Directory talk:Jon Awbrey/Papers/Inquiry Driven Systems : Part 6

Fragments

6.19. Examples of Self-Reference

In previous work I developed a version of propositional calculus based on C.S. Peirce's existential graphs and implemented this calculus in computational form as a sentential calculus interpreter. Taking this calculus as a point of departure, I devised a theory of differential extensions for propositional domains that can be used, figuratively speaking, to put universes of discourse “in motion”, in other words, to provide qualitative descriptions of processes taking place in logical spaces. See (Awbrey, 1989 and 1994) for an account of this calculus, documentation of its computer program, and a detailed treatment of differential extensions.

In previous work (Awbrey, 1989) I described a system of notation for propositional calculus based on C.S. Peirce's existential graphs, documented a computer implementation of this formalism, and showed how to provide this calculus with a differential extension that can be used to describe changing universes of discourse. In subsequent work (Awbrey, 1994) the resulting system of differential logic was applied to give qualitative descriptions of change in discrete dynamical systems. This section draws on that earlier work, summarizing the conceptions that are needed to give logical representations of sign relations and recording a few changes of a minor nature in the typographical conventions used.

Abstractly, a domain of propositions is known by the axioms it satisfies. Concretely, one thinks of a proposition as applying to the objects it is true of.

Logically, a domain of properties or propositions is known by the axioms it is subject to. Concretely, a property or proposition is known by the things or situations it is true of. Typically, the signs of properties and propositions are called terms and sentences, respectively.

6.23. Intensional Representations of Sign Relations

In the formalized examples of IRs to be presented in this work, I will keep to the level of logical reasoning that is usually referred to as propositional calculus or sentential logic.

The contrast between ERs and IRs is strongly correlated with another dimension of interest in the study of inquiry, namely, the tension between empirical and rational modes of inquiry.

This section begins the explicit discussion of ERs by taking a second look at the sign relations \(L(\text{A})\!\) and \(L(\text{B}).\!\) Since the form of these examples no longer presents any novelty, this second presentation of \(L(\text{A})\!\) and \(L(\text{B})\!\) provides a first opportunity to introduce some new material. In the process of reviewing this material, it is useful to anticipate a number of incidental issues that are on the point of becoming critical, and to begin introducing the generic types of technical devices that are needed to deal with them.

Therefore, the easiest way to begin an explicit treatment of ERs is by recollecting the Tables of the sign relations \(L(\text{A})\!\) and \(L(\text{B})\!\) and by finishing the corresponding Tables of their dyadic components. Since the form of the sign relations \(L(\text{A})\!\) and \(L(\text{B})\!\) no longer presents any novelty, I can use the second presentation of these examples as a first opportunity to examine a selection of their finer points, previously overlooked.

Starting from this standpoint, 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 their missing details and expand the abbreviated contents of these forms, and to review their full structures in a more formal light.

Because of the perfect parallelism that the literal coding contrives between individual signs and grammatical categories, this arrangement illustrates not so much a code transformation as a re-interpretation of the original signs under different headings.

6.33. Sign Relational Complexes

I would like to record here, in what is topically the appropriate place, notice of a number of open questions that will have to be addressed if anyone desires to make a consistent calculus out of this link notation. Perhaps it is only because the franker forms of liaison involved in the couple \(a \widehat{~} b\!\) are more subject to the vagaries of syntactic elision than the corresponding bindings of the anglish ligature \((a, b),\!\) but for some reason or other the circumflex character of these diacritical notices are much more liable to suggest various forms of elaboration, including higher order generalizations and information-theoretic partializations of the very idea of \(n\!\)-tuples and sequences.

One way to deal with the problems of partial information …

Relational Complex?

\(L ~=~ L^{(1)} \cup \ldots \cup L^{(k)}\!\)

Sign Relational Complex?

\(L ~=~ L^{(1)} \cup L^{(2)} \cup L^{(3)}\!\)

Linkages can be chained together to form sequences of indications or \(n\!\)-tuples, without worrying too much about the order of collecting terms in the corresponding angle brackets.

\(\begin{matrix} a \widehat{~} b \widehat{~} c & = & (a, b, c) & = & (a, (b, c)) & = & ((a, b), c). \end{matrix}\)

These equivalences depend on the existence of natural isomorphisms between different ways of constructing \(n\!\)-place product spaces, that is, on the associativity of pairwise products, a not altogether trivial result (Mac Lane, CatWorkMath, ch. 7).

Higher Order Indications (HOIs)?

\(\begin{matrix} \widehat{~} x & = & (~, x) & ? \\[4pt] x \widehat{~} & = & (x, ~) & ? \\[4pt] \widehat{~}~\widehat{~} x & = & (~, (~, x)) & ? \\[4pt] x \widehat{~}~\widehat{~} & = & ((x, ~), ~) & ? \end{matrix}\)

In talking about properties and classes of relations, one would like to refer to all relations as forming a topic of potential discussion, and then take it as a background for contemplating …

In talking and thinking, often in just that order, about properties and classes of relations, one is always invoking, explicitly or implicitly, a particular background, a limited field of experience, actual or potential, against which each object of discussion and thought figures. Expressing the matter in the idiom of logical inquiry, one brings to mind a preconceived universe of discourse \(U\!\) or a restricted domain of discussion \(X,\!\) and then contemplates …

This direction of generalization expands the scope of PIRs by means of an analogical extension, and can be charted in the following manner. If the name of a relation can be taken as a PIR to elementary relations, that is, if the formula of an \(n\!\)-place relation can be interpreted as a proposition about \(n\!\)-tuples, then a PIR to relations themselves can be formulated as a proposition about relations and thus as a HOPE about elementary relations or \(n\!\)-tuples.

One way to extend the generic brand of partiality among relations in a non-trivial direction can be charted as follows. If the name or formula of a relation is a PIR to elementary relations, that is, if a sign or expression of an \(n\!\)-place relation is interpreted as a proposition about \(n\!\)-tuples, then a PIR to relations …

Scrap Area

Edit Buffer

When it comes to the subject of systems theory, a particular POV is so widely propagated that it might as well be regarded as the established, received, or traditional POV. The POV in question says that there are dynamic systems and symbolic systems, and never the twain shall meet. I naturally intend to challenge this assumption, preferring to suggest that dynamic …

Table Scraps

Table 37.1  Sign Relational Schema C
	Object	Sign	Interpretant
	x	"x"	"x"
	"x"	"x"	"x"

Table 37.2  Sign Relational Schema D
	Object	Sign	Interpretant
	x	"x"	"x"

Table 37.3  Sign Relational Schema E
	Object	Sign	Interpretant
	"x"	"x"	"x"

Table 37.4  Sign Relational Schema D'
	Object	Sign	Interpretant
	x	"x"	"x"
	x	"x"	<x>
	x	<x>	"x"
	x	<x>	<x>

Work Area

Alternate Text

A semigroup consists of a nonempty set with an associative LOC on it. On formal occasions, a semigroup is introduced by means a formula like \(X = (X, *),\!\) interpreted to mean that a semigroup \(X\!\) is specified by giving two pieces of data, a nonempty set that conventionally, if somewhat ambiguously, goes under the same name \({}^{\backprime\backprime} X {}^{\prime\prime},\!\) plus an associative binary operation denoted by \({}^{\backprime\backprime} * {}^{\prime\prime}.\!\) In contexts where there is only one semigroup being discussed, or where the additional structure is otherwise understood, it is common practice to call the semigroup by the name of the underlying set. In contexts where more than one semigroup is formed on the same set, one may use notations like \(X_i = (X, *_i)\!\) to distinguish them.

Additive Presentation

Version 1

The \(n^\text{th}\!\) multiple of an element \(x\!\) in a semigroup \(\underline{X} = (X, +, 0),\!\) for integer \(n > 0,\!\) is notated as \(nx\!\) and defined as follows. Proceeding recursively, for \(n = 1,\!\) let \(1x = x,\!\) and for \(n > 1,\!\) let \(nx = (n-1)x + x.\!\)

The \(n^\text{th}\!\) multiple of \(x\!\) in a monoid \(\underline{X} = (X, +, 0),\!\) for integer \(n \ge 0,\!\) is defined the same way for \(n > 0,\!\) letting \(0x = 0\!\) when \(n = 0.\!\)

The \(n^\text{th}\!\) multiple of \(x\!\) in a group \(\underline{X} = (X, +, 0),\!\) for any integer \(n,\!\) is defined the same way for \(n \ge 0,\!\) letting \(nx = (-n)(-x)\!\) for \(n < 0.\!\)

Version 2

In a semigroup written additively, the \(n^\text{th}\!\) multiple of an element \(x\!\) is notated as \(nx\!\) and defined for every positive integer \(n\!\) in the following manner. Proceeding recursively, let \(1x = x\!\) and let \(nx = (n-1)x + x\!\) for all \(n > 1.\!\)

In a monoid written additively, the multiple \(nx\!\) is defined for every non-negative integer \(n\!\) by letting \(0x = 0\!\) and proceeding the same way for \(n > 0.\!\)

In a group written additively, the multiple \(nx\!\) is defined for every integer \(n\!\) by letting \(nx = (-n)(-x)\!\) for \(n < 0\!\) and proceeding the same way for \(n \ge 0.\!\)

Set Displays

\(\begin{smallmatrix} \text{A} & = & \{ & (\text{A}, {}^{\backprime\backprime} \text{A} {}^{\prime\prime}, {}^{\backprime\backprime} \text{A} {}^{\prime\prime}), & \ldots, & (\text{A}, {}^{\backprime\backprime} \text{i} {}^{\prime\prime}, {}^{\backprime\backprime} \text{i} {}^{\prime\prime}) & , & (\text{B}, {}^{\backprime\backprime} \text{B} {}^{\prime\prime}, {}^{\backprime\backprime} \text{B} {}^{\prime\prime}), & \ldots, & (\text{B}, {}^{\backprime\backprime} \text{u} {}^{\prime\prime}, {}^{\backprime\backprime} \text{u} {}^{\prime\prime}) & \} \\[10pt] \text{B} & = & \{ & (\text{A}, {}^{\backprime\backprime} \text{A} {}^{\prime\prime}, {}^{\backprime\backprime} \text{A} {}^{\prime\prime}), & \ldots, & (\text{A}, {}^{\backprime\backprime} \text{u} {}^{\prime\prime}, {}^{\backprime\backprime} \text{u} {}^{\prime\prime}) & , & (\text{B}, {}^{\backprime\backprime} \text{B} {}^{\prime\prime}, {}^{\backprime\backprime} \text{B} {}^{\prime\prime}), & \ldots, & (\text{B}, {}^{\backprime\backprime} \text{i} {}^{\prime\prime}, {}^{\backprime\backprime} \text{i} {}^{\prime\prime}) & \} \end{smallmatrix}\)

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

\(\begin{array}{*{15}{c}} W & = & \{ & \text{A} & , & \text{B} & , & {}^{\backprime\backprime} \text{A} {}^{\prime\prime} & , & {}^{\backprime\backprime} \text{B} {}^{\prime\prime} & , & {}^{\backprime\backprime} \text{i} {}^{\prime\prime} & , & {}^{\backprime\backprime} \text{u} {}^{\prime\prime} & \} \\ & = & \{ & w_1 & , & w_2 & , & w_3 & , & w_4 & , & w_5 & , & w_6 & \} \end{array}\)

Relations In General

Next let's re-examine the numerical incidence properties of relations, concentrating on the definitions of the assorted regularity conditions.

For example, \(L\!\) is said to be \(^{\backprime\backprime} c\text{-regular at}~ j \, ^{\prime\prime}\) if and only if the cardinality of the local flag \(L_{x \,\text{at}\, j}\) is equal to \(c\!\) for all \(x \in X_j,\) coded in symbols, if and only if \(|L_{x \,\text{at}\, j}| = c\) for all \(x \in X_j.\)

In a similar fashion, it is possible to define the numerical incidence properties \(^{\backprime\backprime}(< c)\text{-regular at}~ j \, ^{\prime\prime},\) \(^{\backprime\backprime}(> c)\text{-regular at}~ j \, ^{\prime\prime},\) and so on. For ease of reference, a few of these definitions are recorded below.

\(\begin{array}{lll} L ~\text{is}~ c\text{-regular at}~ j & \iff & |L_{x \,\text{at}\, j}| = c ~\text{for all}~ x \in X_j. \\[6pt] L ~\text{is}~ (< c)\text{-regular at}~ j & \iff & |L_{x \,\text{at}\, j}| < c ~\text{for all}~ x \in X_j. \\[6pt] L ~\text{is}~ (> c)\text{-regular at}~ j & \iff & |L_{x \,\text{at}\, j}| > c ~\text{for all}~ x \in X_j. \\[6pt] L ~\text{is}~ (\le c)\text{-regular at}~ j & \iff & |L_{x \,\text{at}\, j}| \le c ~\text{for all}~ x \in X_j. \\[6pt] L ~\text{is}~ (\ge c)\text{-regular at}~ j & \iff & |L_{x \,\text{at}\, j}| \ge c ~\text{for all}~ x \in X_j. \end{array}\)

Clearly, if any relation is \((\le c)\text{-regular}\) on one of its domains \(X_j\!\) and also \((\ge c)\text{-regular}\) on the same domain, then it must be \((= c)\text{-regular}\!\) on that domain, in effect, \(c\text{-regular}\!\) at \(j.\!\)

Among the variety of conceivable regularities affecting 2-adic relations, we pay special attention to the \(c\!\)-regularity conditions where \(c\!\) is equal to 1.

Let \(L \subseteq X \times Y\!\) be an arbitrary 2-adic relation. The following properties of \(L\!\) can then be defined:

\(\begin{array}{lll} L ~\text{is total at}~ X & \iff & L ~\text{is}~ (\ge 1)\text{-regular}~ \text{at}~ X. \\[6pt] L ~\text{is total at}~ Y & \iff & L ~\text{is}~ (\ge 1)\text{-regular}~ \text{at}~ Y. \\[6pt] L ~\text{is tubular at}~ X & \iff & L ~\text{is}~ (\le 1)\text{-regular}~ \text{at}~ X. \\[6pt] L ~\text{is tubular at}~ Y & \iff & L ~\text{is}~ (\le 1)\text{-regular}~ \text{at}~ Y. \end{array}\)

We have already looked at 2-adic relations that separately exemplify each of these regularities. We also introduced a few bits of additional terminology and special-purpose notations for working with tubular relations.

If \(L\!\) is tubular at \(X,\!\) then \(L\!\) is known as a partial function or a prefunction from \(X\!\) to \(Y,\!\) indicated by writing \(L : X \rightharpoonup Y.\!\) We have the following definitions and notations.

\(\begin{array}{lll} L ~\text{is a prefunction}~ L : X \rightharpoonup Y & \iff & L ~\text{is tubular at}~ X. \\[6pt] L ~\text{is a prefunction}~ L : X \leftharpoonup Y & \iff & L ~\text{is tubular at}~ Y. \end{array}\)

We arrive by way of this winding stair at the special stamps of 2-adic relations \(L \subseteq X \times Y\!\) that are variously described as 1-regular, total and tubular, or total prefunctions on specified domains, either \(X\!\) or \(Y\!\) or both, and that are more often celebrated as functions on those domains.

If \(L\!\) is a prefunction \(L : X \rightharpoonup Y\!\) that happens to be total at \(X,\!\) then \(L\!\) is known as a function from \(X\!\) to \(Y,\!\) indicated by writing \(L : X \to Y.\!\) To say that a relation \(L \subseteq X \times Y\!\) is totally tubular at \(X\!\) is to say that \(L\!\) is 1-regular at \(X.\!\) Thus, we may formalize the following definitions.

\(\begin{array}{lll} L ~\text{is a function}~ L : X \to Y & \iff & L ~\text{is}~ 1\text{-regular at}~ X. \\[6pt] L ~\text{is a function}~ L : X \leftarrow Y & \iff & L ~\text{is}~ 1\text{-regular at}~ Y. \end{array}\)

In the case of a 2-adic relation \(L \subseteq X \times Y\!\) that has the qualifications of a function \(f : X \to Y,\!\) there are a number of further differentia that arise.

\(\begin{array}{lll} f ~\text{is surjective} & \iff & f ~\text{is total at}~ Y. \\[6pt] f ~\text{is injective} & \iff & f ~\text{is tubular at}~ Y. \\[6pt] f ~\text{is bijective} & \iff & f ~\text{is}~ 1\text{-regular at}~ Y. \end{array}\)

Table Work

Group Operations

\(\text{Table 32.1}~~\text{Scheme of a Group Operation Table}\)

\(*\!\)	\(x_0\!\)	\(\cdots\!\)	\(x_j\!\)	\(\cdots\!\)
\(x_0\!\)	\(x_0 * x_0\!\)	\(\cdots\!\)	\(x_0 * x_j\!\)	\(\cdots\!\)
\(\cdots\!\)	\(\cdots\!\)	\(\cdots\!\)	\(\cdots\!\)	\(\cdots\!\)
\(x_i\!\)	\(x_i * x_0\!\)	\(\cdots\!\)	\(x_i * x_j\!\)	\(\cdots\!\)
\(\cdots\!\)	\(\cdots\!\)	\(\cdots\!\)	\(\cdots\!\)	\(\cdots\!\)

\(\text{Table 32.2}~~\text{Scheme of the Regular Ante-Representation}\)

\(\text{Element}\!\)	\(\text{Function as Set of Ordered Pairs of Elements}\!\)
\(x_0\!\)	\(\{\!\)	\((x_0 ~,~ x_0 * x_0),\!\)	\(\cdots\!\)	\((x_j ~,~ x_0 * x_j),\!\)	\(\cdots\!\)	\(\}\!\)
\(\cdots\!\)	\(\{\!\)	\(\cdots\!\)	\(\cdots\!\)	\(\cdots\!\)	\(\cdots\!\)	\(\}\!\)
\(x_i\!\)	\(\{\!\)	\((x_0 ~,~ x_i * x_0),\!\)	\(\cdots\!\)	\((x_j ~,~ x_i * x_j),\!\)	\(\cdots\!\)	\(\}\!\)
\(\cdots\!\)	\(\{\!\)	\(\cdots\!\)	\(\cdots\!\)	\(\cdots\!\)	\(\cdots\!\)	\(\}\!\)

\(\text{Table 32.3}~~\text{Scheme of the Regular Post-Representation}\)

\(\text{Element}\!\)	\(\text{Function as Set of Ordered Pairs of Elements}\!\)
\(x_0\!\)	\(\{\!\)	\((x_0 ~,~ x_0 * x_0),\!\)	\(\cdots\!\)	\((x_j ~,~ x_j * x_0),\!\)	\(\cdots\!\)	\(\}\!\)
\(\cdots\!\)	\(\{\!\)	\(\cdots\!\)	\(\cdots\!\)	\(\cdots\!\)	\(\cdots\!\)	\(\}\!\)
\(x_i\!\)	\(\{\!\)	\((x_0 ~,~ x_0 * x_i),\!\)	\(\cdots\!\)	\((x_j ~,~ x_j * x_i),\!\)	\(\cdots\!\)	\(\}\!\)
\(\cdots\!\)	\(\{\!\)	\(\cdots\!\)	\(\cdots\!\)	\(\cdots\!\)	\(\cdots\!\)	\(\}\!\)

\(\text{Table 33.1}~~\text{Multiplication Operation of the Group}~V_4\)

\(\cdot\!\)	\(\operatorname{e}\)	\(\operatorname{f}\)	\(\operatorname{g}\)	\(\operatorname{h}\)
\(\operatorname{e}\)	\(\operatorname{e}\)	\(\operatorname{f}\)	\(\operatorname{g}\)	\(\operatorname{h}\)
\(\operatorname{f}\)	\(\operatorname{f}\)	\(\operatorname{e}\)	\(\operatorname{h}\)	\(\operatorname{g}\)
\(\operatorname{g}\)	\(\operatorname{g}\)	\(\operatorname{h}\)	\(\operatorname{e}\)	\(\operatorname{f}\)
\(\operatorname{h}\)	\(\operatorname{h}\)	\(\operatorname{g}\)	\(\operatorname{f}\)	\(\operatorname{e}\)

\(\text{Table 33.2}~~\text{Regular Representation of the Group}~V_4\)

\(\text{Element}\!\)	\(\text{Function as Set of Ordered Pairs of Elements}\!\)
\(\operatorname{e}\)	\(\{\!\)	\((\operatorname{e}, \operatorname{e}),\)	\((\operatorname{f}, \operatorname{f}),\)	\((\operatorname{g}, \operatorname{g}),\)	\((\operatorname{h}, \operatorname{h})\)	\(\}\!\)
\(\operatorname{f}\)	\(\{\!\)	\((\operatorname{e}, \operatorname{f}),\)	\((\operatorname{f}, \operatorname{e}),\)	\((\operatorname{g}, \operatorname{h}),\)	\((\operatorname{h}, \operatorname{g})\)	\(\}\!\)
\(\operatorname{g}\)	\(\{\!\)	\((\operatorname{e}, \operatorname{g}),\)	\((\operatorname{f}, \operatorname{h}),\)	\((\operatorname{g}, \operatorname{e}),\)	\((\operatorname{h}, \operatorname{f})\)	\(\}\!\)
\(\operatorname{h}\)	\(\{\!\)	\((\operatorname{e}, \operatorname{h}),\)	\((\operatorname{f}, \operatorname{g}),\)	\((\operatorname{g}, \operatorname{f}),\)	\((\operatorname{h}, \operatorname{e})\)	\(\}\!\)

\(\text{Table 33.3}~~\text{Regular Representation of the Group}~V_4\)

\(\text{Element}\!\)	\(\text{Function as Set of Ordered Pairs of Symbols}\!\)
\(\operatorname{e}\)	\(\{\!\)	\(({}^{\backprime\backprime}\text{e}{}^{\prime\prime}, {}^{\backprime\backprime}\text{e}{}^{\prime\prime}),\)	\(({}^{\backprime\backprime}\text{f}{}^{\prime\prime}, {}^{\backprime\backprime}\text{f}{}^{\prime\prime}),\)	\(({}^{\backprime\backprime}\text{g}{}^{\prime\prime}, {}^{\backprime\backprime}\text{g}{}^{\prime\prime}),\)	\(({}^{\backprime\backprime}\text{h}{}^{\prime\prime}, {}^{\backprime\backprime}\text{h}{}^{\prime\prime})\)	\(\}\!\)
\(\operatorname{f}\)	\(\{\!\)	\(({}^{\backprime\backprime}\text{e}{}^{\prime\prime}, {}^{\backprime\backprime}\text{f}{}^{\prime\prime}),\)	\(({}^{\backprime\backprime}\text{f}{}^{\prime\prime}, {}^{\backprime\backprime}\text{e}{}^{\prime\prime}),\)	\(({}^{\backprime\backprime}\text{g}{}^{\prime\prime}, {}^{\backprime\backprime}\text{h}{}^{\prime\prime}),\)	\(({}^{\backprime\backprime}\text{h}{}^{\prime\prime}, {}^{\backprime\backprime}\text{g}{}^{\prime\prime})\)	\(\}\!\)
\(\operatorname{g}\)	\(\{\!\)	\(({}^{\backprime\backprime}\text{e}{}^{\prime\prime}, {}^{\backprime\backprime}\text{g}{}^{\prime\prime}),\)	\(({}^{\backprime\backprime}\text{f}{}^{\prime\prime}, {}^{\backprime\backprime}\text{h}{}^{\prime\prime}),\)	\(({}^{\backprime\backprime}\text{g}{}^{\prime\prime}, {}^{\backprime\backprime}\text{e}{}^{\prime\prime}),\)	\(({}^{\backprime\backprime}\text{h}{}^{\prime\prime}, {}^{\backprime\backprime}\text{f}{}^{\prime\prime})\)	\(\}\!\)
\(\operatorname{h}\)	\(\{\!\)	\(({}^{\backprime\backprime}\text{e}{}^{\prime\prime}, {}^{\backprime\backprime}\text{h}{}^{\prime\prime}),\)	\(({}^{\backprime\backprime}\text{f}{}^{\prime\prime}, {}^{\backprime\backprime}\text{g}{}^{\prime\prime}),\)	\(({}^{\backprime\backprime}\text{g}{}^{\prime\prime}, {}^{\backprime\backprime}\text{f}{}^{\prime\prime}),\)	\(({}^{\backprime\backprime}\text{h}{}^{\prime\prime}, {}^{\backprime\backprime}\text{e}{}^{\prime\prime})\)	\(\}\!\)

\(\text{Table 34.1}~~\text{Multiplicative Presentation of the Group}~Z_4(\cdot)\)

\(\cdot\!\)	\(\operatorname{1}\)	\(\operatorname{a}\)	\(\operatorname{b}\)	\(\operatorname{c}\)
\(\operatorname{1}\)	\(\operatorname{1}\)	\(\operatorname{a}\)	\(\operatorname{b}\)	\(\operatorname{c}\)
\(\operatorname{a}\)	\(\operatorname{a}\)	\(\operatorname{b}\)	\(\operatorname{c}\)	\(\operatorname{1}\)
\(\operatorname{b}\)	\(\operatorname{b}\)	\(\operatorname{c}\)	\(\operatorname{1}\)	\(\operatorname{a}\)
\(\operatorname{c}\)	\(\operatorname{c}\)	\(\operatorname{1}\)	\(\operatorname{a}\)	\(\operatorname{b}\)

\(\text{Table 34.2}~~\text{Regular Representation of the Group}~Z_4(\cdot)\)

\(\text{Element}\!\)	\(\text{Function as Set of Ordered Pairs of Elements}\!\)
\(\operatorname{1}\)	\(\{\!\)	\((\operatorname{1}, \operatorname{1}),\)	\((\operatorname{a}, \operatorname{a}),\)	\((\operatorname{b}, \operatorname{b}),\)	\((\operatorname{c}, \operatorname{c})\)	\(\}\!\)
\(\operatorname{a}\)	\(\{\!\)	\((\operatorname{1}, \operatorname{a}),\)	\((\operatorname{a}, \operatorname{b}),\)	\((\operatorname{b}, \operatorname{c}),\)	\((\operatorname{c}, \operatorname{1})\)	\(\}\!\)
\(\operatorname{b}\)	\(\{\!\)	\((\operatorname{1}, \operatorname{b}),\)	\((\operatorname{a}, \operatorname{c}),\)	\((\operatorname{b}, \operatorname{1}),\)	\((\operatorname{c}, \operatorname{a})\)	\(\}\!\)
\(\operatorname{c}\)	\(\{\!\)	\((\operatorname{1}, \operatorname{c}),\)	\((\operatorname{a}, \operatorname{1}),\)	\((\operatorname{b}, \operatorname{a}),\)	\((\operatorname{c}, \operatorname{b})\)	\(\}\!\)

\(\text{Table 35.1}~~\text{Additive Presentation of the Group}~Z_4(+)\)

\(+\!\)	\(\operatorname{0}\)	\(\operatorname{1}\)	\(\operatorname{2}\)	\(\operatorname{3}\)
\(\operatorname{0}\)	\(\operatorname{0}\)	\(\operatorname{1}\)	\(\operatorname{2}\)	\(\operatorname{3}\)
\(\operatorname{1}\)	\(\operatorname{1}\)	\(\operatorname{2}\)	\(\operatorname{3}\)	\(\operatorname{0}\)
\(\operatorname{2}\)	\(\operatorname{2}\)	\(\operatorname{3}\)	\(\operatorname{0}\)	\(\operatorname{1}\)
\(\operatorname{3}\)	\(\operatorname{3}\)	\(\operatorname{0}\)	\(\operatorname{1}\)	\(\operatorname{2}\)

\(\text{Table 35.2}~~\text{Regular Representation of the Group}~Z_4(+)\)

\(\text{Element}\!\)	\(\text{Function as Set of Ordered Pairs of Elements}\!\)
\(\operatorname{0}\)	\(\{\!\)	\((\operatorname{0}, \operatorname{0}),\)	\((\operatorname{1}, \operatorname{1}),\)	\((\operatorname{2}, \operatorname{2}),\)	\((\operatorname{3}, \operatorname{3})\)	\(\}\!\)
\(\operatorname{1}\)	\(\{\!\)	\((\operatorname{0}, \operatorname{1}),\)	\((\operatorname{1}, \operatorname{2}),\)	\((\operatorname{2}, \operatorname{3}),\)	\((\operatorname{3}, \operatorname{0})\)	\(\}\!\)
\(\operatorname{2}\)	\(\{\!\)	\((\operatorname{0}, \operatorname{2}),\)	\((\operatorname{1}, \operatorname{3}),\)	\((\operatorname{2}, \operatorname{0}),\)	\((\operatorname{3}, \operatorname{1})\)	\(\}\!\)
\(\operatorname{3}\)	\(\{\!\)	\((\operatorname{0}, \operatorname{3}),\)	\((\operatorname{1}, \operatorname{0}),\)	\((\operatorname{2}, \operatorname{1}),\)	\((\operatorname{3}, \operatorname{2})\)	\(\}\!\)

Sign Relations

\(\text{Table 1.} ~~ \text{Sign Relation of Interpreter A}\!\)

\(\text{Object}\!\)	\(\text{Sign}\!\)	\(\text{Interpretant}\!\)
\(\begin{matrix} \text{A} \\ \text{A} \\ \text{A} \\ \text{A} \end{matrix}\)	\(\begin{matrix} {}^{\backprime\backprime} \text{A} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{A} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \end{matrix}\)	\(\begin{matrix} {}^{\backprime\backprime} \text{A} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{A} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \end{matrix}\)
\(\begin{matrix} \text{B} \\ \text{B} \\ \text{B} \\ \text{B} \end{matrix}\)	\(\begin{matrix} {}^{\backprime\backprime} \text{B} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{B} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \end{matrix}\)	\(\begin{matrix} {}^{\backprime\backprime} \text{B} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{B} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \end{matrix}\)

\(\text{Table 2.} ~~ \text{Sign Relation of Interpreter B}\!\)

\(\text{Object}\!\)	\(\text{Sign}\!\)	\(\text{Interpretant}\!\)
\(\begin{matrix} \text{A} \\ \text{A} \\ \text{A} \\ \text{A} \end{matrix}\)	\(\begin{matrix} {}^{\backprime\backprime} \text{A} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{A} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \end{matrix}\)	\(\begin{matrix} {}^{\backprime\backprime} \text{A} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{A} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \end{matrix}\)
\(\begin{matrix} \text{B} \\ \text{B} \\ \text{B} \\ \text{B} \end{matrix}\)	\(\begin{matrix} {}^{\backprime\backprime} \text{B} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{B} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \end{matrix}\)	\(\begin{matrix} {}^{\backprime\backprime} \text{B} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{B} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \end{matrix}\)

\(\text{Table 36.} ~~ \text{Semantics for Higher Order Signs}\!\)

\(\text{Object Denoted}\!\)	\(\text{Equivalent Signs}\!\)
\(\begin{matrix} \text{A} \\ \text{B} \end{matrix}\)	\(\begin{matrix} {}^{\langle} \text{A} {}^{\rangle} & = & {}^{\backprime\backprime} \text{A} {}^{\prime\prime} \\ {}^{\langle} \text{B} {}^{\rangle} & = & {}^{\backprime\backprime} \text{B} {}^{\prime\prime} \end{matrix}\)
\(\begin{matrix} {}^{\backprime\backprime} \text{A} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{B} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \end{matrix}\)	\(\begin{matrix} {}^{\langle\langle} \text{A} {}^{\rangle\rangle} & = & {}^{\langle\backprime\backprime} \text{A} {}^{\prime\prime\rangle} & = & {}^{\backprime\backprime\langle} \text{A} {}^{\rangle\prime\prime} \\ {}^{\langle\langle} \text{B} {}^{\rangle\rangle} & = & {}^{\langle\backprime\backprime} \text{B} {}^{\prime\prime\rangle} & = & {}^{\backprime\backprime\langle} \text{B} {}^{\rangle\prime\prime} \\ {}^{\langle\langle} \text{i} {}^{\rangle\rangle} & = & {}^{\langle\backprime\backprime} \text{i} {}^{\prime\prime\rangle} & = & {}^{\backprime\backprime\langle} \text{i} {}^{\rangle\prime\prime} \\ {}^{\langle\langle} \text{u} {}^{\rangle\rangle} & = & {}^{\langle\backprime\backprime} \text{u} {}^{\prime\prime\rangle} & = & {}^{\backprime\backprime\langle} \text{u} {}^{\rangle\prime\prime} \end{matrix}\)

\(\text{Table 37.} ~~ \text{Sign Relation Containing a Higher Order Sign}\!\)

\(\text{Object}\!\)	\(\text{Sign}\!\)	\(\text{Interpretant}\!\)
\(\begin{matrix} \ldots \\[2pt] \ldots \\[2pt] \text{s} \end{matrix}\)	\(\begin{matrix} \text{s} \\[2pt] \ldots \\[2pt] \text{t} \end{matrix}\)	\(\begin{matrix} \ldots \\[2pt] \ldots \\[2pt] \ldots \end{matrix}\)

\(\text{Table 38.} ~~ \text{Sign Relation for a Succession of Higher Order Signs (1)}\!\)

\(\text{Object}\!\)	\(\text{Sign}\!\)	\(\text{Interpretant}\!\)
\(\begin{matrix} x \\[2pt] {}^{\langle} x {}^{\rangle} \\[2pt] {}^{\langle\langle} x {}^{\rangle\rangle} \\[2pt] \ldots \end{matrix}\)	\(\begin{matrix} {}^{\langle} x {}^{\rangle} \\[2pt] {}^{\langle\langle} x {}^{\rangle\rangle} \\[2pt] {}^{\langle\langle\langle} x {}^{\rangle\rangle\rangle} \\[2pt] \ldots \end{matrix}\)	\(\begin{matrix} \ldots \\[2pt] \ldots \\[2pt] \ldots \\[2pt] \ldots \end{matrix}\)

\(\text{Table 39.} ~~ \text{Sign Relation for a Succession of Higher Order Signs (2)}\!\)

\(\text{Object}\!\)	\(\text{Sign}\!\)	\(\text{Interpretant}\!\)
\(\begin{matrix} x \\[2pt] s_1 \\[2pt] s_2 \\[2pt] \ldots \end{matrix}\)	\(\begin{matrix} s_1 \\[2pt] s_2 \\[2pt] s_3 \\[2pt] \ldots \end{matrix}\)	\(\begin{matrix} \ldots \\[2pt] \ldots \\[2pt] \ldots \\[2pt] \ldots \end{matrix}\)

\(\text{Table 40.} ~~ \text{Reflective Origin} ~ \operatorname{Ref}^0 L(\text{A})\!\)

\(\text{Object}\!\)	\(\text{Sign}\!\)	\(\text{Interpretant}\!\)
\(\begin{matrix} \text{A} \\ \text{A} \\ \text{A} \\ \text{A} \end{matrix}\)	\(\begin{matrix} {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{i} {}^{\rangle} \\ {}^{\langle} \text{i} {}^{\rangle} \end{matrix}\)	\(\begin{matrix} {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{i} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{i} {}^{\rangle} \end{matrix}\)
\(\begin{matrix} \text{B} \\ \text{B} \\ \text{B} \\ \text{B} \end{matrix}\)	\(\begin{matrix} {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{u} {}^{\rangle} \\ {}^{\langle} \text{u} {}^{\rangle} \end{matrix}\)	\(\begin{matrix} {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{u} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{u} {}^{\rangle} \end{matrix}\)

\(\text{Table 41.} ~~ \text{Reflective Origin} ~ \operatorname{Ref}^0 L(\text{B})\!\)

\(\text{Object}\!\)	\(\text{Sign}\!\)	\(\text{Interpretant}\!\)
\(\begin{matrix} \text{A} \\ \text{A} \\ \text{A} \\ \text{A} \end{matrix}\)	\(\begin{matrix} {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{u} {}^{\rangle} \\ {}^{\langle} \text{u} {}^{\rangle} \end{matrix}\)	\(\begin{matrix} {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{u} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{u} {}^{\rangle} \end{matrix}\)
\(\begin{matrix} \text{B} \\ \text{B} \\ \text{B} \\ \text{B} \end{matrix}\)	\(\begin{matrix} {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{i} {}^{\rangle} \\ {}^{\langle} \text{i} {}^{\rangle} \end{matrix}\)	\(\begin{matrix} {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{i} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{i} {}^{\rangle} \end{matrix}\)

\(\text{Table 42.} ~~ \text{Higher Ascent Sign Relation} ~ \operatorname{Ref}^1 L(\text{A})\!\)

\(\text{Object}\!\)	\(\text{Sign}\!\)	\(\text{Interpretant}\!\)
\(\begin{matrix} \text{A} \\ \text{A} \\ \text{A} \\ \text{A} \end{matrix}\)	\(\begin{matrix} {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{i} {}^{\rangle} \\ {}^{\langle} \text{i} {}^{\rangle} \end{matrix}\)	\(\begin{matrix} {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{i} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{i} {}^{\rangle} \end{matrix}\)
\(\begin{matrix} \text{B} \\ \text{B} \\ \text{B} \\ \text{B} \end{matrix}\)	\(\begin{matrix} {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{u} {}^{\rangle} \\ {}^{\langle} \text{u} {}^{\rangle} \end{matrix}\)	\(\begin{matrix} {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{u} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{u} {}^{\rangle} \end{matrix}\)
\(\begin{matrix} {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{i} {}^{\rangle} \\ {}^{\langle} \text{u} {}^{\rangle} \end{matrix}\)	\(\begin{matrix} {}^{\langle\langle} \text{A} {}^{\rangle\rangle} \\ {}^{\langle\langle} \text{B} {}^{\rangle\rangle} \\ {}^{\langle\langle} \text{i} {}^{\rangle\rangle} \\ {}^{\langle\langle} \text{u} {}^{\rangle\rangle} \end{matrix}\)	\(\begin{matrix} {}^{\langle\langle} \text{A} {}^{\rangle\rangle} \\ {}^{\langle\langle} \text{B} {}^{\rangle\rangle} \\ {}^{\langle\langle} \text{i} {}^{\rangle\rangle} \\ {}^{\langle\langle} \text{u} {}^{\rangle\rangle} \end{matrix}\)

\(\text{Table 43.} ~~ \text{Higher Ascent Sign Relation} ~ \operatorname{Ref}^1 L(\text{B})\!\)

\(\text{Object}\!\)	\(\text{Sign}\!\)	\(\text{Interpretant}\!\)
\(\begin{matrix} \text{A} \\ \text{A} \\ \text{A} \\ \text{A} \end{matrix}\)	\(\begin{matrix} {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{u} {}^{\rangle} \\ {}^{\langle} \text{u} {}^{\rangle} \end{matrix}\)	\(\begin{matrix} {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{u} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{u} {}^{\rangle} \end{matrix}\)
\(\begin{matrix} \text{B} \\ \text{B} \\ \text{B} \\ \text{B} \end{matrix}\)	\(\begin{matrix} {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{i} {}^{\rangle} \\ {}^{\langle} \text{i} {}^{\rangle} \end{matrix}\)	\(\begin{matrix} {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{i} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{i} {}^{\rangle} \end{matrix}\)
\(\begin{matrix} {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{i} {}^{\rangle} \\ {}^{\langle} \text{u} {}^{\rangle} \end{matrix}\)	\(\begin{matrix} {}^{\langle\langle} \text{A} {}^{\rangle\rangle} \\ {}^{\langle\langle} \text{B} {}^{\rangle\rangle} \\ {}^{\langle\langle} \text{i} {}^{\rangle\rangle} \\ {}^{\langle\langle} \text{u} {}^{\rangle\rangle} \end{matrix}\)	\(\begin{matrix} {}^{\langle\langle} \text{A} {}^{\rangle\rangle} \\ {}^{\langle\langle} \text{B} {}^{\rangle\rangle} \\ {}^{\langle\langle} \text{i} {}^{\rangle\rangle} \\ {}^{\langle\langle} \text{u} {}^{\rangle\rangle} \end{matrix}\)

\(\text{Table 44.} ~~ \text{Higher Import Sign Relation} ~ \operatorname{HI}^1 L(\text{A})\!\)

\(\text{Object}\!\)	\(\text{Sign}\!\)	\(\text{Interpretant}\!\)
\(\begin{matrix} \text{A} \\ \text{A} \\ \text{A} \\ \text{A} \end{matrix}\)	\(\begin{matrix} {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{i} {}^{\rangle} \\ {}^{\langle} \text{i} {}^{\rangle} \end{matrix}\)	\(\begin{matrix} {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{i} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{i} {}^{\rangle} \end{matrix}\)
\(\begin{matrix} \text{B} \\ \text{B} \\ \text{B} \\ \text{B} \end{matrix}\)	\(\begin{matrix} {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{u} {}^{\rangle} \\ {}^{\langle} \text{u} {}^{\rangle} \end{matrix}\)	\(\begin{matrix} {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{u} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{u} {}^{\rangle} \end{matrix}\)
\(\begin{matrix} ( & \text{A} & , & {}^{\langle} \text{A} {}^{\rangle} & , & {}^{\langle} \text{A} {}^{\rangle} & ) \\ ( & \text{A} & , & {}^{\langle} \text{A} {}^{\rangle} & , & {}^{\langle} \text{i} {}^{\rangle} & ) \\ ( & \text{A} & , & {}^{\langle} \text{i} {}^{\rangle} & , & {}^{\langle} \text{A} {}^{\rangle} & ) \\ ( & \text{A} & , & {}^{\langle} \text{i} {}^{\rangle} & , & {}^{\langle} \text{i} {}^{\rangle} & ) \end{matrix}\)	\(\begin{matrix} {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \end{matrix}\)	\(\begin{matrix} {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \end{matrix}\)
\(\begin{matrix} ( & \text{B} & , & {}^{\langle} \text{B} {}^{\rangle} & , & {}^{\langle} \text{B} {}^{\rangle} & ) \\ ( & \text{B} & , & {}^{\langle} \text{B} {}^{\rangle} & , & {}^{\langle} \text{u} {}^{\rangle} & ) \\ ( & \text{B} & , & {}^{\langle} \text{u} {}^{\rangle} & , & {}^{\langle} \text{B} {}^{\rangle} & ) \\ ( & \text{B} & , & {}^{\langle} \text{u} {}^{\rangle} & , & {}^{\langle} \text{u} {}^{\rangle} & ) \end{matrix}\)	\(\begin{matrix} {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \end{matrix}\)	\(\begin{matrix} {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \end{matrix}\)
\(\begin{matrix} ( & \text{A} & , & {}^{\langle} \text{A} {}^{\rangle} & , & {}^{\langle} \text{A} {}^{\rangle} & ) \\ ( & \text{A} & , & {}^{\langle} \text{A} {}^{\rangle} & , & {}^{\langle} \text{u} {}^{\rangle} & ) \\ ( & \text{A} & , & {}^{\langle} \text{u} {}^{\rangle} & , & {}^{\langle} \text{A} {}^{\rangle} & ) \\ ( & \text{A} & , & {}^{\langle} \text{u} {}^{\rangle} & , & {}^{\langle} \text{u} {}^{\rangle} & ) \end{matrix}\)	\(\begin{matrix} {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \end{matrix}\)	\(\begin{matrix} {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \end{matrix}\)
\(\begin{matrix} ( & \text{B} & , & {}^{\langle} \text{B} {}^{\rangle} & , & {}^{\langle} \text{B} {}^{\rangle} & ) \\ ( & \text{B} & , & {}^{\langle} \text{B} {}^{\rangle} & , & {}^{\langle} \text{i} {}^{\rangle} & ) \\ ( & \text{B} & , & {}^{\langle} \text{i} {}^{\rangle} & , & {}^{\langle} \text{B} {}^{\rangle} & ) \\ ( & \text{B} & , & {}^{\langle} \text{i} {}^{\rangle} & , & {}^{\langle} \text{i} {}^{\rangle} & ) \end{matrix}\)	\(\begin{matrix} {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \end{matrix}\)	\(\begin{matrix} {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \end{matrix}\)

\(\text{Table 45.} ~~ \text{Higher Import Sign Relation} ~ \operatorname{HI}^1 L(\text{B})\!\)

\(\text{Object}\!\)	\(\text{Sign}\!\)	\(\text{Interpretant}\!\)
\(\begin{matrix} \text{A} \\ \text{A} \\ \text{A} \\ \text{A} \end{matrix}\)	\(\begin{matrix} {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{u} {}^{\rangle} \\ {}^{\langle} \text{u} {}^{\rangle} \end{matrix}\)	\(\begin{matrix} {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{u} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{u} {}^{\rangle} \end{matrix}\)
\(\begin{matrix} \text{B} \\ \text{B} \\ \text{B} \\ \text{B} \end{matrix}\)	\(\begin{matrix} {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{i} {}^{\rangle} \\ {}^{\langle} \text{i} {}^{\rangle} \end{matrix}\)	\(\begin{matrix} {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{i} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{i} {}^{\rangle} \end{matrix}\)
\(\begin{matrix} ( & \text{A} & , & {}^{\langle} \text{A} {}^{\rangle} & , & {}^{\langle} \text{A} {}^{\rangle} & ) \\ ( & \text{A} & , & {}^{\langle} \text{A} {}^{\rangle} & , & {}^{\langle} \text{i} {}^{\rangle} & ) \\ ( & \text{A} & , & {}^{\langle} \text{i} {}^{\rangle} & , & {}^{\langle} \text{A} {}^{\rangle} & ) \\ ( & \text{A} & , & {}^{\langle} \text{i} {}^{\rangle} & , & {}^{\langle} \text{i} {}^{\rangle} & ) \end{matrix}\)	\(\begin{matrix} {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \end{matrix}\)	\(\begin{matrix} {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \end{matrix}\)
\(\begin{matrix} ( & \text{B} & , & {}^{\langle} \text{B} {}^{\rangle} & , & {}^{\langle} \text{B} {}^{\rangle} & ) \\ ( & \text{B} & , & {}^{\langle} \text{B} {}^{\rangle} & , & {}^{\langle} \text{u} {}^{\rangle} & ) \\ ( & \text{B} & , & {}^{\langle} \text{u} {}^{\rangle} & , & {}^{\langle} \text{B} {}^{\rangle} & ) \\ ( & \text{B} & , & {}^{\langle} \text{u} {}^{\rangle} & , & {}^{\langle} \text{u} {}^{\rangle} & ) \end{matrix}\)	\(\begin{matrix} {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \end{matrix}\)	\(\begin{matrix} {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \end{matrix}\)
\(\begin{matrix} ( & \text{A} & , & {}^{\langle} \text{A} {}^{\rangle} & , & {}^{\langle} \text{A} {}^{\rangle} & ) \\ ( & \text{A} & , & {}^{\langle} \text{A} {}^{\rangle} & , & {}^{\langle} \text{u} {}^{\rangle} & ) \\ ( & \text{A} & , & {}^{\langle} \text{u} {}^{\rangle} & , & {}^{\langle} \text{A} {}^{\rangle} & ) \\ ( & \text{A} & , & {}^{\langle} \text{u} {}^{\rangle} & , & {}^{\langle} \text{u} {}^{\rangle} & ) \end{matrix}\)	\(\begin{matrix} {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \end{matrix}\)	\(\begin{matrix} {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \end{matrix}\)
\(\begin{matrix} ( & \text{B} & , & {}^{\langle} \text{B} {}^{\rangle} & , & {}^{\langle} \text{B} {}^{\rangle} & ) \\ ( & \text{B} & , & {}^{\langle} \text{B} {}^{\rangle} & , & {}^{\langle} \text{i} {}^{\rangle} & ) \\ ( & \text{B} & , & {}^{\langle} \text{i} {}^{\rangle} & , & {}^{\langle} \text{B} {}^{\rangle} & ) \\ ( & \text{B} & , & {}^{\langle} \text{i} {}^{\rangle} & , & {}^{\langle} \text{i} {}^{\rangle} & ) \end{matrix}\)	\(\begin{matrix} {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \end{matrix}\)	\(\begin{matrix} {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \end{matrix}\)

\(\text{Table 46.} ~~ \text{Higher Order Sign Relation for} ~ Q(\text{A}, \text{B})\!\)

\(\text{Object}\!\)	\(\text{Sign}\!\)	\(\text{Interpretant}\!\)
\(\begin{matrix} \text{A} \\ \text{B} \end{matrix}\)	\(\begin{matrix} {}^{\langle} L {}^{\rangle} \\ {}^{\langle} L {}^{\rangle} \end{matrix}\)	\(\begin{matrix} {}^{\langle} L {}^{\rangle} \\ {}^{\langle} L {}^{\rangle} \end{matrix}\)
\(\begin{matrix} {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{i} {}^{\rangle} \\ {}^{\langle} \text{u} {}^{\rangle} \end{matrix}\)	\(\begin{matrix} {}^{\langle} q {}^{\rangle} \\ {}^{\langle} q {}^{\rangle} \\ {}^{\langle} q {}^{\rangle} \\ {}^{\langle} q {}^{\rangle} \end{matrix}\)	\(\begin{matrix} {}^{\langle} q {}^{\rangle} \\ {}^{\langle} q {}^{\rangle} \\ {}^{\langle} q {}^{\rangle} \\ {}^{\langle} q {}^{\rangle} \end{matrix}\)
\(\begin{matrix} ( & \text{A} & , & {}^{\langle} \text{A} {}^{\rangle} & , & {}^{\langle} \text{A} {}^{\rangle} & ) \\ ( & \text{A} & , & {}^{\langle} \text{A} {}^{\rangle} & , & {}^{\langle} \text{i} {}^{\rangle} & ) \\ ( & \text{A} & , & {}^{\langle} \text{i} {}^{\rangle} & , & {}^{\langle} \text{A} {}^{\rangle} & ) \\ ( & \text{A} & , & {}^{\langle} \text{i} {}^{\rangle} & , & {}^{\langle} \text{i} {}^{\rangle} & ) \\ ( & \text{B} & , & {}^{\langle} \text{B} {}^{\rangle} & , & {}^{\langle} \text{B} {}^{\rangle} & ) \\ ( & \text{B} & , & {}^{\langle} \text{B} {}^{\rangle} & , & {}^{\langle} \text{u} {}^{\rangle} & ) \\ ( & \text{B} & , & {}^{\langle} \text{u} {}^{\rangle} & , & {}^{\langle} \text{B} {}^{\rangle} & ) \\ ( & \text{B} & , & {}^{\langle} \text{u} {}^{\rangle} & , & {}^{\langle} \text{u} {}^{\rangle} & ) \end{matrix}\)	\(\begin{matrix} {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \end{matrix}\)	\(\begin{matrix} {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \end{matrix}\)
\(\begin{matrix} ( & \text{A} & , & {}^{\langle} \text{A} {}^{\rangle} & , & {}^{\langle} \text{A} {}^{\rangle} & ) \\ ( & \text{A} & , & {}^{\langle} \text{A} {}^{\rangle} & , & {}^{\langle} \text{u} {}^{\rangle} & ) \\ ( & \text{A} & , & {}^{\langle} \text{u} {}^{\rangle} & , & {}^{\langle} \text{A} {}^{\rangle} & ) \\ ( & \text{A} & , & {}^{\langle} \text{u} {}^{\rangle} & , & {}^{\langle} \text{u} {}^{\rangle} & ) \\ ( & \text{B} & , & {}^{\langle} \text{B} {}^{\rangle} & , & {}^{\langle} \text{B} {}^{\rangle} & ) \\ ( & \text{B} & , & {}^{\langle} \text{B} {}^{\rangle} & , & {}^{\langle} \text{i} {}^{\rangle} & ) \\ ( & \text{B} & , & {}^{\langle} \text{i} {}^{\rangle} & , & {}^{\langle} \text{B} {}^{\rangle} & ) \\ ( & \text{B} & , & {}^{\langle} \text{i} {}^{\rangle} & , & {}^{\langle} \text{i} {}^{\rangle} & ) \end{matrix}\)	\(\begin{matrix} {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \end{matrix}\)	\(\begin{matrix} {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \end{matrix}\)
\(\begin{matrix} (( & {}^{\langle} \text{A} {}^{\rangle} & , & \text{A} & ), & \text{A} & ) \\ (( & {}^{\langle} \text{A} {}^{\rangle} & , & \text{B} & ), & \text{A} & ) \\ (( & {}^{\langle} \text{B} {}^{\rangle} & , & \text{A} & ), & \text{B} & ) \\ (( & {}^{\langle} \text{B} {}^{\rangle} & , & \text{B} & ), & \text{B} & ) \\ (( & {}^{\langle} \text{i} {}^{\rangle} & , & \text{A} & ), & \text{A} & ) \\ (( & {}^{\langle} \text{i} {}^{\rangle} & , & \text{B} & ), & \text{B} & ) \\ (( & {}^{\langle} \text{u} {}^{\rangle} & , & \text{A} & ), & \text{B} & ) \\ (( & {}^{\langle} \text{u} {}^{\rangle} & , & \text{B} & ), & \text{A} & ) \end{matrix}\)	\(\begin{matrix} {}^{\langle} \operatorname{De} {}^{\rangle} \\ {}^{\langle} \operatorname{De} {}^{\rangle} \\ {}^{\langle} \operatorname{De} {}^{\rangle} \\ {}^{\langle} \operatorname{De} {}^{\rangle} \\ {}^{\langle} \operatorname{De} {}^{\rangle} \\ {}^{\langle} \operatorname{De} {}^{\rangle} \\ {}^{\langle} \operatorname{De} {}^{\rangle} \\ {}^{\langle} \operatorname{De} {}^{\rangle} \end{matrix}\)	\(\begin{matrix} {}^{\langle} \operatorname{De} {}^{\rangle} \\ {}^{\langle} \operatorname{De} {}^{\rangle} \\ {}^{\langle} \operatorname{De} {}^{\rangle} \\ {}^{\langle} \operatorname{De} {}^{\rangle} \\ {}^{\langle} \operatorname{De} {}^{\rangle} \\ {}^{\langle} \operatorname{De} {}^{\rangle} \\ {}^{\langle} \operatorname{De} {}^{\rangle} \\ {}^{\langle} \operatorname{De} {}^{\rangle} \end{matrix}\)

\(\text{Table 48.1} ~~ \operatorname{ER}(L_\text{A}) : \text{Extensional Representation of} ~ L_\text{A}\!\)

\(\text{Object}\!\)	\(\text{Sign}\!\)	\(\text{Interpretant}\!\)
\(\begin{matrix} \text{A} \\ \text{A} \\ \text{A} \\ \text{A} \end{matrix}\)	\(\begin{matrix} {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{i} {}^{\rangle} \\ {}^{\langle} \text{i} {}^{\rangle} \end{matrix}\)	\(\begin{matrix} {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{i} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{i} {}^{\rangle} \end{matrix}\)
\(\begin{matrix} \text{B} \\ \text{B} \\ \text{B} \\ \text{B} \end{matrix}\)	\(\begin{matrix} {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{u} {}^{\rangle} \\ {}^{\langle} \text{u} {}^{\rangle} \end{matrix}\)	\(\begin{matrix} {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{u} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{u} {}^{\rangle} \end{matrix}\)

\(\text{Table 48.2} ~~ \operatorname{ER}(\operatorname{Den}(L_\text{A})) : \text{Denotative Component of} ~ L_\text{A}\!\)

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

\(\text{Table 48.3} ~~ \operatorname{ER}(\operatorname{Con}(L_\text{A})) : \text{Connotative Component of} ~ L_\text{A}\!\)

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

\(\text{Table 49.1} ~~ \operatorname{ER}(L_\text{B}) : \text{Extensional Representation of} ~ L_\text{B}\!\)

\(\text{Object}\!\)	\(\text{Sign}\!\)	\(\text{Interpretant}\!\)
\(\begin{matrix} \text{A} \\ \text{A} \\ \text{A} \\ \text{A} \end{matrix}\)	\(\begin{matrix} {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{u} {}^{\rangle} \\ {}^{\langle} \text{u} {}^{\rangle} \end{matrix}\)	\(\begin{matrix} {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{u} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{u} {}^{\rangle} \end{matrix}\)
\(\begin{matrix} \text{B} \\ \text{B} \\ \text{B} \\ \text{B} \end{matrix}\)	\(\begin{matrix} {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{i} {}^{\rangle} \\ {}^{\langle} \text{i} {}^{\rangle} \end{matrix}\)	\(\begin{matrix} {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{i} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{i} {}^{\rangle} \end{matrix}\)

\(\text{Table 49.2} ~~ \operatorname{ER}(\operatorname{Den}(L_\text{B})) : \text{Denotative Component of} ~ L_\text{B}\!\)

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

\(\text{Table 49.3} ~~ \operatorname{ER}(\operatorname{Con}(L_\text{B})) : \text{Connotative Component of} ~ L_\text{B}\!\)

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

Type Tables

\(\text{Table 47.1} ~~ \text{Basic Types for ERs and IRs of Sign Relations}\!\)

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

\(\text{Table 47.2} ~~ \text{Derived Types for ERs of Sign Relations}\!\)

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

\(\text{Table 47.3} ~~ \text{Derived Types for IRs of Sign Relations}\!\)

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

Completed Work

\(\text{Table 50.} ~~ \text{Notations for Objects and Their Signs}\!\)

\(\text{Object}\!\)

\(\text{Sign of Object}\!\)

\(\begin{matrix} \text{A} & \text{A} & w_1 \\[6pt] \text{B} & \text{B} & w_2 \\[12pt] {}^{\backprime\backprime} \text{A} {}^{\prime\prime} & {}^{\langle} \text{A} {}^{\rangle} & w_3 \\[6pt] {}^{\backprime\backprime} \text{B} {}^{\prime\prime} & {}^{\langle} \text{B} {}^{\rangle} & w_4 \\[6pt] {}^{\backprime\backprime} \text{i} {}^{\prime\prime} & {}^{\langle} \text{i} {}^{\rangle} & w_5 \\[6pt] {}^{\backprime\backprime} \text{u} {}^{\prime\prime} & {}^{\langle} \text{u} {}^{\rangle} & w_6 \end{matrix}\)

\(\begin{matrix} {}^{\langle} \text{A} {}^{\rangle} & {}^{\langle} \text{A} {}^{\rangle} & {}^{\langle} w_1 {}^{\rangle} \\[6pt] {}^{\langle} \text{B} {}^{\rangle} & {}^{\langle} \text{B} {}^{\rangle} & {}^{\langle} w_2 {}^{\rangle} \\[12pt] {}^{\langle\backprime\backprime} \text{A} {}^{\prime\prime\rangle} & {}^{\langle\langle} \text{A} {}^{\rangle\rangle} & {}^{\langle} w_3 {}^{\rangle} \\[6pt] {}^{\langle\backprime\backprime} \text{B} {}^{\prime\prime\rangle} & {}^{\langle\langle} \text{B} {}^{\rangle\rangle} & {}^{\langle} w_4 {}^{\rangle} \\[6pt] {}^{\langle\backprime\backprime} \text{i} {}^{\prime\prime\rangle} & {}^{\langle\langle} \text{i} {}^{\rangle\rangle} & {}^{\langle} w_5 {}^{\rangle} \\[6pt] {}^{\langle\backprime\backprime} \text{u} {}^{\prime\prime\rangle} & {}^{\langle\langle} \text{u} {}^{\rangle\rangle} & {}^{\langle} w_6 {}^{\rangle} \end{matrix}\)

\(\text{Table 51.1} ~~ \text{Notations for Properties and Their Signs (1)}\!\)

\(\text{Property}\!\)

\(\text{Sign of Property}\!\)

\(\begin{matrix} {}^{\lbrace} \text{A} {}^{\rbrace} & {}^{\lbrace} \text{A} {}^{\rbrace} & {}^{\lbrace} w_1 {}^{\rbrace} \\[6pt] {}^{\lbrace} \text{B} {}^{\rbrace} & {}^{\lbrace} \text{B} {}^{\rbrace} & {}^{\lbrace} w_2 {}^{\rbrace} \\[12pt] {}^{\lbrace\backprime\backprime} \text{A} {}^{\prime\prime\rbrace} & {}^{\lbrace\langle} \text{A} {}^{\rangle\rbrace} & {}^{\lbrace} w_3 {}^{\rbrace} \\[6pt] {}^{\lbrace\backprime\backprime} \text{B} {}^{\prime\prime\rbrace} & {}^{\lbrace\langle} \text{B} {}^{\rangle\rbrace} & {}^{\lbrace} w_4 {}^{\rbrace} \\[6pt] {}^{\lbrace\backprime\backprime} \text{i} {}^{\prime\prime\rbrace} & {}^{\lbrace\langle} \text{i} {}^{\rangle\rbrace} & {}^{\lbrace} w_5 {}^{\rbrace} \\[6pt] {}^{\lbrace\backprime\backprime} \text{u} {}^{\prime\prime\rbrace} & {}^{\lbrace\langle} \text{u} {}^{\rangle\rbrace} & {}^{\lbrace} w_6 {}^{\rbrace} \end{matrix}\)

\(\begin{matrix} {}^{\langle\lbrace} \text{A} {}^{\rbrace\rangle} & {}^{\langle\lbrace} \text{A} {}^{\rbrace\rangle} & {}^{\langle\lbrace} w_1 {}^{\rbrace\rangle} \\[6pt] {}^{\langle\lbrace} \text{B} {}^{\rbrace\rangle} & {}^{\langle\lbrace} \text{B} {}^{\rbrace\rangle} & {}^{\langle\lbrace} w_2 {}^{\rbrace\rangle} \\[12pt] {}^{\langle\lbrace\backprime\backprime} \text{A} {}^{\prime\prime\rbrace\rangle} & {}^{\langle\lbrace\langle} \text{A} {}^{\rangle\rbrace\rangle} & {}^{\langle\lbrace} w_3 {}^{\rbrace\rangle} \\[6pt] {}^{\langle\lbrace\backprime\backprime} \text{B} {}^{\prime\prime\rbrace\rangle} & {}^{\langle\lbrace\langle} \text{B} {}^{\rangle\rbrace\rangle} & {}^{\langle\lbrace} w_4 {}^{\rbrace\rangle} \\[6pt] {}^{\langle\lbrace\backprime\backprime} \text{i} {}^{\prime\prime\rbrace\rangle} & {}^{\langle\lbrace\langle} \text{i} {}^{\rangle\rbrace\rangle} & {}^{\langle\lbrace} w_5 {}^{\rbrace\rangle} \\[6pt] {}^{\langle\lbrace\backprime\backprime} \text{u} {}^{\prime\prime\rbrace\rangle} & {}^{\langle\lbrace\langle} \text{u} {}^{\rangle\rbrace\rangle} & {}^{\langle\lbrace} w_6 {}^{\rbrace\rangle} \end{matrix}\)

\(\text{Table 51.2} ~~ \text{Notations for Properties and Their Signs (2)}\!\)

\(\text{Property}\!\)

\(\text{Sign of Property}\!\)

\(\begin{matrix} \underline{\underline{\text{A}}} & \underline{\underline{\text{A}}} & \underline{\underline{w_1}} \\[6pt] \underline{\underline{\text{B}}} & \underline{\underline{\text{B}}} & \underline{\underline{w_2}} \\[12pt] \underline{\underline{{}^{\backprime\backprime} \text{A} {}^{\prime\prime}}} & \underline{\underline{{}^{\langle} \text{A} {}^{\rangle}}} & \underline{\underline{w_3}} \\[6pt] \underline{\underline{{}^{\backprime\backprime} \text{B} {}^{\prime\prime}}} & \underline{\underline{{}^{\langle} \text{B} {}^{\rangle}}} & \underline{\underline{w_4}} \\[6pt] \underline{\underline{{}^{\backprime\backprime} \text{i} {}^{\prime\prime}}} & \underline{\underline{{}^{\langle} \text{i} {}^{\rangle}}} & \underline{\underline{w_5}} \\[6pt] \underline{\underline{{}^{\backprime\backprime} \text{u} {}^{\prime\prime}}} & \underline{\underline{{}^{\langle} \text{u} {}^{\rangle}}} & \underline{\underline{w_6}} \end{matrix}\)

\(\begin{matrix} {}^{\langle} \underline{\underline{\text{A}}} {}^{\rangle} & {}^{\langle} \underline{\underline{\text{A}}} {}^{\rangle} & {}^{\langle} \underline{\underline{w_1}} {}^{\rangle} \\[6pt] {}^{\langle} \underline{\underline{\text{B}}} {}^{\rangle} & {}^{\langle} \underline{\underline{\text{B}}} {}^{\rangle} & {}^{\langle} \underline{\underline{w_2}} {}^{\rangle} \\[12pt] {}^{\langle} \underline{\underline{{}^{\backprime\backprime} \text{A} {}^{\prime\prime}}} {}^{\rangle} & {}^{\langle} \underline{\underline{{}^{\langle} \text{A} {}^{\rangle}}} {}^{\rangle} & {}^{\langle} \underline{\underline{w_3}} {}^{\rangle} \\[6pt] {}^{\langle} \underline{\underline{{}^{\backprime\backprime} \text{B} {}^{\prime\prime}}} {}^{\rangle} & {}^{\langle} \underline{\underline{{}^{\langle} \text{B} {}^{\rangle}}} {}^{\rangle} & {}^{\langle} \underline{\underline{w_4}} {}^{\rangle} \\[6pt] {}^{\langle} \underline{\underline{{}^{\backprime\backprime} \text{i} {}^{\prime\prime}}} {}^{\rangle} & {}^{\langle} \underline{\underline{{}^{\langle} \text{i} {}^{\rangle}}} {}^{\rangle} & {}^{\langle} \underline{\underline{w_5}} {}^{\rangle} \\[6pt] {}^{\langle} \underline{\underline{{}^{\backprime\backprime} \text{u} {}^{\prime\prime}}} {}^{\rangle} & {}^{\langle} \underline{\underline{{}^{\langle} \text{u} {}^{\rangle}}} {}^{\rangle} & {}^{\langle} \underline{\underline{w_6}} {}^{\rangle} \end{matrix}\)

\(\text{Table 51.3} ~~ \text{Notations for Properties and Their Signs (3)}\!\)

\(\text{Property}\!\)

\(\text{Sign of Property}\!\)

\(\begin{matrix} \underline{\underline{\text{A}}} & \underline{\underline{o_1}} & \underline{\underline{w_1}} \\[6pt] \underline{\underline{\text{B}}} & \underline{\underline{o_2}} & \underline{\underline{w_2}} \\[12pt] \underline{\underline{\text{a}}} & \underline{\underline{s_1}} & \underline{\underline{w_3}} \\[6pt] \underline{\underline{\text{b}}} & \underline{\underline{s_2}} & \underline{\underline{w_4}} \\[6pt] \underline{\underline{\text{i}}} & \underline{\underline{s_3}} & \underline{\underline{w_5}} \\[6pt] \underline{\underline{\text{u}}} & \underline{\underline{s_4}} & \underline{\underline{w_6}} \end{matrix}\)

\(\begin{matrix} {}^{\langle} \underline{\underline{\text{A}}} {}^{\rangle} & {}^{\langle} \underline{\underline{o_1}} {}^{\rangle} & {}^{\langle} \underline{\underline{w_1}} {}^{\rangle} \\[6pt] {}^{\langle} \underline{\underline{\text{B}}} {}^{\rangle} & {}^{\langle} \underline{\underline{o_2}} {}^{\rangle} & {}^{\langle} \underline{\underline{w_2}} {}^{\rangle} \\[12pt] {}^{\langle} \underline{\underline{\text{a}}} {}^{\rangle} & {}^{\langle} \underline{\underline{s_1}} {}^{\rangle} & {}^{\langle} \underline{\underline{w_3}} {}^{\rangle} \\[6pt] {}^{\langle} \underline{\underline{\text{b}}} {}^{\rangle} & {}^{\langle} \underline{\underline{s_2}} {}^{\rangle} & {}^{\langle} \underline{\underline{w_4}} {}^{\rangle} \\[6pt] {}^{\langle} \underline{\underline{\text{i}}} {}^{\rangle} & {}^{\langle} \underline{\underline{s_3}} {}^{\rangle} & {}^{\langle} \underline{\underline{w_5}} {}^{\rangle} \\[6pt] {}^{\langle} \underline{\underline{\text{u}}} {}^{\rangle} & {}^{\langle} \underline{\underline{s_4}} {}^{\rangle} & {}^{\langle} \underline{\underline{w_6}} {}^{\rangle} \end{matrix}\)

\(\text{Table 52.1} ~~ \text{Notations for Instances and Their Signs (1)}\!\)

\(\text{Instance}\!\)

\(\text{Sign of Instance}\!\)

\(\begin{matrix} {}^{\lbrack} \text{A} {}^{\rbrack} & {}^{\lbrack} \text{A} {}^{\rbrack} & {}^{\lbrack} w_1 {}^{\rbrack} \\[6pt] {}^{\lbrack} \text{B} {}^{\rbrack} & {}^{\lbrack} \text{B} {}^{\rbrack} & {}^{\lbrack} w_2 {}^{\rbrack} \\[12pt] {}^{\lbrack\backprime\backprime} \text{A} {}^{\prime\prime\rbrack} & {}^{\lbrack\langle} \text{A} {}^{\rangle\rbrack} & {}^{\lbrack} w_3 {}^{\rbrack} \\[6pt] {}^{\lbrack\backprime\backprime} \text{B} {}^{\prime\prime\rbrack} & {}^{\lbrack\langle} \text{B} {}^{\rangle\rbrack} & {}^{\lbrack} w_4 {}^{\rbrack} \\[6pt] {}^{\lbrack\backprime\backprime} \text{i} {}^{\prime\prime\rbrack} & {}^{\lbrack\langle} \text{i} {}^{\rangle\rbrack} & {}^{\lbrack} w_5 {}^{\rbrack} \\[6pt] {}^{\lbrack\backprime\backprime} \text{u} {}^{\prime\prime\rbrack} & {}^{\lbrack\langle} \text{u} {}^{\rangle\rbrack} & {}^{\lbrack} w_6 {}^{\rbrack} \end{matrix}\)

\(\begin{matrix} {}^{\langle\lbrack} \text{A} {}^{\rbrack\rangle} & {}^{\langle\lbrack} \text{A} {}^{\rbrack\rangle} & {}^{\langle\lbrack} w_1 {}^{\rbrack\rangle} \\[6pt] {}^{\langle\lbrack} \text{B} {}^{\rbrack\rangle} & {}^{\langle\lbrack} \text{B} {}^{\rbrack\rangle} & {}^{\langle\lbrack} w_2 {}^{\rbrack\rangle} \\[12pt] {}^{\langle\lbrack\backprime\backprime} \text{A} {}^{\prime\prime\rbrack\rangle} & {}^{\langle\lbrack\langle} \text{A} {}^{\rangle\rbrack\rangle} & {}^{\langle\lbrack} w_3 {}^{\rbrack\rangle} \\[6pt] {}^{\langle\lbrack\backprime\backprime} \text{B} {}^{\prime\prime\rbrack\rangle} & {}^{\langle\lbrack\langle} \text{B} {}^{\rangle\rbrack\rangle} & {}^{\langle\lbrack} w_4 {}^{\rbrack\rangle} \\[6pt] {}^{\langle\lbrack\backprime\backprime} \text{i} {}^{\prime\prime\rbrack\rangle} & {}^{\langle\lbrack\langle} \text{i} {}^{\rangle\rbrack\rangle} & {}^{\langle\lbrack} w_5 {}^{\rbrack\rangle} \\[6pt] {}^{\langle\lbrack\backprime\backprime} \text{u} {}^{\prime\prime\rbrack\rangle} & {}^{\langle\lbrack\langle} \text{u} {}^{\rangle\rbrack\rangle} & {}^{\langle\lbrack} w_6 {}^{\rbrack\rangle} \end{matrix}\)

\(\text{Table 52.2} ~~ \text{Notations for Instances and Their Signs (2)}\!\)

\(\text{Instance}\!\)

\(\text{Sign of Instance}\!\)

\(\begin{matrix} \overline{\text{A}} & \overline{\text{A}} & \overline{w_1} \\[6pt] \overline{\text{B}} & \overline{\text{B}} & \overline{w_2} \\[12pt] \overline{{}^{\backprime\backprime} \text{A} {}^{\prime\prime}} & \overline{{}^{\langle} \text{A} {}^{\rangle}} & \overline{w_3} \\[6pt] \overline{{}^{\backprime\backprime} \text{B} {}^{\prime\prime}} & \overline{{}^{\langle} \text{B} {}^{\rangle}} & \overline{w_4} \\[6pt] \overline{{}^{\backprime\backprime} \text{i} {}^{\prime\prime}} & \overline{{}^{\langle} \text{i} {}^{\rangle}} & \overline{w_5} \\[6pt] \overline{{}^{\backprime\backprime} \text{u} {}^{\prime\prime}} & \overline{{}^{\langle} \text{u} {}^{\rangle}} & \overline{w_6} \end{matrix}\)

\(\begin{matrix} {}^{\langle} \overline{\text{A}} {}^{\rangle} & {}^{\langle} \overline{\text{A}} {}^{\rangle} & {}^{\langle} \overline{w_1} {}^{\rangle} \\[6pt] {}^{\langle} \overline{\text{B}} {}^{\rangle} & {}^{\langle} \overline{\text{B}} {}^{\rangle} & {}^{\langle} \overline{w_2} {}^{\rangle} \\[12pt] {}^{\langle} \overline{{}^{\backprime\backprime} \text{A} {}^{\prime\prime}} {}^{\rangle} & {}^{\langle} \overline{{}^{\langle} \text{A} {}^{\rangle}} {}^{\rangle} & {}^{\langle} \overline{w_3} {}^{\rangle} \\[6pt] {}^{\langle} \overline{{}^{\backprime\backprime} \text{B} {}^{\prime\prime}} {}^{\rangle} & {}^{\langle} \overline{{}^{\langle} \text{B} {}^{\rangle}} {}^{\rangle} & {}^{\langle} \overline{w_4} {}^{\rangle} \\[6pt] {}^{\langle} \overline{{}^{\backprime\backprime} \text{i} {}^{\prime\prime}} {}^{\rangle} & {}^{\langle} \overline{{}^{\langle} \text{i} {}^{\rangle}} {}^{\rangle} & {}^{\langle} \overline{w_5} {}^{\rangle} \\[6pt] {}^{\langle} \overline{{}^{\backprime\backprime} \text{u} {}^{\prime\prime}} {}^{\rangle} & {}^{\langle} \overline{{}^{\langle} \text{u} {}^{\rangle}} {}^{\rangle} & {}^{\langle} \overline{w_6} {}^{\rangle} \end{matrix}\)

\(\text{Table 52.3} ~~ \text{Notations for Instances and Their Signs (3)}\!\)

\(\text{Instance}\!\)

\(\text{Sign of Instance}\!\)

\(\begin{matrix} \overline{\text{A}} & \overline{o_1} & \overline{w_1} \\[6pt] \overline{\text{B}} & \overline{o_2} & \overline{w_2} \\[12pt] \overline{\text{a}} & \overline{s_1} & \overline{w_3} \\[6pt] \overline{\text{b}} & \overline{s_2} & \overline{w_4} \\[6pt] \overline{\text{i}} & \overline{s_3} & \overline{w_5} \\[6pt] \overline{\text{u}} & \overline{s_4} & \overline{w_6} \end{matrix}\)

\(\begin{matrix} {}^{\langle} \overline{\text{A}} {}^{\rangle} & {}^{\langle} \overline{o_1} {}^{\rangle} & {}^{\langle} \overline{w_1} {}^{\rangle} \\[6pt] {}^{\langle} \overline{\text{B}} {}^{\rangle} & {}^{\langle} \overline{o_2} {}^{\rangle} & {}^{\langle} \overline{w_2} {}^{\rangle} \\[12pt] {}^{\langle} \overline{\text{a}} {}^{\rangle} & {}^{\langle} \overline{s_1} {}^{\rangle} & {}^{\langle} \overline{w_3} {}^{\rangle} \\[6pt] {}^{\langle} \overline{\text{b}} {}^{\rangle} & {}^{\langle} \overline{s_2} {}^{\rangle} & {}^{\langle} \overline{w_4} {}^{\rangle} \\[6pt] {}^{\langle} \overline{\text{i}} {}^{\rangle} & {}^{\langle} \overline{s_3} {}^{\rangle} & {}^{\langle} \overline{w_5} {}^{\rangle} \\[6pt] {}^{\langle} \overline{\text{u}} {}^{\rangle} & {}^{\langle} \overline{s_4} {}^{\rangle} & {}^{\langle} \overline{w_6} {}^{\rangle} \end{matrix}\)

\(\text{Table 53.1} ~~ \text{Elements of} ~ \operatorname{ER}(W)\!\)

\(\text{Mnemonic Element}\!\) \(w \in W\!\)	\(\text{Pragmatic Element}\!\) \(w \in W\!\)	\(\text{Abstract Element}\!\) \(w_i \in W\!\)
\(\begin{matrix} \text{A} \\[4pt] \text{B} \\[4pt] {}^{\backprime\backprime} \text{A} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \text{B} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \end{matrix}\)	\(\begin{matrix} o_1 \\[4pt] o_2 \\[4pt] s_1 \\[4pt] s_2 \\[4pt] s_3 \\[4pt] s_4 \end{matrix}\)	\(\begin{matrix} w_1 \\[4pt] w_2 \\[4pt] w_3 \\[4pt] w_4 \\[4pt] w_5 \\[4pt] w_6 \end{matrix}\)

\(\text{Table 53.2} ~~ \text{Features of} ~ \operatorname{LIR}(W)\!\)

\(\text{Mnemonic Feature}\!\) \(\underline{\underline{w}} \in \underline{\underline{W}}\!\)	\(\text{Pragmatic Feature}\!\) \(\underline{\underline{w}} \in \underline{\underline{W}}\!\)	\(\text{Abstract Feature}\!\) \(\underline{\underline{w_i}} \in \underline{\underline{W}}\!\)
\(\begin{matrix} \underline{\underline{\text{A}}} \\[4pt] \underline{\underline{\text{B}}} \\[4pt] \underline{\underline{\text{a}}} \\[4pt] \underline{\underline{\text{b}}} \\[4pt] \underline{\underline{\text{i}}} \\[4pt] \underline{\underline{\text{u}}} \end{matrix}\)	\(\begin{matrix} \underline{\underline{o_1}} \\[4pt] \underline{\underline{o_2}} \\[4pt] \underline{\underline{s_1}} \\[4pt] \underline{\underline{s_2}} \\[4pt] \underline{\underline{s_3}} \\[4pt] \underline{\underline{s_4}} \end{matrix}\)	\(\begin{matrix} \underline{\underline{w_1}} \\[4pt] \underline{\underline{w_2}} \\[4pt] \underline{\underline{w_3}} \\[4pt] \underline{\underline{w_4}} \\[4pt] \underline{\underline{w_5}} \\[4pt] \underline{\underline{w_6}} \end{matrix}\)

\(\text{Table 54.1} ~~ \text{Mnemonic Literal Codes for Interpreters A and B}\!\)

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

\(\text{Table 54.2} ~~ \text{Pragmatic Literal Codes for Interpreters A and B}\!\)

\(\text{Element}\!\)	\(\text{Vector}\!\)	\(\text{Conjunct Term}\!\)	\(\text{Code}\!\)
\(\begin{matrix} \text{A} \\[4pt] \text{B} \\[4pt] {}^{\backprime\backprime} \text{A} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \text{B} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \end{matrix}\)	\(\begin{matrix} 100000 \\[4pt] 010000 \\[4pt] 001000 \\[4pt] 000100 \\[4pt] 000010 \\[4pt] 000001 \end{matrix}\)	\(\begin{matrix} ~\underline{\underline{o_1}}~ (\underline{\underline{o_2}}) (\underline{\underline{s_1}}) (\underline{\underline{s_2}}) (\underline{\underline{s_3}}) (\underline{\underline{s_4}}) \\[4pt] (\underline{\underline{o_1}}) ~\underline{\underline{o_2}}~ (\underline{\underline{s_1}}) (\underline{\underline{s_2}}) (\underline{\underline{s_3}}) (\underline{\underline{s_4}}) \\[4pt] (\underline{\underline{o_1}}) (\underline{\underline{o_2}}) ~\underline{\underline{s_1}}~ (\underline{\underline{s_2}}) (\underline{\underline{s_3}}) (\underline{\underline{s_4}}) \\[4pt] (\underline{\underline{o_1}}) (\underline{\underline{o_2}}) (\underline{\underline{s_1}}) ~\underline{\underline{s_2}}~ (\underline{\underline{s_3}}) (\underline{\underline{s_4}}) \\[4pt] (\underline{\underline{o_1}}) (\underline{\underline{o_2}}) (\underline{\underline{s_1}}) (\underline{\underline{s_2}}) ~\underline{\underline{s_3}}~ (\underline{\underline{s_4}}) \\[4pt] (\underline{\underline{o_1}}) (\underline{\underline{o_2}}) (\underline{\underline{s_1}}) (\underline{\underline{s_2}}) (\underline{\underline{s_3}}) ~\underline{\underline{s_4}}~ \end{matrix}\)	\(\begin{matrix} {\langle\underline{\underline{o_1}}\rangle}_W \\[4pt] {\langle\underline{\underline{o_2}}\rangle}_W \\[4pt] {\langle\underline{\underline{s_1}}\rangle}_W \\[4pt] {\langle\underline{\underline{s_2}}\rangle}_W \\[4pt] {\langle\underline{\underline{s_3}}\rangle}_W \\[4pt] {\langle\underline{\underline{s_4}}\rangle}_W \end{matrix}\)

\(\text{Table 54.3} ~~ \text{Abstract Literal Codes for Interpreters A and B}\!\)

\(\text{Element}\!\)	\(\text{Vector}\!\)	\(\text{Conjunct Term}\!\)	\(\text{Code}\!\)
\(\begin{matrix} \text{A} \\[4pt] \text{B} \\[4pt] {}^{\backprime\backprime} \text{A} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \text{B} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \end{matrix}\)	\(\begin{matrix} 100000 \\[4pt] 010000 \\[4pt] 001000 \\[4pt] 000100 \\[4pt] 000010 \\[4pt] 000001 \end{matrix}\)	\(\begin{matrix} ~\underline{\underline{w_1}}~ (\underline{\underline{w_2}}) (\underline{\underline{w_3}}) (\underline{\underline{w_4}}) (\underline{\underline{w_5}}) (\underline{\underline{w_6}}) \\[4pt] (\underline{\underline{w_1}}) ~\underline{\underline{w_2}}~ (\underline{\underline{w_3}}) (\underline{\underline{w_4}}) (\underline{\underline{w_5}}) (\underline{\underline{w_6}}) \\[4pt] (\underline{\underline{w_1}}) (\underline{\underline{w_2}}) ~\underline{\underline{w_3}}~ (\underline{\underline{w_4}}) (\underline{\underline{w_5}}) (\underline{\underline{w_6}}) \\[4pt] (\underline{\underline{w_1}}) (\underline{\underline{w_2}}) (\underline{\underline{w_3}}) ~\underline{\underline{w_4}}~ (\underline{\underline{w_5}}) (\underline{\underline{w_6}}) \\[4pt] (\underline{\underline{w_1}}) (\underline{\underline{w_2}}) (\underline{\underline{w_3}}) (\underline{\underline{w_4}}) ~\underline{\underline{w_5}}~ (\underline{\underline{w_6}}) \\[4pt] (\underline{\underline{w_1}}) (\underline{\underline{w_2}}) (\underline{\underline{w_3}}) (\underline{\underline{w_4}}) (\underline{\underline{w_5}}) ~\underline{\underline{w_6}}~ \end{matrix}\)	\(\begin{matrix} {\langle\underline{\underline{w_1}}\rangle}_W \\[4pt] {\langle\underline{\underline{w_2}}\rangle}_W \\[4pt] {\langle\underline{\underline{w_3}}\rangle}_W \\[4pt] {\langle\underline{\underline{w_4}}\rangle}_W \\[4pt] {\langle\underline{\underline{w_5}}\rangle}_W \\[4pt] {\langle\underline{\underline{w_6}}\rangle}_W \end{matrix}\)

\(\text{Table 55.1} ~~ \operatorname{LIR}_1 (L_\text{A}) : \text{Literal Representation of} ~ L_\text{A}\!\)

\(\text{Object}\!\)	\(\text{Sign}\!\)	\(\text{Interpretant}\!\)
\(\begin{matrix} {\langle\underline{\underline{\text{A}}}\rangle}_W \\[4pt] {\langle\underline{\underline{\text{A}}}\rangle}_W \\[4pt] {\langle\underline{\underline{\text{A}}}\rangle}_W \\[4pt] {\langle\underline{\underline{\text{A}}}\rangle}_W \end{matrix}\)	\(\begin{matrix} {\langle\underline{\underline{\text{a}}}\rangle}_W \\[4pt] {\langle\underline{\underline{\text{a}}}\rangle}_W \\[4pt] {\langle\underline{\underline{\text{i}}}\rangle}_W \\[4pt] {\langle\underline{\underline{\text{i}}}\rangle}_W \end{matrix}\)	\(\begin{matrix} {\langle\underline{\underline{\text{a}}}\rangle}_W \\[4pt] {\langle\underline{\underline{\text{i}}}\rangle}_W \\[4pt] {\langle\underline{\underline{\text{a}}}\rangle}_W \\[4pt] {\langle\underline{\underline{\text{i}}}\rangle}_W \end{matrix}\)
\(\begin{matrix} {\langle\underline{\underline{\text{B}}}\rangle}_W \\[4pt] {\langle\underline{\underline{\text{B}}}\rangle}_W \\[4pt] {\langle\underline{\underline{\text{B}}}\rangle}_W \\[4pt] {\langle\underline{\underline{\text{B}}}\rangle}_W \end{matrix}\)	\(\begin{matrix} {\langle\underline{\underline{\text{b}}}\rangle}_W \\[4pt] {\langle\underline{\underline{\text{b}}}\rangle}_W \\[4pt] {\langle\underline{\underline{\text{u}}}\rangle}_W \\[4pt] {\langle\underline{\underline{\text{u}}}\rangle}_W \end{matrix}\)	\(\begin{matrix} {\langle\underline{\underline{\text{b}}}\rangle}_W \\[4pt] {\langle\underline{\underline{\text{u}}}\rangle}_W \\[4pt] {\langle\underline{\underline{\text{b}}}\rangle}_W \\[4pt] {\langle\underline{\underline{\text{u}}}\rangle}_W \end{matrix}\)

\(\text{Table 55.2} ~~ \operatorname{LIR}_1 (\operatorname{Den}(L_\text{A})) : \text{Denotative Component of} ~ L_\text{A}\!\)

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

\(\text{Table 55.3} ~~ \operatorname{LIR}_1 (\operatorname{Con}(L_\text{A})) : \text{Connotative Component of} ~ L_\text{A}\!\)

\(\text{Sign}\!\)	\(\text{Interpretant}\!\)	\(\text{Transition}\!\)
\(\begin{matrix} {\langle\underline{\underline{\text{a}}}\rangle}_W \\[4pt] {\langle\underline{\underline{\text{a}}}\rangle}_W \\[4pt] {\langle\underline{\underline{\text{i}}}\rangle}_W \\[4pt] {\langle\underline{\underline{\text{i}}}\rangle}_W \end{matrix}\)	\(\begin{matrix} {\langle\underline{\underline{\text{a}}}\rangle}_W \\[4pt] {\langle\underline{\underline{\text{i}}}\rangle}_W \\[4pt] {\langle\underline{\underline{\text{a}}}\rangle}_W \\[4pt] {\langle\underline{\underline{\text{i}}}\rangle}_W \end{matrix}\)	\(\begin{matrix} 0_{\operatorname{d}W} \\[4pt] {\langle \operatorname{d}\underline{\underline{\text{a}}} ~ \operatorname{d}\underline{\underline{\text{i}}} \rangle}_{\operatorname{d}W} \\[4pt] {\langle \operatorname{d}\underline{\underline{\text{a}}} ~ \operatorname{d}\underline{\underline{\text{i}}} \rangle}_{\operatorname{d}W} \\[4pt] 0_{\operatorname{d}W} \end{matrix}\)
\(\begin{matrix} {\langle\underline{\underline{\text{b}}}\rangle}_W \\[4pt] {\langle\underline{\underline{\text{b}}}\rangle}_W \\[4pt] {\langle\underline{\underline{\text{u}}}\rangle}_W \\[4pt] {\langle\underline{\underline{\text{u}}}\rangle}_W \end{matrix}\)	\(\begin{matrix} {\langle\underline{\underline{\text{b}}}\rangle}_W \\[4pt] {\langle\underline{\underline{\text{u}}}\rangle}_W \\[4pt] {\langle\underline{\underline{\text{b}}}\rangle}_W \\[4pt] {\langle\underline{\underline{\text{u}}}\rangle}_W \end{matrix}\)	\(\begin{matrix} 0_{\operatorname{d}W} \\[4pt] {\langle \operatorname{d}\underline{\underline{\text{b}}} ~ \operatorname{d}\underline{\underline{\text{u}}} \rangle}_{\operatorname{d}W} \\[4pt] {\langle \operatorname{d}\underline{\underline{\text{b}}} ~ \operatorname{d}\underline{\underline{\text{u}}} \rangle}_{\operatorname{d}W} \\[4pt] 0_{\operatorname{d}W} \end{matrix}\)

\(\text{Table 56.1} ~~ \operatorname{LIR}_1 (L_\text{B}) : \text{Literal Representation of} ~ L_\text{B}\!\)

\(\text{Object}\!\)	\(\text{Sign}\!\)	\(\text{Interpretant}\!\)
\(\begin{matrix} {\langle\underline{\underline{\text{A}}}\rangle}_W \\[4pt] {\langle\underline{\underline{\text{A}}}\rangle}_W \\[4pt] {\langle\underline{\underline{\text{A}}}\rangle}_W \\[4pt] {\langle\underline{\underline{\text{A}}}\rangle}_W \end{matrix}\)	\(\begin{matrix} {\langle\underline{\underline{\text{a}}}\rangle}_W \\[4pt] {\langle\underline{\underline{\text{a}}}\rangle}_W \\[4pt] {\langle\underline{\underline{\text{u}}}\rangle}_W \\[4pt] {\langle\underline{\underline{\text{u}}}\rangle}_W \end{matrix}\)	\(\begin{matrix} {\langle\underline{\underline{\text{a}}}\rangle}_W \\[4pt] {\langle\underline{\underline{\text{u}}}\rangle}_W \\[4pt] {\langle\underline{\underline{\text{a}}}\rangle}_W \\[4pt] {\langle\underline{\underline{\text{u}}}\rangle}_W \end{matrix}\)
\(\begin{matrix} {\langle\underline{\underline{\text{B}}}\rangle}_W \\[4pt] {\langle\underline{\underline{\text{B}}}\rangle}_W \\[4pt] {\langle\underline{\underline{\text{B}}}\rangle}_W \\[4pt] {\langle\underline{\underline{\text{B}}}\rangle}_W \end{matrix}\)	\(\begin{matrix} {\langle\underline{\underline{\text{b}}}\rangle}_W \\[4pt] {\langle\underline{\underline{\text{b}}}\rangle}_W \\[4pt] {\langle\underline{\underline{\text{i}}}\rangle}_W \\[4pt] {\langle\underline{\underline{\text{i}}}\rangle}_W \end{matrix}\)	\(\begin{matrix} {\langle\underline{\underline{\text{b}}}\rangle}_W \\[4pt] {\langle\underline{\underline{\text{i}}}\rangle}_W \\[4pt] {\langle\underline{\underline{\text{b}}}\rangle}_W \\[4pt] {\langle\underline{\underline{\text{i}}}\rangle}_W \end{matrix}\)

\(\text{Table 56.2} ~~ \operatorname{LIR}_1 (\operatorname{Den}(L_\text{B})) : \text{Denotative Component of} ~ L_\text{B}\!\)

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

\(\text{Table 56.3} ~~ \operatorname{LIR}_1 (\operatorname{Con}(L_\text{B})) : \text{Connotative Component of} ~ L_\text{B}\!\)

\(\text{Sign}\!\)	\(\text{Interpretant}\!\)	\(\text{Transition}\!\)
\(\begin{matrix} {\langle\underline{\underline{\text{a}}}\rangle}_W \\[4pt] {\langle\underline{\underline{\text{a}}}\rangle}_W \\[4pt] {\langle\underline{\underline{\text{u}}}\rangle}_W \\[4pt] {\langle\underline{\underline{\text{u}}}\rangle}_W \end{matrix}\)	\(\begin{matrix} {\langle\underline{\underline{\text{a}}}\rangle}_W \\[4pt] {\langle\underline{\underline{\text{u}}}\rangle}_W \\[4pt] {\langle\underline{\underline{\text{a}}}\rangle}_W \\[4pt] {\langle\underline{\underline{\text{u}}}\rangle}_W \end{matrix}\)	\(\begin{matrix} 0_{\operatorname{d}W} \\[4pt] {\langle \operatorname{d}\underline{\underline{\text{a}}} ~ \operatorname{d}\underline{\underline{\text{u}}} \rangle}_{\operatorname{d}W} \\[4pt] {\langle \operatorname{d}\underline{\underline{\text{a}}} ~ \operatorname{d}\underline{\underline{\text{u}}} \rangle}_{\operatorname{d}W} \\[4pt] 0_{\operatorname{d}W} \end{matrix}\)
\(\begin{matrix} {\langle\underline{\underline{\text{b}}}\rangle}_W \\[4pt] {\langle\underline{\underline{\text{b}}}\rangle}_W \\[4pt] {\langle\underline{\underline{\text{i}}}\rangle}_W \\[4pt] {\langle\underline{\underline{\text{i}}}\rangle}_W \end{matrix}\)	\(\begin{matrix} {\langle\underline{\underline{\text{b}}}\rangle}_W \\[4pt] {\langle\underline{\underline{\text{i}}}\rangle}_W \\[4pt] {\langle\underline{\underline{\text{b}}}\rangle}_W \\[4pt] {\langle\underline{\underline{\text{i}}}\rangle}_W \end{matrix}\)	\(\begin{matrix} 0_{\operatorname{d}W} \\[4pt] {\langle \operatorname{d}\underline{\underline{\text{b}}} ~ \operatorname{d}\underline{\underline{\text{i}}} \rangle}_{\operatorname{d}W} \\[4pt] {\langle \operatorname{d}\underline{\underline{\text{b}}} ~ \operatorname{d}\underline{\underline{\text{i}}} \rangle}_{\operatorname{d}W} \\[4pt] 0_{\operatorname{d}W} \end{matrix}\)

\(\text{Table 57.1} ~~ \text{Mnemonic Lateral Codes for Interpreters A and B}\!\)

\(\text{Element}\!\)	\(\text{Vector}\!\)	\(\text{Conjunct Term}\!\)	\(\text{Code}\!\)
\(\begin{matrix} \text{A} \\[4pt] \text{B} \\[4pt] {}^{\backprime\backprime} \text{A} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \text{B} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \end{matrix}\)	\(\begin{matrix} {10}_X \\[4pt] {01}_X \\[4pt] {1000}_Y \\[4pt] {0100}_Y \\[4pt] {0010}_Y \\[4pt] {0001}_Y \end{matrix}\)	\(\begin{matrix} ~\underline{\underline{A}}~ (\underline{\underline{B}}) \\[4pt] (\underline{\underline{A}}) ~\underline{\underline{B}}~ \\[4pt] ~\underline{\underline{a}}~ (\underline{\underline{b}}) (\underline{\underline{i}}) (\underline{\underline{u}}) \\[4pt] (\underline{\underline{a}}) ~\underline{\underline{b}}~ (\underline{\underline{i}}) (\underline{\underline{u}}) \\[4pt] (\underline{\underline{a}}) (\underline{\underline{b}}) ~\underline{\underline{i}}~ (\underline{\underline{u}}) \\[4pt] (\underline{\underline{a}}) (\underline{\underline{b}}) (\underline{\underline{i}}) ~\underline{\underline{u}}~ \end{matrix}\)	\(\begin{matrix} {\langle\underline{\underline{A}}\rangle}_X \\[4pt] {\langle\underline{\underline{B}}\rangle}_X \\[4pt] {\langle\underline{\underline{a}}\rangle}_Y \\[4pt] {\langle\underline{\underline{b}}\rangle}_Y \\[4pt] {\langle\underline{\underline{i}}\rangle}_Y \\[4pt] {\langle\underline{\underline{u}}\rangle}_Y \end{matrix}\)

\(\text{Table 57.2} ~~ \text{Pragmatic Lateral Codes for Interpreters A and B}\!\)

\(\text{Element}\!\)	\(\text{Vector}\!\)	\(\text{Conjunct Term}\!\)	\(\text{Code}\!\)
\(\begin{matrix} \text{A} \\[4pt] \text{B} \\[4pt] {}^{\backprime\backprime} \text{A} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \text{B} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \end{matrix}\)	\(\begin{matrix} {10}_X \\[4pt] {01}_X \\[4pt] {1000}_Y \\[4pt] {0100}_Y \\[4pt] {0010}_Y \\[4pt] {0001}_Y \end{matrix}\)	\(\begin{matrix} ~\underline{\underline{o_1}}~ (\underline{\underline{o_2}}) \\[4pt] (\underline{\underline{o_1}}) ~\underline{\underline{o_2}}~ \\[4pt] ~\underline{\underline{s_1}}~ (\underline{\underline{s_2}}) (\underline{\underline{s_3}}) (\underline{\underline{s_4}}) \\[4pt] (\underline{\underline{s_1}}) ~\underline{\underline{s_2}}~ (\underline{\underline{s_3}}) (\underline{\underline{s_4}}) \\[4pt] (\underline{\underline{s_1}}) (\underline{\underline{s_2}}) ~\underline{\underline{s_3}}~ (\underline{\underline{s_4}}) \\[4pt] (\underline{\underline{s_1}}) (\underline{\underline{s_2}}) (\underline{\underline{s_3}}) ~\underline{\underline{s_4}}~ \end{matrix}\)	\(\begin{matrix} {\langle\underline{\underline{o_1}}\rangle}_X \\[4pt] {\langle\underline{\underline{o_2}}\rangle}_X \\[4pt] {\langle\underline{\underline{s_1}}\rangle}_Y \\[4pt] {\langle\underline{\underline{s_2}}\rangle}_Y \\[4pt] {\langle\underline{\underline{s_3}}\rangle}_Y \\[4pt] {\langle\underline{\underline{s_4}}\rangle}_Y \end{matrix}\)

\(\text{Table 57.3} ~~ \text{Abstract Lateral Codes for Interpreters A and B}\!\)

\(\text{Element}\!\)	\(\text{Vector}\!\)	\(\text{Conjunct Term}\!\)	\(\text{Code}\!\)
\(\begin{matrix} \text{A} \\[4pt] \text{B} \\[4pt] {}^{\backprime\backprime} \text{A} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \text{B} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \end{matrix}\)	\(\begin{matrix} {10}_X \\[4pt] {01}_X \\[4pt] {1000}_Y \\[4pt] {0100}_Y \\[4pt] {0010}_Y \\[4pt] {0001}_Y \end{matrix}\)	\(\begin{matrix} ~\underline{\underline{x_1}}~ (\underline{\underline{x_2}}) \\[4pt] (\underline{\underline{x_1}}) ~\underline{\underline{x_2}}~ \\[4pt] ~\underline{\underline{y_1}}~ (\underline{\underline{y_2}}) (\underline{\underline{y_3}}) (\underline{\underline{y_4}}) \\[4pt] (\underline{\underline{y_1}}) ~\underline{\underline{y_2}}~ (\underline{\underline{y_3}}) (\underline{\underline{y_4}}) \\[4pt] (\underline{\underline{y_1}}) (\underline{\underline{y_2}}) ~\underline{\underline{y_3}}~ (\underline{\underline{y_4}}) \\[4pt] (\underline{\underline{y_1}}) (\underline{\underline{y_2}}) (\underline{\underline{y_3}}) ~\underline{\underline{y_4}}~ \end{matrix}\)	\(\begin{matrix} {\langle\underline{\underline{x_1}}\rangle}_X \\[4pt] {\langle\underline{\underline{x_2}}\rangle}_X \\[4pt] {\langle\underline{\underline{y_1}}\rangle}_Y \\[4pt] {\langle\underline{\underline{y_2}}\rangle}_Y \\[4pt] {\langle\underline{\underline{y_3}}\rangle}_Y \\[4pt] {\langle\underline{\underline{y_4}}\rangle}_Y \end{matrix}\)

\(\text{Table 58.1} ~~ \operatorname{LIR}_2 (L_\text{A}) : \text{Lateral Representation of} ~ L_\text{A}\!\)

\(\text{Object}\!\)	\(\text{Sign}\!\)	\(\text{Interpretant}\!\)
\(\begin{matrix} ~\underline{\underline{\text{A}}}~ (\underline{\underline{\text{B}}}) \\[4pt] ~\underline{\underline{\text{A}}}~ (\underline{\underline{\text{B}}}) \\[4pt] ~\underline{\underline{\text{A}}}~ (\underline{\underline{\text{B}}}) \\[4pt] ~\underline{\underline{\text{A}}}~ (\underline{\underline{\text{B}}}) \end{matrix}\)	\(\begin{matrix} ~\underline{\underline{\text{a}}}~ (\underline{\underline{\text{b}}}) (\underline{\underline{\text{i}}}) (\underline{\underline{\text{u}}}) \\[4pt] ~\underline{\underline{\text{a}}}~ (\underline{\underline{\text{b}}}) (\underline{\underline{\text{i}}}) (\underline{\underline{\text{u}}}) \\[4pt] (\underline{\underline{\text{a}}}) (\underline{\underline{\text{b}}}) ~\underline{\underline{\text{i}}}~ (\underline{\underline{\text{u}}}) \\[4pt] (\underline{\underline{\text{a}}}) (\underline{\underline{\text{b}}}) ~\underline{\underline{\text{i}}}~ (\underline{\underline{\text{u}}}) \end{matrix}\)	\(\begin{matrix} ~\underline{\underline{\text{a}}}~ (\underline{\underline{\text{b}}}) (\underline{\underline{\text{i}}}) (\underline{\underline{\text{u}}}) \\[4pt] (\underline{\underline{\text{a}}}) (\underline{\underline{\text{b}}}) ~\underline{\underline{\text{i}}}~ (\underline{\underline{\text{u}}}) \\[4pt] ~\underline{\underline{\text{a}}}~ (\underline{\underline{\text{b}}}) (\underline{\underline{\text{i}}}) (\underline{\underline{\text{u}}}) \\[4pt] (\underline{\underline{\text{a}}}) (\underline{\underline{\text{b}}}) ~\underline{\underline{\text{i}}}~ (\underline{\underline{\text{u}}}) \end{matrix}\)
\(\begin{matrix} (\underline{\underline{\text{A}}}) ~\underline{\underline{\text{B}}}~ \\[4pt] (\underline{\underline{\text{A}}}) ~\underline{\underline{\text{B}}}~ \\[4pt] (\underline{\underline{\text{A}}}) ~\underline{\underline{\text{B}}}~ \\[4pt] (\underline{\underline{\text{A}}}) ~\underline{\underline{\text{B}}}~ \end{matrix}\)	\(\begin{matrix} (\underline{\underline{\text{a}}}) ~\underline{\underline{\text{b}}}~ (\underline{\underline{\text{i}}}) (\underline{\underline{\text{u}}}) \\[4pt] (\underline{\underline{\text{a}}}) ~\underline{\underline{\text{b}}}~ (\underline{\underline{\text{i}}}) (\underline{\underline{\text{u}}}) \\[4pt] (\underline{\underline{\text{a}}}) (\underline{\underline{\text{b}}}) (\underline{\underline{\text{i}}}) ~\underline{\underline{\text{u}}}~ \\[4pt] (\underline{\underline{\text{a}}}) (\underline{\underline{\text{b}}}) (\underline{\underline{\text{i}}}) ~\underline{\underline{\text{u}}}~ \end{matrix}\)	\(\begin{matrix} (\underline{\underline{\text{a}}}) ~\underline{\underline{\text{b}}}~ (\underline{\underline{\text{i}}}) (\underline{\underline{\text{u}}}) \\[4pt] (\underline{\underline{\text{a}}}) (\underline{\underline{\text{b}}}) (\underline{\underline{\text{i}}}) ~\underline{\underline{\text{u}}}~ \\[4pt] (\underline{\underline{\text{a}}}) ~\underline{\underline{\text{b}}}~ (\underline{\underline{\text{i}}}) (\underline{\underline{\text{u}}}) \\[4pt] (\underline{\underline{\text{a}}}) (\underline{\underline{\text{b}}}) (\underline{\underline{\text{i}}}) ~\underline{\underline{\text{u}}}~ \end{matrix}\)

\(\text{Table 58.2} ~~ \operatorname{LIR}_2 (\operatorname{Den}(L_\text{A})) : \text{Denotative Component of} ~ L_\text{A}\!\)

\(\text{Object}\!\)	\(\text{Sign}\!\)	\(\text{Transition}\!\)
\(\begin{matrix} ~\underline{\underline{\text{A}}}~ (\underline{\underline{\text{B}}}) \\[4pt] ~\underline{\underline{\text{A}}}~ (\underline{\underline{\text{B}}}) \end{matrix}\)	\(\begin{matrix} ~\underline{\underline{\text{a}}}~ (\underline{\underline{\text{b}}}) (\underline{\underline{\text{i}}}) (\underline{\underline{\text{u}}}) \\[4pt] (\underline{\underline{\text{a}}}) (\underline{\underline{\text{b}}}) ~\underline{\underline{\text{i}}}~ (\underline{\underline{\text{u}}}) \end{matrix}\)	\(\begin{matrix} ({\langle\underline{\underline{\text{a}}}\rangle}_Y, {\langle\underline{\underline{\text{A}}}\rangle}_X) \\[4pt] ({\langle\underline{\underline{\text{i}}}\rangle}_Y, {\langle\underline{\underline{\text{A}}}\rangle}_X) \end{matrix}\)
\(\begin{matrix} (\underline{\underline{\text{A}}}) ~\underline{\underline{\text{B}}}~ \\[4pt] (\underline{\underline{\text{A}}}) ~\underline{\underline{\text{B}}}~ \end{matrix}\)	\(\begin{matrix} (\underline{\underline{\text{a}}}) ~\underline{\underline{\text{b}}}~ (\underline{\underline{\text{i}}}) (\underline{\underline{\text{u}}}) \\[4pt] (\underline{\underline{\text{a}}}) (\underline{\underline{\text{b}}}) (\underline{\underline{\text{i}}}) ~\underline{\underline{\text{u}}}~ \end{matrix}\)	\(\begin{matrix} ({\langle\underline{\underline{\text{b}}}\rangle}_Y, {\langle\underline{\underline{\text{B}}}\rangle}_X) \\[4pt] ({\langle\underline{\underline{\text{u}}}\rangle}_Y, {\langle\underline{\underline{\text{B}}}\rangle}_X) \end{matrix}\)

\(\text{Table 58.3} ~~ \operatorname{LIR}_2 (\operatorname{Con}(L_\text{A})) : \text{Connotative Component of} ~ L_\text{A}\!\)

\(\text{Sign}\!\)	\(\text{Interpretant}\!\)	\(\text{Transition}\!\)
\(\begin{matrix} ~\underline{\underline{\text{a}}}~ (\underline{\underline{\text{b}}}) (\underline{\underline{\text{i}}}) (\underline{\underline{\text{u}}}) \\[4pt] ~\underline{\underline{\text{a}}}~ (\underline{\underline{\text{b}}}) (\underline{\underline{\text{i}}}) (\underline{\underline{\text{u}}}) \\[4pt] (\underline{\underline{\text{a}}}) (\underline{\underline{\text{b}}}) ~\underline{\underline{\text{i}}}~ (\underline{\underline{\text{u}}}) \\[4pt] (\underline{\underline{\text{a}}}) (\underline{\underline{\text{b}}}) ~\underline{\underline{\text{i}}}~ (\underline{\underline{\text{u}}}) \end{matrix}\)	\(\begin{matrix} ~\underline{\underline{\text{a}}}~ (\underline{\underline{\text{b}}}) (\underline{\underline{\text{i}}}) (\underline{\underline{\text{u}}}) \\[4pt] (\underline{\underline{\text{a}}}) (\underline{\underline{\text{b}}}) ~\underline{\underline{\text{i}}}~ (\underline{\underline{\text{u}}}) \\[4pt] ~\underline{\underline{\text{a}}}~ (\underline{\underline{\text{b}}}) (\underline{\underline{\text{i}}}) (\underline{\underline{\text{u}}}) \\[4pt] (\underline{\underline{\text{a}}}) (\underline{\underline{\text{b}}}) ~\underline{\underline{\text{i}}}~ (\underline{\underline{\text{u}}}) \end{matrix}\)	\(\begin{matrix} (\underline{\underline{\text{da}}}) (\underline{\underline{\text{db}}}) (\underline{\underline{\text{di}}}) (\underline{\underline{\text{du}}}) \\[4pt] ~\underline{\underline{\text{da}}}~ (\underline{\underline{\text{db}}}) ~\underline{\underline{\text{di}}}~ (\underline{\underline{\text{du}}}) \\[4pt] ~\underline{\underline{\text{da}}}~ (\underline{\underline{\text{db}}}) ~\underline{\underline{\text{di}}}~ (\underline{\underline{\text{du}}}) \\[4pt] (\underline{\underline{\text{da}}}) (\underline{\underline{\text{db}}}) (\underline{\underline{\text{di}}}) (\underline{\underline{\text{du}}}) \end{matrix}\)
\(\begin{matrix} (\underline{\underline{\text{a}}}) ~\underline{\underline{\text{b}}}~ (\underline{\underline{\text{i}}}) (\underline{\underline{\text{u}}}) \\[4pt] (\underline{\underline{\text{a}}}) ~\underline{\underline{\text{b}}}~ (\underline{\underline{\text{i}}}) (\underline{\underline{\text{u}}}) \\[4pt] (\underline{\underline{\text{a}}}) (\underline{\underline{\text{b}}}) (\underline{\underline{\text{i}}}) ~\underline{\underline{\text{u}}}~ \\[4pt] (\underline{\underline{\text{a}}}) (\underline{\underline{\text{b}}}) (\underline{\underline{\text{i}}}) ~\underline{\underline{\text{u}}}~ \end{matrix}\)	\(\begin{matrix} (\underline{\underline{\text{a}}}) ~\underline{\underline{\text{b}}}~ (\underline{\underline{\text{i}}}) (\underline{\underline{\text{u}}}) \\[4pt] (\underline{\underline{\text{a}}}) (\underline{\underline{\text{b}}}) (\underline{\underline{\text{i}}}) ~\underline{\underline{\text{u}}}~ \\[4pt] (\underline{\underline{\text{a}}}) ~\underline{\underline{\text{b}}}~ (\underline{\underline{\text{i}}}) (\underline{\underline{\text{u}}}) \\[4pt] (\underline{\underline{\text{a}}}) (\underline{\underline{\text{b}}}) (\underline{\underline{\text{i}}}) ~\underline{\underline{\text{u}}}~ \end{matrix}\)	\(\begin{matrix} (\underline{\underline{\text{da}}}) (\underline{\underline{\text{db}}}) (\underline{\underline{\text{di}}}) (\underline{\underline{\text{du}}}) \\[4pt] (\underline{\underline{\text{da}}}) ~\underline{\underline{\text{db}}}~ (\underline{\underline{\text{di}}}) ~\underline{\underline{\text{du}}}~ \\[4pt] (\underline{\underline{\text{da}}}) ~\underline{\underline{\text{db}}}~ (\underline{\underline{\text{di}}}) ~\underline{\underline{\text{du}}}~ \\[4pt] (\underline{\underline{\text{da}}}) (\underline{\underline{\text{db}}}) (\underline{\underline{\text{di}}}) (\underline{\underline{\text{du}}}) \end{matrix}\)

\(\text{Table 59.1} ~~ \operatorname{LIR}_2 (L_\text{B}) : \text{Lateral Representation of} ~ L_\text{B}\!\)

\(\text{Object}\!\)	\(\text{Sign}\!\)	\(\text{Interpretant}\!\)
\(\begin{matrix} ~\underline{\underline{\text{A}}}~ (\underline{\underline{\text{B}}}) \\[4pt] ~\underline{\underline{\text{A}}}~ (\underline{\underline{\text{B}}}) \\[4pt] ~\underline{\underline{\text{A}}}~ (\underline{\underline{\text{B}}}) \\[4pt] ~\underline{\underline{\text{A}}}~ (\underline{\underline{\text{B}}}) \end{matrix}\)	\(\begin{matrix} ~\underline{\underline{\text{a}}}~ (\underline{\underline{\text{b}}}) (\underline{\underline{\text{i}}}) (\underline{\underline{\text{u}}}) \\[4pt] ~\underline{\underline{\text{a}}}~ (\underline{\underline{\text{b}}}) (\underline{\underline{\text{i}}}) (\underline{\underline{\text{u}}}) \\[4pt] (\underline{\underline{\text{a}}}) (\underline{\underline{\text{b}}}) (\underline{\underline{\text{i}}}) ~\underline{\underline{\text{u}}}~ \\[4pt] (\underline{\underline{\text{a}}}) (\underline{\underline{\text{b}}}) (\underline{\underline{\text{i}}}) ~\underline{\underline{\text{u}}}~ \end{matrix}\)	\(\begin{matrix} ~\underline{\underline{\text{a}}}~ (\underline{\underline{\text{b}}}) (\underline{\underline{\text{i}}}) (\underline{\underline{\text{u}}}) \\[4pt] (\underline{\underline{\text{a}}}) (\underline{\underline{\text{b}}}) (\underline{\underline{\text{i}}}) ~\underline{\underline{\text{u}}}~ \\[4pt] ~\underline{\underline{\text{a}}}~ (\underline{\underline{\text{b}}}) (\underline{\underline{\text{i}}}) (\underline{\underline{\text{u}}}) \\[4pt] (\underline{\underline{\text{a}}}) (\underline{\underline{\text{b}}}) (\underline{\underline{\text{i}}}) ~\underline{\underline{\text{u}}}~ \end{matrix}\)
\(\begin{matrix} (\underline{\underline{\text{A}}}) ~\underline{\underline{\text{B}}}~ \\[4pt] (\underline{\underline{\text{A}}}) ~\underline{\underline{\text{B}}}~ \\[4pt] (\underline{\underline{\text{A}}}) ~\underline{\underline{\text{B}}}~ \\[4pt] (\underline{\underline{\text{A}}}) ~\underline{\underline{\text{B}}}~ \end{matrix}\)	\(\begin{matrix} (\underline{\underline{\text{a}}}) ~\underline{\underline{\text{b}}}~ (\underline{\underline{\text{i}}}) (\underline{\underline{\text{u}}}) \\[4pt] (\underline{\underline{\text{a}}}) ~\underline{\underline{\text{b}}}~ (\underline{\underline{\text{i}}}) (\underline{\underline{\text{u}}}) \\[4pt] (\underline{\underline{\text{a}}}) (\underline{\underline{\text{b}}}) ~\underline{\underline{\text{i}}}~ (\underline{\underline{\text{u}}}) \\[4pt] (\underline{\underline{\text{a}}}) (\underline{\underline{\text{b}}}) ~\underline{\underline{\text{i}}}~ (\underline{\underline{\text{u}}}) \end{matrix}\)	\(\begin{matrix} (\underline{\underline{\text{a}}}) ~\underline{\underline{\text{b}}}~ (\underline{\underline{\text{i}}}) (\underline{\underline{\text{u}}}) \\[4pt] (\underline{\underline{\text{a}}}) (\underline{\underline{\text{b}}}) ~\underline{\underline{\text{i}}}~ (\underline{\underline{\text{u}}}) \\[4pt] (\underline{\underline{\text{a}}}) ~\underline{\underline{\text{b}}}~ (\underline{\underline{\text{i}}}) (\underline{\underline{\text{u}}}) \\[4pt] (\underline{\underline{\text{a}}}) (\underline{\underline{\text{b}}}) ~\underline{\underline{\text{i}}}~ (\underline{\underline{\text{u}}}) \end{matrix}\)

\(\text{Table 59.2} ~~ \operatorname{LIR}_2 (\operatorname{Den}(L_\text{B})) : \text{Denotative Component of} ~ L_\text{B}\!\)

\(\text{Object}\!\)	\(\text{Sign}\!\)	\(\text{Transition}\!\)
\(\begin{matrix} ~\underline{\underline{\text{A}}}~ (\underline{\underline{\text{B}}}) \\[4pt] ~\underline{\underline{\text{A}}}~ (\underline{\underline{\text{B}}}) \end{matrix}\)	\(\begin{matrix} ~\underline{\underline{\text{a}}}~ (\underline{\underline{\text{b}}}) (\underline{\underline{\text{i}}}) (\underline{\underline{\text{u}}}) \\[4pt] (\underline{\underline{\text{a}}}) (\underline{\underline{\text{b}}}) (\underline{\underline{\text{i}}}) ~\underline{\underline{\text{u}}}~ \end{matrix}\)	\(\begin{matrix} ({\langle\underline{\underline{\text{a}}}\rangle}_Y, {\langle\underline{\underline{\text{A}}}\rangle}_X) \\[4pt] ({\langle\underline{\underline{\text{u}}}\rangle}_Y, {\langle\underline{\underline{\text{A}}}\rangle}_X) \end{matrix}\)
\(\begin{matrix} (\underline{\underline{\text{A}}}) ~\underline{\underline{\text{B}}}~ \\[4pt] (\underline{\underline{\text{A}}}) ~\underline{\underline{\text{B}}}~ \end{matrix}\)	\(\begin{matrix} (\underline{\underline{\text{a}}}) ~\underline{\underline{\text{b}}}~ (\underline{\underline{\text{i}}}) (\underline{\underline{\text{u}}}) \\[4pt] (\underline{\underline{\text{a}}}) (\underline{\underline{\text{b}}}) ~\underline{\underline{\text{i}}}~ (\underline{\underline{\text{u}}}) \end{matrix}\)	\(\begin{matrix} ({\langle\underline{\underline{\text{b}}}\rangle}_Y, {\langle\underline{\underline{\text{B}}}\rangle}_X) \\[4pt] ({\langle\underline{\underline{\text{i}}}\rangle}_Y, {\langle\underline{\underline{\text{B}}}\rangle}_X) \end{matrix}\)

\(\text{Table 59.3} ~~ \operatorname{LIR}_2 (\operatorname{Con}(L_\text{B})) : \text{Connotative Component of} ~ L_\text{B}\!\)

\(\text{Sign}\!\)	\(\text{Interpretant}\!\)	\(\text{Transition}\!\)
\(\begin{matrix} ~\underline{\underline{\text{a}}}~ (\underline{\underline{\text{b}}}) (\underline{\underline{\text{i}}}) (\underline{\underline{\text{u}}}) \\[4pt] ~\underline{\underline{\text{a}}}~ (\underline{\underline{\text{b}}}) (\underline{\underline{\text{i}}}) (\underline{\underline{\text{u}}}) \\[4pt] (\underline{\underline{\text{a}}}) (\underline{\underline{\text{b}}}) (\underline{\underline{\text{i}}}) ~\underline{\underline{\text{u}}}~ \\[4pt] (\underline{\underline{\text{a}}}) (\underline{\underline{\text{b}}}) (\underline{\underline{\text{i}}}) ~\underline{\underline{\text{u}}}~ \end{matrix}\)	\(\begin{matrix} ~\underline{\underline{\text{a}}}~ (\underline{\underline{\text{b}}}) (\underline{\underline{\text{i}}}) (\underline{\underline{\text{u}}}) \\[4pt] (\underline{\underline{\text{a}}}) (\underline{\underline{\text{b}}}) (\underline{\underline{\text{i}}}) ~\underline{\underline{\text{u}}}~ \\[4pt] ~\underline{\underline{\text{a}}}~ (\underline{\underline{\text{b}}}) (\underline{\underline{\text{i}}}) (\underline{\underline{\text{u}}}) \\[4pt] (\underline{\underline{\text{a}}}) (\underline{\underline{\text{b}}}) (\underline{\underline{\text{i}}}) ~\underline{\underline{\text{u}}}~ \end{matrix}\)	\(\begin{matrix} (\underline{\underline{\text{da}}}) (\underline{\underline{\text{db}}}) (\underline{\underline{\text{di}}}) (\underline{\underline{\text{du}}}) \\[4pt] ~\underline{\underline{\text{da}}}~ (\underline{\underline{\text{db}}}) (\underline{\underline{\text{di}}}) ~\underline{\underline{\text{du}}}~ \\[4pt] ~\underline{\underline{\text{da}}}~ (\underline{\underline{\text{db}}}) (\underline{\underline{\text{di}}}) ~\underline{\underline{\text{du}}}~ \\[4pt] (\underline{\underline{\text{da}}}) (\underline{\underline{\text{db}}}) (\underline{\underline{\text{di}}}) (\underline{\underline{\text{du}}}) \end{matrix}\)
\(\begin{matrix} (\underline{\underline{\text{a}}}) ~\underline{\underline{\text{b}}}~ (\underline{\underline{\text{i}}}) (\underline{\underline{\text{u}}}) \\[4pt] (\underline{\underline{\text{a}}}) ~\underline{\underline{\text{b}}}~ (\underline{\underline{\text{i}}}) (\underline{\underline{\text{u}}}) \\[4pt] (\underline{\underline{\text{a}}}) (\underline{\underline{\text{b}}}) ~\underline{\underline{\text{i}}}~ (\underline{\underline{\text{u}}}) \\[4pt] (\underline{\underline{\text{a}}}) (\underline{\underline{\text{b}}}) ~\underline{\underline{\text{i}}}~ (\underline{\underline{\text{u}}}) \end{matrix}\)	\(\begin{matrix} (\underline{\underline{\text{a}}}) ~\underline{\underline{\text{b}}}~ (\underline{\underline{\text{i}}}) (\underline{\underline{\text{u}}}) \\[4pt] (\underline{\underline{\text{a}}}) (\underline{\underline{\text{b}}}) ~\underline{\underline{\text{i}}}~ (\underline{\underline{\text{u}}}) \\[4pt] (\underline{\underline{\text{a}}}) ~\underline{\underline{\text{b}}}~ (\underline{\underline{\text{i}}}) (\underline{\underline{\text{u}}}) \\[4pt] (\underline{\underline{\text{a}}}) (\underline{\underline{\text{b}}}) ~\underline{\underline{\text{i}}}~ (\underline{\underline{\text{u}}}) \end{matrix}\)	\(\begin{matrix} (\underline{\underline{\text{da}}}) (\underline{\underline{\text{db}}}) (\underline{\underline{\text{di}}}) (\underline{\underline{\text{du}}}) \\[4pt] (\underline{\underline{\text{da}}}) ~\underline{\underline{\text{db}}}~ ~\underline{\underline{\text{di}}}~ (\underline{\underline{\text{du}}}) \\[4pt] (\underline{\underline{\text{da}}}) ~\underline{\underline{\text{db}}}~ ~\underline{\underline{\text{di}}}~ (\underline{\underline{\text{du}}}) \\[4pt] (\underline{\underline{\text{da}}}) (\underline{\underline{\text{db}}}) (\underline{\underline{\text{di}}}) (\underline{\underline{\text{du}}}) \end{matrix}\)

\(\text{Table 60.1} ~~ \operatorname{LIR}_3 (L_\text{A}) : \text{Lateral Representation of} ~ L_\text{A}\!\)

\(\text{Object}\!\)	\(\text{Sign}\!\)	\(\text{Interpretant}\!\)
\(\begin{matrix} {\langle\underline{\underline{\text{A}}}\rangle}_X \\[4pt] {\langle\underline{\underline{\text{A}}}\rangle}_X \\[4pt] {\langle\underline{\underline{\text{A}}}\rangle}_X \\[4pt] {\langle\underline{\underline{\text{A}}}\rangle}_X \end{matrix}\)	\(\begin{matrix} {\langle\underline{\underline{\text{a}}}\rangle}_Y \\[4pt] {\langle\underline{\underline{\text{a}}}\rangle}_Y \\[4pt] {\langle\underline{\underline{\text{i}}}\rangle}_Y \\[4pt] {\langle\underline{\underline{\text{i}}}\rangle}_Y \end{matrix}\)	\(\begin{matrix} {\langle\underline{\underline{\text{a}}}\rangle}_Y \\[4pt] {\langle\underline{\underline{\text{i}}}\rangle}_Y \\[4pt] {\langle\underline{\underline{\text{a}}}\rangle}_Y \\[4pt] {\langle\underline{\underline{\text{i}}}\rangle}_Y \end{matrix}\)
\(\begin{matrix} {\langle\underline{\underline{\text{B}}}\rangle}_X \\[4pt] {\langle\underline{\underline{\text{B}}}\rangle}_X \\[4pt] {\langle\underline{\underline{\text{B}}}\rangle}_X \\[4pt] {\langle\underline{\underline{\text{B}}}\rangle}_X \end{matrix}\)	\(\begin{matrix} {\langle\underline{\underline{\text{b}}}\rangle}_Y \\[4pt] {\langle\underline{\underline{\text{b}}}\rangle}_Y \\[4pt] {\langle\underline{\underline{\text{u}}}\rangle}_Y \\[4pt] {\langle\underline{\underline{\text{u}}}\rangle}_Y \end{matrix}\)	\(\begin{matrix} {\langle\underline{\underline{\text{b}}}\rangle}_Y \\[4pt] {\langle\underline{\underline{\text{u}}}\rangle}_Y \\[4pt] {\langle\underline{\underline{\text{b}}}\rangle}_Y \\[4pt] {\langle\underline{\underline{\text{u}}}\rangle}_Y \end{matrix}\)

\(\text{Table 60.2} ~~ \operatorname{LIR}_3 (\operatorname{Den}(L_\text{A})) : \text{Denotative Component of} ~ L_\text{A}\!\)

\(\text{Object}\!\)	\(\text{Sign}\!\)	\(\text{Transition}\!\)
\(\begin{matrix} {\langle\underline{\underline{\text{A}}}\rangle}_X \\[4pt] {\langle\underline{\underline{\text{A}}}\rangle}_X \end{matrix}\)	\(\begin{matrix} {\langle\underline{\underline{\text{a}}}\rangle}_Y \\[4pt] {\langle\underline{\underline{\text{i}}}\rangle}_Y \end{matrix}\)	\(\begin{matrix} ({\langle\underline{\underline{\text{a}}}\rangle}_Y, {\langle\underline{\underline{\text{A}}}\rangle}_X) \\[4pt] ({\langle\underline{\underline{\text{i}}}\rangle}_Y, {\langle\underline{\underline{\text{A}}}\rangle}_X) \end{matrix}\)
\(\begin{matrix} {\langle\underline{\underline{\text{B}}}\rangle}_X \\[4pt] {\langle\underline{\underline{\text{B}}}\rangle}_X \end{matrix}\)	\(\begin{matrix} {\langle\underline{\underline{\text{b}}}\rangle}_Y \\[4pt] {\langle\underline{\underline{\text{u}}}\rangle}_Y \end{matrix}\)	\(\begin{matrix} ({\langle\underline{\underline{\text{b}}}\rangle}_Y, {\langle\underline{\underline{\text{B}}}\rangle}_X) \\[4pt] ({\langle\underline{\underline{\text{u}}}\rangle}_Y, {\langle\underline{\underline{\text{B}}}\rangle}_X) \end{matrix}\)

\(\text{Table 60.3} ~~ \operatorname{LIR}_3 (\operatorname{Con}(L_\text{A})) : \text{Connotative Component of} ~ L_\text{A}\!\)

\(\text{Sign}\!\)	\(\text{Interpretant}\!\)	\(\text{Transition}\!\)
\(\begin{matrix} {\langle\underline{\underline{\text{a}}}\rangle}_Y \\[4pt] {\langle\underline{\underline{\text{a}}}\rangle}_Y \\[4pt] {\langle\underline{\underline{\text{i}}}\rangle}_Y \\[4pt] {\langle\underline{\underline{\text{i}}}\rangle}_Y \end{matrix}\)	\(\begin{matrix} {\langle\underline{\underline{\text{a}}}\rangle}_Y \\[4pt] {\langle\underline{\underline{\text{i}}}\rangle}_Y \\[4pt] {\langle\underline{\underline{\text{a}}}\rangle}_Y \\[4pt] {\langle\underline{\underline{\text{i}}}\rangle}_Y \end{matrix}\)	\(\begin{matrix} 0_{\operatorname{d}Y} \\[4pt] {\langle \operatorname{d}\underline{\underline{\text{a}}} ~ \operatorname{d}\underline{\underline{\text{i}}} \rangle}_{\operatorname{d}Y} \\[4pt] {\langle \operatorname{d}\underline{\underline{\text{a}}} ~ \operatorname{d}\underline{\underline{\text{i}}} \rangle}_{\operatorname{d}Y} \\[4pt] 0_{\operatorname{d}Y} \end{matrix}\)
\(\begin{matrix} {\langle\underline{\underline{\text{b}}}\rangle}_Y \\[4pt] {\langle\underline{\underline{\text{b}}}\rangle}_Y \\[4pt] {\langle\underline{\underline{\text{u}}}\rangle}_Y \\[4pt] {\langle\underline{\underline{\text{u}}}\rangle}_Y \end{matrix}\)	\(\begin{matrix} {\langle\underline{\underline{\text{b}}}\rangle}_Y \\[4pt] {\langle\underline{\underline{\text{u}}}\rangle}_Y \\[4pt] {\langle\underline{\underline{\text{b}}}\rangle}_Y \\[4pt] {\langle\underline{\underline{\text{u}}}\rangle}_Y \end{matrix}\)	\(\begin{matrix} 0_{\operatorname{d}Y} \\[4pt] {\langle \operatorname{d}\underline{\underline{\text{b}}} ~ \operatorname{d}\underline{\underline{\text{u}}} \rangle}_{\operatorname{d}Y} \\[4pt] {\langle \operatorname{d}\underline{\underline{\text{b}}} ~ \operatorname{d}\underline{\underline{\text{u}}} \rangle}_{\operatorname{d}Y} \\[4pt] 0_{\operatorname{d}Y} \end{matrix}\)

\(\text{Table 61.1} ~~ \operatorname{LIR}_3 (L_\text{B}) : \text{Lateral Representation of} ~ L_\text{B}\!\)

\(\text{Object}\!\)	\(\text{Sign}\!\)	\(\text{Interpretant}\!\)
\(\begin{matrix} {\langle\underline{\underline{\text{A}}}\rangle}_X \\[4pt] {\langle\underline{\underline{\text{A}}}\rangle}_X \\[4pt] {\langle\underline{\underline{\text{A}}}\rangle}_X \\[4pt] {\langle\underline{\underline{\text{A}}}\rangle}_X \end{matrix}\)	\(\begin{matrix} {\langle\underline{\underline{\text{a}}}\rangle}_Y \\[4pt] {\langle\underline{\underline{\text{a}}}\rangle}_Y \\[4pt] {\langle\underline{\underline{\text{u}}}\rangle}_Y \\[4pt] {\langle\underline{\underline{\text{u}}}\rangle}_Y \end{matrix}\)	\(\begin{matrix} {\langle\underline{\underline{\text{a}}}\rangle}_Y \\[4pt] {\langle\underline{\underline{\text{u}}}\rangle}_Y \\[4pt] {\langle\underline{\underline{\text{a}}}\rangle}_Y \\[4pt] {\langle\underline{\underline{\text{u}}}\rangle}_Y \end{matrix}\)
\(\begin{matrix} {\langle\underline{\underline{\text{B}}}\rangle}_X \\[4pt] {\langle\underline{\underline{\text{B}}}\rangle}_X \\[4pt] {\langle\underline{\underline{\text{B}}}\rangle}_X \\[4pt] {\langle\underline{\underline{\text{B}}}\rangle}_X \end{matrix}\)	\(\begin{matrix} {\langle\underline{\underline{\text{b}}}\rangle}_Y \\[4pt] {\langle\underline{\underline{\text{b}}}\rangle}_Y \\[4pt] {\langle\underline{\underline{\text{i}}}\rangle}_Y \\[4pt] {\langle\underline{\underline{\text{i}}}\rangle}_Y \end{matrix}\)	\(\begin{matrix} {\langle\underline{\underline{\text{b}}}\rangle}_Y \\[4pt] {\langle\underline{\underline{\text{i}}}\rangle}_Y \\[4pt] {\langle\underline{\underline{\text{b}}}\rangle}_Y \\[4pt] {\langle\underline{\underline{\text{i}}}\rangle}_Y \end{matrix}\)

\(\text{Table 61.2} ~~ \operatorname{LIR}_3 (\operatorname{Den}(L_\text{B})) : \text{Denotative Component of} ~ L_\text{B}\!\)

\(\text{Object}\!\)	\(\text{Sign}\!\)	\(\text{Transition}\!\)
\(\begin{matrix} {\langle\underline{\underline{\text{A}}}\rangle}_X \\[4pt] {\langle\underline{\underline{\text{A}}}\rangle}_X \end{matrix}\)	\(\begin{matrix} {\langle\underline{\underline{\text{a}}}\rangle}_Y \\[4pt] {\langle\underline{\underline{\text{u}}}\rangle}_Y \end{matrix}\)	\(\begin{matrix} ({\langle\underline{\underline{\text{a}}}\rangle}_Y, {\langle\underline{\underline{\text{A}}}\rangle}_X) \\[4pt] ({\langle\underline{\underline{\text{u}}}\rangle}_Y, {\langle\underline{\underline{\text{A}}}\rangle}_X) \end{matrix}\)
\(\begin{matrix} {\langle\underline{\underline{\text{B}}}\rangle}_X \\[4pt] {\langle\underline{\underline{\text{B}}}\rangle}_X \end{matrix}\)	\(\begin{matrix} {\langle\underline{\underline{\text{b}}}\rangle}_Y \\[4pt] {\langle\underline{\underline{\text{i}}}\rangle}_Y \end{matrix}\)	\(\begin{matrix} ({\langle\underline{\underline{\text{b}}}\rangle}_Y, {\langle\underline{\underline{\text{B}}}\rangle}_X) \\[4pt] ({\langle\underline{\underline{\text{i}}}\rangle}_Y, {\langle\underline{\underline{\text{B}}}\rangle}_X) \end{matrix}\)

\(\text{Table 61.3} ~~ \operatorname{LIR}_3 (\operatorname{Con}(L_\text{B})) : \text{Connotative Component of} ~ L_\text{B}\!\)

\(\text{Sign}\!\)	\(\text{Interpretant}\!\)	\(\text{Transition}\!\)
\(\begin{matrix} {\langle\underline{\underline{\text{a}}}\rangle}_Y \\[4pt] {\langle\underline{\underline{\text{a}}}\rangle}_Y \\[4pt] {\langle\underline{\underline{\text{u}}}\rangle}_Y \\[4pt] {\langle\underline{\underline{\text{u}}}\rangle}_Y \end{matrix}\)	\(\begin{matrix} {\langle\underline{\underline{\text{a}}}\rangle}_Y \\[4pt] {\langle\underline{\underline{\text{u}}}\rangle}_Y \\[4pt] {\langle\underline{\underline{\text{a}}}\rangle}_Y \\[4pt] {\langle\underline{\underline{\text{u}}}\rangle}_Y \end{matrix}\)	\(\begin{matrix} 0_{\operatorname{d}Y} \\[4pt] {\langle \operatorname{d}\underline{\underline{\text{a}}} ~ \operatorname{d}\underline{\underline{\text{u}}} \rangle}_{\operatorname{d}Y} \\[4pt] {\langle \operatorname{d}\underline{\underline{\text{a}}} ~ \operatorname{d}\underline{\underline{\text{u}}} \rangle}_{\operatorname{d}Y} \\[4pt] 0_{\operatorname{d}Y} \end{matrix}\)
\(\begin{matrix} {\langle\underline{\underline{\text{b}}}\rangle}_Y \\[4pt] {\langle\underline{\underline{\text{b}}}\rangle}_Y \\[4pt] {\langle\underline{\underline{\text{i}}}\rangle}_Y \\[4pt] {\langle\underline{\underline{\text{i}}}\rangle}_Y \end{matrix}\)	\(\begin{matrix} {\langle\underline{\underline{\text{b}}}\rangle}_Y \\[4pt] {\langle\underline{\underline{\text{i}}}\rangle}_Y \\[4pt] {\langle\underline{\underline{\text{b}}}\rangle}_Y \\[4pt] {\langle\underline{\underline{\text{i}}}\rangle}_Y \end{matrix}\)	\(\begin{matrix} 0_{\operatorname{d}Y} \\[4pt] {\langle \operatorname{d}\underline{\underline{\text{b}}} ~ \operatorname{d}\underline{\underline{\text{i}}} \rangle}_{\operatorname{d}Y} \\[4pt] {\langle \operatorname{d}\underline{\underline{\text{b}}} ~ \operatorname{d}\underline{\underline{\text{i}}} \rangle}_{\operatorname{d}Y} \\[4pt] 0_{\operatorname{d}Y} \end{matrix}\)

\(\text{Table 62.1} ~~ \text{Analytic Codes for Object Features}\!\)

\(\text{Category}\!\)	\(\text{Mnemonic}\!\)	\(\text{Code}\!\)
\(\begin{array}{l} \text{Self} \\[4pt] \text{Other} \end{array}\)	\(\begin{matrix} \text{self} \\[4pt] \text{(self)} \end{matrix}\)	\(\begin{matrix} \text{s} \\[4pt] \text{(s)} \end{matrix}\)

\(\text{Table 62.2} ~~ \text{Analytic Codes for Semantic Features}\!\)

\(\text{Category}\!\)	\(\text{Mnemonic}\!\)	\(\text{Code}\!\)
\(\begin{array}{l} \text{1st Person} \\[4pt] \text{2nd Person} \end{array}\)	\(\begin{matrix} \text{my} \\[4pt] \text{(my)} \end{matrix}\)	\(\begin{matrix} \text{m} \\[4pt] \text{(m)} \end{matrix}\)

\(\text{Table 62.3} ~~ \text{Analytic Codes for Syntactic Features}\!\)

\(\text{Category}\!\)	\(\text{Mnemonic}\!\)	\(\text{Code}\!\)
\(\begin{array}{l} \text{Noun} \\[4pt] \text{Pronoun} \end{array}\)	\(\begin{matrix} \text{name} \\[4pt] \text{(name)} \end{matrix}\)	\(\begin{matrix} \text{n} \\[4pt] \text{(n)} \end{matrix}\)

\(\text{Table 63.} ~~ \text{Analytic Codes for Interpreter A}\!\)

\(\text{Name}\!\)	\(\text{Vector}\!\)	\(\text{Conjunct Term}\!\)	\(\text{Mnemonic}\!\)	\(\text{Code}\!\)
\(\begin{matrix} \text{A} \\[4pt] \text{B} \\[4pt] {}^{\backprime\backprime} \text{A} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \text{B} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \end{matrix}\)	\(\begin{matrix} {1}_X \\[4pt] {0}_X \\[4pt] {11}_Y \\[4pt] {01}_Y \\[4pt] {10}_Y \\[4pt] {00}_Y \end{matrix}\)	\(\begin{matrix} ~x_1~ \\[4pt] (x_1) \\[4pt] ~y_1~~y_2~ \\[4pt] (y_1)~y_2~ \\[4pt] ~y_1~(y_2) \\[4pt] (y_1)(y_2) \end{matrix}\)	\(\begin{matrix} ~\text{self}~ \\[4pt] (\text{self}) \\[4pt] ~\text{my}~~\text{name}~ \\[4pt] (\text{my})~\text{name}~ \\[4pt] ~\text{my}~(\text{name}) \\[4pt] (\text{my})(\text{name}) \end{matrix}\)	\(\begin{matrix} ~\text{s}~ \\[4pt] (\text{s}) \\[4pt] ~\text{m}~~\text{n}~ \\[4pt] (\text{m})~\text{n}~ \\[4pt] ~\text{m}~(\text{n}) \\[4pt] (\text{m})(\text{n}) \end{matrix}\)

\(\text{Table 64.} ~~ \text{Analytic Codes for Interpreter B}\!\)

\(\text{Name}\!\)	\(\text{Vector}\!\)	\(\text{Conjunct Term}\!\)	\(\text{Mnemonic}\!\)	\(\text{Code}\!\)
\(\begin{matrix} \text{A} \\[4pt] \text{B} \\[4pt] {}^{\backprime\backprime} \text{A} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \text{B} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \end{matrix}\)	\(\begin{matrix} {0}_X \\[4pt] {1}_X \\[4pt] {01}_Y \\[4pt] {11}_Y \\[4pt] {10}_Y \\[4pt] {00}_Y \end{matrix}\)	\(\begin{matrix} (x_1) \\[4pt] ~x_1~ \\[4pt] (y_1)~y_2~ \\[4pt] ~y_1~~y_2~ \\[4pt] ~y_1~(y_2) \\[4pt] (y_1)(y_2) \end{matrix}\)	\(\begin{matrix} (\text{self}) \\[4pt] ~\text{self}~ \\[4pt] (\text{my})~\text{name}~ \\[4pt] ~\text{my}~~\text{name}~ \\[4pt] ~\text{my}~(\text{name}) \\[4pt] (\text{my})(\text{name}) \end{matrix}\)	\(\begin{matrix} (\text{s}) \\[4pt] ~\text{s}~ \\[4pt] (\text{m})~\text{n}~ \\[4pt] ~\text{m}~~\text{n}~ \\[4pt] ~\text{m}~(\text{n}) \\[4pt] (\text{m})(\text{n}) \end{matrix}\)

\(\text{Table 65.1} ~~ \operatorname{AIR}_1 (L_\text{A}) : \text{Analytic Representation of} ~ L_\text{A}\!\)

\(\text{Object}\!\)	\(\text{Sign}\!\)	\(\text{Interpretant}\!\)
\(\begin{matrix} \text{s} \\[4pt] \text{s} \\[4pt] \text{s} \\[4pt] \text{s} \end{matrix}\)	\(\begin{matrix} ~\text{m}~~\text{n}~ \\[4pt] ~\text{m}~~\text{n}~ \\[4pt] ~\text{m}~(\text{n}) \\[4pt] ~\text{m}~(\text{n}) \end{matrix}\)	\(\begin{matrix} ~\text{m}~~\text{n}~ \\[4pt] ~\text{m}~(\text{n}) \\[4pt] ~\text{m}~~\text{n}~ \\[4pt] ~\text{m}~(\text{n}) \end{matrix}\)
\(\begin{matrix} (\text{s}) \\[4pt] (\text{s}) \\[4pt] (\text{s}) \\[4pt] (\text{s}) \end{matrix}\)	\(\begin{matrix} (\text{m})~\text{n}~ \\[4pt] (\text{m})~\text{n}~ \\[4pt] (\text{m})(\text{n}) \\[4pt] (\text{m})(\text{n}) \end{matrix}\)	\(\begin{matrix} (\text{m})~\text{n}~ \\[4pt] (\text{m})(\text{n}) \\[4pt] (\text{m})~\text{n}~ \\[4pt] (\text{m})(\text{n}) \end{matrix}\)

\(\text{Table 65.2} ~~ \operatorname{AIR}_1 (\operatorname{Den}(L_\text{A})) : \text{Denotative Component of} ~ L_\text{A}\!\)

\(\text{Object}\!\)	\(\text{Sign}\!\)	\(\text{Transition}\!\)
\(\begin{matrix} \text{s} \\[4pt] \text{s} \end{matrix}\)	\(\begin{matrix} ~\text{m}~~\text{n}~ \\[4pt] ~\text{m}~(\text{n}) \end{matrix}\)	\(\begin{matrix} ~\text{m}~~\text{n}~ \mapsto ~\text{s}~ \\[4pt] ~\text{m}~(\text{n}) \mapsto ~\text{s}~ \end{matrix}\)
\(\begin{matrix} (\text{s}) \\[4pt] (\text{s}) \end{matrix}\)	\(\begin{matrix} (\text{m})~\text{n}~ \\[4pt] (\text{m})(\text{n}) \end{matrix}\)	\(\begin{matrix} (\text{m})~\text{n}~ \mapsto (\text{s}) \\[4pt] (\text{m})(\text{n}) \mapsto (\text{s}) \end{matrix}\)

\(\text{Table 65.3} ~~ \operatorname{AIR}_1 (\operatorname{Con}(L_\text{A})) : \text{Connotative Component of} ~ L_\text{A}\!\)

\(\text{Sign}\!\)	\(\text{Interpretant}\!\)	\(\text{Transition}\!\)
\(\begin{matrix} ~\text{m}~~\text{n}~ \\[4pt] ~\text{m}~~\text{n}~ \\[4pt] ~\text{m}~(\text{n}) \\[4pt] ~\text{m}~(\text{n}) \end{matrix}\)	\(\begin{matrix} ~\text{m}~~\text{n}~ \\[4pt] ~\text{m}~(\text{n}) \\[4pt] ~\text{m}~~\text{n}~ \\[4pt] ~\text{m}~(\text{n}) \end{matrix}\)	\(\begin{matrix} (\text{dm})(\text{dn}) \\[4pt] (\text{dm})~\text{dn}~ \\[4pt] (\text{dm})~\text{dn}~ \\[4pt] (\text{dm})(\text{dn}) \end{matrix}\)
\(\begin{matrix} (\text{m})~\text{n}~ \\[4pt] (\text{m})~\text{n}~ \\[4pt] (\text{m})(\text{n}) \\[4pt] (\text{m})(\text{n}) \end{matrix}\)	\(\begin{matrix} (\text{m})~\text{n}~ \\[4pt] (\text{m})(\text{n}) \\[4pt] (\text{m})~\text{n}~ \\[4pt] (\text{m})(\text{n}) \end{matrix}\)	\(\begin{matrix} (\text{dm})(\text{dn}) \\[4pt] (\text{dm})~\text{dn}~ \\[4pt] (\text{dm})~\text{dn}~ \\[4pt] (\text{dm})(\text{dn}) \end{matrix}\)

\(\text{Table 66.1} ~~ \operatorname{AIR}_1 (L_\text{B}) : \text{Analytic Representation of} ~ L_\text{B}\!\)

\(\text{Object}\!\)	\(\text{Sign}\!\)	\(\text{Interpretant}\!\)
\(\begin{matrix} (\text{s}) \\[4pt] (\text{s}) \\[4pt] (\text{s}) \\[4pt] (\text{s}) \end{matrix}\)	\(\begin{matrix} (\text{m})~\text{n}~ \\[4pt] (\text{m})~\text{n}~ \\[4pt] (\text{m})(\text{n}) \\[4pt] (\text{m})(\text{n}) \end{matrix}\)	\(\begin{matrix} (\text{m})~\text{n}~ \\[4pt] (\text{m})(\text{n}) \\[4pt] (\text{m})~\text{n}~ \\[4pt] (\text{m})(\text{n}) \end{matrix}\)
\(\begin{matrix} \text{s} \\[4pt] \text{s} \\[4pt] \text{s} \\[4pt] \text{s} \end{matrix}\)	\(\begin{matrix} ~\text{m}~~\text{n}~ \\[4pt] ~\text{m}~~\text{n}~ \\[4pt] ~\text{m}~(\text{n}) \\[4pt] ~\text{m}~(\text{n}) \end{matrix}\)	\(\begin{matrix} ~\text{m}~~\text{n}~ \\[4pt] ~\text{m}~(\text{n}) \\[4pt] ~\text{m}~~\text{n}~ \\[4pt] ~\text{m}~(\text{n}) \end{matrix}\)

\(\text{Table 66.2} ~~ \operatorname{AIR}_1 (\operatorname{Den}(L_\text{B})) : \text{Denotative Component of} ~ L_\text{B}\!\)

\(\text{Object}\!\)	\(\text{Sign}\!\)	\(\text{Transition}\!\)
\(\begin{matrix} (\text{s}) \\[4pt] (\text{s}) \end{matrix}\)	\(\begin{matrix} (\text{m})~\text{n}~ \\[4pt] (\text{m})(\text{n}) \end{matrix}\)	\(\begin{matrix} (\text{m})~\text{n}~ \mapsto (\text{s}) \\[4pt] (\text{m})(\text{n}) \mapsto (\text{s}) \end{matrix}\)
\(\begin{matrix} \text{s} \\[4pt] \text{s} \end{matrix}\)	\(\begin{matrix} ~\text{m}~~\text{n}~ \\[4pt] ~\text{m}~(\text{n}) \end{matrix}\)	\(\begin{matrix} ~\text{m}~~\text{n}~ \mapsto ~\text{s}~ \\[4pt] ~\text{m}~(\text{n}) \mapsto ~\text{s}~ \end{matrix}\)

\(\text{Table 66.3} ~~ \operatorname{AIR}_1 (\operatorname{Con}(L_\text{B})) : \text{Connotative Component of} ~ L_\text{B}\!\)

\(\text{Sign}\!\)	\(\text{Interpretant}\!\)	\(\text{Transition}\!\)
\(\begin{matrix} (\text{m})~\text{n}~ \\[4pt] (\text{m})~\text{n}~ \\[4pt] (\text{m})(\text{n}) \\[4pt] (\text{m})(\text{n}) \end{matrix}\)	\(\begin{matrix} (\text{m})~\text{n}~ \\[4pt] (\text{m})(\text{n}) \\[4pt] (\text{m})~\text{n}~ \\[4pt] (\text{m})(\text{n}) \end{matrix}\)	\(\begin{matrix} (\text{dm})(\text{dn}) \\[4pt] (\text{dm})~\text{dn}~ \\[4pt] (\text{dm})~\text{dn}~ \\[4pt] (\text{dm})(\text{dn}) \end{matrix}\)
\(\begin{matrix} ~\text{m}~~\text{n}~ \\[4pt] ~\text{m}~~\text{n}~ \\[4pt] ~\text{m}~(\text{n}) \\[4pt] ~\text{m}~(\text{n}) \end{matrix}\)	\(\begin{matrix} ~\text{m}~~\text{n}~ \\[4pt] ~\text{m}~(\text{n}) \\[4pt] ~\text{m}~~\text{n}~ \\[4pt] ~\text{m}~(\text{n}) \end{matrix}\)	\(\begin{matrix} (\text{dm})(\text{dn}) \\[4pt] (\text{dm})~\text{dn}~ \\[4pt] (\text{dm})~\text{dn}~ \\[4pt] (\text{dm})(\text{dn}) \end{matrix}\)

\(\text{Table 67.1} ~~ \operatorname{AIR}_2 (L_\text{A}) : \text{Analytic Representation of} ~ L_\text{A}\!\)

\(\text{Object}\!\)	\(\text{Sign}\!\)	\(\text{Interpretant}\!\)
\(\begin{matrix} {\langle * \rangle}_X \\[4pt] {\langle * \rangle}_X \\[4pt] {\langle * \rangle}_X \\[4pt] {\langle * \rangle}_X \end{matrix}\)	\(\begin{matrix} {\langle * \rangle}_Y \\[4pt] {\langle * \rangle}_Y \\[4pt] {\langle\text{m}\rangle}_Y \\[4pt] {\langle\text{m}\rangle}_Y \end{matrix}\)	\(\begin{matrix} {\langle * \rangle}_Y \\[4pt] {\langle\text{m}\rangle}_Y \\[4pt] {\langle * \rangle}_Y \\[4pt] {\langle\text{m}\rangle}_Y \end{matrix}\)
\(\begin{matrix} {\langle ! \rangle}_X \\[4pt] {\langle ! \rangle}_X \\[4pt] {\langle ! \rangle}_X \\[4pt] {\langle ! \rangle}_X \end{matrix}\)	\(\begin{matrix} {\langle\text{n}\rangle}_Y \\[4pt] {\langle\text{n}\rangle}_Y \\[4pt] {\langle ! \rangle}_Y \\[4pt] {\langle ! \rangle}_Y \end{matrix}\)	\(\begin{matrix} {\langle\text{n}\rangle}_Y \\[4pt] {\langle ! \rangle}_Y \\[4pt] {\langle\text{n}\rangle}_Y \\[4pt] {\langle ! \rangle}_Y \end{matrix}\)

\(\text{Table 67.2} ~~ \operatorname{AIR}_2 (\operatorname{Den}(L_\text{A})) : \text{Denotative Component of} ~ L_\text{A}\!\)

\(\text{Object}\!\)	\(\text{Sign}\!\)	\(\text{Transition}\!\)
\(\begin{matrix} {\langle * \rangle}_X \\[4pt] {\langle * \rangle}_X \end{matrix}\)	\(\begin{matrix} {\langle * \rangle}_Y \\[4pt] {\langle\text{m}\rangle}_Y \end{matrix}\)	\(\begin{array}{r} {\langle * \rangle}_Y \mapsto {\langle * \rangle}_X \\[4pt] {\langle\text{m}\rangle}_Y \mapsto {\langle * \rangle}_X \end{array}\)
\(\begin{matrix} {\langle ! \rangle}_X \\[4pt] {\langle ! \rangle}_X \end{matrix}\)	\(\begin{matrix} {\langle\text{n}\rangle}_Y \\[4pt] {\langle ! \rangle}_Y \end{matrix}\)	\(\begin{array}{r} {\langle\text{n}\rangle}_Y \mapsto {\langle ! \rangle}_X \\[4pt] {\langle ! \rangle}_Y \mapsto {\langle ! \rangle}_X \end{array}\)

\(\text{Table 67.3} ~~ \operatorname{AIR}_2 (\operatorname{Con}(L_\text{A})) : \text{Connotative Component of} ~ L_\text{A}\!\)

\(\text{Sign}\!\)	\(\text{Interpretant}\!\)	\(\text{Transition}\!\)
\(\begin{matrix} {\langle * \rangle}_Y \\[4pt] {\langle * \rangle}_Y \\[4pt] {\langle\text{m}\rangle}_Y \\[4pt] {\langle\text{m}\rangle}_Y \end{matrix}\)	\(\begin{matrix} {\langle * \rangle}_Y \\[4pt] {\langle\text{m}\rangle}_Y \\[4pt] {\langle * \rangle}_Y \\[4pt] {\langle\text{m}\rangle}_Y \end{matrix}\)	\(\begin{matrix} {\langle\operatorname{d}!\rangle}_{\operatorname{d}Y} \\[4pt] {\langle\operatorname{d}\text{n}\rangle}_{\operatorname{d}Y} \\[4pt] {\langle\operatorname{d}\text{n}\rangle}_{\operatorname{d}Y} \\[4pt] {\langle\operatorname{d}!\rangle}_{\operatorname{d}Y} \end{matrix}\)
\(\begin{matrix} {\langle\text{n}\rangle}_Y \\[4pt] {\langle\text{n}\rangle}_Y \\[4pt] {\langle ! \rangle}_Y \\[4pt] {\langle ! \rangle}_Y \end{matrix}\)	\(\begin{matrix} {\langle\text{n}\rangle}_Y \\[4pt] {\langle ! \rangle}_Y \\[4pt] {\langle\text{n}\rangle}_Y \\[4pt] {\langle ! \rangle}_Y \end{matrix}\)	\(\begin{matrix} {\langle\operatorname{d}!\rangle}_{\operatorname{d}Y} \\[4pt] {\langle\operatorname{d}\text{n}\rangle}_{\operatorname{d}Y} \\[4pt] {\langle\operatorname{d}\text{n}\rangle}_{\operatorname{d}Y} \\[4pt] {\langle\operatorname{d}!\rangle}_{\operatorname{d}Y} \end{matrix}\)

\(\text{Table 68.1} ~~ \operatorname{AIR}_2 (L_\text{B}) : \text{Analytic Representation of} ~ L_\text{B}\!\)

\(\text{Object}\!\)	\(\text{Sign}\!\)	\(\text{Interpretant}\!\)
\(\begin{matrix} {\langle ! \rangle}_X \\[4pt] {\langle ! \rangle}_X \\[4pt] {\langle ! \rangle}_X \\[4pt] {\langle ! \rangle}_X \end{matrix}\)	\(\begin{matrix} {\langle\text{n}\rangle}_Y \\[4pt] {\langle\text{n}\rangle}_Y \\[4pt] {\langle ! \rangle}_Y \\[4pt] {\langle ! \rangle}_Y \end{matrix}\)	\(\begin{matrix} {\langle\text{n}\rangle}_Y \\[4pt] {\langle ! \rangle}_Y \\[4pt] {\langle\text{n}\rangle}_Y \\[4pt] {\langle ! \rangle}_Y \end{matrix}\)
\(\begin{matrix} {\langle * \rangle}_X \\[4pt] {\langle * \rangle}_X \\[4pt] {\langle * \rangle}_X \\[4pt] {\langle * \rangle}_X \end{matrix}\)	\(\begin{matrix} {\langle * \rangle}_Y \\[4pt] {\langle * \rangle}_Y \\[4pt] {\langle\text{m}\rangle}_Y \\[4pt] {\langle\text{m}\rangle}_Y \end{matrix}\)	\(\begin{matrix} {\langle * \rangle}_Y \\[4pt] {\langle\text{m}\rangle}_Y \\[4pt] {\langle * \rangle}_Y \\[4pt] {\langle\text{m}\rangle}_Y \end{matrix}\)

\(\text{Table 68.2} ~~ \operatorname{AIR}_2 (\operatorname{Den}(L_\text{B})) : \text{Denotative Component of} ~ L_\text{B}\!\)

\(\text{Object}\!\)	\(\text{Sign}\!\)	\(\text{Transition}\!\)
\(\begin{matrix} {\langle ! \rangle}_X \\[4pt] {\langle ! \rangle}_X \end{matrix}\)	\(\begin{matrix} {\langle\text{n}\rangle}_Y \\[4pt] {\langle ! \rangle}_Y \end{matrix}\)	\(\begin{array}{r} {\langle\text{n}\rangle}_Y \mapsto {\langle ! \rangle}_X \\[4pt] {\langle ! \rangle}_Y \mapsto {\langle ! \rangle}_X \end{array}\)
\(\begin{matrix} {\langle * \rangle}_X \\[4pt] {\langle * \rangle}_X \end{matrix}\)	\(\begin{matrix} {\langle * \rangle}_Y \\[4pt] {\langle\text{m}\rangle}_Y \end{matrix}\)	\(\begin{array}{r} {\langle * \rangle}_Y \mapsto {\langle * \rangle}_X \\[4pt] {\langle\text{m}\rangle}_Y \mapsto {\langle * \rangle}_X \end{array}\)

\(\text{Table 68.3} ~~ \operatorname{AIR}_2 (\operatorname{Con}(L_\text{B})) : \text{Connotative Component of} ~ L_\text{B}\!\)

\(\text{Sign}\!\)	\(\text{Interpretant}\!\)	\(\text{Transition}\!\)
\(\begin{matrix} {\langle\text{n}\rangle}_Y \\[4pt] {\langle\text{n}\rangle}_Y \\[4pt] {\langle ! \rangle}_Y \\[4pt] {\langle ! \rangle}_Y \end{matrix}\)	\(\begin{matrix} {\langle\text{n}\rangle}_Y \\[4pt] {\langle ! \rangle}_Y \\[4pt] {\langle\text{n}\rangle}_Y \\[4pt] {\langle ! \rangle}_Y \end{matrix}\)	\(\begin{matrix} {\langle\operatorname{d}!\rangle}_{\operatorname{d}Y} \\[4pt] {\langle\operatorname{d}\text{n}\rangle}_{\operatorname{d}Y} \\[4pt] {\langle\operatorname{d}\text{n}\rangle}_{\operatorname{d}Y} \\[4pt] {\langle\operatorname{d}!\rangle}_{\operatorname{d}Y} \end{matrix}\)
\(\begin{matrix} {\langle * \rangle}_Y \\[4pt] {\langle * \rangle}_Y \\[4pt] {\langle\text{m}\rangle}_Y \\[4pt] {\langle\text{m}\rangle}_Y \end{matrix}\)	\(\begin{matrix} {\langle * \rangle}_Y \\[4pt] {\langle\text{m}\rangle}_Y \\[4pt] {\langle * \rangle}_Y \\[4pt] {\langle\text{m}\rangle}_Y \end{matrix}\)	\(\begin{matrix} {\langle\operatorname{d}!\rangle}_{\operatorname{d}Y} \\[4pt] {\langle\operatorname{d}\text{n}\rangle}_{\operatorname{d}Y} \\[4pt] {\langle\operatorname{d}\text{n}\rangle}_{\operatorname{d}Y} \\[4pt] {\langle\operatorname{d}!\rangle}_{\operatorname{d}Y} \end{matrix}\)

\(\text{Table 69.} ~~ \text{Schematism of Sequential Inference}\!\)

\(\text{Initial Premiss}\!\)	\(\text{Differential Premiss}\!\)	\(\text{Inferred Sequel}\!\)
\(\begin{matrix} ~x~ ~\operatorname{at}~ t \\[4pt] ~x~ ~\operatorname{at}~ t \\[4pt] (x) ~\operatorname{at}~ t \\[4pt] (x) ~\operatorname{at}~ t \end{matrix}\)	\(\begin{matrix} ~\operatorname{d}x~ ~\operatorname{at}~ t \\[4pt] (\operatorname{d}x) ~\operatorname{at}~ t \\[4pt] ~\operatorname{d}x~ ~\operatorname{at}~ t \\[4pt] (\operatorname{d}x) ~\operatorname{at}~ t \end{matrix}\)	\(\begin{matrix} (x) ~\operatorname{at}~ t' \\[4pt] ~x~ ~\operatorname{at}~ t' \\[4pt] ~x~ ~\operatorname{at}~ t' \\[4pt] (x) ~\operatorname{at}~ t' \end{matrix}\)

\(\text{Table 70.1} ~~ \text{Group Representation} ~ \operatorname{Rep}^\text{A} (V_4)\!\)

\(\begin{matrix} \text{Abstract} \\ \text{Element} \end{matrix}\)	\(\begin{matrix} \text{Logical} \\ \text{Element} \end{matrix}\)	\(\begin{matrix} \text{Active} \\ \text{List} \end{matrix}\)	\(\begin{matrix} \text{Active} \\ \text{Term} \end{matrix}\)	\(\begin{matrix} \text{Genetic} \\ \text{Element} \end{matrix}\)
\(\begin{matrix} 1 \\[4pt] r \\[4pt] s \\[4pt] t \end{matrix}\)	\(\begin{matrix} (\operatorname{d}\underline{\underline{\text{a}}}) (\operatorname{d}\underline{\underline{\text{b}}}) (\operatorname{d}\underline{\underline{\text{i}}}) (\operatorname{d}\underline{\underline{\text{u}}}) \\[4pt] ~\operatorname{d}\underline{\underline{\text{a}}}~ (\operatorname{d}\underline{\underline{\text{b}}}) ~\operatorname{d}\underline{\underline{\text{i}}}~ (\operatorname{d}\underline{\underline{\text{u}}}) \\[4pt] (\operatorname{d}\underline{\underline{\text{a}}}) ~\operatorname{d}\underline{\underline{\text{b}}}~ (\operatorname{d}\underline{\underline{\text{i}}}) ~\operatorname{d}\underline{\underline{\text{u}}}~ \\[4pt] ~\operatorname{d}\underline{\underline{\text{a}}}~ ~\operatorname{d}\underline{\underline{\text{b}}}~ ~\operatorname{d}\underline{\underline{\text{i}}}~ ~\operatorname{d}\underline{\underline{\text{u}}}~ \end{matrix}\)	\(\begin{matrix} \langle \operatorname{d}! \rangle \\[4pt] \langle \operatorname{d}\underline{\underline{\text{a}}} ~ \operatorname{d}\underline{\underline{\text{i}}} \rangle \\[4pt] \langle \operatorname{d}\underline{\underline{\text{b}}} ~ \operatorname{d}\underline{\underline{\text{u}}} \rangle \\[4pt] \langle \operatorname{d}* \rangle \end{matrix}\)	\(\begin{matrix} \operatorname{d}! \\[4pt] \operatorname{d}\underline{\underline{\text{a}}} \cdot \operatorname{d}\underline{\underline{\text{i}}} ~ ! \\[4pt] \operatorname{d}\underline{\underline{\text{b}}} \cdot \operatorname{d}\underline{\underline{\text{u}}} ~ ! \\[4pt] \operatorname{d}* \end{matrix}\)	\(\begin{matrix} 1 \\[4pt] \operatorname{d}_{\text{ai}} \\[4pt] \operatorname{d}_{\text{bu}} \\[4pt] \operatorname{d}_{\text{ai}} * \operatorname{d}_{\text{bu}} \end{matrix}\)

\(\text{Table 70.2} ~~ \text{Group Representation} ~ \operatorname{Rep}^\text{B} (V_4)\!\)

\(\begin{matrix} \text{Abstract} \\ \text{Element} \end{matrix}\)	\(\begin{matrix} \text{Logical} \\ \text{Element} \end{matrix}\)	\(\begin{matrix} \text{Active} \\ \text{List} \end{matrix}\)	\(\begin{matrix} \text{Active} \\ \text{Term} \end{matrix}\)	\(\begin{matrix} \text{Genetic} \\ \text{Element} \end{matrix}\)
\(\begin{matrix} 1 \\[4pt] r \\[4pt] s \\[4pt] t \end{matrix}\)	\(\begin{matrix} (\operatorname{d}\underline{\underline{\text{a}}}) (\operatorname{d}\underline{\underline{\text{b}}}) (\operatorname{d}\underline{\underline{\text{i}}}) (\operatorname{d}\underline{\underline{\text{u}}}) \\[4pt] ~\operatorname{d}\underline{\underline{\text{a}}}~ (\operatorname{d}\underline{\underline{\text{b}}}) (\operatorname{d}\underline{\underline{\text{i}}}) ~\operatorname{d}\underline{\underline{\text{u}}}~ \\[4pt] (\operatorname{d}\underline{\underline{\text{a}}}) ~\operatorname{d}\underline{\underline{\text{b}}}~ ~\operatorname{d}\underline{\underline{\text{i}}}~ (\operatorname{d}\underline{\underline{\text{u}}}) \\[4pt] ~\operatorname{d}\underline{\underline{\text{a}}}~ ~\operatorname{d}\underline{\underline{\text{b}}}~ ~\operatorname{d}\underline{\underline{\text{i}}}~ ~\operatorname{d}\underline{\underline{\text{u}}}~ \end{matrix}\)	\(\begin{matrix} \langle \operatorname{d}! \rangle \\[4pt] \langle \operatorname{d}\underline{\underline{\text{a}}} ~ \operatorname{d}\underline{\underline{\text{u}}} \rangle \\[4pt] \langle \operatorname{d}\underline{\underline{\text{b}}} ~ \operatorname{d}\underline{\underline{\text{i}}} \rangle \\[4pt] \langle \operatorname{d}* \rangle \end{matrix}\)	\(\begin{matrix} \operatorname{d}! \\[4pt] \operatorname{d}\underline{\underline{\text{a}}} \cdot \operatorname{d}\underline{\underline{\text{u}}} ~ ! \\[4pt] \operatorname{d}\underline{\underline{\text{b}}} \cdot \operatorname{d}\underline{\underline{\text{i}}} ~ ! \\[4pt] \operatorname{d}* \end{matrix}\)	\(\begin{matrix} 1 \\[4pt] \operatorname{d}_{\text{au}} \\[4pt] \operatorname{d}_{\text{bi}} \\[4pt] \operatorname{d}_{\text{au}} * \operatorname{d}_{\text{bi}} \end{matrix}\)

\(\text{Table 70.3} ~~ \text{Group Representation} ~ \operatorname{Rep}^\text{C} (V_4)\!\)

\(\begin{matrix} \text{Abstract} \\ \text{Element} \end{matrix}\)	\(\begin{matrix} \text{Logical} \\ \text{Element} \end{matrix}\)	\(\begin{matrix} \text{Active} \\ \text{List} \end{matrix}\)	\(\begin{matrix} \text{Active} \\ \text{Term} \end{matrix}\)	\(\begin{matrix} \text{Genetic} \\ \text{Element} \end{matrix}\)
\(\begin{matrix} 1 \\[4pt] r \\[4pt] s \\[4pt] t \end{matrix}\)	\(\begin{matrix} (\operatorname{d}\text{m}) (\operatorname{d}\text{n}) \\[4pt] ~\operatorname{d}\text{m}~ (\operatorname{d}\text{n}) \\[4pt] (\operatorname{d}\text{m}) ~\operatorname{d}\text{n}~ \\[4pt] ~\operatorname{d}\text{m}~ ~\operatorname{d}\text{n}~ \end{matrix}\)	\(\begin{matrix} \langle\operatorname{d}!\rangle \\[4pt] \langle\operatorname{d}\text{m}\rangle \\[4pt] \langle\operatorname{d}\text{n}\rangle \\[4pt] \langle\operatorname{d}*\rangle \end{matrix}\)	\(\begin{matrix} \operatorname{d}! \\[4pt] \operatorname{d}\text{m}! \\[4pt] \operatorname{d}\text{n}! \\[4pt] \operatorname{d}* \end{matrix}\)	\(\begin{matrix} 1 \\[4pt] \operatorname{d}_{\text{m}} \\[4pt] \operatorname{d}_{\text{n}} \\[4pt] \operatorname{d}_{\text{m}} * \operatorname{d}_{\text{n}} \end{matrix}\)

\(\text{Table 71.1} ~~ \text{The Differential Group} ~ G = V_4\!\)

\(\begin{matrix} \text{Abstract} \\ \text{Element} \end{matrix}\)	\(\begin{matrix} \text{Logical} \\ \text{Element} \end{matrix}\)	\(\begin{matrix} \text{Active} \\ \text{List} \end{matrix}\)	\(\begin{matrix} \text{Active} \\ \text{Term} \end{matrix}\)	\(\begin{matrix} \text{Genetic} \\ \text{Element} \end{matrix}\)
\(\begin{matrix} 1 \\[4pt] r \\[4pt] s \\[4pt] t \end{matrix}\)	\(\begin{matrix} (\operatorname{d}\text{m}) (\operatorname{d}\text{n}) \\[4pt] ~\operatorname{d}\text{m}~ (\operatorname{d}\text{n}) \\[4pt] (\operatorname{d}\text{m}) ~\operatorname{d}\text{n}~ \\[4pt] ~\operatorname{d}\text{m}~ ~\operatorname{d}\text{n}~ \end{matrix}\)	\(\begin{matrix} \langle\operatorname{d}!\rangle \\[4pt] \langle\operatorname{d}\text{m}\rangle \\[4pt] \langle\operatorname{d}\text{n}\rangle \\[4pt] \langle\operatorname{d}*\rangle \end{matrix}\)	\(\begin{matrix} \operatorname{d}! \\[4pt] \operatorname{d}\text{m}! \\[4pt] \operatorname{d}\text{n}! \\[4pt] \operatorname{d}* \end{matrix}\)	\(\begin{matrix} 1 \\[4pt] \operatorname{d}_{\text{m}} \\[4pt] \operatorname{d}_{\text{n}} \\[4pt] \operatorname{d}_{\text{m}} * \operatorname{d}_{\text{n}} \end{matrix}\)

\(\text{Table 71.2} ~~ \text{Cosets of} ~ G_\text{m} ~ \text{in} ~ G\!\)

\(\text{Group Coset}\!\)	\(\text{Logical Coset}\!\)	\(\text{Logical Element}\!\)	\(\text{Group Element}\!\)
\(G_\text{m}\!\)	\((\operatorname{d}\text{m})\!\)	\(\begin{matrix} (\operatorname{d}\text{m})(\operatorname{d}\text{n}) \\[4pt] (\operatorname{d}\text{m})~\operatorname{d}\text{n}~ \end{matrix}\)	\(\begin{matrix} 1 \\[4pt] \operatorname{d}_\text{n} \end{matrix}\)
\(G_\text{m} * \operatorname{d}_\text{m}\!\)	\(\operatorname{d}\text{m}\!\)	\(\begin{matrix} ~\operatorname{d}\text{m}~(\operatorname{d}\text{n}) \\[4pt] ~\operatorname{d}\text{m}~~\operatorname{d}\text{n}~ \end{matrix}\)	\(\begin{matrix} \operatorname{d}_\text{m} \\[4pt] \operatorname{d}_\text{n} * \operatorname{d}_\text{m} \end{matrix}\)

\(\text{Table 71.3} ~~ \text{Cosets of} ~ G_\text{n} ~ \text{in} ~ G\!\)

\(\text{Group Coset}\!\)	\(\text{Logical Coset}\!\)	\(\text{Logical Element}\!\)	\(\text{Group Element}\!\)
\(G_\text{n}\!\)	\((\operatorname{d}\text{n})\!\)	\(\begin{matrix} (\operatorname{d}\text{m})(\operatorname{d}\text{n}) \\[4pt] ~\operatorname{d}\text{m}~(\operatorname{d}\text{n}) \end{matrix}\)	\(\begin{matrix} 1 \\[4pt] \operatorname{d}_\text{m} \end{matrix}\)
\(G_\text{n} * \operatorname{d}_\text{n}\!\)	\(\operatorname{d}\text{n}\!\)	\(\begin{matrix} (\operatorname{d}\text{m})~\operatorname{d}\text{n}~ \\[4pt] ~\operatorname{d}\text{m}~~\operatorname{d}\text{n}~ \end{matrix}\)	\(\begin{matrix} \operatorname{d}_\text{n} \\[4pt] \operatorname{d}_\text{m} * \operatorname{d}_\text{n} \end{matrix}\)

\(\text{Table 72.1} ~~ \text{Sign Relation of Interpreter A}\!\)

\(\text{Object}\!\)	\(\text{Sign}\!\)	\(\text{Interpretant}\!\)
\(\begin{matrix} \text{A} \\ \text{A} \\ \text{A} \\ \text{A} \end{matrix}\)	\(\begin{matrix} {}^{\backprime\backprime} \text{A} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{A} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \end{matrix}\)	\(\begin{matrix} {}^{\backprime\backprime} \text{A} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{A} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \end{matrix}\)
\(\begin{matrix} \text{B} \\ \text{B} \\ \text{B} \\ \text{B} \end{matrix}\)	\(\begin{matrix} {}^{\backprime\backprime} \text{B} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{B} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \end{matrix}\)	\(\begin{matrix} {}^{\backprime\backprime} \text{B} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{B} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \end{matrix}\)

\(\text{Table 72.2} ~~ \text{Dyadic Projection} ~ L(\text{A})_{OS}\!\)

\(\text{Object}\!\)	\(\text{Sign}\!\)
\(\begin{matrix} \text{A} \\ \text{A} \end{matrix}\)	\(\begin{matrix} {}^{\backprime\backprime} \text{A} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \end{matrix}\)
\(\begin{matrix} \text{B} \\ \text{B} \end{matrix}\)	\(\begin{matrix} {}^{\backprime\backprime} \text{B} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \end{matrix}\)

\(\text{Table 72.3} ~~ \text{Dyadic Projection} ~ L(\text{A})_{OI}\!\)

\(\text{Object}\!\)	\(\text{Interpretant}\!\)
\(\begin{matrix} \text{A} \\ \text{A} \end{matrix}\)	\(\begin{matrix} {}^{\backprime\backprime} \text{A} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \end{matrix}\)
\(\begin{matrix} \text{B} \\ \text{B} \end{matrix}\)	\(\begin{matrix} {}^{\backprime\backprime} \text{B} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \end{matrix}\)

\(\text{Table 72.4} ~~ \text{Dyadic Projection} ~ L(\text{A})_{SI}\!\)

\(\text{Sign}\!\)	\(\text{Interpretant}\!\)
\(\begin{matrix} {}^{\backprime\backprime} \text{A} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{A} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \end{matrix}\)	\(\begin{matrix} {}^{\backprime\backprime} \text{A} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{A} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \end{matrix}\)
\(\begin{matrix} {}^{\backprime\backprime} \text{B} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{B} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \end{matrix}\)	\(\begin{matrix} {}^{\backprime\backprime} \text{B} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{B} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \end{matrix}\)

\(\text{Table 73.1} ~~ \text{Sign Relation of Interpreter B}\!\)

\(\text{Object}\!\)	\(\text{Sign}\!\)	\(\text{Interpretant}\!\)
\(\begin{matrix} \text{A} \\ \text{A} \\ \text{A} \\ \text{A} \end{matrix}\)	\(\begin{matrix} {}^{\backprime\backprime} \text{A} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{A} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \end{matrix}\)	\(\begin{matrix} {}^{\backprime\backprime} \text{A} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{A} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \end{matrix}\)
\(\begin{matrix} \text{B} \\ \text{B} \\ \text{B} \\ \text{B} \end{matrix}\)	\(\begin{matrix} {}^{\backprime\backprime} \text{B} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{B} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \end{matrix}\)	\(\begin{matrix} {}^{\backprime\backprime} \text{B} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{B} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \end{matrix}\)

\(\text{Table 73.2} ~~ \text{Dyadic Projection} ~ L(\text{B})_{OS}\!\)

\(\text{Object}\!\)	\(\text{Sign}\!\)
\(\begin{matrix} \text{A} \\ \text{A} \end{matrix}\)	\(\begin{matrix} {}^{\backprime\backprime} \text{A} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \end{matrix}\)
\(\begin{matrix} \text{B} \\ \text{B} \end{matrix}\)	\(\begin{matrix} {}^{\backprime\backprime} \text{B} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \end{matrix}\)

\(\text{Table 73.3} ~~ \text{Dyadic Projection} ~ L(\text{B})_{OI}\!\)

\(\text{Object}\!\)	\(\text{Interpretant}\!\)
\(\begin{matrix} \text{A} \\ \text{A} \end{matrix}\)	\(\begin{matrix} {}^{\backprime\backprime} \text{A} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \end{matrix}\)
\(\begin{matrix} \text{B} \\ \text{B} \end{matrix}\)	\(\begin{matrix} {}^{\backprime\backprime} \text{B} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \end{matrix}\)

\(\text{Table 73.4} ~~ \text{Dyadic Projection} ~ L(\text{B})_{SI}\!\)

\(\text{Sign}\!\)	\(\text{Interpretant}\!\)
\(\begin{matrix} {}^{\backprime\backprime} \text{A} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{A} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \end{matrix}\)	\(\begin{matrix} {}^{\backprime\backprime} \text{A} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{A} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \end{matrix}\)
\(\begin{matrix} {}^{\backprime\backprime} \text{B} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{B} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \end{matrix}\)	\(\begin{matrix} {}^{\backprime\backprime} \text{B} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{B} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \end{matrix}\)

Current Work

\(\text{Table 74.1} ~~ \text{Relation} ~ L_0 =\{ (x, y, z) \in \mathbb{B}^3 : x + y + z = 0 \}\!\)

\(x\!\)	\(y\!\)	\(z\!\)
\(\begin{matrix}0\\0\\1\\1\end{matrix}\)	\(\begin{matrix}0\\1\\0\\1\end{matrix}\)	\(\begin{matrix}0\\1\\1\\0\end{matrix}\)

Table 74.2  Dyadic Projection R012
	x	y
	0	0
	0	1
	1	0
	1	1

Table 74.3  Dyadic Projection R013
	x	z
	0	0
	0	1
	1	1
	1	0

Table 74.4  Dyadic Projection R023
	y	z
	0	0
	1	1
	0	1
	1	0

\(\text{Table 75.1} ~~ \text{Relation} ~ L_1 =\{ (x, y, z) \in \mathbb{B}^3 : x + y + z = 1 \}\!\)

\(x\!\)	\(y\!\)	\(z\!\)
\(\begin{matrix}0\\0\\1\\1\end{matrix}\)	\(\begin{matrix}0\\1\\0\\1\end{matrix}\)	\(\begin{matrix}1\\0\\0\\1\end{matrix}\)

Table 75.2  Dyadic Projection R112
	x	y
	0	0
	0	1
	1	0
	1	1

Table 75.3  Dyadic Projection R113
	x	z
	0	1
	0	0
	1	0
	1	1

Table 75.4  Dyadic Projection R123
	y	z
	0	1
	1	0
	0	0
	1	1