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

Deletions

6.38. Considering the Source

There is one remaining form of useful continuity that can be established between these newly formalized inventions and the ordinary conventions of common practice that are customary to apply in the informal context. Conforming to the ascriptions made above, I revive an old usage for framing interjections and enunciate the quotation \({}^{\backprime\backprime} \text{X} {}^{\prime\prime\text{I}}\!\) as \({}^{\backprime\backprime} \text{X} {}^{\prime\prime} ~ \text{quotha}.\!\) Readers who find this custom too curious for words might consider the twofold origins of inquiry and interpretation, one in the virtue of addressing uncertainty and another in the acknowledgment of surprise.

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) & ? \'"`UNIQ-MathJax1-QINU`"' In contrast, the SER for interpreter \(\text{B}\!\) yields the semiotic equations:   \([{}^{\backprime\backprime} \text{A} {}^{\prime\prime}]_\text{B}\!\) \(=\!\) \([{}^{\backprime\backprime} \text{u} {}^{\prime\prime}]_\text{B}\!\)   \([{}^{\backprime\backprime} \text{B} {}^{\prime\prime}]_\text{B}\!\) \(=\!\) \([{}^{\backprime\backprime} \text{i} {}^{\prime\prime}]_\text{B}\!\) or  \({}^{\backprime\backprime} \text{A} {}^{\prime\prime}\!\) \(=_\text{B}\!\)  \({}^{\backprime\backprime} \text{u} {}^{\prime\prime}\!\)    \({}^{\backprime\backprime} \text{B} {}^{\prime\prime}\!\) \(=_\text{B}\!\)  \({}^{\backprime\backprime} \text{i} {}^{\prime\prime}\!\) and the semiotic partition\[\{ \{ {}^{\backprime\backprime} \text{A} {}^{\prime\prime}, {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \} , \{ {}^{\backprime\backprime} \text{B} {}^{\prime\prime}, {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \} \}.\!\] 6.38. Considering the Source Attributed Sign Relation \(\begin{array}{ccl} O & = & \{ \text{A}, \text{B} \} \\[6pt] S & = & \{ {}^{\backprime\backprime} \text{A} {}^{\prime\prime\text{A}}, {}^{\backprime\backprime} \text{B} {}^{\prime\prime\text{A}}, {}^{\backprime\backprime} \text{i} {}^{\prime\prime\text{A}}, {}^{\backprime\backprime} \text{u} {}^{\prime\prime\text{A}}, {}^{\backprime\backprime} \text{A} {}^{\prime\prime\text{B}}, {}^{\backprime\backprime} \text{B} {}^{\prime\prime\text{B}}, {}^{\backprime\backprime} \text{i} {}^{\prime\prime\text{B}}, {}^{\backprime\backprime} \text{u} {}^{\prime\prime\text{B}} \} \\[6pt] I & = & \{ {}^{\backprime\backprime} \text{A} {}^{\prime\prime\text{A}}, {}^{\backprime\backprime} \text{B} {}^{\prime\prime\text{A}}, {}^{\backprime\backprime} \text{i} {}^{\prime\prime\text{A}}, {}^{\backprime\backprime} \text{u} {}^{\prime\prime\text{A}}, {}^{\backprime\backprime} \text{A} {}^{\prime\prime\text{B}}, {}^{\backprime\backprime} \text{B} {}^{\prime\prime\text{B}}, {}^{\backprime\backprime} \text{i} {}^{\prime\prime\text{B}}, {}^{\backprime\backprime} \text{u} {}^{\prime\prime\text{B}} \} \end{array}\) Thus informed, the semiotic equivalence relation for interpreter \(\text{A}\!\) yields the following semiotic equations.   \([{}^{\backprime\backprime} \text{A} {}^{\prime\prime\text{A}}]_\text{A}\!\) \(=\!\) \([{}^{\backprime\backprime} \text{A} {}^{\prime\prime\text{B}}]_\text{A}\!\) \(=\!\) \([{}^{\backprime\backprime} \text{i} {}^{\prime\prime\text{A}}]_\text{A}\!\) \(=\!\) \([{}^{\backprime\backprime} \text{u} {}^{\prime\prime\text{B}}]_\text{A}\!\) or  \({}^{\backprime\backprime} \text{A} {}^{\prime\prime\text{A}}\!\) \(=_\text{A}\!\)  \({}^{\backprime\backprime} \text{A} {}^{\prime\prime\text{B}}\!\) \(=_\text{A}\!\)  \({}^{\backprime\backprime} \text{i} {}^{\prime\prime\text{A}}\!\) \(=_\text{A}\!\)  \({}^{\backprime\backprime} \text{u} {}^{\prime\prime\text{B}}\!\) In comparison, the semiotic equivalence relation for interpreter \(\text{B}\!\) yields the following semiotic equations.   \([{}^{\backprime\backprime} \text{A} {}^{\prime\prime\text{A}}]_\text{B}\!\) \(=\!\) \([{}^{\backprime\backprime} \text{A} {}^{\prime\prime\text{B}}]_\text{B}\!\) \(=\!\) \([{}^{\backprime\backprime} \text{i} {}^{\prime\prime\text{A}}]_\text{B}\!\) \(=\!\) \([{}^{\backprime\backprime} \text{u} {}^{\prime\prime\text{B}}]_\text{B}\!\) or  \({}^{\backprime\backprime} \text{A} {}^{\prime\prime\text{A}}\!\) \(=_\text{B}\!\)  \({}^{\backprime\backprime} \text{A} {}^{\prime\prime\text{B}}\!\) \(=_\text{B}\!\)  \({}^{\backprime\backprime} \text{i} {}^{\prime\prime\text{A}}\!\) \(=_\text{B}\!\)  \({}^{\backprime\backprime} \text{u} {}^{\prime\prime\text{B}}\!\) Consequently, the semiotic equivalence relations for \(\text{A}\!\) and \(\text{B}\!\) both induce the same semiotic partition on \(S,\!\) namely, the following. \( \{ \{ {}^{\backprime\backprime} \text{A} {}^{\prime\prime\text{A}}, {}^{\backprime\backprime} \text{A} {}^{\prime\prime\text{B}}, {}^{\backprime\backprime} \text{i} {}^{\prime\prime\text{A}}, {}^{\backprime\backprime} \text{u} {}^{\prime\prime\text{B}} \}~,~\{ {}^{\backprime\backprime} \text{B} {}^{\prime\prime\text{A}}, {}^{\backprime\backprime} \text{B} {}^{\prime\prime\text{B}}, {}^{\backprime\backprime} \text{i} {}^{\prime\prime\text{B}}, {}^{\backprime\backprime} \text{u} {}^{\prime\prime\text{A}} \} \}.\! \) Augmented Sign Relation \(\begin{array}{ccl} O & = & \{ \text{A}, \text{B} \} \\[8pt] S & = & \{ {}^{\backprime\backprime} [ {}^\langle \text{A} {}^\rangle ]_\text{A} {}^{\prime\prime}, {}^{\backprime\backprime} [ {}^\langle \text{B} {}^\rangle ]_\text{A} {}^{\prime\prime}, {}^{\backprime\backprime} [ {}^\langle \text{i} {}^\rangle ]_\text{A} {}^{\prime\prime}, {}^{\backprime\backprime} [ {}^\langle \text{u} {}^\rangle ]_\text{A} {}^{\prime\prime}, {}^{\backprime\backprime} [ {}^\langle \text{A} {}^\rangle ]_\text{B} {}^{\prime\prime}, {}^{\backprime\backprime} [ {}^\langle \text{B} {}^\rangle ]_\text{B} {}^{\prime\prime}, {}^{\backprime\backprime} [ {}^\langle \text{i} {}^\rangle ]_\text{B} {}^{\prime\prime}, {}^{\backprime\backprime} [ {}^\langle \text{u} {}^\rangle ]_\text{B} {}^{\prime\prime} \} \\[8pt] I & = & \{ {}^{\backprime\backprime} [ {}^\langle \text{A} {}^\rangle ]_\text{A} {}^{\prime\prime}, {}^{\backprime\backprime} [ {}^\langle \text{B} {}^\rangle ]_\text{A} {}^{\prime\prime}, {}^{\backprime\backprime} [ {}^\langle \text{i} {}^\rangle ]_\text{A} {}^{\prime\prime}, {}^{\backprime\backprime} [ {}^\langle \text{u} {}^\rangle ]_\text{A} {}^{\prime\prime}, {}^{\backprime\backprime} [ {}^\langle \text{A} {}^\rangle ]_\text{B} {}^{\prime\prime}, {}^{\backprime\backprime} [ {}^\langle \text{B} {}^\rangle ]_\text{B} {}^{\prime\prime}, {}^{\backprime\backprime} [ {}^\langle \text{i} {}^\rangle ]_\text{B} {}^{\prime\prime}, {}^{\backprime\backprime} [ {}^\langle \text{u} {}^\rangle ]_\text{B} {}^{\prime\prime} \} \end{array}\) \(\begin{array}{lll} O & = & \{ \text{A}, \text{B} \} \end{array}\) \(\begin{array}{lllllll} S & = & \{ & {}^{\backprime\backprime} [ {}^\langle \text{A} {}^\rangle ]_\text{A} {}^{\prime\prime}, {}^{\backprime\backprime} [ {}^\langle \text{B} {}^\rangle ]_\text{A} {}^{\prime\prime}, {}^{\backprime\backprime} [ {}^\langle \text{i} {}^\rangle ]_\text{A} {}^{\prime\prime}, {}^{\backprime\backprime} [ {}^\langle \text{u} {}^\rangle ]_\text{A} {}^{\prime\prime}, & \\[4pt] & & & {}^{\backprime\backprime} [ {}^\langle \text{A} {}^\rangle ]_\text{B} {}^{\prime\prime}, {}^{\backprime\backprime} [ {}^\langle \text{B} {}^\rangle ]_\text{B} {}^{\prime\prime}, {}^{\backprime\backprime} [ {}^\langle \text{i} {}^\rangle ]_\text{B} {}^{\prime\prime}, {}^{\backprime\backprime} [ {}^\langle \text{u} {}^\rangle ]_\text{B} {}^{\prime\prime} & \} \\[10pt] I & = & \{ & {}^{\backprime\backprime} [ {}^\langle \text{A} {}^\rangle ]_\text{A} {}^{\prime\prime}, {}^{\backprime\backprime} [ {}^\langle \text{B} {}^\rangle ]_\text{A} {}^{\prime\prime}, {}^{\backprime\backprime} [ {}^\langle \text{i} {}^\rangle ]_\text{A} {}^{\prime\prime}, {}^{\backprime\backprime} [ {}^\langle \text{u} {}^\rangle ]_\text{A} {}^{\prime\prime}, & \\[4pt] & & & {}^{\backprime\backprime} [ {}^\langle \text{A} {}^\rangle ]_\text{B} {}^{\prime\prime}, {}^{\backprime\backprime} [ {}^\langle \text{B} {}^\rangle ]_\text{B} {}^{\prime\prime}, {}^{\backprime\backprime} [ {}^\langle \text{i} {}^\rangle ]_\text{B} {}^{\prime\prime}, {}^{\backprime\backprime} [ {}^\langle \text{u} {}^\rangle ]_\text{B} {}^{\prime\prime} & \} \end{array}\) 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 \(\text{Table 1.} ~~ \text{Sign Relation of Interpreter A}\!\) \(\text{Object}\!\) \(\text{Sign}\!\) \(\text{Interpretant}\!\) \(\begin{matrix} \text{A} \\ \text{A} \\ \text{A} \\ \text{A} \end{matrix}\) \(\begin{matrix} {}^{\backprime\backprime} \text{A} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{A} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \end{matrix}\) \(\begin{matrix} {}^{\backprime\backprime} \text{A} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{A} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \end{matrix}\) \(\begin{matrix} \text{B} \\ \text{B} \\ \text{B} \\ \text{B} \end{matrix}\) \(\begin{matrix} {}^{\backprime\backprime} \text{B} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{B} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \end{matrix}\) \(\begin{matrix} {}^{\backprime\backprime} \text{B} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{B} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \end{matrix}\) \(\text{Table 2.} ~~ \text{Sign Relation of Interpreter B}\!\) \(\text{Object}\!\) \(\text{Sign}\!\) \(\text{Interpretant}\!\) \(\begin{matrix} \text{A} \\ \text{A} \\ \text{A} \\ \text{A} \end{matrix}\) \(\begin{matrix} {}^{\backprime\backprime} \text{A} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{A} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \end{matrix}\) \(\begin{matrix} {}^{\backprime\backprime} \text{A} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{A} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \end{matrix}\) \(\begin{matrix} \text{B} \\ \text{B} \\ \text{B} \\ \text{B} \end{matrix}\) \(\begin{matrix} {}^{\backprime\backprime} \text{B} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{B} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \end{matrix}\) \(\begin{matrix} {}^{\backprime\backprime} \text{B} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{B} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \end{matrix}\) \(\text{Table 36.} ~~ \text{Semantics for Higher Order Signs}\!\) \(\text{Object Denoted}\!\) \(\text{Equivalent Signs}\!\) \(\begin{matrix} \text{A} \\ \text{B} \end{matrix}\) \(\begin{matrix} {}^{\langle} \text{A} {}^{\rangle} & = & {}^{\backprime\backprime} \text{A} {}^{\prime\prime} \\ {}^{\langle} \text{B} {}^{\rangle} & = & {}^{\backprime\backprime} \text{B} {}^{\prime\prime} \end{matrix}\) \(\begin{matrix} {}^{\backprime\backprime} \text{A} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{B} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \end{matrix}\) \(\begin{matrix} {}^{\langle\langle} \text{A} {}^{\rangle\rangle} & = & {}^{\langle\backprime\backprime} \text{A} {}^{\prime\prime\rangle} & = & {}^{\backprime\backprime\langle} \text{A} {}^{\rangle\prime\prime} \\ {}^{\langle\langle} \text{B} {}^{\rangle\rangle} & = & {}^{\langle\backprime\backprime} \text{B} {}^{\prime\prime\rangle} & = & {}^{\backprime\backprime\langle} \text{B} {}^{\rangle\prime\prime} \\ {}^{\langle\langle} \text{i} {}^{\rangle\rangle} & = & {}^{\langle\backprime\backprime} \text{i} {}^{\prime\prime\rangle} & = & {}^{\backprime\backprime\langle} \text{i} {}^{\rangle\prime\prime} \\ {}^{\langle\langle} \text{u} {}^{\rangle\rangle} & = & {}^{\langle\backprime\backprime} \text{u} {}^{\prime\prime\rangle} & = & {}^{\backprime\backprime\langle} \text{u} {}^{\rangle\prime\prime} \end{matrix}\) \(\text{Table 37.} ~~ \text{Sign Relation Containing a Higher Order Sign}\!\) \(\text{Object}\!\) \(\text{Sign}\!\) \(\text{Interpretant}\!\) \(\begin{matrix} \ldots \\[2pt] \ldots \\[2pt] \text{s} \end{matrix}\) \(\begin{matrix} \text{s} \\[2pt] \ldots \\[2pt] \text{t} \end{matrix}\) \(\begin{matrix} \ldots \\[2pt] \ldots \\[2pt] \ldots \end{matrix}\) \(\text{Table 38.} ~~ \text{Sign Relation for a Succession of Higher Order Signs (1)}\!\) \(\text{Object}\!\) \(\text{Sign}\!\) \(\text{Interpretant}\!\) \(\begin{matrix} x \\[2pt] {}^{\langle} x {}^{\rangle} \\[2pt] {}^{\langle\langle} x {}^{\rangle\rangle} \\[2pt] \ldots \end{matrix}\) \(\begin{matrix} {}^{\langle} x {}^{\rangle} \\[2pt] {}^{\langle\langle} x {}^{\rangle\rangle} \\[2pt] {}^{\langle\langle\langle} x {}^{\rangle\rangle\rangle} \\[2pt] \ldots \end{matrix}\) \(\begin{matrix} \ldots \\[2pt] \ldots \\[2pt] \ldots \\[2pt] \ldots \end{matrix}\) \(\text{Table 39.} ~~ \text{Sign Relation for a Succession of Higher Order Signs (2)}\!\) \(\text{Object}\!\) \(\text{Sign}\!\) \(\text{Interpretant}\!\) \(\begin{matrix} x \\[2pt] s_1 \\[2pt] s_2 \\[2pt] \ldots \end{matrix}\) \(\begin{matrix} s_1 \\[2pt] s_2 \\[2pt] s_3 \\[2pt] \ldots \end{matrix}\) \(\begin{matrix} \ldots \\[2pt] \ldots \\[2pt] \ldots \\[2pt] \ldots \end{matrix}\) \(\text{Table 40.} ~~ \text{Reflective Origin} ~ \operatorname{Ref}^0 L(\text{A})\!\) \(\text{Object}\!\) \(\text{Sign}\!\) \(\text{Interpretant}\!\) \(\begin{matrix} \text{A} \\ \text{A} \\ \text{A} \\ \text{A} \end{matrix}\) \(\begin{matrix} {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{i} {}^{\rangle} \\ {}^{\langle} \text{i} {}^{\rangle} \end{matrix}\) \(\begin{matrix} {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{i} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{i} {}^{\rangle} \end{matrix}\) \(\begin{matrix} \text{B} \\ \text{B} \\ \text{B} \\ \text{B} \end{matrix}\) \(\begin{matrix} {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{u} {}^{\rangle} \\ {}^{\langle} \text{u} {}^{\rangle} \end{matrix}\) \(\begin{matrix} {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{u} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{u} {}^{\rangle} \end{matrix}\) \(\text{Table 41.} ~~ \text{Reflective Origin} ~ \operatorname{Ref}^0 L(\text{B})\!\) \(\text{Object}\!\) \(\text{Sign}\!\) \(\text{Interpretant}\!\) \(\begin{matrix} \text{A} \\ \text{A} \\ \text{A} \\ \text{A} \end{matrix}\) \(\begin{matrix} {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{u} {}^{\rangle} \\ {}^{\langle} \text{u} {}^{\rangle} \end{matrix}\) \(\begin{matrix} {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{u} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{u} {}^{\rangle} \end{matrix}\) \(\begin{matrix} \text{B} \\ \text{B} \\ \text{B} \\ \text{B} \end{matrix}\) \(\begin{matrix} {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{i} {}^{\rangle} \\ {}^{\langle} \text{i} {}^{\rangle} \end{matrix}\) \(\begin{matrix} {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{i} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{i} {}^{\rangle} \end{matrix}\) \(\text{Table 42.} ~~ \text{Higher Ascent Sign Relation} ~ \operatorname{Ref}^1 L(\text{A})\!\) \(\text{Object}\!\) \(\text{Sign}\!\) \(\text{Interpretant}\!\) \(\begin{matrix} \text{A} \\ \text{A} \\ \text{A} \\ \text{A} \end{matrix}\) \(\begin{matrix} {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{i} {}^{\rangle} \\ {}^{\langle} \text{i} {}^{\rangle} \end{matrix}\) \(\begin{matrix} {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{i} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{i} {}^{\rangle} \end{matrix}\) \(\begin{matrix} \text{B} \\ \text{B} \\ \text{B} \\ \text{B} \end{matrix}\) \(\begin{matrix} {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{u} {}^{\rangle} \\ {}^{\langle} \text{u} {}^{\rangle} \end{matrix}\) \(\begin{matrix} {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{u} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{u} {}^{\rangle} \end{matrix}\) \(\begin{matrix} {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{i} {}^{\rangle} \\ {}^{\langle} \text{u} {}^{\rangle} \end{matrix}\) \(\begin{matrix} {}^{\langle\langle} \text{A} {}^{\rangle\rangle} \\ {}^{\langle\langle} \text{B} {}^{\rangle\rangle} \\ {}^{\langle\langle} \text{i} {}^{\rangle\rangle} \\ {}^{\langle\langle} \text{u} {}^{\rangle\rangle} \end{matrix}\) \(\begin{matrix} {}^{\langle\langle} \text{A} {}^{\rangle\rangle} \\ {}^{\langle\langle} \text{B} {}^{\rangle\rangle} \\ {}^{\langle\langle} \text{i} {}^{\rangle\rangle} \\ {}^{\langle\langle} \text{u} {}^{\rangle\rangle} \end{matrix}\) \(\text{Table 43.} ~~ \text{Higher Ascent Sign Relation} ~ \operatorname{Ref}^1 L(\text{B})\!\) \(\text{Object}\!\) \(\text{Sign}\!\) \(\text{Interpretant}\!\) \(\begin{matrix} \text{A} \\ \text{A} \\ \text{A} \\ \text{A} \end{matrix}\) \(\begin{matrix} {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{u} {}^{\rangle} \\ {}^{\langle} \text{u} {}^{\rangle} \end{matrix}\) \(\begin{matrix} {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{u} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{u} {}^{\rangle} \end{matrix}\) \(\begin{matrix} \text{B} \\ \text{B} \\ \text{B} \\ \text{B} \end{matrix}\) \(\begin{matrix} {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{i} {}^{\rangle} \\ {}^{\langle} \text{i} {}^{\rangle} \end{matrix}\) \(\begin{matrix} {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{i} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{i} {}^{\rangle} \end{matrix}\) \(\begin{matrix} {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{i} {}^{\rangle} \\ {}^{\langle} \text{u} {}^{\rangle} \end{matrix}\) \(\begin{matrix} {}^{\langle\langle} \text{A} {}^{\rangle\rangle} \\ {}^{\langle\langle} \text{B} {}^{\rangle\rangle} \\ {}^{\langle\langle} \text{i} {}^{\rangle\rangle} \\ {}^{\langle\langle} \text{u} {}^{\rangle\rangle} \end{matrix}\) \(\begin{matrix} {}^{\langle\langle} \text{A} {}^{\rangle\rangle} \\ {}^{\langle\langle} \text{B} {}^{\rangle\rangle} \\ {}^{\langle\langle} \text{i} {}^{\rangle\rangle} \\ {}^{\langle\langle} \text{u} {}^{\rangle\rangle} \end{matrix}\) \(\text{Table 44.} ~~ \text{Higher Import Sign Relation} ~ \operatorname{HI}^1 L(\text{A})\!\) \(\text{Object}\!\) \(\text{Sign}\!\) \(\text{Interpretant}\!\) \(\begin{matrix} \text{A} \\ \text{A} \\ \text{A} \\ \text{A} \end{matrix}\) \(\begin{matrix} {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{i} {}^{\rangle} \\ {}^{\langle} \text{i} {}^{\rangle} \end{matrix}\) \(\begin{matrix} {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{i} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{i} {}^{\rangle} \end{matrix}\) \(\begin{matrix} \text{B} \\ \text{B} \\ \text{B} \\ \text{B} \end{matrix}\) \(\begin{matrix} {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{u} {}^{\rangle} \\ {}^{\langle} \text{u} {}^{\rangle} \end{matrix}\) \(\begin{matrix} {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{u} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{u} {}^{\rangle} \end{matrix}\) \(\begin{matrix} ( & \text{A} & , & {}^{\langle} \text{A} {}^{\rangle} & , & {}^{\langle} \text{A} {}^{\rangle} & ) \\ ( & \text{A} & , & {}^{\langle} \text{A} {}^{\rangle} & , & {}^{\langle} \text{i} {}^{\rangle} & ) \\ ( & \text{A} & , & {}^{\langle} \text{i} {}^{\rangle} & , & {}^{\langle} \text{A} {}^{\rangle} & ) \\ ( & \text{A} & , & {}^{\langle} \text{i} {}^{\rangle} & , & {}^{\langle} \text{i} {}^{\rangle} & ) \end{matrix}\) \(\begin{matrix} {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \end{matrix}\) \(\begin{matrix} {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \end{matrix}\) \(\begin{matrix} ( & \text{B} & , & {}^{\langle} \text{B} {}^{\rangle} & , & {}^{\langle} \text{B} {}^{\rangle} & ) \\ ( & \text{B} & , & {}^{\langle} \text{B} {}^{\rangle} & , & {}^{\langle} \text{u} {}^{\rangle} & ) \\ ( & \text{B} & , & {}^{\langle} \text{u} {}^{\rangle} & , & {}^{\langle} \text{B} {}^{\rangle} & ) \\ ( & \text{B} & , & {}^{\langle} \text{u} {}^{\rangle} & , & {}^{\langle} \text{u} {}^{\rangle} & ) \end{matrix}\) \(\begin{matrix} {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \end{matrix}\) \(\begin{matrix} {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \end{matrix}\) \(\begin{matrix} ( & \text{A} & , & {}^{\langle} \text{A} {}^{\rangle} & , & {}^{\langle} \text{A} {}^{\rangle} & ) \\ ( & \text{A} & , & {}^{\langle} \text{A} {}^{\rangle} & , & {}^{\langle} \text{u} {}^{\rangle} & ) \\ ( & \text{A} & , & {}^{\langle} \text{u} {}^{\rangle} & , & {}^{\langle} \text{A} {}^{\rangle} & ) \\ ( & \text{A} & , & {}^{\langle} \text{u} {}^{\rangle} & , & {}^{\langle} \text{u} {}^{\rangle} & ) \end{matrix}\) \(\begin{matrix} {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \end{matrix}\) \(\begin{matrix} {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \end{matrix}\) \(\begin{matrix} ( & \text{B} & , & {}^{\langle} \text{B} {}^{\rangle} & , & {}^{\langle} \text{B} {}^{\rangle} & ) \\ ( & \text{B} & , & {}^{\langle} \text{B} {}^{\rangle} & , & {}^{\langle} \text{i} {}^{\rangle} & ) \\ ( & \text{B} & , & {}^{\langle} \text{i} {}^{\rangle} & , & {}^{\langle} \text{B} {}^{\rangle} & ) \\ ( & \text{B} & , & {}^{\langle} \text{i} {}^{\rangle} & , & {}^{\langle} \text{i} {}^{\rangle} & ) \end{matrix}\) \(\begin{matrix} {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \end{matrix}\) \(\begin{matrix} {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \end{matrix}\) \(\text{Table 45.} ~~ \text{Higher Import Sign Relation} ~ \operatorname{HI}^1 L(\text{B})\!\) \(\text{Object}\!\) \(\text{Sign}\!\) \(\text{Interpretant}\!\) \(\begin{matrix} \text{A} \\ \text{A} \\ \text{A} \\ \text{A} \end{matrix}\) \(\begin{matrix} {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{u} {}^{\rangle} \\ {}^{\langle} \text{u} {}^{\rangle} \end{matrix}\) \(\begin{matrix} {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{u} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{u} {}^{\rangle} \end{matrix}\) \(\begin{matrix} \text{B} \\ \text{B} \\ \text{B} \\ \text{B} \end{matrix}\) \(\begin{matrix} {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{i} {}^{\rangle} \\ {}^{\langle} \text{i} {}^{\rangle} \end{matrix}\) \(\begin{matrix} {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{i} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{i} {}^{\rangle} \end{matrix}\) \(\begin{matrix} ( & \text{A} & , & {}^{\langle} \text{A} {}^{\rangle} & , & {}^{\langle} \text{A} {}^{\rangle} & ) \\ ( & \text{A} & , & {}^{\langle} \text{A} {}^{\rangle} & , & {}^{\langle} \text{i} {}^{\rangle} & ) \\ ( & \text{A} & , & {}^{\langle} \text{i} {}^{\rangle} & , & {}^{\langle} \text{A} {}^{\rangle} & ) \\ ( & \text{A} & , & {}^{\langle} \text{i} {}^{\rangle} & , & {}^{\langle} \text{i} {}^{\rangle} & ) \end{matrix}\) \(\begin{matrix} {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \end{matrix}\) \(\begin{matrix} {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \end{matrix}\) \(\begin{matrix} ( & \text{B} & , & {}^{\langle} \text{B} {}^{\rangle} & , & {}^{\langle} \text{B} {}^{\rangle} & ) \\ ( & \text{B} & , & {}^{\langle} \text{B} {}^{\rangle} & , & {}^{\langle} \text{u} {}^{\rangle} & ) \\ ( & \text{B} & , & {}^{\langle} \text{u} {}^{\rangle} & , & {}^{\langle} \text{B} {}^{\rangle} & ) \\ ( & \text{B} & , & {}^{\langle} \text{u} {}^{\rangle} & , & {}^{\langle} \text{u} {}^{\rangle} & ) \end{matrix}\) \(\begin{matrix} {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \end{matrix}\) \(\begin{matrix} {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \end{matrix}\) \(\begin{matrix} ( & \text{A} & , & {}^{\langle} \text{A} {}^{\rangle} & , & {}^{\langle} \text{A} {}^{\rangle} & ) \\ ( & \text{A} & , & {}^{\langle} \text{A} {}^{\rangle} & , & {}^{\langle} \text{u} {}^{\rangle} & ) \\ ( & \text{A} & , & {}^{\langle} \text{u} {}^{\rangle} & , & {}^{\langle} \text{A} {}^{\rangle} & ) \\ ( & \text{A} & , & {}^{\langle} \text{u} {}^{\rangle} & , & {}^{\langle} \text{u} {}^{\rangle} & ) \end{matrix}\) \(\begin{matrix} {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \end{matrix}\) \(\begin{matrix} {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \end{matrix}\) \(\begin{matrix} ( & \text{B} & , & {}^{\langle} \text{B} {}^{\rangle} & , & {}^{\langle} \text{B} {}^{\rangle} & ) \\ ( & \text{B} & , & {}^{\langle} \text{B} {}^{\rangle} & , & {}^{\langle} \text{i} {}^{\rangle} & ) \\ ( & \text{B} & , & {}^{\langle} \text{i} {}^{\rangle} & , & {}^{\langle} \text{B} {}^{\rangle} & ) \\ ( & \text{B} & , & {}^{\langle} \text{i} {}^{\rangle} & , & {}^{\langle} \text{i} {}^{\rangle} & ) \end{matrix}\) \(\begin{matrix} {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \end{matrix}\) \(\begin{matrix} {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \end{matrix}\) \(\text{Table 46.} ~~ \text{Higher Order Sign Relation for} ~ Q(\text{A}, \text{B})\!\) \(\text{Object}\!\) \(\text{Sign}\!\) \(\text{Interpretant}\!\) \(\begin{matrix} \text{A} \\ \text{B} \end{matrix}\) \(\begin{matrix} {}^{\langle} L {}^{\rangle} \\ {}^{\langle} L {}^{\rangle} \end{matrix}\) \(\begin{matrix} {}^{\langle} L {}^{\rangle} \\ {}^{\langle} L {}^{\rangle} \end{matrix}\) \(\begin{matrix} {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{i} {}^{\rangle} \\ {}^{\langle} \text{u} {}^{\rangle} \end{matrix}\) \(\begin{matrix} {}^{\langle} q {}^{\rangle} \\ {}^{\langle} q {}^{\rangle} \\ {}^{\langle} q {}^{\rangle} \\ {}^{\langle} q {}^{\rangle} \end{matrix}\) \(\begin{matrix} {}^{\langle} q {}^{\rangle} \\ {}^{\langle} q {}^{\rangle} \\ {}^{\langle} q {}^{\rangle} \\ {}^{\langle} q {}^{\rangle} \end{matrix}\) \(\begin{matrix} ( & \text{A} & , & {}^{\langle} \text{A} {}^{\rangle} & , & {}^{\langle} \text{A} {}^{\rangle} & ) \\ ( & \text{A} & , & {}^{\langle} \text{A} {}^{\rangle} & , & {}^{\langle} \text{i} {}^{\rangle} & ) \\ ( & \text{A} & , & {}^{\langle} \text{i} {}^{\rangle} & , & {}^{\langle} \text{A} {}^{\rangle} & ) \\ ( & \text{A} & , & {}^{\langle} \text{i} {}^{\rangle} & , & {}^{\langle} \text{i} {}^{\rangle} & ) \\ ( & \text{B} & , & {}^{\langle} \text{B} {}^{\rangle} & , & {}^{\langle} \text{B} {}^{\rangle} & ) \\ ( & \text{B} & , & {}^{\langle} \text{B} {}^{\rangle} & , & {}^{\langle} \text{u} {}^{\rangle} & ) \\ ( & \text{B} & , & {}^{\langle} \text{u} {}^{\rangle} & , & {}^{\langle} \text{B} {}^{\rangle} & ) \\ ( & \text{B} & , & {}^{\langle} \text{u} {}^{\rangle} & , & {}^{\langle} \text{u} {}^{\rangle} & ) \end{matrix}\) \(\begin{matrix} {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \end{matrix}\) \(\begin{matrix} {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \end{matrix}\) \(\begin{matrix} ( & \text{A} & , & {}^{\langle} \text{A} {}^{\rangle} & , & {}^{\langle} \text{A} {}^{\rangle} & ) \\ ( & \text{A} & , & {}^{\langle} \text{A} {}^{\rangle} & , & {}^{\langle} \text{u} {}^{\rangle} & ) \\ ( & \text{A} & , & {}^{\langle} \text{u} {}^{\rangle} & , & {}^{\langle} \text{A} {}^{\rangle} & ) \\ ( & \text{A} & , & {}^{\langle} \text{u} {}^{\rangle} & , & {}^{\langle} \text{u} {}^{\rangle} & ) \\ ( & \text{B} & , & {}^{\langle} \text{B} {}^{\rangle} & , & {}^{\langle} \text{B} {}^{\rangle} & ) \\ ( & \text{B} & , & {}^{\langle} \text{B} {}^{\rangle} & , & {}^{\langle} \text{i} {}^{\rangle} & ) \\ ( & \text{B} & , & {}^{\langle} \text{i} {}^{\rangle} & , & {}^{\langle} \text{B} {}^{\rangle} & ) \\ ( & \text{B} & , & {}^{\langle} \text{i} {}^{\rangle} & , & {}^{\langle} \text{i} {}^{\rangle} & ) \end{matrix}\) \(\begin{matrix} {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \end{matrix}\) \(\begin{matrix} {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \end{matrix}\) \(\begin{matrix} (( & {}^{\langle} \text{A} {}^{\rangle} & , & \text{A} & ), & \text{A} & ) \\ (( & {}^{\langle} \text{A} {}^{\rangle} & , & \text{B} & ), & \text{A} & ) \\ (( & {}^{\langle} \text{B} {}^{\rangle} & , & \text{A} & ), & \text{B} & ) \\ (( & {}^{\langle} \text{B} {}^{\rangle} & , & \text{B} & ), & \text{B} & ) \\ (( & {}^{\langle} \text{i} {}^{\rangle} & , & \text{A} & ), & \text{A} & ) \\ (( & {}^{\langle} \text{i} {}^{\rangle} & , & \text{B} & ), & \text{B} & ) \\ (( & {}^{\langle} \text{u} {}^{\rangle} & , & \text{A} & ), & \text{B} & ) \\ (( & {}^{\langle} \text{u} {}^{\rangle} & , & \text{B} & ), & \text{A} & ) \end{matrix}\) \(\begin{matrix} {}^{\langle} \operatorname{De} {}^{\rangle} \\ {}^{\langle} \operatorname{De} {}^{\rangle} \\ {}^{\langle} \operatorname{De} {}^{\rangle} \\ {}^{\langle} \operatorname{De} {}^{\rangle} \\ {}^{\langle} \operatorname{De} {}^{\rangle} \\ {}^{\langle} \operatorname{De} {}^{\rangle} \\ {}^{\langle} \operatorname{De} {}^{\rangle} \\ {}^{\langle} \operatorname{De} {}^{\rangle} \end{matrix}\) \(\begin{matrix} {}^{\langle} \operatorname{De} {}^{\rangle} \\ {}^{\langle} \operatorname{De} {}^{\rangle} \\ {}^{\langle} \operatorname{De} {}^{\rangle} \\ {}^{\langle} \operatorname{De} {}^{\rangle} \\ {}^{\langle} \operatorname{De} {}^{\rangle} \\ {}^{\langle} \operatorname{De} {}^{\rangle} \\ {}^{\langle} \operatorname{De} {}^{\rangle} \\ {}^{\langle} \operatorname{De} {}^{\rangle} \end{matrix}\) \(\text{Table 48.1} ~~ \operatorname{ER}(L_\text{A}) : \text{Extensional Representation of} ~ L_\text{A}\!\) \(\text{Object}\!\) \(\text{Sign}\!\) \(\text{Interpretant}\!\) \(\begin{matrix} \text{A} \\ \text{A} \\ \text{A} \\ \text{A} \end{matrix}\) \(\begin{matrix} {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{i} {}^{\rangle} \\ {}^{\langle} \text{i} {}^{\rangle} \end{matrix}\) \(\begin{matrix} {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{i} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{i} {}^{\rangle} \end{matrix}\) \(\begin{matrix} \text{B} \\ \text{B} \\ \text{B} \\ \text{B} \end{matrix}\) \(\begin{matrix} {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{u} {}^{\rangle} \\ {}^{\langle} \text{u} {}^{\rangle} \end{matrix}\) \(\begin{matrix} {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{u} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{u} {}^{\rangle} \end{matrix}\) \(\text{Table 48.2} ~~ \operatorname{ER}(\operatorname{Den}(L_\text{A})) : \text{Denotative Component of} ~ L_\text{A}\!\) \(\text{Object}\!\) \(\text{Sign}\!\) \(\text{Transition}\!\) \(\begin{matrix} \text{A} \\ \text{A} \end{matrix}\) \(\begin{matrix} {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{i} {}^{\rangle} \end{matrix}\) \(\begin{matrix} ({}^{\langle} \text{A} {}^{\rangle}, \text{A}) \\ ({}^{\langle} \text{i} {}^{\rangle}, \text{A}) \end{matrix}\) \(\begin{matrix} \text{B} \\ \text{B} \end{matrix}\) \(\begin{matrix} {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{u} {}^{\rangle} \end{matrix}\) \(\begin{matrix} ({}^{\langle} \text{B} {}^{\rangle}, \text{B}) \\ ({}^{\langle} \text{u} {}^{\rangle}, \text{B}) \end{matrix}\) \(\text{Table 48.3} ~~ \operatorname{ER}(\operatorname{Con}(L_\text{A})) : \text{Connotative Component of} ~ L_\text{A}\!\) \(\text{Sign}\!\) \(\text{Interpretant}\!\) \(\text{Transition}\!\) \(\begin{matrix} {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{i} {}^{\rangle} \\ {}^{\langle} \text{i} {}^{\rangle} \end{matrix}\) \(\begin{matrix} {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{i} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{i} {}^{\rangle} \end{matrix}\) \(\begin{matrix} ({}^{\langle} \text{A} {}^{\rangle}, {}^{\langle} \text{A} {}^{\rangle}) \\ ({}^{\langle} \text{A} {}^{\rangle}, {}^{\langle} \text{i} {}^{\rangle}) \\ ({}^{\langle} \text{i} {}^{\rangle}, {}^{\langle} \text{A} {}^{\rangle}) \\ ({}^{\langle} \text{i} {}^{\rangle}, {}^{\langle} \text{i} {}^{\rangle}) \end{matrix}\) \(\begin{matrix} {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{u} {}^{\rangle} \\ {}^{\langle} \text{u} {}^{\rangle} \end{matrix}\) \(\begin{matrix} {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{u} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{u} {}^{\rangle} \end{matrix}\) \(\begin{matrix} ({}^{\langle} \text{B} {}^{\rangle}, {}^{\langle} \text{B} {}^{\rangle}) \\ ({}^{\langle} \text{B} {}^{\rangle}, {}^{\langle} \text{u} {}^{\rangle}) \\ ({}^{\langle} \text{u} {}^{\rangle}, {}^{\langle} \text{B} {}^{\rangle}) \\ ({}^{\langle} \text{u} {}^{\rangle}, {}^{\langle} \text{u} {}^{\rangle}) \end{matrix}\) \(\text{Table 49.1} ~~ \operatorname{ER}(L_\text{B}) : \text{Extensional Representation of} ~ L_\text{B}\!\) \(\text{Object}\!\) \(\text{Sign}\!\) \(\text{Interpretant}\!\) \(\begin{matrix} \text{A} \\ \text{A} \\ \text{A} \\ \text{A} \end{matrix}\) \(\begin{matrix} {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{u} {}^{\rangle} \\ {}^{\langle} \text{u} {}^{\rangle} \end{matrix}\) \(\begin{matrix} {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{u} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{u} {}^{\rangle} \end{matrix}\) \(\begin{matrix} \text{B} \\ \text{B} \\ \text{B} \\ \text{B} \end{matrix}\) \(\begin{matrix} {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{i} {}^{\rangle} \\ {}^{\langle} \text{i} {}^{\rangle} \end{matrix}\) \(\begin{matrix} {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{i} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{i} {}^{\rangle} \end{matrix}\) \(\text{Table 49.2} ~~ \operatorname{ER}(\operatorname{Den}(L_\text{B})) : \text{Denotative Component of} ~ L_\text{B}\!\) \(\text{Object}\!\) \(\text{Sign}\!\) \(\text{Transition}\!\) \(\begin{matrix} \text{A} \\ \text{A} \end{matrix}\) \(\begin{matrix} {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{u} {}^{\rangle} \end{matrix}\) \(\begin{matrix} ({}^{\langle} \text{A} {}^{\rangle}, \text{A}) \\ ({}^{\langle} \text{u} {}^{\rangle}, \text{A}) \end{matrix}\) \(\begin{matrix} \text{B} \\ \text{B} \end{matrix}\) \(\begin{matrix} {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{i} {}^{\rangle} \end{matrix}\) \(\begin{matrix} ({}^{\langle} \text{B} {}^{\rangle}, \text{B}) \\ ({}^{\langle} \text{i} {}^{\rangle}, \text{B}) \end{matrix}\) \(\text{Table 49.3} ~~ \operatorname{ER}(\operatorname{Con}(L_\text{B})) : \text{Connotative Component of} ~ L_\text{B}\!\) \(\text{Sign}\!\) \(\text{Interpretant}\!\) \(\text{Transition}\!\) \(\begin{matrix} {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{u} {}^{\rangle} \\ {}^{\langle} \text{u} {}^{\rangle} \end{matrix}\) \(\begin{matrix} {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{u} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{u} {}^{\rangle} \end{matrix}\) \(\begin{matrix} ({}^{\langle} \text{A} {}^{\rangle}, {}^{\langle} \text{A} {}^{\rangle}) \\ ({}^{\langle} \text{A} {}^{\rangle}, {}^{\langle} \text{u} {}^{\rangle}) \\ ({}^{\langle} \text{u} {}^{\rangle}, {}^{\langle} \text{A} {}^{\rangle}) \\ ({}^{\langle} \text{u} {}^{\rangle}, {}^{\langle} \text{u} {}^{\rangle}) \end{matrix}\) \(\begin{matrix} {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{i} {}^{\rangle} \\ {}^{\langle} \text{i} {}^{\rangle} \end{matrix}\) \(\begin{matrix} {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{i} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{i} {}^{\rangle} \end{matrix}\) \(\begin{matrix} ({}^{\langle} \text{B} {}^{\rangle}, {}^{\langle} \text{B} {}^{\rangle}) \\ ({}^{\langle} \text{B} {}^{\rangle}, {}^{\langle} \text{i} {}^{\rangle}) \\ ({}^{\langle} \text{i} {}^{\rangle}, {}^{\langle} \text{B} {}^{\rangle}) \\ ({}^{\langle} \text{i} {}^{\rangle}, {}^{\langle} \text{i} {}^{\rangle}) \end{matrix}\) Sign Processes Blocked Version \(\text{Table 78.} ~~ \text{Sign Process of Interpreter A}\!\) \(\text{Object}\!\) \(\text{Sign}\!\) \(\text{Interpretant}\!\) \(\begin{matrix} \text{A} \\ \text{A} \\ \text{A} \\ \text{A} \end{matrix}\) \(\begin{matrix} {}^{\backprime\backprime} \text{A} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{A} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \end{matrix}\) \(\begin{matrix} {}^{\backprime\backprime} \text{A} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{A} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \end{matrix}\) \(\begin{matrix} \text{A} \\ \text{A} \\ \text{A} \\ \text{A} \end{matrix}\) \(\begin{matrix} {}^{\backprime\backprime} \text{B} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{B} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \end{matrix}\) \(\begin{matrix} {}^{\backprime\backprime} \text{A} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{A} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \end{matrix}\) \(\begin{matrix} \text{B} \\ \text{B} \\ \text{B} \\ \text{B} \end{matrix}\) \(\begin{matrix} {}^{\backprime\backprime} \text{A} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{A} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \end{matrix}\) \(\begin{matrix} {}^{\backprime\backprime} \text{B} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{B} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \end{matrix}\) \(\begin{matrix} \text{B} \\ \text{B} \\ \text{B} \\ \text{B} \end{matrix}\) \(\begin{matrix} {}^{\backprime\backprime} \text{B} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{B} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \end{matrix}\) \(\begin{matrix} {}^{\backprime\backprime} \text{B} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{B} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \end{matrix}\) \(\text{Table 79.} ~~ \text{Sign Process of Interpreter B}\!\) \(\text{Object}\!\) \(\text{Sign}\!\) \(\text{Interpretant}\!\) \(\begin{matrix} \text{A} \\ \text{A} \\ \text{A} \\ \text{A} \end{matrix}\) \(\begin{matrix} {}^{\backprime\backprime} \text{A} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{A} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \end{matrix}\) \(\begin{matrix} {}^{\backprime\backprime} \text{A} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{A} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \end{matrix}\) \(\begin{matrix} \text{A} \\ \text{A} \\ \text{A} \\ \text{A} \end{matrix}\) \(\begin{matrix} {}^{\backprime\backprime} \text{B} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{B} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \end{matrix}\) \(\begin{matrix} {}^{\backprime\backprime} \text{A} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{A} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \end{matrix}\) \(\begin{matrix} \text{B} \\ \text{B} \\ \text{B} \\ \text{B} \end{matrix}\) \(\begin{matrix} {}^{\backprime\backprime} \text{A} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{A} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \end{matrix}\) \(\begin{matrix} {}^{\backprime\backprime} \text{B} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{B} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \end{matrix}\) \(\begin{matrix} \text{B} \\ \text{B} \\ \text{B} \\ \text{B} \end{matrix}\) \(\begin{matrix} {}^{\backprime\backprime} \text{B} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{B} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \end{matrix}\) \(\begin{matrix} {}^{\backprime\backprime} \text{B} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{B} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \end{matrix}\) Sorted Version \(\text{Table 78.} ~~ \text{Sign Process of Interpreter A}\!\) \(\text{Object}\!\) \(\text{Sign}\!\) \(\text{Interpretant}\!\) \(\begin{matrix} \text{A} \\ \text{A} \\ \text{A} \\ \text{A} \\ \text{A} \\ \text{A} \\ \text{A} \\ \text{A} \end{matrix}\) \(\begin{matrix} {}^{\backprime\backprime} \text{A} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{A} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{B} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{B} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \end{matrix}\) \(\begin{matrix} {}^{\backprime\backprime} \text{A} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{A} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{A} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{A} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \end{matrix}\) \(\begin{matrix} \text{B} \\ \text{B} \\ \text{B} \\ \text{B} \\ \text{B} \\ \text{B} \\ \text{B} \\ \text{B} \end{matrix}\) \(\begin{matrix} {}^{\backprime\backprime} \text{A} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{A} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{B} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{B} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \end{matrix}\) \(\begin{matrix} {}^{\backprime\backprime} \text{B} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{B} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{B} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{B} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \end{matrix}\) \(\text{Table 79.} ~~ \text{Sign Process of Interpreter B}\!\) \(\text{Object}\!\) \(\text{Sign}\!\) \(\text{Interpretant}\!\) \(\begin{matrix} \text{A} \\ \text{A} \\ \text{A} \\ \text{A} \\ \text{A} \\ \text{A} \\ \text{A} \\ \text{A} \end{matrix}\) \(\begin{matrix} {}^{\backprime\backprime} \text{A} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{A} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{B} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{B} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \end{matrix}\) \(\begin{matrix} {}^{\backprime\backprime} \text{A} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{A} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{A} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{A} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \end{matrix}\) \(\begin{matrix} \text{B} \\ \text{B} \\ \text{B} \\ \text{B} \\ \text{B} \\ \text{B} \\ \text{B} \\ \text{B} \end{matrix}\) \(\begin{matrix} {}^{\backprime\backprime} \text{A} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{A} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{B} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{B} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \end{matrix}\) \(\begin{matrix} {}^{\backprime\backprime} \text{B} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{B} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{B} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{B} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \end{matrix}\) 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 \(\text{Table 50.} ~~ \text{Notations for Objects and Their Signs}\!\) \(\text{Object}\!\) \(\text{Sign of Object}\!\) \(\begin{matrix} \text{A} & \text{A} & w_1 \\[6pt] \text{B} & \text{B} & w_2 \\[12pt] {}^{\backprime\backprime} \text{A} {}^{\prime\prime} & {}^{\langle} \text{A} {}^{\rangle} & w_3 \\[6pt] {}^{\backprime\backprime} \text{B} {}^{\prime\prime} & {}^{\langle} \text{B} {}^{\rangle} & w_4 \\[6pt] {}^{\backprime\backprime} \text{i} {}^{\prime\prime} & {}^{\langle} \text{i} {}^{\rangle} & w_5 \\[6pt] {}^{\backprime\backprime} \text{u} {}^{\prime\prime} & {}^{\langle} \text{u} {}^{\rangle} & w_6 \end{matrix}\) \(\begin{matrix} {}^{\langle} \text{A} {}^{\rangle} & {}^{\langle} \text{A} {}^{\rangle} & {}^{\langle} w_1 {}^{\rangle} \\[6pt] {}^{\langle} \text{B} {}^{\rangle} & {}^{\langle} \text{B} {}^{\rangle} & {}^{\langle} w_2 {}^{\rangle} \\[12pt] {}^{\langle\backprime\backprime} \text{A} {}^{\prime\prime\rangle} & {}^{\langle\langle} \text{A} {}^{\rangle\rangle} & {}^{\langle} w_3 {}^{\rangle} \\[6pt] {}^{\langle\backprime\backprime} \text{B} {}^{\prime\prime\rangle} & {}^{\langle\langle} \text{B} {}^{\rangle\rangle} & {}^{\langle} w_4 {}^{\rangle} \\[6pt] {}^{\langle\backprime\backprime} \text{i} {}^{\prime\prime\rangle} & {}^{\langle\langle} \text{i} {}^{\rangle\rangle} & {}^{\langle} w_5 {}^{\rangle} \\[6pt] {}^{\langle\backprime\backprime} \text{u} {}^{\prime\prime\rangle} & {}^{\langle\langle} \text{u} {}^{\rangle\rangle} & {}^{\langle} w_6 {}^{\rangle} \end{matrix}\) \(\text{Table 51.1} ~~ \text{Notations for Properties and Their Signs (1)}\!\) \(\text{Property}\!\) \(\text{Sign of Property}\!\) \(\begin{matrix} {}^{\lbrace} \text{A} {}^{\rbrace} & {}^{\lbrace} \text{A} {}^{\rbrace} & {}^{\lbrace} w_1 {}^{\rbrace} \\[6pt] {}^{\lbrace} \text{B} {}^{\rbrace} & {}^{\lbrace} \text{B} {}^{\rbrace} & {}^{\lbrace} w_2 {}^{\rbrace} \\[12pt] {}^{\lbrace\backprime\backprime} \text{A} {}^{\prime\prime\rbrace} & {}^{\lbrace\langle} \text{A} {}^{\rangle\rbrace} & {}^{\lbrace} w_3 {}^{\rbrace} \\[6pt] {}^{\lbrace\backprime\backprime} \text{B} {}^{\prime\prime\rbrace} & {}^{\lbrace\langle} \text{B} {}^{\rangle\rbrace} & {}^{\lbrace} w_4 {}^{\rbrace} \\[6pt] {}^{\lbrace\backprime\backprime} \text{i} {}^{\prime\prime\rbrace} & {}^{\lbrace\langle} \text{i} {}^{\rangle\rbrace} & {}^{\lbrace} w_5 {}^{\rbrace} \\[6pt] {}^{\lbrace\backprime\backprime} \text{u} {}^{\prime\prime\rbrace} & {}^{\lbrace\langle} \text{u} {}^{\rangle\rbrace} & {}^{\lbrace} w_6 {}^{\rbrace} \end{matrix}\) \(\begin{matrix} {}^{\langle\lbrace} \text{A} {}^{\rbrace\rangle} & {}^{\langle\lbrace} \text{A} {}^{\rbrace\rangle} & {}^{\langle\lbrace} w_1 {}^{\rbrace\rangle} \\[6pt] {}^{\langle\lbrace} \text{B} {}^{\rbrace\rangle} & {}^{\langle\lbrace} \text{B} {}^{\rbrace\rangle} & {}^{\langle\lbrace} w_2 {}^{\rbrace\rangle} \\[12pt] {}^{\langle\lbrace\backprime\backprime} \text{A} {}^{\prime\prime\rbrace\rangle} & {}^{\langle\lbrace\langle} \text{A} {}^{\rangle\rbrace\rangle} & {}^{\langle\lbrace} w_3 {}^{\rbrace\rangle} \\[6pt] {}^{\langle\lbrace\backprime\backprime} \text{B} {}^{\prime\prime\rbrace\rangle} & {}^{\langle\lbrace\langle} \text{B} {}^{\rangle\rbrace\rangle} & {}^{\langle\lbrace} w_4 {}^{\rbrace\rangle} \\[6pt] {}^{\langle\lbrace\backprime\backprime} \text{i} {}^{\prime\prime\rbrace\rangle} & {}^{\langle\lbrace\langle} \text{i} {}^{\rangle\rbrace\rangle} & {}^{\langle\lbrace} w_5 {}^{\rbrace\rangle} \\[6pt] {}^{\langle\lbrace\backprime\backprime} \text{u} {}^{\prime\prime\rbrace\rangle} & {}^{\langle\lbrace\langle} \text{u} {}^{\rangle\rbrace\rangle} & {}^{\langle\lbrace} w_6 {}^{\rbrace\rangle} \end{matrix}\) \(\text{Table 51.2} ~~ \text{Notations for Properties and Their Signs (2)}\!\) \(\text{Property}\!\) \(\text{Sign of Property}\!\) \(\begin{matrix} \underline{\underline{\text{A}}} & \underline{\underline{\text{A}}} & \underline{\underline{w_1}} \\[6pt] \underline{\underline{\text{B}}} & \underline{\underline{\text{B}}} & \underline{\underline{w_2}} \\[12pt] \underline{\underline{{}^{\backprime\backprime} \text{A} {}^{\prime\prime}}} & \underline{\underline{{}^{\langle} \text{A} {}^{\rangle}}} & \underline{\underline{w_3}} \\[6pt] \underline{\underline{{}^{\backprime\backprime} \text{B} {}^{\prime\prime}}} & \underline{\underline{{}^{\langle} \text{B} {}^{\rangle}}} & \underline{\underline{w_4}} \\[6pt] \underline{\underline{{}^{\backprime\backprime} \text{i} {}^{\prime\prime}}} & \underline{\underline{{}^{\langle} \text{i} {}^{\rangle}}} & \underline{\underline{w_5}} \\[6pt] \underline{\underline{{}^{\backprime\backprime} \text{u} {}^{\prime\prime}}} & \underline{\underline{{}^{\langle} \text{u} {}^{\rangle}}} & \underline{\underline{w_6}} \end{matrix}\) \(\begin{matrix} {}^{\langle} \underline{\underline{\text{A}}} {}^{\rangle} & {}^{\langle} \underline{\underline{\text{A}}} {}^{\rangle} & {}^{\langle} \underline{\underline{w_1}} {}^{\rangle} \\[6pt] {}^{\langle} \underline{\underline{\text{B}}} {}^{\rangle} & {}^{\langle} \underline{\underline{\text{B}}} {}^{\rangle} & {}^{\langle} \underline{\underline{w_2}} {}^{\rangle} \\[12pt] {}^{\langle} \underline{\underline{{}^{\backprime\backprime} \text{A} {}^{\prime\prime}}} {}^{\rangle} & {}^{\langle} \underline{\underline{{}^{\langle} \text{A} {}^{\rangle}}} {}^{\rangle} & {}^{\langle} \underline{\underline{w_3}} {}^{\rangle} \\[6pt] {}^{\langle} \underline{\underline{{}^{\backprime\backprime} \text{B} {}^{\prime\prime}}} {}^{\rangle} & {}^{\langle} \underline{\underline{{}^{\langle} \text{B} {}^{\rangle}}} {}^{\rangle} & {}^{\langle} \underline{\underline{w_4}} {}^{\rangle} \\[6pt] {}^{\langle} \underline{\underline{{}^{\backprime\backprime} \text{i} {}^{\prime\prime}}} {}^{\rangle} & {}^{\langle} \underline{\underline{{}^{\langle} \text{i} {}^{\rangle}}} {}^{\rangle} & {}^{\langle} \underline{\underline{w_5}} {}^{\rangle} \\[6pt] {}^{\langle} \underline{\underline{{}^{\backprime\backprime} \text{u} {}^{\prime\prime}}} {}^{\rangle} & {}^{\langle} \underline{\underline{{}^{\langle} \text{u} {}^{\rangle}}} {}^{\rangle} & {}^{\langle} \underline{\underline{w_6}} {}^{\rangle} \end{matrix}\) \(\text{Table 51.3} ~~ \text{Notations for Properties and Their Signs (3)}\!\) \(\text{Property}\!\) \(\text{Sign of Property}\!\) \(\begin{matrix} \underline{\underline{\text{A}}} & \underline{\underline{o_1}} & \underline{\underline{w_1}} \\[6pt] \underline{\underline{\text{B}}} & \underline{\underline{o_2}} & \underline{\underline{w_2}} \\[12pt] \underline{\underline{\text{a}}} & \underline{\underline{s_1}} & \underline{\underline{w_3}} \\[6pt] \underline{\underline{\text{b}}} & \underline{\underline{s_2}} & \underline{\underline{w_4}} \\[6pt] \underline{\underline{\text{i}}} & \underline{\underline{s_3}} & \underline{\underline{w_5}} \\[6pt] \underline{\underline{\text{u}}} & \underline{\underline{s_4}} & \underline{\underline{w_6}} \end{matrix}\) \(\begin{matrix} {}^{\langle} \underline{\underline{\text{A}}} {}^{\rangle} & {}^{\langle} \underline{\underline{o_1}} {}^{\rangle} & {}^{\langle} \underline{\underline{w_1}} {}^{\rangle} \\[6pt] {}^{\langle} \underline{\underline{\text{B}}} {}^{\rangle} & {}^{\langle} \underline{\underline{o_2}} {}^{\rangle} & {}^{\langle} \underline{\underline{w_2}} {}^{\rangle} \\[12pt] {}^{\langle} \underline{\underline{\text{a}}} {}^{\rangle} & {}^{\langle} \underline{\underline{s_1}} {}^{\rangle} & {}^{\langle} \underline{\underline{w_3}} {}^{\rangle} \\[6pt] {}^{\langle} \underline{\underline{\text{b}}} {}^{\rangle} & {}^{\langle} \underline{\underline{s_2}} {}^{\rangle} & {}^{\langle} \underline{\underline{w_4}} {}^{\rangle} \\[6pt] {}^{\langle} \underline{\underline{\text{i}}} {}^{\rangle} & {}^{\langle} \underline{\underline{s_3}} {}^{\rangle} & {}^{\langle} \underline{\underline{w_5}} {}^{\rangle} \\[6pt] {}^{\langle} \underline{\underline{\text{u}}} {}^{\rangle} & {}^{\langle} \underline{\underline{s_4}} {}^{\rangle} & {}^{\langle} \underline{\underline{w_6}} {}^{\rangle} \end{matrix}\) \(\text{Table 52.1} ~~ \text{Notations for Instances and Their Signs (1)}\!\) \(\text{Instance}\!\) \(\text{Sign of Instance}\!\) \(\begin{matrix} {}^{\lbrack} \text{A} {}^{\rbrack} & {}^{\lbrack} \text{A} {}^{\rbrack} & {}^{\lbrack} w_1 {}^{\rbrack} \\[6pt] {}^{\lbrack} \text{B} {}^{\rbrack} & {}^{\lbrack} \text{B} {}^{\rbrack} & {}^{\lbrack} w_2 {}^{\rbrack} \\[12pt] {}^{\lbrack\backprime\backprime} \text{A} {}^{\prime\prime\rbrack} & {}^{\lbrack\langle} \text{A} {}^{\rangle\rbrack} & {}^{\lbrack} w_3 {}^{\rbrack} \\[6pt] {}^{\lbrack\backprime\backprime} \text{B} {}^{\prime\prime\rbrack} & {}^{\lbrack\langle} \text{B} {}^{\rangle\rbrack} & {}^{\lbrack} w_4 {}^{\rbrack} \\[6pt] {}^{\lbrack\backprime\backprime} \text{i} {}^{\prime\prime\rbrack} & {}^{\lbrack\langle} \text{i} {}^{\rangle\rbrack} & {}^{\lbrack} w_5 {}^{\rbrack} \\[6pt] {}^{\lbrack\backprime\backprime} \text{u} {}^{\prime\prime\rbrack} & {}^{\lbrack\langle} \text{u} {}^{\rangle\rbrack} & {}^{\lbrack} w_6 {}^{\rbrack} \end{matrix}\) \(\begin{matrix} {}^{\langle\lbrack} \text{A} {}^{\rbrack\rangle} & {}^{\langle\lbrack} \text{A} {}^{\rbrack\rangle} & {}^{\langle\lbrack} w_1 {}^{\rbrack\rangle} \\[6pt] {}^{\langle\lbrack} \text{B} {}^{\rbrack\rangle} & {}^{\langle\lbrack} \text{B} {}^{\rbrack\rangle} & {}^{\langle\lbrack} w_2 {}^{\rbrack\rangle} \\[12pt] {}^{\langle\lbrack\backprime\backprime} \text{A} {}^{\prime\prime\rbrack\rangle} & {}^{\langle\lbrack\langle} \text{A} {}^{\rangle\rbrack\rangle} & {}^{\langle\lbrack} w_3 {}^{\rbrack\rangle} \\[6pt] {}^{\langle\lbrack\backprime\backprime} \text{B} {}^{\prime\prime\rbrack\rangle} & {}^{\langle\lbrack\langle} \text{B} {}^{\rangle\rbrack\rangle} & {}^{\langle\lbrack} w_4 {}^{\rbrack\rangle} \\[6pt] {}^{\langle\lbrack\backprime\backprime} \text{i} {}^{\prime\prime\rbrack\rangle} & {}^{\langle\lbrack\langle} \text{i} {}^{\rangle\rbrack\rangle} & {}^{\langle\lbrack} w_5 {}^{\rbrack\rangle} \\[6pt] {}^{\langle\lbrack\backprime\backprime} \text{u} {}^{\prime\prime\rbrack\rangle} & {}^{\langle\lbrack\langle} \text{u} {}^{\rangle\rbrack\rangle} & {}^{\langle\lbrack} w_6 {}^{\rbrack\rangle} \end{matrix}\) \(\text{Table 52.2} ~~ \text{Notations for Instances and Their Signs (2)}\!\) \(\text{Instance}\!\) \(\text{Sign of Instance}\!\) \(\begin{matrix} \overline{\text{A}} & \overline{\text{A}} & \overline{w_1} \\[6pt] \overline{\text{B}} & \overline{\text{B}} & \overline{w_2} \\[12pt] \overline{{}^{\backprime\backprime} \text{A} {}^{\prime\prime}} & \overline{{}^{\langle} \text{A} {}^{\rangle}} & \overline{w_3} \\[6pt] \overline{{}^{\backprime\backprime} \text{B} {}^{\prime\prime}} & \overline{{}^{\langle} \text{B} {}^{\rangle}} & \overline{w_4} \\[6pt] \overline{{}^{\backprime\backprime} \text{i} {}^{\prime\prime}} & \overline{{}^{\langle} \text{i} {}^{\rangle}} & \overline{w_5} \\[6pt] \overline{{}^{\backprime\backprime} \text{u} {}^{\prime\prime}} & \overline{{}^{\langle} \text{u} {}^{\rangle}} & \overline{w_6} \end{matrix}\) \(\begin{matrix} {}^{\langle} \overline{\text{A}} {}^{\rangle} & {}^{\langle} \overline{\text{A}} {}^{\rangle} & {}^{\langle} \overline{w_1} {}^{\rangle} \\[6pt] {}^{\langle} \overline{\text{B}} {}^{\rangle} & {}^{\langle} \overline{\text{B}} {}^{\rangle} & {}^{\langle} \overline{w_2} {}^{\rangle} \\[12pt] {}^{\langle} \overline{{}^{\backprime\backprime} \text{A} {}^{\prime\prime}} {}^{\rangle} & {}^{\langle} \overline{{}^{\langle} \text{A} {}^{\rangle}} {}^{\rangle} & {}^{\langle} \overline{w_3} {}^{\rangle} \\[6pt] {}^{\langle} \overline{{}^{\backprime\backprime} \text{B} {}^{\prime\prime}} {}^{\rangle} & {}^{\langle} \overline{{}^{\langle} \text{B} {}^{\rangle}} {}^{\rangle} & {}^{\langle} \overline{w_4} {}^{\rangle} \\[6pt] {}^{\langle} \overline{{}^{\backprime\backprime} \text{i} {}^{\prime\prime}} {}^{\rangle} & {}^{\langle} \overline{{}^{\langle} \text{i} {}^{\rangle}} {}^{\rangle} & {}^{\langle} \overline{w_5} {}^{\rangle} \\[6pt] {}^{\langle} \overline{{}^{\backprime\backprime} \text{u} {}^{\prime\prime}} {}^{\rangle} & {}^{\langle} \overline{{}^{\langle} \text{u} {}^{\rangle}} {}^{\rangle} & {}^{\langle} \overline{w_6} {}^{\rangle} \end{matrix}\) \(\text{Table 52.3} ~~ \text{Notations for Instances and Their Signs (3)}\!\) \(\text{Instance}\!\) \(\text{Sign of Instance}\!\) \(\begin{matrix} \overline{\text{A}} & \overline{o_1} & \overline{w_1} \\[6pt] \overline{\text{B}} & \overline{o_2} & \overline{w_2} \\[12pt] \overline{\text{a}} & \overline{s_1} & \overline{w_3} \\[6pt] \overline{\text{b}} & \overline{s_2} & \overline{w_4} \\[6pt] \overline{\text{i}} & \overline{s_3} & \overline{w_5} \\[6pt] \overline{\text{u}} & \overline{s_4} & \overline{w_6} \end{matrix}\) \(\begin{matrix} {}^{\langle} \overline{\text{A}} {}^{\rangle} & {}^{\langle} \overline{o_1} {}^{\rangle} & {}^{\langle} \overline{w_1} {}^{\rangle} \\[6pt] {}^{\langle} \overline{\text{B}} {}^{\rangle} & {}^{\langle} \overline{o_2} {}^{\rangle} & {}^{\langle} \overline{w_2} {}^{\rangle} \\[12pt] {}^{\langle} \overline{\text{a}} {}^{\rangle} & {}^{\langle} \overline{s_1} {}^{\rangle} & {}^{\langle} \overline{w_3} {}^{\rangle} \\[6pt] {}^{\langle} \overline{\text{b}} {}^{\rangle} & {}^{\langle} \overline{s_2} {}^{\rangle} & {}^{\langle} \overline{w_4} {}^{\rangle} \\[6pt] {}^{\langle} \overline{\text{i}} {}^{\rangle} & {}^{\langle} \overline{s_3} {}^{\rangle} & {}^{\langle} \overline{w_5} {}^{\rangle} \\[6pt] {}^{\langle} \overline{\text{u}} {}^{\rangle} & {}^{\langle} \overline{s_4} {}^{\rangle} & {}^{\langle} \overline{w_6} {}^{\rangle} \end{matrix}\) \(\text{Table 53.1} ~~ \text{Elements of} ~ \operatorname{ER}(W)\!\) \(\text{Mnemonic Element}\!\) \(w \in W\!\) \(\text{Pragmatic Element}\!\) \(w \in W\!\) \(\text{Abstract Element}\!\) \(w_i \in W\!\) \(\begin{matrix} \text{A} \\[4pt] \text{B} \\[4pt] {}^{\backprime\backprime} \text{A} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \text{B} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \end{matrix}\) \(\begin{matrix} o_1 \\[4pt] o_2 \\[4pt] s_1 \\[4pt] s_2 \\[4pt] s_3 \\[4pt] s_4 \end{matrix}\) \(\begin{matrix} w_1 \\[4pt] w_2 \\[4pt] w_3 \\[4pt] w_4 \\[4pt] w_5 \\[4pt] w_6 \end{matrix}\) \(\text{Table 53.2} ~~ \text{Features of} ~ \operatorname{LIR}(W)\!\) \(\text{Mnemonic Feature}\!\) \(\underline{\underline{w}} \in \underline{\underline{W}}\!\) \(\text{Pragmatic Feature}\!\) \(\underline{\underline{w}} \in \underline{\underline{W}}\!\) \(\text{Abstract Feature}\!\) \(\underline{\underline{w_i}} \in \underline{\underline{W}}\!\) \(\begin{matrix} \underline{\underline{\text{A}}} \\[4pt] \underline{\underline{\text{B}}} \\[4pt] \underline{\underline{\text{a}}} \\[4pt] \underline{\underline{\text{b}}} \\[4pt] \underline{\underline{\text{i}}} \\[4pt] \underline{\underline{\text{u}}} \end{matrix}\) \(\begin{matrix} \underline{\underline{o_1}} \\[4pt] \underline{\underline{o_2}} \\[4pt] \underline{\underline{s_1}} \\[4pt] \underline{\underline{s_2}} \\[4pt] \underline{\underline{s_3}} \\[4pt] \underline{\underline{s_4}} \end{matrix}\) \(\begin{matrix} \underline{\underline{w_1}} \\[4pt] \underline{\underline{w_2}} \\[4pt] \underline{\underline{w_3}} \\[4pt] \underline{\underline{w_4}} \\[4pt] \underline{\underline{w_5}} \\[4pt] \underline{\underline{w_6}} \end{matrix}\) \(\text{Table 54.1} ~~ \text{Mnemonic Literal Codes for Interpreters A and B}\!\) \(\text{Element}\!\) \(\text{Vector}\!\) \(\text{Conjunct Term}\!\) \(\text{Code}\!\) \(\begin{matrix} \text{A} \\[4pt] \text{B} \\[4pt] {}^{\backprime\backprime} \text{A} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \text{B} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \end{matrix}\) \(\begin{matrix} 100000 \\[4pt] 010000 \\[4pt] 001000 \\[4pt] 000100 \\[4pt] 000010 \\[4pt] 000001 \end{matrix}\) \(\begin{matrix} ~\underline{\underline{A}}~ (\underline{\underline{B}}) (\underline{\underline{a}}) (\underline{\underline{b}}) (\underline{\underline{i}}) (\underline{\underline{u}}) \\[4pt] (\underline{\underline{A}}) ~\underline{\underline{B}}~ (\underline{\underline{a}}) (\underline{\underline{b}}) (\underline{\underline{i}}) (\underline{\underline{u}}) \\[4pt] (\underline{\underline{A}}) (\underline{\underline{B}}) ~\underline{\underline{a}}~ (\underline{\underline{b}}) (\underline{\underline{i}}) (\underline{\underline{u}}) \\[4pt] (\underline{\underline{A}}) (\underline{\underline{B}}) (\underline{\underline{a}}) ~\underline{\underline{b}}~ (\underline{\underline{i}}) (\underline{\underline{u}}) \\[4pt] (\underline{\underline{A}}) (\underline{\underline{B}}) (\underline{\underline{a}}) (\underline{\underline{b}}) ~\underline{\underline{i}}~ (\underline{\underline{u}}) \\[4pt] (\underline{\underline{A}}) (\underline{\underline{B}}) (\underline{\underline{a}}) (\underline{\underline{b}}) (\underline{\underline{i}}) ~\underline{\underline{u}}~ \end{matrix}\) \(\begin{matrix} {\langle\underline{\underline{A}}\rangle}_W \\[4pt] {\langle\underline{\underline{B}}\rangle}_W \\[4pt] {\langle\underline{\underline{a}}\rangle}_W \\[4pt] {\langle\underline{\underline{b}}\rangle}_W \\[4pt] {\langle\underline{\underline{i}}\rangle}_W \\[4pt] {\langle\underline{\underline{u}}\rangle}_W \end{matrix}\) \(\text{Table 54.2} ~~ \text{Pragmatic Literal Codes for Interpreters A and B}\!\) \(\text{Element}\!\) \(\text{Vector}\!\) \(\text{Conjunct Term}\!\) \(\text{Code}\!\) \(\begin{matrix} \text{A} \\[4pt] \text{B} \\[4pt] {}^{\backprime\backprime} \text{A} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \text{B} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \end{matrix}\) \(\begin{matrix} 100000 \\[4pt] 010000 \\[4pt] 001000 \\[4pt] 000100 \\[4pt] 000010 \\[4pt] 000001 \end{matrix}\) \(\begin{matrix} ~\underline{\underline{o_1}}~ (\underline{\underline{o_2}}) (\underline{\underline{s_1}}) (\underline{\underline{s_2}}) (\underline{\underline{s_3}}) (\underline{\underline{s_4}}) \\[4pt] (\underline{\underline{o_1}}) ~\underline{\underline{o_2}}~ (\underline{\underline{s_1}}) (\underline{\underline{s_2}}) (\underline{\underline{s_3}}) (\underline{\underline{s_4}}) \\[4pt] (\underline{\underline{o_1}}) (\underline{\underline{o_2}}) ~\underline{\underline{s_1}}~ (\underline{\underline{s_2}}) (\underline{\underline{s_3}}) (\underline{\underline{s_4}}) \\[4pt] (\underline{\underline{o_1}}) (\underline{\underline{o_2}}) (\underline{\underline{s_1}}) ~\underline{\underline{s_2}}~ (\underline{\underline{s_3}}) (\underline{\underline{s_4}}) \\[4pt] (\underline{\underline{o_1}}) (\underline{\underline{o_2}}) (\underline{\underline{s_1}}) (\underline{\underline{s_2}}) ~\underline{\underline{s_3}}~ (\underline{\underline{s_4}}) \\[4pt] (\underline{\underline{o_1}}) (\underline{\underline{o_2}}) (\underline{\underline{s_1}}) (\underline{\underline{s_2}}) (\underline{\underline{s_3}}) ~\underline{\underline{s_4}}~ \end{matrix}\) \(\begin{matrix} {\langle\underline{\underline{o_1}}\rangle}_W \\[4pt] {\langle\underline{\underline{o_2}}\rangle}_W \\[4pt] {\langle\underline{\underline{s_1}}\rangle}_W \\[4pt] {\langle\underline{\underline{s_2}}\rangle}_W \\[4pt] {\langle\underline{\underline{s_3}}\rangle}_W \\[4pt] {\langle\underline{\underline{s_4}}\rangle}_W \end{matrix}\) \(\text{Table 54.3} ~~ \text{Abstract Literal Codes for Interpreters A and B}\!\) \(\text{Element}\!\) \(\text{Vector}\!\) \(\text{Conjunct Term}\!\) \(\text{Code}\!\) \(\begin{matrix} \text{A} \\[4pt] \text{B} \\[4pt] {}^{\backprime\backprime} \text{A} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \text{B} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \end{matrix}\) \(\begin{matrix} 100000 \\[4pt] 010000 \\[4pt] 001000 \\[4pt] 000100 \\[4pt] 000010 \\[4pt] 000001 \end{matrix}\) \(\begin{matrix} ~\underline{\underline{w_1}}~ (\underline{\underline{w_2}}) (\underline{\underline{w_3}}) (\underline{\underline{w_4}}) (\underline{\underline{w_5}}) (\underline{\underline{w_6}}) \\[4pt] (\underline{\underline{w_1}}) ~\underline{\underline{w_2}}~ (\underline{\underline{w_3}}) (\underline{\underline{w_4}}) (\underline{\underline{w_5}}) (\underline{\underline{w_6}}) \\[4pt] (\underline{\underline{w_1}}) (\underline{\underline{w_2}}) ~\underline{\underline{w_3}}~ (\underline{\underline{w_4}}) (\underline{\underline{w_5}}) (\underline{\underline{w_6}}) \\[4pt] (\underline{\underline{w_1}}) (\underline{\underline{w_2}}) (\underline{\underline{w_3}}) ~\underline{\underline{w_4}}~ (\underline{\underline{w_5}}) (\underline{\underline{w_6}}) \\[4pt] (\underline{\underline{w_1}}) (\underline{\underline{w_2}}) (\underline{\underline{w_3}}) (\underline{\underline{w_4}}) ~\underline{\underline{w_5}}~ (\underline{\underline{w_6}}) \\[4pt] (\underline{\underline{w_1}}) (\underline{\underline{w_2}}) (\underline{\underline{w_3}}) (\underline{\underline{w_4}}) (\underline{\underline{w_5}}) ~\underline{\underline{w_6}}~ \end{matrix}\) \(\begin{matrix} {\langle\underline{\underline{w_1}}\rangle}_W \\[4pt] {\langle\underline{\underline{w_2}}\rangle}_W \\[4pt] {\langle\underline{\underline{w_3}}\rangle}_W \\[4pt] {\langle\underline{\underline{w_4}}\rangle}_W \\[4pt] {\langle\underline{\underline{w_5}}\rangle}_W \\[4pt] {\langle\underline{\underline{w_6}}\rangle}_W \end{matrix}\) \(\text{Table 55.1} ~~ \operatorname{LIR}_1 (L_\text{A}) : \text{Literal Representation of} ~ L_\text{A}\!\) \(\text{Object}\!\) \(\text{Sign}\!\) \(\text{Interpretant}\!\) \(\begin{matrix} {\langle\underline{\underline{\text{A}}}\rangle}_W \\[4pt] {\langle\underline{\underline{\text{A}}}\rangle}_W \\[4pt] {\langle\underline{\underline{\text{A}}}\rangle}_W \\[4pt] {\langle\underline{\underline{\text{A}}}\rangle}_W \end{matrix}\) \(\begin{matrix} {\langle\underline{\underline{\text{a}}}\rangle}_W \\[4pt] {\langle\underline{\underline{\text{a}}}\rangle}_W \\[4pt] {\langle\underline{\underline{\text{i}}}\rangle}_W \\[4pt] {\langle\underline{\underline{\text{i}}}\rangle}_W \end{matrix}\) \(\begin{matrix} {\langle\underline{\underline{\text{a}}}\rangle}_W \\[4pt] {\langle\underline{\underline{\text{i}}}\rangle}_W \\[4pt] {\langle\underline{\underline{\text{a}}}\rangle}_W \\[4pt] {\langle\underline{\underline{\text{i}}}\rangle}_W \end{matrix}\) \(\begin{matrix} {\langle\underline{\underline{\text{B}}}\rangle}_W \\[4pt] {\langle\underline{\underline{\text{B}}}\rangle}_W \\[4pt] {\langle\underline{\underline{\text{B}}}\rangle}_W \\[4pt] {\langle\underline{\underline{\text{B}}}\rangle}_W \end{matrix}\) \(\begin{matrix} {\langle\underline{\underline{\text{b}}}\rangle}_W \\[4pt] {\langle\underline{\underline{\text{b}}}\rangle}_W \\[4pt] {\langle\underline{\underline{\text{u}}}\rangle}_W \\[4pt] {\langle\underline{\underline{\text{u}}}\rangle}_W \end{matrix}\) \(\begin{matrix} {\langle\underline{\underline{\text{b}}}\rangle}_W \\[4pt] {\langle\underline{\underline{\text{u}}}\rangle}_W \\[4pt] {\langle\underline{\underline{\text{b}}}\rangle}_W \\[4pt] {\langle\underline{\underline{\text{u}}}\rangle}_W \end{matrix}\) \(\text{Table 55.2} ~~ \operatorname{LIR}_1 (\operatorname{Den}(L_\text{A})) : \text{Denotative Component of} ~ L_\text{A}\!\) \(\text{Object}\!\) \(\text{Sign}\!\) \(\text{Transition}\!\) \(\begin{matrix} {\langle\underline{\underline{\text{A}}}\rangle}_W \\[4pt] {\langle\underline{\underline{\text{A}}}\rangle}_W \end{matrix}\) \(\begin{matrix} {\langle\underline{\underline{\text{a}}}\rangle}_W \\[4pt] {\langle\underline{\underline{\text{i}}}\rangle}_W \end{matrix}\) \(\begin{matrix} ({\langle\underline{\underline{\text{a}}}\rangle}_W, {\langle\underline{\underline{\text{A}}}\rangle}_W) \\[4pt] ({\langle\underline{\underline{\text{i}}}\rangle}_W, {\langle\underline{\underline{\text{A}}}\rangle}_W) \end{matrix}\) \(\begin{matrix} {\langle\underline{\underline{\text{B}}}\rangle}_W \\[4pt] {\langle\underline{\underline{\text{B}}}\rangle}_W \end{matrix}\) \(\begin{matrix} {\langle\underline{\underline{\text{b}}}\rangle}_W \\[4pt] {\langle\underline{\underline{\text{u}}}\rangle}_W \end{matrix}\) \(\begin{matrix} ({\langle\underline{\underline{\text{b}}}\rangle}_W, {\langle\underline{\underline{\text{B}}}\rangle}_W) \\[4pt] ({\langle\underline{\underline{\text{u}}}\rangle}_W, {\langle\underline{\underline{\text{B}}}\rangle}_W) \end{matrix}\) \(\text{Table 55.3} ~~ \operatorname{LIR}_1 (\operatorname{Con}(L_\text{A})) : \text{Connotative Component of} ~ L_\text{A}\!\) \(\text{Sign}\!\) \(\text{Interpretant}\!\) \(\text{Transition}\!\) \(\begin{matrix} {\langle\underline{\underline{\text{a}}}\rangle}_W \\[4pt] {\langle\underline{\underline{\text{a}}}\rangle}_W \\[4pt] {\langle\underline{\underline{\text{i}}}\rangle}_W \\[4pt] {\langle\underline{\underline{\text{i}}}\rangle}_W \end{matrix}\) \(\begin{matrix} {\langle\underline{\underline{\text{a}}}\rangle}_W \\[4pt] {\langle\underline{\underline{\text{i}}}\rangle}_W \\[4pt] {\langle\underline{\underline{\text{a}}}\rangle}_W \\[4pt] {\langle\underline{\underline{\text{i}}}\rangle}_W \end{matrix}\) \(\begin{matrix} 0_{\operatorname{d}W} \\[4pt] {\langle \operatorname{d}\underline{\underline{\text{a}}} ~ \operatorname{d}\underline{\underline{\text{i}}} \rangle}_{\operatorname{d}W} \\[4pt] {\langle \operatorname{d}\underline{\underline{\text{a}}} ~ \operatorname{d}\underline{\underline{\text{i}}} \rangle}_{\operatorname{d}W} \\[4pt] 0_{\operatorname{d}W} \end{matrix}\) \(\begin{matrix} {\langle\underline{\underline{\text{b}}}\rangle}_W \\[4pt] {\langle\underline{\underline{\text{b}}}\rangle}_W \\[4pt] {\langle\underline{\underline{\text{u}}}\rangle}_W \\[4pt] {\langle\underline{\underline{\text{u}}}\rangle}_W \end{matrix}\) \(\begin{matrix} {\langle\underline{\underline{\text{b}}}\rangle}_W \\[4pt] {\langle\underline{\underline{\text{u}}}\rangle}_W \\[4pt] {\langle\underline{\underline{\text{b}}}\rangle}_W \\[4pt] {\langle\underline{\underline{\text{u}}}\rangle}_W \end{matrix}\) \(\begin{matrix} 0_{\operatorname{d}W} \\[4pt] {\langle \operatorname{d}\underline{\underline{\text{b}}} ~ \operatorname{d}\underline{\underline{\text{u}}} \rangle}_{\operatorname{d}W} \\[4pt] {\langle \operatorname{d}\underline{\underline{\text{b}}} ~ \operatorname{d}\underline{\underline{\text{u}}} \rangle}_{\operatorname{d}W} \\[4pt] 0_{\operatorname{d}W} \end{matrix}\) \(\text{Table 56.1} ~~ \operatorname{LIR}_1 (L_\text{B}) : \text{Literal Representation of} ~ L_\text{B}\!\) \(\text{Object}\!\) \(\text{Sign}\!\) \(\text{Interpretant}\!\) \(\begin{matrix} {\langle\underline{\underline{\text{A}}}\rangle}_W \\[4pt] {\langle\underline{\underline{\text{A}}}\rangle}_W \\[4pt] {\langle\underline{\underline{\text{A}}}\rangle}_W \\[4pt] {\langle\underline{\underline{\text{A}}}\rangle}_W \end{matrix}\) \(\begin{matrix} {\langle\underline{\underline{\text{a}}}\rangle}_W \\[4pt] {\langle\underline{\underline{\text{a}}}\rangle}_W \\[4pt] {\langle\underline{\underline{\text{u}}}\rangle}_W \\[4pt] {\langle\underline{\underline{\text{u}}}\rangle}_W \end{matrix}\) \(\begin{matrix} {\langle\underline{\underline{\text{a}}}\rangle}_W \\[4pt] {\langle\underline{\underline{\text{u}}}\rangle}_W \\[4pt] {\langle\underline{\underline{\text{a}}}\rangle}_W \\[4pt] {\langle\underline{\underline{\text{u}}}\rangle}_W \end{matrix}\) \(\begin{matrix} {\langle\underline{\underline{\text{B}}}\rangle}_W \\[4pt] {\langle\underline{\underline{\text{B}}}\rangle}_W \\[4pt] {\langle\underline{\underline{\text{B}}}\rangle}_W \\[4pt] {\langle\underline{\underline{\text{B}}}\rangle}_W \end{matrix}\) \(\begin{matrix} {\langle\underline{\underline{\text{b}}}\rangle}_W \\[4pt] {\langle\underline{\underline{\text{b}}}\rangle}_W \\[4pt] {\langle\underline{\underline{\text{i}}}\rangle}_W \\[4pt] {\langle\underline{\underline{\text{i}}}\rangle}_W \end{matrix}\) \(\begin{matrix} {\langle\underline{\underline{\text{b}}}\rangle}_W \\[4pt] {\langle\underline{\underline{\text{i}}}\rangle}_W \\[4pt] {\langle\underline{\underline{\text{b}}}\rangle}_W \\[4pt] {\langle\underline{\underline{\text{i}}}\rangle}_W \end{matrix}\) \(\text{Table 56.2} ~~ \operatorname{LIR}_1 (\operatorname{Den}(L_\text{B})) : \text{Denotative Component of} ~ L_\text{B}\!\) \(\text{Object}\!\) \(\text{Sign}\!\) \(\text{Transition}\!\) \(\begin{matrix} {\langle\underline{\underline{\text{A}}}\rangle}_W \\[4pt] {\langle\underline{\underline{\text{A}}}\rangle}_W \end{matrix}\) \(\begin{matrix} {\langle\underline{\underline{\text{a}}}\rangle}_W \\[4pt] {\langle\underline{\underline{\text{u}}}\rangle}_W \end{matrix}\) \(\begin{matrix} ({\langle\underline{\underline{\text{a}}}\rangle}_W, {\langle\underline{\underline{\text{A}}}\rangle}_W) \\[4pt] ({\langle\underline{\underline{\text{u}}}\rangle}_W, {\langle\underline{\underline{\text{A}}}\rangle}_W) \end{matrix}\) \(\begin{matrix} {\langle\underline{\underline{\text{B}}}\rangle}_W \\[4pt] {\langle\underline{\underline{\text{B}}}\rangle}_W \end{matrix}\) \(\begin{matrix} {\langle\underline{\underline{\text{b}}}\rangle}_W \\[4pt] {\langle\underline{\underline{\text{i}}}\rangle}_W \end{matrix}\) \(\begin{matrix} ({\langle\underline{\underline{\text{b}}}\rangle}_W, {\langle\underline{\underline{\text{B}}}\rangle}_W) \\[4pt] ({\langle\underline{\underline{\text{i}}}\rangle}_W, {\langle\underline{\underline{\text{B}}}\rangle}_W) \end{matrix}\) \(\text{Table 56.3} ~~ \operatorname{LIR}_1 (\operatorname{Con}(L_\text{B})) : \text{Connotative Component of} ~ L_\text{B}\!\) \(\text{Sign}\!\) \(\text{Interpretant}\!\) \(\text{Transition}\!\) \(\begin{matrix} {\langle\underline{\underline{\text{a}}}\rangle}_W \\[4pt] {\langle\underline{\underline{\text{a}}}\rangle}_W \\[4pt] {\langle\underline{\underline{\text{u}}}\rangle}_W \\[4pt] {\langle\underline{\underline{\text{u}}}\rangle}_W \end{matrix}\) \(\begin{matrix} {\langle\underline{\underline{\text{a}}}\rangle}_W \\[4pt] {\langle\underline{\underline{\text{u}}}\rangle}_W \\[4pt] {\langle\underline{\underline{\text{a}}}\rangle}_W \\[4pt] {\langle\underline{\underline{\text{u}}}\rangle}_W \end{matrix}\) \(\begin{matrix} 0_{\operatorname{d}W} \\[4pt] {\langle \operatorname{d}\underline{\underline{\text{a}}} ~ \operatorname{d}\underline{\underline{\text{u}}} \rangle}_{\operatorname{d}W} \\[4pt] {\langle \operatorname{d}\underline{\underline{\text{a}}} ~ \operatorname{d}\underline{\underline{\text{u}}} \rangle}_{\operatorname{d}W} \\[4pt] 0_{\operatorname{d}W} \end{matrix}\) \(\begin{matrix} {\langle\underline{\underline{\text{b}}}\rangle}_W \\[4pt] {\langle\underline{\underline{\text{b}}}\rangle}_W \\[4pt] {\langle\underline{\underline{\text{i}}}\rangle}_W \\[4pt] {\langle\underline{\underline{\text{i}}}\rangle}_W \end{matrix}\) \(\begin{matrix} {\langle\underline{\underline{\text{b}}}\rangle}_W \\[4pt] {\langle\underline{\underline{\text{i}}}\rangle}_W \\[4pt] {\langle\underline{\underline{\text{b}}}\rangle}_W \\[4pt] {\langle\underline{\underline{\text{i}}}\rangle}_W \end{matrix}\) \(\begin{matrix} 0_{\operatorname{d}W} \\[4pt] {\langle \operatorname{d}\underline{\underline{\text{b}}} ~ \operatorname{d}\underline{\underline{\text{i}}} \rangle}_{\operatorname{d}W} \\[4pt] {\langle \operatorname{d}\underline{\underline{\text{b}}} ~ \operatorname{d}\underline{\underline{\text{i}}} \rangle}_{\operatorname{d}W} \\[4pt] 0_{\operatorname{d}W} \end{matrix}\) \(\text{Table 57.1} ~~ \text{Mnemonic Lateral Codes for Interpreters A and B}\!\) \(\text{Element}\!\) \(\text{Vector}\!\) \(\text{Conjunct Term}\!\) \(\text{Code}\!\) \(\begin{matrix} \text{A} \\[4pt] \text{B} \\[4pt] {}^{\backprime\backprime} \text{A} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \text{B} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \end{matrix}\) \(\begin{matrix} {10}_X \\[4pt] {01}_X \\[4pt] {1000}_Y \\[4pt] {0100}_Y \\[4pt] {0010}_Y \\[4pt] {0001}_Y \end{matrix}\) \(\begin{matrix} ~\underline{\underline{A}}~ (\underline{\underline{B}}) \\[4pt] (\underline{\underline{A}}) ~\underline{\underline{B}}~ \\[4pt] ~\underline{\underline{a}}~ (\underline{\underline{b}}) (\underline{\underline{i}}) (\underline{\underline{u}}) \\[4pt] (\underline{\underline{a}}) ~\underline{\underline{b}}~ (\underline{\underline{i}}) (\underline{\underline{u}}) \\[4pt] (\underline{\underline{a}}) (\underline{\underline{b}}) ~\underline{\underline{i}}~ (\underline{\underline{u}}) \\[4pt] (\underline{\underline{a}}) (\underline{\underline{b}}) (\underline{\underline{i}}) ~\underline{\underline{u}}~ \end{matrix}\) \(\begin{matrix} {\langle\underline{\underline{A}}\rangle}_X \\[4pt] {\langle\underline{\underline{B}}\rangle}_X \\[4pt] {\langle\underline{\underline{a}}\rangle}_Y \\[4pt] {\langle\underline{\underline{b}}\rangle}_Y \\[4pt] {\langle\underline{\underline{i}}\rangle}_Y \\[4pt] {\langle\underline{\underline{u}}\rangle}_Y \end{matrix}\) \(\text{Table 57.2} ~~ \text{Pragmatic Lateral Codes for Interpreters A and B}\!\) \(\text{Element}\!\) \(\text{Vector}\!\) \(\text{Conjunct Term}\!\) \(\text{Code}\!\) \(\begin{matrix} \text{A} \\[4pt] \text{B} \\[4pt] {}^{\backprime\backprime} \text{A} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \text{B} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \end{matrix}\) \(\begin{matrix} {10}_X \\[4pt] {01}_X \\[4pt] {1000}_Y \\[4pt] {0100}_Y \\[4pt] {0010}_Y \\[4pt] {0001}_Y \end{matrix}\) \(\begin{matrix} ~\underline{\underline{o_1}}~ (\underline{\underline{o_2}}) \\[4pt] (\underline{\underline{o_1}}) ~\underline{\underline{o_2}}~ \\[4pt] ~\underline{\underline{s_1}}~ (\underline{\underline{s_2}}) (\underline{\underline{s_3}}) (\underline{\underline{s_4}}) \\[4pt] (\underline{\underline{s_1}}) ~\underline{\underline{s_2}}~ (\underline{\underline{s_3}}) (\underline{\underline{s_4}}) \\[4pt] (\underline{\underline{s_1}}) (\underline{\underline{s_2}}) ~\underline{\underline{s_3}}~ (\underline{\underline{s_4}}) \\[4pt] (\underline{\underline{s_1}}) (\underline{\underline{s_2}}) (\underline{\underline{s_3}}) ~\underline{\underline{s_4}}~ \end{matrix}\) \(\begin{matrix} {\langle\underline{\underline{o_1}}\rangle}_X \\[4pt] {\langle\underline{\underline{o_2}}\rangle}_X \\[4pt] {\langle\underline{\underline{s_1}}\rangle}_Y \\[4pt] {\langle\underline{\underline{s_2}}\rangle}_Y \\[4pt] {\langle\underline{\underline{s_3}}\rangle}_Y \\[4pt] {\langle\underline{\underline{s_4}}\rangle}_Y \end{matrix}\) \(\text{Table 57.3} ~~ \text{Abstract Lateral Codes for Interpreters A and B}\!\) \(\text{Element}\!\) \(\text{Vector}\!\) \(\text{Conjunct Term}\!\) \(\text{Code}\!\) \(\begin{matrix} \text{A} \\[4pt] \text{B} \\[4pt] {}^{\backprime\backprime} \text{A} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \text{B} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \end{matrix}\) \(\begin{matrix} {10}_X \\[4pt] {01}_X \\[4pt] {1000}_Y \\[4pt] {0100}_Y \\[4pt] {0010}_Y \\[4pt] {0001}_Y \end{matrix}\) \(\begin{matrix} ~\underline{\underline{x_1}}~ (\underline{\underline{x_2}}) \\[4pt] (\underline{\underline{x_1}}) ~\underline{\underline{x_2}}~ \\[4pt] ~\underline{\underline{y_1}}~ (\underline{\underline{y_2}}) (\underline{\underline{y_3}}) (\underline{\underline{y_4}}) \\[4pt] (\underline{\underline{y_1}}) ~\underline{\underline{y_2}}~ (\underline{\underline{y_3}}) (\underline{\underline{y_4}}) \\[4pt] (\underline{\underline{y_1}}) (\underline{\underline{y_2}}) ~\underline{\underline{y_3}}~ (\underline{\underline{y_4}}) \\[4pt] (\underline{\underline{y_1}}) (\underline{\underline{y_2}}) (\underline{\underline{y_3}}) ~\underline{\underline{y_4}}~ \end{matrix}\) \(\begin{matrix} {\langle\underline{\underline{x_1}}\rangle}_X \\[4pt] {\langle\underline{\underline{x_2}}\rangle}_X \\[4pt] {\langle\underline{\underline{y_1}}\rangle}_Y \\[4pt] {\langle\underline{\underline{y_2}}\rangle}_Y \\[4pt] {\langle\underline{\underline{y_3}}\rangle}_Y \\[4pt] {\langle\underline{\underline{y_4}}\rangle}_Y \end{matrix}\) \(\text{Table 58.1} ~~ \operatorname{LIR}_2 (L_\text{A}) : \text{Lateral Representation of} ~ L_\text{A}\!\) \(\text{Object}\!\) \(\text{Sign}\!\) \(\text{Interpretant}\!\) \(\begin{matrix} ~\underline{\underline{\text{A}}}~ (\underline{\underline{\text{B}}}) \\[4pt] ~\underline{\underline{\text{A}}}~ (\underline{\underline{\text{B}}}) \\[4pt] ~\underline{\underline{\text{A}}}~ (\underline{\underline{\text{B}}}) \\[4pt] ~\underline{\underline{\text{A}}}~ (\underline{\underline{\text{B}}}) \end{matrix}\) \(\begin{matrix} ~\underline{\underline{\text{a}}}~ (\underline{\underline{\text{b}}}) (\underline{\underline{\text{i}}}) (\underline{\underline{\text{u}}}) \\[4pt] ~\underline{\underline{\text{a}}}~ (\underline{\underline{\text{b}}}) (\underline{\underline{\text{i}}}) (\underline{\underline{\text{u}}}) \\[4pt] (\underline{\underline{\text{a}}}) (\underline{\underline{\text{b}}}) ~\underline{\underline{\text{i}}}~ (\underline{\underline{\text{u}}}) \\[4pt] (\underline{\underline{\text{a}}}) (\underline{\underline{\text{b}}}) ~\underline{\underline{\text{i}}}~ (\underline{\underline{\text{u}}}) \end{matrix}\) \(\begin{matrix} ~\underline{\underline{\text{a}}}~ (\underline{\underline{\text{b}}}) (\underline{\underline{\text{i}}}) (\underline{\underline{\text{u}}}) \\[4pt] (\underline{\underline{\text{a}}}) (\underline{\underline{\text{b}}}) ~\underline{\underline{\text{i}}}~ (\underline{\underline{\text{u}}}) \\[4pt] ~\underline{\underline{\text{a}}}~ (\underline{\underline{\text{b}}}) (\underline{\underline{\text{i}}}) (\underline{\underline{\text{u}}}) \\[4pt] (\underline{\underline{\text{a}}}) (\underline{\underline{\text{b}}}) ~\underline{\underline{\text{i}}}~ (\underline{\underline{\text{u}}}) \end{matrix}\) \(\begin{matrix} (\underline{\underline{\text{A}}}) ~\underline{\underline{\text{B}}}~ \\[4pt] (\underline{\underline{\text{A}}}) ~\underline{\underline{\text{B}}}~ \\[4pt] (\underline{\underline{\text{A}}}) ~\underline{\underline{\text{B}}}~ \\[4pt] (\underline{\underline{\text{A}}}) ~\underline{\underline{\text{B}}}~ \end{matrix}\) \(\begin{matrix} (\underline{\underline{\text{a}}}) ~\underline{\underline{\text{b}}}~ (\underline{\underline{\text{i}}}) (\underline{\underline{\text{u}}}) \\[4pt] (\underline{\underline{\text{a}}}) ~\underline{\underline{\text{b}}}~ (\underline{\underline{\text{i}}}) (\underline{\underline{\text{u}}}) \\[4pt] (\underline{\underline{\text{a}}}) (\underline{\underline{\text{b}}}) (\underline{\underline{\text{i}}}) ~\underline{\underline{\text{u}}}~ \\[4pt] (\underline{\underline{\text{a}}}) (\underline{\underline{\text{b}}}) (\underline{\underline{\text{i}}}) ~\underline{\underline{\text{u}}}~ \end{matrix}\) \(\begin{matrix} (\underline{\underline{\text{a}}}) ~\underline{\underline{\text{b}}}~ (\underline{\underline{\text{i}}}) (\underline{\underline{\text{u}}}) \\[4pt] (\underline{\underline{\text{a}}}) (\underline{\underline{\text{b}}}) (\underline{\underline{\text{i}}}) ~\underline{\underline{\text{u}}}~ \\[4pt] (\underline{\underline{\text{a}}}) ~\underline{\underline{\text{b}}}~ (\underline{\underline{\text{i}}}) (\underline{\underline{\text{u}}}) \\[4pt] (\underline{\underline{\text{a}}}) (\underline{\underline{\text{b}}}) (\underline{\underline{\text{i}}}) ~\underline{\underline{\text{u}}}~ \end{matrix}\) \(\text{Table 58.2} ~~ \operatorname{LIR}_2 (\operatorname{Den}(L_\text{A})) : \text{Denotative Component of} ~ L_\text{A}\!\) \(\text{Object}\!\) \(\text{Sign}\!\) \(\text{Transition}\!\) \(\begin{matrix} ~\underline{\underline{\text{A}}}~ (\underline{\underline{\text{B}}}) \\[4pt] ~\underline{\underline{\text{A}}}~ (\underline{\underline{\text{B}}}) \end{matrix}\) \(\begin{matrix} ~\underline{\underline{\text{a}}}~ (\underline{\underline{\text{b}}}) (\underline{\underline{\text{i}}}) (\underline{\underline{\text{u}}}) \\[4pt] (\underline{\underline{\text{a}}}) (\underline{\underline{\text{b}}}) ~\underline{\underline{\text{i}}}~ (\underline{\underline{\text{u}}}) \end{matrix}\) \(\begin{matrix} ({\langle\underline{\underline{\text{a}}}\rangle}_Y, {\langle\underline{\underline{\text{A}}}\rangle}_X) \\[4pt] ({\langle\underline{\underline{\text{i}}}\rangle}_Y, {\langle\underline{\underline{\text{A}}}\rangle}_X) \end{matrix}\) \(\begin{matrix} (\underline{\underline{\text{A}}}) ~\underline{\underline{\text{B}}}~ \\[4pt] (\underline{\underline{\text{A}}}) ~\underline{\underline{\text{B}}}~ \end{matrix}\) \(\begin{matrix} (\underline{\underline{\text{a}}}) ~\underline{\underline{\text{b}}}~ (\underline{\underline{\text{i}}}) (\underline{\underline{\text{u}}}) \\[4pt] (\underline{\underline{\text{a}}}) (\underline{\underline{\text{b}}}) (\underline{\underline{\text{i}}}) ~\underline{\underline{\text{u}}}~ \end{matrix}\) \(\begin{matrix} ({\langle\underline{\underline{\text{b}}}\rangle}_Y, {\langle\underline{\underline{\text{B}}}\rangle}_X) \\[4pt] ({\langle\underline{\underline{\text{u}}}\rangle}_Y, {\langle\underline{\underline{\text{B}}}\rangle}_X) \end{matrix}\) \(\text{Table 58.3} ~~ \operatorname{LIR}_2 (\operatorname{Con}(L_\text{A})) : \text{Connotative Component of} ~ L_\text{A}\!\) \(\text{Sign}\!\) \(\text{Interpretant}\!\) \(\text{Transition}\!\) \(\begin{matrix} ~\underline{\underline{\text{a}}}~ (\underline{\underline{\text{b}}}) (\underline{\underline{\text{i}}}) (\underline{\underline{\text{u}}}) \\[4pt] ~\underline{\underline{\text{a}}}~ (\underline{\underline{\text{b}}}) (\underline{\underline{\text{i}}}) (\underline{\underline{\text{u}}}) \\[4pt] (\underline{\underline{\text{a}}}) (\underline{\underline{\text{b}}}) ~\underline{\underline{\text{i}}}~ (\underline{\underline{\text{u}}}) \\[4pt] (\underline{\underline{\text{a}}}) (\underline{\underline{\text{b}}}) ~\underline{\underline{\text{i}}}~ (\underline{\underline{\text{u}}}) \end{matrix}\) \(\begin{matrix} ~\underline{\underline{\text{a}}}~ (\underline{\underline{\text{b}}}) (\underline{\underline{\text{i}}}) (\underline{\underline{\text{u}}}) \\[4pt] (\underline{\underline{\text{a}}}) (\underline{\underline{\text{b}}}) ~\underline{\underline{\text{i}}}~ (\underline{\underline{\text{u}}}) \\[4pt] ~\underline{\underline{\text{a}}}~ (\underline{\underline{\text{b}}}) (\underline{\underline{\text{i}}}) (\underline{\underline{\text{u}}}) \\[4pt] (\underline{\underline{\text{a}}}) (\underline{\underline{\text{b}}}) ~\underline{\underline{\text{i}}}~ (\underline{\underline{\text{u}}}) \end{matrix}\) \(\begin{matrix} (\underline{\underline{\text{da}}}) (\underline{\underline{\text{db}}}) (\underline{\underline{\text{di}}}) (\underline{\underline{\text{du}}}) \\[4pt] ~\underline{\underline{\text{da}}}~ (\underline{\underline{\text{db}}}) ~\underline{\underline{\text{di}}}~ (\underline{\underline{\text{du}}}) \\[4pt] ~\underline{\underline{\text{da}}}~ (\underline{\underline{\text{db}}}) ~\underline{\underline{\text{di}}}~ (\underline{\underline{\text{du}}}) \\[4pt] (\underline{\underline{\text{da}}}) (\underline{\underline{\text{db}}}) (\underline{\underline{\text{di}}}) (\underline{\underline{\text{du}}}) \end{matrix}\) \(\begin{matrix} (\underline{\underline{\text{a}}}) ~\underline{\underline{\text{b}}}~ (\underline{\underline{\text{i}}}) (\underline{\underline{\text{u}}}) \\[4pt] (\underline{\underline{\text{a}}}) ~\underline{\underline{\text{b}}}~ (\underline{\underline{\text{i}}}) (\underline{\underline{\text{u}}}) \\[4pt] (\underline{\underline{\text{a}}}) (\underline{\underline{\text{b}}}) (\underline{\underline{\text{i}}}) ~\underline{\underline{\text{u}}}~ \\[4pt] (\underline{\underline{\text{a}}}) (\underline{\underline{\text{b}}}) (\underline{\underline{\text{i}}}) ~\underline{\underline{\text{u}}}~ \end{matrix}\) \(\begin{matrix} (\underline{\underline{\text{a}}}) ~\underline{\underline{\text{b}}}~ (\underline{\underline{\text{i}}}) (\underline{\underline{\text{u}}}) \\[4pt] (\underline{\underline{\text{a}}}) (\underline{\underline{\text{b}}}) (\underline{\underline{\text{i}}}) ~\underline{\underline{\text{u}}}~ \\[4pt] (\underline{\underline{\text{a}}}) ~\underline{\underline{\text{b}}}~ (\underline{\underline{\text{i}}}) (\underline{\underline{\text{u}}}) \\[4pt] (\underline{\underline{\text{a}}}) (\underline{\underline{\text{b}}}) (\underline{\underline{\text{i}}}) ~\underline{\underline{\text{u}}}~ \end{matrix}\) \(\begin{matrix} (\underline{\underline{\text{da}}}) (\underline{\underline{\text{db}}}) (\underline{\underline{\text{di}}}) (\underline{\underline{\text{du}}}) \\[4pt] (\underline{\underline{\text{da}}}) ~\underline{\underline{\text{db}}}~ (\underline{\underline{\text{di}}}) ~\underline{\underline{\text{du}}}~ \\[4pt] (\underline{\underline{\text{da}}}) ~\underline{\underline{\text{db}}}~ (\underline{\underline{\text{di}}}) ~\underline{\underline{\text{du}}}~ \\[4pt] (\underline{\underline{\text{da}}}) (\underline{\underline{\text{db}}}) (\underline{\underline{\text{di}}}) (\underline{\underline{\text{du}}}) \end{matrix}\) \(\text{Table 59.1} ~~ \operatorname{LIR}_2 (L_\text{B}) : \text{Lateral Representation of} ~ L_\text{B}\!\) \(\text{Object}\!\) \(\text{Sign}\!\) \(\text{Interpretant}\!\) \(\begin{matrix} ~\underline{\underline{\text{A}}}~ (\underline{\underline{\text{B}}}) \\[4pt] ~\underline{\underline{\text{A}}}~ (\underline{\underline{\text{B}}}) \\[4pt] ~\underline{\underline{\text{A}}}~ (\underline{\underline{\text{B}}}) \\[4pt] ~\underline{\underline{\text{A}}}~ (\underline{\underline{\text{B}}}) \end{matrix}\) \(\begin{matrix} ~\underline{\underline{\text{a}}}~ (\underline{\underline{\text{b}}}) (\underline{\underline{\text{i}}}) (\underline{\underline{\text{u}}}) \\[4pt] ~\underline{\underline{\text{a}}}~ (\underline{\underline{\text{b}}}) (\underline{\underline{\text{i}}}) (\underline{\underline{\text{u}}}) \\[4pt] (\underline{\underline{\text{a}}}) (\underline{\underline{\text{b}}}) (\underline{\underline{\text{i}}}) ~\underline{\underline{\text{u}}}~ \\[4pt] (\underline{\underline{\text{a}}}) (\underline{\underline{\text{b}}}) (\underline{\underline{\text{i}}}) ~\underline{\underline{\text{u}}}~ \end{matrix}\) \(\begin{matrix} ~\underline{\underline{\text{a}}}~ (\underline{\underline{\text{b}}}) (\underline{\underline{\text{i}}}) (\underline{\underline{\text{u}}}) \\[4pt] (\underline{\underline{\text{a}}}) (\underline{\underline{\text{b}}}) (\underline{\underline{\text{i}}}) ~\underline{\underline{\text{u}}}~ \\[4pt] ~\underline{\underline{\text{a}}}~ (\underline{\underline{\text{b}}}) (\underline{\underline{\text{i}}}) (\underline{\underline{\text{u}}}) \\[4pt] (\underline{\underline{\text{a}}}) (\underline{\underline{\text{b}}}) (\underline{\underline{\text{i}}}) ~\underline{\underline{\text{u}}}~ \end{matrix}\) \(\begin{matrix} (\underline{\underline{\text{A}}}) ~\underline{\underline{\text{B}}}~ \\[4pt] (\underline{\underline{\text{A}}}) ~\underline{\underline{\text{B}}}~ \\[4pt] (\underline{\underline{\text{A}}}) ~\underline{\underline{\text{B}}}~ \\[4pt] (\underline{\underline{\text{A}}}) ~\underline{\underline{\text{B}}}~ \end{matrix}\) \(\begin{matrix} (\underline{\underline{\text{a}}}) ~\underline{\underline{\text{b}}}~ (\underline{\underline{\text{i}}}) (\underline{\underline{\text{u}}}) \\[4pt] (\underline{\underline{\text{a}}}) ~\underline{\underline{\text{b}}}~ (\underline{\underline{\text{i}}}) (\underline{\underline{\text{u}}}) \\[4pt] (\underline{\underline{\text{a}}}) (\underline{\underline{\text{b}}}) ~\underline{\underline{\text{i}}}~ (\underline{\underline{\text{u}}}) \\[4pt] (\underline{\underline{\text{a}}}) (\underline{\underline{\text{b}}}) ~\underline{\underline{\text{i}}}~ (\underline{\underline{\text{u}}}) \end{matrix}\) \(\begin{matrix} (\underline{\underline{\text{a}}}) ~\underline{\underline{\text{b}}}~ (\underline{\underline{\text{i}}}) (\underline{\underline{\text{u}}}) \\[4pt] (\underline{\underline{\text{a}}}) (\underline{\underline{\text{b}}}) ~\underline{\underline{\text{i}}}~ (\underline{\underline{\text{u}}}) \\[4pt] (\underline{\underline{\text{a}}}) ~\underline{\underline{\text{b}}}~ (\underline{\underline{\text{i}}}) (\underline{\underline{\text{u}}}) \\[4pt] (\underline{\underline{\text{a}}}) (\underline{\underline{\text{b}}}) ~\underline{\underline{\text{i}}}~ (\underline{\underline{\text{u}}}) \end{matrix}\) \(\text{Table 59.2} ~~ \operatorname{LIR}_2 (\operatorname{Den}(L_\text{B})) : \text{Denotative Component of} ~ L_\text{B}\!\) \(\text{Object}\!\) \(\text{Sign}\!\) \(\text{Transition}\!\) \(\begin{matrix} ~\underline{\underline{\text{A}}}~ (\underline{\underline{\text{B}}}) \\[4pt] ~\underline{\underline{\text{A}}}~ (\underline{\underline{\text{B}}}) \end{matrix}\) \(\begin{matrix} ~\underline{\underline{\text{a}}}~ (\underline{\underline{\text{b}}}) (\underline{\underline{\text{i}}}) (\underline{\underline{\text{u}}}) \\[4pt] (\underline{\underline{\text{a}}}) (\underline{\underline{\text{b}}}) (\underline{\underline{\text{i}}}) ~\underline{\underline{\text{u}}}~ \end{matrix}\) \(\begin{matrix} ({\langle\underline{\underline{\text{a}}}\rangle}_Y, {\langle\underline{\underline{\text{A}}}\rangle}_X) \\[4pt] ({\langle\underline{\underline{\text{u}}}\rangle}_Y, {\langle\underline{\underline{\text{A}}}\rangle}_X) \end{matrix}\) \(\begin{matrix} (\underline{\underline{\text{A}}}) ~\underline{\underline{\text{B}}}~ \\[4pt] (\underline{\underline{\text{A}}}) ~\underline{\underline{\text{B}}}~ \end{matrix}\) \(\begin{matrix} (\underline{\underline{\text{a}}}) ~\underline{\underline{\text{b}}}~ (\underline{\underline{\text{i}}}) (\underline{\underline{\text{u}}}) \\[4pt] (\underline{\underline{\text{a}}}) (\underline{\underline{\text{b}}}) ~\underline{\underline{\text{i}}}~ (\underline{\underline{\text{u}}}) \end{matrix}\) \(\begin{matrix} ({\langle\underline{\underline{\text{b}}}\rangle}_Y, {\langle\underline{\underline{\text{B}}}\rangle}_X) \\[4pt] ({\langle\underline{\underline{\text{i}}}\rangle}_Y, {\langle\underline{\underline{\text{B}}}\rangle}_X) \end{matrix}\) \(\text{Table 59.3} ~~ \operatorname{LIR}_2 (\operatorname{Con}(L_\text{B})) : \text{Connotative Component of} ~ L_\text{B}\!\) \(\text{Sign}\!\) \(\text{Interpretant}\!\) \(\text{Transition}\!\) \(\begin{matrix} ~\underline{\underline{\text{a}}}~ (\underline{\underline{\text{b}}}) (\underline{\underline{\text{i}}}) (\underline{\underline{\text{u}}}) \\[4pt] ~\underline{\underline{\text{a}}}~ (\underline{\underline{\text{b}}}) (\underline{\underline{\text{i}}}) (\underline{\underline{\text{u}}}) \\[4pt] (\underline{\underline{\text{a}}}) (\underline{\underline{\text{b}}}) (\underline{\underline{\text{i}}}) ~\underline{\underline{\text{u}}}~ \\[4pt] (\underline{\underline{\text{a}}}) (\underline{\underline{\text{b}}}) (\underline{\underline{\text{i}}}) ~\underline{\underline{\text{u}}}~ \end{matrix}\) \(\begin{matrix} ~\underline{\underline{\text{a}}}~ (\underline{\underline{\text{b}}}) (\underline{\underline{\text{i}}}) (\underline{\underline{\text{u}}}) \\[4pt] (\underline{\underline{\text{a}}}) (\underline{\underline{\text{b}}}) (\underline{\underline{\text{i}}}) ~\underline{\underline{\text{u}}}~ \\[4pt] ~\underline{\underline{\text{a}}}~ (\underline{\underline{\text{b}}}) (\underline{\underline{\text{i}}}) (\underline{\underline{\text{u}}}) \\[4pt] (\underline{\underline{\text{a}}}) (\underline{\underline{\text{b}}}) (\underline{\underline{\text{i}}}) ~\underline{\underline{\text{u}}}~ \end{matrix}\) \(\begin{matrix} (\underline{\underline{\text{da}}}) (\underline{\underline{\text{db}}}) (\underline{\underline{\text{di}}}) (\underline{\underline{\text{du}}}) \\[4pt] ~\underline{\underline{\text{da}}}~ (\underline{\underline{\text{db}}}) (\underline{\underline{\text{di}}}) ~\underline{\underline{\text{du}}}~ \\[4pt] ~\underline{\underline{\text{da}}}~ (\underline{\underline{\text{db}}}) (\underline{\underline{\text{di}}}) ~\underline{\underline{\text{du}}}~ \\[4pt] (\underline{\underline{\text{da}}}) (\underline{\underline{\text{db}}}) (\underline{\underline{\text{di}}}) (\underline{\underline{\text{du}}}) \end{matrix}\) \(\begin{matrix} (\underline{\underline{\text{a}}}) ~\underline{\underline{\text{b}}}~ (\underline{\underline{\text{i}}}) (\underline{\underline{\text{u}}}) \\[4pt] (\underline{\underline{\text{a}}}) ~\underline{\underline{\text{b}}}~ (\underline{\underline{\text{i}}}) (\underline{\underline{\text{u}}}) \\[4pt] (\underline{\underline{\text{a}}}) (\underline{\underline{\text{b}}}) ~\underline{\underline{\text{i}}}~ (\underline{\underline{\text{u}}}) \\[4pt] (\underline{\underline{\text{a}}}) (\underline{\underline{\text{b}}}) ~\underline{\underline{\text{i}}}~ (\underline{\underline{\text{u}}}) \end{matrix}\) \(\begin{matrix} (\underline{\underline{\text{a}}}) ~\underline{\underline{\text{b}}}~ (\underline{\underline{\text{i}}}) (\underline{\underline{\text{u}}}) \\[4pt] (\underline{\underline{\text{a}}}) (\underline{\underline{\text{b}}}) ~\underline{\underline{\text{i}}}~ (\underline{\underline{\text{u}}}) \\[4pt] (\underline{\underline{\text{a}}}) ~\underline{\underline{\text{b}}}~ (\underline{\underline{\text{i}}}) (\underline{\underline{\text{u}}}) \\[4pt] (\underline{\underline{\text{a}}}) (\underline{\underline{\text{b}}}) ~\underline{\underline{\text{i}}}~ (\underline{\underline{\text{u}}}) \end{matrix}\) \(\begin{matrix} (\underline{\underline{\text{da}}}) (\underline{\underline{\text{db}}}) (\underline{\underline{\text{di}}}) (\underline{\underline{\text{du}}}) \\[4pt] (\underline{\underline{\text{da}}}) ~\underline{\underline{\text{db}}}~ ~\underline{\underline{\text{di}}}~ (\underline{\underline{\text{du}}}) \\[4pt] (\underline{\underline{\text{da}}}) ~\underline{\underline{\text{db}}}~ ~\underline{\underline{\text{di}}}~ (\underline{\underline{\text{du}}}) \\[4pt] (\underline{\underline{\text{da}}}) (\underline{\underline{\text{db}}}) (\underline{\underline{\text{di}}}) (\underline{\underline{\text{du}}}) \end{matrix}\) \(\text{Table 60.1} ~~ \operatorname{LIR}_3 (L_\text{A}) : \text{Lateral Representation of} ~ L_\text{A}\!\) \(\text{Object}\!\) \(\text{Sign}\!\) \(\text{Interpretant}\!\) \(\begin{matrix} {\langle\underline{\underline{\text{A}}}\rangle}_X \\[4pt] {\langle\underline{\underline{\text{A}}}\rangle}_X \\[4pt] {\langle\underline{\underline{\text{A}}}\rangle}_X \\[4pt] {\langle\underline{\underline{\text{A}}}\rangle}_X \end{matrix}\) \(\begin{matrix} {\langle\underline{\underline{\text{a}}}\rangle}_Y \\[4pt] {\langle\underline{\underline{\text{a}}}\rangle}_Y \\[4pt] {\langle\underline{\underline{\text{i}}}\rangle}_Y \\[4pt] {\langle\underline{\underline{\text{i}}}\rangle}_Y \end{matrix}\) \(\begin{matrix} {\langle\underline{\underline{\text{a}}}\rangle}_Y \\[4pt] {\langle\underline{\underline{\text{i}}}\rangle}_Y \\[4pt] {\langle\underline{\underline{\text{a}}}\rangle}_Y \\[4pt] {\langle\underline{\underline{\text{i}}}\rangle}_Y \end{matrix}\) \(\begin{matrix} {\langle\underline{\underline{\text{B}}}\rangle}_X \\[4pt] {\langle\underline{\underline{\text{B}}}\rangle}_X \\[4pt] {\langle\underline{\underline{\text{B}}}\rangle}_X \\[4pt] {\langle\underline{\underline{\text{B}}}\rangle}_X \end{matrix}\) \(\begin{matrix} {\langle\underline{\underline{\text{b}}}\rangle}_Y \\[4pt] {\langle\underline{\underline{\text{b}}}\rangle}_Y \\[4pt] {\langle\underline{\underline{\text{u}}}\rangle}_Y \\[4pt] {\langle\underline{\underline{\text{u}}}\rangle}_Y \end{matrix}\) \(\begin{matrix} {\langle\underline{\underline{\text{b}}}\rangle}_Y \\[4pt] {\langle\underline{\underline{\text{u}}}\rangle}_Y \\[4pt] {\langle\underline{\underline{\text{b}}}\rangle}_Y \\[4pt] {\langle\underline{\underline{\text{u}}}\rangle}_Y \end{matrix}\) \(\text{Table 60.2} ~~ \operatorname{LIR}_3 (\operatorname{Den}(L_\text{A})) : \text{Denotative Component of} ~ L_\text{A}\!\) \(\text{Object}\!\) \(\text{Sign}\!\) \(\text{Transition}\!\) \(\begin{matrix} {\langle\underline{\underline{\text{A}}}\rangle}_X \\[4pt] {\langle\underline{\underline{\text{A}}}\rangle}_X \end{matrix}\) \(\begin{matrix} {\langle\underline{\underline{\text{a}}}\rangle}_Y \\[4pt] {\langle\underline{\underline{\text{i}}}\rangle}_Y \end{matrix}\) \(\begin{matrix} ({\langle\underline{\underline{\text{a}}}\rangle}_Y, {\langle\underline{\underline{\text{A}}}\rangle}_X) \\[4pt] ({\langle\underline{\underline{\text{i}}}\rangle}_Y, {\langle\underline{\underline{\text{A}}}\rangle}_X) \end{matrix}\) \(\begin{matrix} {\langle\underline{\underline{\text{B}}}\rangle}_X \\[4pt] {\langle\underline{\underline{\text{B}}}\rangle}_X \end{matrix}\) \(\begin{matrix} {\langle\underline{\underline{\text{b}}}\rangle}_Y \\[4pt] {\langle\underline{\underline{\text{u}}}\rangle}_Y \end{matrix}\) \(\begin{matrix} ({\langle\underline{\underline{\text{b}}}\rangle}_Y, {\langle\underline{\underline{\text{B}}}\rangle}_X) \\[4pt] ({\langle\underline{\underline{\text{u}}}\rangle}_Y, {\langle\underline{\underline{\text{B}}}\rangle}_X) \end{matrix}\) \(\text{Table 60.3} ~~ \operatorname{LIR}_3 (\operatorname{Con}(L_\text{A})) : \text{Connotative Component of} ~ L_\text{A}\!\) \(\text{Sign}\!\) \(\text{Interpretant}\!\) \(\text{Transition}\!\) \(\begin{matrix} {\langle\underline{\underline{\text{a}}}\rangle}_Y \\[4pt] {\langle\underline{\underline{\text{a}}}\rangle}_Y \\[4pt] {\langle\underline{\underline{\text{i}}}\rangle}_Y \\[4pt] {\langle\underline{\underline{\text{i}}}\rangle}_Y \end{matrix}\) \(\begin{matrix} {\langle\underline{\underline{\text{a}}}\rangle}_Y \\[4pt] {\langle\underline{\underline{\text{i}}}\rangle}_Y \\[4pt] {\langle\underline{\underline{\text{a}}}\rangle}_Y \\[4pt] {\langle\underline{\underline{\text{i}}}\rangle}_Y \end{matrix}\) \(\begin{matrix} 0_{\operatorname{d}Y} \\[4pt] {\langle \operatorname{d}\underline{\underline{\text{a}}} ~ \operatorname{d}\underline{\underline{\text{i}}} \rangle}_{\operatorname{d}Y} \\[4pt] {\langle \operatorname{d}\underline{\underline{\text{a}}} ~ \operatorname{d}\underline{\underline{\text{i}}} \rangle}_{\operatorname{d}Y} \\[4pt] 0_{\operatorname{d}Y} \end{matrix}\) \(\begin{matrix} {\langle\underline{\underline{\text{b}}}\rangle}_Y \\[4pt] {\langle\underline{\underline{\text{b}}}\rangle}_Y \\[4pt] {\langle\underline{\underline{\text{u}}}\rangle}_Y \\[4pt] {\langle\underline{\underline{\text{u}}}\rangle}_Y \end{matrix}\) \(\begin{matrix} {\langle\underline{\underline{\text{b}}}\rangle}_Y \\[4pt] {\langle\underline{\underline{\text{u}}}\rangle}_Y \\[4pt] {\langle\underline{\underline{\text{b}}}\rangle}_Y \\[4pt] {\langle\underline{\underline{\text{u}}}\rangle}_Y \end{matrix}\) \(\begin{matrix} 0_{\operatorname{d}Y} \\[4pt] {\langle \operatorname{d}\underline{\underline{\text{b}}} ~ \operatorname{d}\underline{\underline{\text{u}}} \rangle}_{\operatorname{d}Y} \\[4pt] {\langle \operatorname{d}\underline{\underline{\text{b}}} ~ \operatorname{d}\underline{\underline{\text{u}}} \rangle}_{\operatorname{d}Y} \\[4pt] 0_{\operatorname{d}Y} \end{matrix}\) \(\text{Table 61.1} ~~ \operatorname{LIR}_3 (L_\text{B}) : \text{Lateral Representation of} ~ L_\text{B}\!\) \(\text{Object}\!\) \(\text{Sign}\!\) \(\text{Interpretant}\!\) \(\begin{matrix} {\langle\underline{\underline{\text{A}}}\rangle}_X \\[4pt] {\langle\underline{\underline{\text{A}}}\rangle}_X \\[4pt] {\langle\underline{\underline{\text{A}}}\rangle}_X \\[4pt] {\langle\underline{\underline{\text{A}}}\rangle}_X \end{matrix}\) \(\begin{matrix} {\langle\underline{\underline{\text{a}}}\rangle}_Y \\[4pt] {\langle\underline{\underline{\text{a}}}\rangle}_Y \\[4pt] {\langle\underline{\underline{\text{u}}}\rangle}_Y \\[4pt] {\langle\underline{\underline{\text{u}}}\rangle}_Y \end{matrix}\) \(\begin{matrix} {\langle\underline{\underline{\text{a}}}\rangle}_Y \\[4pt] {\langle\underline{\underline{\text{u}}}\rangle}_Y \\[4pt] {\langle\underline{\underline{\text{a}}}\rangle}_Y \\[4pt] {\langle\underline{\underline{\text{u}}}\rangle}_Y \end{matrix}\) \(\begin{matrix} {\langle\underline{\underline{\text{B}}}\rangle}_X \\[4pt] {\langle\underline{\underline{\text{B}}}\rangle}_X \\[4pt] {\langle\underline{\underline{\text{B}}}\rangle}_X \\[4pt] {\langle\underline{\underline{\text{B}}}\rangle}_X \end{matrix}\) \(\begin{matrix} {\langle\underline{\underline{\text{b}}}\rangle}_Y \\[4pt] {\langle\underline{\underline{\text{b}}}\rangle}_Y \\[4pt] {\langle\underline{\underline{\text{i}}}\rangle}_Y \\[4pt] {\langle\underline{\underline{\text{i}}}\rangle}_Y \end{matrix}\) \(\begin{matrix} {\langle\underline{\underline{\text{b}}}\rangle}_Y \\[4pt] {\langle\underline{\underline{\text{i}}}\rangle}_Y \\[4pt] {\langle\underline{\underline{\text{b}}}\rangle}_Y \\[4pt] {\langle\underline{\underline{\text{i}}}\rangle}_Y \end{matrix}\) \(\text{Table 61.2} ~~ \operatorname{LIR}_3 (\operatorname{Den}(L_\text{B})) : \text{Denotative Component of} ~ L_\text{B}\!\) \(\text{Object}\!\) \(\text{Sign}\!\) \(\text{Transition}\!\) \(\begin{matrix} {\langle\underline{\underline{\text{A}}}\rangle}_X \\[4pt] {\langle\underline{\underline{\text{A}}}\rangle}_X \end{matrix}\) \(\begin{matrix} {\langle\underline{\underline{\text{a}}}\rangle}_Y \\[4pt] {\langle\underline{\underline{\text{u}}}\rangle}_Y \end{matrix}\) \(\begin{matrix} ({\langle\underline{\underline{\text{a}}}\rangle}_Y, {\langle\underline{\underline{\text{A}}}\rangle}_X) \\[4pt] ({\langle\underline{\underline{\text{u}}}\rangle}_Y, {\langle\underline{\underline{\text{A}}}\rangle}_X) \end{matrix}\) \(\begin{matrix} {\langle\underline{\underline{\text{B}}}\rangle}_X \\[4pt] {\langle\underline{\underline{\text{B}}}\rangle}_X \end{matrix}\) \(\begin{matrix} {\langle\underline{\underline{\text{b}}}\rangle}_Y \\[4pt] {\langle\underline{\underline{\text{i}}}\rangle}_Y \end{matrix}\) \(\begin{matrix} ({\langle\underline{\underline{\text{b}}}\rangle}_Y, {\langle\underline{\underline{\text{B}}}\rangle}_X) \\[4pt] ({\langle\underline{\underline{\text{i}}}\rangle}_Y, {\langle\underline{\underline{\text{B}}}\rangle}_X) \end{matrix}\) \(\text{Table 61.3} ~~ \operatorname{LIR}_3 (\operatorname{Con}(L_\text{B})) : \text{Connotative Component of} ~ L_\text{B}\!\) \(\text{Sign}\!\) \(\text{Interpretant}\!\) \(\text{Transition}\!\) \(\begin{matrix} {\langle\underline{\underline{\text{a}}}\rangle}_Y \\[4pt] {\langle\underline{\underline{\text{a}}}\rangle}_Y \\[4pt] {\langle\underline{\underline{\text{u}}}\rangle}_Y \\[4pt] {\langle\underline{\underline{\text{u}}}\rangle}_Y \end{matrix}\) \(\begin{matrix} {\langle\underline{\underline{\text{a}}}\rangle}_Y \\[4pt] {\langle\underline{\underline{\text{u}}}\rangle}_Y \\[4pt] {\langle\underline{\underline{\text{a}}}\rangle}_Y \\[4pt] {\langle\underline{\underline{\text{u}}}\rangle}_Y \end{matrix}\) \(\begin{matrix} 0_{\operatorname{d}Y} \\[4pt] {\langle \operatorname{d}\underline{\underline{\text{a}}} ~ \operatorname{d}\underline{\underline{\text{u}}} \rangle}_{\operatorname{d}Y} \\[4pt] {\langle \operatorname{d}\underline{\underline{\text{a}}} ~ \operatorname{d}\underline{\underline{\text{u}}} \rangle}_{\operatorname{d}Y} \\[4pt] 0_{\operatorname{d}Y} \end{matrix}\) \(\begin{matrix} {\langle\underline{\underline{\text{b}}}\rangle}_Y \\[4pt] {\langle\underline{\underline{\text{b}}}\rangle}_Y \\[4pt] {\langle\underline{\underline{\text{i}}}\rangle}_Y \\[4pt] {\langle\underline{\underline{\text{i}}}\rangle}_Y \end{matrix}\) \(\begin{matrix} {\langle\underline{\underline{\text{b}}}\rangle}_Y \\[4pt] {\langle\underline{\underline{\text{i}}}\rangle}_Y \\[4pt] {\langle\underline{\underline{\text{b}}}\rangle}_Y \\[4pt] {\langle\underline{\underline{\text{i}}}\rangle}_Y \end{matrix}\) \(\begin{matrix} 0_{\operatorname{d}Y} \\[4pt] {\langle \operatorname{d}\underline{\underline{\text{b}}} ~ \operatorname{d}\underline{\underline{\text{i}}} \rangle}_{\operatorname{d}Y} \\[4pt] {\langle \operatorname{d}\underline{\underline{\text{b}}} ~ \operatorname{d}\underline{\underline{\text{i}}} \rangle}_{\operatorname{d}Y} \\[4pt] 0_{\operatorname{d}Y} \end{matrix}\) \(\text{Table 62.1} ~~ \text{Analytic Codes for Object Features}\!\) \(\text{Category}\!\) \(\text{Mnemonic}\!\) \(\text{Code}\!\) \(\begin{array}{l} \text{Self} \\[4pt] \text{Other} \end{array}\) \(\begin{matrix} \text{self} \\[4pt] \text{(self)} \end{matrix}\) \(\begin{matrix} \text{s} \\[4pt] \text{(s)} \end{matrix}\) \(\text{Table 62.2} ~~ \text{Analytic Codes for Semantic Features}\!\) \(\text{Category}\!\) \(\text{Mnemonic}\!\) \(\text{Code}\!\) \(\begin{array}{l} \text{1st Person} \\[4pt] \text{2nd Person} \end{array}\) \(\begin{matrix} \text{my} \\[4pt] \text{(my)} \end{matrix}\) \(\begin{matrix} \text{m} \\[4pt] \text{(m)} \end{matrix}\) \(\text{Table 62.3} ~~ \text{Analytic Codes for Syntactic Features}\!\) \(\text{Category}\!\) \(\text{Mnemonic}\!\) \(\text{Code}\!\) \(\begin{array}{l} \text{Noun} \\[4pt] \text{Pronoun} \end{array}\) \(\begin{matrix} \text{name} \\[4pt] \text{(name)} \end{matrix}\) \(\begin{matrix} \text{n} \\[4pt] \text{(n)} \end{matrix}\) \(\text{Table 63.} ~~ \text{Analytic Codes for Interpreter A}\!\) \(\text{Name}\!\) \(\text{Vector}\!\) \(\text{Conjunct Term}\!\) \(\text{Mnemonic}\!\) \(\text{Code}\!\) \(\begin{matrix} \text{A} \\[4pt] \text{B} \\[4pt] {}^{\backprime\backprime} \text{A} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \text{B} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \end{matrix}\) \(\begin{matrix} {1}_X \\[4pt] {0}_X \\[4pt] {11}_Y \\[4pt] {01}_Y \\[4pt] {10}_Y \\[4pt] {00}_Y \end{matrix}\) \(\begin{matrix} ~x_1~ \\[4pt] (x_1) \\[4pt] ~y_1~~y_2~ \\[4pt] (y_1)~y_2~ \\[4pt] ~y_1~(y_2) \\[4pt] (y_1)(y_2) \end{matrix}\) \(\begin{matrix} ~\text{self}~ \\[4pt] (\text{self}) \\[4pt] ~\text{my}~~\text{name}~ \\[4pt] (\text{my})~\text{name}~ \\[4pt] ~\text{my}~(\text{name}) \\[4pt] (\text{my})(\text{name}) \end{matrix}\) \(\begin{matrix} ~\text{s}~ \\[4pt] (\text{s}) \\[4pt] ~\text{m}~~\text{n}~ \\[4pt] (\text{m})~\text{n}~ \\[4pt] ~\text{m}~(\text{n}) \\[4pt] (\text{m})(\text{n}) \end{matrix}\) \(\text{Table 64.} ~~ \text{Analytic Codes for Interpreter B}\!\) \(\text{Name}\!\) \(\text{Vector}\!\) \(\text{Conjunct Term}\!\) \(\text{Mnemonic}\!\) \(\text{Code}\!\) \(\begin{matrix} \text{A} \\[4pt] \text{B} \\[4pt] {}^{\backprime\backprime} \text{A} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \text{B} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \\[4pt] {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \end{matrix}\) \(\begin{matrix} {0}_X \\[4pt] {1}_X \\[4pt] {01}_Y \\[4pt] {11}_Y \\[4pt] {10}_Y \\[4pt] {00}_Y \end{matrix}\) \(\begin{matrix} (x_1) \\[4pt] ~x_1~ \\[4pt] (y_1)~y_2~ \\[4pt] ~y_1~~y_2~ \\[4pt] ~y_1~(y_2) \\[4pt] (y_1)(y_2) \end{matrix}\) \(\begin{matrix} (\text{self}) \\[4pt] ~\text{self}~ \\[4pt] (\text{my})~\text{name}~ \\[4pt] ~\text{my}~~\text{name}~ \\[4pt] ~\text{my}~(\text{name}) \\[4pt] (\text{my})(\text{name}) \end{matrix}\) \(\begin{matrix} (\text{s}) \\[4pt] ~\text{s}~ \\[4pt] (\text{m})~\text{n}~ \\[4pt] ~\text{m}~~\text{n}~ \\[4pt] ~\text{m}~(\text{n}) \\[4pt] (\text{m})(\text{n}) \end{matrix}\) \(\text{Table 65.1} ~~ \operatorname{AIR}_1 (L_\text{A}) : \text{Analytic Representation of} ~ L_\text{A}\!\) \(\text{Object}\!\) \(\text{Sign}\!\) \(\text{Interpretant}\!\) \(\begin{matrix} \text{s} \\[4pt] \text{s} \\[4pt] \text{s} \\[4pt] \text{s} \end{matrix}\) \(\begin{matrix} ~\text{m}~~\text{n}~ \\[4pt] ~\text{m}~~\text{n}~ \\[4pt] ~\text{m}~(\text{n}) \\[4pt] ~\text{m}~(\text{n}) \end{matrix}\) \(\begin{matrix} ~\text{m}~~\text{n}~ \\[4pt] ~\text{m}~(\text{n}) \\[4pt] ~\text{m}~~\text{n}~ \\[4pt] ~\text{m}~(\text{n}) \end{matrix}\) \(\begin{matrix} (\text{s}) \\[4pt] (\text{s}) \\[4pt] (\text{s}) \\[4pt] (\text{s}) \end{matrix}\) \(\begin{matrix} (\text{m})~\text{n}~ \\[4pt] (\text{m})~\text{n}~ \\[4pt] (\text{m})(\text{n}) \\[4pt] (\text{m})(\text{n}) \end{matrix}\) \(\begin{matrix} (\text{m})~\text{n}~ \\[4pt] (\text{m})(\text{n}) \\[4pt] (\text{m})~\text{n}~ \\[4pt] (\text{m})(\text{n}) \end{matrix}\) \(\text{Table 65.2} ~~ \operatorname{AIR}_1 (\operatorname{Den}(L_\text{A})) : \text{Denotative Component of} ~ L_\text{A}\!\) \(\text{Object}\!\) \(\text{Sign}\!\) \(\text{Transition}\!\) \(\begin{matrix} \text{s} \\[4pt] \text{s} \end{matrix}\) \(\begin{matrix} ~\text{m}~~\text{n}~ \\[4pt] ~\text{m}~(\text{n}) \end{matrix}\) \(\begin{matrix} ~\text{m}~~\text{n}~ \mapsto ~\text{s}~ \\[4pt] ~\text{m}~(\text{n}) \mapsto ~\text{s}~ \end{matrix}\) \(\begin{matrix} (\text{s}) \\[4pt] (\text{s}) \end{matrix}\) \(\begin{matrix} (\text{m})~\text{n}~ \\[4pt] (\text{m})(\text{n}) \end{matrix}\) \(\begin{matrix} (\text{m})~\text{n}~ \mapsto (\text{s}) \\[4pt] (\text{m})(\text{n}) \mapsto (\text{s}) \end{matrix}\) \(\text{Table 65.3} ~~ \operatorname{AIR}_1 (\operatorname{Con}(L_\text{A})) : \text{Connotative Component of} ~ L_\text{A}\!\) \(\text{Sign}\!\) \(\text{Interpretant}\!\) \(\text{Transition}\!\) \(\begin{matrix} ~\text{m}~~\text{n}~ \\[4pt] ~\text{m}~~\text{n}~ \\[4pt] ~\text{m}~(\text{n}) \\[4pt] ~\text{m}~(\text{n}) \end{matrix}\) \(\begin{matrix} ~\text{m}~~\text{n}~ \\[4pt] ~\text{m}~(\text{n}) \\[4pt] ~\text{m}~~\text{n}~ \\[4pt] ~\text{m}~(\text{n}) \end{matrix}\) \(\begin{matrix} (\text{dm})(\text{dn}) \\[4pt] (\text{dm})~\text{dn}~ \\[4pt] (\text{dm})~\text{dn}~ \\[4pt] (\text{dm})(\text{dn}) \end{matrix}\) \(\begin{matrix} (\text{m})~\text{n}~ \\[4pt] (\text{m})~\text{n}~ \\[4pt] (\text{m})(\text{n}) \\[4pt] (\text{m})(\text{n}) \end{matrix}\) \(\begin{matrix} (\text{m})~\text{n}~ \\[4pt] (\text{m})(\text{n}) \\[4pt] (\text{m})~\text{n}~ \\[4pt] (\text{m})(\text{n}) \end{matrix}\) \(\begin{matrix} (\text{dm})(\text{dn}) \\[4pt] (\text{dm})~\text{dn}~ \\[4pt] (\text{dm})~\text{dn}~ \\[4pt] (\text{dm})(\text{dn}) \end{matrix}\) \(\text{Table 66.1} ~~ \operatorname{AIR}_1 (L_\text{B}) : \text{Analytic Representation of} ~ L_\text{B}\!\) \(\text{Object}\!\) \(\text{Sign}\!\) \(\text{Interpretant}\!\) \(\begin{matrix} (\text{s}) \\[4pt] (\text{s}) \\[4pt] (\text{s}) \\[4pt] (\text{s}) \end{matrix}\) \(\begin{matrix} (\text{m})~\text{n}~ \\[4pt] (\text{m})~\text{n}~ \\[4pt] (\text{m})(\text{n}) \\[4pt] (\text{m})(\text{n}) \end{matrix}\) \(\begin{matrix} (\text{m})~\text{n}~ \\[4pt] (\text{m})(\text{n}) \\[4pt] (\text{m})~\text{n}~ \\[4pt] (\text{m})(\text{n}) \end{matrix}\) \(\begin{matrix} \text{s} \\[4pt] \text{s} \\[4pt] \text{s} \\[4pt] \text{s} \end{matrix}\) \(\begin{matrix} ~\text{m}~~\text{n}~ \\[4pt] ~\text{m}~~\text{n}~ \\[4pt] ~\text{m}~(\text{n}) \\[4pt] ~\text{m}~(\text{n}) \end{matrix}\) \(\begin{matrix} ~\text{m}~~\text{n}~ \\[4pt] ~\text{m}~(\text{n}) \\[4pt] ~\text{m}~~\text{n}~ \\[4pt] ~\text{m}~(\text{n}) \end{matrix}\) \(\text{Table 66.2} ~~ \operatorname{AIR}_1 (\operatorname{Den}(L_\text{B})) : \text{Denotative Component of} ~ L_\text{B}\!\) \(\text{Object}\!\) \(\text{Sign}\!\) \(\text{Transition}\!\) \(\begin{matrix} (\text{s}) \\[4pt] (\text{s}) \end{matrix}\) \(\begin{matrix} (\text{m})~\text{n}~ \\[4pt] (\text{m})(\text{n}) \end{matrix}\) \(\begin{matrix} (\text{m})~\text{n}~ \mapsto (\text{s}) \\[4pt] (\text{m})(\text{n}) \mapsto (\text{s}) \end{matrix}\) \(\begin{matrix} \text{s} \\[4pt] \text{s} \end{matrix}\) \(\begin{matrix} ~\text{m}~~\text{n}~ \\[4pt] ~\text{m}~(\text{n}) \end{matrix}\) \(\begin{matrix} ~\text{m}~~\text{n}~ \mapsto ~\text{s}~ \\[4pt] ~\text{m}~(\text{n}) \mapsto ~\text{s}~ \end{matrix}\) \(\text{Table 66.3} ~~ \operatorname{AIR}_1 (\operatorname{Con}(L_\text{B})) : \text{Connotative Component of} ~ L_\text{B}\!\) \(\text{Sign}\!\) \(\text{Interpretant}\!\) \(\text{Transition}\!\) \(\begin{matrix} (\text{m})~\text{n}~ \\[4pt] (\text{m})~\text{n}~ \\[4pt] (\text{m})(\text{n}) \\[4pt] (\text{m})(\text{n}) \end{matrix}\) \(\begin{matrix} (\text{m})~\text{n}~ \\[4pt] (\text{m})(\text{n}) \\[4pt] (\text{m})~\text{n}~ \\[4pt] (\text{m})(\text{n}) \end{matrix}\) \(\begin{matrix} (\text{dm})(\text{dn}) \\[4pt] (\text{dm})~\text{dn}~ \\[4pt] (\text{dm})~\text{dn}~ \\[4pt] (\text{dm})(\text{dn}) \end{matrix}\) \(\begin{matrix} ~\text{m}~~\text{n}~ \\[4pt] ~\text{m}~~\text{n}~ \\[4pt] ~\text{m}~(\text{n}) \\[4pt] ~\text{m}~(\text{n}) \end{matrix}\) \(\begin{matrix} ~\text{m}~~\text{n}~ \\[4pt] ~\text{m}~(\text{n}) \\[4pt] ~\text{m}~~\text{n}~ \\[4pt] ~\text{m}~(\text{n}) \end{matrix}\) \(\begin{matrix} (\text{dm})(\text{dn}) \\[4pt] (\text{dm})~\text{dn}~ \\[4pt] (\text{dm})~\text{dn}~ \\[4pt] (\text{dm})(\text{dn}) \end{matrix}\) \(\text{Table 67.1} ~~ \operatorname{AIR}_2 (L_\text{A}) : \text{Analytic Representation of} ~ L_\text{A}\!\) \(\text{Object}\!\) \(\text{Sign}\!\) \(\text{Interpretant}\!\) \(\begin{matrix} {\langle * \rangle}_X \\[4pt] {\langle * \rangle}_X \\[4pt] {\langle * \rangle}_X \\[4pt] {\langle * \rangle}_X \end{matrix}\) \(\begin{matrix} {\langle * \rangle}_Y \\[4pt] {\langle * \rangle}_Y \\[4pt] {\langle\text{m}\rangle}_Y \\[4pt] {\langle\text{m}\rangle}_Y \end{matrix}\) \(\begin{matrix} {\langle * \rangle}_Y \\[4pt] {\langle\text{m}\rangle}_Y \\[4pt] {\langle * \rangle}_Y \\[4pt] {\langle\text{m}\rangle}_Y \end{matrix}\) \(\begin{matrix} {\langle ! \rangle}_X \\[4pt] {\langle ! \rangle}_X \\[4pt] {\langle ! \rangle}_X \\[4pt] {\langle ! \rangle}_X \end{matrix}\) \(\begin{matrix} {\langle\text{n}\rangle}_Y \\[4pt] {\langle\text{n}\rangle}_Y \\[4pt] {\langle ! \rangle}_Y \\[4pt] {\langle ! \rangle}_Y \end{matrix}\) \(\begin{matrix} {\langle\text{n}\rangle}_Y \\[4pt] {\langle ! \rangle}_Y \\[4pt] {\langle\text{n}\rangle}_Y \\[4pt] {\langle ! \rangle}_Y \end{matrix}\) \(\text{Table 67.2} ~~ \operatorname{AIR}_2 (\operatorname{Den}(L_\text{A})) : \text{Denotative Component of} ~ L_\text{A}\!\) \(\text{Object}\!\) \(\text{Sign}\!\) \(\text{Transition}\!\) \(\begin{matrix} {\langle * \rangle}_X \\[4pt] {\langle * \rangle}_X \end{matrix}\) \(\begin{matrix} {\langle * \rangle}_Y \\[4pt] {\langle\text{m}\rangle}_Y \end{matrix}\) \(\begin{array}{r} {\langle * \rangle}_Y \mapsto {\langle * \rangle}_X \\[4pt] {\langle\text{m}\rangle}_Y \mapsto {\langle * \rangle}_X \end{array}\) \(\begin{matrix} {\langle ! \rangle}_X \\[4pt] {\langle ! \rangle}_X \end{matrix}\) \(\begin{matrix} {\langle\text{n}\rangle}_Y \\[4pt] {\langle ! \rangle}_Y \end{matrix}\) \(\begin{array}{r} {\langle\text{n}\rangle}_Y \mapsto {\langle ! \rangle}_X \\[4pt] {\langle ! \rangle}_Y \mapsto {\langle ! \rangle}_X \end{array}\) \(\text{Table 67.3} ~~ \operatorname{AIR}_2 (\operatorname{Con}(L_\text{A})) : \text{Connotative Component of} ~ L_\text{A}\!\) \(\text{Sign}\!\) \(\text{Interpretant}\!\) \(\text{Transition}\!\) \(\begin{matrix} {\langle * \rangle}_Y \\[4pt] {\langle * \rangle}_Y \\[4pt] {\langle\text{m}\rangle}_Y \\[4pt] {\langle\text{m}\rangle}_Y \end{matrix}\) \(\begin{matrix} {\langle * \rangle}_Y \\[4pt] {\langle\text{m}\rangle}_Y \\[4pt] {\langle * \rangle}_Y \\[4pt] {\langle\text{m}\rangle}_Y \end{matrix}\) \(\begin{matrix} {\langle\operatorname{d}!\rangle}_{\operatorname{d}Y} \\[4pt] {\langle\operatorname{d}\text{n}\rangle}_{\operatorname{d}Y} \\[4pt] {\langle\operatorname{d}\text{n}\rangle}_{\operatorname{d}Y} \\[4pt] {\langle\operatorname{d}!\rangle}_{\operatorname{d}Y} \end{matrix}\) \(\begin{matrix} {\langle\text{n}\rangle}_Y \\[4pt] {\langle\text{n}\rangle}_Y \\[4pt] {\langle ! \rangle}_Y \\[4pt] {\langle ! \rangle}_Y \end{matrix}\) \(\begin{matrix} {\langle\text{n}\rangle}_Y \\[4pt] {\langle ! \rangle}_Y \\[4pt] {\langle\text{n}\rangle}_Y \\[4pt] {\langle ! \rangle}_Y \end{matrix}\) \(\begin{matrix} {\langle\operatorname{d}!\rangle}_{\operatorname{d}Y} \\[4pt] {\langle\operatorname{d}\text{n}\rangle}_{\operatorname{d}Y} \\[4pt] {\langle\operatorname{d}\text{n}\rangle}_{\operatorname{d}Y} \\[4pt] {\langle\operatorname{d}!\rangle}_{\operatorname{d}Y} \end{matrix}\) \(\text{Table 68.1} ~~ \operatorname{AIR}_2 (L_\text{B}) : \text{Analytic Representation of} ~ L_\text{B}\!\) \(\text{Object}\!\) \(\text{Sign}\!\) \(\text{Interpretant}\!\) \(\begin{matrix} {\langle ! \rangle}_X \\[4pt] {\langle ! \rangle}_X \\[4pt] {\langle ! \rangle}_X \\[4pt] {\langle ! \rangle}_X \end{matrix}\) \(\begin{matrix} {\langle\text{n}\rangle}_Y \\[4pt] {\langle\text{n}\rangle}_Y \\[4pt] {\langle ! \rangle}_Y \\[4pt] {\langle ! \rangle}_Y \end{matrix}\) \(\begin{matrix} {\langle\text{n}\rangle}_Y \\[4pt] {\langle ! \rangle}_Y \\[4pt] {\langle\text{n}\rangle}_Y \\[4pt] {\langle ! \rangle}_Y \end{matrix}\) \(\begin{matrix} {\langle * \rangle}_X \\[4pt] {\langle * \rangle}_X \\[4pt] {\langle * \rangle}_X \\[4pt] {\langle * \rangle}_X \end{matrix}\) \(\begin{matrix} {\langle * \rangle}_Y \\[4pt] {\langle * \rangle}_Y \\[4pt] {\langle\text{m}\rangle}_Y \\[4pt] {\langle\text{m}\rangle}_Y \end{matrix}\) \(\begin{matrix} {\langle * \rangle}_Y \\[4pt] {\langle\text{m}\rangle}_Y \\[4pt] {\langle * \rangle}_Y \\[4pt] {\langle\text{m}\rangle}_Y \end{matrix}\) \(\text{Table 68.2} ~~ \operatorname{AIR}_2 (\operatorname{Den}(L_\text{B})) : \text{Denotative Component of} ~ L_\text{B}\!\) \(\text{Object}\!\) \(\text{Sign}\!\) \(\text{Transition}\!\) \(\begin{matrix} {\langle ! \rangle}_X \\[4pt] {\langle ! \rangle}_X \end{matrix}\) \(\begin{matrix} {\langle\text{n}\rangle}_Y \\[4pt] {\langle ! \rangle}_Y \end{matrix}\) \(\begin{array}{r} {\langle\text{n}\rangle}_Y \mapsto {\langle ! \rangle}_X \\[4pt] {\langle ! \rangle}_Y \mapsto {\langle ! \rangle}_X \end{array}\) \(\begin{matrix} {\langle * \rangle}_X \\[4pt] {\langle * \rangle}_X \end{matrix}\) \(\begin{matrix} {\langle * \rangle}_Y \\[4pt] {\langle\text{m}\rangle}_Y \end{matrix}\) \(\begin{array}{r} {\langle * \rangle}_Y \mapsto {\langle * \rangle}_X \\[4pt] {\langle\text{m}\rangle}_Y \mapsto {\langle * \rangle}_X \end{array}\) \(\text{Table 68.3} ~~ \operatorname{AIR}_2 (\operatorname{Con}(L_\text{B})) : \text{Connotative Component of} ~ L_\text{B}\!\) \(\text{Sign}\!\) \(\text{Interpretant}\!\) \(\text{Transition}\!\) \(\begin{matrix} {\langle\text{n}\rangle}_Y \\[4pt] {\langle\text{n}\rangle}_Y \\[4pt] {\langle ! \rangle}_Y \\[4pt] {\langle ! \rangle}_Y \end{matrix}\) \(\begin{matrix} {\langle\text{n}\rangle}_Y \\[4pt] {\langle ! \rangle}_Y \\[4pt] {\langle\text{n}\rangle}_Y \\[4pt] {\langle ! \rangle}_Y \end{matrix}\) \(\begin{matrix} {\langle\operatorname{d}!\rangle}_{\operatorname{d}Y} \\[4pt] {\langle\operatorname{d}\text{n}\rangle}_{\operatorname{d}Y} \\[4pt] {\langle\operatorname{d}\text{n}\rangle}_{\operatorname{d}Y} \\[4pt] {\langle\operatorname{d}!\rangle}_{\operatorname{d}Y} \end{matrix}\) \(\begin{matrix} {\langle * \rangle}_Y \\[4pt] {\langle * \rangle}_Y \\[4pt] {\langle\text{m}\rangle}_Y \\[4pt] {\langle\text{m}\rangle}_Y \end{matrix}\) \(\begin{matrix} {\langle * \rangle}_Y \\[4pt] {\langle\text{m}\rangle}_Y \\[4pt] {\langle * \rangle}_Y \\[4pt] {\langle\text{m}\rangle}_Y \end{matrix}\) \(\begin{matrix} {\langle\operatorname{d}!\rangle}_{\operatorname{d}Y} \\[4pt] {\langle\operatorname{d}\text{n}\rangle}_{\operatorname{d}Y} \\[4pt] {\langle\operatorname{d}\text{n}\rangle}_{\operatorname{d}Y} \\[4pt] {\langle\operatorname{d}!\rangle}_{\operatorname{d}Y} \end{matrix}\) \(\text{Table 69.} ~~ \text{Schematism of Sequential Inference}\!\) \(\text{Initial Premiss}\!\) \(\text{Differential Premiss}\!\) \(\text{Inferred Sequel}\!\) \(\begin{matrix} ~x~ ~\operatorname{at}~ t \\[4pt] ~x~ ~\operatorname{at}~ t \\[4pt] (x) ~\operatorname{at}~ t \\[4pt] (x) ~\operatorname{at}~ t \end{matrix}\) \(\begin{matrix} ~\operatorname{d}x~ ~\operatorname{at}~ t \\[4pt] (\operatorname{d}x) ~\operatorname{at}~ t \\[4pt] ~\operatorname{d}x~ ~\operatorname{at}~ t \\[4pt] (\operatorname{d}x) ~\operatorname{at}~ t \end{matrix}\) \(\begin{matrix} (x) ~\operatorname{at}~ t' \\[4pt] ~x~ ~\operatorname{at}~ t' \\[4pt] ~x~ ~\operatorname{at}~ t' \\[4pt] (x) ~\operatorname{at}~ t' \end{matrix}\) \(\text{Table 70.1} ~~ \text{Group Representation} ~ \operatorname{Rep}^\text{A} (V_4)\!\) \(\begin{matrix} \text{Abstract} \\ \text{Element} \end{matrix}\) \(\begin{matrix} \text{Logical} \\ \text{Element} \end{matrix}\) \(\begin{matrix} \text{Active} \\ \text{List} \end{matrix}\) \(\begin{matrix} \text{Active} \\ \text{Term} \end{matrix}\) \(\begin{matrix} \text{Genetic} \\ \text{Element} \end{matrix}\) \(\begin{matrix} 1 \\[4pt] r \\[4pt] s \\[4pt] t \end{matrix}\) \(\begin{matrix} (\operatorname{d}\underline{\underline{\text{a}}}) (\operatorname{d}\underline{\underline{\text{b}}}) (\operatorname{d}\underline{\underline{\text{i}}}) (\operatorname{d}\underline{\underline{\text{u}}}) \\[4pt] ~\operatorname{d}\underline{\underline{\text{a}}}~ (\operatorname{d}\underline{\underline{\text{b}}}) ~\operatorname{d}\underline{\underline{\text{i}}}~ (\operatorname{d}\underline{\underline{\text{u}}}) \\[4pt] (\operatorname{d}\underline{\underline{\text{a}}}) ~\operatorname{d}\underline{\underline{\text{b}}}~ (\operatorname{d}\underline{\underline{\text{i}}}) ~\operatorname{d}\underline{\underline{\text{u}}}~ \\[4pt] ~\operatorname{d}\underline{\underline{\text{a}}}~ ~\operatorname{d}\underline{\underline{\text{b}}}~ ~\operatorname{d}\underline{\underline{\text{i}}}~ ~\operatorname{d}\underline{\underline{\text{u}}}~ \end{matrix}\) \(\begin{matrix} \langle \operatorname{d}! \rangle \\[4pt] \langle \operatorname{d}\underline{\underline{\text{a}}} ~ \operatorname{d}\underline{\underline{\text{i}}} \rangle \\[4pt] \langle \operatorname{d}\underline{\underline{\text{b}}} ~ \operatorname{d}\underline{\underline{\text{u}}} \rangle \\[4pt] \langle \operatorname{d}* \rangle \end{matrix}\) \(\begin{matrix} \operatorname{d}! \\[4pt] \operatorname{d}\underline{\underline{\text{a}}} \cdot \operatorname{d}\underline{\underline{\text{i}}} ~ ! \\[4pt] \operatorname{d}\underline{\underline{\text{b}}} \cdot \operatorname{d}\underline{\underline{\text{u}}} ~ ! \\[4pt] \operatorname{d}* \end{matrix}\) \(\begin{matrix} 1 \\[4pt] \operatorname{d}_{\text{ai}} \\[4pt] \operatorname{d}_{\text{bu}} \\[4pt] \operatorname{d}_{\text{ai}} * \operatorname{d}_{\text{bu}} \end{matrix}\) \(\text{Table 70.2} ~~ \text{Group Representation} ~ \operatorname{Rep}^\text{B} (V_4)\!\) \(\begin{matrix} \text{Abstract} \\ \text{Element} \end{matrix}\) \(\begin{matrix} \text{Logical} \\ \text{Element} \end{matrix}\) \(\begin{matrix} \text{Active} \\ \text{List} \end{matrix}\) \(\begin{matrix} \text{Active} \\ \text{Term} \end{matrix}\) \(\begin{matrix} \text{Genetic} \\ \text{Element} \end{matrix}\) \(\begin{matrix} 1 \\[4pt] r \\[4pt] s \\[4pt] t \end{matrix}\) \(\begin{matrix} (\operatorname{d}\underline{\underline{\text{a}}}) (\operatorname{d}\underline{\underline{\text{b}}}) (\operatorname{d}\underline{\underline{\text{i}}}) (\operatorname{d}\underline{\underline{\text{u}}}) \\[4pt] ~\operatorname{d}\underline{\underline{\text{a}}}~ (\operatorname{d}\underline{\underline{\text{b}}}) (\operatorname{d}\underline{\underline{\text{i}}}) ~\operatorname{d}\underline{\underline{\text{u}}}~ \\[4pt] (\operatorname{d}\underline{\underline{\text{a}}}) ~\operatorname{d}\underline{\underline{\text{b}}}~ ~\operatorname{d}\underline{\underline{\text{i}}}~ (\operatorname{d}\underline{\underline{\text{u}}}) \\[4pt] ~\operatorname{d}\underline{\underline{\text{a}}}~ ~\operatorname{d}\underline{\underline{\text{b}}}~ ~\operatorname{d}\underline{\underline{\text{i}}}~ ~\operatorname{d}\underline{\underline{\text{u}}}~ \end{matrix}\) \(\begin{matrix} \langle \operatorname{d}! \rangle \\[4pt] \langle \operatorname{d}\underline{\underline{\text{a}}} ~ \operatorname{d}\underline{\underline{\text{u}}} \rangle \\[4pt] \langle \operatorname{d}\underline{\underline{\text{b}}} ~ \operatorname{d}\underline{\underline{\text{i}}} \rangle \\[4pt] \langle \operatorname{d}* \rangle \end{matrix}\) \(\begin{matrix} \operatorname{d}! \\[4pt] \operatorname{d}\underline{\underline{\text{a}}} \cdot \operatorname{d}\underline{\underline{\text{u}}} ~ ! \\[4pt] \operatorname{d}\underline{\underline{\text{b}}} \cdot \operatorname{d}\underline{\underline{\text{i}}} ~ ! \\[4pt] \operatorname{d}* \end{matrix}\) \(\begin{matrix} 1 \\[4pt] \operatorname{d}_{\text{au}} \\[4pt] \operatorname{d}_{\text{bi}} \\[4pt] \operatorname{d}_{\text{au}} * \operatorname{d}_{\text{bi}} \end{matrix}\) \(\text{Table 70.3} ~~ \text{Group Representation} ~ \operatorname{Rep}^\text{C} (V_4)\!\) \(\begin{matrix} \text{Abstract} \\ \text{Element} \end{matrix}\) \(\begin{matrix} \text{Logical} \\ \text{Element} \end{matrix}\) \(\begin{matrix} \text{Active} \\ \text{List} \end{matrix}\) \(\begin{matrix} \text{Active} \\ \text{Term} \end{matrix}\) \(\begin{matrix} \text{Genetic} \\ \text{Element} \end{matrix}\) \(\begin{matrix} 1 \\[4pt] r \\[4pt] s \\[4pt] t \end{matrix}\) \(\begin{matrix} (\operatorname{d}\text{m}) (\operatorname{d}\text{n}) \\[4pt] ~\operatorname{d}\text{m}~ (\operatorname{d}\text{n}) \\[4pt] (\operatorname{d}\text{m}) ~\operatorname{d}\text{n}~ \\[4pt] ~\operatorname{d}\text{m}~ ~\operatorname{d}\text{n}~ \end{matrix}\) \(\begin{matrix} \langle\operatorname{d}!\rangle \\[4pt] \langle\operatorname{d}\text{m}\rangle \\[4pt] \langle\operatorname{d}\text{n}\rangle \\[4pt] \langle\operatorname{d}*\rangle \end{matrix}\) \(\begin{matrix} \operatorname{d}! \\[4pt] \operatorname{d}\text{m}! \\[4pt] \operatorname{d}\text{n}! \\[4pt] \operatorname{d}* \end{matrix}\) \(\begin{matrix} 1 \\[4pt] \operatorname{d}_{\text{m}} \\[4pt] \operatorname{d}_{\text{n}} \\[4pt] \operatorname{d}_{\text{m}} * \operatorname{d}_{\text{n}} \end{matrix}\) \(\text{Table 71.1} ~~ \text{The Differential Group} ~ G = V_4\!\) \(\begin{matrix} \text{Abstract} \\ \text{Element} \end{matrix}\) \(\begin{matrix} \text{Logical} \\ \text{Element} \end{matrix}\) \(\begin{matrix} \text{Active} \\ \text{List} \end{matrix}\) \(\begin{matrix} \text{Active} \\ \text{Term} \end{matrix}\) \(\begin{matrix} \text{Genetic} \\ \text{Element} \end{matrix}\) \(\begin{matrix} 1 \\[4pt] r \\[4pt] s \\[4pt] t \end{matrix}\) \(\begin{matrix} (\operatorname{d}\text{m}) (\operatorname{d}\text{n}) \\[4pt] ~\operatorname{d}\text{m}~ (\operatorname{d}\text{n}) \\[4pt] (\operatorname{d}\text{m}) ~\operatorname{d}\text{n}~ \\[4pt] ~\operatorname{d}\text{m}~ ~\operatorname{d}\text{n}~ \end{matrix}\) \(\begin{matrix} \langle\operatorname{d}!\rangle \\[4pt] \langle\operatorname{d}\text{m}\rangle \\[4pt] \langle\operatorname{d}\text{n}\rangle \\[4pt] \langle\operatorname{d}*\rangle \end{matrix}\) \(\begin{matrix} \operatorname{d}! \\[4pt] \operatorname{d}\text{m}! \\[4pt] \operatorname{d}\text{n}! \\[4pt] \operatorname{d}* \end{matrix}\) \(\begin{matrix} 1 \\[4pt] \operatorname{d}_{\text{m}} \\[4pt] \operatorname{d}_{\text{n}} \\[4pt] \operatorname{d}_{\text{m}} * \operatorname{d}_{\text{n}} \end{matrix}\) \(\text{Table 71.2} ~~ \text{Cosets of} ~ G_\text{m} ~ \text{in} ~ G\!\) \(\text{Group Coset}\!\) \(\text{Logical Coset}\!\) \(\text{Logical Element}\!\) \(\text{Group Element}\!\) \(G_\text{m}\!\) \((\operatorname{d}\text{m})\!\) \(\begin{matrix} (\operatorname{d}\text{m})(\operatorname{d}\text{n}) \\[4pt] (\operatorname{d}\text{m})~\operatorname{d}\text{n}~ \end{matrix}\) \(\begin{matrix} 1 \\[4pt] \operatorname{d}_\text{n} \end{matrix}\) \(G_\text{m} * \operatorname{d}_\text{m}\!\) \(\operatorname{d}\text{m}\!\) \(\begin{matrix} ~\operatorname{d}\text{m}~(\operatorname{d}\text{n}) \\[4pt] ~\operatorname{d}\text{m}~~\operatorname{d}\text{n}~ \end{matrix}\) \(\begin{matrix} \operatorname{d}_\text{m} \\[4pt] \operatorname{d}_\text{n} * \operatorname{d}_\text{m} \end{matrix}\) \(\text{Table 71.3} ~~ \text{Cosets of} ~ G_\text{n} ~ \text{in} ~ G\!\) \(\text{Group Coset}\!\) \(\text{Logical Coset}\!\) \(\text{Logical Element}\!\) \(\text{Group Element}\!\) \(G_\text{n}\!\) \((\operatorname{d}\text{n})\!\) \(\begin{matrix} (\operatorname{d}\text{m})(\operatorname{d}\text{n}) \\[4pt] ~\operatorname{d}\text{m}~(\operatorname{d}\text{n}) \end{matrix}\) \(\begin{matrix} 1 \\[4pt] \operatorname{d}_\text{m} \end{matrix}\) \(G_\text{n} * \operatorname{d}_\text{n}\!\) \(\operatorname{d}\text{n}\!\) \(\begin{matrix} (\operatorname{d}\text{m})~\operatorname{d}\text{n}~ \\[4pt] ~\operatorname{d}\text{m}~~\operatorname{d}\text{n}~ \end{matrix}\) \(\begin{matrix} \operatorname{d}_\text{n} \\[4pt] \operatorname{d}_\text{m} * \operatorname{d}_\text{n} \end{matrix}\) \(\text{Table 72.1} ~~ \text{Sign Relation of Interpreter A}\!\) \(\text{Object}\!\) \(\text{Sign}\!\) \(\text{Interpretant}\!\) \(\begin{matrix} \text{A} \\ \text{A} \\ \text{A} \\ \text{A} \end{matrix}\) \(\begin{matrix} {}^{\backprime\backprime} \text{A} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{A} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \end{matrix}\) \(\begin{matrix} {}^{\backprime\backprime} \text{A} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{A} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \end{matrix}\) \(\begin{matrix} \text{B} \\ \text{B} \\ \text{B} \\ \text{B} \end{matrix}\) \(\begin{matrix} {}^{\backprime\backprime} \text{B} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{B} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \end{matrix}\) \(\begin{matrix} {}^{\backprime\backprime} \text{B} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{B} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \end{matrix}\) \(\text{Table 72.2} ~~ \text{Dyadic Projection} ~ L(\text{A})_{OS}\!\) \(\text{Object}\!\) \(\text{Sign}\!\) \(\begin{matrix} \text{A} \\ \text{A} \end{matrix}\) \(\begin{matrix} {}^{\backprime\backprime} \text{A} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \end{matrix}\) \(\begin{matrix} \text{B} \\ \text{B} \end{matrix}\) \(\begin{matrix} {}^{\backprime\backprime} \text{B} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \end{matrix}\) \(\text{Table 72.3} ~~ \text{Dyadic Projection} ~ L(\text{A})_{OI}\!\) \(\text{Object}\!\) \(\text{Interpretant}\!\) \(\begin{matrix} \text{A} \\ \text{A} \end{matrix}\) \(\begin{matrix} {}^{\backprime\backprime} \text{A} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \end{matrix}\) \(\begin{matrix} \text{B} \\ \text{B} \end{matrix}\) \(\begin{matrix} {}^{\backprime\backprime} \text{B} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \end{matrix}\) \(\text{Table 72.4} ~~ \text{Dyadic Projection} ~ L(\text{A})_{SI}\!\) \(\text{Sign}\!\) \(\text{Interpretant}\!\) \(\begin{matrix} {}^{\backprime\backprime} \text{A} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{A} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \end{matrix}\) \(\begin{matrix} {}^{\backprime\backprime} \text{A} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{A} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \end{matrix}\) \(\begin{matrix} {}^{\backprime\backprime} \text{B} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{B} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \end{matrix}\) \(\begin{matrix} {}^{\backprime\backprime} \text{B} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{B} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \end{matrix}\) \(\text{Table 73.1} ~~ \text{Sign Relation of Interpreter B}\!\) \(\text{Object}\!\) \(\text{Sign}\!\) \(\text{Interpretant}\!\) \(\begin{matrix} \text{A} \\ \text{A} \\ \text{A} \\ \text{A} \end{matrix}\) \(\begin{matrix} {}^{\backprime\backprime} \text{A} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{A} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \end{matrix}\) \(\begin{matrix} {}^{\backprime\backprime} \text{A} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{A} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \end{matrix}\) \(\begin{matrix} \text{B} \\ \text{B} \\ \text{B} \\ \text{B} \end{matrix}\) \(\begin{matrix} {}^{\backprime\backprime} \text{B} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{B} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \end{matrix}\) \(\begin{matrix} {}^{\backprime\backprime} \text{B} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{B} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \end{matrix}\) \(\text{Table 73.2} ~~ \text{Dyadic Projection} ~ L(\text{B})_{OS}\!\) \(\text{Object}\!\) \(\text{Sign}\!\) \(\begin{matrix} \text{A} \\ \text{A} \end{matrix}\) \(\begin{matrix} {}^{\backprime\backprime} \text{A} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \end{matrix}\) \(\begin{matrix} \text{B} \\ \text{B} \end{matrix}\) \(\begin{matrix} {}^{\backprime\backprime} \text{B} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \end{matrix}\) \(\text{Table 73.3} ~~ \text{Dyadic Projection} ~ L(\text{B})_{OI}\!\) \(\text{Object}\!\) \(\text{Interpretant}\!\) \(\begin{matrix} \text{A} \\ \text{A} \end{matrix}\) \(\begin{matrix} {}^{\backprime\backprime} \text{A} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \end{matrix}\) \(\begin{matrix} \text{B} \\ \text{B} \end{matrix}\) \(\begin{matrix} {}^{\backprime\backprime} \text{B} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \end{matrix}\) \(\text{Table 73.4} ~~ \text{Dyadic Projection} ~ L(\text{B})_{SI}\!\) \(\text{Sign}\!\) \(\text{Interpretant}\!\) \(\begin{matrix} {}^{\backprime\backprime} \text{A} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{A} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \end{matrix}\) \(\begin{matrix} {}^{\backprime\backprime} \text{A} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{A} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \end{matrix}\) \(\begin{matrix} {}^{\backprime\backprime} \text{B} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{B} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \end{matrix}\) \(\begin{matrix} {}^{\backprime\backprime} \text{B} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{B} {}^{\prime\prime} \\ {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \end{matrix}\) \(\text{Table 74.1} ~~ \text{Relation} ~ L_0 =\{ (x, y, z) \in \mathbb{B}^3 : x + y + z = 0 \}\!\) \(x\!\) \(y\!\) \(z\!\) \(\begin{matrix}0\\0\\1\\1\end{matrix}\) \(\begin{matrix}0\\1\\0\\1\end{matrix}\) \(\begin{matrix}0\\1\\1\\0\end{matrix}\) \(\text{Table 74.2} ~~ \text{Dyadic Projection} ~ (L_0)_{12}\!\) \(x\!\) \(y\!\) \(\begin{matrix}0\\0\\1\\1\end{matrix}\) \(\begin{matrix}0\\1\\0\\1\end{matrix}\) \(\text{Table 74.3} ~~ \text{Dyadic Projection} ~ (L_0)_{13}\!\) \(x\!\) \(z\!\) \(\begin{matrix}0\\0\\1\\1\end{matrix}\) \(\begin{matrix}0\\1\\1\\0\end{matrix}\) \(\text{Table 74.4} ~~ \text{Dyadic Projection} ~ (L_0)_{23}\!\) \(y\!\) \(z\!\) \(\begin{matrix}0\\1\\0\\1\end{matrix}\) \(\begin{matrix}0\\1\\1\\0\end{matrix}\) \(\text{Table 75.1} ~~ \text{Relation} ~ L_1 =\{ (x, y, z) \in \mathbb{B}^3 : x + y + z = 1 \}\!\) \(x\!\) \(y\!\) \(z\!\) \(\begin{matrix}0\\0\\1\\1\end{matrix}\) \(\begin{matrix}0\\1\\0\\1\end{matrix}\) \(\begin{matrix}1\\0\\0\\1\end{matrix}\) \(\text{Table 75.2} ~~ \text{Dyadic Projection} ~ (L_1)_{12}\!\) \(x\!\) \(y\!\) \(\begin{matrix}0\\0\\1\\1\end{matrix}\) \(\begin{matrix}0\\1\\0\\1\end{matrix}\) \(\text{Table 75.3} ~~ \text{Dyadic Projection} ~ (L_1)_{13}\!\) \(x\!\) \(z\!\) \(\begin{matrix}0\\0\\1\\1\end{matrix}\) \(\begin{matrix}1\\0\\0\\1\end{matrix}\) \(\text{Table 75.4} ~~ \text{Dyadic Projection} ~ (L_1)_{23}\!\) \(y\!\) \(z\!\) \(\begin{matrix}0\\1\\0\\1\end{matrix}\) \(\begin{matrix}1\\0\\0\\1\end{matrix}\) \(\text{Table 76.} ~~ \text{Attributed Sign Relation for Interpreters A and B}\!\) \(\text{Object}\!\) \(\text{Sign}\!\) \(\text{Interpretant}\!\) \(\begin{matrix} \text{A} \\ \text{A} \\ \text{A} \\ \text{A} \end{matrix}\) \(\begin{matrix} {}^{\backprime\backprime} \text{A} {}^{\prime\prime\text{A}} \\ {}^{\backprime\backprime} \text{A} {}^{\prime\prime\text{A}} \\ {}^{\backprime\backprime} \text{A} {}^{\prime\prime\text{A}} \\ {}^{\backprime\backprime} \text{A} {}^{\prime\prime\text{A}} \end{matrix}\) \(\begin{matrix} {}^{\backprime\backprime} \text{A} {}^{\prime\prime\text{A}} \\ {}^{\backprime\backprime} \text{A} {}^{\prime\prime\text{B}} \\ {}^{\backprime\backprime} \text{i} {}^{\prime\prime\text{A}} \\ {}^{\backprime\backprime} \text{u} {}^{\prime\prime\text{B}} \end{matrix}\) \(\begin{matrix} \text{A} \\ \text{A} \\ \text{A} \\ \text{A} \end{matrix}\) \(\begin{matrix} {}^{\backprime\backprime} \text{A} {}^{\prime\prime\text{B}} \\ {}^{\backprime\backprime} \text{A} {}^{\prime\prime\text{B}} \\ {}^{\backprime\backprime} \text{A} {}^{\prime\prime\text{B}} \\ {}^{\backprime\backprime} \text{A} {}^{\prime\prime\text{B}} \end{matrix}\) \(\begin{matrix} {}^{\backprime\backprime} \text{A} {}^{\prime\prime\text{A}} \\ {}^{\backprime\backprime} \text{A} {}^{\prime\prime\text{B}} \\ {}^{\backprime\backprime} \text{i} {}^{\prime\prime\text{A}} \\ {}^{\backprime\backprime} \text{u} {}^{\prime\prime\text{B}} \end{matrix}\) \(\begin{matrix} \text{A} \\ \text{A} \\ \text{A} \\ \text{A} \end{matrix}\) \(\begin{matrix} {}^{\backprime\backprime} \text{i} {}^{\prime\prime\text{A}} \\ {}^{\backprime\backprime} \text{i} {}^{\prime\prime\text{A}} \\ {}^{\backprime\backprime} \text{i} {}^{\prime\prime\text{A}} \\ {}^{\backprime\backprime} \text{i} {}^{\prime\prime\text{A}} \end{matrix}\) \(\begin{matrix} {}^{\backprime\backprime} \text{A} {}^{\prime\prime\text{A}} \\ {}^{\backprime\backprime} \text{A} {}^{\prime\prime\text{B}} \\ {}^{\backprime\backprime} \text{i} {}^{\prime\prime\text{A}} \\ {}^{\backprime\backprime} \text{u} {}^{\prime\prime\text{B}} \end{matrix}\) \(\begin{matrix} \text{A} \\ \text{A} \\ \text{A} \\ \text{A} \end{matrix}\) \(\begin{matrix} {}^{\backprime\backprime} \text{u} {}^{\prime\prime\text{B}} \\ {}^{\backprime\backprime} \text{u} {}^{\prime\prime\text{B}} \\ {}^{\backprime\backprime} \text{u} {}^{\prime\prime\text{B}} \\ {}^{\backprime\backprime} \text{u} {}^{\prime\prime\text{B}} \end{matrix}\) \(\begin{matrix} {}^{\backprime\backprime} \text{A} {}^{\prime\prime\text{A}} \\ {}^{\backprime\backprime} \text{A} {}^{\prime\prime\text{B}} \\ {}^{\backprime\backprime} \text{i} {}^{\prime\prime\text{A}} \\ {}^{\backprime\backprime} \text{u} {}^{\prime\prime\text{B}} \end{matrix}\) \(\begin{matrix} \text{B} \\ \text{B} \\ \text{B} \\ \text{B} \end{matrix}\) \(\begin{matrix} {}^{\backprime\backprime} \text{B} {}^{\prime\prime\text{A}} \\ {}^{\backprime\backprime} \text{B} {}^{\prime\prime\text{A}} \\ {}^{\backprime\backprime} \text{B} {}^{\prime\prime\text{A}} \\ {}^{\backprime\backprime} \text{B} {}^{\prime\prime\text{A}} \end{matrix}\) \(\begin{matrix} {}^{\backprime\backprime} \text{B} {}^{\prime\prime\text{A}} \\ {}^{\backprime\backprime} \text{B} {}^{\prime\prime\text{B}} \\ {}^{\backprime\backprime} \text{i} {}^{\prime\prime\text{B}} \\ {}^{\backprime\backprime} \text{u} {}^{\prime\prime\text{A}} \end{matrix}\) \(\begin{matrix} \text{B} \\ \text{B} \\ \text{B} \\ \text{B} \end{matrix}\) \(\begin{matrix} {}^{\backprime\backprime} \text{B} {}^{\prime\prime\text{B}} \\ {}^{\backprime\backprime} \text{B} {}^{\prime\prime\text{B}} \\ {}^{\backprime\backprime} \text{B} {}^{\prime\prime\text{B}} \\ {}^{\backprime\backprime} \text{B} {}^{\prime\prime\text{B}} \end{matrix}\) \(\begin{matrix} {}^{\backprime\backprime} \text{B} {}^{\prime\prime\text{A}} \\ {}^{\backprime\backprime} \text{B} {}^{\prime\prime\text{B}} \\ {}^{\backprime\backprime} \text{i} {}^{\prime\prime\text{B}} \\ {}^{\backprime\backprime} \text{u} {}^{\prime\prime\text{A}} \end{matrix}\) \(\begin{matrix} \text{B} \\ \text{B} \\ \text{B} \\ \text{B} \end{matrix}\) \(\begin{matrix} {}^{\backprime\backprime} \text{i} {}^{\prime\prime\text{B}} \\ {}^{\backprime\backprime} \text{i} {}^{\prime\prime\text{B}} \\ {}^{\backprime\backprime} \text{i} {}^{\prime\prime\text{B}} \\ {}^{\backprime\backprime} \text{i} {}^{\prime\prime\text{B}} \end{matrix}\) \(\begin{matrix} {}^{\backprime\backprime} \text{B} {}^{\prime\prime\text{A}} \\ {}^{\backprime\backprime} \text{B} {}^{\prime\prime\text{B}} \\ {}^{\backprime\backprime} \text{i} {}^{\prime\prime\text{B}} \\ {}^{\backprime\backprime} \text{u} {}^{\prime\prime\text{A}} \end{matrix}\) \(\begin{matrix} \text{B} \\ \text{B} \\ \text{B} \\ \text{B} \end{matrix}\) \(\begin{matrix} {}^{\backprime\backprime} \text{u} {}^{\prime\prime\text{A}} \\ {}^{\backprime\backprime} \text{u} {}^{\prime\prime\text{A}} \\ {}^{\backprime\backprime} \text{u} {}^{\prime\prime\text{A}} \\ {}^{\backprime\backprime} \text{u} {}^{\prime\prime\text{A}} \end{matrix}\) \(\begin{matrix} {}^{\backprime\backprime} \text{B} {}^{\prime\prime\text{A}} \\ {}^{\backprime\backprime} \text{B} {}^{\prime\prime\text{B}} \\ {}^{\backprime\backprime} \text{i} {}^{\prime\prime\text{B}} \\ {}^{\backprime\backprime} \text{u} {}^{\prime\prime\text{A}} \end{matrix}\) \(\text{Table 77.} ~~ \text{Augmented Sign Relation for Interpreters A and B}\!\) \(\text{Object}\!\) \(\text{Sign}\!\) \(\text{Interpretant}\!\) \(\begin{matrix} \text{A} \\ \text{A} \\ \text{A} \\ \text{A} \end{matrix}\) \(\begin{matrix} {}^{\backprime\backprime} [ {}^\langle \text{A} {}^\rangle ]_\text{A} {}^{\prime\prime} \\ {}^{\backprime\backprime} [ {}^\langle \text{A} {}^\rangle ]_\text{A} {}^{\prime\prime} \\ {}^{\backprime\backprime} [ {}^\langle \text{A} {}^\rangle ]_\text{A} {}^{\prime\prime} \\ {}^{\backprime\backprime} [ {}^\langle \text{A} {}^\rangle ]_\text{A} {}^{\prime\prime} \end{matrix}\) \(\begin{matrix} {}^{\backprime\backprime} [ {}^\langle \text{A} {}^\rangle ]_\text{A} {}^{\prime\prime} \\ {}^{\backprime\backprime} [ {}^\langle \text{A} {}^\rangle ]_\text{B} {}^{\prime\prime} \\ {}^{\backprime\backprime} [ {}^\langle \text{i} {}^\rangle ]_\text{A} {}^{\prime\prime} \\ {}^{\backprime\backprime} [ {}^\langle \text{u} {}^\rangle ]_\text{B} {}^{\prime\prime} \end{matrix}\) \(\begin{matrix} \text{A} \\ \text{A} \\ \text{A} \\ \text{A} \end{matrix}\) \(\begin{matrix} {}^{\backprime\backprime} [ {}^\langle \text{A} {}^\rangle ]_\text{B} {}^{\prime\prime} \\ {}^{\backprime\backprime} [ {}^\langle \text{A} {}^\rangle ]_\text{B} {}^{\prime\prime} \\ {}^{\backprime\backprime} [ {}^\langle \text{A} {}^\rangle ]_\text{B} {}^{\prime\prime} \\ {}^{\backprime\backprime} [ {}^\langle \text{A} {}^\rangle ]_\text{B} {}^{\prime\prime} \end{matrix}\) \(\begin{matrix} {}^{\backprime\backprime} [ {}^\langle \text{A} {}^\rangle ]_\text{A} {}^{\prime\prime} \\ {}^{\backprime\backprime} [ {}^\langle \text{A} {}^\rangle ]_\text{B} {}^{\prime\prime} \\ {}^{\backprime\backprime} [ {}^\langle \text{i} {}^\rangle ]_\text{A} {}^{\prime\prime} \\ {}^{\backprime\backprime} [ {}^\langle \text{u} {}^\rangle ]_\text{B} {}^{\prime\prime} \end{matrix}\) \(\begin{matrix} \text{A} \\ \text{A} \\ \text{A} \\ \text{A} \end{matrix}\) \(\begin{matrix} {}^{\backprime\backprime} [ {}^\langle \text{i} {}^\rangle ]_\text{A} {}^{\prime\prime} \\ {}^{\backprime\backprime} [ {}^\langle \text{i} {}^\rangle ]_\text{A} {}^{\prime\prime} \\ {}^{\backprime\backprime} [ {}^\langle \text{i} {}^\rangle ]_\text{A} {}^{\prime\prime} \\ {}^{\backprime\backprime} [ {}^\langle \text{i} {}^\rangle ]_\text{A} {}^{\prime\prime} \end{matrix}\) \(\begin{matrix} {}^{\backprime\backprime} [ {}^\langle \text{A} {}^\rangle ]_\text{A} {}^{\prime\prime} \\ {}^{\backprime\backprime} [ {}^\langle \text{A} {}^\rangle ]_\text{B} {}^{\prime\prime} \\ {}^{\backprime\backprime} [ {}^\langle \text{i} {}^\rangle ]_\text{A} {}^{\prime\prime} \\ {}^{\backprime\backprime} [ {}^\langle \text{u} {}^\rangle ]_\text{B} {}^{\prime\prime} \end{matrix}\) \(\begin{matrix} \text{A} \\ \text{A} \\ \text{A} \\ \text{A} \end{matrix}\) \(\begin{matrix} {}^{\backprime\backprime} [ {}^\langle \text{u} {}^\rangle ]_\text{B} {}^{\prime\prime} \\ {}^{\backprime\backprime} [ {}^\langle \text{u} {}^\rangle ]_\text{B} {}^{\prime\prime} \\ {}^{\backprime\backprime} [ {}^\langle \text{u} {}^\rangle ]_\text{B} {}^{\prime\prime} \\ {}^{\backprime\backprime} [ {}^\langle \text{u} {}^\rangle ]_\text{B} {}^{\prime\prime} \end{matrix}\) \(\begin{matrix} {}^{\backprime\backprime} [ {}^\langle \text{A} {}^\rangle ]_\text{A} {}^{\prime\prime} \\ {}^{\backprime\backprime} [ {}^\langle \text{A} {}^\rangle ]_\text{B} {}^{\prime\prime} \\ {}^{\backprime\backprime} [ {}^\langle \text{i} {}^\rangle ]_\text{A} {}^{\prime\prime} \\ {}^{\backprime\backprime} [ {}^\langle \text{u} {}^\rangle ]_\text{B} {}^{\prime\prime} \end{matrix}\) \(\begin{matrix} \text{B} \\ \text{B} \\ \text{B} \\ \text{B} \end{matrix}\) \(\begin{matrix} {}^{\backprime\backprime} [ {}^\langle \text{B} {}^\rangle ]_\text{A} {}^{\prime\prime} \\ {}^{\backprime\backprime} [ {}^\langle \text{B} {}^\rangle ]_\text{A} {}^{\prime\prime} \\ {}^{\backprime\backprime} [ {}^\langle \text{B} {}^\rangle ]_\text{A} {}^{\prime\prime} \\ {}^{\backprime\backprime} [ {}^\langle \text{B} {}^\rangle ]_\text{A} {}^{\prime\prime} \end{matrix}\) \(\begin{matrix} {}^{\backprime\backprime} [ {}^\langle \text{B} {}^\rangle ]_\text{A} {}^{\prime\prime} \\ {}^{\backprime\backprime} [ {}^\langle \text{B} {}^\rangle ]_\text{B} {}^{\prime\prime} \\ {}^{\backprime\backprime} [ {}^\langle \text{i} {}^\rangle ]_\text{B} {}^{\prime\prime} \\ {}^{\backprime\backprime} [ {}^\langle \text{u} {}^\rangle ]_\text{A} {}^{\prime\prime} \end{matrix}\) \(\begin{matrix} \text{B} \\ \text{B} \\ \text{B} \\ \text{B} \end{matrix}\) \(\begin{matrix} {}^{\backprime\backprime} [ {}^\langle \text{B} {}^\rangle ]_\text{B} {}^{\prime\prime} \\ {}^{\backprime\backprime} [ {}^\langle \text{B} {}^\rangle ]_\text{B} {}^{\prime\prime} \\ {}^{\backprime\backprime} [ {}^\langle \text{B} {}^\rangle ]_\text{B} {}^{\prime\prime} \\ {}^{\backprime\backprime} [ {}^\langle \text{B} {}^\rangle ]_\text{B} {}^{\prime\prime} \end{matrix}\) \(\begin{matrix} {}^{\backprime\backprime} [ {}^\langle \text{B} {}^\rangle ]_\text{A} {}^{\prime\prime} \\ {}^{\backprime\backprime} [ {}^\langle \text{B} {}^\rangle ]_\text{B} {}^{\prime\prime} \\ {}^{\backprime\backprime} [ {}^\langle \text{i} {}^\rangle ]_\text{B} {}^{\prime\prime} \\ {}^{\backprime\backprime} [ {}^\langle \text{u} {}^\rangle ]_\text{A} {}^{\prime\prime} \end{matrix}\) \(\begin{matrix} \text{B} \\ \text{B} \\ \text{B} \\ \text{B} \end{matrix}\) \(\begin{matrix} {}^{\backprime\backprime} [ {}^\langle \text{i} {}^\rangle ]_\text{B} {}^{\prime\prime} \\ {}^{\backprime\backprime} [ {}^\langle \text{i} {}^\rangle ]_\text{B} {}^{\prime\prime} \\ {}^{\backprime\backprime} [ {}^\langle \text{i} {}^\rangle ]_\text{B} {}^{\prime\prime} \\ {}^{\backprime\backprime} [ {}^\langle \text{i} {}^\rangle ]_\text{B} {}^{\prime\prime} \end{matrix}\) \(\begin{matrix} {}^{\backprime\backprime} [ {}^\langle \text{B} {}^\rangle ]_\text{A} {}^{\prime\prime} \\ {}^{\backprime\backprime} [ {}^\langle \text{B} {}^\rangle ]_\text{B} {}^{\prime\prime} \\ {}^{\backprime\backprime} [ {}^\langle \text{i} {}^\rangle ]_\text{B} {}^{\prime\prime} \\ {}^{\backprime\backprime} [ {}^\langle \text{u} {}^\rangle ]_\text{A} {}^{\prime\prime} \end{matrix}\) \(\begin{matrix} \text{B} \\ \text{B} \\ \text{B} \\ \text{B} \end{matrix}\) \(\begin{matrix} {}^{\backprime\backprime} [ {}^\langle \text{u} {}^\rangle ]_\text{A} {}^{\prime\prime} \\ {}^{\backprime\backprime} [ {}^\langle \text{u} {}^\rangle ]_\text{A} {}^{\prime\prime} \\ {}^{\backprime\backprime} [ {}^\langle \text{u} {}^\rangle ]_\text{A} {}^{\prime\prime} \\ {}^{\backprime\backprime} [ {}^\langle \text{u} {}^\rangle ]_\text{A} {}^{\prime\prime} \end{matrix}\) \(\begin{matrix} {}^{\backprime\backprime} [ {}^\langle \text{B} {}^\rangle ]_\text{A} {}^{\prime\prime} \\ {}^{\backprime\backprime} [ {}^\langle \text{B} {}^\rangle ]_\text{B} {}^{\prime\prime} \\ {}^{\backprime\backprime} [ {}^\langle \text{i} {}^\rangle ]_\text{B} {}^{\prime\prime} \\ {}^{\backprime\backprime} [ {}^\langle \text{u} {}^\rangle ]_\text{A} {}^{\prime\prime} \end{matrix}\) \(\text{Table 80.} ~~ \text{Reflective Extension} ~ \operatorname{Ref}^1 (\text{A})\!\) \(\text{Object}\!\) \(\text{Sign}\!\) \(\text{Interpretant}\!\) \(\begin{matrix} \text{A} \\ \text{A} \\ \text{A} \\ \text{A} \end{matrix}\) \(\begin{matrix} {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{i} {}^{\rangle} \\ {}^{\langle} \text{i} {}^{\rangle} \end{matrix}\) \(\begin{matrix} {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{i} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{i} {}^{\rangle} \end{matrix}\) \(\begin{matrix} \text{B} \\ \text{B} \\ \text{B} \\ \text{B} \end{matrix}\) \(\begin{matrix} {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{u} {}^{\rangle} \\ {}^{\langle} \text{u} {}^{\rangle} \end{matrix}\) \(\begin{matrix} {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{u} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{u} {}^{\rangle} \end{matrix}\) \(\begin{matrix} {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{i} {}^{\rangle} \\ {}^{\langle} \text{u} {}^{\rangle} \end{matrix}\) \(\begin{matrix} {}^{\langle\langle} \text{A} {}^{\rangle\rangle} \\ {}^{\langle\langle} \text{B} {}^{\rangle\rangle} \\ {}^{\langle\langle} \text{i} {}^{\rangle\rangle} \\ {}^{\langle\langle} \text{u} {}^{\rangle\rangle} \end{matrix}\) \(\begin{matrix} {}^{\langle\langle} \text{A} {}^{\rangle\rangle} \\ {}^{\langle\langle} \text{B} {}^{\rangle\rangle} \\ {}^{\langle\langle} \text{i} {}^{\rangle\rangle} \\ {}^{\langle\langle} \text{u} {}^{\rangle\rangle} \end{matrix}\) \(\text{Table 81.} ~~ \text{Reflective Extension} ~ \operatorname{Ref}^1 (\text{B})\!\) \(\text{Object}\!\) \(\text{Sign}\!\) \(\text{Interpretant}\!\) \(\begin{matrix} \text{A} \\ \text{A} \\ \text{A} \\ \text{A} \end{matrix}\) \(\begin{matrix} {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{u} {}^{\rangle} \\ {}^{\langle} \text{u} {}^{\rangle} \end{matrix}\) \(\begin{matrix} {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{u} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{u} {}^{\rangle} \end{matrix}\) \(\begin{matrix} \text{B} \\ \text{B} \\ \text{B} \\ \text{B} \end{matrix}\) \(\begin{matrix} {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{i} {}^{\rangle} \\ {}^{\langle} \text{i} {}^{\rangle} \end{matrix}\) \(\begin{matrix} {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{i} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{i} {}^{\rangle} \end{matrix}\) \(\begin{matrix} {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{i} {}^{\rangle} \\ {}^{\langle} \text{u} {}^{\rangle} \end{matrix}\) \(\begin{matrix} {}^{\langle\langle} \text{A} {}^{\rangle\rangle} \\ {}^{\langle\langle} \text{B} {}^{\rangle\rangle} \\ {}^{\langle\langle} \text{i} {}^{\rangle\rangle} \\ {}^{\langle\langle} \text{u} {}^{\rangle\rangle} \end{matrix}\) \(\begin{matrix} {}^{\langle\langle} \text{A} {}^{\rangle\rangle} \\ {}^{\langle\langle} \text{B} {}^{\rangle\rangle} \\ {}^{\langle\langle} \text{i} {}^{\rangle\rangle} \\ {}^{\langle\langle} \text{u} {}^{\rangle\rangle} \end{matrix}\) \(\text{Table 82.} ~~ \text{Reflective Extension} ~ \operatorname{Ref}^1 (\text{A} | E_1)\!\) \(\text{Object}\!\) \(\text{Sign}\!\) \(\text{Interpretant}\!\) \(\begin{matrix} \text{A} \\ \text{A} \\ \text{A} \\ \text{A} \end{matrix}\) \(\begin{matrix} {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{i} {}^{\rangle} \\ {}^{\langle} \text{i} {}^{\rangle} \end{matrix}\) \(\begin{matrix} {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{i} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{i} {}^{\rangle} \end{matrix}\) \(\begin{matrix} \text{B} \\ \text{B} \\ \text{B} \\ \text{B} \end{matrix}\) \(\begin{matrix} {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{u} {}^{\rangle} \\ {}^{\langle} \text{u} {}^{\rangle} \end{matrix}\) \(\begin{matrix} {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{u} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{u} {}^{\rangle} \end{matrix}\) \(\begin{matrix} {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{i} {}^{\rangle} \\ {}^{\langle} \text{u} {}^{\rangle} \end{matrix}\) \(\begin{matrix} {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{i} {}^{\rangle} \\ {}^{\langle} \text{u} {}^{\rangle} \end{matrix}\) \(\begin{matrix} {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{i} {}^{\rangle} \\ {}^{\langle} \text{u} {}^{\rangle} \end{matrix}\) \(\text{Table 83.} ~~ \text{Reflective Extension} ~ \operatorname{Ref}^1 (\text{B} | E_1)\!\) \(\text{Object}\!\) \(\text{Sign}\!\) \(\text{Interpretant}\!\) \(\begin{matrix} \text{A} \\ \text{A} \\ \text{A} \\ \text{A} \end{matrix}\) \(\begin{matrix} {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{u} {}^{\rangle} \\ {}^{\langle} \text{u} {}^{\rangle} \end{matrix}\) \(\begin{matrix} {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{u} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{u} {}^{\rangle} \end{matrix}\) \(\begin{matrix} \text{B} \\ \text{B} \\ \text{B} \\ \text{B} \end{matrix}\) \(\begin{matrix} {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{i} {}^{\rangle} \\ {}^{\langle} \text{i} {}^{\rangle} \end{matrix}\) \(\begin{matrix} {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{i} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{i} {}^{\rangle} \end{matrix}\) \(\begin{matrix} {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{i} {}^{\rangle} \\ {}^{\langle} \text{u} {}^{\rangle} \end{matrix}\) \(\begin{matrix} {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{i} {}^{\rangle} \\ {}^{\langle} \text{u} {}^{\rangle} \end{matrix}\) \(\begin{matrix} {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{i} {}^{\rangle} \\ {}^{\langle} \text{u} {}^{\rangle} \end{matrix}\) \(\text{Table 84.} ~~ \text{Reflective Extension} ~ \operatorname{Ref}^1 (\text{A} | E_2)\!\) \(\text{Object}\!\) \(\text{Sign}\!\) \(\text{Interpretant}\!\) \(\begin{matrix} \text{A} \\ \text{A} \\ \text{A} \\ \text{A} \end{matrix}\) \(\begin{matrix} {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{i} {}^{\rangle} \\ {}^{\langle} \text{i} {}^{\rangle} \end{matrix}\) \(\begin{matrix} {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{i} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{i} {}^{\rangle} \end{matrix}\) \(\begin{matrix} \text{B} \\ \text{B} \\ \text{B} \\ \text{B} \end{matrix}\) \(\begin{matrix} {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{u} {}^{\rangle} \\ {}^{\langle} \text{u} {}^{\rangle} \end{matrix}\) \(\begin{matrix} {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{u} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{u} {}^{\rangle} \end{matrix}\) \(\begin{matrix} {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{i} {}^{\rangle} \\ {}^{\langle} \text{u} {}^{\rangle} \end{matrix}\) \(\begin{matrix} \text{A} \\ \text{B} \\ \text{A} \\ \text{B} \end{matrix}\) \(\begin{matrix} \text{A} \\ \text{B} \\ \text{A} \\ \text{B} \end{matrix}\) \(\text{Table 85.} ~~ \text{Reflective Extension} ~ \operatorname{Ref}^1 (\text{B} | E_2)\!\) \(\text{Object}\!\) \(\text{Sign}\!\) \(\text{Interpretant}\!\) \(\begin{matrix} \text{A} \\ \text{A} \\ \text{A} \\ \text{A} \end{matrix}\) \(\begin{matrix} {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{u} {}^{\rangle} \\ {}^{\langle} \text{u} {}^{\rangle} \end{matrix}\) \(\begin{matrix} {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{u} {}^{\rangle} \\ {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{u} {}^{\rangle} \end{matrix}\) \(\begin{matrix} \text{B} \\ \text{B} \\ \text{B} \\ \text{B} \end{matrix}\) \(\begin{matrix} {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{i} {}^{\rangle} \\ {}^{\langle} \text{i} {}^{\rangle} \end{matrix}\) \(\begin{matrix} {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{i} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{i} {}^{\rangle} \end{matrix}\) \(\begin{matrix} {}^{\langle} \text{A} {}^{\rangle} \\ {}^{\langle} \text{B} {}^{\rangle} \\ {}^{\langle} \text{i} {}^{\rangle} \\ {}^{\langle} \text{u} {}^{\rangle} \end{matrix}\) \(\begin{matrix} \text{A} \\ \text{B} \\ \text{B} \\ \text{A} \end{matrix}\) \(\begin{matrix} \text{A} \\ \text{B} \\ \text{B} \\ \text{A} \end{matrix}\) Current Work The semiotic equivalence relation given by \(\operatorname{Con}^1 (L_\text{A})\!\) for interpreter \(\text{A}\!\) has the following semiotic equations.   \([ {}^{\langle} \text{A} {}^{\rangle} ]_\text{A}\!\) \(=\!\) \([ {}^{\langle} \text{i} {}^{\rangle} ]_\text{A}\!\)   \([ {}^{\langle} \text{B} {}^{\rangle} ]_\text{A}\!\) \(=\!\) \([ {}^{\langle} \text{u} {}^{\rangle} ]_\text{A}\!\) or  \({}^{\langle} \text{A} {}^{\rangle}\!\) \(=_\text{A}\!\)  \({}^{\langle} \text{i} {}^{\rangle}\!\)    \({}^{\langle} \text{B} {}^{\rangle}\!\) \(=_\text{A}\!\)  \({}^{\langle} \text{u} {}^{\rangle}\!\) These equations induce the following semiotic partition. \( \{ \{ {}^{\langle} \text{A} {}^{\rangle}, {}^{\langle} \text{i} {}^{\rangle} \}, \{ {}^{\langle} \text{B} {}^{\rangle}, {}^{\langle} \text{u} {}^{\rangle} \}, \{ {}^{\langle\langle} \text{A} {}^{\rangle\rangle} \}, \{ {}^{\langle\langle} \text{i} {}^{\rangle\rangle} \}, \{ {}^{\langle\langle} \text{B} {}^{\rangle\rangle} \}, \{ {}^{\langle\langle} \text{u} {}^{\rangle\rangle} \} \}.\! \) The semiotic equivalence relation given by \(\operatorname{Con}^1 (L_\text{B})\!\) for interpreter \(\text{B}\!\) has the following semiotic equations.   \([ {}^{\langle} \text{A} {}^{\rangle} ]_\text{B}\!\) \(=\!\) \([ {}^{\langle} \text{u} {}^{\rangle} ]_\text{B}\!\)   \([ {}^{\langle} \text{B} {}^{\rangle} ]_\text{B}\!\) \(=\!\) \([ {}^{\langle} \text{i} {}^{\rangle} ]_\text{B}\!\) or  \({}^{\langle} \text{A} {}^{\rangle}\!\) \(=_\text{B}\!\)  \({}^{\langle} \text{u} {}^{\rangle}\!\)    \({}^{\langle} \text{B} {}^{\rangle}\!\) \(=_\text{B}\!\)  \({}^{\langle} \text{i} {}^{\rangle}\!\) These equations induce the following semiotic partition. \( \{ \{ {}^{\langle} \text{A} {}^{\rangle}, {}^{\langle} \text{u} {}^{\rangle} \}, \{ {}^{\langle} \text{B} {}^{\rangle}, {}^{\langle} \text{i} {}^{\rangle} \}, \{ {}^{\langle\langle} \text{A} {}^{\rangle\rangle} \}, \{ {}^{\langle\langle} \text{i} {}^{\rangle\rangle} \}, \{ {}^{\langle\langle} \text{B} {}^{\rangle\rangle} \}, \{ {}^{\langle\langle} \text{u} {}^{\rangle\rangle} \} \}.\! \) Table 86. Confounded Sign Relation C Object Sign Interpretant A "A" "A" A "A" "i" A "A" "u" A "i" "A" A "i" "i" A "u" "A" A "u" "u" B "B" "B" B "B" "i" B "B" "u" B "i" "B" B "i" "i" B "u" "B" B "u" "u" Table 87. Disjointed Sign Relation D Object Sign Interpretant AA "A"A "A"A AA "A"A "i"A AA "i"A "A"A AA "i"A "i"A AB "A"B "A"B AB "A"B "u"B AB "u"B "A"B AB "u"B "u"B BA "B"A "B"A BA "B"A "u"A BA "u"A "B"A BA "u"A "u"A BB "B"B "B"B BB "B"B "i"B BB "i"B "B"B BB "i"B "i"B