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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.
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.
===6.40. Dynamic and Evaluative Frameworks===
The sign relations <math>L(\text{A})\!</math> and <math>L(\text{B})\!</math> are lacking in several dimensions of realistic properties that would ordinarily be more fully developed in the kinds of sign relations that are found to be involved in inquiry. This section initiates a discussion of two such dimensions, the ''dynamic'' and the ''evaluative'' aspects of sign relations, and it treats the materials that are organized along these lines at two broad levels, either ''within'' or ''between'' particular examples of sign relations.
# The ''dynamic dimension'' deals with change. Thus, it details the forms of diversity that sign relations distribute in a temporal process. It is concerned with the transitions that take place from element to element within a sign relation and also with the changes that take place from one whole sign relation to another, thereby generating various types and levels of ''sign process''.
# The ''evaluative dimension'' deals with goals. Thus, it details the forms of diversity that sign relations contribute to a definite purpose. It is concerned with the comparisons that can be made on a scale of values between the elements within a sign relation and also between whole sign relations themselves, with a view toward deciding which is better for a ''designated purpose''.
At the primary level of analysis, one is concerned with the application of these two dimensions ''within'' particular sign relations. At every subsequent level of analysis, one deals with the dynamic transitions and evaluative comparisons that can be contemplated ''between'' particular sign relations. In order to cover all these dimensions, types, and levels of diversity in a unified way, there is need for a substantive term that can allow one to indicate any of the above objects of discussion and thought — including elements of sign relations, particular sign relations, and states of systems — and to regard it as an “object, sign, or state in a certain stage of construction”. I will use the word ''station'' for this purpose.
In order to organize the discussion of these two dimensions, both within and between particular sign relations, and to coordinate their ordinary relation to each other in practical situations, it pays to develop a combined form of ''dynamic evaluative framework'' (DEF), similar in design and utility to the objective frameworks set up earlier.
A ''dynamic evaluative framework'' (DEF) encompasses two dimensions of comparison between stations:
<ol style="list-style-type:decimal">
<li>
<p>A dynamic dimension, as swept out by a process of changing stations, permits comparison between stations in terms of before and after on a scale of temporal order.</p>
<p>A terminal station on a dynamic dimension is called a ''stable station''.</p></li>
<li>
<p>An evaluative dimension permits comparison between stations on a scale of values.</p>
<p>A terminal station on an evaluative dimension is called a ''canonical station'' or a ''standard station''.</p></li></ol>
A station that is both stable and standard is called a ''normal station''.
Consider the following analogies or correspondences that exist between different orders of sign relational structure:
# Just as a sign represents its object and becomes associated with more or less equivalent signs in the minds of interpretive agents, the corpus of signs that embodies a SOI represents in a collective way its own proper object, intended objective, or ''try at objectivity'' (TAO).
# Just as the relationship of a sign to its semantic objects and interpretive associates can be formalized within a single sign relation, the relation of a dynamically changing SOI to its reference environment, developmental goals, and desired characteristics of interpretive performance can be formalized by means of a higher order sign relation, one that further establishes a grounds of comparison for relating the growing SOI, not only to its former and future selves, but to a diverse company of other SOIs.
From an outside perspective the distinction between a sign and its object is usually regarded as obvious, though agents operating in the thick of a SOI often act as though they cannot see the difference. Nevertheless, as a rule in practice, a sign is not a good thing to be confused with its object. Even in the rare and usually controversial cases where an identity of substance is contemplated, usually only for the sake of argument, there is still a distinction of roles to be maintained between the sign and its object. Just so, …
Although there are aspects of inquiry processes that operate within the single sign relation, the characteristic features of inquiry do not come into full bloom until one considers the whole diversity of dynamically developing sign relations. Because it will be some time before this discussion acquires the formal power it needs to deal with higher order sign relations, these issues will need to be treated on an informal basis as they arise, and often in cursory and ''ad hoc'' manner.
===6.41. Elective and Motive Forces===
The <math>\text{A}\!</math> and <math>\text{B}\!</math> example, in the fragmentary aspects of its sign relations presented so far, is unrealistic in its simplification of semantic issues, lacking a full development of many kinds of attributes that almost always become significant in situations of practical interest. Just to mention two related features of importance to inquiry that are missing from this example, there is no sense of directional process and no dimension of differential value defined either within or between the semantic equivalence classes.
When there is a clear sense of dynamic tendency or purposeful direction driving the passage from signs to interpretants in the connotative project of a sign relation, then the study moves from sign relations, statically viewed, to genuine sign processes. In the pragmatic theory of signs, such processes are usually dignified with the name ''semiosis'' and their systematic investigation is called ''semiotics''.
Further, when this dynamism or purpose is consistent and confluent with a differential value system defined on the syntactic domain, then the sign process in question becomes a candidate for the kind of clarity-gaining, canon-seeking process, capable of supporting learning and reasoning, that I classify as an ''inquiry driven system''.
There is a mathematical turn of thought that I will often take in discussing these kinds of issues. Instead of saying that a system has no attribute of a particular type, I will say that it has the attribute, but in a degenerate or trivial sense. This is merely a strategy of classification that allows one to include null cases in a taxonomy and to make use of continuity arguments in passing from case to case in a class of examples. Viewed in this way, each of the sign relations <math>L(\text{A})\!</math> and <math>L(\text{B})\!</math> can be taken to exhibit a trivial dynamic process and a trivial standard of value defined on the syntactic domain.
===6.42. Sign Processes : A Start===
To articulate the dynamic aspects of a sign relation, one can interpret it as determining a discrete or finite state transition system. In the usual ways of doing this, the states of the system are given by the elements of the syntactic domain, while the elements of the object domain correspond to input data or control parameters that affect transitions from signs to interpretant signs in the syntactic state space.
Working from these principles alone, there are numerous ways that a plausible dynamics can be invented for a given sign relation. I will concentrate on two principal forms of dynamic realization, or two ways of interpreting and augmenting sign relations as sign processes.
One form of realization lets each element of the object domain <math>O\!</math> correspond to the observed presence of an object in the environment of the systematic agent. In this interpretation, the object <math>x\!</math> acts as an input datum that causes the system <math>Y\!</math> to shift from whatever sign state it happens to occupy at a given moment to a random sign state in <math>[x]_Y.\!</math> Expressed in a cognitive vein, <math>{}^{\backprime\backprime} Y ~\mathrm{notes}~ x {}^{\prime\prime}.</math>
Another form of realization lets each element of the object domain <math>O\!</math> correspond to the autonomous intention of the systematic agent to denote an object, achieve an objective, or broadly speaking to accomplish any other purpose with respect to an object in its domain. In this interpretation, the object <math>x\!</math> is a control parameter that brings the system <math>Y\!</math> into line with realizing a target set <math>[x]_Y.\!</math>
Tables 78 and 79 show how the sign relations for <math>\text{A}\!</math> and <math>\text{B}\!</math> can be filled out as finite state processes in conformity with the interpretive principles just described. Rather than letting the actions go undefined for some combinations of inputs in <math>O\!</math> and states in <math>S,\!</math> transitions have been added that take the interpreters from whatever else they might have been thinking about to the semantic equivalence classes of their objects. In either modality of realization, cognitive-oriented or control-oriented, the abstract structure of the resulting sign process is exactly the same.
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
|+ style="height:30px" | <math>\text{Table 78.} ~~ \text{Sign Process of Interpreter A}\!</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}
{}^{\backprime\backprime} \text{A} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{A} {}^{\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{A} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{i} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{A} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{i} {}^{\prime\prime}
\\
{}^{\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}</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{A} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{A} {}^{\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{u} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{B} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{u} {}^{\prime\prime}
\\
{}^{\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}</math>
|}
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
|+ style="height:30px" | <math>\text{Table 79.} ~~ \text{Sign Process of Interpreter 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}
{}^{\backprime\backprime} \text{A} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{A} {}^{\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{A} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{u} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{A} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{u} {}^{\prime\prime}
\\
{}^{\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}</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{A} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{A} {}^{\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{B} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{i} {}^{\prime\prime}
\\
{}^{\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}</math>
|}
<br>
Treated in accord with these interpretations, the sign relations <math>L(\text{A})\!</math> and <math>L(\text{B})\!</math> constitute partially degenerate cases of dynamic processes, in which the transitions are totally non-deterministic up to semantic equivalence classes but still manage to preserve those classes. Whether construed as present observation or projective speculation, the most significant feature to note about a sign process is how the contemplation of an object or objective leads the system from a less determined to a more determined condition.
On reflection, one observes that these processes are not completely trivial since they preserve the structure of their semantic partitions. In fact, each sign process preserves the entire topology — the family of sets closed under finite intersections and arbitrary unions — that is generated by its semantic equivalence classes. These topologies, <math>\mathrm{Top}(\text{A})\!</math> and <math>\mathrm{Top}(\text{B}),\!</math> can be viewed as partially ordered sets, <math>\mathrm{Poset}(\text{A})\!</math> and <math>\mathrm{Poset}(\text{B}),\!</math> by taking the inclusion ordering <math>(\subseteq)\!</math> as <math>(\le).\!</math> For each of the interpreters <math>\text{A}\!</math> and <math>\text{B},\!</math> as things stand in their respective orderings <math>\mathrm{Poset}(\text{A})\!</math> and <math>\mathrm{Poset}(\text{B}),\!</math> the semantic equivalence classes of <math>{}^{\backprime\backprime} \text{A} {}^{\prime\prime}\!</math> and <math>{}^{\backprime\backprime} \text{B} {}^{\prime\prime}\!</math> are situated as intermediate elements that are incomparable to each other.
{| align="center" cellspacing="6" width="90%"
|
<math>\begin{array}{lllll}
\mathrm{Top}(\text{A})
& = &
\mathrm{Poset}(\text{A})
& = &
\{
\varnothing,
\{
{}^{\backprime\backprime} \text{A} {}^{\prime\prime},
{}^{\backprime\backprime} \text{i} {}^{\prime\prime}
\},
\{
{}^{\backprime\backprime} \text{B} {}^{\prime\prime},
{}^{\backprime\backprime} \text{u} {}^{\prime\prime}
\},
S
\}.
\\[6pt]
\mathrm{Top}(\text{B})
& = &
\mathrm{Poset}(\text{B})
& = &
\{ \varnothing,
\{
{}^{\backprime\backprime} \text{A} {}^{\prime\prime},
{}^{\backprime\backprime} \text{u} {}^{\prime\prime}
\},
\{
{}^{\backprime\backprime} \text{B} {}^{\prime\prime},
{}^{\backprime\backprime} \text{i} {}^{\prime\prime}
\},
S
\}.
\end{array}</math>
|}
In anticipation of things to come, these orderings are germinal versions of the kinds of semantic hierarchies that will be used in this project to define the ''ontologies'', ''perspectives'', or ''world views'' corresponding to individual interpreters.
When it comes to discussing the stability properties of dynamic systems, the sets that remain invariant under iterated applications of a process are called its ''attractors'' or ''basins of attraction''.
'''Note.''' More care needed here. Strongly and weakly connected components of digraphs?
The dynamic realizations of the sign relations <math>L(\text{A})\!</math> and <math>L(\text{B})\!</math> augment their semantic equivalence relations in an “attractive” way. To describe this additional structure, I introduce a set of graph-theoretical concepts and notations.
The ''attractor'' of <math>x\!</math> in <math>Y.\!</math>
{| align="center" cellspacing="6" width="90%"
|
<math>Y ~\text{at}~ x ~=~ \mathrm{At}[x]_Y ~=~ [x]_Y \cup \{ \text{arcs into}~ [x]_Y \}.</math>
|}
In effect, this discussion of dynamic realizations of sign relations has advanced from considering semiotic partitions as partitioning the set of points in <math>S\!</math> to considering attractors as partitioning the set of arcs in <math>S \times I = S \times S.\!</math>
===6.43. Reflective Extensions===
This section takes up the topic of reflective extensions in a more systematic fashion, starting from the sign relations <math>L(\text{A})\!</math> and <math>L(\text{B})\!</math> once again and keeping its focus within their vicinity, but exploring the space of nearby extensions in greater detail.
Tables 80 and 81 show one way that the sign relations <math>L(\text{A})\!</math> and <math>L(\text{B})\!</math> can be extended in a reflective sense through the use of quotational devices, yielding the ''first order reflective extensions'', <math>\mathrm{Ref}^1 (\text{A})\!</math> and <math>\mathrm{Ref}^1 (\text{B}).\!</math>
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
|+ style="height:30px" |
<math>{\text{Table 80.} ~~ \text{Reflective Extension} ~ \mathrm{Ref}^1 (\text{A})}\!</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}
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\end{matrix}</math>
|-
| valign="bottom" |
<math>\begin{matrix}
\text{B}
\\
\text{B}
\\
\text{B}
\\
\text{B}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\end{matrix}</math>
|-
| valign="bottom" |
<math>\begin{matrix}
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\end{matrix}</math>
| valign="bottom" |
<math>\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}</math>
| valign="bottom" |
<math>\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}</math>
|}
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
|+ style="height:30px" |
<math>{\text{Table 81.} ~~ \text{Reflective Extension} ~ \mathrm{Ref}^1 (\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}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\end{matrix}</math>
|-
| valign="bottom" |
<math>\begin{matrix}
\text{B}
\\
\text{B}
\\
\text{B}
\\
\text{B}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\end{matrix}</math>
|-
| valign="bottom" |
<math>\begin{matrix}
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\end{matrix}</math>
| valign="bottom" |
<math>\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}</math>
| valign="bottom" |
<math>\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}</math>
|}
<br>
The common ''world'' <math>W\!</math> of the reflective extensions <math>\mathrm{Ref}^1 (\text{A})\!</math> and <math>\mathrm{Ref}^1 (\text{B})\!</math> is the totality of objects and signs they contain, namely, the following set of 10 elements.
{| align="center" cellspacing="8" width="90%"
| <math>W = \{ \text{A}, \text{B}, {}^{\langle} \text{A} {}^{\rangle}, {}^{\langle} \text{B} {}^{\rangle}, {}^{\langle} \text{i} {}^{\rangle}, {}^{\langle} \text{u} {}^{\rangle}, {}^{\langle\langle} \text{A} {}^{\rangle\rangle}, {}^{\langle\langle} \text{B} {}^{\rangle\rangle}, {}^{\langle\langle} \text{i} {}^{\rangle\rangle}, {}^{\langle\langle} \text{u} {}^{\rangle\rangle} \}.</math>
|}
Raised angle brackets or ''supercilia'' <math>({}^{\langle} \ldots {}^{\rangle})\!</math> are here being used on a par with ordinary quotation marks <math>({}^{\backprime\backprime} \ldots {}^{\prime\prime})\!</math> to construct a new sign whose object is precisely the sign they enclose.
Regarded as sign relations in their own right, <math>\mathrm{Ref}^1 (\text{A})\!</math> and <math>\mathrm{Ref}^1 (\text{B})\!</math> are formed on the following relational domains.
{| align="center" cellspacing="6" width="90%"
|
<math>\begin{array}{ccccl}
O & = & O^{(1)} \cup O^{(2)} & = &
\{ \text{A}, \text{B} \}
~ \cup ~
\{
{}^{\langle} \text{A} {}^{\rangle},
{}^{\langle} \text{B} {}^{\rangle},
{}^{\langle} \text{i} {}^{\rangle},
{}^{\langle} \text{u} {}^{\rangle}
\}
\\[8pt]
S & = & S^{(1)} \cup S^{(2)} & = &
\{
{}^{\langle} \text{A} {}^{\rangle},
{}^{\langle} \text{B} {}^{\rangle},
{}^{\langle} \text{i} {}^{\rangle},
{}^{\langle} \text{u} {}^{\rangle}
\}
~ \cup ~
\{
{}^{\langle\langle} \text{A} {}^{\rangle\rangle},
{}^{\langle\langle} \text{B} {}^{\rangle\rangle},
{}^{\langle\langle} \text{i} {}^{\rangle\rangle},
{}^{\langle\langle} \text{u} {}^{\rangle\rangle}
\}
\\[8pt]
I & = & I^{(1)} \cup I^{(2)} & = &
\{
{}^{\langle} \text{A} {}^{\rangle},
{}^{\langle} \text{B} {}^{\rangle},
{}^{\langle} \text{i} {}^{\rangle},
{}^{\langle} \text{u} {}^{\rangle}
\}
~ \cup ~
\{
{}^{\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{array}</math>
|}
It may be observed that <math>S\!</math> overlaps with <math>O\!</math> in the set of first-order signs or second-order objects, <math>S^{(1)} = O^{(2)},\!</math> exemplifying the extent to which signs have become objects in the new sign relations.
To discuss how the denotative and connotative aspects of a sign related are affected by its reflective extension it is useful to introduce a few abbreviations. For each sign relation <math>L\!</math> in <math>\{ L_\text{A}, L_\text{B} \}\!</math> the following operations may be defined.
{| align="center" cellspacing="6" width="90%"
|
<math>\begin{array}{lllll}
\mathrm{Den}^1 (L)
& = &
(\mathrm{Ref}^1 (L))_{SO}
& = &
\mathrm{proj}_{OS} (\mathrm{Ref}^1 (L))
\\[6pt]
\mathrm{Con}^1 (L)
& = &
(\mathrm{Ref}^1 (L))_{SI}
& = &
\mathrm{proj}_{SI} (\mathrm{Ref}^1 (L))
\end{array}\!</math>
|}
The dyadic components of sign relations can be given graph-theoretic representations, namely, as ''digraphs'' (directed graphs), that provide concise pictures of their structural and potential dynamic properties. By way of terminology, a directed edge <math>(x, y)\!</math> is called an ''arc'' from point <math>x\!</math> to point <math>y,\!</math> and a self-loop <math>(x, x)\!</math> is called a ''sling'' at <math>x.\!</math>
The denotative components <math>\mathrm{Den}^1 (L_\text{A})\!</math> and <math>\mathrm{Den}^1 (L_\text{B})\!</math> can be viewed as digraphs on the 10 points of the world set <math>W.\!</math> The arcs of these digraphs are given as follows.
<ol>
<li><math>\mathrm{Den}^1 (L_\text{A})\!</math> has an arc from each point of <math>[\text{A}]_\text{A} = \{ {}^{\langle} \text{A} {}^{\rangle}, {}^{\langle} \text{i}{}^{\rangle} \}\!</math> to <math>\text{A}\!</math> and from each point of <math>[\text{B}]_\text{A} = \{ {}^{\langle} \text{B} {}^{\rangle}, {}^{\langle} \text{u} {}^{\rangle} \}\!</math> to <math>\text{B}.\!</math></li>
<li><math>\mathrm{Den}^1 (L_\text{B})\!</math> has an arc from each point of <math>[\text{A}]_\text{B} = \{ {}^{\langle} \text{A} {}^{\rangle}, {}^{\langle} \text{u}{}^{\rangle} \}\!</math> to <math>\text{A}\!</math> and from each point of <math>[\text{B}]_\text{B} = \{ {}^{\langle} \text{B} {}^{\rangle}, {}^{\langle} \text{i} {}^{\rangle} \}\!</math> to <math>\text{B}.\!</math></li>
<li>In the parts added by reflective extension <math>\mathrm{Den}^1 (L_\text{A})\!</math> and <math>\mathrm{Den}^1 (L_\text{B})\!</math> both have arcs from <math>{}^{\langle} s {}^{\rangle}\!</math> to <math>s,\!</math> for each <math>s \in S^{(1)}.\!</math></li>
</ol>
Taken as transition digraphs, <math>\mathrm{Den}^1 (L_\text{A})\!</math> and <math>\mathrm{Den}^1 (L_\text{B})\!</math> summarize the upshots, end results, or effective steps of computation that are involved in the respective evaluations of signs in <math>S\!</math> by <math>\mathrm{Ref}^1 (\text{A})\!</math> and <math>\mathrm{Ref}^1 (\text{B}).\!</math>
The connotative components <math>\mathrm{Con}^1 (L_\text{A})~\!</math> and <math>\mathrm{Con}^1 (L_\text{B})~\!</math> can be viewed as digraphs on the eight points of the syntactic domain <math>S.\!</math> The arcs of these digraphs are given as follows.
<ol>
<li><math>\mathrm{Con}^1 (L_\text{A})\!</math> inherits from <math>L_\text{A}\!</math> the structure of a semiotic equivalence relation on <math>S^{(1)},\!</math> having a sling on each point of <math>S^{(1)},\!</math> arcs in both directions between <math>{}^{\langle} \text{A} {}^{\rangle}\!</math> and <math>{}^{\langle} \text{i}{}^{\rangle},\!</math> and arcs in both directions between <math>{}^{\langle} \text{B} {}^{\rangle}~\!</math> and <math>{}^{\langle} \text{u}{}^{\rangle}.~\!</math> The reflective extension <math>\mathrm{Ref}^1 (L_\text{A})\!</math> adds a sling on each point of <math>S^{(2)},\!</math> creating a semiotic equivalence relation on <math>S.\!</math></li>
<li><math>\mathrm{Con}^1 (L_\text{B})~\!</math> inherits from <math>L_\text{B}\!</math> the structure of a semiotic equivalence relation on <math>S^{(1)},\!</math> having a sling on each point of <math>S^{(1)},\!</math> arcs in both directions between <math>{}^{\langle} \text{A} {}^{\rangle}\!</math> and <math>{}^{\langle} \text{u}{}^{\rangle},\!</math> and arcs in both directions between <math>{}^{\langle} \text{B} {}^{\rangle}~\!</math> and <math>{}^{\langle} \text{i}{}^{\rangle}.~\!</math> The reflective extension <math>\mathrm{Ref}^1 (L_\text{B})\!</math> adds a sling on each point of <math>S^{(2)},\!</math> creating a semiotic equivalence relation on <math>S.\!</math></li>
</ol>
Taken as transition digraphs, <math>\mathrm{Con}^1 (L_\text{A})~\!</math> and <math>\mathrm{Con}^1 (L_\text{B})~\!</math> highlight the associations between signs in <math>\mathrm{Ref}^1 (L_\text{A})\!</math> and <math>\mathrm{Ref}^1 (L_\text{B}),\!</math> respectively.
The semiotic equivalence relation given by <math>\mathrm{Con}^1 (L_\text{A})\!</math> for interpreter <math>\text{A}\!</math> has the following semiotic equations.
{| cellpadding="10"
| width="10%" |
| <math>[ {}^{\langle} \text{A} {}^{\rangle} ]_\text{A}\!</math>
| <math>=\!</math>
| <math>[ {}^{\langle} \text{i} {}^{\rangle} ]_\text{A}\!</math>
| width="20%" |
| <math>[ {}^{\langle} \text{B} {}^{\rangle} ]_\text{A}\!</math>
| <math>=\!</math>
| <math>[ {}^{\langle} \text{u} {}^{\rangle} ]_\text{A}\!</math>
|-
| width="10%" | or
| <math>{}^{\langle} \text{A} {}^{\rangle}~\!</math>
| <math>=_\text{A}\!</math>
| <math>{}^{\langle} \text{i} {}^{\rangle}~\!</math>
| width="20%" |
| <math>{}^{\langle} \text{B} {}^{\rangle}~\!</math>
| <math>=_\text{A}\!</math>
| <math>{}^{\langle} \text{u} {}^{\rangle}~\!</math>
|}
These equations induce the following semiotic partition.
{| align="center" cellspacing="6" width="90%"
|
<math>
\{
\{ {}^{\langle} \text{A} {}^{\rangle}, {}^{\langle} \text{i} {}^{\rangle} \},
\{ {}^{\langle} \text{B} {}^{\rangle}, {}^{\langle} \text{u} {}^{\rangle} \},
\{ {}^{\langle\langle} \text{A} {}^{\rangle\rangle} \},
\{ {}^{\langle\langle} \text{i} {}^{\rangle\rangle} \},
\{ {}^{\langle\langle} \text{B} {}^{\rangle\rangle} \},
\{ {}^{\langle\langle} \text{u} {}^{\rangle\rangle} \}
\}.\!
</math>
|}
The semiotic equivalence relation given by <math>\mathrm{Con}^1 (L_\text{B})~\!</math> for interpreter <math>\text{B}\!</math> has the following semiotic equations.
{| cellpadding="10"
| width="10%" |
| <math>[ {}^{\langle} \text{A} {}^{\rangle} ]_\text{B}\!</math>
| <math>=\!</math>
| <math>[ {}^{\langle} \text{u} {}^{\rangle} ]_\text{B}\!</math>
| width="20%" |
| <math>[ {}^{\langle} \text{B} {}^{\rangle} ]_\text{B}\!</math>
| <math>=\!</math>
| <math>[ {}^{\langle} \text{i} {}^{\rangle} ]_\text{B}\!</math>
|-
| width="10%" | or
| <math>{}^{\langle} \text{A} {}^{\rangle}~\!</math>
| <math>=_\text{B}\!</math>
| <math>{}^{\langle} \text{u} {}^{\rangle}~\!</math>
| width="20%" |
| <math>{}^{\langle} \text{B} {}^{\rangle}~\!</math>
| <math>=_\text{B}\!</math>
| <math>{}^{\langle} \text{i} {}^{\rangle}~\!</math>
|}
These equations induce the following semiotic partition.
{| align="center" cellspacing="6" width="90%"
|
<math>
\{
\{ {}^{\langle} \text{A} {}^{\rangle}, {}^{\langle} \text{u} {}^{\rangle} \},
\{ {}^{\langle} \text{B} {}^{\rangle}, {}^{\langle} \text{i} {}^{\rangle} \},
\{ {}^{\langle\langle} \text{A} {}^{\rangle\rangle} \},
\{ {}^{\langle\langle} \text{i} {}^{\rangle\rangle} \},
\{ {}^{\langle\langle} \text{B} {}^{\rangle\rangle} \},
\{ {}^{\langle\langle} \text{u} {}^{\rangle\rangle} \}
\}.\!
</math>
|}
Notice that the semiotic equivalences of nouns and pronouns for each interpreter do not extend to equivalences of their second-order signs, exactly as demanded by the literal character of quotations. Moreover, the new sign relations for interpreters <math>\text{A}\!</math> and <math>\text{B}\!</math> coincide in their reflective parts, since exactly the same triples are added to each set.
There are many ways to extend sign relations in an effort to increase their reflective capacities. The implicit goal of a reflective project is to achieve ''reflective closure'', <math>S \subseteq O,\!</math> where every sign is an object.
Considered as reflective extensions, there is nothing unique about the constructions of <math>\mathrm{Ref}^1 (\text{A})\!</math> and <math>\mathrm{Ref}^1 (\text{B})\!</math> but their common pattern of development illustrates a typical approach toward reflective closure. In a sense it epitomizes the project of ''free'', ''naive'', or ''uncritical'' reflection, since continuing this mode of production to its closure would generate an infinite sign relation, passing through infinitely many higher orders of signs, but without examining critically to what purpose the effort is directed or evaluating alternative constraints that might be imposed on the initial generators toward this end.
At first sight it seems as though the imposition of reflective closure has multiplied a finite sign relation into an infinite profusion of highly distracting and largely redundant signs, all by itself and all in one step. But this explosion of orders happens only with the complicity of another requirement, that of deterministic interpretation.
There are two types of non-determinism, denotative and connotative, that can affect a sign relation.
<ol>
<li>A sign relation <math>L\!</math> has a non-deterministic denotation if its dyadic component <math>{L_{SO}}\!</math> is not a function <math>L_{SO} : S \to O,\!</math> in other words, if there are signs in <math>S\!</math> with missing or multiple objects in <math>O.\!</math></li>
<li>A sign relation <math>L\!</math> has a non-deterministic connotation if its dyadic component <math>L_{SI}\!</math> is not a function <math>L_{SI} : S \to I,\!</math> in other words, if there are signs in <math>S\!</math> with missing or multiple interpretants in <math>I.\!</math> As a rule, sign relations are rife with this variety of non-determinism, but it is usually felt to be under control so long as <math>L_{SI}\!</math> remains close to being an equivalence relation.</li>
</ol>
Thus, it is really the denotative type of indeterminacy that is felt to be a problem in this context.
The next two pairs of reflective extensions demonstrate that there are ways of achieving reflective closure that do not generate infinite sign relations.
As a flexible and fairly general strategy for describing reflective extensions, it is convenient to take the following tack. Given a syntactic domain <math>S,\!</math> there is an independent formal language <math>F = F(S) = S \langle {}^{\langle\rangle} \rangle,\!</math> called the ''free quotational extension of <math>S,\!</math>'' that can be generated from <math>S\!</math> by embedding each of its signs to any depth of quotation marks. Within <math>F,\!</math> the quoting operation can be regarded as a syntactic generator that is inherently free of constraining relations. In other words, for every <math>s \in S,\!</math> the sequence <math>s, {}^{\langle} s {}^{\rangle}, {}^{\langle\langle} s {}^{\rangle\rangle}, \ldots\!</math> contains nothing but pairwise distinct elements in <math>F\!</math> no matter how far it is produced. The set <math>F(s) = s \langle {}^{\langle\rangle} \rangle \subseteq F\!</math> that collects the elements of this sequence is called the ''subset of <math>F\!</math> generated from <math>s\!</math> by quotation''.
Against this background, other varieties of reflective extension can be specified by means of semantic equations that are considered to be imposed on the elements of <math>F.\!</math> Taking the reflective extensions <math>\mathrm{Ref}^1 (\text{A})\!</math> and <math>\mathrm{Ref}^1 (\text{B})\!</math> as the first orders of a “free” project toward reflective closure, variant extensions can be described by relating their entries with those of comparable members in the standard sequences <math>\mathrm{Ref}^n (\text{A})\!</math> and <math>\mathrm{Ref}^n (\text{B}).\!</math>
A variant pair of reflective extensions, <math>\mathrm{Ref}^1 (\text{A} | E_1)\!</math> and <math>\mathrm{Ref}^1 (\text{B} | E_1),\!</math> is presented in Tables 82 and 83, respectively. These are identical to the corresponding free variants, <math>\mathrm{Ref}^1 (\text{A})~\!</math> and <math>\mathrm{Ref}^1 (\text{B}),~\!</math> with the exception of those entries that are constrained by the following system of semantic equations.
{| align="center" cellspacing="8" width="90%"
|
<math>\begin{matrix}
E_1 :
&
{}^{\langle\langle} \text{A} {}^{\rangle\rangle} = {}^{\langle} \text{A} {}^{\rangle},
&
{}^{\langle\langle} \text{B} {}^{\rangle\rangle} = {}^{\langle} \text{B} {}^{\rangle},
&
{}^{\langle\langle} \text{i} {}^{\rangle\rangle} = {}^{\langle} \text{i} {}^{\rangle},
&
{}^{\langle\langle} \text{u} {}^{\rangle\rangle} = {}^{\langle} \text{u} {}^{\rangle}.
\end{matrix}</math>
|}
This has the effect of making all levels of quotation equivalent.
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
|+ style="height:30px" | <math>\text{Table 82.} ~~ \text{Reflective Extension} ~ \mathrm{Ref}^1 (\text{A} | E_1)\!</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}
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\end{matrix}</math>
|-
| valign="bottom" |
<math>\begin{matrix}
\text{B}
\\
\text{B}
\\
\text{B}
\\
\text{B}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\end{matrix}</math>
|-
| valign="bottom" |
<math>\begin{matrix}
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\end{matrix}</math>
|}
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
|+ style="height:30px" | <math>\text{Table 83.} ~~ \text{Reflective Extension} ~ \mathrm{Ref}^1 (\text{B} | E_1)\!</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}
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\end{matrix}</math>
|-
| valign="bottom" |
<math>\begin{matrix}
\text{B}
\\
\text{B}
\\
\text{B}
\\
\text{B}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\end{matrix}</math>
|-
| valign="bottom" |
<math>\begin{matrix}
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\end{matrix}</math>
|}
<br>
Another pair of reflective extensions, <math>\mathrm{Ref}^1 (\text{A} | E_2)\!</math> and <math>\mathrm{Ref}^1 (\text{B} | E_2),\!</math> is presented in Tables 84 and 85, respectively. These are identical to the corresponding free variants, <math>\mathrm{Ref}^1 (\text{A})~\!</math> and <math>\mathrm{Ref}^1 (\text{B}),~\!</math> except for the entries constrained by the following semantic equations.
{| align="center" cellspacing="8" width="90%"
|
<math>\begin{matrix}
E_2 :
&
{}^{\langle\langle} \text{A} {}^{\rangle\rangle} = \text{A},
&
{}^{\langle\langle} \text{B} {}^{\rangle\rangle} = \text{B},
&
{}^{\langle\langle} \text{i} {}^{\rangle\rangle} = \text{i},
&
{}^{\langle\langle} \text{u} {}^{\rangle\rangle} = \text{u}.
\end{matrix}</math>
|}
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
|+ style="height:30px" | <math>\text{Table 84.} ~~ \text{Reflective Extension} ~ \mathrm{Ref}^1 (\text{A} | E_2)\!</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}
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\end{matrix}</math>
|-
| valign="bottom" |
<math>\begin{matrix}
\text{B}
\\
\text{B}
\\
\text{B}
\\
\text{B}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\end{matrix}</math>
|-
| valign="bottom" |
<math>\begin{matrix}
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
\text{A}
\\
\text{B}
\\
\text{A}
\\
\text{B}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
\text{A}
\\
\text{B}
\\
\text{A}
\\
\text{B}
\end{matrix}</math>
|}
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
|+ style="height:30px" | <math>\text{Table 85.} ~~ \text{Reflective Extension} ~ \mathrm{Ref}^1 (\text{B} | E_2)\!</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}
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\end{matrix}</math>
|-
| valign="bottom" |
<math>\begin{matrix}
\text{B}
\\
\text{B}
\\
\text{B}
\\
\text{B}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\end{matrix}</math>
|-
| valign="bottom" |
<math>\begin{matrix}
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
\text{A}
\\
\text{B}
\\
\text{B}
\\
\text{A}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
\text{A}
\\
\text{B}
\\
\text{B}
\\
\text{A}
\end{matrix}</math>
|}
<br>
By calling attention to their intended status as ''semantic'' equations, meaning that signs are being set equal in the semantic equivalence classes they inhabit or the objects they denote, I hope to emphasize that these equations are able to say something significant about objects.
'''Question.''' Redo <math>F(S)\!</math> over <math>W\!</math>? Use <math>W_F = O \cup F\!</math>?
===6.44. Reflections on Closure===
The previous section dealt with a formal operation that was dubbed ''reflection'' and found that it was closely associated with the device of ''quotation'' that makes it possible to treat signs as objects by making or finding other signs that refer to them. Clearly, an ability to take signs as objects is one component of a cognitive capacity for reflection. But a genuine and less superficial species of reflection can do more than grasp just the isolated signs and the separate interpretants of the thinking process as objects — it can pause the fleeting procession of signs upon signs and seize their generic patterns of transition as valid objects of discussion. This involves the conception and composition of not just ''higher order'' signs but also ''higher type'' signs, orders of signs that aspire to catch whole sign relations up in one breath.
…
