the paradigm structure, treating the sign relations ${\displaystyle L({\text{A}})}$ and ${\displaystyle L({\text{B}})}$ as sundry aspects of a single, unitary, but still uninterpreted object. Then I return at various successive stages to differentiate and individualize the two interpreters, to arrange more functional flesh on the basis provided by their structural bones, and to illustrate how their bare forms can be arrayed in many different styles of qualitative detail.

In building connections between ERs and IRs of sign relations the discussion turns on two types of partially ordered sets, or posets. Suppose that ${\displaystyle L}$ is one of the sign relations ${\displaystyle L({\text{A}})}$ and ${\displaystyle L({\text{B}}),}$ and let ${\displaystyle \mathrm {ER} (L)}$ be an ER of ${\displaystyle L.}$

In the sign relations ${\displaystyle L({\text{A}})}$ and ${\displaystyle L({\text{B}}),}$ both of their ERs are based on a common world set:

 ${\displaystyle {\begin{array}{*{15}{c}}W&=&\{&{\text{A}}&,&{\text{B}}&,&{}^{\backprime \backprime }{\text{A}}{}^{\prime \prime }&,&{}^{\backprime \backprime }{\text{B}}{}^{\prime \prime }&,&{}^{\backprime \backprime }{\text{i}}{}^{\prime \prime }&,&{}^{\backprime \backprime }{\text{u}}{}^{\prime \prime }&\}\\&=&\{&w_{1}&,&w_{2}&,&w_{3}&,&w_{4}&,&w_{5}&,&w_{6}&\}\end{array}}}$

An IR of any object is a description of that object in terms of its properties. A successful description of a particular object usually involves a selection of properties, those that are relevant to a particular purpose. An IR of ${\displaystyle L({\text{A}})}$ or ${\displaystyle L({\text{B}})}$ involves properties of its elementary points ${\displaystyle w\in W}$ and properties of its elementary relations ${\displaystyle \ell \in O\times S\times I.}$

To devise an IR of any relation ${\displaystyle L}$ one needs to describe ${\displaystyle L}$ in terms of properties of its ingredients. Broadly speaking, the ingredients of a relation include its elementary relations or ${\displaystyle n}$-tuples and the elementary components of these ${\displaystyle n}$-tuples that reside in the relational domains.

The poset ${\displaystyle \mathrm {Pos} (W)}$ of interest here is the power set ${\displaystyle {\mathcal {P}}(W)=\mathrm {Pow} (W).}$

The elements of these posets are abstractly regarded as properties or propositions that apply to the elements of ${\displaystyle W.}$ These properties and propositions are independently given entities. In other words, they are primitive elements in their own right, and cannot in general be defined in terms of points, but they exist in relation to these points, and their extensions can be represented as sets of points.

[Variant] For a variety of foundational reasons that I do not fully understand, perhaps most of all because theoretically given structures have their real foundations outside the realm of theory, in empirically given structures, it is best to regard points, properties, and propositions as equally primitive elements, related to each other but not defined in terms of each other, analogous to the undefined elements of a geometry.

[Variant] There is a foundational issue arising in this context that I do not pretend to fully understand and cannot attempt to finally dispatch. What I do understand I will try to express in terms of an aesthetic principle: On balance, it seems best to regard extensional elements and intensional features as independently given entities. This involves treating points and properties as fundamental realities in their own rights, placing them on an equal basis with each other, and seeking their relation to each other, but not trying to reduce one to the other.

The discussion is now specialized to consider the IRs of the sign relations ${\displaystyle L({\text{A}})}$ and ${\displaystyle L({\text{B}}),}$ their denotative projections as the digraphs ${\displaystyle \mathrm {Den} (L_{\text{A}})}$ and ${\displaystyle \mathrm {Den} (L_{\text{B}}),}$ and their connotative projections as the digraphs ${\displaystyle \mathrm {Con} (L_{\text{A}})}$ and ${\displaystyle \mathrm {Con} (L_{\text{B}}).}$ In doing this I take up two different strategies of representation:

1. The first strategy is called the literal coding, because it sticks to obvious features of each syntactic element to contrive its code, or the ${\displaystyle {{\mathcal {O}}(n)}}$ coding, because it uses a number on the order of ${\displaystyle n}$ logical features to represent a domain of ${\displaystyle n}$ elements.
2. The second strategy is called the analytic coding, because it attends to the nuances of each sign's interpretation to fashion its code, or the ${\displaystyle \log(n)}$ coding, because it uses roughly ${\displaystyle \log _{2}(n)}$ binary features to represent a domain of ${\displaystyle n}$ elements.

### Literal Intensional Representations

In this section I prepare the grounds for building bridges between ERs and IRs of sign relations. To establish an initial foothold on either side of the distinction and to gain a first march on connecting the two sites of the intended construction, I introduce an intermediate mode of description called a literal intensional representation (LIR).

Any LIR is a nominal form of IR that has exactly the same level of detail as an ER, merely shifting the interpretation of primitive terms from an extensional to an intensional modality, namely, from a frame of reference terminating in points, atomic elements, elementary objects, or real particulars to a frame of reference terminating in qualities, basic features, fundamental properties, or simple propositions. This modification, that translates the entire set of elementary objects in an ER into a parallel set of fundamental properties in a LIR, constitutes a form of modulation that can be subtle or trivial, depending on one's point of view. Regarded as trivial, it tends to go unmarked, leaving it up to the judgment of the interpreter to decide whether the same sign is meant to denote a point, a particular, a property, or a proposition. An interpretive variance that goes unstated tends to be treated as final. It is always possible to bring in more signs in an attempt to signify the variants intended, but it needs to be noted that every effort to control the interpretive variance by means of these epithets and expletives only increases the level of liability for accidental errors, if not the actual probability of misinterpretation. For the sake of this introduction, and in spite of these risks, I treat the distinction between extensional and intensional modes of interpretation as worthy of note and deserving of an explicit notation.

 ${\displaystyle {\text{Object}}}$ ${\displaystyle {\text{Sign of Object}}}$ ${\displaystyle {\begin{matrix}{\text{A}}&{\text{A}}&w_{1}\\[6pt]{\text{B}}&{\text{B}}&w_{2}\\[12pt]{}^{\backprime \backprime }{\text{A}}{}^{\prime \prime }&{}^{\langle }{\text{A}}{}^{\rangle }&w_{3}\\[6pt]{}^{\backprime \backprime }{\text{B}}{}^{\prime \prime }&{}^{\langle }{\text{B}}{}^{\rangle }&w_{4}\\[6pt]{}^{\backprime \backprime }{\text{i}}{}^{\prime \prime }&{}^{\langle }{\text{i}}{}^{\rangle }&w_{5}\\[6pt]{}^{\backprime \backprime }{\text{u}}{}^{\prime \prime }&{}^{\langle }{\text{u}}{}^{\rangle }&w_{6}\end{matrix}}}$ ${\displaystyle {\begin{matrix}{}^{\langle }{\text{A}}{}^{\rangle }&{}^{\langle }{\text{A}}{}^{\rangle }&{}^{\langle }w_{1}{}^{\rangle }\\[6pt]{}^{\langle }{\text{B}}{}^{\rangle }&{}^{\langle }{\text{B}}{}^{\rangle }&{}^{\langle }w_{2}{}^{\rangle }\\[12pt]{}^{\langle \backprime \backprime }{\text{A}}{}^{\prime \prime \rangle }&{}^{\langle \langle }{\text{A}}{}^{\rangle \rangle }&{}^{\langle }w_{3}{}^{\rangle }\\[6pt]{}^{\langle \backprime \backprime }{\text{B}}{}^{\prime \prime \rangle }&{}^{\langle \langle }{\text{B}}{}^{\rangle \rangle }&{}^{\langle }w_{4}{}^{\rangle }\\[6pt]{}^{\langle \backprime \backprime }{\text{i}}{}^{\prime \prime \rangle }&{}^{\langle \langle }{\text{i}}{}^{\rangle \rangle }&{}^{\langle }w_{5}{}^{\rangle }\\[6pt]{}^{\langle \backprime \backprime }{\text{u}}{}^{\prime \prime \rangle }&{}^{\langle \langle }{\text{u}}{}^{\rangle \rangle }&{}^{\langle }w_{6}{}^{\rangle }\end{matrix}}}$

 ${\displaystyle {\text{Property}}}$ ${\displaystyle {\text{Sign of Property}}}$ ${\displaystyle {\begin{matrix}{}^{\lbrace }{\text{A}}{}^{\rbrace }&{}^{\lbrace }{\text{A}}{}^{\rbrace }&{}^{\lbrace }w_{1}{}^{\rbrace }\\[6pt]{}^{\lbrace }{\text{B}}{}^{\rbrace }&{}^{\lbrace }{\text{B}}{}^{\rbrace }&{}^{\lbrace }w_{2}{}^{\rbrace }\\[12pt]{}^{\lbrace \backprime \backprime }{\text{A}}{}^{\prime \prime \rbrace }&{}^{\lbrace \langle }{\text{A}}{}^{\rangle \rbrace }&{}^{\lbrace }w_{3}{}^{\rbrace }\\[6pt]{}^{\lbrace \backprime \backprime }{\text{B}}{}^{\prime \prime \rbrace }&{}^{\lbrace \langle }{\text{B}}{}^{\rangle \rbrace }&{}^{\lbrace }w_{4}{}^{\rbrace }\\[6pt]{}^{\lbrace \backprime \backprime }{\text{i}}{}^{\prime \prime \rbrace }&{}^{\lbrace \langle }{\text{i}}{}^{\rangle \rbrace }&{}^{\lbrace }w_{5}{}^{\rbrace }\\[6pt]{}^{\lbrace \backprime \backprime }{\text{u}}{}^{\prime \prime \rbrace }&{}^{\lbrace \langle }{\text{u}}{}^{\rangle \rbrace }&{}^{\lbrace }w_{6}{}^{\rbrace }\end{matrix}}}$ ${\displaystyle {\begin{matrix}{}^{\langle \lbrace }{\text{A}}{}^{\rbrace \rangle }&{}^{\langle \lbrace }{\text{A}}{}^{\rbrace \rangle }&{}^{\langle \lbrace }w_{1}{}^{\rbrace \rangle }\\[6pt]{}^{\langle \lbrace }{\text{B}}{}^{\rbrace \rangle }&{}^{\langle \lbrace }{\text{B}}{}^{\rbrace \rangle }&{}^{\langle \lbrace }w_{2}{}^{\rbrace \rangle }\\[12pt]{}^{\langle \lbrace \backprime \backprime }{\text{A}}{}^{\prime \prime \rbrace \rangle }&{}^{\langle \lbrace \langle }{\text{A}}{}^{\rangle \rbrace \rangle }&{}^{\langle \lbrace }w_{3}{}^{\rbrace \rangle }\\[6pt]{}^{\langle \lbrace \backprime \backprime }{\text{B}}{}^{\prime \prime \rbrace \rangle }&{}^{\langle \lbrace \langle }{\text{B}}{}^{\rangle \rbrace \rangle }&{}^{\langle \lbrace }w_{4}{}^{\rbrace \rangle }\\[6pt]{}^{\langle \lbrace \backprime \backprime }{\text{i}}{}^{\prime \prime \rbrace \rangle }&{}^{\langle \lbrace \langle }{\text{i}}{}^{\rangle \rbrace \rangle }&{}^{\langle \lbrace }w_{5}{}^{\rbrace \rangle }\\[6pt]{}^{\langle \lbrace \backprime \backprime }{\text{u}}{}^{\prime \prime \rbrace \rangle }&{}^{\langle \lbrace \langle }{\text{u}}{}^{\rangle \rbrace \rangle }&{}^{\langle \lbrace }w_{6}{}^{\rbrace \rangle }\end{matrix}}}$

 ${\displaystyle {\text{Property}}}$ ${\displaystyle {\text{Sign of Property}}}$ ${\displaystyle {\begin{matrix}{\underline {\underline {\text{A}}}}&{\underline {\underline {\text{A}}}}&{\underline {\underline {w_{1}}}}\\[6pt]{\underline {\underline {\text{B}}}}&{\underline {\underline {\text{B}}}}&{\underline {\underline {w_{2}}}}\\[12pt]{\underline {\underline {{}^{\backprime \backprime }{\text{A}}{}^{\prime \prime }}}}&{\underline {\underline {{}^{\langle }{\text{A}}{}^{\rangle }}}}&{\underline {\underline {w_{3}}}}\\[6pt]{\underline {\underline {{}^{\backprime \backprime }{\text{B}}{}^{\prime \prime }}}}&{\underline {\underline {{}^{\langle }{\text{B}}{}^{\rangle }}}}&{\underline {\underline {w_{4}}}}\\[6pt]{\underline {\underline {{}^{\backprime \backprime }{\text{i}}{}^{\prime \prime }}}}&{\underline {\underline {{}^{\langle }{\text{i}}{}^{\rangle }}}}&{\underline {\underline {w_{5}}}}\\[6pt]{\underline {\underline {{}^{\backprime \backprime }{\text{u}}{}^{\prime \prime }}}}&{\underline {\underline {{}^{\langle }{\text{u}}{}^{\rangle }}}}&{\underline {\underline {w_{6}}}}\end{matrix}}}$ ${\displaystyle {\begin{matrix}{}^{\langle }{\underline {\underline {\text{A}}}}{}^{\rangle }&{}^{\langle }{\underline {\underline {\text{A}}}}{}^{\rangle }&{}^{\langle }{\underline {\underline {w_{1}}}}{}^{\rangle }\\[6pt]{}^{\langle }{\underline {\underline {\text{B}}}}{}^{\rangle }&{}^{\langle }{\underline {\underline {\text{B}}}}{}^{\rangle }&{}^{\langle }{\underline {\underline {w_{2}}}}{}^{\rangle }\\[12pt]{}^{\langle }{\underline {\underline {{}^{\backprime \backprime }{\text{A}}{}^{\prime \prime }}}}{}^{\rangle }&{}^{\langle }{\underline {\underline {{}^{\langle }{\text{A}}{}^{\rangle }}}}{}^{\rangle }&{}^{\langle }{\underline {\underline {w_{3}}}}{}^{\rangle }\\[6pt]{}^{\langle }{\underline {\underline {{}^{\backprime \backprime }{\text{B}}{}^{\prime \prime }}}}{}^{\rangle }&{}^{\langle }{\underline {\underline {{}^{\langle }{\text{B}}{}^{\rangle }}}}{}^{\rangle }&{}^{\langle }{\underline {\underline {w_{4}}}}{}^{\rangle }\\[6pt]{}^{\langle }{\underline {\underline {{}^{\backprime \backprime }{\text{i}}{}^{\prime \prime }}}}{}^{\rangle }&{}^{\langle }{\underline {\underline {{}^{\langle }{\text{i}}{}^{\rangle }}}}{}^{\rangle }&{}^{\langle }{\underline {\underline {w_{5}}}}{}^{\rangle }\\[6pt]{}^{\langle }{\underline {\underline {{}^{\backprime \backprime }{\text{u}}{}^{\prime \prime }}}}{}^{\rangle }&{}^{\langle }{\underline {\underline {{}^{\langle }{\text{u}}{}^{\rangle }}}}{}^{\rangle }&{}^{\langle }{\underline {\underline {w_{6}}}}{}^{\rangle }\end{matrix}}}$

 ${\displaystyle {\text{Property}}}$ ${\displaystyle {\text{Sign of Property}}}$ ${\displaystyle {\begin{matrix}{\underline {\underline {\text{A}}}}&{\underline {\underline {o_{1}}}}&{\underline {\underline {w_{1}}}}\\[6pt]{\underline {\underline {\text{B}}}}&{\underline {\underline {o_{2}}}}&{\underline {\underline {w_{2}}}}\\[12pt]{\underline {\underline {\text{a}}}}&{\underline {\underline {s_{1}}}}&{\underline {\underline {w_{3}}}}\\[6pt]{\underline {\underline {\text{b}}}}&{\underline {\underline {s_{2}}}}&{\underline {\underline {w_{4}}}}\\[6pt]{\underline {\underline {\text{i}}}}&{\underline {\underline {s_{3}}}}&{\underline {\underline {w_{5}}}}\\[6pt]{\underline {\underline {\text{u}}}}&{\underline {\underline {s_{4}}}}&{\underline {\underline {w_{6}}}}\end{matrix}}}$ ${\displaystyle {\begin{matrix}{}^{\langle }{\underline {\underline {\text{A}}}}{}^{\rangle }&{}^{\langle }{\underline {\underline {o_{1}}}}{}^{\rangle }&{}^{\langle }{\underline {\underline {w_{1}}}}{}^{\rangle }\\[6pt]{}^{\langle }{\underline {\underline {\text{B}}}}{}^{\rangle }&{}^{\langle }{\underline {\underline {o_{2}}}}{}^{\rangle }&{}^{\langle }{\underline {\underline {w_{2}}}}{}^{\rangle }\\[12pt]{}^{\langle }{\underline {\underline {\text{a}}}}{}^{\rangle }&{}^{\langle }{\underline {\underline {s_{1}}}}{}^{\rangle }&{}^{\langle }{\underline {\underline {w_{3}}}}{}^{\rangle }\\[6pt]{}^{\langle }{\underline {\underline {\text{b}}}}{}^{\rangle }&{}^{\langle }{\underline {\underline {s_{2}}}}{}^{\rangle }&{}^{\langle }{\underline {\underline {w_{4}}}}{}^{\rangle }\\[6pt]{}^{\langle }{\underline {\underline {\text{i}}}}{}^{\rangle }&{}^{\langle }{\underline {\underline {s_{3}}}}{}^{\rangle }&{}^{\langle }{\underline {\underline {w_{5}}}}{}^{\rangle }\\[6pt]{}^{\langle }{\underline {\underline {\text{u}}}}{}^{\rangle }&{}^{\langle }{\underline {\underline {s_{4}}}}{}^{\rangle }&{}^{\langle }{\underline {\underline {w_{6}}}}{}^{\rangle }\end{matrix}}}$

 ${\displaystyle {\text{Instance}}}$ ${\displaystyle {\text{Sign of Instance}}}$ ${\displaystyle {\begin{matrix}{}^{\lbrack }{\text{A}}{}^{\rbrack }&{}^{\lbrack }{\text{A}}{}^{\rbrack }&{}^{\lbrack }w_{1}{}^{\rbrack }\\[6pt]{}^{\lbrack }{\text{B}}{}^{\rbrack }&{}^{\lbrack }{\text{B}}{}^{\rbrack }&{}^{\lbrack }w_{2}{}^{\rbrack }\\[12pt]{}^{\lbrack \backprime \backprime }{\text{A}}{}^{\prime \prime \rbrack }&{}^{\lbrack \langle }{\text{A}}{}^{\rangle \rbrack }&{}^{\lbrack }w_{3}{}^{\rbrack }\\[6pt]{}^{\lbrack \backprime \backprime }{\text{B}}{}^{\prime \prime \rbrack }&{}^{\lbrack \langle }{\text{B}}{}^{\rangle \rbrack }&{}^{\lbrack }w_{4}{}^{\rbrack }\\[6pt]{}^{\lbrack \backprime \backprime }{\text{i}}{}^{\prime \prime \rbrack }&{}^{\lbrack \langle }{\text{i}}{}^{\rangle \rbrack }&{}^{\lbrack }w_{5}{}^{\rbrack }\\[6pt]{}^{\lbrack \backprime \backprime }{\text{u}}{}^{\prime \prime \rbrack }&{}^{\lbrack \langle }{\text{u}}{}^{\rangle \rbrack }&{}^{\lbrack }w_{6}{}^{\rbrack }\end{matrix}}}$ ${\displaystyle {\begin{matrix}{}^{\langle \lbrack }{\text{A}}{}^{\rbrack \rangle }&{}^{\langle \lbrack }{\text{A}}{}^{\rbrack \rangle }&{}^{\langle \lbrack }w_{1}{}^{\rbrack \rangle }\\[6pt]{}^{\langle \lbrack }{\text{B}}{}^{\rbrack \rangle }&{}^{\langle \lbrack }{\text{B}}{}^{\rbrack \rangle }&{}^{\langle \lbrack }w_{2}{}^{\rbrack \rangle }\\[12pt]{}^{\langle \lbrack \backprime \backprime }{\text{A}}{}^{\prime \prime \rbrack \rangle }&{}^{\langle \lbrack \langle }{\text{A}}{}^{\rangle \rbrack \rangle }&{}^{\langle \lbrack }w_{3}{}^{\rbrack \rangle }\\[6pt]{}^{\langle \lbrack \backprime \backprime }{\text{B}}{}^{\prime \prime \rbrack \rangle }&{}^{\langle \lbrack \langle }{\text{B}}{}^{\rangle \rbrack \rangle }&{}^{\langle \lbrack }w_{4}{}^{\rbrack \rangle }\\[6pt]{}^{\langle \lbrack \backprime \backprime }{\text{i}}{}^{\prime \prime \rbrack \rangle }&{}^{\langle \lbrack \langle }{\text{i}}{}^{\rangle \rbrack \rangle }&{}^{\langle \lbrack }w_{5}{}^{\rbrack \rangle }\\[6pt]{}^{\langle \lbrack \backprime \backprime }{\text{u}}{}^{\prime \prime \rbrack \rangle }&{}^{\langle \lbrack \langle }{\text{u}}{}^{\rangle \rbrack \rangle }&{}^{\langle \lbrack }w_{6}{}^{\rbrack \rangle }\end{matrix}}}$

 ${\displaystyle {\text{Instance}}}$ ${\displaystyle {\text{Sign of Instance}}}$ ${\displaystyle {\begin{matrix}{\overline {\text{A}}}&{\overline {\text{A}}}&{\overline {w_{1}}}\\[6pt]{\overline {\text{B}}}&{\overline {\text{B}}}&{\overline {w_{2}}}\\[12pt]{\overline {{}^{\backprime \backprime }{\text{A}}{}^{\prime \prime }}}&{\overline {{}^{\langle }{\text{A}}{}^{\rangle }}}&{\overline {w_{3}}}\\[6pt]{\overline {{}^{\backprime \backprime }{\text{B}}{}^{\prime \prime }}}&{\overline {{}^{\langle }{\text{B}}{}^{\rangle }}}&{\overline {w_{4}}}\\[6pt]{\overline {{}^{\backprime \backprime }{\text{i}}{}^{\prime \prime }}}&{\overline {{}^{\langle }{\text{i}}{}^{\rangle }}}&{\overline {w_{5}}}\\[6pt]{\overline {{}^{\backprime \backprime }{\text{u}}{}^{\prime \prime }}}&{\overline {{}^{\langle }{\text{u}}{}^{\rangle }}}&{\overline {w_{6}}}\end{matrix}}}$ ${\displaystyle {\begin{matrix}{}^{\langle }{\overline {\text{A}}}{}^{\rangle }&{}^{\langle }{\overline {\text{A}}}{}^{\rangle }&{}^{\langle }{\overline {w_{1}}}{}^{\rangle }\\[6pt]{}^{\langle }{\overline {\text{B}}}{}^{\rangle }&{}^{\langle }{\overline {\text{B}}}{}^{\rangle }&{}^{\langle }{\overline {w_{2}}}{}^{\rangle }\\[12pt]{}^{\langle }{\overline {{}^{\backprime \backprime }{\text{A}}{}^{\prime \prime }}}{}^{\rangle }&{}^{\langle }{\overline {{}^{\langle }{\text{A}}{}^{\rangle }}}{}^{\rangle }&{}^{\langle }{\overline {w_{3}}}{}^{\rangle }\\[6pt]{}^{\langle }{\overline {{}^{\backprime \backprime }{\text{B}}{}^{\prime \prime }}}{}^{\rangle }&{}^{\langle }{\overline {{}^{\langle }{\text{B}}{}^{\rangle }}}{}^{\rangle }&{}^{\langle }{\overline {w_{4}}}{}^{\rangle }\\[6pt]{}^{\langle }{\overline {{}^{\backprime \backprime }{\text{i}}{}^{\prime \prime }}}{}^{\rangle }&{}^{\langle }{\overline {{}^{\langle }{\text{i}}{}^{\rangle }}}{}^{\rangle }&{}^{\langle }{\overline {w_{5}}}{}^{\rangle }\\[6pt]{}^{\langle }{\overline {{}^{\backprime \backprime }{\text{u}}{}^{\prime \prime }}}{}^{\rangle }&{}^{\langle }{\overline {{}^{\langle }{\text{u}}{}^{\rangle }}}{}^{\rangle }&{}^{\langle }{\overline {w_{6}}}{}^{\rangle }\end{matrix}}}$

 ${\displaystyle {\text{Instance}}}$ ${\displaystyle {\text{Sign of Instance}}}$ ${\displaystyle {\begin{matrix}{\overline {\text{A}}}&{\overline {o_{1}}}&{\overline {w_{1}}}\\[6pt]{\overline {\text{B}}}&{\overline {o_{2}}}&{\overline {w_{2}}}\\[12pt]{\overline {\text{a}}}&{\overline {s_{1}}}&{\overline {w_{3}}}\\[6pt]{\overline {\text{b}}}&{\overline {s_{2}}}&{\overline {w_{4}}}\\[6pt]{\overline {\text{i}}}&{\overline {s_{3}}}&{\overline {w_{5}}}\\[6pt]{\overline {\text{u}}}&{\overline {s_{4}}}&{\overline {w_{6}}}\end{matrix}}}$ ${\displaystyle {\begin{matrix}{}^{\langle }{\overline {\text{A}}}{}^{\rangle }&{}^{\langle }{\overline {o_{1}}}{}^{\rangle }&{}^{\langle }{\overline {w_{1}}}{}^{\rangle }\\[6pt]{}^{\langle }{\overline {\text{B}}}{}^{\rangle }&{}^{\langle }{\overline {o_{2}}}{}^{\rangle }&{}^{\langle }{\overline {w_{2}}}{}^{\rangle }\\[12pt]{}^{\langle }{\overline {\text{a}}}{}^{\rangle }&{}^{\langle }{\overline {s_{1}}}{}^{\rangle }&{}^{\langle }{\overline {w_{3}}}{}^{\rangle }\\[6pt]{}^{\langle }{\overline {\text{b}}}{}^{\rangle }&{}^{\langle }{\overline {s_{2}}}{}^{\rangle }&{}^{\langle }{\overline {w_{4}}}{}^{\rangle }\\[6pt]{}^{\langle }{\overline {\text{i}}}{}^{\rangle }&{}^{\langle }{\overline {s_{3}}}{}^{\rangle }&{}^{\langle }{\overline {w_{5}}}{}^{\rangle }\\[6pt]{}^{\langle }{\overline {\text{u}}}{}^{\rangle }&{}^{\langle }{\overline {s_{4}}}{}^{\rangle }&{}^{\langle }{\overline {w_{6}}}{}^{\rangle }\end{matrix}}}$

Using two different strategies of representation:

Literal Coding. The first strategy is called the literal coding because it sticks to obvious features of each syntactic element to contrive its code, or the ${\displaystyle {{\mathcal {O}}(n)}}$ coding, because it uses a number on the order of ${\displaystyle n}$ logical features to represent a domain of ${\displaystyle n}$ elements.

Being superficial as a matter of principle, or adhering to the surface appearances of signs, enjoys the initial advantage that the very same codes can be used by any interpreter that is capable of observing them. The down side of resorting to this technique is that it typically uses an excessive number of logical dimensions to get each point of the intended space across.

Even while operating within the general lines of the literal, superficial, or ${\displaystyle {{\mathcal {O}}(n)}}$ strategy, there are still a number of choices to be made in the style of coding to be employed. For example, if there is an obvious distinction between different components of the world, like that between the objects in ${\displaystyle O=\{{\text{A}},{\text{B}}\}}$ and the signs in ${\displaystyle S=\{{}^{\backprime \backprime }{\text{A}}{}^{\prime \prime },{}^{\backprime \backprime }{\text{B}}{}^{\prime \prime },{}^{\backprime \backprime }{\text{i}}{}^{\prime \prime },{}^{\backprime \backprime }{\text{u}}{}^{\prime \prime }\},}$ then it is common to let this distinction go formally unmarked in the LIR, that is, to omit the requirement of declaring an explicit logical feature to make a note of it in the formal coding. The distinction itself, as a property of reality, is in no danger of being obliterated or permanently erased, but it can be obscured and temporarily ignored. In practice, the distinction is not so much ignored as it is casually observed and informally attended to, usually being marked by incidental indices in the context of the representation.

Literal Coding

For the domain ${\displaystyle W=\{{\text{A}},{\text{B}},{}^{\backprime \backprime }{\text{A}}{}^{\prime \prime },{}^{\backprime \backprime }{\text{B}}{}^{\prime \prime },{}^{\backprime \backprime }{\text{i}}{}^{\prime \prime },{}^{\backprime \backprime }{\text{u}}{}^{\prime \prime }\}}$ of six elements one needs to use six logical features, in effect, elevating each individual object to the status of an exclusive ontological category in its own right. The easiest way to do this is simply to reuse the world domain ${\displaystyle W}$ as a logical alphabet ${\displaystyle {\underline {\underline {W}}},}$ taking element-wise identifications as follows:

 ${\displaystyle {\begin{array}{*{15}{c}}W&=&\{&o_{1}&,&o_{2}&,&s_{1}&,&s_{2}&,&s_{3}&,&s_{4}&\}\\&=&\{&{\text{A}}&,&{\text{B}}&,&{}^{\backprime \backprime }{\text{A}}{}^{\prime \prime }&,&{}^{\backprime \backprime }{\text{B}}{}^{\prime \prime }&,&{}^{\backprime \backprime }{\text{i}}{}^{\prime \prime }&,&{}^{\backprime \backprime }{\text{u}}{}^{\prime \prime }&\}\\[10pt]{\underline {\underline {W}}}&=&\{&{\underline {\underline {w_{1}}}}&,&{\underline {\underline {w_{2}}}}&,&{\underline {\underline {w_{3}}}}&,&{\underline {\underline {w_{4}}}}&,&{\underline {\underline {w_{5}}}}&,&{\underline {\underline {w_{6}}}}&\}\\&=&\{&{\underline {\underline {\text{A}}}}&,&{\underline {\underline {\text{B}}}}&,&{\underline {\underline {\text{a}}}}&,&{\underline {\underline {\text{b}}}}&,&{\underline {\underline {\text{i}}}}&,&{\underline {\underline {\text{u}}}}&\}\end{array}}}$

Tables 53.1 and 53.2 show three different ways of coding the elements of an ER and the features of a LIR, respectively, for the world set ${\displaystyle W=W({\text{A}},{\text{B}}),}$ that is, for the set of objects, signs, and interpretants that are common to the sign relations ${\displaystyle L(A)}$ and ${\displaystyle L(B).}$ Successive columns of these Tables give the mnemonic code, the pragmatic code, and the abstract code, respectively, for each element.

 ${\displaystyle {\text{Mnemonic Element}}}$ ${\displaystyle w\in W}$ ${\displaystyle {\text{Pragmatic Element}}}$ ${\displaystyle w\in W}$ ${\displaystyle {\text{Abstract Element}}}$ ${\displaystyle w_{i}\in W}$ ${\displaystyle {\begin{matrix}{\text{A}}\\[4pt]{\text{B}}\\[4pt]{}^{\backprime \backprime }{\text{A}}{}^{\prime \prime }\\[4pt]{}^{\backprime \backprime }{\text{B}}{}^{\prime \prime }\\[4pt]{}^{\backprime \backprime }{\text{i}}{}^{\prime \prime }\\[4pt]{}^{\backprime \backprime }{\text{u}}{}^{\prime \prime }\end{matrix}}}$ ${\displaystyle {\begin{matrix}o_{1}\\[4pt]o_{2}\\[4pt]s_{1}\\[4pt]s_{2}\\[4pt]s_{3}\\[4pt]s_{4}\end{matrix}}}$ ${\displaystyle {\begin{matrix}w_{1}\\[4pt]w_{2}\\[4pt]w_{3}\\[4pt]w_{4}\\[4pt]w_{5}\\[4pt]w_{6}\end{matrix}}}$

 ${\displaystyle {\text{Mnemonic Feature}}}$ ${\displaystyle {\underline {\underline {w}}}\in {\underline {\underline {W}}}}$ ${\displaystyle {\text{Pragmatic Feature}}}$ ${\displaystyle {\underline {\underline {w}}}\in {\underline {\underline {W}}}}$ ${\displaystyle {\text{Abstract Feature}}}$ ${\displaystyle {\underline {\underline {w_{i}}}}\in {\underline {\underline {W}}}}$ ${\displaystyle {\begin{matrix}{\underline {\underline {\text{A}}}}\\[4pt]{\underline {\underline {\text{B}}}}\\[4pt]{\underline {\underline {\text{a}}}}\\[4pt]{\underline {\underline {\text{b}}}}\\[4pt]{\underline {\underline {\text{i}}}}\\[4pt]{\underline {\underline {\text{u}}}}\end{matrix}}}$ ${\displaystyle {\begin{matrix}{\underline {\underline {o_{1}}}}\\[4pt]{\underline {\underline {o_{2}}}}\\[4pt]{\underline {\underline {s_{1}}}}\\[4pt]{\underline {\underline {s_{2}}}}\\[4pt]{\underline {\underline {s_{3}}}}\\[4pt]{\underline {\underline {s_{4}}}}\end{matrix}}}$ ${\displaystyle {\begin{matrix}{\underline {\underline {w_{1}}}}\\[4pt]{\underline {\underline {w_{2}}}}\\[4pt]{\underline {\underline {w_{3}}}}\\[4pt]{\underline {\underline {w_{4}}}}\\[4pt]{\underline {\underline {w_{5}}}}\\[4pt]{\underline {\underline {w_{6}}}}\end{matrix}}}$

If the world of ${\displaystyle {\text{A}}}$ and ${\displaystyle {\text{B}},}$ the set ${\displaystyle W=\{{\text{A}},{\text{B}},{}^{\backprime \backprime }{\text{A}}{}^{\prime \prime },{}^{\backprime \backprime }{\text{B}}{}^{\prime \prime },{}^{\backprime \backprime }{\text{i}}{}^{\prime \prime },{}^{\backprime \backprime }{\text{u}}{}^{\prime \prime }\},}$ is viewed abstractly, as an arbitrary set of six atomic points, then there are exactly ${\displaystyle 2^{6}=64}$ abstract properties or potential attributes that might be applied to or recognized in these points. The elements of ${\displaystyle W}$ that possess a given property form a subset of ${\displaystyle W}$ called the extension of that property. Thus the extensions of abstract properties are exactly the subsets of ${\displaystyle W.}$ The set of all subsets of ${\displaystyle W}$ is called the power set of ${\displaystyle W,}$ notated as ${\displaystyle \mathrm {Pow} (W)}$ or ${\displaystyle {\mathcal {P}}(W).}$ In order to make this way of talking about properties consistent with the previous definition of reality, it is necessary to say that one potential property is never realized, since no point has it, and its extension is the empty set ${\displaystyle \varnothing =\{\}.}$ All the natural properties of points that one observes in a concrete situation, properties whose extensions are known as natural kinds, can be recognized among the abstract, arbitrary, or set-theoretic properties that are systematically generated in this way. Typically, however, many of these abstract properties will not be recognized as falling among the more natural kinds.

Tables 54.1, 54.2, and 54.3 show three different ways of representing the elements of the world set ${\displaystyle W}$ as vectors in the coordinate space ${\displaystyle {\underline {W}}}$ and as singular propositions in the universe of discourse ${\displaystyle W^{\Box }.}$ Altogether, these Tables present the literal codes for the elements of ${\displaystyle {\underline {W}}}$ and ${\displaystyle W^{\circ }}$ in their mnemonic, pragmatic, and abstract versions, respectively. In each Table, Column 1 lists the element ${\displaystyle w\in W,}$ while Column 2 gives the corresponding coordinate vector ${\displaystyle {\underline {w}}\in {\underline {W}}}$ in the form of a bit string. The next two Columns represent each ${\displaystyle w\in W}$ as a proposition in ${\displaystyle W^{\circ }\!,}$ in effect, reconstituting it as a function ${\displaystyle w:{\underline {W}}\to \mathbb {B} .}$ Column 3 shows the propositional expression of each element in the form of a conjunct term, in other words, as a logical product of positive and negative features. Column 4 gives the compact code for each element, using a conjunction of positive features in subscripted angle brackets to represent the singular proposition corresponding to each element.

 ${\displaystyle {\text{Element}}}$ ${\displaystyle {\text{Vector}}}$ ${\displaystyle {\text{Conjunct Term}}}$ ${\displaystyle {\text{Code}}}$ ${\displaystyle {\begin{matrix}{\text{A}}\\[4pt]{\text{B}}\\[4pt]{}^{\backprime \backprime }{\text{A}}{}^{\prime \prime }\\[4pt]{}^{\backprime \backprime }{\text{B}}{}^{\prime \prime }\\[4pt]{}^{\backprime \backprime }{\text{i}}{}^{\prime \prime }\\[4pt]{}^{\backprime \backprime }{\text{u}}{}^{\prime \prime }\end{matrix}}}$ ${\displaystyle {\begin{matrix}100000\\[4pt]010000\\[4pt]001000\\[4pt]000100\\[4pt]000010\\[4pt]000001\end{matrix}}}$ ${\displaystyle {\begin{matrix}~{\underline {\underline {A}}}~({\underline {\underline {B}}})({\underline {\underline {a}}})({\underline {\underline {b}}})({\underline {\underline {i}}})({\underline {\underline {u}}})\\[4pt]({\underline {\underline {A}}})~{\underline {\underline {B}}}~({\underline {\underline {a}}})({\underline {\underline {b}}})({\underline {\underline {i}}})({\underline {\underline {u}}})\\[4pt]({\underline {\underline {A}}})({\underline {\underline {B}}})~{\underline {\underline {a}}}~({\underline {\underline {b}}})({\underline {\underline {i}}})({\underline {\underline {u}}})\\[4pt]({\underline {\underline {A}}})({\underline {\underline {B}}})({\underline {\underline {a}}})~{\underline {\underline {b}}}~({\underline {\underline {i}}})({\underline {\underline {u}}})\\[4pt]({\underline {\underline {A}}})({\underline {\underline {B}}})({\underline {\underline {a}}})({\underline {\underline {b}}})~{\underline {\underline {i}}}~({\underline {\underline {u}}})\\[4pt]({\underline {\underline {A}}})({\underline {\underline {B}}})({\underline {\underline {a}}})({\underline {\underline {b}}})({\underline {\underline {i}}})~{\underline {\underline {u}}}~\end{matrix}}}$ ${\displaystyle {\begin{matrix}{\langle {\underline {\underline {A}}}\rangle }_{W}\\[4pt]{\langle {\underline {\underline {B}}}\rangle }_{W}\\[4pt]{\langle {\underline {\underline {a}}}\rangle }_{W}\\[4pt]{\langle {\underline {\underline {b}}}\rangle }_{W}\\[4pt]{\langle {\underline {\underline {i}}}\rangle }_{W}\\[4pt]{\langle {\underline {\underline {u}}}\rangle }_{W}\end{matrix}}}$

 ${\displaystyle {\text{Element}}}$ ${\displaystyle {\text{Vector}}}$ ${\displaystyle {\text{Conjunct Term}}}$ ${\displaystyle {\text{Code}}}$ ${\displaystyle {\begin{matrix}{\text{A}}\\[4pt]{\text{B}}\\[4pt]{}^{\backprime \backprime }{\text{A}}{}^{\prime \prime }\\[4pt]{}^{\backprime \backprime }{\text{B}}{}^{\prime \prime }\\[4pt]{}^{\backprime \backprime }{\text{i}}{}^{\prime \prime }\\[4pt]{}^{\backprime \backprime }{\text{u}}{}^{\prime \prime }\end{matrix}}}$ ${\displaystyle {\begin{matrix}100000\\[4pt]010000\\[4pt]001000\\[4pt]000100\\[4pt]000010\\[4pt]000001\end{matrix}}}$ ${\displaystyle {\begin{matrix}~{\underline {\underline {o_{1}}}}~({\underline {\underline {o_{2}}}})({\underline {\underline {s_{1}}}})({\underline {\underline {s_{2}}}})({\underline {\underline {s_{3}}}})({\underline {\underline {s_{4}}}})\\[4pt]({\underline {\underline {o_{1}}}})~{\underline {\underline {o_{2}}}}~({\underline {\underline {s_{1}}}})({\underline {\underline {s_{2}}}})({\underline {\underline {s_{3}}}})({\underline {\underline {s_{4}}}})\\[4pt]({\underline {\underline {o_{1}}}})({\underline {\underline {o_{2}}}})~{\underline {\underline {s_{1}}}}~({\underline {\underline {s_{2}}}})({\underline {\underline {s_{3}}}})({\underline {\underline {s_{4}}}})\\[4pt]({\underline {\underline {o_{1}}}})({\underline {\underline {o_{2}}}})({\underline {\underline {s_{1}}}})~{\underline {\underline {s_{2}}}}~({\underline {\underline {s_{3}}}})({\underline {\underline {s_{4}}}})\\[4pt]({\underline {\underline {o_{1}}}})({\underline {\underline {o_{2}}}})({\underline {\underline {s_{1}}}})({\underline {\underline {s_{2}}}})~{\underline {\underline {s_{3}}}}~({\underline {\underline {s_{4}}}})\\[4pt]({\underline {\underline {o_{1}}}})({\underline {\underline {o_{2}}}})({\underline {\underline {s_{1}}}})({\underline {\underline {s_{2}}}})({\underline {\underline {s_{3}}}})~{\underline {\underline {s_{4}}}}~\end{matrix}}}$ ${\displaystyle {\begin{matrix}{\langle {\underline {\underline {o_{1}}}}\rangle }_{W}\\[4pt]{\langle {\underline {\underline {o_{2}}}}\rangle }_{W}\\[4pt]{\langle {\underline {\underline {s_{1}}}}\rangle }_{W}\\[4pt]{\langle {\underline {\underline {s_{2}}}}\rangle }_{W}\\[4pt]{\langle {\underline {\underline {s_{3}}}}\rangle }_{W}\\[4pt]{\langle {\underline {\underline {s_{4}}}}\rangle }_{W}\end{matrix}}}$

 ${\displaystyle {\text{Element}}}$ ${\displaystyle {\text{Vector}}}$ ${\displaystyle {\text{Conjunct Term}}}$ ${\displaystyle {\text{Code}}}$ ${\displaystyle {\begin{matrix}{\text{A}}\\[4pt]{\text{B}}\\[4pt]{}^{\backprime \backprime }{\text{A}}{}^{\prime \prime }\\[4pt]{}^{\backprime \backprime }{\text{B}}{}^{\prime \prime }\\[4pt]{}^{\backprime \backprime }{\text{i}}{}^{\prime \prime }\\[4pt]{}^{\backprime \backprime }{\text{u}}{}^{\prime \prime }\end{matrix}}}$ ${\displaystyle {\begin{matrix}100000\\[4pt]010000\\[4pt]001000\\[4pt]000100\\[4pt]000010\\[4pt]000001\end{matrix}}}$ ${\displaystyle {\begin{matrix}~{\underline {\underline {w_{1}}}}~({\underline {\underline {w_{2}}}})({\underline {\underline {w_{3}}}})({\underline {\underline {w_{4}}}})({\underline {\underline {w_{5}}}})({\underline {\underline {w_{6}}}})\\[4pt]({\underline {\underline {w_{1}}}})~{\underline {\underline {w_{2}}}}~({\underline {\underline {w_{3}}}})({\underline {\underline {w_{4}}}})({\underline {\underline {w_{5}}}})({\underline {\underline {w_{6}}}})\\[4pt]({\underline {\underline {w_{1}}}})({\underline {\underline {w_{2}}}})~{\underline {\underline {w_{3}}}}~({\underline {\underline {w_{4}}}})({\underline {\underline {w_{5}}}})({\underline {\underline {w_{6}}}})\\[4pt]({\underline {\underline {w_{1}}}})({\underline {\underline {w_{2}}}})({\underline {\underline {w_{3}}}})~{\underline {\underline {w_{4}}}}~({\underline {\underline {w_{5}}}})({\underline {\underline {w_{6}}}})\\[4pt]({\underline {\underline {w_{1}}}})({\underline {\underline {w_{2}}}})({\underline {\underline {w_{3}}}})({\underline {\underline {w_{4}}}})~{\underline {\underline {w_{5}}}}~({\underline {\underline {w_{6}}}})\\[4pt]({\underline {\underline {w_{1}}}})({\underline {\underline {w_{2}}}})({\underline {\underline {w_{3}}}})({\underline {\underline {w_{4}}}})({\underline {\underline {w_{5}}}})~{\underline {\underline {w_{6}}}}~\end{matrix}}}$ ${\displaystyle {\begin{matrix}{\langle {\underline {\underline {w_{1}}}}\rangle }_{W}\\[4pt]{\langle {\underline {\underline {w_{2}}}}\rangle }_{W}\\[4pt]{\langle {\underline {\underline {w_{3}}}}\rangle }_{W}\\[4pt]{\langle {\underline {\underline {w_{4}}}}\rangle }_{W}\\[4pt]{\langle {\underline {\underline {w_{5}}}}\rangle }_{W}\\[4pt]{\langle {\underline {\underline {w_{6}}}}\rangle }_{W}\end{matrix}}}$

 ${\displaystyle {\text{Object}}}$ ${\displaystyle {\text{Sign}}}$ ${\displaystyle {\text{Interpretant}}}$ ${\displaystyle {\begin{matrix}{\langle {\underline {\underline {\text{A}}}}\rangle }_{W}\\[4pt]{\langle {\underline {\underline {\text{A}}}}\rangle }_{W}\\[4pt]{\langle {\underline {\underline {\text{A}}}}\rangle }_{W}\\[4pt]{\langle {\underline {\underline {\text{A}}}}\rangle }_{W}\end{matrix}}}$ ${\displaystyle {\begin{matrix}{\langle {\underline {\underline {\text{a}}}}\rangle }_{W}\\[4pt]{\langle {\underline {\underline {\text{a}}}}\rangle }_{W}\\[4pt]{\langle {\underline {\underline {\text{i}}}}\rangle }_{W}\\[4pt]{\langle {\underline {\underline {\text{i}}}}\rangle }_{W}\end{matrix}}}$ ${\displaystyle {\begin{matrix}{\langle {\underline {\underline {\text{a}}}}\rangle }_{W}\\[4pt]{\langle {\underline {\underline {\text{i}}}}\rangle }_{W}\\[4pt]{\langle {\underline {\underline {\text{a}}}}\rangle }_{W}\\[4pt]{\langle {\underline {\underline {\text{i}}}}\rangle }_{W}\end{matrix}}}$ ${\displaystyle {\begin{matrix}{\langle {\underline {\underline {\text{B}}}}\rangle }_{W}\\[4pt]{\langle {\underline {\underline {\text{B}}}}\rangle }_{W}\\[4pt]{\langle {\underline {\underline {\text{B}}}}\rangle }_{W}\\[4pt]{\langle {\underline {\underline {\text{B}}}}\rangle }_{W}\end{matrix}}}$ ${\displaystyle {\begin{matrix}{\langle {\underline {\underline {\text{b}}}}\rangle }_{W}\\[4pt]{\langle {\underline {\underline {\text{b}}}}\rangle }_{W}\\[4pt]{\langle {\underline {\underline {\text{u}}}}\rangle }_{W}\\[4pt]{\langle {\underline {\underline {\text{u}}}}\rangle }_{W}\end{matrix}}}$ ${\displaystyle {\begin{matrix}{\langle {\underline {\underline {\text{b}}}}\rangle }_{W}\\[4pt]{\langle {\underline {\underline {\text{u}}}}\rangle }_{W}\\[4pt]{\langle {\underline {\underline {\text{b}}}}\rangle }_{W}\\[4pt]{\langle {\underline {\underline {\text{u}}}}\rangle }_{W}\end{matrix}}}$

 ${\displaystyle {\text{Object}}}$ ${\displaystyle {\text{Sign}}}$ ${\displaystyle {\text{Transition}}}$ ${\displaystyle {\begin{matrix}{\langle {\underline {\underline {\text{A}}}}\rangle }_{W}\\[4pt]{\langle {\underline {\underline {\text{A}}}}\rangle }_{W}\end{matrix}}}$ ${\displaystyle {\begin{matrix}{\langle {\underline {\underline {\text{a}}}}\rangle }_{W}\\[4pt]{\langle {\underline {\underline {\text{i}}}}\rangle }_{W}\end{matrix}}}$ ${\displaystyle {\begin{matrix}({\langle {\underline {\underline {\text{a}}}}\rangle }_{W},{\langle {\underline {\underline {\text{A}}}}\rangle }_{W})\\[4pt]({\langle {\underline {\underline {\text{i}}}}\rangle }_{W},{\langle {\underline {\underline {\text{A}}}}\rangle }_{W})\end{matrix}}}$ ${\displaystyle {\begin{matrix}{\langle {\underline {\underline {\text{B}}}}\rangle }_{W}\\[4pt]{\langle {\underline {\underline {\text{B}}}}\rangle }_{W}\end{matrix}}}$ ${\displaystyle {\begin{matrix}{\langle {\underline {\underline {\text{b}}}}\rangle }_{W}\\[4pt]{\langle {\underline {\underline {\text{u}}}}\rangle }_{W}\end{matrix}}}$ ${\displaystyle {\begin{matrix}({\langle {\underline {\underline {\text{b}}}}\rangle }_{W},{\langle {\underline {\underline {\text{B}}}}\rangle }_{W})\\[4pt]({\langle {\underline {\underline {\text{u}}}}\rangle }_{W},{\langle {\underline {\underline {\text{B}}}}\rangle }_{W})\end{matrix}}}$

 ${\displaystyle {\text{Sign}}}$ ${\displaystyle {\text{Interpretant}}}$ ${\displaystyle {\text{Transition}}}$ ${\displaystyle {\begin{matrix}{\langle {\underline {\underline {\text{a}}}}\rangle }_{W}\\[4pt]{\langle {\underline {\underline {\text{a}}}}\rangle }_{W}\\[4pt]{\langle {\underline {\underline {\text{i}}}}\rangle }_{W}\\[4pt]{\langle {\underline {\underline {\text{i}}}}\rangle }_{W}\end{matrix}}}$ ${\displaystyle {\begin{matrix}{\langle {\underline {\underline {\text{a}}}}\rangle }_{W}\\[4pt]{\langle {\underline {\underline {\text{i}}}}\rangle }_{W}\\[4pt]{\langle {\underline {\underline {\text{a}}}}\rangle }_{W}\\[4pt]{\langle {\underline {\underline {\text{i}}}}\rangle }_{W}\end{matrix}}}$ ${\displaystyle {\begin{matrix}0_{\mathrm {d} W}\\[4pt]{\langle \mathrm {d} {\underline {\underline {\text{a}}}}~\mathrm {d} {\underline {\underline {\text{i}}}}\rangle }_{\mathrm {d} W}\\[4pt]{\langle \mathrm {d} {\underline {\underline {\text{a}}}}~\mathrm {d} {\underline {\underline {\text{i}}}}\rangle }_{\mathrm {d} W}\\[4pt]0_{\mathrm {d} W}\end{matrix}}}$ ${\displaystyle {\begin{matrix}{\langle {\underline {\underline {\text{b}}}}\rangle }_{W}\\[4pt]{\langle {\underline {\underline {\text{b}}}}\rangle }_{W}\\[4pt]{\langle {\underline {\underline {\text{u}}}}\rangle }_{W}\\[4pt]{\langle {\underline {\underline {\text{u}}}}\rangle }_{W}\end{matrix}}}$ ${\displaystyle {\begin{matrix}{\langle {\underline {\underline {\text{b}}}}\rangle }_{W}\\[4pt]{\langle {\underline {\underline {\text{u}}}}\rangle }_{W}\\[4pt]{\langle {\underline {\underline {\text{b}}}}\rangle }_{W}\\[4pt]{\langle {\underline {\underline {\text{u}}}}\rangle }_{W}\end{matrix}}}$ ${\displaystyle {\begin{matrix}0_{\mathrm {d} W}\\[4pt]{\langle \mathrm {d} {\underline {\underline {\text{b}}}}~\mathrm {d} {\underline {\underline {\text{u}}}}\rangle }_{\mathrm {d} W}\\[4pt]{\langle \mathrm {d} {\underline {\underline {\text{b}}}}~\mathrm {d} {\underline {\underline {\text{u}}}}\rangle }_{\mathrm {d} W}\\[4pt]0_{\mathrm {d} W}\end{matrix}}}$

 ${\displaystyle {\text{Object}}}$ ${\displaystyle {\text{Sign}}}$ ${\displaystyle {\text{Interpretant}}}$ ${\displaystyle {\begin{matrix}{\langle {\underline {\underline {\text{A}}}}\rangle }_{W}\\[4pt]{\langle {\underline {\underline {\text{A}}}}\rangle }_{W}\\[4pt]{\langle {\underline {\underline {\text{A}}}}\rangle }_{W}\\[4pt]{\langle {\underline {\underline {\text{A}}}}\rangle }_{W}\end{matrix}}}$