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Directory talk:Jon Awbrey/Papers/Inquiry Driven Systems : Part 6 (view source)
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'''MathJaX SuXs ❢❢❢'''
<p align="center"><font face="curlz mt" size="7">'''MathJaX SuXs ❢❢❢'''</font></p>
<div class="nonumtoc">__TOC__</div>
<div class="nonumtoc">__TOC__</div>
==6. Reflective Interpretive Frameworks (cont.)==
==6. Reflective Interpretive Frameworks (cont.)==
===6.47. Mutually Intelligible Codes===
Before this complex of relationships can be formalized in much detail, I must introduce linguistic devices for generating ''higher order signs'', used to indicate other signs, and ''situated signs'', indexed by the names of their users, their contexts of use, and other types of information incidental to their usage in general. This leads to the consideration of ''systems of interpretation'' (SOIs) that maintain recursive mechanisms for naming everything within their purview. This “nominal generosity” gives them a new order of generative capacity, producing a sufficient number of distinctive signs to name all the objects and then name the names that are needed in a given discussion.
Symbolic systems for quoting inscriptions and ascribing quotations are associated in metamathematics with ''gödel numberings'' of formal objects, enumerative functions that provide systematic but ostensibly arbitrary reference numbers for the signs and expressions in a formal language. Assuming these signs and expressions denote anything at all, their formal enumerations become the ''codes'' of formal objects, just as programs taken literally are code names for certain mathematical objects known as computable functions. Partial forms of specification notwithstanding, these codes are the only complete modes of representation that formal objects can have in the medium of mechanical activity.
In the dialogue of <math>\text{A}\!</math> and <math>\text{B}\!</math> there happens to be an exact coincidence between signs and states. That is, the states of the interpretive systems <math>\text{A}\!</math> and <math>\text{B}\!</math> are not distinguished from the signs in <math>S\!</math> that are imagined to be mediating, moment by moment, the attentions of the interpretive agents <math>\text{A}\!</math> and <math>\text{B}\!</math> toward their respective objects in <math>O.\!</math> So the question arises: Is this identity bound to be a general property of all useful sign relations, or is it only a degenerate feature occurring by chance or unconscious design in the immediate example?
To move toward a resolution of this question I reason as follows. In one direction, it seems obvious that a ''sign in use'' (SIU) by a particular interpreter constitutes a component of that agent's state. In other words, the very notion of an identifiable SIU refers to numerous instances of a particular interpreter's state that share in the abstract property of being such instances, whether or not anyone can give a more concise or illuminating characterization of the concept under which these momentary states are gathered. Conversely, it is at least conceivable that the whole state of a system, constituting its transitory response to the entirety of its environment, history, and goals, can be interpreted as a sign of something to someone. In sum, there remains an outside chance of signs and states being precisely the same things, since nothing precludes the existence of an ''interpretive framework'' (IF) that could make it so.
Still, if the question about the distinction or coincidence between signs and states is restricted to the domains where existential realizations are conceivable, no matter whether in biological or computational media, then the prerequisites of the task become more severe, due to the narrower scope of materials that are admitted to answer them. In focusing on this arena the problem is threefold:
# The crucial point is not just whether it is possible to imagine an ideal SOI, an external perspective or an independent POV, for which all states are signs, but whether this is so for the prospective SOI of the very agent that passes through these states.
# To what extent can the transient states and persistent conduct of each agent in a community of interpretation take on a moderately public and objective aspect in relation to the other participants?
# How far in this respect, in the common regard for this species of outward demeanor, can each agent's behavior act as a sign of genuine objects in the eyes of other interpreters?
The special task of a nuanced hermeneutic approach to computational interpretation is to realize the relativity of all formal codes to their formal coders, and to seek ways of facilitating mutual intelligibility among interpreters whose internal codes can be thoroughly private, synchronistically keyed to external events, and even a bit idiosyncratic.
Ultimately, working through this maze of “meta” questions, as posed on the tentative grounds of the present project, leads to a question about the ''logical reference frames'' or ''metamathematical coordinate systems'' that are supposed to distinguish “objective” from “symbolic” entities and are imagined to discriminate a range of gradations along their lines. The question is: Whether any gauge of objectivity or scale of virtuality has invariant properties discoverable by all independent interpreters, or whether all is vanity and inane relativism, and everything concerning a subjective point of view is sheer caprice?
Thus, the problem of mutual intelligibility turns on the question of ''common significance'': How can there be signs that are truly public, when the most natural signs that distinct agents can know, their own internal states, have no guarantee and very little likelihood of being related in systematically fathomable ways? As a partial answer to this, I am willing to contemplate certain forms of pre-established harmony, like the common evolution of a biological species or the shared culture of an interpretive community, but my experience has been that harmony, once established, quickly corrupts unless active means are available to maintain it. So there still remains the task of identifying these means. With or without the benefit of a prior consensus, or the assumption of an initial but possibly fragile equilibrium, an explanation of robust harmony must detail the modes of maintaining communication that enable coordinated action to persist in the meanest of times.
The formal character of these questions, in the potential complexities that can be forced on contemplation in the pursuit of their answers, is independent of the species of interpreters that are chosen for the termini of comparison, whether person to person, person to computer, or computer to computer. As always, the truth of this kind of thesis is formal, all too formal. What it brings is a new refrain of an old motif: Are there meaningful, if necessarily formal series of analogies that can be strung from the patterns of whizzing electrons and humming protons, whose controlled modes of collective excitation form and inform the conducts of computers, all the way to the rather different patterns of wizened electrons and humbled protons, whose deliberate energies of communal striving substantiate the forms of life known to be intelligible?
A full consideration of the geometries available for the spaces in which these levels of reflective abstraction are commonly imagined to reside leads to the conclusion that familiar distinctions of “top down” versus “bottom up” are being taken for granted in an arena that has not even been established to be orientable. Thus, it needs to be recognized that the distinction between objects and signs is relative to a definite system of interpretation. The pragmatic theory of signs is designed, in part, precisely to deal with the circumstance that thoroughly objective states of systems can be signs of each other, undermining any pretended distinction between objects and signs that one might propose to draw on essential grounds.
From now on, I will reuse the ancient term ''gnomon'' in a technical sense to refer to the gödel numbers or code names of formal objects. In other words, a gnomon is a gödel numbering or enumeration function that maps a domain of objects into a domain of signs, <math>\mathrm{Gno} : O \to S.\!</math> When the syntactic domain <math>S\!</math> is contained within the object domain <math>O,\!</math> then the part of the gnomon that maps <math>S\!</math> into <math>S,\!</math> providing names for signs and expressions, is usually regarded as a ''quoting function''.
In the pluralistic contexts that go with pragmatic theories of signs, it is no longer entirely appropriate to refer to ''the'' gnomon of any object. At any moment of discussion, I can only have so-and-so's gnomon or code word for each thing under the sun. Thus, apparent references to a uniquely determined gnomon only make sense if taken as enthymemic invocations of the ordinary context and all that is comprehended to be implied in it, promising to convert tacit common sense into definite articulations of what is understood. Actually achieving this requires each elliptic reference to the gnomon to be explicitly grounded in the context of informal discussion, interpreted with respect to the conventional basis of understanding assumed in it, and relayed to the indexing function taken for granted by all parties to it.
In computational terms, this brand of pluralism means that neither the gnomon nor the quoting function that forms a part of it can be viewed as well-defined unless it is indexed, explicitly or implicitly, by the name of a particular interpreter. I will use either one of the equivalent notations <math>{}^{\backprime\backprime} \mathrm{Gno}_i (x) {}^{\prime\prime}\!</math> or <math>{}^{\backprime\backprime\langle} x, i {}^{\rangle\prime\prime}\!</math> to indicate the gnomon of the object <math>x\!</math> with respect to the interpreter <math>i.\!</math> The value <math>\mathrm{Gno}_i (x) = {}^{\langle} x, i {}^{\rangle} \in S\!</math> is the ''nominal sign in use'' or the ''name in use'' (NIU) of the object <math>x\!</math> with respect to the interpreter <math>i,\!</math> and thus it constitutes a component of <math>i\!</math>'s state.
In the special case where <math>x\!</math> is a sign or expression in the syntactic domain, then <math>\mathrm{Gno}_i (x) = {}^{\langle} x, i {}^{\rangle}\!</math> is tantamount to the quotation of <math>x\!</math> by and for the use of the interpreter <math>i,\!</math> in short, the nominal sign to <math>i\!</math> that makes <math>x\!</math> an object for <math>i.\!</math> For signs and expressions, it is usually only the quoting function that makes them objects. But nothing is an object in any sense for an interpreter unless it is an object of a sign relation for that interpreter. Therefore, …
If it is now asked what measure of invariant understanding can be enjoyed by diverse parties of interpretive agents, then the discussion has come upon an issue with a familiar echo in mathematical analysis. The organization of many local coordinate frames into systems capable of supporting communicative references to relatively “objective” objects is usually handled by means of the concept of a ''manifold''. Therefore, the analogous task that is suggested for this project is to arrive at a workable definition of ''sign relational manifolds''.
The discrete nature of the <math>\text{A}\!</math> and <math>\text{B}\!</math> dialogue renders moot the larger share of issues of interest in continuous and differentiable manifolds. However, it is still possible to get things moving in this direction by looking at simple structural analogies that connect the pragmatic theory of sign relations with the basic notions of analysis on manifolds.
===6.48. Discourse Analysis : Ways and Means===
Before the discussion of the <math>\text{A}\!</math> and <math>\text{B}\!</math> dialogue can proceed to richer veins of semantic structure it will be necessary to extract the relevant traces of embedded sign relations from their environments of informally interpreted syntax.
On the substantive front, sign relations serving as raw materials of discourse need to be refined and their content assayed, but first their identifying signatures must be sounded out, carved out, and lifted from their embroiling inclusions in the dense strata of obscure intuitions that sediment ordinary discussion. On the instrumental front, sign relations serving as primitive tools of discourse analysis need to be identified and improved by a deliberate examination of their designs and purposes.
So far, the models and methods made available to formal treatment were borrowed outright, with little hesitation and less recognition, from the context of casual discussion. Thus, these materials and mechanisms have come to the threshold of critical reflection already in play, devoid of concern for the presuppositions and consequences associated with their use, and only belatedly turned to the effortful work and tedious formalities of self-conscious exposition.
To reflect on the properties of complex and higher order sign relations with any degree of clarity it is necessary to arrange a clearer field of investigation and a less cluttered staging area for analytic work than is commonly provided. Habitual processes of interpretation that typically operate as automatic routines and uncritical defaults in the informal context of discussion have to be selectively inhibited, slowed down, and critically examined as objective possibilities, instead of being taken for granted as absolute necessities.
In other words, an apparatus for critical reflection does not merely add more mirrors to the kaleidoscopic fun-house of interpretive discourse, but it provides transient moments of equanimity, or balanced neutrality, and a moderately detached perspective on alternative points of view. A scope so limited does not by any means grant a god's eye view, but permits a sufficient quantity of light to consider how the original array of sights and reflections might have been created otherwise.
Ordinarily, the extra degree of attention to syntax that is needed for critical reflection on interpretive processes is called into play by means of syntactic operators and diacritical devices acting at the level of individual signs and elementary expressions. For example, quotation marks are used to force one type of “semantic ascent”, causing signs to be treated as objects and marking points of interpretive shift as they occur in the syntactic medium. But these operators and devices must be symbolized, and these symbols must be interpreted. Consequently, there is no way to avoid the invocation of a cohering interpretive framework, one that needs to be specialized for analytic purposes.
The best way to achieve the desired type of reflective capacity is by attaching a parameter to the interpretive framework used as an instrument of formal study, specifying certain choices or interpretive presumptions that affect the entire context of discussion. The aesthetic distance needed to arrive at a formal perspective on sign relations is maintained, not by jury-rigging ordinary discussion with locally effective syntactic devices, but by asking the reader to consider certain dimensions of parametric variation in the global interpretive frameworks used to comprehend the sign relations under study.
The interpretive parameter of paramount importance to this work is one that is critical to reflection. It can be presented as a choice between two alternative conventions, affecting the way one reflexively regards each sign in a text: (1) as a sign provoking interest only in passing, exchanged for the sake of a meaningful object it is always taken for granted to have, or (2) as a sign comprising an interest in and of itself, a state of a system or a modification of a medium that can signify an external value but does not necessarily denote anything else at all. I will name these options for responding to signs according to the aspects of character that are most appreciated in their net effects, whether signs for the sake of objects, or signs for their own sake, respectively.
The first option I call the ''object convention'', recognizing it as the natural default of informal language use. In the ordinary language context it is the automatic assumption that signs and expressions are intended to denote something external to themselves, and even though it is quite obvious to all interpreters that the medium is filled with the appearances of signs and not with the objects themselves, this fact passes for little more than transitory interest in the rush to cash out tokens for their indicated values.
The object convention, as appropriate to an introduction that needs to begin in the context of ordinary discussion, is the parametric choice that was left in force throughout the treatment of the A and B example. Doing things this way is like trying to roller skate in a buffalo herd, that is, it attempts to formalize a fragment of discussion on a patchwork of local scales without interrupting the automatic routines and default assumptions that prevail on a global basis in the informal context. Ultimately, one cannot avoid stumbling over the hoofprints <math>( {}^{\backprime\backprime} \, {}^{\prime\prime} )\!</math> of overly cited and opaquely enthymematic textual deposits.
The second option I call the ''sign convention'', observing it to be the treatment of choice in programming and formal language studies. In the formal language context it is necessary to consider the possibility that not all signs and expressions are assured to denote or even connote much of anything at all. This danger is amplified in computational frameworks where it resonates with a related theme, that not all programs are guaranteed to terminate normally with a definite result. In order to deal with these eventualities, a more cautious approach to sign relations is demanded to cover the risk of generating nonsense, in other words, to guard against degenerate forms of sign relations that fail to serve any significant purpose in communication or inquiry.
Whenever a greater degree of care is required, it becomes necessary to replace the object convention with the sign convention, which presumes to take for granted only what can be obvious to all observers, namely, the phenomenal appearances and temporal occurrences of objectified states of systems. To be sure, these modulations of media are still presented as signs, but only potentially as signs of other things. It goes with the territory of the formal language context to constantly check the inveterate impulses of the literate mind, to reflect on its automatic reflex toward meaning, to inhibit its uncontrolled operation, and to pause long enough in the rush to judgment to question whether its constant presumption of a motive is itself innocent.
In order to deal with these issues of discourse analysis in an explicit way, it is necessary to have in place a technical notation for marking the very kinds of interpretive assumptions that normally go unmarked. Thus, I will describe a set of devices for annotating certain kinds of interpretive contingencies, namely, the ''discourse analysis frames'' or the ''global interpretive frames'' that may be operative at any given moment in a particular context of discussion.
To mark a context of discussion where a particular set <math>J\!</math> of interpretive conventions is being maintained, I use labeled brackets of the following two forms: “unitary”, as <math>\{ J | \ldots | J \},\!</math> or “divided”, as <math>\{ J | \ldots | \ldots | J \}.\!</math> The unitary form encloses a context of discussion by delimiting a range of text whose reading is subject to the interpretive constraints <math>J.\!</math> The divided form specifies the objects, signs, and interpretive information in accord with which a species of discussion is generated. Labeled brackets enclosing contexts can be nested in their scopes, with interpretive data on each outer envelope applying to every inclusion. Labeled brackets arranging the ''conversation pieces'' or the ''generators and relations'' of a topic can lead to discussions that spill outside their frames, and thus are permitted to constitute overlapping contexts.
For the present, I will consider two types of interpretive parameters to be used as indices of labeled brackets.
<ol style="list-style-type:decimal">
<li>Names of interpreters or other references to context can be used to indicate the provenance of the objects and signs that make up the assorted contents of brackets. On occasion, I will use the first person singular pronoun to signify the immediate context of informal discussion, as in <math>\{ I | \ldots | I \},\!</math> but more often than not this context goes unmarked.</li>
<li>Two other modifiers can be used to toggle between the options of the object convention, more common in casual or ordinary contexts, and the sign convention, more useful in formal or sign theoretic contexts.</li>
<ol style="list-style-type:lower-alpha">
<li>
<p>The brackets <math>\{ o | \ldots | o \}\!</math> mark a context of informal language use or ordinary discussion, where the object convention applies. To specify the elements of a sign relation under these conditions, I use a form of presentation like the following:</p>
{| align="center" cellpadding="8" width="90%"
|
<math>\{ o |~ \text{A}, \text{B} ~|||~ {}^{\backprime\backprime} \text{A} {}^{\prime\prime}, {}^{\backprime\backprime} \text{B} {}^{\prime\prime}, {}^{\backprime\backprime} \text{i} {}^{\prime\prime}, {}^{\backprime\backprime} \text{u} {}^{\prime\prime} ~| o \}.\!</math>
|}
<p>Here, the names of objects are placed on the left side and the names of signs on the right side of the central divide, and the outer brackets stipulate that the object convention is in force throughout the discussion of a sign relation that is generated on these elements.</p></li>
<li>
<p>The brackets <math>\{ s | \ldots | s \}\!</math> mark a context of formal language use or controlled discussion, where the sign convention applies. To specify the elements of a sign relation in this case, I use a form like:</p>
{| align="center" cellpadding="8" width="90%"
|
<math>\{ s |~ [\text{A}], [\text{B}] ~|||~ \text{A}, \text{B}, \text{i}, \text{u} ~| s \}.</math>
|}
<p>Again, expressions for objects are placed on the left and expressions of signs on the right, but formal language conventions are now invoked to let the alphabet letters and the lexical items of a formal vocabulary stand for themselves, and denotation brackets <math>{}^{\backprime\backprime} [ \dots ] {}^{\prime\prime}\!</math> are placed around signs to indicate the corresponding objects, when they exist.</p></li>
</ol></ol>
When the information carried by labeled brackets becomes more involved and more extensive, a set of convenient abbreviations and suggestions for “pretty printing” can be followed. When the bracket labels become too long to bother repeating, I will leave the last label blank or use ditto marks, as with <math>\{ a, b, c ~|~ \ldots ~| {}^{\prime\prime} \}.\!</math> When it is necessary to break labeled brackets over several lines, multiple dividers and dittos can be used to fill out corresponding columns, as in the following text:
{| align="center" cellpadding="8" width="90%"
|
<math>\begin{array}{*{12}{c}}
\{ & I & , & o & | & \text{A} & , & \text{B} & & & &
\\
| & | & | & | & | &
{}^{\backprime\backprime} \text{A} {}^{\prime\prime} & , &
{}^{\backprime\backprime} \text{B} {}^{\prime\prime} & , &
{}^{\backprime\backprime} \text{i} {}^{\prime\prime} & , &
{}^{\backprime\backprime} \text{u} {}^{\prime\prime}
\\
| & {}^{\prime\prime} & {}^{\prime\prime} & {}^{\prime\prime} & \} & & & & & & &
\end{array}</math>
|}
A notation for discourse analysis ought to find a crucial test of its usefulness in whether it can help to disclose structural properties of interpretive frameworks that would otherwise escape the attention due. If the dimensions of interpretive choice that are represented by these devices are to serve a useful function, then …
Although these devices for discourse analysis are bound to seem a bit ''ad hoc'' at this point, they have been designed with a sign relational bootstrap in mind, that is, with a view to being formalized and recognized as a species within the domain of sign relations itself, where this is the very domain that is laid out as their field of application.
One note of caution may help to prevent a common misunderstanding. It is futile to imagine that any system of interpretive markers for discourse can become totally self sufficient, like the Worm Uroboros, determining all aspects of interpretation and eliminating all ambiguity. The ultimate appeal of signs, and signs upon signs, is always to an intelligent interpreter, a reader who knows there are more interpretive choices to make than could ever be surrendered to signs, and whose free responsibility to appropriate interpretations cannot be abdicated to any text or abridged by any gloss on it, no matter how fit or finished.
In a sense, at least at first, nothing is being created that could not have been noticed without signs. It is merely that actions are being articulated that were not articulated before, and hopefully in ways that make transient insights easier to remember and reuse on new occasions. Instead, the requirement here is to devise a language, the marks of which can reflect the ambient light of observation on its own process. It is not unusual to succeed at this in artificial environments crafted especially for the purpose, but to achieve the critical angle ''in vivo'', in the living context of a natural language, takes more art.
===6.49. Combinations of Sign Relations===
At a point like this in the development of a formal subject matter, it is customary to introduce elements of a logical calculus that can be used to describe relevant aspects of the formal structures involved and to expedite reasoning about their manifold combinations and decompositions. I will hold off from doing this for sign relations in any formal way at present. Instead, I consider the informal requirements and the foreseeable ends that a suitable calculus for sign relations might be expected to meet, and I present as tentative alternatives a few different ways of proceeding to formalize these intentions.
The first order of business in the “comparative anatomy” and “developmental biology” of sign relations is to undertake a pair of closely related tasks: (1) to examine the structural articulation of highly complex sign relations in terms of the primitive constituents that are found available, and (2) to explain the functional genesis of formal (that is, reflectively considered and critically regarded) sign relations as they naturally arise within the informal context of representational and communicational activities.
Converting to a political metaphor, how does the “republic” constituted by a sign relation — the representational community of agents invested with a congeries of legislative, executive, and interpretive powers, employing a consensual body of conventional languages, encompassing a commonwealth of comprehensible meanings, diversely but flexibly manifested in the practical administration of abiding and shared representations — how does all of this first come into being?
… and their development from primitive/ rudimentary to highly structured …
The grasp of the discussion between <math>\text{A}\!</math> and <math>\text{B}\!</math> that is represented in their separate sign relations can best be described as fragmentary. It fails to capture what everyone knows <math>\text{A}\!</math> and <math>\text{B}\!</math> would know about each other's language use.
How can the fragmentary system of interpretation (SOI) constituted by the juxtaposition of individual sign relations <math>L(\text{A})\!</math> and <math>L(\text{B})\!</math> be combined or developed into a new SOI that represents what agents like <math>\text{A}\!</math> and <math>\text{B}\!</math> are sure to know about each other's language use? In order to make it clear that this is a non-trivial question, and in the process to illustrate different ways of combining sign relations, I begin by considering a couple of obvious suggestions for their integration that immediate reflection will show to miss the mark.
The first thing to try is the set-theoretic union of the sign relations. This leads to a “confused” or “confounded” combination of the component sign relations. For example, the sign relation defined as <math>L_\text{C} = L_\text{A} \cup L_\text{B}\!</math> is shown in Table 86. Interpreted as a transition digraph on the four points of the syntactic domain <math>S = \{ {}^{\backprime\backprime} \text{A} {}^{\prime\prime}, {}^{\backprime\backprime} \text{B} {}^{\prime\prime}, {}^{\backprime\backprime} \text{i} {}^{\prime\prime}, {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \},\!</math> the sign relation <math>L_\text{C}\!</math> specifies the following behavior for the conduct of its interpreter:
# <math>\text{A}\!\cdot\!L_\text{C}\!</math> has a sling at each point of <math>\{ {}^{\backprime\backprime} \text{A} {}^{\prime\prime}, {}^{\backprime\backprime} \text{i} {}^{\prime\prime}, {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \}\!</math> and two-way arcs on the pairs <math>\{ {}^{\backprime\backprime} \text{A} {}^{\prime\prime}, {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \}\!</math> and <math>\{ {}^{\backprime\backprime} \text{A} {}^{\prime\prime}, {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \}.\!</math>
# <math>\text{B}\!\cdot\!L_\text{C}\!</math> has a sling at each point of <math>\{ {}^{\backprime\backprime} \text{B} {}^{\prime\prime}, {}^{\backprime\backprime} \text{i} {}^{\prime\prime}, {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \}\!</math> and two-way arcs on the pairs <math>\{ {}^{\backprime\backprime} \text{B} {}^{\prime\prime}, {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \}\!</math> and <math>\{ {}^{\backprime\backprime} \text{B} {}^{\prime\prime}, {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \}.\!</math>
These sub-relations do not form equivalence relations on the relevant sets of signs. If closed up under transitive compositions, then <math>\{ {}^{\backprime\backprime} \text{A} {}^{\prime\prime}, {}^{\backprime\backprime} \text{i} {}^{\prime\prime}, {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \}\!</math> are all equivalent in the presence of object <math>\text{A},\!</math> but <math>\{ {}^{\backprime\backprime} \text{B} {}^{\prime\prime}, {}^{\backprime\backprime} \text{i} {}^{\prime\prime}, {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \}\!</math> are all equivalent in the presence of object <math>\text{B}.\!</math> This may accurately represent certain types of political thinking, but it does not constitute the kind of sign relation that is wanted here.
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
|+ style="height:30px" | <math>\text{Table 86.} ~~ \text{Confounded Sign Relation} ~ L_\text{C} = L_\text{A} \cup L_\text{B} ~ \!</math>
|- style="height:40px; background:#f0f0ff"
| width="33%" | <math>\text{Object}\!</math>
| width="33%" | <math>\text{Sign}\!</math>
| width="33%" | <math>\text{Interpretant}\!</math>
|-
| valign="bottom" |
<math>\begin{matrix}
\text{A}
\\
\text{A}
\\
\text{A}
\\
\text{A}
\\
\text{A}
\\
\text{A}
\\
\text{A}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{}^{\backprime\backprime} \text{A} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{A} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{A} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{i} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{i} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{u} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{u} {}^{\prime\prime}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{}^{\backprime\backprime} \text{A} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{i} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{u} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{A} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{i} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{A} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{u} {}^{\prime\prime}
\end{matrix}</math>
|-
| valign="bottom" |
<math>\begin{matrix}
\text{B}
\\
\text{B}
\\
\text{B}
\\
\text{B}
\\
\text{B}
\\
\text{B}
\\
\text{B}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{}^{\backprime\backprime} \text{B} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{B} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{B} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{i} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{i} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{u} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{u} {}^{\prime\prime}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{}^{\backprime\backprime} \text{B} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{i} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{u} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{B} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{i} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{B} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{u} {}^{\prime\prime}
\end{matrix}</math>
|}
<br>
Reflecting on this disappointing experience with using simple unions to combine sign relations, it appears that some type of indexed union or categorical co-product might be demanded. Table 87 presents the results of taking the disjoint union <math>\textstyle L_\text{D} = L_\text{A} \coprod L_\text{B}\!</math> to constitute a new sign relation.
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
|+ style="height:30px" | <math>\text{Table 87.} ~~ \text{Disjointed Sign Relation} ~ L_\text{D} = L_\text{A} \textstyle\coprod L_\text{B}\!</math>
|- style="height:40px; background:#f0f0ff"
| width="33%" | <math>\text{Object}\!</math>
| width="33%" | <math>\text{Sign}\!</math>
| width="33%" | <math>\text{Interpretant}\!</math>
|-
| valign="bottom" |
<math>\begin{matrix}
\text{A}_\text{A}
\\
\text{A}_\text{A}
\\
\text{A}_\text{A}
\\
\text{A}_\text{A}
\end{matrix}\!</math>
| valign="bottom" |
<math>\begin{matrix}
{{}^{\backprime\backprime} \text{A} {}^{\prime\prime}}_\text{A}
\\
{{}^{\backprime\backprime} \text{A} {}^{\prime\prime}}_\text{A}
\\
{{}^{\backprime\backprime} \text{i} {}^{\prime\prime}}_\text{A}
\\
{{}^{\backprime\backprime} \text{i} {}^{\prime\prime}}_\text{A}
\end{matrix}\!</math>
| valign="bottom" |
<math>\begin{matrix}
{{}^{\backprime\backprime} \text{A} {}^{\prime\prime}}_\text{A}
\\
{{}^{\backprime\backprime} \text{i} {}^{\prime\prime}}_\text{A}
\\
{{}^{\backprime\backprime} \text{A} {}^{\prime\prime}}_\text{A}
\\
{{}^{\backprime\backprime} \text{i} {}^{\prime\prime}}_\text{A}
\end{matrix}\!</math>
|-
| valign="bottom" |
<math>\begin{matrix}
\text{A}_\text{B}
\\
\text{A}_\text{B}
\\
\text{A}_\text{B}
\\
\text{A}_\text{B}
\end{matrix}\!</math>
| valign="bottom" |
<math>\begin{matrix}
{{}^{\backprime\backprime} \text{A} {}^{\prime\prime}}_\text{B}
\\
{{}^{\backprime\backprime} \text{A} {}^{\prime\prime}}_\text{B}
\\
{{}^{\backprime\backprime} \text{u} {}^{\prime\prime}}_\text{B}
\\
{{}^{\backprime\backprime} \text{u} {}^{\prime\prime}}_\text{B}
\end{matrix}\!</math>
| valign="bottom" |
<math>\begin{matrix}
{{}^{\backprime\backprime} \text{A} {}^{\prime\prime}}_\text{B}
\\
{{}^{\backprime\backprime} \text{u} {}^{\prime\prime}}_\text{B}
\\
{{}^{\backprime\backprime} \text{A} {}^{\prime\prime}}_\text{B}
\\
{{}^{\backprime\backprime} \text{u} {}^{\prime\prime}}_\text{B}
\end{matrix}\!</math>
|-
| valign="bottom" |
<math>\begin{matrix}
\text{B}_\text{A}
\\
\text{B}_\text{A}
\\
\text{B}_\text{A}
\\
\text{B}_\text{A}
\end{matrix}\!</math>
| valign="bottom" |
<math>\begin{matrix}
{{}^{\backprime\backprime} \text{B} {}^{\prime\prime}}_\text{A}
\\
{{}^{\backprime\backprime} \text{B} {}^{\prime\prime}}_\text{A}
\\
{{}^{\backprime\backprime} \text{u} {}^{\prime\prime}}_\text{A}
\\
{{}^{\backprime\backprime} \text{u} {}^{\prime\prime}}_\text{A}
\end{matrix}\!</math>
| valign="bottom" |
<math>\begin{matrix}
{{}^{\backprime\backprime} \text{B} {}^{\prime\prime}}_\text{A}
\\
{{}^{\backprime\backprime} \text{u} {}^{\prime\prime}}_\text{A}
\\
{{}^{\backprime\backprime} \text{B} {}^{\prime\prime}}_\text{A}
\\
{{}^{\backprime\backprime} \text{u} {}^{\prime\prime}}_\text{A}
\end{matrix}\!</math>
|-
| valign="bottom" |
<math>\begin{matrix}
\text{B}_\text{B}
\\
\text{B}_\text{B}
\\
\text{B}_\text{B}
\\
\text{B}_\text{B}
\end{matrix}\!</math>
| valign="bottom" |
<math>\begin{matrix}
{{}^{\backprime\backprime} \text{B} {}^{\prime\prime}}_\text{B}
\\
{{}^{\backprime\backprime} \text{B} {}^{\prime\prime}}_\text{B}
\\
{{}^{\backprime\backprime} \text{i} {}^{\prime\prime}}_\text{B}
\\
{{}^{\backprime\backprime} \text{i} {}^{\prime\prime}}_\text{B}
\end{matrix}\!</math>
| valign="bottom" |
<math>\begin{matrix}
{{}^{\backprime\backprime} \text{B} {}^{\prime\prime}}_\text{B}
\\
{{}^{\backprime\backprime} \text{i} {}^{\prime\prime}}_\text{B}
\\
{{}^{\backprime\backprime} \text{B} {}^{\prime\prime}}_\text{B}
\\
{{}^{\backprime\backprime} \text{i} {}^{\prime\prime}}_\text{B}
\end{matrix}\!</math>
|}
<br>
===6.50. Revisiting the Source===
'''…'''
==Deletions==
==Deletions==