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

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

Table Scraps

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

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

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

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

Work Area

Alternate Text

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

Additive Presentation

Version 1

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

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

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

Version 2

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

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

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

Set Displays

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

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

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

Table Work

Group Operations

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Sign Relations

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Type Tables

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

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

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

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

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

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

Completed Work

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

\(\text{Object}\!\)

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

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

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

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

\(\text{Property}\!\)

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

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

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

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

\(\text{Property}\!\)

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

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

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

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

\(\text{Property}\!\)

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

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

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

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

\(\text{Instance}\!\)

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

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

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

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

\(\text{Instance}\!\)

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

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

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

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

\(\text{Instance}\!\)

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

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

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

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

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

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

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

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

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

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

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

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

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

Current Work

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

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

Table 55.2  LIR1 (Den A):  Denotative Component of A
	Object	Sign	Transition
	<A>W	<a>W	<<a>W, <A>W>
	<A>W	<i>W	<<i>W, <A>W>
	<B>W	<b>W	<<b>W, <B>W>
	<B>W	<u>W	<<u>W, <B>W>

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

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

Table 55.3  LIR1 (Con A):  Connotative Component of A
	Sign	Interpretant	Transition
	<a>W	<a>W	<>dW
	<a>W	<i>W	<da di>dW
	<i>W	<a>W	<da di>dW
	<i>W	<i>W	<>dW
	<b>W	<b>W	<>dW
	<b>W	<u>W	<db du>dW
	<u>W	<b>W	<db du>dW
	<u>W	<u>W	<>dW

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

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

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

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

Table 56.2  LIR1 (Den B):  Denotative Component of B
	Object	Sign	Transition
	<A>W	<a>W	<<a>W, <A>W>
	<A>W	<u>W	<<u>W, <A>W>
	<B>W	<b>W	<<b>W, <B>W>
	<B>W	<i>W	<<i>W, <B>W>

Table 56.3  LIR1 (Con B):  Connotative Component of B
	Sign	Interpretant	Transition
	<a>W	<a>W	<>dW
	<a>W	<u>W	<da du>dW
	<u>W	<a>W	<da du>dW
	<u>W	<u>W	<>dW
	<b>W	<b>W	<>dW
	<b>W	<i>W	<db di>dW
	<i>W	<b>W	<db di>dW
	<i>W	<i>W	<>dW

Table 57.1  Mnemonic Lateral Codes for Interpreters A & B
	Element	Vector	Conjunct Term	Code
	A	10X	 A (B)	<A>X
	B	01X	(A) B 	<B>X
	"A"	1000Y	 a (b)(i)(u)	<a>Y
	"B"	0100Y	(a) b (i)(u)	<b>Y
	"i"	0010Y	(a)(b) i (u)	<i>Y
	"u"	0001Y	(a)(b)(i) u 	<u>Y

Table 57.2  Pragmatic Lateral Codes for Interpreters A & B
	Element	Vector	Conjunct Term	Code
	A	10X	 o1 (o2)	<o1>X
	B	01X	(o1) o2 	<o2>X
	"A"	1000Y	 s1 (s2)(s3)(s4)	<s1>Y
	"B"	0100Y	(s1) s2 (s3)(s4)	<s2>Y
	"i"	0010Y	(s1)(s2) s3 (s4)	<s3>Y
	"u"	0001Y	(s1)(s2)(s3) s4 	<s4>Y

Table 57.3  Abstract Lateral Codes for Interpreters A & B
	Element	Vector	Conjunct Term	Code
	A	10X	 x1 (x2)	<x1>X
	B	01X	(x1) x2 	<x2>X
	"A"	1000Y	 y1 (y2)(y3)(y4)	<y1>Y
	"B"	0100Y	(y1) y2 (y3)(y4)	<y2>Y
	"i"	0010Y	(y1)(y2) y3 (y4)	<y3>Y
	"u"	0001Y	(y1)(y2)(y3) y4 	<y4>Y

Table 58.1  LIR2 (A):  Lateral Representation of A
	Object	Sign	Interpretant
	 A (B)	 a (b)(i)(u)	 a (b)(i)(u)
	 A (B)	 a (b)(i)(u)	(a)(b) i (u)
	 A (B)	(a)(b) i (u)	 a (b)(i)(u)
	 A (B)	(a)(b) i (u)	(a)(b) i (u)
	(A) B 	(a) b (i)(u)	(a) b (i)(u)
	(A) B 	(a) b (i)(u)	(a)(b)(i) u 
	(A) B 	(a)(b)(i) u 	(a) b (i)(u)
	(A) B 	(a)(b)(i) u 	(a)(b)(i) u 

Table 58.2  LIR2 (Den A):  Denotative Component of A
	Object	Sign	Transition
	 A (B)	 a (b)(i)(u)	<<a>Y, <A>X>
	 A (B)	(a)(b) i (u)	<<i>Y, <A>X>
	(A) B 	(a) b (i)(u)	<<b>Y, <B>X>
	(A) B 	(a)(b)(i) u 	<<u>Y, <B>X>

Table 58.3  LIR2 (Con A):  Connotative Component of A
	Sign	Interpretant	Transition
	 a (b)(i)(u)	 a (b)(i)(u)	(da)(db)(di)(du)
	 a (b)(i)(u)	(a)(b) i (u)	 da (db) di (du)
	(a)(b) i (u)	 a (b)(i)(u)	 da (db) di (du)
	(a)(b) i (u)	(a)(b) i (u)	(da)(db)(di)(du)
	(a) b (i)(u)	(a) b (i)(u)	(da)(db)(di)(du)
	(a) b (i)(u)	(a)(b)(i) u 	(da) db (di) du 
	(a)(b)(i) u 	(a) b (i)(u)	(da) db (di) du 
	(a)(b)(i) u 	(a)(b)(i) u 	(da)(db)(di)(du)

Table 59.1  LIR2 (B):  Lateral Representation of B
	Object	Sign	Interpretant
	 A (B)	 a (b)(i)(u)	 a (b)(i)(u)
	 A (B)	 a (b)(i)(u)	(a)(b)(i) u 
	 A (B)	(a)(b)(i) u 	 a (b)(i)(u)
	 A (B)	(a)(b)(i) u 	(a)(b)(i) u 
	(A) B 	(a) b (i)(u)	(a) b (i)(u)
	(A) B 	(a) b (i)(u)	(a)(b) i (u)
	(A) B 	(a)(b) i (u)	(a) b (i)(u)
	(A) B 	(a)(b) i (u)	(a)(b) i (u)

Table 59.2  LIR2 (Den B):  Denotative Component of B
	Object	Sign	Transition
	 A (B)	 a (b)(i)(u)	<<a>Y, <A>X>
	 A (B)	(a)(b)(i) u 	<<u>Y, <A>X>
	(A) B 	(a) b (i)(u)	<<b>Y, <B>X>
	(A) B 	(a)(b) i (u)	<<i>Y, <B>X>

Table 59.3  LIR2 (Con B):  Connotative Component of B
	Sign	Interpretant	Transition
	 a (b)(i)(u)	 a (b)(i)(u)	(da)(db)(di)(du)
	 a (b)(i)(u)	(a)(b)(i) u 	 da (db)(di) du 
	(a)(b)(i) u 	 a (b)(i)(u)	 da (db)(di) du 
	(a)(b)(i) u 	(a)(b)(i) u 	(da)(db)(di)(du)
	(a) b (i)(u)	(a) b (i)(u)	(da)(db)(di)(du)
	(a) b (i)(u)	(a)(b) i (u)	(da) db  di (du)
	(a)(b) i (u)	(a) b (i)(u)	(da) db  di (du)
	(a)(b) i (u)	(a)(b) i (u)	(da)(db)(di)(du)

Table 60.1  LIR3 (A):  Lateral Representation of A
	Object	Sign	Interpretant
	<A>X	<a>Y	<a>Y
	<A>X	<a>Y	<i>Y
	<A>X	<i>Y	<a>Y
	<A>X	<i>Y	<i>Y
	<B>X	<b>Y	<b>Y
	<B>X	<b>Y	<u>Y
	<B>X	<u>Y	<b>Y
	<B>X	<u>Y	<u>Y

Table 60.2  LIR3 (Den A):  Denotative Component of A
	Object	Sign	Transition
	<A>X	<a>Y	<<a>Y, <A>X>
	<A>X	<i>Y	<<i>Y, <A>X>
	<B>X	<b>Y	<<b>Y, <B>X>
	<B>X	<u>Y	<<u>Y, <B>X>

Table 60.3  LIR3 (Con A):  Connotative Component of A
	Sign	Interpretant	Transition
	<a>Y	<a>Y	<>dY
	<a>Y	<i>Y	<da di>dY
	<i>Y	<a>Y	<da di>dY
	<i>Y	<i>Y	<>dY
	<b>Y	<b>Y	<>dY
	<b>Y	<u>Y	<db du>dY
	<u>Y	<b>Y	<db du>dY
	<u>Y	<u>Y	<>dY

Table 61.1  LIR3 (B):  Lateral Representation of B
	Object	Sign	Interpretant
	<A>X	<a>Y	<a>Y
	<A>X	<a>Y	<u>Y
	<A>X	<u>Y	<a>Y
	<A>X	<u>Y	<u>Y
	<B>X	<b>Y	<b>Y
	<B>X	<b>Y	<i>Y
	<B>X	<i>Y	<b>Y
	<B>X	<i>Y	<i>Y

Table 61.2  LIR3 (Den B):  Denotative Component of B
	Object	Sign	Transition
	<A>X	<a>Y	<<a>Y, <A>X>
	<A>X	<u>Y	<<u>Y, <A>X>
	<B>X	<b>Y	<<b>Y, <B>X>
	<B>X	<i>Y	<<i>Y, <B>X>

Table 61.3  LIR3 (Con B):  Connotative Component of B
	Sign	Interpretant	Transition
	<a>Y	<a>Y	<>dY
	<a>Y	<u>Y	<da du>dY
	<u>Y	<a>Y	<da du>dY
	<u>Y	<u>Y	<>dY
	<b>Y	<b>Y	<>dY
	<b>Y	<i>Y	<db di>dY
	<i>Y	<b>Y	<db di>dY
	<i>Y	<i>Y	<>dY