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===6.21. Building Bridges Between Representations===
===6.21. Building Bridges Between Representations===
<pre>
On the way to integrating dynamic and symbolic approaches to systems there is one important watershed that has to be crossed and recrossed, time and time again. This is a form of continental divide that decides between two alternative and exclusive "modes of description" (MOD's) or "categories of representation" (COR's), and marks a writer's moment to moment selection of "extensional representation" (ER), on the one side, or "intensional representation" (IR), on the other. To apply the theme, in this section I address the task of building conceptual bridges between two different ways of describing or representing sign relations: (1) the ER that describes a sign relation in terms of its instances, and (2) the IR that describes a sign relation in terms of its properties.
It is best to begin the work of bridge building on informal grounds, using concrete examples of ER's and IR's and taking advantage of basic ideas about their relationship that are readily available to every reader. After the overall scheme of construction is roughed out in this fashion, I plan to revisit the concept of representation in a more formal style, examining the balance of its in and ex "tensions" with a sharper eye to the relevant details and a greater chance of compassing the depths of form that arise between the two points of view.
The task of building this bridge is not trivial. In places, the basic elements of construction are yet to be forged from the available stocks, in others, the needed materials still lie in their ores, awaiting a suitable process to extract them, refine them, and bring them to a usable state. Due to the difficulties of this task and the length of time it will take to carry it out, I think it is advisable to establish two points of reference before setting to work.
1. As a way of providing sufficient motivation for the effort, I will indicate the importance of this bridge with respect to the aims of inquiry in general.
2. As a guard against a host of precipitous shortcuts that have been tried in the past, I will point out as clearly as possible a few of the obstacles that need to be surmounted. Once their structures are rightly understood, the obstructions that lie in the path of this bridge can be chalked up to experience with the reality of its construction, turned to use as stepping stones in the advance of its ultimate course, and given a fitting place in the progress of instruction.
Terms referring to properties of sign relations make it possible to formulate propositions about sign relations, either as occasioned by a clear and present example or in abstraction from any concrete instance. In turn, this makes it possible to carry on chains of reasoning about the properties of sign relations in detachment from the presence of actual cases that may or may not come to mind in the immediate present. This mode of abstraction, invoking the kind of IR that is involved in mediating every form of propositional reasoning, gives logic its wings and can lead to theories of great conceptual power, but it incurs the risk of leading reasoning astray into realms of irreferent pretension, eventually degenerating into spurious sounds that signify nothing.
It is only by means of an IR that logical reasoning, properly speaking, is able to begin. The stringency of this precept, if it is taken too strictly as a starting condition and applied solely in absolute terms, would be correctly perceived as demanding a provision that is jarring to every brand of good sense. But it was never meant to be taken this severely. In practice, the starkness of this tentative stipulation is moderated by the degree of fuzziness that still continues to reside in the interpretive distinction between ER's and IR's.
The alleged distinction between ER's and IR's, when it is projected to have a global application, remains arbitrary so long as it is taken at that level of abstraction, and it comes to take on the semblance of a definition only in relation to the interpretive conduct of a particular arbiter. No representation in actual practice is purely of one sort or the other, nor fails to have the characters of both types as a part of its mix. In other words, extensions and intensions are only abstractions from a profounder "tension" that is logically prior but functionally intermediate to them both, and every representation of any use will have its aspect of extensional particularity permeated by its aspect of intensional generality.
Toward the end of this construction I hope it will become clear that this bridge is a project intermediate in scale between the elementary linkage of signs to interpretants that is built into every sign relation and all the courses of conduct that go to span the gulf and build communication between vastly different systems of interpretation. In the meantime, there are strong analogies that make the architecture of this bridge parallel in form to the structures existing at both ends of the scale, shaping it in congruence with patterns of action that reside at both the micro and the macro levels. Observing these similarities and their lines of potential use as they arise will serve to guide the current work.
A sign relation is a complex object and its representations, insofar as they faithfully preserve its structure, are complex signs. Accordingly, the problems of translating between ER's and IR's of sign relations, of detecting when representations alleged to be of sign relations do indeed represent objects of the specified character, and of recognizing whether different representations do or do not represent the same sign relation as their common object — these are the familiar questions that would be asked of the signs and interpretants in a simple sign relation, but this time asked at a higher level, in regard to the complex signs and complex interpretants that are posed by the different stripes of representation. At the same time, it should be obvious that these are also the natural questions to be faced in building a bridge between representations.
How many different sorts of entities are conceivably involved in translating between ER's and IR's of sign relations? To address this question it helps to introduce a system of type notations that can be used to keep track of the various sorts of things, or the varieties of objects of thought, that are generated in the process of answering it. Table 47.1 summarizes the basic types of things that are needed in this pursuit, while the rest can be derived by constructions of the form "X of Y", notated "X(Y)" or just "XY", for any basic types X and Y. The constructed types of things involved in the ER's and IR's of sign relations are listed in Tables 47.2 and 47.3, respectively.
Table 47.1 Basic Types for ER's & IR's of Sign Relations
Type Symbol
Property P
Sign S
Set S
Triple T
Underlying Element U
Table 47.2 Derived Types for ER's of Sign Relations
Type Symbol Construction
Relation R S(T(U))
Table 47.3 Derived Types for IR's of Sign Relations
Type Symbol Construction
Relation P(R) P(S(T(U)))
Nothing as yet in this scheme of types says that all of the entities playing a part in the discussion are necessarily distinct, but only that there are this many roles to fill.
Let S be the type of signs, S the type of sets, T the type of triples, and U the type of underlying objects. Now consider the various sorts of things, or the varieties of objects of thought, that are invoked on each side, annotating each type as it is mentioned:
1. ER's of sign relations describe them as sets (S's) of triples (T's) of underlying elements (U's). This makes for three levels of objective structure that must be put in coordination with each other, a task that is projected to be carried out in the appropriate OF of sign relations. Corresponding to this aspect of structure in the OF, there is a parallel aspect of structure in the IF of sign relations. Namely, the accessory sign relations that are used to discuss a targeted sign relation need to have signs for sets (SS's), signs for triples (ST's), and signs for the underlying elements (SU's). This accounts for three levels of syntactic structure in the IF of sign relations that must be coordinated with each other and also with the targeted levels of objective structure.
2. IR's of sign relations describe them in terms of properties (P's) that are taken as primitive entities in their own right. / refer to properties (P's) of transactions (T's) of underlying elements (U's).
2. IR's of sign relations refer to properties of sets (PS's), properties of triples (PT's), and properties of underlying elements (PU's). This amounts to three more levels of objective structure in the OF of the IR that need to be coordinated with each other and interlaced with the OF of the ER if the two are to be brought into the same discussion, possibly for the purpose of translating either into the other. Accordingly, the accessory sign relations that are used to discuss an IR of a targeted sign relation need to have SPS's, SPT's, and SPU's.
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===6.22. Extensional Representations of Sign Relations===
===6.22. Extensional Representations of Sign Relations===
