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

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

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

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

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

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

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

Current Work

Table 50.  Notation for Objects & Their Signs
	Object	Sign of Object
	A	A	w1	<A>	<A>	<w1>
	B	B	w2	<B>	<B>	<w2>
	"A"	<A>	w3	<"A">	<<A>>	<w3>
	"B"	<B>	w4	<"B">	<<B>>	<w4>
	"i"	<i>	w5	<"i">	<<i>>	<w5>
	"u"	<u>	w6	<"u">	<<u>>	<w6>

Table 51.1  Notation for Properties & Their Signs (1)
	Property	Sign of Property
	{A}	{A}	{w1}	<{A}>	<{A}>	<{w1}>
	{B}	{B}	{w2}	<{B}>	<{B}>	<{w2}>
	{"A"}	{<A>}	{w3}	<{"A"}>	<{<A>}>	<{w3}>
	{"B"}	{<B>}	{w4}	<{"B"}>	<{<B>}>	<{w4}>
	{"i"}	{<i>}	{w5}	<{"i"}>	<{<i>}>	<{w5}>
	{"u"}	{<u>}	{w6}	<{"u"}>	<{<u>}>	<{w6}>

Table 51.2  Notation for Properties & Their Signs (2)
	Property	Sign of Property
	A	A	w1	<A>	<A>	<w1>
	B	B	w2	<B>	<B>	<w2>
	"A"	<A>	w3	<"A">	<<A>>	<w3>
	"B"	<B>	w4	<"B">	<<B>>	<w4>
	"i"	<i>	w5	<"i">	<<i>>	<w5>
	"u"	<u>	w6	<"u">	<<u>>	<w6>

Table 51.3  Notation for Properties & Their Signs (3)
	Property	Sign of Property
	A	o1	w1	<A>	<o1>	<w1>
	B	o2	w2	<B>	<o2>	<w2>
	a	s1	w3	<a>	<s1>	<w3>
	b	s2	w4	<b>	<s2>	<w4>
	i	s3	w5	<i>	<s3>	<w5>
	u	s4	w6	<u>	<s4>	<w6>

Table 52.1  Notation for Instances & Their Signs (1)
	Instance	Sign of Instance
	[A]	[A]	[w1]	<[A]>	<[A]>	<[w1]>
	[B]	[B]	[w2]	<[B]>	<[B]>	<[w2]>
	["A"]	[<A>]	[w3]	<["A"]>	<[<A>]>	<[w3]>
	["B"]	[<B>]	[w4]	<["B"]>	<[<B>]>	<[w4]>
	["i"]	[<i>]	[w5]	<["i"]>	<[<i>]>	<[w5]>
	["u"]	[<u>]	[w6]	<["u"]>	<[<u>]>	<[w6]>

Table 52.2  Notation for Instances & Their Signs (2)
	Instance	Sign of Instance
	A	A	w1	<A>	<A>	<w1>
	B	B	w2	<B>	<B>	<w2>
	"A"	<A>	w3	<"A">	<<A>>	<w3>
	"B"	<B>	w4	<"B">	<<B>>	<w4>
	"i"	<i>	w5	<"i">	<<i>>	<w5>
	"u"	<u>	w6	<"u">	<<u>>	<w6>

Table 52.3  Notation for Instances & Their Signs (3)
	Instance	Sign of Instance
	A	o1	w1	<A>	<o1>	<w1>
	B	o2	w2	<B>	<o2>	<w2>
	a	s1	w3	<a>	<s1>	<w3>
	b	s2	w4	<b>	<s2>	<w4>
	i	s3	w5	<i>	<s3>	<w5>
	u	s4	w6	<u>	<s4>	<w6>