Sign Relations • Semiotic Equivalence Relations 2

A few items of notation are useful in discussing equivalence relations in general and semiotic equivalence relations in particular.

In general, if $E$ is an equivalence relation on a set $X$ then every element $x$ of $X$ belongs to a unique equivalence class under $E$ called the equivalence class of $x$ under $E$ . Convention provides the square bracket notation for denoting such equivalence classes, in either the form $[x]_E$ or the simpler form $[x]$ when the subscript $E$ is understood. A statement that the elements $x$ and $y$ are equivalent under $E$ is called an equation or an equivalence and may be expressed in any of the following ways.

$\begin{array}{clc} (x, y) & \in & E \\[4pt] x & \in & [y]_E \\[4pt] y & \in & [x]_E \\[4pt] [x]_E & = & [y]_E \\[4pt] x & =_E & y \end{array}$

Thus we have the following definitions.

$\begin{array}{ccc} [x]_E & = & \{ y \in X : (x, y) \in E \} \\[6pt] x =_E y & \Leftrightarrow & (x, y) \in E \end{array}$

In the application to sign relations it is useful to extend the square bracket notation in the following ways. If $L$ is a sign relation whose connotative component $L_{SI}$ is an equivalence relation on $S = I,$ let $[s]_L$ be the equivalence class of $s$ under $L_{SI}.$ In short, $[s]_L = [s]_{L_{SI}}.$ A statement that the signs $x$ and $y$ belong to the same equivalence class under a semiotic equivalence relation $L_{SI}$ is called a semiotic equation (SEQ) and may be written in either of the following forms.

$\begin{array}{clc} [x]_L & = & [y]_L \\[6pt] x & =_L & y \end{array}$

In many situations there is one further adaptation of the square bracket notation for semiotic equivalence classes which can be useful. Namely, when there is known to exist a particular triple $(o, s, i)$ in a sign relation $L,$ it is permissible to let $[o]_L$ be defined as $[s]_L.$ This lets the notation for semiotic equivalence classes harmonize more smoothly with the frequent use of similar devices for the denotations of signs and expressions.

Applying the array of equivalence notations to the sign relations for $A$ and $B$ will serve to illustrate their use and utility.

The semiotic equivalence relation for interpreter $\mathrm{A}$ yields the following semiotic equations.

$\begin{matrix} [ {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime} ]_{L_\mathrm{A}} & = & [ {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime} ]_{L_\mathrm{A}} \\[6pt] [ {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime} ]_{L_\mathrm{A}} & = & [ {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} ]_{L_\mathrm{A}} \end{matrix}$

$\begin{matrix} {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime} & =_{L_\mathrm{A}} & {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime} \\[6pt] {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime} & =_{L_\mathrm{A}} & {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \end{matrix}$

Thus it induces the semiotic partition:

$\{ \{ {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime} \}, \{ {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \} \}.$

The semiotic equivalence relation for interpreter $\mathrm{B}$ yields the following semiotic equations.

$\begin{matrix} [ {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime} ]_{L_\mathrm{B}} & = & [ {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} ]_{L_\mathrm{B}} \\[6pt] [ {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime} ]_{L_\mathrm{B}} & = & [ {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime} ]_{L_\mathrm{B}} \end{matrix}$

$\begin{matrix} {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime} & =_{L_\mathrm{B}} & {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \\[6pt] {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime} & =_{L_\mathrm{B}} & {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime} \end{matrix}$

Thus it induces the semiotic partition:

$\{ \{ {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \}, \{ {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime} \} \}.$

References

Charles S. Peirce (1902), “Parts of Carnegie Application” (L 75), in Carolyn Eisele (ed., 1976), The New Elements of Mathematics by Charles S. Peirce, vol. 4, 13–73. Online.
Awbrey, J.L., and Awbrey, S.M. (1995), “Interpretation as Action : The Risk of Inquiry”, Inquiry : Critical Thinking Across the Disciplines 15(1), pp. 40–52. Archive. Journal. Online (doc) (pdf).

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