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At a point like this in the development of a formal subject matter, it is customary to introduce elements of a logical calculus that can be used to describe relevant aspects of the formal structures involved and to expedite reasoning about their manifold combinations and decompositions. I will hold off from doing this for sign relations in any formal way at present. Instead, I consider the informal requirements and the foreseeable ends that a suitable calculus for sign relations might be expected to meet, and I present as tentative alternatives a few different ways of proceeding to formalize these intentions.
At a point like this in the development of a formal subject matter, it is customary to introduce elements of a logical calculus that can be used to describe relevant aspects of the formal structures involved and to expedite reasoning about their manifold combinations and decompositions. I will hold off from doing this for sign relations in any formal way at present. Instead, I consider the informal requirements and the foreseeable ends that a suitable calculus for sign relations might be expected to meet, and I present as tentative alternatives a few different ways of proceeding to formalize these intentions.
The first order of business in the “comparative anatomy” and “developmental biology” of sign relations is to undertake a pair of closely related tasks: (1) to examine the structural articulation of highly complex sign relations in terms of the primitive constituents that are found available, and (2) to explain the functional genesis of formal (that is, reflectively considered and critically regarded) sign relations as they naturally arise within the informal context of representational and communicational activities.
The first order of business for the "comparative anatomy" and the "developmental biology" of sign relations is to undertake a pair of closely related tasks: (1) to examine the structural articulation of highly complex sign relations in terms of the primitive constituents that are found available, and (2) to explain the functional genesis of formal (that is, reflectively considered and critically regarded) sign relations as they naturally arise within the informal context of representational and communicational activities.
Converting to a political metaphor, how does the "republic" constituted by a sign relation — the representational community of agents invested with a congeries of legislative, executive, and interpretive powers, employing a consensual body of conventional languages, encompassing a commonwealth of comprehensible meanings, diversely but flexibly manifested in the practical administration of abiding and shared representations — how does all of this first come into being?
Converting to a political metaphor, how does the “republic” constituted by a sign relation — the representational community of agents invested with a congeries of legislative, executive, and interpretive powers, employing a consensual body of conventional languages, encompassing a commonwealth of comprehensible meanings, diversely but flexibly manifested in the practical administration of abiding and shared representations — how does all of this first come into being?
… and their development from primitive/ rudimentary to highly structured …
The grasp of the discussion between A and B that is represented in the separate sign relations given for them can best be described as fragmentary. It fails to capture what everyone knows A and B would know about each other's language use.
The grasp of the discussion between <math>\text{A}\!</math> and <math>\text{B}\!</math> that is represented in their separate sign relations can best be described as fragmentary. It fails to capture what everyone knows <math>\text{A}\!</math> and <math>\text{B}\!</math> would know about each other's language use.
How can the fragmentary system of interpretation (SOI) constituted by the juxtaposition of individual sign relations A and B be combined or developed into a new SOI that represents what agents like A and B are sure to know about each other's language use? In order to make it clear that this is a non trivial question, and in the process to illustrate different ways of combining sign relations, I begin by considering a couple of obvious suggestions for their integration that immediate reflection will show to miss the mark.
How can the fragmentary system of interpretation (SOI) constituted by the juxtaposition of individual sign relations <math>L(\text{A})\!</math> and <math>L(\text{B})\!</math> be combined or developed into a new SOI that represents what agents like <math>\text{A}\!</math> and <math>\text{B}\!</math> are sure to know about each other's language use? In order to make it clear that this is a non-trivial question, and in the process to illustrate different ways of combining sign relations, I begin by considering a couple of obvious suggestions for their integration that immediate reflection will show to miss the mark.
The first thing to try is the set theoretic union of the sign relations. This commonly leads to a "confused" or "confounded" combination of the component sign relations. For example, the sign relation defined as C = A U B is shown in Table 86. Interpreted as a transition digraph on the four points of the syntactic domain S = {"A", "B", "i", "u"}, the sign relation C specifies the following behavior for the conduct of its interpreter:
The first thing to try is the set-theoretic union of the sign relations. This leads to a “confused” or “confounded” combination of the component sign relations. For example, the sign relation defined as <math>L_\text{C} = L_\text{A} \cup L_\text{B}\!</math> is shown in Table 86. Interpreted as a transition digraph on the four points of the syntactic domain <math>S = \{ {}^{\backprime\backprime} \text{A} {}^{\prime\prime}, {}^{\backprime\backprime} \text{B} {}^{\prime\prime}, {}^{\backprime\backprime} \text{i} {}^{\prime\prime}, {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \},\!</math> the sign relation <math>L_\text{C}\!</math> specifies the following behavior for the conduct of its interpreter:
# <math>\text{A}\!\cdot\!L_\text{C}\!</math> has a sling at each point of <math>\{ {}^{\backprime\backprime} \text{A} {}^{\prime\prime}, {}^{\backprime\backprime} \text{i} {}^{\prime\prime}, {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \}\!</math> and two-way arcs on the pairs <math>\{ {}^{\backprime\backprime} \text{A} {}^{\prime\prime}, {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \}\!</math> and <math>\{ {}^{\backprime\backprime} \text{A} {}^{\prime\prime}, {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \}.\!</math>
# <math>\text{B}\!\cdot\!L_\text{C}\!</math> has a sling at each point of <math>\{ {}^{\backprime\backprime} \text{B} {}^{\prime\prime}, {}^{\backprime\backprime} \text{i} {}^{\prime\prime}, {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \}\!</math> and two-way arcs on the pairs <math>\{ {}^{\backprime\backprime} \text{B} {}^{\prime\prime}, {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \}\!</math> and <math>\{ {}^{\backprime\backprime} \text{B} {}^{\prime\prime}, {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \}.\!</math>
These sub-relations do not form equivalence relations on the relevant sets of signs. If closed up under transitive compositions, then <math>\{ {}^{\backprime\backprime} \text{A} {}^{\prime\prime}, {}^{\backprime\backprime} \text{i} {}^{\prime\prime}, {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \}\!</math> are all equivalent in the presence of object <math>\text{A},\!</math> but <math>\{ {}^{\backprime\backprime} \text{B} {}^{\prime\prime}, {}^{\backprime\backprime} \text{i} {}^{\prime\prime}, {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \}\!</math> are all equivalent in the presence of object <math>\text{B}.\!</math> This may accurately represent certain types of political thinking, but it does not constitute the kind of sign relation that is wanted here.
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
</pre>
|+ style="height:30px" | <math>\text{Table 86.} ~~ \text{Confounded Sign Relation} ~ L_\text{C} = L_\text{A} \cup L_\text{B} ~ \!</math>
|- style="height:40px; background:#f0f0ff"
| width="33%" | <math>\text{Object}\!</math>
| width="33%" | <math>\text{Sign}\!</math>
| width="33%" | <math>\text{Interpretant}\!</math>
|-
| valign="bottom" |
<math>\begin{matrix}
\text{A}
\\
\text{A}
\\
\text{A}
\\
\text{A}
\\
\text{A}
\\
\text{A}
\\
\text{A}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{}^{\backprime\backprime} \text{A} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{A} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{A} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{i} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{i} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{u} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{u} {}^{\prime\prime}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{}^{\backprime\backprime} \text{A} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{i} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{u} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{A} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{i} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{A} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{u} {}^{\prime\prime}
\end{matrix}</math>
|-
| valign="bottom" |
<math>\begin{matrix}
\text{B}
\\
\text{B}
\\
\text{B}
\\
\text{B}
\\
\text{B}
\\
\text{B}
\\
\text{B}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{}^{\backprime\backprime} \text{B} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{B} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{B} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{i} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{i} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{u} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{u} {}^{\prime\prime}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{}^{\backprime\backprime} \text{B} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{i} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{u} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{B} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{i} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{B} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{u} {}^{\prime\prime}
\end{matrix}</math>
|}
<br>
<br>
<pre>
Reflecting on this disappointing experience with using simple unions to combine sign relations, it appears that some type of indexed union or categorical co-product might be demanded. Table 87 presents the results of taking the disjoint union <math>\textstyle L_\text{D} = L_\text{A} \coprod L_\text{B}\!</math> to constitute a new sign relation.
</pre>
<br>
<br>
<pre>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
Table 87. Disjointed Sign Relation D
|+ style="height:30px" | <math>\text{Table 87.} ~~ \text{Disjointed Sign Relation} ~ L_\text{D} = L_\text{A} \textstyle\coprod L_\text{B}\!</math>
|- style="height:40px; background:#f0f0ff"
| width="33%" | <math>\text{Object}\!</math>
| width="33%" | <math>\text{Sign}\!</math>
| width="33%" | <math>\text{Interpretant}\!</math>
|-
| valign="bottom" |
<math>\begin{matrix}
\text{A}_\text{A}
\\
\text{A}_\text{A}
\\
\text{A}_\text{A}
\\
\text{A}_\text{A}
\end{matrix}\!</math>
| valign="bottom" |
<math>\begin{matrix}
</pre>
{{}^{\backprime\backprime} \text{A} {}^{\prime\prime}}_\text{A}
\\
{{}^{\backprime\backprime} \text{A} {}^{\prime\prime}}_\text{A}
\\
{{}^{\backprime\backprime} \text{i} {}^{\prime\prime}}_\text{A}
\\
{{}^{\backprime\backprime} \text{i} {}^{\prime\prime}}_\text{A}
\end{matrix}\!</math>
| valign="bottom" |
<math>\begin{matrix}
{{}^{\backprime\backprime} \text{A} {}^{\prime\prime}}_\text{A}
\\
{{}^{\backprime\backprime} \text{i} {}^{\prime\prime}}_\text{A}
\\
{{}^{\backprime\backprime} \text{A} {}^{\prime\prime}}_\text{A}
\\
{{}^{\backprime\backprime} \text{i} {}^{\prime\prime}}_\text{A}
\end{matrix}\!</math>
|-
| valign="bottom" |
<math>\begin{matrix}
\text{A}_\text{B}
\\
\text{A}_\text{B}
\\
\text{A}_\text{B}
\\
\text{A}_\text{B}
\end{matrix}\!</math>
| valign="bottom" |
<math>\begin{matrix}
{{}^{\backprime\backprime} \text{A} {}^{\prime\prime}}_\text{B}
\\
{{}^{\backprime\backprime} \text{A} {}^{\prime\prime}}_\text{B}
\\
{{}^{\backprime\backprime} \text{u} {}^{\prime\prime}}_\text{B}
\\
{{}^{\backprime\backprime} \text{u} {}^{\prime\prime}}_\text{B}
\end{matrix}\!</math>
| valign="bottom" |
<math>\begin{matrix}
{{}^{\backprime\backprime} \text{A} {}^{\prime\prime}}_\text{B}
\\
{{}^{\backprime\backprime} \text{u} {}^{\prime\prime}}_\text{B}
\\
{{}^{\backprime\backprime} \text{A} {}^{\prime\prime}}_\text{B}
\\
{{}^{\backprime\backprime} \text{u} {}^{\prime\prime}}_\text{B}
\end{matrix}\!</math>
|-
| valign="bottom" |
<math>\begin{matrix}
\text{B}_\text{A}
\\
\text{B}_\text{A}
\\
\text{B}_\text{A}
\\
\text{B}_\text{A}
\end{matrix}\!</math>
| valign="bottom" |
<math>\begin{matrix}
{{}^{\backprime\backprime} \text{B} {}^{\prime\prime}}_\text{A}
\\
{{}^{\backprime\backprime} \text{B} {}^{\prime\prime}}_\text{A}
\\
{{}^{\backprime\backprime} \text{u} {}^{\prime\prime}}_\text{A}
\\
{{}^{\backprime\backprime} \text{u} {}^{\prime\prime}}_\text{A}
\end{matrix}\!</math>
| valign="bottom" |
<math>\begin{matrix}
{{}^{\backprime\backprime} \text{B} {}^{\prime\prime}}_\text{A}
\\
{{}^{\backprime\backprime} \text{u} {}^{\prime\prime}}_\text{A}
\\
{{}^{\backprime\backprime} \text{B} {}^{\prime\prime}}_\text{A}
\\
{{}^{\backprime\backprime} \text{u} {}^{\prime\prime}}_\text{A}
\end{matrix}\!</math>
|-
| valign="bottom" |
<math>\begin{matrix}
\text{B}_\text{B}
\\
\text{B}_\text{B}
\\
\text{B}_\text{B}
\\
\text{B}_\text{B}
\end{matrix}\!</math>
| valign="bottom" |
<math>\begin{matrix}
{{}^{\backprime\backprime} \text{B} {}^{\prime\prime}}_\text{B}
\\
{{}^{\backprime\backprime} \text{B} {}^{\prime\prime}}_\text{B}
\\
{{}^{\backprime\backprime} \text{i} {}^{\prime\prime}}_\text{B}
\\
{{}^{\backprime\backprime} \text{i} {}^{\prime\prime}}_\text{B}
\end{matrix}\!</math>
| valign="bottom" |
<math>\begin{matrix}
{{}^{\backprime\backprime} \text{B} {}^{\prime\prime}}_\text{B}
\\
{{}^{\backprime\backprime} \text{i} {}^{\prime\prime}}_\text{B}
\\
{{}^{\backprime\backprime} \text{B} {}^{\prime\prime}}_\text{B}
\\
{{}^{\backprime\backprime} \text{i} {}^{\prime\prime}}_\text{B}
\end{matrix}\!</math>
|}
<br>
===6.50. Revisiting the Source===
===6.50. Revisiting the Source===
