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Revision as of 13:51, 28 August 2014
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Consider any coordinate position <math>(s, i)\!</math> in the plane <math>S \times I.\!</math> If <math>(s, i)\!</math> is not in <math>L_{SI}\!</math> then there can be no element <math>(o, s, i)\!</math> in <math>L,\!</math> therefore we may restrict our attention to positions <math>(s, i)\!</math> in <math>L_{SI},\!</math> knowing that there exist at least <math>|L_{SI}| = 8\!</math> elements in <math>L,\!</math> and seeking only to determine what objects <math>o\!</math> exist such that <math>(o, s, i)\!</math> is an element in the objective ''fiber'' of <math>(s, i).\!</math> In other words, for what <math>{o \in O}\!</math> is <math>(o, s, i) \in \mathrm{proj}_{SI}^{-1}((s, i))?\!</math> The fact that <math>L_{OS}\!</math> has exactly one element <math>(o, s)\!</math> for each coordinate <math>s \in S\!</math> and that <math>L_{OI}\!</math> has exactly one element <math>(o, i)\!</math> for each coordinate <math>i \in I,\!</math> plus the “coincidence” of it being the same <math>o\!</math> at any one choice for <math>(s, i),\!</math> tells us that <math>L\!</math> has just the one element <math>(o, s, i)\!</math> over each point of <math>S \times I.\!</math> This proves that both <math>L(\text{A})\!</math> and <math>L(\text{B})\!</math> are reducible in an informational sense to triples of dyadic relations, that is, they are ''dyadically reducible''.
Consider any coordinate position <math>(s, i)\!</math> in the plane <math>S \times I.\!</math> If <math>(s, i)\!</math> is not in <math>L_{SI}\!</math> then there can be no element <math>(o, s, i)\!</math> in <math>L,\!</math> therefore we may restrict our attention to positions <math>(s, i)\!</math> in <math>L_{SI},\!</math> knowing that there exist at least <math>|L_{SI}| = 8\!</math> elements in <math>L,\!</math> and seeking only to determine what objects <math>o\!</math> exist such that <math>(o, s, i)\!</math> is an element in the objective ''fiber'' of <math>(s, i).\!</math> In other words, for what <math>{o \in O}\!</math> is <math>(o, s, i) \in \mathrm{proj}_{SI}^{-1}((s, i))?\!</math> The fact that <math>L_{OS}\!</math> has exactly one element <math>(o, s)\!</math> for each coordinate <math>s \in S\!</math> and that <math>L_{OI}\!</math> has exactly one element <math>(o, i)\!</math> for each coordinate <math>i \in I,\!</math> plus the “coincidence” of it being the same <math>o\!</math> at any one choice for <math>(s, i),\!</math> tells us that <math>L\!</math> has just the one element <math>(o, s, i)\!</math> over each point of <math>S \times I.\!</math> This proves that both <math>L(\text{A})\!</math> and <math>L(\text{B})\!</math> are reducible in an informational sense to triples of dyadic relations, that is, they are ''dyadically reducible''.
===6.36. Irreducibly Triadic Relations===
Most likely, any triadic relation <math>L \subseteq X \times Y \times Z\!</math> imposed on arbitrary domains <math>X, Y, Z\!</math> could find use as a sign relation, provided it embodies any constraint at all, in other words, so long as it forms a proper subset of its total space, a relationship symbolized by writing <math>L \subset X \times Y \times Z.\!</math> However, triadic relations of this sort are not guaranteed to form the most natural examples of sign relations.
In order to show what an irreducibly triadic relation looks like, this Section presents a pair of triadic relations that have the same dyadic projections, and thus cannot be distinguished from each other on this basis alone. As it happens, these examples of triadic relations can be discussed independently of sign relational concerns, but structures of their general ilk are frequently found arising in signal-theoretic applications, and they are undoubtedly closely associated with problems of reliable coding and communication.
Tables 74.1 and 75.1 show a pair of irreducibly triadic relations <math>L_0\!</math> and <math>L_1,\!</math> respectively. Tables 74.2 to 74.4 and Tables 75.2 to 75.4 show the dyadic relations comprising <math>\mathrm{Proj}^{(2)} L_0\!</math> and <math>\mathrm{Proj}^{(2)} L_1,\!</math> respectively.
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"
|+ style="height:30px" | <math>\text{Table 74.1} ~~ \text{Relation} ~ L_0 =\{ (x, y, z) \in \mathbb{B}^3 : x + y + z = 0 \}\!</math>
|- style="height:40px; background:#f0f0ff"
| width="33%" | <math>x\!</math>
| width="33%" | <math>y\!</math>
| width="33%" | <math>z\!</math>
|-
| valign="bottom" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}</math>
| valign="bottom" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}</math>
| valign="bottom" | <math>\begin{matrix}0\\1\\1\\0\end{matrix}</math>
|}
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:50%"
|+ style="height:30px" | <math>\text{Table 74.2} ~~ \text{Dyadic Projection} ~ (L_0)_{12}\!</math>
|- style="height:40px; background:#f0f0ff"
| width="33%" | <math>x\!</math>
| width="33%" | <math>y\!</math>
|-
| valign="bottom" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}</math>
| valign="bottom" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}</math>
|}
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:50%"
|+ style="height:30px" | <math>\text{Table 74.3} ~~ \text{Dyadic Projection} ~ (L_0)_{13}\!</math>
|- style="height:40px; background:#f0f0ff"
| width="33%" | <math>x\!</math>
| width="33%" | <math>z\!</math>
|-
| valign="bottom" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}</math>
| valign="bottom" | <math>\begin{matrix}0\\1\\1\\0\end{matrix}</math>
|}
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:50%"
|+ style="height:30px" | <math>\text{Table 74.4} ~~ \text{Dyadic Projection} ~ (L_0)_{23}\!</math>
|- style="height:40px; background:#f0f0ff"
| width="33%" | <math>y\!</math>
| width="33%" | <math>z\!</math>
|-
| valign="bottom" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}</math>
| valign="bottom" | <math>\begin{matrix}0\\1\\1\\0\end{matrix}</math>
|}
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"
|+ style="height:30px" | <math>\text{Table 75.1} ~~ \text{Relation} ~ L_1 =\{ (x, y, z) \in \mathbb{B}^3 : x + y + z = 1 \}\!</math>
|- style="height:40px; background:#f0f0ff"
| width="33%" | <math>x\!</math>
| width="33%" | <math>y\!</math>
| width="33%" | <math>z\!</math>
|-
| valign="bottom" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}</math>
| valign="bottom" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}</math>
| valign="bottom" | <math>\begin{matrix}1\\0\\0\\1\end{matrix}</math>
|}
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:50%"
|+ style="height:30px" | <math>\text{Table 75.2} ~~ \text{Dyadic Projection} ~ (L_1)_{12}\!</math>
|- style="height:40px; background:#f0f0ff"
| width="33%" | <math>x\!</math>
| width="33%" | <math>y\!</math>
|-
| valign="bottom" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}</math>
| valign="bottom" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}</math>
|}
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:50%"
|+ style="height:30px" | <math>\text{Table 75.3} ~~ \text{Dyadic Projection} ~ (L_1)_{13}\!</math>
|- style="height:40px; background:#f0f0ff"
| width="33%" | <math>x\!</math>
| width="33%" | <math>z\!</math>
|-
| valign="bottom" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}</math>
| valign="bottom" | <math>\begin{matrix}1\\0\\0\\1\end{matrix}</math>
|}
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:50%"
|+ style="height:30px" | <math>\text{Table 75.4} ~~ \text{Dyadic Projection} ~ (L_1)_{23}\!</math>
|- style="height:40px; background:#f0f0ff"
| width="33%" | <math>y\!</math>
| width="33%" | <math>z\!</math>
|-
| valign="bottom" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}</math>
| valign="bottom" | <math>\begin{matrix}1\\0\\0\\1\end{matrix}</math>
|}
<br>
The relations <math>L_0, L_1 \subseteq \mathbb{B}^3\!</math> are defined by the following equations, with algebraic operations taking place as in <math>\text{GF}(2),\!</math> that is, with <math>1 + 1 = 0.\!</math>
# The triple <math>(x, y, z)\!</math> in <math>\mathbb{B}^3\!</math> belongs to <math>L_0\!</math> if and only if <math>{x + y + z = 0}.\!</math> Thus, <math>L_0\!</math> is the set of even-parity bit vectors, with <math>x + y = z.\!</math>
# The triple <math>(x, y, z)\!</math> in <math>\mathbb{B}^3\!</math> belongs to <math>L_1\!</math> if and only if <math>{x + y + z = 1}.\!</math> Thus, <math>L_1\!</math> is the set of odd-parity bit vectors, with <math>x + y = z + 1.\!</math>
The corresponding projections of <math>\mathrm{Proj}^{(2)} L_0\!</math> and <math>\mathrm{Proj}^{(2)} L_1\!</math> are identical. In fact, all six projections, taken at the level of logical abstraction, constitute precisely the same dyadic relation, isomorphic to the whole of <math>\mathbb{B} \times \mathbb{B}\!</math> and expressed by the universal constant proposition <math>1 : \mathbb{B} \times \mathbb{B} \to \mathbb{B}.\!</math> In summary:
{| align="center" cellspacing="8" width="90%"
|
<math>\begin{array}{lllll}
(L_0)_{12} & = & (L_1)_{12} & \cong & \mathbb{B}^2
\\[4pt]
(L_0)_{13} & = & (L_1)_{13} & \cong & \mathbb{B}^2
\\[4pt]
(L_0)_{23} & = & (L_1)_{23} & \cong & \mathbb{B}^2
\end{array}</math>
|}
Thus, <math>L_0\!</math> and <math>L_1\!</math> are both examples of irreducibly triadic relations.
===6.37. Propositional Types===
This Section describes a formal system of ''type expressions'' that are analogous to formulas of propositional logic and discusses their use as a calculus of predicates for classifying, analyzing, and drawing typical inferences about <math>k\!</math>-place relations, in particular, for reasoning about the results of operations on relations and about the properties of their transformations and combinations.
'''Definition.''' Given a cartesian product <math>X \times Y,\!</math> an ordered pair <math>(x, y) \in X \times Y\!</math> has the type <math>S \cdot T,\!</math> written <math>(x, y) : S \cdot T,\!</math> if and only if <math>x \in S \subseteq X\!</math> and <math>y \in T \subseteq Y.\!</math> Notice that an ordered pair may have many types.
'''Definition.''' A relation <math>L \subseteq X \times Y\!</math> has type <math>S \cdot T,\!</math> written <math>L : S \cdot T,\!</math> if and only if every <math>(x, y) \in L\!</math> has type <math>S \cdot T,\!</math> that is, if and only if <math>L \subseteq S \times T\!</math> for some <math>S \subseteq X\!</math> and <math>T \subseteq Y.\!</math>
'''Notation.''' Parentheses in the Courier or Teletype font, <math>\texttt{( ... )},\!</math> are used to indicate the negations of propositions and the complements of sets. When a <math>k\!</math>-place relation <math>L\!</math> is initially given relative to the domains <math>X_1, \ldots, X_k\!</math> and a set <math>S\!</math> is mentioned as a subset of one of them, say <math>S \subseteq X_j,\!</math> then the ''relevant complement'' of <math>S\!</math> in such a context is the one taken relative to <math>X_j.\!</math> Thus we have the following equivalents.
{| align="center" cellspacing="8" width="90%"
| <math>\texttt{(} S \texttt{)} ~=~ -\!S ~=~ X_j - S\!</math>
|}
In case of ambiguities that are not resolved by context, indices may be used as follows.
{| align="center" cellspacing="8" width="90%"
| <math>\texttt{(} S \texttt{)}_j ~=~ X_j - S\!</math>
|}
In any case, the intended term can always be written out in full, as <math>X_j - S.\!</math>
<br>
<center>'''Fragments'''</center>
Consider a relation <math>L\!</math> of the following type.
{| align="center" cellspacing="8" width="90%"
| <math>L : \texttt{(} S \texttt{(} T \texttt{))}\!</math>
|}
[The following piece occurs in § 6.35.]
The set of triples of dyadic relations, with pairwise cartesian products chosen in a pre-arranged order from a triple of three sets <math>(X, Y, Z),\!</math> is called the ''dyadic explosion'' of <math>X \times Y \times Z.\!</math> This object is denoted <math>\mathrm{Explo}(X, Y, Z ~|~ 2),\!</math> read as the ''explosion of <math>X \times Y \times Z\!</math> by twos'', or more simply as <math>X, Y, Z ~\mathrm{choose}~ 2,\!</math> and defined as follows:
{| align="center" cellspacing="8" width="90%"
| <math>\mathrm{Explo}(X, Y, Z ~|~ 2) ~=~ \mathrm{Pow}(X \times Y) \times \mathrm{Pow}(X \times Z) \times \mathrm{Pow}(Y \times Z)\!</math>
|}
This domain is defined well enough to serve the immediate purposes of this section, but later it will become necessary to examine its construction more closely.
[Maybe the following piece belongs there, too.]
Just to provide a hint of what's at stake, consider the following suggestive identity:
{| align="center" cellspacing="8" width="90%"
| <math>2^{XY} \times 2^{XZ} \times 2^{YZ} ~=~ 2^{(XY + XY + YZ)}\!</math>
|}
What sense would have to be found for the sums on the right in order to interpret this equation as a set theoretic isomorphism? Answering this question requires the concept of a ''co-product'', roughly speaking, a “disjointed union” of sets. By the time this discussion has detailed the forms of indexing necessary to maintain these constructions, it should have become patently obvious that the forms of analysis and synthesis that are called on to achieve the putative reductions to and reconstructions from dyadic relations in actual fact never really leave the realm of genuinely triadic relations, but merely reshuffle its contents in various convenient fashions.
===6.38. Considering the Source===
There are several ways to contemplate the supplementation of signs, the sorts of augmentation that are crucial to meaning in the case of indices. Some approaches are analytic, in the sense that they regard signs as derivative compounds and try to break up the unitary concept of an individual sign into a congeries of seemingly more real, more actual, or more determinate sign instances. Other approaches are synthetic, in the sense that they accept a given collection of signs at face value and try to reconstruct more objective realities through the formation of abstract categories on this basis.
====6.38.1. Attributed Signs====
One type of analytic method takes it as a maxim for the logic of context that “Every sign or text is indexed by the context in which it occurs”. This means that all signs, including indices, are themselves indexed, though initially only tacitly, by the objective situation, the syntactic context, and the actual interpreter that makes use of them.
To begin formalizing this brand of supplementation, it is necessary to mark salient aspects of the situational, contextual, and inclusively interpretive features of sign usage that were previously held tacit. In effect, signs once regarded as primitive objects need to be newly analyzed as categorical abstractions that cover multitudes of existential sign instances or ''signs in use''.
One way to develop these dimensions of the <math>\text{A}\!</math> and <math>\text{B}\!</math> example is to articulate the interpretive parameters of signs by means of subscripts or superscripts attached to the signs or their quotations, in this way forming a corresponding set of ''situated signs'' or ''attributed remarks''.
The attribution of signs to their interpreters preserves the original object domain but produces an expanded syntactic domain, a corresponding set of ''attributed signs''. In our <math>\text{A}\!</math> and <math>\text{B}\!</math> example this gives the following domains.
{| align="center" cellspacing="6" width="90%"
|
<math>\begin{array}{ccl}
O & = &
\{ \text{A}, \text{B} \}
\\[6pt]
S & = &
\{
{}^{\backprime\backprime} \text{A} {}^{\prime\prime\text{A}},
{}^{\backprime\backprime} \text{B} {}^{\prime\prime\text{A}},
{}^{\backprime\backprime} \text{i} {}^{\prime\prime\text{A}},
{}^{\backprime\backprime} \text{u} {}^{\prime\prime\text{A}},
{}^{\backprime\backprime} \text{A} {}^{\prime\prime\text{B}},
{}^{\backprime\backprime} \text{B} {}^{\prime\prime\text{B}},
{}^{\backprime\backprime} \text{i} {}^{\prime\prime\text{B}},
{}^{\backprime\backprime} \text{u} {}^{\prime\prime\text{B}}
\}
\\[6pt]
I & = &
\{
{}^{\backprime\backprime} \text{A} {}^{\prime\prime\text{A}},
{}^{\backprime\backprime} \text{B} {}^{\prime\prime\text{A}},
{}^{\backprime\backprime} \text{i} {}^{\prime\prime\text{A}},
{}^{\backprime\backprime} \text{u} {}^{\prime\prime\text{A}},
{}^{\backprime\backprime} \text{A} {}^{\prime\prime\text{B}},
{}^{\backprime\backprime} \text{B} {}^{\prime\prime\text{B}},
{}^{\backprime\backprime} \text{i} {}^{\prime\prime\text{B}},
{}^{\backprime\backprime} \text{u} {}^{\prime\prime\text{B}}
\}
\end{array}</math>
|}
Table 76 displays the results of indexing every sign of the <math>\text{A}\!</math> and <math>\text{B}\!</math> example with a superscript indicating its source or ''exponent'', namely, the interpreter who actively communicates or transmits the sign. The operation of attribution produces two new sign relations, but it turns out that both sign relations have the same form and content, so a single Table will do. The new sign relation generated by this operation will be denoted <math>\mathrm{At} (\text{A}, \text{B})\!</math> and called the ''attributed sign relation'' for the <math>\text{A}\!</math> and <math>\text{B}\!</math> example.
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
|+ style="height:30px" | <math>\text{Table 76.} ~~ \text{Attributed Sign Relation for Interpreters A and 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}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{}^{\backprime\backprime} \text{A} {}^{\prime\prime\text{A}}
\\
{}^{\backprime\backprime} \text{A} {}^{\prime\prime\text{A}}
\\
{}^{\backprime\backprime} \text{A} {}^{\prime\prime\text{A}}
\\
{}^{\backprime\backprime} \text{A} {}^{\prime\prime\text{A}}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{}^{\backprime\backprime} \text{A} {}^{\prime\prime\text{A}}
\\
{}^{\backprime\backprime} \text{A} {}^{\prime\prime\text{B}}
\\
{}^{\backprime\backprime} \text{i} {}^{\prime\prime\text{A}}
\\
{}^{\backprime\backprime} \text{u} {}^{\prime\prime\text{B}}
\end{matrix}</math>
|-
| valign="bottom" |
<math>\begin{matrix}
\text{A}
\\
\text{A}
\\
\text{A}
\\
\text{A}
\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{A} {}^{\prime\prime\text{B}}
\\
{}^{\backprime\backprime} \text{A} {}^{\prime\prime\text{B}}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{}^{\backprime\backprime} \text{A} {}^{\prime\prime\text{A}}
\\
{}^{\backprime\backprime} \text{A} {}^{\prime\prime\text{B}}
\\
{}^{\backprime\backprime} \text{i} {}^{\prime\prime\text{A}}
\\
{}^{\backprime\backprime} \text{u} {}^{\prime\prime\text{B}}
\end{matrix}</math>
|-
| valign="bottom" |
<math>\begin{matrix}
\text{A}
\\
\text{A}
\\
\text{A}
\\
\text{A}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{}^{\backprime\backprime} \text{i} {}^{\prime\prime\text{A}}
\\
{}^{\backprime\backprime} \text{i} {}^{\prime\prime\text{A}}
\\
{}^{\backprime\backprime} \text{i} {}^{\prime\prime\text{A}}
\\
{}^{\backprime\backprime} \text{i} {}^{\prime\prime\text{A}}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{}^{\backprime\backprime} \text{A} {}^{\prime\prime\text{A}}
\\
{}^{\backprime\backprime} \text{A} {}^{\prime\prime\text{B}}
\\
{}^{\backprime\backprime} \text{i} {}^{\prime\prime\text{A}}
\\
{}^{\backprime\backprime} \text{u} {}^{\prime\prime\text{B}}
\end{matrix}</math>
|-
| valign="bottom" |
<math>\begin{matrix}
\text{A}
\\
\text{A}
\\
\text{A}
\\
\text{A}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{}^{\backprime\backprime} \text{u} {}^{\prime\prime\text{B}}
\\
{}^{\backprime\backprime} \text{u} {}^{\prime\prime\text{B}}
\\
{}^{\backprime\backprime} \text{u} {}^{\prime\prime\text{B}}
\\
{}^{\backprime\backprime} \text{u} {}^{\prime\prime\text{B}}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{}^{\backprime\backprime} \text{A} {}^{\prime\prime\text{A}}
\\
{}^{\backprime\backprime} \text{A} {}^{\prime\prime\text{B}}
\\
{}^{\backprime\backprime} \text{i} {}^{\prime\prime\text{A}}
\\
{}^{\backprime\backprime} \text{u} {}^{\prime\prime\text{B}}
\end{matrix}</math>
|-
| valign="bottom" |
<math>\begin{matrix}
\text{B}
\\
\text{B}
\\
\text{B}
\\
\text{B}
\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{B} {}^{\prime\prime\text{A}}
\\
{}^{\backprime\backprime} \text{B} {}^{\prime\prime\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{B}}
\\
{}^{\backprime\backprime} \text{i} {}^{\prime\prime\text{B}}
\\
{}^{\backprime\backprime} \text{u} {}^{\prime\prime\text{A}}
\end{matrix}</math>
|-
| valign="bottom" |
<math>\begin{matrix}
\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{B} {}^{\prime\prime\text{B}}
\\
{}^{\backprime\backprime} \text{B} {}^{\prime\prime\text{B}}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{}^{\backprime\backprime} \text{B} {}^{\prime\prime\text{A}}
\\
{}^{\backprime\backprime} \text{B} {}^{\prime\prime\text{B}}
\\
{}^{\backprime\backprime} \text{i} {}^{\prime\prime\text{B}}
\\
{}^{\backprime\backprime} \text{u} {}^{\prime\prime\text{A}}
\end{matrix}</math>
|-
| valign="bottom" |
<math>\begin{matrix}
\text{B}
\\
\text{B}
\\
\text{B}
\\
\text{B}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{}^{\backprime\backprime} \text{i} {}^{\prime\prime\text{B}}
\\
{}^{\backprime\backprime} \text{i} {}^{\prime\prime\text{B}}
\\
{}^{\backprime\backprime} \text{i} {}^{\prime\prime\text{B}}
\\
{}^{\backprime\backprime} \text{i} {}^{\prime\prime\text{B}}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{}^{\backprime\backprime} \text{B} {}^{\prime\prime\text{A}}
\\
{}^{\backprime\backprime} \text{B} {}^{\prime\prime\text{B}}
\\
{}^{\backprime\backprime} \text{i} {}^{\prime\prime\text{B}}
\\
{}^{\backprime\backprime} \text{u} {}^{\prime\prime\text{A}}
\end{matrix}</math>
|-
| valign="bottom" |
<math>\begin{matrix}
\text{B}
\\
\text{B}
\\
\text{B}
\\
\text{B}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{}^{\backprime\backprime} \text{u} {}^{\prime\prime\text{A}}
\\
{}^{\backprime\backprime} \text{u} {}^{\prime\prime\text{A}}
\\
{}^{\backprime\backprime} \text{u} {}^{\prime\prime\text{A}}
\\
{}^{\backprime\backprime} \text{u} {}^{\prime\prime\text{A}}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{}^{\backprime\backprime} \text{B} {}^{\prime\prime\text{A}}
\\
{}^{\backprime\backprime} \text{B} {}^{\prime\prime\text{B}}
\\
{}^{\backprime\backprime} \text{i} {}^{\prime\prime\text{B}}
\\
{}^{\backprime\backprime} \text{u} {}^{\prime\prime\text{A}}
\end{matrix}</math>
|}
<br>
Thus informed, the semiotic equivalence relation for interpreter <math>\text{A}\!</math> yields the following semiotic equations.
{| cellpadding="10"
| width="10%" |
| <math>[{}^{\backprime\backprime} \text{A} {}^{\prime\prime\text{A}}]_\text{A}\!</math>
| <math>=\!</math>
| <math>[{}^{\backprime\backprime} \text{A} {}^{\prime\prime\text{B}}]_\text{A}\!</math>
| <math>=\!</math>
| <math>[{}^{\backprime\backprime} \text{i} {}^{\prime\prime\text{A}}]_\text{A}\!</math>
| <math>=\!</math>
| <math>[{}^{\backprime\backprime} \text{u} {}^{\prime\prime\text{B}}]_\text{A}\!</math>
|-
| width="10%" | or
| <math>{}^{\backprime\backprime} \text{A} {}^{\prime\prime\text{A}}\!</math>
| valign="bottom" | <math>=_\text{A}\!</math>
| <math>{}^{\backprime\backprime} \text{A} {}^{\prime\prime\text{B}}\!</math>
| valign="bottom" | <math>=_\text{A}\!</math>
| <math>{}^{\backprime\backprime} \text{i} {}^{\prime\prime\text{A}}\!</math>
| valign="bottom" | <math>=_\text{A}\!</math>
| <math>{}^{\backprime\backprime} \text{u} {}^{\prime\prime\text{B}}\!</math>
|}
In comparison, the semiotic equivalence relation for interpreter <math>\text{B}\!</math> yields the following semiotic equations.
{| cellpadding="10"
| width="10%" |
| <math>[{}^{\backprime\backprime} \text{A} {}^{\prime\prime\text{A}}]_\text{B}\!</math>
| <math>=\!</math>
| <math>[{}^{\backprime\backprime} \text{A} {}^{\prime\prime\text{B}}]_\text{B}\!</math>
| <math>=\!</math>
| <math>[{}^{\backprime\backprime} \text{i} {}^{\prime\prime\text{A}}]_\text{B}\!</math>
| <math>=\!</math>
| <math>[{}^{\backprime\backprime} \text{u} {}^{\prime\prime\text{B}}]_\text{B}\!</math>
|-
| width="10%" | or
| <math>{}^{\backprime\backprime} \text{A} {}^{\prime\prime\text{A}}\!</math>
| valign="bottom" | <math>=_\text{B}\!</math>
| <math>{}^{\backprime\backprime} \text{A} {}^{\prime\prime\text{B}}\!</math>
| valign="bottom" | <math>=_\text{B}\!</math>
| <math>{}^{\backprime\backprime} \text{i} {}^{\prime\prime\text{A}}\!</math>
| valign="bottom" | <math>=_\text{B}\!</math>
| <math>{}^{\backprime\backprime} \text{u} {}^{\prime\prime\text{B}}\!</math>
|}
Consequently, the semiotic equivalence relations for <math>\text{A}\!</math> and <math>\text{B}\!</math> both induce the same semiotic partition on <math>S,\!</math> namely, the following.
{| align="center" cellspacing="6" width="90%"
|
<math>
\{ \{
{}^{\backprime\backprime} \text{A} {}^{\prime\prime\text{A}},
{}^{\backprime\backprime} \text{A} {}^{\prime\prime\text{B}},
{}^{\backprime\backprime} \text{i} {}^{\prime\prime\text{A}},
{}^{\backprime\backprime} \text{u} {}^{\prime\prime\text{B}}
\}~,~\{
{}^{\backprime\backprime} \text{B} {}^{\prime\prime\text{A}},
{}^{\backprime\backprime} \text{B} {}^{\prime\prime\text{B}},
{}^{\backprime\backprime} \text{i} {}^{\prime\prime\text{B}},
{}^{\backprime\backprime} \text{u} {}^{\prime\prime\text{A}}
\} \}.\!
</math>
|}
By means of a simple attribution step a certain level of congruity has been reached in the community of interpretation comprised of <math>\text{A}\!</math> and <math>\text{B}.\!</math> This new-found agreement on what is abstractly a single semiotic equivalence relation means that its equivalence classes reconstruct the structure of the object domain within the parts of the corresponding semiotic partition. This allows a measure of objectivity or inter-subjectivity to be predicated of the sign relation's representation.
An instance of <math>\text{Y}\!</math> using <math>{}^{\backprime\backprime} \text{X} {}^{\prime\prime}\!</math> is considered to be an objective event, the kind of happening to which all suitably placed observers can point, and adverting to an occurrence of <math>{}^{\backprime\backprime} \text{X} {}^{\prime\prime\text{Y}}\!</math> is more specific and less vague than resorting to instances of <math>{}^{\backprime\backprime} \text{X} {}^{\prime\prime}\!</math> as if being issued by anonymous sources. The situated sign <math>{}^{\backprime\backprime} \text{X} {}^{\prime\prime\text{Y}}\!</math> is a ''wider sign'' than <math>{}^{\backprime\backprime} \text{X} {}^{\prime\prime}\!</math> in the sense that it takes in a broader field of view on the interpretive situation and provides more information about the context of use. As to the reception of attributed remarks, the interpreter that can recognize signs of the form <math>{}^{\backprime\backprime} \text{X} {}^{\prime\prime\text{Y}}\!</math> is one that knows what it means to ''consider the source''.
It is best to read the superscripts on attributed signs as accentuations and integral parts of the quotation marks, taking <math>{}^{\backprime\backprime} \ldots {}^{\prime\prime\text{A}}\!</math> and <math>{}^{\backprime\backprime} \ldots {}^{\prime\prime\text{B}}\!</math> as variant inflections of <math>{}^{\backprime\backprime} \ldots {}^{\prime\prime}.\!</math> Thus, I can refer to the sign <math>{}^{\backprime\backprime} \text{X} {}^{\prime\prime\text{Y}}\!</math> just as I would refer to the sign <math>{}^{\backprime\backprime} \text{X} {}^{\prime\prime}\!</math> in the present informal context, without any additional marks of quotation.
Taking a cue from this usage, the ordinary quotes that I use to mark salient relationships of signs and expressions with respect to the informal context can now be regarded as quotes that I myself, operating as a casual interpreter, tacitly index. Even without knowing the complete sign relation that I have in mind, the one that I presumably use to conduct this discussion, the sign relation that <math>{}^{\backprime\backprime} \text{I} {}^{\prime\prime}\!</math> represents can nevertheless be partially formalized by means of a certain functional equation, namely, the following equation between semantic functions:
{| align="center" cellspacing="8" width="90%"
| <math>{}^{\backprime\backprime} \ldots {}^{\prime\prime} ~=~ {}^{\backprime\backprime} \ldots {}^{\prime\prime\text{I}}\!</math>
|}
By way of vocal expression, the attributed sign <math>{}^{\backprime\backprime} \text{X} {}^{\prime\prime\text{Y}}\!</math> can be pronounced in any of the following ways.
{| align="center" cellspacing="8" width="90%"
|
<math>\begin{array}{l}
{}^{\backprime\backprime} \text{X} {}^{\prime\prime} ~\text{quoth}~ \text{Y}
\\[4pt]
{}^{\backprime\backprime} \text{X} {}^{\prime\prime} ~\text{said by}~ \text{Y}
\\[4pt]
{}^{\backprime\backprime} \text{X} {}^{\prime\prime} ~\text{used by}~ \text{Y}
\end{array}</math>
|}
To facilitate visual imagery, each token of the type <math>{}^{\backprime\backprime} \text{X} {}^{\prime\prime\text{Y}}\!</math> can be pictured as a specific occasion where the sign <math>{}^{\backprime\backprime} \text{X} {}^{\prime\prime}\!</math> is being used or issued by the interpreter <math>\text{Y}.\!</math>
The construal of objects as classes of attributed signs leads to a measure of inter-subjective agreement between the interpreters <math>\text{A}\!</math> and <math>\text{B}.\!</math> Something like this must be the goal of any system of communication, and analogous forms of congruity and gregarity are likely to be found in any system for establishing mutually intelligible responses and maintaining socially coordinated practices.
Nevertheless, the particular types of “analytic” solutions that were proposed for resolving the conflict of interpretations between <math>\text{A}\!</math> and <math>\text{B}\!</math> are conceptually unsatisfactory in several ways. The constructions instituted retain the quality of hypotheses, especially due to the level of speculation about fundamental objects that is required to support them. There remains something fictional and imaginary about the nature of the object instances that are posited to form the ontological infrastructure, the supposedly more determinate strata of being that are presumed to anchor the initial objects of discussion.
Founding objects on a particular selection of object instances is always initially an arbitrary choice, a meet response to a judgment call and a responsibility that cannot be avoided, but still a bit of guesswork that needs to be tested for its reality in practice.
This means that the postulated objects of objects cannot have their reality probed and proved in detail but evaluated only in terms of their conceivable practical effects.
====6.38.2. Augmented Signs====
One synthetic method …
Suppose now that each of the agents <math>\text{A}\!</math> and <math>\text{B}\!</math> reflects on the situational context of their discussion and observes on every occasion of utterance exactly who is saying what. By this critically reflective operation of ''considering the source'' each interpreter is empowered to create, in effect, an ''extended token'' or ''situated sign'' out of each utterance by indexing it with the proper name of its utterer. Though it arises by reflection, the augmented sign is not a higher order of abstraction so much as a restoration or reconstitution of what was lost by abstracting the sign from the signer in the first instance.
In order to continue the development of this example, I need to employ a more precise system of marking quotations in order to keep track of who says what and in what kinds of context. To help with this, I use raised angle brackets <math>{}^\langle \ldots {}^\rangle\!</math> on a par with ordinary quotation marks <math>{}^{\backprime\backprime} \ldots {}^{\prime\prime}\!</math> to call attention to pieces of text as signs or expressions. The angle quotes are especially useful for embedded quotations and for text regarded as used or mentioned by interpreters other than myself, for instance, by the fictional characters <math>\text{A}\!</math> and <math>\text{B}.\!</math> Whenever possible, I save ordinary quotes for the outermost level, the one that interfaces with the context of informal discussion.
A notation like <math>{}^{\backprime\backprime ~ \langle\langle} \text{A} {}^\rangle, \text{B}, \text{C} {}^{\rangle ~ \prime\prime}\!</math> is intended to indicate the construction of an extended (attributed, indexed, or situated) sign, in this case, by enclosing an initial sign <math>{}^{\backprime\backprime} \text{A} {}^{\prime\prime}\!</math> in a contextual envelope <math>{}^{\backprime\backprime ~ \langle\langle} ~\underline{~}~ {}^\rangle, ~\underline{~}~, ~\underline{~}~ {}^{\rangle ~ \prime\prime}\!</math> and inscribing it with relevant items of situational data, as represented by the signs <math>{}^{\backprime\backprime} \text{B} {}^{\prime\prime}\!</math> and <math>{}^{\backprime\backprime} \text{C} {}^{\prime\prime}.\!</math>
# When a salient component of the situational data represents an observation of the agent <math>\text{B}\!</math> communicating the sign <math>{}^{\backprime\backprime} \text{A} {}^{\prime\prime},\!</math> then the compressed form <math>{}^{\backprime\backprime ~ \langle\langle} \text{A} {}^\rangle \text{B}, \text{C} {}^{\rangle ~ \prime\prime}\!</math> can be used to mark that fact.
# When there is no additional contextual information beyond the marking of a sign's source, the form <math>{}^{\backprime\backprime ~ \langle\langle} \text{A} {}^\rangle \text{B} {}^{\rangle ~ \prime\prime}\!</math> suffices to say that <math>\text{B}\!</math> said <math>{}^{\backprime\backprime} \text{A} {}^{\prime\prime}.\!</math>
With this last modification, angle quotes become like ascribed quotes or attributed remarks, indexed with the name of the interpretive agent that issued the message in question. In sum, the notation <math>{}^{\backprime\backprime ~ \langle\langle} \text{A} {}^\rangle \text{B} {}^{\rangle ~ \prime\prime}\!</math> is intended to situate the sign <math>{}^{\backprime\backprime} \text{A} {}^{\prime\prime}\!</math> in the context of its contemplated use and to index the sign <math>{}^{\backprime\backprime} \text{A} {}^{\prime\prime}\!</math> with the name of the interpreter that is considered to be using it on a given occasion.
The notation <math>{}^{\backprime\backprime ~ \langle\langle} \text{A} {}^\rangle \text{B} {}^{\rangle ~ \prime\prime},~\!</math> read <math>{}^{\backprime\backprime ~ \langle} \text{A} {}^\rangle ~\text{quoth}~ \text{B} {}^{\prime\prime}\!</math> or <math>{}^{\backprime\backprime ~ \langle} \text{A} {}^\rangle ~\text{used by}~ \text{B} {}^{\prime\prime},\!</math> is an expression that indicates the use of the sign <math>{}^{\backprime\backprime} \text{A} {}^{\prime\prime}\!</math> by the interpreter <math>\text{B}.\!</math> The expression inside the outer quotes is referred to as an ''indexed quotation'', since it is indexed by the name of the interpreter to which it is referred.
Since angle quotes with a blank index are equivalent to ordinary quotes, we have the following equivalence. [Not sure about this.]
{| align="center" cellspacing="6" width="90%"
|
<math>{}^{\backprime\backprime} ~ {}^\langle \text{A} {}^\rangle \text{B} ~ {}^{\prime\prime} ~=~ {}^{\langle\langle} \text{A} {}^\rangle \text{B} {}^\rangle\!</math>
|}
Enclosing a piece of text with raised angle brackets and following it with the name of an interpreter is intended to call to mind …
The augmentation of signs by the names of their interpreters preserves the original object domain but produces an extended syntactic domain. In our <math>\text{A}\!</math> and <math>\text{B}\!</math> example this gives the following domains.
{| align="center" cellspacing="8" width="90%"
|
<math>\begin{array}{lll}
O & = & \{ \text{A}, \text{B} \}
\end{array}</math>
|}
{| align="center" cellspacing="8" width="90%"
|
<math>\begin{array}{lllllll}
S
& = &
\{ &
{}^{\backprime\backprime} [ {}^\langle \text{A} {}^\rangle ]_\text{A} {}^{\prime\prime},
{}^{\backprime\backprime} [ {}^\langle \text{B} {}^\rangle ]_\text{A} {}^{\prime\prime},
{}^{\backprime\backprime} [ {}^\langle \text{i} {}^\rangle ]_\text{A} {}^{\prime\prime},
{}^{\backprime\backprime} [ {}^\langle \text{u} {}^\rangle ]_\text{A} {}^{\prime\prime},
&
\\[4pt]
& & &
{}^{\backprime\backprime} [ {}^\langle \text{A} {}^\rangle ]_\text{B} {}^{\prime\prime},
{}^{\backprime\backprime} [ {}^\langle \text{B} {}^\rangle ]_\text{B} {}^{\prime\prime},
{}^{\backprime\backprime} [ {}^\langle \text{i} {}^\rangle ]_\text{B} {}^{\prime\prime},
{}^{\backprime\backprime} [ {}^\langle \text{u} {}^\rangle ]_\text{B} {}^{\prime\prime}
& \}
\\[10pt]
I
& = &
\{ &
{}^{\backprime\backprime} [ {}^\langle \text{A} {}^\rangle ]_\text{A} {}^{\prime\prime},
{}^{\backprime\backprime} [ {}^\langle \text{B} {}^\rangle ]_\text{A} {}^{\prime\prime},
{}^{\backprime\backprime} [ {}^\langle \text{i} {}^\rangle ]_\text{A} {}^{\prime\prime},
{}^{\backprime\backprime} [ {}^\langle \text{u} {}^\rangle ]_\text{A} {}^{\prime\prime},
&
\\[4pt]
& & &
{}^{\backprime\backprime} [ {}^\langle \text{A} {}^\rangle ]_\text{B} {}^{\prime\prime},
{}^{\backprime\backprime} [ {}^\langle \text{B} {}^\rangle ]_\text{B} {}^{\prime\prime},
{}^{\backprime\backprime} [ {}^\langle \text{i} {}^\rangle ]_\text{B} {}^{\prime\prime},
{}^{\backprime\backprime} [ {}^\langle \text{u} {}^\rangle ]_\text{B} {}^{\prime\prime}
& \}
\end{array}</math>
|}
The situated sign or indexed expression <math>{}^{\backprime\backprime} [ {}^\langle \text{A} {}^\rangle ]_\text{B} {}^{\prime\prime}\!</math> presents the sign or expression <math>{}^{\backprime\backprime} \text{A} {}^{\prime\prime}\!</math> as used by the interpreter <math>\text{B}.\!</math> In other words, the sign is indexed by the name of an interpreter to indicate a use of that sign by that interpreter. Thus, <math>{}^{\backprime\backprime} [ {}^\langle \text{A} {}^\rangle ]_\text{B} {}^{\prime\prime}\!</math> augments <math>{}^{\backprime\backprime} \text{A} {}^{\prime\prime}\!</math> to form a new and more complete sign by including additional information about the context of its transmission, in particular, by the consideration of its source.
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
|+ style="height:30px" | <math>\text{Table 77.} ~~ \text{Augmented Sign Relation for Interpreters A and 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}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{}^{\backprime\backprime} [ {}^\langle \text{A} {}^\rangle ]_\text{A} {}^{\prime\prime}
\\
{}^{\backprime\backprime} [ {}^\langle \text{A} {}^\rangle ]_\text{A} {}^{\prime\prime}
\\
{}^{\backprime\backprime} [ {}^\langle \text{A} {}^\rangle ]_\text{A} {}^{\prime\prime}
\\
{}^{\backprime\backprime} [ {}^\langle \text{A} {}^\rangle ]_\text{A} {}^{\prime\prime}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{}^{\backprime\backprime} [ {}^\langle \text{A} {}^\rangle ]_\text{A} {}^{\prime\prime}
\\
{}^{\backprime\backprime} [ {}^\langle \text{A} {}^\rangle ]_\text{B} {}^{\prime\prime}
\\
{}^{\backprime\backprime} [ {}^\langle \text{i} {}^\rangle ]_\text{A} {}^{\prime\prime}
\\
{}^{\backprime\backprime} [ {}^\langle \text{u} {}^\rangle ]_\text{B} {}^{\prime\prime}
\end{matrix}</math>
|-
| valign="bottom" |
<math>\begin{matrix}
\text{A}
\\
\text{A}
\\
\text{A}
\\
\text{A}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{}^{\backprime\backprime} [ {}^\langle \text{A} {}^\rangle ]_\text{B} {}^{\prime\prime}
\\
{}^{\backprime\backprime} [ {}^\langle \text{A} {}^\rangle ]_\text{B} {}^{\prime\prime}
\\
{}^{\backprime\backprime} [ {}^\langle \text{A} {}^\rangle ]_\text{B} {}^{\prime\prime}
\\
{}^{\backprime\backprime} [ {}^\langle \text{A} {}^\rangle ]_\text{B} {}^{\prime\prime}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{}^{\backprime\backprime} [ {}^\langle \text{A} {}^\rangle ]_\text{A} {}^{\prime\prime}
\\
{}^{\backprime\backprime} [ {}^\langle \text{A} {}^\rangle ]_\text{B} {}^{\prime\prime}
\\
{}^{\backprime\backprime} [ {}^\langle \text{i} {}^\rangle ]_\text{A} {}^{\prime\prime}
\\
{}^{\backprime\backprime} [ {}^\langle \text{u} {}^\rangle ]_\text{B} {}^{\prime\prime}
\end{matrix}</math>
|-
| valign="bottom" |
<math>\begin{matrix}
\text{A}
\\
\text{A}
\\
\text{A}
\\
\text{A}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{}^{\backprime\backprime} [ {}^\langle \text{i} {}^\rangle ]_\text{A} {}^{\prime\prime}
\\
{}^{\backprime\backprime} [ {}^\langle \text{i} {}^\rangle ]_\text{A} {}^{\prime\prime}
\\
{}^{\backprime\backprime} [ {}^\langle \text{i} {}^\rangle ]_\text{A} {}^{\prime\prime}
\\
{}^{\backprime\backprime} [ {}^\langle \text{i} {}^\rangle ]_\text{A} {}^{\prime\prime}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{}^{\backprime\backprime} [ {}^\langle \text{A} {}^\rangle ]_\text{A} {}^{\prime\prime}
\\
{}^{\backprime\backprime} [ {}^\langle \text{A} {}^\rangle ]_\text{B} {}^{\prime\prime}
\\
{}^{\backprime\backprime} [ {}^\langle \text{i} {}^\rangle ]_\text{A} {}^{\prime\prime}
\\
{}^{\backprime\backprime} [ {}^\langle \text{u} {}^\rangle ]_\text{B} {}^{\prime\prime}
\end{matrix}</math>
|-
| valign="bottom" |
<math>\begin{matrix}
\text{A}
\\
\text{A}
\\
\text{A}
\\
\text{A}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{}^{\backprime\backprime} [ {}^\langle \text{u} {}^\rangle ]_\text{B} {}^{\prime\prime}
\\
{}^{\backprime\backprime} [ {}^\langle \text{u} {}^\rangle ]_\text{B} {}^{\prime\prime}
\\
{}^{\backprime\backprime} [ {}^\langle \text{u} {}^\rangle ]_\text{B} {}^{\prime\prime}
\\
{}^{\backprime\backprime} [ {}^\langle \text{u} {}^\rangle ]_\text{B} {}^{\prime\prime}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{}^{\backprime\backprime} [ {}^\langle \text{A} {}^\rangle ]_\text{A} {}^{\prime\prime}
\\
{}^{\backprime\backprime} [ {}^\langle \text{A} {}^\rangle ]_\text{B} {}^{\prime\prime}
\\
{}^{\backprime\backprime} [ {}^\langle \text{i} {}^\rangle ]_\text{A} {}^{\prime\prime}
\\
{}^{\backprime\backprime} [ {}^\langle \text{u} {}^\rangle ]_\text{B} {}^{\prime\prime}
\end{matrix}</math>
|-
| valign="bottom" |
<math>\begin{matrix}
\text{B}
\\
\text{B}
\\
\text{B}
\\
\text{B}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{}^{\backprime\backprime} [ {}^\langle \text{B} {}^\rangle ]_\text{A} {}^{\prime\prime}
\\
{}^{\backprime\backprime} [ {}^\langle \text{B} {}^\rangle ]_\text{A} {}^{\prime\prime}
\\
{}^{\backprime\backprime} [ {}^\langle \text{B} {}^\rangle ]_\text{A} {}^{\prime\prime}
\\
{}^{\backprime\backprime} [ {}^\langle \text{B} {}^\rangle ]_\text{A} {}^{\prime\prime}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{}^{\backprime\backprime} [ {}^\langle \text{B} {}^\rangle ]_\text{A} {}^{\prime\prime}
\\
{}^{\backprime\backprime} [ {}^\langle \text{B} {}^\rangle ]_\text{B} {}^{\prime\prime}
\\
{}^{\backprime\backprime} [ {}^\langle \text{i} {}^\rangle ]_\text{B} {}^{\prime\prime}
\\
{}^{\backprime\backprime} [ {}^\langle \text{u} {}^\rangle ]_\text{A} {}^{\prime\prime}
\end{matrix}</math>
|-
| valign="bottom" |
<math>\begin{matrix}
\text{B}
\\
\text{B}
\\
\text{B}
\\
\text{B}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{}^{\backprime\backprime} [ {}^\langle \text{B} {}^\rangle ]_\text{B} {}^{\prime\prime}
\\
{}^{\backprime\backprime} [ {}^\langle \text{B} {}^\rangle ]_\text{B} {}^{\prime\prime}
\\
{}^{\backprime\backprime} [ {}^\langle \text{B} {}^\rangle ]_\text{B} {}^{\prime\prime}
\\
{}^{\backprime\backprime} [ {}^\langle \text{B} {}^\rangle ]_\text{B} {}^{\prime\prime}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{}^{\backprime\backprime} [ {}^\langle \text{B} {}^\rangle ]_\text{A} {}^{\prime\prime}
\\
{}^{\backprime\backprime} [ {}^\langle \text{B} {}^\rangle ]_\text{B} {}^{\prime\prime}
\\
{}^{\backprime\backprime} [ {}^\langle \text{i} {}^\rangle ]_\text{B} {}^{\prime\prime}
\\
{}^{\backprime\backprime} [ {}^\langle \text{u} {}^\rangle ]_\text{A} {}^{\prime\prime}
\end{matrix}</math>
|-
| valign="bottom" |
<math>\begin{matrix}
\text{B}
\\
\text{B}
\\
\text{B}
\\
\text{B}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{}^{\backprime\backprime} [ {}^\langle \text{i} {}^\rangle ]_\text{B} {}^{\prime\prime}
\\
{}^{\backprime\backprime} [ {}^\langle \text{i} {}^\rangle ]_\text{B} {}^{\prime\prime}
\\
{}^{\backprime\backprime} [ {}^\langle \text{i} {}^\rangle ]_\text{B} {}^{\prime\prime}
\\
{}^{\backprime\backprime} [ {}^\langle \text{i} {}^\rangle ]_\text{B} {}^{\prime\prime}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{}^{\backprime\backprime} [ {}^\langle \text{B} {}^\rangle ]_\text{A} {}^{\prime\prime}
\\
{}^{\backprime\backprime} [ {}^\langle \text{B} {}^\rangle ]_\text{B} {}^{\prime\prime}
\\
{}^{\backprime\backprime} [ {}^\langle \text{i} {}^\rangle ]_\text{B} {}^{\prime\prime}
\\
{}^{\backprime\backprime} [ {}^\langle \text{u} {}^\rangle ]_\text{A} {}^{\prime\prime}
\end{matrix}</math>
|-
| valign="bottom" |
<math>\begin{matrix}
\text{B}
\\
\text{B}
\\
\text{B}
\\
\text{B}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{}^{\backprime\backprime} [ {}^\langle \text{u} {}^\rangle ]_\text{A} {}^{\prime\prime}
\\
{}^{\backprime\backprime} [ {}^\langle \text{u} {}^\rangle ]_\text{A} {}^{\prime\prime}
\\
{}^{\backprime\backprime} [ {}^\langle \text{u} {}^\rangle ]_\text{A} {}^{\prime\prime}
\\
{}^{\backprime\backprime} [ {}^\langle \text{u} {}^\rangle ]_\text{A} {}^{\prime\prime}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{}^{\backprime\backprime} [ {}^\langle \text{B} {}^\rangle ]_\text{A} {}^{\prime\prime}
\\
{}^{\backprime\backprime} [ {}^\langle \text{B} {}^\rangle ]_\text{B} {}^{\prime\prime}
\\
{}^{\backprime\backprime} [ {}^\langle \text{i} {}^\rangle ]_\text{B} {}^{\prime\prime}
\\
{}^{\backprime\backprime} [ {}^\langle \text{u} {}^\rangle ]_\text{A} {}^{\prime\prime}
\end{matrix}</math>
|}
<br>
===6.39. Prospective Indices : Pointers to Future Work===
In the effort to unify dynamical, connectionist, and symbolic approaches to intelligent systems, indices supply important stepping stones between the sorts of signs that remain bound to circumscribed theaters of action and the kinds of signs that can function globally as generic symbols. Current technology presents an array of largely accidental discoveries that have been brought into being for implementing indexical systems. Bringing systematic study to bear on this variety of accessory devices and trying to discern within the wealth of incidental features their essential principles and effective ingredients could help to improve the traction this form of bridge affords.
In the points where this project addresses work on the indexical front, a primary task is to show how the ''actual connections'' promised by the definition of indexical signs can be translated into system-theoretic terms and implemented by means of the class of ''dynamic connections'' that can persist in realistic systems.
An offshoot of this investigation would be to explore how indices like pointer variables could be realized within “connectionist” systems. There is no reason in principle why this cannot be done, but I think that pragmatic reasons and practical success will force the contemplation of higher orders of connectivity than those currently fashioned in two-dimensional arrays of connections. To be specific, further advances will require the generative power of genuinely triadic relations to be exploited to the fullest possible degree.
To avert one potential misunderstanding of what this entails, computing with triadic relations is not really a live option unless the algebraic tools and logical calculi needed to do so are developed to greater levels of facility than they are at present. Merely officiating over the storage of “dead letters” in higher dimensional arrays will not do the trick. Turning static sign relations into the orders of dynamic sign processes that can support live inquiries will demand new means of representation and new methods of computation.
To fulfill their intended roles, a formal calculus for sign relations and the associated implementation must be able to address and restore the full dimensionalities of the existential and social matrices in which inquiry takes place. Informational constraints that define objective situations of interest need to be freed from the locally linear confines of the “dia-matrix” and reposted within the realm of the “tri-matrix”, that is, reconstituted in a manner that allows critical reflection on their form and content.
The descriptive and conceptual architectures needed to frame this task must allow space for interlacing forms of “open work”, projects that anticipate the desirability of higher order relations and build in the capability for higher order reflections at the very beginning, and do not merely hope against hope to arrange these capacities as afterthoughts.
