12,080
edits
Changes
Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Part 6 (view source)
Revision as of 04:10, 28 August 2014
, 04:10, 28 August 2014add more
'''[Variant]''' IRs of sign relations refer to properties of sets <math>(PSs),\!</math> properties of triples <math>(PTs),\!</math> and properties of underlying elements <math>(PUs).\!</math> 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 <math>\underline{S}PSs,\!</math> <math>\underline{S}PTs,\!</math> and <math>\underline{S}PUs.\!</math>
'''[Variant]''' IRs of sign relations refer to properties of sets <math>(PSs),\!</math> properties of triples <math>(PTs),\!</math> and properties of underlying elements <math>(PUs).\!</math> 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 <math>\underline{S}PSs,\!</math> <math>\underline{S}PTs,\!</math> and <math>\underline{S}PUs.\!</math>
===6.22. 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 <math>L(\text{A})\!</math> and <math>L(\text{B})\!</math> 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 <math>\text{A}\!</math> and <math>\text{B}.\!</math> 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 <math>\text{A}\!</math> and <math>\text{B},\!</math> recollecting the Tables of their sign relations and finishing up the Tables of their dyadic components. Since the form of the sign relations <math>L(\text{A})\!</math> and <math>L(\text{B})\!</math> 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 <math>L(\text{A})\!</math> and <math>L(\text{B}).\!</math> 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 <math>({}^{\backprime\backprime} \ldots {}^{\prime\prime})\!</math> are transcribed as invocations of the ''arch operator'' <math>({}^{\langle} \ldots {}^{\rangle}).\!</math> The reason for this slight change of notation will be explained shortly. The denotative components <math>\operatorname{Den}(\text{A})\!</math> and <math>\operatorname{Den}(\text{B})\!</math> 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 <math>(s, o).\!</math> The connotative components <math>\operatorname{Con}(\text{A})\!</math> and <math>\operatorname{Con}(\text{B})\!</math> 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 <math>(s, i).\!</math>
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
|+ style="height:30px" |
<math>\text{Table 48.1} ~~ \operatorname{ER}(L_\text{A}) : \text{Extensional Representation of} ~ L_\text{A}\!</math>
|- style="height:40px; background:#f0f0ff"
| <math>\text{Object}\!</math>
| <math>\text{Sign}\!</math>
| <math>\text{Interpretant}\!</math>
|-
| valign="bottom" width="33%" |
<math>\begin{matrix}
\text{A}
\\
\text{A}
\\
\text{A}
\\
\text{A}
\end{matrix}</math>
| valign="bottom" width="33%" |
<math>\begin{matrix}
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\end{matrix}</math>
| valign="bottom" width="33%" |
<math>\begin{matrix}
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\end{matrix}</math>
|-
| valign="bottom" width="33%" |
<math>\begin{matrix}
\text{B}
\\
\text{B}
\\
\text{B}
\\
\text{B}
\end{matrix}</math>
| valign="bottom" width="33%" |
<math>\begin{matrix}
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\end{matrix}</math>
| valign="bottom" width="33%" |
<math>\begin{matrix}
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\end{matrix}</math>
|}
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
|+ style="height:30px" |
<math>\text{Table 48.2} ~~ \operatorname{ER}(\operatorname{Den}(L_\text{A})) : \text{Denotative Component of} ~ L_\text{A}\!</math>
|- style="height:40px; background:#f0f0ff"
| <math>\text{Object}\!</math>
| <math>\text{Sign}\!</math>
| <math>\text{Transition}\!</math>
|-
| valign="bottom" width="33%" |
<math>\begin{matrix}
\text{A}
\\
\text{A}
\end{matrix}</math>
| valign="bottom" width="33%" |
<math>\begin{matrix}
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\end{matrix}</math>
| valign="bottom" width="33%" |
<math>\begin{matrix}
({}^{\langle} \text{A} {}^{\rangle}, \text{A})
\\
({}^{\langle} \text{i} {}^{\rangle}, \text{A})
\end{matrix}</math>
|-
| valign="bottom" width="33%" |
<math>\begin{matrix}
\text{B}
\\
\text{B}
\end{matrix}</math>
| valign="bottom" width="33%" |
<math>\begin{matrix}
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\end{matrix}</math>
| valign="bottom" width="33%" |
<math>\begin{matrix}
({}^{\langle} \text{B} {}^{\rangle}, \text{B})
\\
({}^{\langle} \text{u} {}^{\rangle}, \text{B})
\end{matrix}</math>
|}
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
|+ style="height:30px" |
<math>\text{Table 48.3} ~~ \operatorname{ER}(\operatorname{Con}(L_\text{A})) : \text{Connotative Component of} ~ L_\text{A}\!</math>
|- style="height:40px; background:#f0f0ff"
| <math>\text{Sign}\!</math>
| <math>\text{Interpretant}\!</math>
| <math>\text{Transition}\!</math>
|-
| valign="bottom" width="33%" |
<math>\begin{matrix}
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\end{matrix}</math>
| valign="bottom" width="33%" |
<math>\begin{matrix}
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\end{matrix}</math>
| valign="bottom" width="33%" |
<math>\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}</math>
|-
| valign="bottom" width="33%" |
<math>\begin{matrix}
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\end{matrix}</math>
| valign="bottom" width="33%" |
<math>\begin{matrix}
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\end{matrix}</math>
| valign="bottom" width="33%" |
<math>\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}</math>
|}
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
|+ style="height:30px" |
<math>\text{Table 49.1} ~~ \operatorname{ER}(L_\text{B}) : \text{Extensional Representation of} ~ L_\text{B}\!</math>
|- style="height:40px; background:#f0f0ff"
| <math>\text{Object}\!</math>
| <math>\text{Sign}\!</math>
| <math>\text{Interpretant}\!</math>
|-
| valign="bottom" width="33%" |
<math>\begin{matrix}
\text{A}
\\
\text{A}
\\
\text{A}
\\
\text{A}
\end{matrix}</math>
| valign="bottom" width="33%" |
<math>\begin{matrix}
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\end{matrix}</math>
| valign="bottom" width="33%" |
<math>\begin{matrix}
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\end{matrix}</math>
|-
| valign="bottom" width="33%" |
<math>\begin{matrix}
\text{B}
\\
\text{B}
\\
\text{B}
\\
\text{B}
\end{matrix}</math>
| valign="bottom" width="33%" |
<math>\begin{matrix}
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\end{matrix}</math>
| valign="bottom" width="33%" |
<math>\begin{matrix}
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\end{matrix}</math>
|}
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
|+ style="height:30px" |
<math>\text{Table 49.2} ~~ \operatorname{ER}(\operatorname{Den}(L_\text{B})) : \text{Denotative Component of} ~ L_\text{B}~\!</math>
|- style="height:40px; background:#f0f0ff"
| <math>\text{Object}\!</math>
| <math>\text{Sign}\!</math>
| <math>\text{Transition}\!</math>
|-
| valign="bottom" width="33%" |
<math>\begin{matrix}
\text{A}
\\
\text{A}
\end{matrix}</math>
| valign="bottom" width="33%" |
<math>\begin{matrix}
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\end{matrix}</math>
| valign="bottom" width="33%" |
<math>\begin{matrix}
({}^{\langle} \text{A} {}^{\rangle}, \text{A})
\\
({}^{\langle} \text{u} {}^{\rangle}, \text{A})
\end{matrix}\!</math>
|-
| valign="bottom" width="33%" |
<math>\begin{matrix}
\text{B}
\\
\text{B}
\end{matrix}</math>
| valign="bottom" width="33%" |
<math>\begin{matrix}
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\end{matrix}</math>
| valign="bottom" width="33%" |
<math>\begin{matrix}
({}^{\langle} \text{B} {}^{\rangle}, \text{B})
\\
({}^{\langle} \text{i} {}^{\rangle}, \text{B})
\end{matrix}</math>
|}
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
|+ style="height:30px" |
<math>\text{Table 49.3} ~~ \operatorname{ER}(\operatorname{Con}(L_\text{B})) : \text{Connotative Component of} ~ L_\text{B}\!</math>
|- style="height:40px; background:#f0f0ff"
| <math>\text{Sign}\!</math>
| <math>\text{Interpretant}\!</math>
| <math>\text{Transition}\!</math>
|-
| valign="bottom" width="33%" |
<math>\begin{matrix}
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\end{matrix}</math>
| valign="bottom" width="33%" |
<math>\begin{matrix}
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\end{matrix}</math>
| valign="bottom" width="33%" |
<math>\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}</math>
|-
| valign="bottom" width="33%" |
<math>\begin{matrix}
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\end{matrix}</math>
| valign="bottom" width="33%" |
<math>\begin{matrix}
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\end{matrix}</math>
| valign="bottom" width="33%" |
<math>\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}</math>
|}
<br>
===6.23. Intensional Representations of Sign Relations===
The next three sections consider how the ERs of <math>L(\text{A})\!</math> and <math>L(\text{B})\!</math> 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 <math>L(\text{A})\!</math> and <math>L(\text{B}),\!</math> 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 <math>L(\text{A})\!</math> and <math>L(\text{B})\!</math> 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 <math>L\!</math> is one of the sign relations <math>L(\text{A})\!</math> and <math>L(\text{B}),\!</math> and let <math>\operatorname{ER}(L)\!</math> be an ER of <math>L.\!</math>
In the sign relations <math>L(\text{A})\!</math> and <math>L(\text{B}),\!</math> both of their ERs are based on a common world set:
{| align="center" cellspacing="8" width="90%"
|
<math>\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}</math>
|}
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 <math>L(\text{A})\!</math> or <math>L(\text{B})\!</math> involves properties of its elementary points <math>w \in W\!</math> and properties of its elementary relations <math>\ell \in O \times S \times I.\!</math>
To devise an IR of any relation <math>L\!</math> one needs to describe <math>L\!</math> in terms of properties of its ingredients. Broadly speaking, the ingredients of a relation include its elementary relations or <math>n\!</math>-tuples and the elementary components of these <math>n\!</math>-tuples that reside in the relational domains.
The poset <math>\operatorname{Pos}(W)\!</math> of interest here is the power set <math>\mathcal{P}(W) = \operatorname{Pow}(W).\!</math>
The elements of these posets are abstractly regarded as ''properties'' or ''propositions'' that apply to the elements of <math>W.\!</math> 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 <math>L(\text{A})\!</math> and <math>L(\text{B}),\!</math> their denotative projections as the digraphs <math>\operatorname{Den}(L_\text{A})\!</math> and <math>\operatorname{Den}(L_\text{B}),\!</math> and their connotative projections as the digraphs <math>\operatorname{Con}(L_\text{A})\!</math> and <math>\operatorname{Con}(L_\text{B}).\!</math> In doing this I take up two different strategies of representation:
# The first strategy is called the ''literal coding'', because it sticks to obvious features of each syntactic element to contrive its code, or the ''<math>{\mathcal{O}(n)}\!</math> coding'', because it uses a number on the order of <math>n\!</math> logical features to represent a domain of <math>n\!</math> elements.
# 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 ''<math>\log (n)\!</math> coding'', because it uses roughly <math>\log_2 (n)\!</math> binary features to represent a domain of <math>n\!</math> elements.
===6.24. 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.
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:70%"
|+ style="height:30px" |
<math>\text{Table 50.} ~~ \text{Notations for Objects and Their Signs}\!</math>
|- style="height:40px; background:#f0f0ff"
| <math>\text{Object}\!</math>
| <math>\text{Sign of Object}\!</math>
|-
| valign="bottom" width="50%" |
<math>\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}</math>
| valign="bottom" width="50%" |
<math>\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}</math>
|}
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:70%"
|+ style="height:30px" |
<math>\text{Table 51.1} ~~ \text{Notations for Properties and Their Signs (1)}\!</math>
|- style="height:40px; background:#f0f0ff"
| <math>\text{Property}\!</math>
| <math>\text{Sign of Property}\!</math>
|-
| valign="bottom" width="50%" |
<math>\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}</math>
| valign="bottom" width="50%" |
<math>\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}</math>
|}
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:70%"
|+ style="height:30px" |
<math>\text{Table 51.2} ~~ \text{Notations for Properties and Their Signs (2)}\!</math>
|- style="height:40px; background:#f0f0ff"
| <math>\text{Property}\!</math>
| <math>\text{Sign of Property}\!</math>
|-
| valign="bottom" width="50%" |
<math>\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}</math>
| valign="bottom" width="50%" |
<math>\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}</math>
|}
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:70%"
|+ style="height:30px" |
<math>\text{Table 51.3} ~~ \text{Notations for Properties and Their Signs (3)}\!</math>
|- style="height:40px; background:#f0f0ff"
| <math>\text{Property}\!</math>
| <math>\text{Sign of Property}\!</math>
|-
| valign="bottom" width="50%" |
<math>\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}</math>
| valign="bottom" width="50%" |
<math>\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}</math>
|}
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:70%"
|+ style="height:30px" |
<math>\text{Table 52.1} ~~ \text{Notations for Instances and Their Signs (1)}\!</math>
|- style="height:40px; background:#f0f0ff"
| <math>\text{Instance}\!</math>
| <math>\text{Sign of Instance}\!</math>
|-
| valign="bottom" width="50%" |
<math>\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}</math>
| valign="bottom" width="50%" |
<math>\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}</math>
|}
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:70%"
|+ style="height:30px" |
<math>\text{Table 52.2} ~~ \text{Notations for Instances and Their Signs (2)}\!</math>
|- style="height:40px; background:#f0f0ff"
| <math>\text{Instance}\!</math>
| <math>\text{Sign of Instance}\!</math>
|-
| valign="bottom" width="50%" |
<math>\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}</math>
| valign="bottom" width="50%" |
<math>\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}</math>
|}
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:70%"
|+ style="height:30px" |
<math>\text{Table 52.3} ~~ \text{Notations for Instances and Their Signs (3)}\!</math>
|- style="height:40px; background:#f0f0ff"
| <math>\text{Instance}\!</math>
| <math>\text{Sign of Instance}\!</math>
|-
| valign="bottom" width="50%" |
<math>\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}</math>
| valign="bottom" width="50%" |
<math>\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}</math>
|}
<br>
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 ''<math>{\mathcal{O}(n)}\!</math> coding'', because it uses a number on the order of <math>n\!</math> logical features to represent a domain of <math>n\!</math> 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 <math>{\mathcal{O}(n)}\!</math> 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 <math>O = \{ \text{A}, \text{B} \}\!</math> and the signs in <math>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} \},\!</math> 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 <math>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} \}\!</math> 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 <math>W\!</math> as a logical alphabet <math>\underline{\underline{W}},\!</math> taking element-wise identifications as follows:
{| align="center" cellspacing="8" width="90%"
|
<math>\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}</math>
|}
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 <math>W = W(\text{A}, \text{B}),\!</math> that is, for the set of objects, signs, and interpretants that are common to the sign relations <math>L(A)\!</math> and <math>L(B).\!</math> Successive columns of these Tables give the ''mnemonic code'', the ''pragmatic code'', and the ''abstract code'', respectively, for each element.
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"
|+ style="height:30px" |
<math>\text{Table 53.1} ~~ \text{Elements of} ~ \operatorname{ER}(W)\!</math>
|- style="background:#f0f0ff"
| <math>\text{Mnemonic Element}\!</math> <br><br> <math>w \in W\!</math>
| <math>\text{Pragmatic Element}\!</math> <br><br> <math>w \in W\!</math>
| <math>\text{Abstract Element}\!</math> <br><br> <math>w_i \in W\!</math>
|-
| valign="bottom" width="33%" |
<math>\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}</math>
| valign="bottom" width="33%" |
<math>\begin{matrix}
o_1
\\[4pt]
o_2
\\[4pt]
s_1
\\[4pt]
s_2
\\[4pt]
s_3
\\[4pt]
s_4
\end{matrix}</math>
| valign="bottom" width="33%" |
<math>\begin{matrix}
w_1
\\[4pt]
w_2
\\[4pt]
w_3
\\[4pt]
w_4
\\[4pt]
w_5
\\[4pt]
w_6
\end{matrix}</math>
|}
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"
|+ style="height:30px" |
<math>\text{Table 53.2} ~~ \text{Features of} ~ \operatorname{LIR}(W)\!</math>
|- style="background:#f0f0ff"
|
<math>\text{Mnemonic Feature}\!</math><br><br>
<math>\underline{\underline{w}} \in \underline{\underline{W}}\!</math>
|
<math>\text{Pragmatic Feature}\!</math><br><br>
<math>\underline{\underline{w}} \in \underline{\underline{W}}\!</math>
|
<math>\text{Abstract Feature}\!</math><br><br>
<math>\underline{\underline{w_i}} \in \underline{\underline{W}}\!</math>
|-
| valign="bottom" width="33%" |
<math>\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}</math>
| valign="bottom" width="33%" |
<math>\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}</math>
| valign="bottom" width="33%" |
<math>\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}</math>
|}
<br>
If the world of <math>\text{A}\!</math> and <math>\text{B},\!</math> the set <math>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} \},\!</math> is viewed abstractly, as an arbitrary set of six atomic points, then there are exactly <math>2^6 = 64\!</math> ''abstract properties'' or ''potential attributes'' that might be applied to or recognized in these points. The elements of <math>W\!</math> that possess a given property form a subset of <math>W\!</math> called the ''extension'' of that property. Thus the extensions of abstract properties are exactly the subsets of <math>W.\!</math> The set of all subsets of <math>W\!</math> is called the ''power set'' of <math>W,\!</math> notated as <math>\operatorname{Pow}(W)\!</math> or <math>\mathcal{P}(W).\!</math> 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 <math>\varnothing = \{ \}.\!</math> 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 <math>W\!</math> as vectors in the coordinate space <math>\underline{W}\!</math> and as singular propositions in the universe of discourse <math>W^\Box.\!</math> Altogether, these Tables present the ''literal'' codes for the elements of <math>\underline{W}\!</math> and <math>W^\circ\!</math> in their ''mnemonic'', ''pragmatic'', and ''abstract'' versions, respectively. In each Table, Column 1 lists the element <math>w \in W,\!</math> while Column 2 gives the corresponding coordinate vector <math>\underline{w} \in \underline{W}\!</math> in the form of a bit string. The next two Columns represent each <math>w \in W\!</math> as a proposition in <math>W^\circ\!,</math> in effect, reconstituting it as a function <math>w : \underline{W} \to \mathbb{B}.</math> 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.
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
|+ style="height:30px" |
<math>\text{Table 54.1} ~~ \text{Mnemonic Literal Codes for Interpreters A and B}\!</math>
|- style="background:#f0f0ff"
| <math>\text{Element}\!</math>
| <math>\text{Vector}\!</math>
| <math>\text{Conjunct Term}\!</math>
| <math>\text{Code}\!</math>
|-
| valign="bottom" width="20%" |
<math>\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}</math>
| valign="bottom" width="20%" |
<math>\begin{matrix}
100000
\\[4pt]
010000
\\[4pt]
001000
\\[4pt]
000100
\\[4pt]
000010
\\[4pt]
000001
\end{matrix}</math>
| valign="bottom" width="40%" |
<math>\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}</math>
| valign="bottom" width="20%" |
<math>\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}</math>
|}
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
|+ style="height:30px" |
<math>\text{Table 54.2} ~~ \text{Pragmatic Literal Codes for Interpreters A and B}\!</math>
|- style="background:#f0f0ff"
| <math>\text{Element}\!</math>
| <math>\text{Vector}\!</math>
| <math>\text{Conjunct Term}\!</math>
| <math>\text{Code}\!</math>
|-
| valign="bottom" width="20%" |
<math>\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}</math>
| valign="bottom" width="20%" |
<math>\begin{matrix}
100000
\\[4pt]
010000
\\[4pt]
001000
\\[4pt]
000100
\\[4pt]
000010
\\[4pt]
000001
\end{matrix}</math>
| valign="bottom" width="40%" |
<math>\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}</math>
| valign="bottom" width="20%" |
<math>\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}</math>
|}
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
|+ style="height:30px" |
<math>\text{Table 54.3} ~~ \text{Abstract Literal Codes for Interpreters A and B}\!</math>
|- style="background:#f0f0ff"
| <math>\text{Element}\!</math>
| <math>\text{Vector}\!</math>
| <math>\text{Conjunct Term}\!</math>
| <math>\text{Code}\!</math>
|-
| valign="bottom" width="20%" |
<math>\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}</math>
| valign="bottom" width="20%" |
<math>\begin{matrix}
100000
\\[4pt]
010000
\\[4pt]
001000
\\[4pt]
000100
\\[4pt]
000010
\\[4pt]
000001
\end{matrix}</math>
| valign="bottom" width="40%" |
<math>\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}</math>
| valign="bottom" width="20%" |
<math>\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}</math>
|}
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
|+ style="height:30px" |
<math>\text{Table 55.1} ~~ \operatorname{LIR}_1 (L_\text{A}) : \text{Literal Representation of} ~ L_\text{A}\!</math>
|- style="height:40px; background:#f0f0ff"
| width="33%" | <math>\text{Object}\!</math>
| width="33%" | <math>\text{Sign}\!</math>
| width="33%" | <math>\text{Interpretant}\!</math>
|-
| valign="bottom" |
<math>\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}</math>
| valign="bottom" |
<math>\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}</math>
| valign="bottom" |
<math>\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}</math>
|-
| valign="bottom" |
<math>\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}</math>
| valign="bottom" |
<math>\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}</math>
| valign="bottom" |
<math>\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}\!</math>
|}
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
|+ style="height:30px" |
<math>\text{Table 55.2} ~~ \operatorname{LIR}_1 (\operatorname{Den}(L_\text{A})) : \text{Denotative Component of} ~ L_\text{A}\!</math>
|- style="height:40px; background:#f0f0ff"
| width="33%" | <math>\text{Object}\!</math>
| width="33%" | <math>\text{Sign}\!</math>
| width="33%" | <math>\text{Transition}\!</math>
|-
| valign="bottom" |
<math>\begin{matrix}
{\langle\underline{\underline{\text{A}}}\rangle}_W
\\[4pt]
{\langle\underline{\underline{\text{A}}}\rangle}_W
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{\langle\underline{\underline{\text{a}}}\rangle}_W
\\[4pt]
{\langle\underline{\underline{\text{i}}}\rangle}_W
\end{matrix}\!</math>
| valign="bottom" |
<math>\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}</math>
|-
| valign="bottom" |
<math>\begin{matrix}
{\langle\underline{\underline{\text{B}}}\rangle}_W
\\[4pt]
{\langle\underline{\underline{\text{B}}}\rangle}_W
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{\langle\underline{\underline{\text{b}}}\rangle}_W
\\[4pt]
{\langle\underline{\underline{\text{u}}}\rangle}_W
\end{matrix}</math>
| valign="bottom" |
<math>\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}</math>
|}
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
|+ style="height:30px" |
<math>\text{Table 55.3} ~~ \operatorname{LIR}_1 (\operatorname{Con}(L_\text{A})) : \text{Connotative Component of} ~ L_\text{A}\!</math>
|- style="height:40px; background:#f0f0ff"
| width="33%" | <math>\text{Sign}\!</math>
| width="33%" | <math>\text{Interpretant}\!</math>
| width="33%" | <math>\text{Transition}\!</math>
|-
| valign="bottom" |
<math>\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}</math>
| valign="bottom" |
<math>\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}</math>
| valign="bottom" |
<math>\begin{matrix}
0_{\operatorname{d}W}
\\[4pt]
{\langle
\operatorname{d}\underline{\underline{\text{a}}}
~
\operatorname{d}\underline{\underline{\text{i}}}
\rangle}_{\operatorname{d}W}
\\[4pt]
{\langle
\operatorname{d}\underline{\underline{\text{a}}}
~
\operatorname{d}\underline{\underline{\text{i}}}
\rangle}_{\operatorname{d}W}
\\[4pt]
0_{\operatorname{d}W}
\end{matrix}</math>
|-
| valign="bottom" |
<math>\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}</math>
| valign="bottom" |
<math>\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}\!</math>
| valign="bottom" |
<math>\begin{matrix}
0_{\operatorname{d}W}
\\[4pt]
{\langle
\operatorname{d}\underline{\underline{\text{b}}}
~
\operatorname{d}\underline{\underline{\text{u}}}
\rangle}_{\operatorname{d}W}
\\[4pt]
{\langle
\operatorname{d}\underline{\underline{\text{b}}}
~
\operatorname{d}\underline{\underline{\text{u}}}
\rangle}_{\operatorname{d}W}
\\[4pt]
0_{\operatorname{d}W}
\end{matrix}</math>
|}
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
|+ style="height:30px" |
<math>\text{Table 56.1} ~~ \operatorname{LIR}_1 (L_\text{B}) : \text{Literal Representation of} ~ L_\text{B}\!</math>
|- style="height:40px; background:#f0f0ff"
| width="33%" | <math>\text{Object}\!</math>
| width="33%" | <math>\text{Sign}\!</math>
| width="33%" | <math>\text{Interpretant}\!</math>
|-
| valign="bottom" |
<math>\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}</math>
| valign="bottom" |
<math>\begin{matrix}
{\langle\underline{\underline{\text{a}}}\rangle}_W
\\[4pt]
{\langle\underline{\underline{\text{a}}}\rangle}_W
\\[4pt]
{\langle\underline{\underline{\text{u}}}\rangle}_W
\\[4pt]
{\langle\underline{\underline{\text{u}}}\rangle}_W
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{\langle\underline{\underline{\text{a}}}\rangle}_W
\\[4pt]
{\langle\underline{\underline{\text{u}}}\rangle}_W
\\[4pt]
{\langle\underline{\underline{\text{a}}}\rangle}_W
\\[4pt]
{\langle\underline{\underline{\text{u}}}\rangle}_W
\end{matrix}</math>
|-
| valign="bottom" |
<math>\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}</math>
| valign="bottom" |
<math>\begin{matrix}
{\langle\underline{\underline{\text{b}}}\rangle}_W
\\[4pt]
{\langle\underline{\underline{\text{b}}}\rangle}_W
\\[4pt]
{\langle\underline{\underline{\text{i}}}\rangle}_W
\\[4pt]
{\langle\underline{\underline{\text{i}}}\rangle}_W
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{\langle\underline{\underline{\text{b}}}\rangle}_W
\\[4pt]
{\langle\underline{\underline{\text{i}}}\rangle}_W
\\[4pt]
{\langle\underline{\underline{\text{b}}}\rangle}_W
\\[4pt]
{\langle\underline{\underline{\text{i}}}\rangle}_W
\end{matrix}\!</math>
|}
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
|+ style="height:30px" |
<math>\text{Table 56.2} ~~ \operatorname{LIR}_1 (\operatorname{Den}(L_\text{B})) : \text{Denotative Component of} ~ L_\text{B}\!</math>
|- style="height:40px; background:#f0f0ff"
| width="33%" | <math>\text{Object}\!</math>
| width="33%" | <math>\text{Sign}\!</math>
| width="33%" | <math>\text{Transition}\!</math>
|-
| valign="bottom" |
<math>\begin{matrix}
{\langle\underline{\underline{\text{A}}}\rangle}_W
\\[4pt]
{\langle\underline{\underline{\text{A}}}\rangle}_W
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{\langle\underline{\underline{\text{a}}}\rangle}_W
\\[4pt]
{\langle\underline{\underline{\text{u}}}\rangle}_W
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
({\langle\underline{\underline{\text{a}}}\rangle}_W,
{\langle\underline{\underline{\text{A}}}\rangle}_W)
\\[4pt]
({\langle\underline{\underline{\text{u}}}\rangle}_W,
{\langle\underline{\underline{\text{A}}}\rangle}_W)
\end{matrix}</math>
|-
| valign="bottom" |
<math>\begin{matrix}
{\langle\underline{\underline{\text{B}}}\rangle}_W
\\[4pt]
{\langle\underline{\underline{\text{B}}}\rangle}_W
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{\langle\underline{\underline{\text{b}}}\rangle}_W
\\[4pt]
{\langle\underline{\underline{\text{i}}}\rangle}_W
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
({\langle\underline{\underline{\text{b}}}\rangle}_W,
{\langle\underline{\underline{\text{B}}}\rangle}_W)
\\[4pt]
({\langle\underline{\underline{\text{i}}}\rangle}_W,
{\langle\underline{\underline{\text{B}}}\rangle}_W)
\end{matrix}</math>
|}
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
|+ style="height:30px" |
<math>\text{Table 56.3} ~~ \operatorname{LIR}_1 (\operatorname{Con}(L_\text{B})) : \text{Connotative Component of} ~ L_\text{B}\!</math>
|- style="height:40px; background:#f0f0ff"
| width="33%" | <math>\text{Sign}\!</math>
| width="33%" | <math>\text{Interpretant}\!</math>
| width="33%" | <math>\text{Transition}\!</math>
|-
| valign="bottom" |
<math>\begin{matrix}
{\langle\underline{\underline{\text{a}}}\rangle}_W
\\[4pt]
{\langle\underline{\underline{\text{a}}}\rangle}_W
\\[4pt]
{\langle\underline{\underline{\text{u}}}\rangle}_W
\\[4pt]
{\langle\underline{\underline{\text{u}}}\rangle}_W
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{\langle\underline{\underline{\text{a}}}\rangle}_W
\\[4pt]
{\langle\underline{\underline{\text{u}}}\rangle}_W
\\[4pt]
{\langle\underline{\underline{\text{a}}}\rangle}_W
\\[4pt]
{\langle\underline{\underline{\text{u}}}\rangle}_W
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
0_{\operatorname{d}W}
\\[4pt]
{\langle
\operatorname{d}\underline{\underline{\text{a}}}
~
\operatorname{d}\underline{\underline{\text{u}}}
\rangle}_{\operatorname{d}W}
\\[4pt]
{\langle
\operatorname{d}\underline{\underline{\text{a}}}
~
\operatorname{d}\underline{\underline{\text{u}}}
\rangle}_{\operatorname{d}W}
\\[4pt]
0_{\operatorname{d}W}
\end{matrix}</math>
|-
| valign="bottom" |
<math>\begin{matrix}
{\langle\underline{\underline{\text{b}}}\rangle}_W
\\[4pt]
{\langle\underline{\underline{\text{b}}}\rangle}_W
\\[4pt]
{\langle\underline{\underline{\text{i}}}\rangle}_W
\\[4pt]
{\langle\underline{\underline{\text{i}}}\rangle}_W
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{\langle\underline{\underline{\text{b}}}\rangle}_W
\\[4pt]
{\langle\underline{\underline{\text{i}}}\rangle}_W
\\[4pt]
{\langle\underline{\underline{\text{b}}}\rangle}_W
\\[4pt]
{\langle\underline{\underline{\text{i}}}\rangle}_W
\end{matrix}\!</math>
| valign="bottom" |
<math>\begin{matrix}
0_{\operatorname{d}W}
\\[4pt]
{\langle
\operatorname{d}\underline{\underline{\text{b}}}
~
\operatorname{d}\underline{\underline{\text{i}}}
\rangle}_{\operatorname{d}W}
\\[4pt]
{\langle
\operatorname{d}\underline{\underline{\text{b}}}
~
\operatorname{d}\underline{\underline{\text{i}}}
\rangle}_{\operatorname{d}W}
\\[4pt]
0_{\operatorname{d}W}
\end{matrix}</math>
|}
<br>
'''Lateral Coding'''
For the domain <math>O = \{ \text{A}, \text{B} \}\!</math> of two elements:
{| align="center" cellspacing="8" width="90%"
| <math>X = \{ o_1, o_2 \} = \{ \text{A}, \text{B} \}\!</math>
|}
'''…'''
For the domain <math>S = I = \{ {}^{\backprime\backprime} \text{A} {}^{\prime\prime}, {}^{\backprime\backprime} \text{B} {}^{\prime\prime}, {}^{\backprime\backprime} \text{i} {}^{\prime\prime}, {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \}\!</math> of four elements one needs to use four logical features, in effect, elevating each individual sign to the status of an exclusive grammatical category in its own right. The easiest way to do this is simply to reuse the syntactic domain <math>S = I\!</math> as a logical alphabet <math>\underline{\underline{Y}},\!</math> taking element-wise identifications as follows:
{| align="center" cellspacing="8" width="90%"
|
<math>\begin{array}{*{11}{c}}
Y
& = &
\{ &
s_1
& , &
s_2
& , &
s_3
& , &
s_4
& \}
\\
& = &
\{ &
{}^{\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{Y}}
& = &
\{ &
\underline{\underline{s_1}}
& , &
\underline{\underline{s_2}}
& , &
\underline{\underline{s_3}}
& , &
\underline{\underline{s_4}}
& \}
\\
& = &
\{ &
\underline{\underline{{}^{\backprime\backprime} \text{A} {}^{\prime\prime}}}
& , &
\underline{\underline{{}^{\backprime\backprime} \text{B} {}^{\prime\prime}}}
& , &
\underline{\underline{{}^{\backprime\backprime} \text{i} {}^{\prime\prime}}}
& , &
\underline{\underline{{}^{\backprime\backprime} \text{u} {}^{\prime\prime}}}
& \}
\end{array}\!</math>
|}
Tables 57.1, 57.2, and 57.3 show several ways of representing the elements of <math>O\!</math> and <math>S,\!</math> presenting the ''lateral'' codes for world elements in their ''mnemonic'', ''pragmatic'', and ''abstract'' versions, respectively. In each Table, Column 2 gives the coordinate vector <math>\underline{x} \in \underline{X}\!</math> or <math>\underline{y} \in \underline{Y}\!</math> as a bit string, using a subscript to indicate the relevant space, <math>\underline{X}\!</math> or <math>\underline{Y}.\!</math> Column 3 lists 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, using doubly underlined capital letters for literal features of objects and doubly underlined lower case letters for literal features of quoted signs. Finally, Column 4 shows the compact code for each element, using a conjunction of positive features in subscripted angle brackets to represent the corresponding conjunct term as a singular proposition.
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
|+ style="height:30px" |
<math>\text{Table 57.1} ~~ \text{Mnemonic Lateral Codes for Interpreters A and B}\!</math>
|- style="background:#f0f0ff"
| <math>\text{Element}\!</math>
| <math>\text{Vector}\!</math>
| <math>\text{Conjunct Term}\!</math>
| <math>\text{Code}\!</math>
|-
| valign="bottom" width="20%" |
<math>\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}</math>
| valign="bottom" width="20%" |
<math>\begin{matrix}
{10}_X
\\[4pt]
{01}_X
\\[4pt]
{1000}_Y
\\[4pt]
{0100}_Y
\\[4pt]
{0010}_Y
\\[4pt]
{0001}_Y
\end{matrix}</math>
| valign="bottom" width="40%" |
<math>\begin{matrix}
~\underline{\underline{A}}~
(\underline{\underline{B}})
\\[4pt]
(\underline{\underline{A}})
~\underline{\underline{B}}~
\\[4pt]
~\underline{\underline{a}}~
(\underline{\underline{b}})
(\underline{\underline{i}})
(\underline{\underline{u}})
\\[4pt]
(\underline{\underline{a}})
~\underline{\underline{b}}~
(\underline{\underline{i}})
(\underline{\underline{u}})
\\[4pt]
(\underline{\underline{a}})
(\underline{\underline{b}})
~\underline{\underline{i}}~
(\underline{\underline{u}})
\\[4pt]
(\underline{\underline{a}})
(\underline{\underline{b}})
(\underline{\underline{i}})
~\underline{\underline{u}}~
\end{matrix}</math>
| valign="bottom" width="20%" |
<math>\begin{matrix}
{\langle\underline{\underline{A}}\rangle}_X
\\[4pt]
{\langle\underline{\underline{B}}\rangle}_X
\\[4pt]
{\langle\underline{\underline{a}}\rangle}_Y
\\[4pt]
{\langle\underline{\underline{b}}\rangle}_Y
\\[4pt]
{\langle\underline{\underline{i}}\rangle}_Y
\\[4pt]
{\langle\underline{\underline{u}}\rangle}_Y
\end{matrix}</math>
|}
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
|+ style="height:30px" |
<math>\text{Table 57.2} ~~ \text{Pragmatic Lateral Codes for Interpreters A and B}\!</math>
|- style="background:#f0f0ff"
| <math>\text{Element}\!</math>
| <math>\text{Vector}\!</math>
| <math>\text{Conjunct Term}\!</math>
| <math>\text{Code}\!</math>
|-
| valign="bottom" width="20%" |
<math>\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}</math>
| valign="bottom" width="20%" |
<math>\begin{matrix}
{10}_X
\\[4pt]
{01}_X
\\[4pt]
{1000}_Y
\\[4pt]
{0100}_Y
\\[4pt]
{0010}_Y
\\[4pt]
{0001}_Y
\end{matrix}</math>
| valign="bottom" width="40%" |
<math>\begin{matrix}
~\underline{\underline{o_1}}~
(\underline{\underline{o_2}})
\\[4pt]
(\underline{\underline{o_1}})
~\underline{\underline{o_2}}~
\\[4pt]
~\underline{\underline{s_1}}~
(\underline{\underline{s_2}})
(\underline{\underline{s_3}})
(\underline{\underline{s_4}})
\\[4pt]
(\underline{\underline{s_1}})
~\underline{\underline{s_2}}~
(\underline{\underline{s_3}})
(\underline{\underline{s_4}})
\\[4pt]
(\underline{\underline{s_1}})
(\underline{\underline{s_2}})
~\underline{\underline{s_3}}~
(\underline{\underline{s_4}})
\\[4pt]
(\underline{\underline{s_1}})
(\underline{\underline{s_2}})
(\underline{\underline{s_3}})
~\underline{\underline{s_4}}~
\end{matrix}</math>
| valign="bottom" width="20%" |
<math>\begin{matrix}
{\langle\underline{\underline{o_1}}\rangle}_X
\\[4pt]
{\langle\underline{\underline{o_2}}\rangle}_X
\\[4pt]
{\langle\underline{\underline{s_1}}\rangle}_Y
\\[4pt]
{\langle\underline{\underline{s_2}}\rangle}_Y
\\[4pt]
{\langle\underline{\underline{s_3}}\rangle}_Y
\\[4pt]
{\langle\underline{\underline{s_4}}\rangle}_Y
\end{matrix}</math>
|}
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
|+ style="height:30px" |
<math>\text{Table 57.3} ~~ \text{Abstract Lateral Codes for Interpreters A and B}\!</math>
|- style="background:#f0f0ff"
| <math>\text{Element}\!</math>
| <math>\text{Vector}\!</math>
| <math>\text{Conjunct Term}\!</math>
| <math>\text{Code}\!</math>
|-
| valign="bottom" width="20%" |
<math>\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}</math>
| valign="bottom" width="20%" |
<math>\begin{matrix}
{10}_X
\\[4pt]
{01}_X
\\[4pt]
{1000}_Y
\\[4pt]
{0100}_Y
\\[4pt]
{0010}_Y
\\[4pt]
{0001}_Y
\end{matrix}</math>
| valign="bottom" width="40%" |
<math>\begin{matrix}
~\underline{\underline{x_1}}~
(\underline{\underline{x_2}})
\\[4pt]
(\underline{\underline{x_1}})
~\underline{\underline{x_2}}~
\\[4pt]
~\underline{\underline{y_1}}~
(\underline{\underline{y_2}})
(\underline{\underline{y_3}})
(\underline{\underline{y_4}})
\\[4pt]
(\underline{\underline{y_1}})
~\underline{\underline{y_2}}~
(\underline{\underline{y_3}})
(\underline{\underline{y_4}})
\\[4pt]
(\underline{\underline{y_1}})
(\underline{\underline{y_2}})
~\underline{\underline{y_3}}~
(\underline{\underline{y_4}})
\\[4pt]
(\underline{\underline{y_1}})
(\underline{\underline{y_2}})
(\underline{\underline{y_3}})
~\underline{\underline{y_4}}~
\end{matrix}</math>
| valign="bottom" width="20%" |
<math>\begin{matrix}
{\langle\underline{\underline{x_1}}\rangle}_X
\\[4pt]
{\langle\underline{\underline{x_2}}\rangle}_X
\\[4pt]
{\langle\underline{\underline{y_1}}\rangle}_Y
\\[4pt]
{\langle\underline{\underline{y_2}}\rangle}_Y
\\[4pt]
{\langle\underline{\underline{y_3}}\rangle}_Y
\\[4pt]
{\langle\underline{\underline{y_4}}\rangle}_Y
\end{matrix}</math>
|}
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
|+ style="height:30px" |
<math>\text{Table 58.1} ~~ \operatorname{LIR}_2 (L_\text{A}) : \text{Lateral Representation of} ~ L_\text{A}\!</math>
|- style="height:40px; background:#f0f0ff"
| width="33%" | <math>\text{Object}\!</math>
| width="33%" | <math>\text{Sign}\!</math>
| width="33%" | <math>\text{Interpretant}\!</math>
|-
| valign="bottom" |
<math>\begin{matrix}
~\underline{\underline{\text{A}}}~
(\underline{\underline{\text{B}}})
\\[4pt]
~\underline{\underline{\text{A}}}~
(\underline{\underline{\text{B}}})
\\[4pt]
~\underline{\underline{\text{A}}}~
(\underline{\underline{\text{B}}})
\\[4pt]
~\underline{\underline{\text{A}}}~
(\underline{\underline{\text{B}}})
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
~\underline{\underline{\text{a}}}~
(\underline{\underline{\text{b}}})
(\underline{\underline{\text{i}}})
(\underline{\underline{\text{u}}})
\\[4pt]
~\underline{\underline{\text{a}}}~
(\underline{\underline{\text{b}}})
(\underline{\underline{\text{i}}})
(\underline{\underline{\text{u}}})
\\[4pt]
(\underline{\underline{\text{a}}})
(\underline{\underline{\text{b}}})
~\underline{\underline{\text{i}}}~
(\underline{\underline{\text{u}}})
\\[4pt]
(\underline{\underline{\text{a}}})
(\underline{\underline{\text{b}}})
~\underline{\underline{\text{i}}}~
(\underline{\underline{\text{u}}})
\end{matrix}\!</math>
| valign="bottom" |
<math>\begin{matrix}
~\underline{\underline{\text{a}}}~
(\underline{\underline{\text{b}}})
(\underline{\underline{\text{i}}})
(\underline{\underline{\text{u}}})
\\[4pt]
(\underline{\underline{\text{a}}})
(\underline{\underline{\text{b}}})
~\underline{\underline{\text{i}}}~
(\underline{\underline{\text{u}}})
\\[4pt]
~\underline{\underline{\text{a}}}~
(\underline{\underline{\text{b}}})
(\underline{\underline{\text{i}}})
(\underline{\underline{\text{u}}})
\\[4pt]
(\underline{\underline{\text{a}}})
(\underline{\underline{\text{b}}})
~\underline{\underline{\text{i}}}~
(\underline{\underline{\text{u}}})
\end{matrix}</math>
|-
| valign="bottom" |
<math>\begin{matrix}
(\underline{\underline{\text{A}}})
~\underline{\underline{\text{B}}}~
\\[4pt]
(\underline{\underline{\text{A}}})
~\underline{\underline{\text{B}}}~
\\[4pt]
(\underline{\underline{\text{A}}})
~\underline{\underline{\text{B}}}~
\\[4pt]
(\underline{\underline{\text{A}}})
~\underline{\underline{\text{B}}}~
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
(\underline{\underline{\text{a}}})
~\underline{\underline{\text{b}}}~
(\underline{\underline{\text{i}}})
(\underline{\underline{\text{u}}})
\\[4pt]
(\underline{\underline{\text{a}}})
~\underline{\underline{\text{b}}}~
(\underline{\underline{\text{i}}})
(\underline{\underline{\text{u}}})
\\[4pt]
(\underline{\underline{\text{a}}})
(\underline{\underline{\text{b}}})
(\underline{\underline{\text{i}}})
~\underline{\underline{\text{u}}}~
\\[4pt]
(\underline{\underline{\text{a}}})
(\underline{\underline{\text{b}}})
(\underline{\underline{\text{i}}})
~\underline{\underline{\text{u}}}~
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
(\underline{\underline{\text{a}}})
~\underline{\underline{\text{b}}}~
(\underline{\underline{\text{i}}})
(\underline{\underline{\text{u}}})
\\[4pt]
(\underline{\underline{\text{a}}})
(\underline{\underline{\text{b}}})
(\underline{\underline{\text{i}}})
~\underline{\underline{\text{u}}}~
\\[4pt]
(\underline{\underline{\text{a}}})
~\underline{\underline{\text{b}}}~
(\underline{\underline{\text{i}}})
(\underline{\underline{\text{u}}})
\\[4pt]
(\underline{\underline{\text{a}}})
(\underline{\underline{\text{b}}})
(\underline{\underline{\text{i}}})
~\underline{\underline{\text{u}}}~
\end{matrix}</math>
|}
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
|+ style="height:30px" |
<math>\text{Table 58.2} ~~ \operatorname{LIR}_2 (\operatorname{Den}(L_\text{A})) : \text{Denotative Component of} ~ L_\text{A}\!</math>
|- style="height:40px; background:#f0f0ff"
| width="33%" | <math>\text{Object}\!</math>
| width="33%" | <math>\text{Sign}\!</math>
| width="33%" | <math>\text{Transition}\!</math>
|-
| valign="bottom" |
<math>\begin{matrix}
~\underline{\underline{\text{A}}}~
(\underline{\underline{\text{B}}})
\\[4pt]
~\underline{\underline{\text{A}}}~
(\underline{\underline{\text{B}}})
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
~\underline{\underline{\text{a}}}~
(\underline{\underline{\text{b}}})
(\underline{\underline{\text{i}}})
(\underline{\underline{\text{u}}})
\\[4pt]
(\underline{\underline{\text{a}}})
(\underline{\underline{\text{b}}})
~\underline{\underline{\text{i}}}~
(\underline{\underline{\text{u}}})
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
({\langle\underline{\underline{\text{a}}}\rangle}_Y,
{\langle\underline{\underline{\text{A}}}\rangle}_X)
\\[4pt]
({\langle\underline{\underline{\text{i}}}\rangle}_Y,
{\langle\underline{\underline{\text{A}}}\rangle}_X)
\end{matrix}</math>
|-
| valign="bottom" |
<math>\begin{matrix}
(\underline{\underline{\text{A}}})
~\underline{\underline{\text{B}}}~
\\[4pt]
(\underline{\underline{\text{A}}})
~\underline{\underline{\text{B}}}~
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
(\underline{\underline{\text{a}}})
~\underline{\underline{\text{b}}}~
(\underline{\underline{\text{i}}})
(\underline{\underline{\text{u}}})
\\[4pt]
(\underline{\underline{\text{a}}})
(\underline{\underline{\text{b}}})
(\underline{\underline{\text{i}}})
~\underline{\underline{\text{u}}}~
\end{matrix}\!</math>
| valign="bottom" |
<math>\begin{matrix}
({\langle\underline{\underline{\text{b}}}\rangle}_Y,
{\langle\underline{\underline{\text{B}}}\rangle}_X)
\\[4pt]
({\langle\underline{\underline{\text{u}}}\rangle}_Y,
{\langle\underline{\underline{\text{B}}}\rangle}_X)
\end{matrix}</math>
|}
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
|+ style="height:30px" |
<math>\text{Table 58.3} ~~ \operatorname{LIR}_2 (\operatorname{Con}(L_\text{A})) : \text{Connotative Component of} ~ L_\text{A}\!</math>
|- style="height:40px; background:#f0f0ff"
| width="33%" | <math>\text{Sign}\!</math>
| width="33%" | <math>\text{Interpretant}\!</math>
| width="33%" | <math>\text{Transition}\!</math>
|-
| valign="bottom" |
<math>\begin{matrix}
~\underline{\underline{\text{a}}}~
(\underline{\underline{\text{b}}})
(\underline{\underline{\text{i}}})
(\underline{\underline{\text{u}}})
\\[4pt]
~\underline{\underline{\text{a}}}~
(\underline{\underline{\text{b}}})
(\underline{\underline{\text{i}}})
(\underline{\underline{\text{u}}})
\\[4pt]
(\underline{\underline{\text{a}}})
(\underline{\underline{\text{b}}})
~\underline{\underline{\text{i}}}~
(\underline{\underline{\text{u}}})
\\[4pt]
(\underline{\underline{\text{a}}})
(\underline{\underline{\text{b}}})
~\underline{\underline{\text{i}}}~
(\underline{\underline{\text{u}}})
\end{matrix}\!</math>
| valign="bottom" |
<math>\begin{matrix}
~\underline{\underline{\text{a}}}~
(\underline{\underline{\text{b}}})
(\underline{\underline{\text{i}}})
(\underline{\underline{\text{u}}})
\\[4pt]
(\underline{\underline{\text{a}}})
(\underline{\underline{\text{b}}})
~\underline{\underline{\text{i}}}~
(\underline{\underline{\text{u}}})
\\[4pt]
~\underline{\underline{\text{a}}}~
(\underline{\underline{\text{b}}})
(\underline{\underline{\text{i}}})
(\underline{\underline{\text{u}}})
\\[4pt]
(\underline{\underline{\text{a}}})
(\underline{\underline{\text{b}}})
~\underline{\underline{\text{i}}}~
(\underline{\underline{\text{u}}})
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
(\underline{\underline{\text{da}}})
(\underline{\underline{\text{db}}})
(\underline{\underline{\text{di}}})
(\underline{\underline{\text{du}}})
\\[4pt]
~\underline{\underline{\text{da}}}~
(\underline{\underline{\text{db}}})
~\underline{\underline{\text{di}}}~
(\underline{\underline{\text{du}}})
\\[4pt]
~\underline{\underline{\text{da}}}~
(\underline{\underline{\text{db}}})
~\underline{\underline{\text{di}}}~
(\underline{\underline{\text{du}}})
\\[4pt]
(\underline{\underline{\text{da}}})
(\underline{\underline{\text{db}}})
(\underline{\underline{\text{di}}})
(\underline{\underline{\text{du}}})
\end{matrix}</math>
|-
| valign="bottom" |
<math>\begin{matrix}
(\underline{\underline{\text{a}}})
~\underline{\underline{\text{b}}}~
(\underline{\underline{\text{i}}})
(\underline{\underline{\text{u}}})
\\[4pt]
(\underline{\underline{\text{a}}})
~\underline{\underline{\text{b}}}~
(\underline{\underline{\text{i}}})
(\underline{\underline{\text{u}}})
\\[4pt]
(\underline{\underline{\text{a}}})
(\underline{\underline{\text{b}}})
(\underline{\underline{\text{i}}})
~\underline{\underline{\text{u}}}~
\\[4pt]
(\underline{\underline{\text{a}}})
(\underline{\underline{\text{b}}})
(\underline{\underline{\text{i}}})
~\underline{\underline{\text{u}}}~
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
(\underline{\underline{\text{a}}})
~\underline{\underline{\text{b}}}~
(\underline{\underline{\text{i}}})
(\underline{\underline{\text{u}}})
\\[4pt]
(\underline{\underline{\text{a}}})
(\underline{\underline{\text{b}}})
(\underline{\underline{\text{i}}})
~\underline{\underline{\text{u}}}~
\\[4pt]
(\underline{\underline{\text{a}}})
~\underline{\underline{\text{b}}}~
(\underline{\underline{\text{i}}})
(\underline{\underline{\text{u}}})
\\[4pt]
(\underline{\underline{\text{a}}})
(\underline{\underline{\text{b}}})
(\underline{\underline{\text{i}}})
~\underline{\underline{\text{u}}}~
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
(\underline{\underline{\text{da}}})
(\underline{\underline{\text{db}}})
(\underline{\underline{\text{di}}})
(\underline{\underline{\text{du}}})
\\[4pt]
(\underline{\underline{\text{da}}})
~\underline{\underline{\text{db}}}~
(\underline{\underline{\text{di}}})
~\underline{\underline{\text{du}}}~
\\[4pt]
(\underline{\underline{\text{da}}})
~\underline{\underline{\text{db}}}~
(\underline{\underline{\text{di}}})
~\underline{\underline{\text{du}}}~
\\[4pt]
(\underline{\underline{\text{da}}})
(\underline{\underline{\text{db}}})
(\underline{\underline{\text{di}}})
(\underline{\underline{\text{du}}})
\end{matrix}</math>
|}
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
|+ style="height:30px" |
<math>\text{Table 59.1} ~~ \operatorname{LIR}_2 (L_\text{B}) : \text{Lateral Representation of} ~ L_\text{B}\!</math>
|- style="height:40px; background:#f0f0ff"
| width="33%" | <math>\text{Object}\!</math>
| width="33%" | <math>\text{Sign}\!</math>
| width="33%" | <math>\text{Interpretant}\!</math>
|-
| valign="bottom" |
<math>\begin{matrix}
~\underline{\underline{\text{A}}}~
(\underline{\underline{\text{B}}})
\\[4pt]
~\underline{\underline{\text{A}}}~
(\underline{\underline{\text{B}}})
\\[4pt]
~\underline{\underline{\text{A}}}~
(\underline{\underline{\text{B}}})
\\[4pt]
~\underline{\underline{\text{A}}}~
(\underline{\underline{\text{B}}})
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
~\underline{\underline{\text{a}}}~
(\underline{\underline{\text{b}}})
(\underline{\underline{\text{i}}})
(\underline{\underline{\text{u}}})
\\[4pt]
~\underline{\underline{\text{a}}}~
(\underline{\underline{\text{b}}})
(\underline{\underline{\text{i}}})
(\underline{\underline{\text{u}}})
\\[4pt]
(\underline{\underline{\text{a}}})
(\underline{\underline{\text{b}}})
(\underline{\underline{\text{i}}})
~\underline{\underline{\text{u}}}~
\\[4pt]
(\underline{\underline{\text{a}}})
(\underline{\underline{\text{b}}})
(\underline{\underline{\text{i}}})
~\underline{\underline{\text{u}}}~
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
~\underline{\underline{\text{a}}}~
(\underline{\underline{\text{b}}})
(\underline{\underline{\text{i}}})
(\underline{\underline{\text{u}}})
\\[4pt]
(\underline{\underline{\text{a}}})
(\underline{\underline{\text{b}}})
(\underline{\underline{\text{i}}})
~\underline{\underline{\text{u}}}~
\\[4pt]
~\underline{\underline{\text{a}}}~
(\underline{\underline{\text{b}}})
(\underline{\underline{\text{i}}})
(\underline{\underline{\text{u}}})
\\[4pt]
(\underline{\underline{\text{a}}})
(\underline{\underline{\text{b}}})
(\underline{\underline{\text{i}}})
~\underline{\underline{\text{u}}}~
\end{matrix}</math>
|-
| valign="bottom" |
<math>\begin{matrix}
(\underline{\underline{\text{A}}})
~\underline{\underline{\text{B}}}~
\\[4pt]
(\underline{\underline{\text{A}}})
~\underline{\underline{\text{B}}}~
\\[4pt]
(\underline{\underline{\text{A}}})
~\underline{\underline{\text{B}}}~
\\[4pt]
(\underline{\underline{\text{A}}})
~\underline{\underline{\text{B}}}~
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
(\underline{\underline{\text{a}}})
~\underline{\underline{\text{b}}}~
(\underline{\underline{\text{i}}})
(\underline{\underline{\text{u}}})
\\[4pt]
(\underline{\underline{\text{a}}})
~\underline{\underline{\text{b}}}~
(\underline{\underline{\text{i}}})
(\underline{\underline{\text{u}}})
\\[4pt]
(\underline{\underline{\text{a}}})
(\underline{\underline{\text{b}}})
~\underline{\underline{\text{i}}}~
(\underline{\underline{\text{u}}})
\\[4pt]
(\underline{\underline{\text{a}}})
(\underline{\underline{\text{b}}})
~\underline{\underline{\text{i}}}~
(\underline{\underline{\text{u}}})
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
(\underline{\underline{\text{a}}})
~\underline{\underline{\text{b}}}~
(\underline{\underline{\text{i}}})
(\underline{\underline{\text{u}}})
\\[4pt]
(\underline{\underline{\text{a}}})
(\underline{\underline{\text{b}}})
~\underline{\underline{\text{i}}}~
(\underline{\underline{\text{u}}})
\\[4pt]
(\underline{\underline{\text{a}}})
~\underline{\underline{\text{b}}}~
(\underline{\underline{\text{i}}})
(\underline{\underline{\text{u}}})
\\[4pt]
(\underline{\underline{\text{a}}})
(\underline{\underline{\text{b}}})
~\underline{\underline{\text{i}}}~
(\underline{\underline{\text{u}}})
\end{matrix}</math>
|}
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
|+ style="height:30px" |
<math>\text{Table 59.2} ~~ \operatorname{LIR}_2 (\operatorname{Den}(L_\text{B})) : \text{Denotative Component of} ~ L_\text{B}\!</math>
|- style="height:40px; background:#f0f0ff"
| width="33%" | <math>\text{Object}\!</math>
| width="33%" | <math>\text{Sign}\!</math>
| width="33%" | <math>\text{Transition}\!</math>
|-
| valign="bottom" |
<math>\begin{matrix}
~\underline{\underline{\text{A}}}~
(\underline{\underline{\text{B}}})
\\[4pt]
~\underline{\underline{\text{A}}}~
(\underline{\underline{\text{B}}})
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
~\underline{\underline{\text{a}}}~
(\underline{\underline{\text{b}}})
(\underline{\underline{\text{i}}})
(\underline{\underline{\text{u}}})
\\[4pt]
(\underline{\underline{\text{a}}})
(\underline{\underline{\text{b}}})
(\underline{\underline{\text{i}}})
~\underline{\underline{\text{u}}}~
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
({\langle\underline{\underline{\text{a}}}\rangle}_Y,
{\langle\underline{\underline{\text{A}}}\rangle}_X)
\\[4pt]
({\langle\underline{\underline{\text{u}}}\rangle}_Y,
{\langle\underline{\underline{\text{A}}}\rangle}_X)
\end{matrix}</math>
|-
| valign="bottom" |
<math>\begin{matrix}
(\underline{\underline{\text{A}}})
~\underline{\underline{\text{B}}}~
\\[4pt]
(\underline{\underline{\text{A}}})
~\underline{\underline{\text{B}}}~
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
(\underline{\underline{\text{a}}})
~\underline{\underline{\text{b}}}~
(\underline{\underline{\text{i}}})
(\underline{\underline{\text{u}}})
\\[4pt]
(\underline{\underline{\text{a}}})
(\underline{\underline{\text{b}}})
~\underline{\underline{\text{i}}}~
(\underline{\underline{\text{u}}})
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
({\langle\underline{\underline{\text{b}}}\rangle}_Y,
{\langle\underline{\underline{\text{B}}}\rangle}_X)
\\[4pt]
({\langle\underline{\underline{\text{i}}}\rangle}_Y,
{\langle\underline{\underline{\text{B}}}\rangle}_X)
\end{matrix}</math>
|}
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
|+ style="height:30px" |
<math>\text{Table 59.3} ~~ \operatorname{LIR}_2 (\operatorname{Con}(L_\text{B})) : \text{Connotative Component of} ~ L_\text{B}\!</math>
|- style="height:40px; background:#f0f0ff"
| width="33%" | <math>\text{Sign}\!</math>
| width="33%" | <math>\text{Interpretant}\!</math>
| width="33%" | <math>\text{Transition}\!</math>
|-
| valign="bottom" |
<math>\begin{matrix}
~\underline{\underline{\text{a}}}~
(\underline{\underline{\text{b}}})
(\underline{\underline{\text{i}}})
(\underline{\underline{\text{u}}})
\\[4pt]
~\underline{\underline{\text{a}}}~
(\underline{\underline{\text{b}}})
(\underline{\underline{\text{i}}})
(\underline{\underline{\text{u}}})
\\[4pt]
(\underline{\underline{\text{a}}})
(\underline{\underline{\text{b}}})
(\underline{\underline{\text{i}}})
~\underline{\underline{\text{u}}}~
\\[4pt]
(\underline{\underline{\text{a}}})
(\underline{\underline{\text{b}}})
(\underline{\underline{\text{i}}})
~\underline{\underline{\text{u}}}~
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
~\underline{\underline{\text{a}}}~
(\underline{\underline{\text{b}}})
(\underline{\underline{\text{i}}})
(\underline{\underline{\text{u}}})
\\[4pt]
(\underline{\underline{\text{a}}})
(\underline{\underline{\text{b}}})
(\underline{\underline{\text{i}}})
~\underline{\underline{\text{u}}}~
\\[4pt]
~\underline{\underline{\text{a}}}~
(\underline{\underline{\text{b}}})
(\underline{\underline{\text{i}}})
(\underline{\underline{\text{u}}})
\\[4pt]
(\underline{\underline{\text{a}}})
(\underline{\underline{\text{b}}})
(\underline{\underline{\text{i}}})
~\underline{\underline{\text{u}}}~
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
(\underline{\underline{\text{da}}})
(\underline{\underline{\text{db}}})
(\underline{\underline{\text{di}}})
(\underline{\underline{\text{du}}})
\\[4pt]
~\underline{\underline{\text{da}}}~
(\underline{\underline{\text{db}}})
(\underline{\underline{\text{di}}})
~\underline{\underline{\text{du}}}~
\\[4pt]
~\underline{\underline{\text{da}}}~
(\underline{\underline{\text{db}}})
(\underline{\underline{\text{di}}})
~\underline{\underline{\text{du}}}~
\\[4pt]
(\underline{\underline{\text{da}}})
(\underline{\underline{\text{db}}})
(\underline{\underline{\text{di}}})
(\underline{\underline{\text{du}}})
\end{matrix}\!</math>
|-
| valign="bottom" |
<math>\begin{matrix}
(\underline{\underline{\text{a}}})
~\underline{\underline{\text{b}}}~
(\underline{\underline{\text{i}}})
(\underline{\underline{\text{u}}})
\\[4pt]
(\underline{\underline{\text{a}}})
~\underline{\underline{\text{b}}}~
(\underline{\underline{\text{i}}})
(\underline{\underline{\text{u}}})
\\[4pt]
(\underline{\underline{\text{a}}})
(\underline{\underline{\text{b}}})
~\underline{\underline{\text{i}}}~
(\underline{\underline{\text{u}}})
\\[4pt]
(\underline{\underline{\text{a}}})
(\underline{\underline{\text{b}}})
~\underline{\underline{\text{i}}}~
(\underline{\underline{\text{u}}})
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
(\underline{\underline{\text{a}}})
~\underline{\underline{\text{b}}}~
(\underline{\underline{\text{i}}})
(\underline{\underline{\text{u}}})
\\[4pt]
(\underline{\underline{\text{a}}})
(\underline{\underline{\text{b}}})
~\underline{\underline{\text{i}}}~
(\underline{\underline{\text{u}}})
\\[4pt]
(\underline{\underline{\text{a}}})
~\underline{\underline{\text{b}}}~
(\underline{\underline{\text{i}}})
(\underline{\underline{\text{u}}})
\\[4pt]
(\underline{\underline{\text{a}}})
(\underline{\underline{\text{b}}})
~\underline{\underline{\text{i}}}~
(\underline{\underline{\text{u}}})
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
(\underline{\underline{\text{da}}})
(\underline{\underline{\text{db}}})
(\underline{\underline{\text{di}}})
(\underline{\underline{\text{du}}})
\\[4pt]
(\underline{\underline{\text{da}}})
~\underline{\underline{\text{db}}}~
~\underline{\underline{\text{di}}}~
(\underline{\underline{\text{du}}})
\\[4pt]
(\underline{\underline{\text{da}}})
~\underline{\underline{\text{db}}}~
~\underline{\underline{\text{di}}}~
(\underline{\underline{\text{du}}})
\\[4pt]
(\underline{\underline{\text{da}}})
(\underline{\underline{\text{db}}})
(\underline{\underline{\text{di}}})
(\underline{\underline{\text{du}}})
\end{matrix}</math>
|}
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
|+ style="height:30px" |
<math>\text{Table 60.1} ~~ \operatorname{LIR}_3 (L_\text{A}) : \text{Lateral Representation of} ~ L_\text{A}\!</math>
|- style="height:40px; background:#f0f0ff"
| width="33%" | <math>\text{Object}\!</math>
| width="33%" | <math>\text{Sign}\!</math>
| width="33%" | <math>\text{Interpretant}\!</math>
|-
| valign="bottom" |
<math>\begin{matrix}
{\langle\underline{\underline{\text{A}}}\rangle}_X
\\[4pt]
{\langle\underline{\underline{\text{A}}}\rangle}_X
\\[4pt]
{\langle\underline{\underline{\text{A}}}\rangle}_X
\\[4pt]
{\langle\underline{\underline{\text{A}}}\rangle}_X
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{\langle\underline{\underline{\text{a}}}\rangle}_Y
\\[4pt]
{\langle\underline{\underline{\text{a}}}\rangle}_Y
\\[4pt]
{\langle\underline{\underline{\text{i}}}\rangle}_Y
\\[4pt]
{\langle\underline{\underline{\text{i}}}\rangle}_Y
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{\langle\underline{\underline{\text{a}}}\rangle}_Y
\\[4pt]
{\langle\underline{\underline{\text{i}}}\rangle}_Y
\\[4pt]
{\langle\underline{\underline{\text{a}}}\rangle}_Y
\\[4pt]
{\langle\underline{\underline{\text{i}}}\rangle}_Y
\end{matrix}</math>
|-
| valign="bottom" |
<math>\begin{matrix}
{\langle\underline{\underline{\text{B}}}\rangle}_X
\\[4pt]
{\langle\underline{\underline{\text{B}}}\rangle}_X
\\[4pt]
{\langle\underline{\underline{\text{B}}}\rangle}_X
\\[4pt]
{\langle\underline{\underline{\text{B}}}\rangle}_X
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{\langle\underline{\underline{\text{b}}}\rangle}_Y
\\[4pt]
{\langle\underline{\underline{\text{b}}}\rangle}_Y
\\[4pt]
{\langle\underline{\underline{\text{u}}}\rangle}_Y
\\[4pt]
{\langle\underline{\underline{\text{u}}}\rangle}_Y
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{\langle\underline{\underline{\text{b}}}\rangle}_Y
\\[4pt]
{\langle\underline{\underline{\text{u}}}\rangle}_Y
\\[4pt]
{\langle\underline{\underline{\text{b}}}\rangle}_Y
\\[4pt]
{\langle\underline{\underline{\text{u}}}\rangle}_Y
\end{matrix}</math>
|}
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
|+ style="height:30px" |
<math>\text{Table 60.2} ~~ \operatorname{LIR}_3 (\operatorname{Den}(L_\text{A})) : \text{Denotative Component of} ~ L_\text{A}\!</math>
|- style="height:40px; background:#f0f0ff"
| width="33%" | <math>\text{Object}\!</math>
| width="33%" | <math>\text{Sign}\!</math>
| width="33%" | <math>\text{Transition}\!</math>
|-
| valign="bottom" |
<math>\begin{matrix}
{\langle\underline{\underline{\text{A}}}\rangle}_X
\\[4pt]
{\langle\underline{\underline{\text{A}}}\rangle}_X
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{\langle\underline{\underline{\text{a}}}\rangle}_Y
\\[4pt]
{\langle\underline{\underline{\text{i}}}\rangle}_Y
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
({\langle\underline{\underline{\text{a}}}\rangle}_Y,
{\langle\underline{\underline{\text{A}}}\rangle}_X)
\\[4pt]
({\langle\underline{\underline{\text{i}}}\rangle}_Y,
{\langle\underline{\underline{\text{A}}}\rangle}_X)
\end{matrix}</math>
|-
| valign="bottom" |
<math>\begin{matrix}
{\langle\underline{\underline{\text{B}}}\rangle}_X
\\[4pt]
{\langle\underline{\underline{\text{B}}}\rangle}_X
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{\langle\underline{\underline{\text{b}}}\rangle}_Y
\\[4pt]
{\langle\underline{\underline{\text{u}}}\rangle}_Y
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
({\langle\underline{\underline{\text{b}}}\rangle}_Y,
{\langle\underline{\underline{\text{B}}}\rangle}_X)
\\[4pt]
({\langle\underline{\underline{\text{u}}}\rangle}_Y,
{\langle\underline{\underline{\text{B}}}\rangle}_X)
\end{matrix}</math>
|}
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
|+ style="height:30px" |
<math>\text{Table 60.3} ~~ \operatorname{LIR}_3 (\operatorname{Con}(L_\text{A})) : \text{Connotative Component of} ~ L_\text{A}\!</math>
|- style="height:40px; background:#f0f0ff"
| width="33%" | <math>\text{Sign}\!</math>
| width="33%" | <math>\text{Interpretant}\!</math>
| width="33%" | <math>\text{Transition}\!</math>
|-
| valign="bottom" |
<math>\begin{matrix}
{\langle\underline{\underline{\text{a}}}\rangle}_Y
\\[4pt]
{\langle\underline{\underline{\text{a}}}\rangle}_Y
\\[4pt]
{\langle\underline{\underline{\text{i}}}\rangle}_Y
\\[4pt]
{\langle\underline{\underline{\text{i}}}\rangle}_Y
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{\langle\underline{\underline{\text{a}}}\rangle}_Y
\\[4pt]
{\langle\underline{\underline{\text{i}}}\rangle}_Y
\\[4pt]
{\langle\underline{\underline{\text{a}}}\rangle}_Y
\\[4pt]
{\langle\underline{\underline{\text{i}}}\rangle}_Y
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
0_{\operatorname{d}Y}
\\[4pt]
{\langle
\operatorname{d}\underline{\underline{\text{a}}}
~
\operatorname{d}\underline{\underline{\text{i}}}
\rangle}_{\operatorname{d}Y}
\\[4pt]
{\langle
\operatorname{d}\underline{\underline{\text{a}}}
~
\operatorname{d}\underline{\underline{\text{i}}}
\rangle}_{\operatorname{d}Y}
\\[4pt]
0_{\operatorname{d}Y}
\end{matrix}</math>
|-
| valign="bottom" |
<math>\begin{matrix}
{\langle\underline{\underline{\text{b}}}\rangle}_Y
\\[4pt]
{\langle\underline{\underline{\text{b}}}\rangle}_Y
\\[4pt]
{\langle\underline{\underline{\text{u}}}\rangle}_Y
\\[4pt]
{\langle\underline{\underline{\text{u}}}\rangle}_Y
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{\langle\underline{\underline{\text{b}}}\rangle}_Y
\\[4pt]
{\langle\underline{\underline{\text{u}}}\rangle}_Y
\\[4pt]
{\langle\underline{\underline{\text{b}}}\rangle}_Y
\\[4pt]
{\langle\underline{\underline{\text{u}}}\rangle}_Y
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
0_{\operatorname{d}Y}
\\[4pt]
{\langle
\operatorname{d}\underline{\underline{\text{b}}}
~
\operatorname{d}\underline{\underline{\text{u}}}
\rangle}_{\operatorname{d}Y}
\\[4pt]
{\langle
\operatorname{d}\underline{\underline{\text{b}}}
~
\operatorname{d}\underline{\underline{\text{u}}}
\rangle}_{\operatorname{d}Y}
\\[4pt]
0_{\operatorname{d}Y}
\end{matrix}</math>
|}
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
|+ style="height:30px" |
<math>\text{Table 61.1} ~~ \operatorname{LIR}_3 (L_\text{B}) : \text{Lateral Representation of} ~ L_\text{B}\!</math>
|- style="height:40px; background:#f0f0ff"
| width="33%" | <math>\text{Object}\!</math>
| width="33%" | <math>\text{Sign}\!</math>
| width="33%" | <math>\text{Interpretant}\!</math>
|-
| valign="bottom" |
<math>\begin{matrix}
{\langle\underline{\underline{\text{A}}}\rangle}_X
\\[4pt]
{\langle\underline{\underline{\text{A}}}\rangle}_X
\\[4pt]
{\langle\underline{\underline{\text{A}}}\rangle}_X
\\[4pt]
{\langle\underline{\underline{\text{A}}}\rangle}_X
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{\langle\underline{\underline{\text{a}}}\rangle}_Y
\\[4pt]
{\langle\underline{\underline{\text{a}}}\rangle}_Y
\\[4pt]
{\langle\underline{\underline{\text{u}}}\rangle}_Y
\\[4pt]
{\langle\underline{\underline{\text{u}}}\rangle}_Y
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{\langle\underline{\underline{\text{a}}}\rangle}_Y
\\[4pt]
{\langle\underline{\underline{\text{u}}}\rangle}_Y
\\[4pt]
{\langle\underline{\underline{\text{a}}}\rangle}_Y
\\[4pt]
{\langle\underline{\underline{\text{u}}}\rangle}_Y
\end{matrix}</math>
|-
| valign="bottom" |
<math>\begin{matrix}
{\langle\underline{\underline{\text{B}}}\rangle}_X
\\[4pt]
{\langle\underline{\underline{\text{B}}}\rangle}_X
\\[4pt]
{\langle\underline{\underline{\text{B}}}\rangle}_X
\\[4pt]
{\langle\underline{\underline{\text{B}}}\rangle}_X
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{\langle\underline{\underline{\text{b}}}\rangle}_Y
\\[4pt]
{\langle\underline{\underline{\text{b}}}\rangle}_Y
\\[4pt]
{\langle\underline{\underline{\text{i}}}\rangle}_Y
\\[4pt]
{\langle\underline{\underline{\text{i}}}\rangle}_Y
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{\langle\underline{\underline{\text{b}}}\rangle}_Y
\\[4pt]
{\langle\underline{\underline{\text{i}}}\rangle}_Y
\\[4pt]
{\langle\underline{\underline{\text{b}}}\rangle}_Y
\\[4pt]
{\langle\underline{\underline{\text{i}}}\rangle}_Y
\end{matrix}</math>
|}
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
|+ style="height:30px" |
<math>\text{Table 61.2} ~~ \operatorname{LIR}_3 (\operatorname{Den}(L_\text{B})) : \text{Denotative Component of} ~ L_\text{B}\!</math>
|- style="height:40px; background:#f0f0ff"
| width="33%" | <math>\text{Object}\!</math>
| width="33%" | <math>\text{Sign}\!</math>
| width="33%" | <math>\text{Transition}\!</math>
|-
| valign="bottom" |
<math>\begin{matrix}
{\langle\underline{\underline{\text{A}}}\rangle}_X
\\[4pt]
{\langle\underline{\underline{\text{A}}}\rangle}_X
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{\langle\underline{\underline{\text{a}}}\rangle}_Y
\\[4pt]
{\langle\underline{\underline{\text{u}}}\rangle}_Y
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
({\langle\underline{\underline{\text{a}}}\rangle}_Y,
{\langle\underline{\underline{\text{A}}}\rangle}_X)
\\[4pt]
({\langle\underline{\underline{\text{u}}}\rangle}_Y,
{\langle\underline{\underline{\text{A}}}\rangle}_X)
\end{matrix}</math>
|-
| valign="bottom" |
<math>\begin{matrix}
{\langle\underline{\underline{\text{B}}}\rangle}_X
\\[4pt]
{\langle\underline{\underline{\text{B}}}\rangle}_X
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{\langle\underline{\underline{\text{b}}}\rangle}_Y
\\[4pt]
{\langle\underline{\underline{\text{i}}}\rangle}_Y
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
({\langle\underline{\underline{\text{b}}}\rangle}_Y,
{\langle\underline{\underline{\text{B}}}\rangle}_X)
\\[4pt]
({\langle\underline{\underline{\text{i}}}\rangle}_Y,
{\langle\underline{\underline{\text{B}}}\rangle}_X)
\end{matrix}</math>
|}
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
|+ style="height:30px" |
<math>\text{Table 61.3} ~~ \operatorname{LIR}_3 (\operatorname{Con}(L_\text{B})) : \text{Connotative Component of} ~ L_\text{B}\!</math>
|- style="height:40px; background:#f0f0ff"
| width="33%" | <math>\text{Sign}\!</math>
| width="33%" | <math>\text{Interpretant}\!</math>
| width="33%" | <math>\text{Transition}\!</math>
|-
| valign="bottom" |
<math>\begin{matrix}
{\langle\underline{\underline{\text{a}}}\rangle}_Y
\\[4pt]
{\langle\underline{\underline{\text{a}}}\rangle}_Y
\\[4pt]
{\langle\underline{\underline{\text{u}}}\rangle}_Y
\\[4pt]
{\langle\underline{\underline{\text{u}}}\rangle}_Y
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{\langle\underline{\underline{\text{a}}}\rangle}_Y
\\[4pt]
{\langle\underline{\underline{\text{u}}}\rangle}_Y
\\[4pt]
{\langle\underline{\underline{\text{a}}}\rangle}_Y
\\[4pt]
{\langle\underline{\underline{\text{u}}}\rangle}_Y
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
0_{\operatorname{d}Y}
\\[4pt]
{\langle
\operatorname{d}\underline{\underline{\text{a}}}
~
\operatorname{d}\underline{\underline{\text{u}}}
\rangle}_{\operatorname{d}Y}
\\[4pt]
{\langle
\operatorname{d}\underline{\underline{\text{a}}}
~
\operatorname{d}\underline{\underline{\text{u}}}
\rangle}_{\operatorname{d}Y}
\\[4pt]
0_{\operatorname{d}Y}
\end{matrix}</math>
|-
| valign="bottom" |
<math>\begin{matrix}
{\langle\underline{\underline{\text{b}}}\rangle}_Y
\\[4pt]
{\langle\underline{\underline{\text{b}}}\rangle}_Y
\\[4pt]
{\langle\underline{\underline{\text{i}}}\rangle}_Y
\\[4pt]
{\langle\underline{\underline{\text{i}}}\rangle}_Y
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{\langle\underline{\underline{\text{b}}}\rangle}_Y
\\[4pt]
{\langle\underline{\underline{\text{i}}}\rangle}_Y
\\[4pt]
{\langle\underline{\underline{\text{b}}}\rangle}_Y
\\[4pt]
{\langle\underline{\underline{\text{i}}}\rangle}_Y
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
0_{\operatorname{d}Y}
\\[4pt]
{\langle
\operatorname{d}\underline{\underline{\text{b}}}
~
\operatorname{d}\underline{\underline{\text{i}}}
\rangle}_{\operatorname{d}Y}
\\[4pt]
{\langle
\operatorname{d}\underline{\underline{\text{b}}}
~
\operatorname{d}\underline{\underline{\text{i}}}
\rangle}_{\operatorname{d}Y}
\\[4pt]
0_{\operatorname{d}Y}
\end{matrix}</math>
|}
<br>
===6.25. Analytic Intensional Representations===
In this section the ERs of <math>L(\text{A})\!</math> and <math>L(\text{B})\!</math> are translated into a variety of different IRs that actually accomplish some measure of analytic work. These are referred to as ''analytic intensional representations'' (AIRs). This strategy of representation is referred to as a ''structural coding'' or a ''sensitive coding'', because it pays attention to the structure of its object domain and attends to the nuances of each sign's interpretation to fashion its code. It may also be characterized as a <math>\log(n)\!</math> coding, because it uses roughly <math>\log_2(n)\!</math> binary features to represent a domain of <math>n\!</math> elements.
For the domain <math>O = \{ \text{A}, \text{B} \}\!</math> of two elements one needs to use a single logical feature. It is often convenient to use an object feature that is relative to the interpreter using it, for instance, telling whether the object described is the self or the other.
For the domain <math>S = I = \{ {}^{\backprime\backprime} \text{A} {}^{\prime\prime}, {}^{\backprime\backprime} \text{B} {}^{\prime\prime}, {}^{\backprime\backprime} \text{i} {}^{\prime\prime}, {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \}\!</math> of four elements one needs to use two logical features. One possibility is to classify each element according to its syntactic category, as being a noun or a pronoun, and according to its semantic category, as denoting the self or the other.
Tables 62.1, 62.2, and 62.3 show several ways of representing these categories in terms of feature-value pairs and propositional codes. In each Table, Column 1 describes the category in question, Column 2 gives the mnemonic form of a propositional expression for that category, and Column 3 gives the abbreviated form of that expression, using a notation for propositional calculus where parentheses circumscribing a term or expression are interpreted as forming its logical negation.
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
|+ style="height:30px" |
<math>\text{Table 62.1} ~~ \text{Analytic Codes for Object Features}\!</math>
|- style="height:40px; background:#f0f0ff"
| width="33%" | <math>\text{Category}\!</math>
| width="33%" | <math>\text{Mnemonic}\!</math>
| width="33%" | <math>\text{Code}\!</math>
|-
|
<math>\begin{array}{l}
\text{Self}
\\[4pt]
\text{Other}
\end{array}</math>
|
<math>\begin{matrix}
\text{self}
\\[4pt]
\text{(self)}
\end{matrix}</math>
|
<math>\begin{matrix}
\text{s}
\\[4pt]
\text{(s)}
\end{matrix}</math>
|}
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
|+ style="height:30px" |
<math>\text{Table 62.2} ~~ \text{Analytic Codes for Semantic Features}\!</math>
|- style="height:40px; background:#f0f0ff"
| width="33%" | <math>\text{Category}\!</math>
| width="33%" | <math>\text{Mnemonic}\!</math>
| width="33%" | <math>\text{Code}\!</math>
|-
|
<math>\begin{array}{l}
\text{1st Person}
\\[4pt]
\text{2nd Person}
\end{array}</math>
|
<math>\begin{matrix}
\text{my}
\\[4pt]
\text{(my)}
\end{matrix}</math>
|
<math>\begin{matrix}
\text{m}
\\[4pt]
\text{(m)}
\end{matrix}</math>
|}
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
|+ style="height:30px" |
<math>\text{Table 62.3} ~~ \text{Analytic Codes for Syntactic Features}\!</math>
|- style="height:40px; background:#f0f0ff"
| width="33%" | <math>\text{Category}\!</math>
| width="33%" | <math>\text{Mnemonic}\!</math>
| width="33%" | <math>\text{Code}\!</math>
|-
|
<math>\begin{array}{l}
\text{Noun}
\\[4pt]
\text{Pronoun}
\end{array}</math>
|
<math>\begin{matrix}
\text{name}
\\[4pt]
\text{(name)}
\end{matrix}</math>
|
<math>\begin{matrix}
\text{n}
\\[4pt]
\text{(n)}
\end{matrix}</math>
|}
<br>
Tables 63 and 64 list the codes for each element of the world domain <math>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} \},\!</math> giving all features relative to the interpreters <math>\text{A}\!</math> and <math>\text{B},\!</math> respectively.
</pre>
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
|+ style="height:30px" |
<math>\text{Table 63.} ~~ \text{Analytic Codes for Interpreter A}\!</math>
|- style="background:#f0f0ff"
| width="16%" | <math>\text{Name}\!</math>
| width="16%" | <math>\text{Vector}\!</math>
| width="26%" | <math>\text{Conjunct Term}\!</math>
| width="26%" | <math>\text{Mnemonic}\!</math>
| width="16%" | <math>\text{Code}\!</math>
|-
| valign="bottom" |
<math>\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}</math>
| valign="bottom" |
<math>\begin{matrix}
{1}_X
\\[4pt]
{0}_X
\\[4pt]
{11}_Y
\\[4pt]
{01}_Y
\\[4pt]
{10}_Y
\\[4pt]
{00}_Y
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
~x_1~
\\[4pt]
(x_1)
\\[4pt]
~y_1~~y_2~
\\[4pt]
(y_1)~y_2~
\\[4pt]
~y_1~(y_2)
\\[4pt]
(y_1)(y_2)
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
~\text{self}~
\\[4pt]
(\text{self})
\\[4pt]
~\text{my}~~\text{name}~
\\[4pt]
(\text{my})~\text{name}~
\\[4pt]
~\text{my}~(\text{name})
\\[4pt]
(\text{my})(\text{name})
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
~\text{s}~
\\[4pt]
(\text{s})
\\[4pt]
~\text{m}~~\text{n}~
\\[4pt]
(\text{m})~\text{n}~
\\[4pt]
~\text{m}~(\text{n})
\\[4pt]
(\text{m})(\text{n})
\end{matrix}</math>
|}
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
|+ style="height:30px" |
<math>\text{Table 64.} ~~ \text{Analytic Codes for Interpreter B}\!</math>
|- style="background:#f0f0ff"
| width="16%" | <math>\text{Name}\!</math>
| width="16%" | <math>\text{Vector}\!</math>
| width="26%" | <math>\text{Conjunct Term}\!</math>
| width="26%" | <math>\text{Mnemonic}\!</math>
| width="16%" | <math>\text{Code}\!</math>
|-
| valign="bottom" |
<math>\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}</math>
| valign="bottom" |
<math>\begin{matrix}
{0}_X
\\[4pt]
{1}_X
\\[4pt]
{01}_Y
\\[4pt]
{11}_Y
\\[4pt]
{10}_Y
\\[4pt]
{00}_Y
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
(x_1)
\\[4pt]
~x_1~
\\[4pt]
(y_1)~y_2~
\\[4pt]
~y_1~~y_2~
\\[4pt]
~y_1~(y_2)
\\[4pt]
(y_1)(y_2)
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
(\text{self})
\\[4pt]
~\text{self}~
\\[4pt]
(\text{my})~\text{name}~
\\[4pt]
~\text{my}~~\text{name}~
\\[4pt]
~\text{my}~(\text{name})
\\[4pt]
(\text{my})(\text{name})
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
(\text{s})
\\[4pt]
~\text{s}~
\\[4pt]
(\text{m})~\text{n}~
\\[4pt]
~\text{m}~~\text{n}~
\\[4pt]
~\text{m}~(\text{n})
\\[4pt]
(\text{m})(\text{n})
\end{matrix}</math>
|}
<br>
Tables 65.1 and 66.1 transcribe the sign relations <math>L(\text{A})\!</math> and <math>L(\text{B}),\!</math> respectively, into the forms of the AIR just suggested. Tables 65.2 and 66.2 extract the denotative components of <math>L(\text{A})\!</math> and <math>L(\text{B}),\!</math> respectively, and isolate the transitions from signs to objects as ordered pairs of the form <math>(s, o).\!</math> Tables 65.3 and 66.3 extract the connotative components of <math>L(\text{A})\!</math> and <math>L(\text{B}),\!</math> respectively, and represent the transitions from signs to interpretants in terms of differential features, in other words, as propositions in the differential extension of the syntactic domain.
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
|+ style="height:30px" |
<math>\text{Table 65.1} ~~ \operatorname{AIR}_1 (L_\text{A}) : \text{Analytic Representation of} ~ L_\text{A}\!</math>
|- style="height:40px; background:#f0f0ff"
| width="33%" | <math>\text{Object}\!</math>
| width="33%" | <math>\text{Sign}\!</math>
| width="33%" | <math>\text{Interpretant}\!</math>
|-
| valign="bottom" |
<math>\begin{matrix}
\text{s}
\\[4pt]
\text{s}
\\[4pt]
\text{s}
\\[4pt]
\text{s}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
~\text{m}~~\text{n}~
\\[4pt]
~\text{m}~~\text{n}~
\\[4pt]
~\text{m}~(\text{n})
\\[4pt]
~\text{m}~(\text{n})
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
~\text{m}~~\text{n}~
\\[4pt]
~\text{m}~(\text{n})
\\[4pt]
~\text{m}~~\text{n}~
\\[4pt]
~\text{m}~(\text{n})
\end{matrix}</math>
|-
| valign="bottom" |
<math>\begin{matrix}
(\text{s})
\\[4pt]
(\text{s})
\\[4pt]
(\text{s})
\\[4pt]
(\text{s})
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
(\text{m})~\text{n}~
\\[4pt]
(\text{m})~\text{n}~
\\[4pt]
(\text{m})(\text{n})
\\[4pt]
(\text{m})(\text{n})
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
(\text{m})~\text{n}~
\\[4pt]
(\text{m})(\text{n})
\\[4pt]
(\text{m})~\text{n}~
\\[4pt]
(\text{m})(\text{n})
\end{matrix}</math>
|}
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
|+ style="height:30px" |
<math>\text{Table 65.2} ~~ \operatorname{AIR}_1 (\operatorname{Den}(L_\text{A})) : \text{Denotative Component of} ~ L_\text{A}\!</math>
|- style="height:40px; background:#f0f0ff"
| width="33%" | <math>\text{Object}\!</math>
| width="33%" | <math>\text{Sign}\!</math>
| width="33%" | <math>\text{Transition}\!</math>
|-
| valign="bottom" |
<math>\begin{matrix}
\text{s}
\\[4pt]
\text{s}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
~\text{m}~~\text{n}~
\\[4pt]
~\text{m}~(\text{n})
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
~\text{m}~~\text{n}~ \mapsto ~\text{s}~
\\[4pt]
~\text{m}~(\text{n}) \mapsto ~\text{s}~
\end{matrix}</math>
|-
| valign="bottom" |
<math>\begin{matrix}
(\text{s})
\\[4pt]
(\text{s})
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
(\text{m})~\text{n}~
\\[4pt]
(\text{m})(\text{n})
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
(\text{m})~\text{n}~ \mapsto (\text{s})
\\[4pt]
(\text{m})(\text{n}) \mapsto (\text{s})
\end{matrix}</math>
|}
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
|+ style="height:30px" |
<math>\text{Table 65.3} ~~ \operatorname{AIR}_1 (\operatorname{Con}(L_\text{A})) : \text{Connotative Component of} ~ L_\text{A}\!</math>
|- style="height:40px; background:#f0f0ff"
| width="33%" | <math>\text{Sign}\!</math>
| width="33%" | <math>\text{Interpretant}\!</math>
| width="33%" | <math>\text{Transition}\!</math>
|-
| valign="bottom" |
<math>\begin{matrix}
~\text{m}~~\text{n}~
\\[4pt]
~\text{m}~~\text{n}~
\\[4pt]
~\text{m}~(\text{n})
\\[4pt]
~\text{m}~(\text{n})
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
~\text{m}~~\text{n}~
\\[4pt]
~\text{m}~(\text{n})
\\[4pt]
~\text{m}~~\text{n}~
\\[4pt]
~\text{m}~(\text{n})
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
(\text{dm})(\text{dn})
\\[4pt]
(\text{dm})~\text{dn}~
\\[4pt]
(\text{dm})~\text{dn}~
\\[4pt]
(\text{dm})(\text{dn})
\end{matrix}</math>
|-
| valign="bottom" |
<math>\begin{matrix}
(\text{m})~\text{n}~
\\[4pt]
(\text{m})~\text{n}~
\\[4pt]
(\text{m})(\text{n})
\\[4pt]
(\text{m})(\text{n})
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
(\text{m})~\text{n}~
\\[4pt]
(\text{m})(\text{n})
\\[4pt]
(\text{m})~\text{n}~
\\[4pt]
(\text{m})(\text{n})
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
(\text{dm})(\text{dn})
\\[4pt]
(\text{dm})~\text{dn}~
\\[4pt]
(\text{dm})~\text{dn}~
\\[4pt]
(\text{dm})(\text{dn})
\end{matrix}</math>
|}
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
|+ style="height:30px" |
<math>\text{Table 66.1} ~~ \operatorname{AIR}_1 (L_\text{B}) : \text{Analytic Representation of} ~ L_\text{B}\!</math>
|- style="height:40px; background:#f0f0ff"
| width="33%" | <math>\text{Object}\!</math>
| width="33%" | <math>\text{Sign}\!</math>
| width="33%" | <math>\text{Interpretant}\!</math>
|-
| valign="bottom" |
<math>\begin{matrix}
(\text{s})
\\[4pt]
(\text{s})
\\[4pt]
(\text{s})
\\[4pt]
(\text{s})
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
(\text{m})~\text{n}~
\\[4pt]
(\text{m})~\text{n}~
\\[4pt]
(\text{m})(\text{n})
\\[4pt]
(\text{m})(\text{n})
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
(\text{m})~\text{n}~
\\[4pt]
(\text{m})(\text{n})
\\[4pt]
(\text{m})~\text{n}~
\\[4pt]
(\text{m})(\text{n})
\end{matrix}</math>
|-
| valign="bottom" |
<math>\begin{matrix}
\text{s}
\\[4pt]
\text{s}
\\[4pt]
\text{s}
\\[4pt]
\text{s}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
~\text{m}~~\text{n}~
\\[4pt]
~\text{m}~~\text{n}~
\\[4pt]
~\text{m}~(\text{n})
\\[4pt]
~\text{m}~(\text{n})
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
~\text{m}~~\text{n}~
\\[4pt]
~\text{m}~(\text{n})
\\[4pt]
~\text{m}~~\text{n}~
\\[4pt]
~\text{m}~(\text{n})
\end{matrix}</math>
|}
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
|+ style="height:30px" |
<math>\text{Table 66.2} ~~ \operatorname{AIR}_1 (\operatorname{Den}(L_\text{B})) : \text{Denotative Component of} ~ L_\text{B}\!</math>
|- style="height:40px; background:#f0f0ff"
| width="33%" | <math>\text{Object}\!</math>
| width="33%" | <math>\text{Sign}\!</math>
| width="33%" | <math>\text{Transition}\!</math>
|-
| valign="bottom" |
<math>\begin{matrix}
(\text{s})
\\[4pt]
(\text{s})
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
(\text{m})~\text{n}~
\\[4pt]
(\text{m})(\text{n})
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
(\text{m})~\text{n}~ \mapsto (\text{s})
\\[4pt]
(\text{m})(\text{n}) \mapsto (\text{s})
\end{matrix}</math>
|-
| valign="bottom" |
<math>\begin{matrix}
\text{s}
\\[4pt]
\text{s}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
~\text{m}~~\text{n}~
\\[4pt]
~\text{m}~(\text{n})
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
~\text{m}~~\text{n}~ \mapsto ~\text{s}~
\\[4pt]
~\text{m}~(\text{n}) \mapsto ~\text{s}~
\end{matrix}</math>
|}
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
|+ style="height:30px" |
<math>\text{Table 66.3} ~~ \operatorname{AIR}_1 (\operatorname{Con}(L_\text{B})) : \text{Connotative Component of} ~ L_\text{B}\!</math>
|- style="height:40px; background:#f0f0ff"
| width="33%" | <math>\text{Sign}\!</math>
| width="33%" | <math>\text{Interpretant}\!</math>
| width="33%" | <math>\text{Transition}\!</math>
|-
| valign="bottom" |
<math>\begin{matrix}
(\text{m})~\text{n}~
\\[4pt]
(\text{m})~\text{n}~
\\[4pt]
(\text{m})(\text{n})
\\[4pt]
(\text{m})(\text{n})
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
(\text{m})~\text{n}~
\\[4pt]
(\text{m})(\text{n})
\\[4pt]
(\text{m})~\text{n}~
\\[4pt]
(\text{m})(\text{n})
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
(\text{dm})(\text{dn})
\\[4pt]
(\text{dm})~\text{dn}~
\\[4pt]
(\text{dm})~\text{dn}~
\\[4pt]
(\text{dm})(\text{dn})
\end{matrix}</math>
|-
| valign="bottom" |
<math>\begin{matrix}
~\text{m}~~\text{n}~
\\[4pt]
~\text{m}~~\text{n}~
\\[4pt]
~\text{m}~(\text{n})
\\[4pt]
~\text{m}~(\text{n})
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
~\text{m}~~\text{n}~
\\[4pt]
~\text{m}~(\text{n})
\\[4pt]
~\text{m}~~\text{n}~
\\[4pt]
~\text{m}~(\text{n})
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
(\text{dm})(\text{dn})
\\[4pt]
(\text{dm})~\text{dn}~
\\[4pt]
(\text{dm})~\text{dn}~
\\[4pt]
(\text{dm})(\text{dn})
\end{matrix}</math>
|}
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
|+ style="height:30px" |
<math>\text{Table 67.1} ~~ \operatorname{AIR}_2 (L_\text{A}) : \text{Analytic Representation of} ~ L_\text{A}\!</math>
|- style="height:40px; background:#f0f0ff"
| width="33%" | <math>\text{Object}\!</math>
| width="33%" | <math>\text{Sign}\!</math>
| width="33%" | <math>\text{Interpretant}\!</math>
|-
| valign="bottom" |
<math>\begin{matrix}
{\langle * \rangle}_X
\\[4pt]
{\langle * \rangle}_X
\\[4pt]
{\langle * \rangle}_X
\\[4pt]
{\langle * \rangle}_X
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{\langle * \rangle}_Y
\\[4pt]
{\langle * \rangle}_Y
\\[4pt]
{\langle\text{m}\rangle}_Y
\\[4pt]
{\langle\text{m}\rangle}_Y
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{\langle * \rangle}_Y
\\[4pt]
{\langle\text{m}\rangle}_Y
\\[4pt]
{\langle * \rangle}_Y
\\[4pt]
{\langle\text{m}\rangle}_Y
\end{matrix}</math>
|-
| valign="bottom" |
<math>\begin{matrix}
{\langle ! \rangle}_X
\\[4pt]
{\langle ! \rangle}_X
\\[4pt]
{\langle ! \rangle}_X
\\[4pt]
{\langle ! \rangle}_X
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{\langle\text{n}\rangle}_Y
\\[4pt]
{\langle\text{n}\rangle}_Y
\\[4pt]
{\langle ! \rangle}_Y
\\[4pt]
{\langle ! \rangle}_Y
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{\langle\text{n}\rangle}_Y
\\[4pt]
{\langle ! \rangle}_Y
\\[4pt]
{\langle\text{n}\rangle}_Y
\\[4pt]
{\langle ! \rangle}_Y
\end{matrix}</math>
|}
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
|+ style="height:30px" |
<math>\text{Table 67.2} ~~ \operatorname{AIR}_2 (\operatorname{Den}(L_\text{A})) : \text{Denotative Component of} ~ L_\text{A}\!</math>
|- style="height:40px; background:#f0f0ff"
| width="33%" | <math>\text{Object}\!</math>
| width="33%" | <math>\text{Sign}\!</math>
| width="33%" | <math>\text{Transition}\!</math>
|-
| valign="bottom" |
<math>\begin{matrix}
{\langle * \rangle}_X
\\[4pt]
{\langle * \rangle}_X
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{\langle * \rangle}_Y
\\[4pt]
{\langle\text{m}\rangle}_Y
\end{matrix}</math>
| valign="bottom" |
<math>\begin{array}{r}
{\langle * \rangle}_Y \mapsto {\langle * \rangle}_X
\\[4pt]
{\langle\text{m}\rangle}_Y \mapsto {\langle * \rangle}_X
\end{array}</math>
|-
| valign="bottom" |
<math>\begin{matrix}
{\langle ! \rangle}_X
\\[4pt]
{\langle ! \rangle}_X
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{\langle\text{n}\rangle}_Y
\\[4pt]
{\langle ! \rangle}_Y
\end{matrix}</math>
| valign="bottom" |
<math>\begin{array}{r}
{\langle\text{n}\rangle}_Y \mapsto {\langle ! \rangle}_X
\\[4pt]
{\langle ! \rangle}_Y \mapsto {\langle ! \rangle}_X
\end{array}</math>
|}
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
|+ style="height:30px" |
<math>\text{Table 67.3} ~~ \operatorname{AIR}_2 (\operatorname{Con}(L_\text{A})) : \text{Connotative Component of} ~ L_\text{A}\!</math>
|- style="height:40px; background:#f0f0ff"
| width="33%" | <math>\text{Sign}\!</math>
| width="33%" | <math>\text{Interpretant}\!</math>
| width="33%" | <math>\text{Transition}\!</math>
|-
| valign="bottom" |
<math>\begin{matrix}
{\langle * \rangle}_Y
\\[4pt]
{\langle * \rangle}_Y
\\[4pt]
{\langle\text{m}\rangle}_Y
\\[4pt]
{\langle\text{m}\rangle}_Y
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{\langle * \rangle}_Y
\\[4pt]
{\langle\text{m}\rangle}_Y
\\[4pt]
{\langle * \rangle}_Y
\\[4pt]
{\langle\text{m}\rangle}_Y
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{\langle\operatorname{d}!\rangle}_{\operatorname{d}Y}
\\[4pt]
{\langle\operatorname{d}\text{n}\rangle}_{\operatorname{d}Y}
\\[4pt]
{\langle\operatorname{d}\text{n}\rangle}_{\operatorname{d}Y}
\\[4pt]
{\langle\operatorname{d}!\rangle}_{\operatorname{d}Y}
\end{matrix}</math>
|-
| valign="bottom" |
<math>\begin{matrix}
{\langle\text{n}\rangle}_Y
\\[4pt]
{\langle\text{n}\rangle}_Y
\\[4pt]
{\langle ! \rangle}_Y
\\[4pt]
{\langle ! \rangle}_Y
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{\langle\text{n}\rangle}_Y
\\[4pt]
{\langle ! \rangle}_Y
\\[4pt]
{\langle\text{n}\rangle}_Y
\\[4pt]
{\langle ! \rangle}_Y
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{\langle\operatorname{d}!\rangle}_{\operatorname{d}Y}
\\[4pt]
{\langle\operatorname{d}\text{n}\rangle}_{\operatorname{d}Y}
\\[4pt]
{\langle\operatorname{d}\text{n}\rangle}_{\operatorname{d}Y}
\\[4pt]
{\langle\operatorname{d}!\rangle}_{\operatorname{d}Y}
\end{matrix}</math>
|}
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
|+ style="height:30px" |
<math>\text{Table 68.1} ~~ \operatorname{AIR}_2 (L_\text{B}) : \text{Analytic Representation of} ~ L_\text{B}\!</math>
|- style="height:40px; background:#f0f0ff"
| width="33%" | <math>\text{Object}\!</math>
| width="33%" | <math>\text{Sign}\!</math>
| width="33%" | <math>\text{Interpretant}\!</math>
|-
| valign="bottom" |
<math>\begin{matrix}
{\langle ! \rangle}_X
\\[4pt]
{\langle ! \rangle}_X
\\[4pt]
{\langle ! \rangle}_X
\\[4pt]
{\langle ! \rangle}_X
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{\langle\text{n}\rangle}_Y
\\[4pt]
{\langle\text{n}\rangle}_Y
\\[4pt]
{\langle ! \rangle}_Y
\\[4pt]
{\langle ! \rangle}_Y
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{\langle\text{n}\rangle}_Y
\\[4pt]
{\langle ! \rangle}_Y
\\[4pt]
{\langle\text{n}\rangle}_Y
\\[4pt]
{\langle ! \rangle}_Y
\end{matrix}</math>
|-
| valign="bottom" |
<math>\begin{matrix}
{\langle * \rangle}_X
\\[4pt]
{\langle * \rangle}_X
\\[4pt]
{\langle * \rangle}_X
\\[4pt]
{\langle * \rangle}_X
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{\langle * \rangle}_Y
\\[4pt]
{\langle * \rangle}_Y
\\[4pt]
{\langle\text{m}\rangle}_Y
\\[4pt]
{\langle\text{m}\rangle}_Y
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{\langle * \rangle}_Y
\\[4pt]
{\langle\text{m}\rangle}_Y
\\[4pt]
{\langle * \rangle}_Y
\\[4pt]
{\langle\text{m}\rangle}_Y
\end{matrix}</math>
|}
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
|+ style="height:30px" |
<math>\text{Table 68.2} ~~ \operatorname{AIR}_2 (\operatorname{Den}(L_\text{B})) : \text{Denotative Component of} ~ L_\text{B}\!</math>
|- style="height:40px; background:#f0f0ff"
| width="33%" | <math>\text{Object}\!</math>
| width="33%" | <math>\text{Sign}\!</math>
| width="33%" | <math>\text{Transition}\!</math>
|-
| valign="bottom" |
<math>\begin{matrix}
{\langle ! \rangle}_X
\\[4pt]
{\langle ! \rangle}_X
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{\langle\text{n}\rangle}_Y
\\[4pt]
{\langle ! \rangle}_Y
\end{matrix}</math>
| valign="bottom" |
<math>\begin{array}{r}
{\langle\text{n}\rangle}_Y \mapsto {\langle ! \rangle}_X
\\[4pt]
{\langle ! \rangle}_Y \mapsto {\langle ! \rangle}_X
\end{array}</math>
|-
| valign="bottom" |
<math>\begin{matrix}
{\langle * \rangle}_X
\\[4pt]
{\langle * \rangle}_X
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{\langle * \rangle}_Y
\\[4pt]
{\langle\text{m}\rangle}_Y
\end{matrix}</math>
| valign="bottom" |
<math>\begin{array}{r}
{\langle * \rangle}_Y \mapsto {\langle * \rangle}_X
\\[4pt]
{\langle\text{m}\rangle}_Y \mapsto {\langle * \rangle}_X
\end{array}</math>
|}
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
|+ style="height:30px" |
<math>\text{Table 68.3} ~~ \operatorname{AIR}_2 (\operatorname{Con}(L_\text{B})) : \text{Connotative Component of} ~ L_\text{B}\!</math>
|- style="height:40px; background:#f0f0ff"
| width="33%" | <math>\text{Sign}\!</math>
| width="33%" | <math>\text{Interpretant}\!</math>
| width="33%" | <math>\text{Transition}\!</math>
|-
| valign="bottom" |
<math>\begin{matrix}
{\langle\text{n}\rangle}_Y
\\[4pt]
{\langle\text{n}\rangle}_Y
\\[4pt]
{\langle ! \rangle}_Y
\\[4pt]
{\langle ! \rangle}_Y
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{\langle\text{n}\rangle}_Y
\\[4pt]
{\langle ! \rangle}_Y
\\[4pt]
{\langle\text{n}\rangle}_Y
\\[4pt]
{\langle ! \rangle}_Y
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{\langle\operatorname{d}!\rangle}_{\operatorname{d}Y}
\\[4pt]
{\langle\operatorname{d}\text{n}\rangle}_{\operatorname{d}Y}
\\[4pt]
{\langle\operatorname{d}\text{n}\rangle}_{\operatorname{d}Y}
\\[4pt]
{\langle\operatorname{d}!\rangle}_{\operatorname{d}Y}
\end{matrix}</math>
|-
| valign="bottom" |
<math>\begin{matrix}
{\langle * \rangle}_Y
\\[4pt]
{\langle * \rangle}_Y
\\[4pt]
{\langle\text{m}\rangle}_Y
\\[4pt]
{\langle\text{m}\rangle}_Y
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{\langle * \rangle}_Y
\\[4pt]
{\langle\text{m}\rangle}_Y
\\[4pt]
{\langle * \rangle}_Y
\\[4pt]
{\langle\text{m}\rangle}_Y
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{\langle\operatorname{d}!\rangle}_{\operatorname{d}Y}
\\[4pt]
{\langle\operatorname{d}\text{n}\rangle}_{\operatorname{d}Y}
\\[4pt]
{\langle\operatorname{d}\text{n}\rangle}_{\operatorname{d}Y}
\\[4pt]
{\langle\operatorname{d}!\rangle}_{\operatorname{d}Y}
\end{matrix}</math>
|}
<br>