12,080
edits
Changes
Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Part 6 (view source)
Revision as of 21:00, 15 August 2011
, 21:00, 15 August 2011→6.23. Intensional Representations of Sign Relations: + raw text
===6.23. Intensional Representations of Sign Relations===
===6.23. Intensional Representations of Sign Relations===
<pre>
The next three sections beginning with this one consider how the ER's of A and B can be translated into a variety of different IR's. For the purposes of this introduction, only "faithful" translations between the different categories of representation are contemplated. This means that the conversion from ER to IR is intended to convey what is essentially the same information about A and B, to preserve all the relevant structural details that are implied by their various modes of description, but to do it in a way that brings selected aspects of their objective forms to light. General considerations surrounding the task of translation are taken up in this section, while the next two sections lay out different ways of carrying it through.
The larger purpose of this discussion is to serve as an introduction, not just to the special topic of devising IR's for sign relations, but to the general issue of producing, using, and comprehending IR's for any kind of relation or any domain of formal objects. It is hoped that a careful study of these simple IR's can inaugurate a degree of insight into the broader arenas of formalism of which they occupy an initial niche and into the wider landscapes of discourse of which they inhabit a natural corner, in time progressing up to the axiomatic presentation of formal theories about combinatorial domains and other mathematical objects.
For the sake of maximum clarity and reusability of results, I begin by articulating the abstract skeleton of the paradigm structure, treating the sign relations A and B as sundry aspects of a single, unitary, but still uninterpreted object. Then I return at various successive stages to differentiate and individualize the two interpreters, to arrange more functional flesh on the basis provided by their structural bones, and to illustrate how their bare forms can be arrayed in many different styles of qualitative detail.
In building connections between ER's and IR's of sign relations the discussion turns on two types of partially ordered sets, or "posets". Suppose that R is one of the sign relations A or B, and let ER (R) be an ER of R.
In the sign relations A and B, both of their ER's are based on a common world set:
W = { A, B, "A", "B", "i", "u"}.
W = {w1, w2, w3, w4, w5, w6).
W = {w1, w2, w3, w4, w5, w6).
An IR of any object is a description of that object in terms of its properties. A successful description of a particular object usually involves a selection of properties, those that are relevant to a particular purpose. An IR of A or B involves properties of its elementary points w C W and properties of its elementary relations r C OxSxI.
To devise an IR of any relation R one needs to describe R in terms of properties of its ingredients. Broadly speaking, the ingredients of a relation include its elementary relations or n tuples and the elementary components of these n tuples that reside in the relational domains.
The poset Pos (W) of interest here is the power set Pow (W).
The elements of these posets are abstractly regarded as "properties" or "propositions" that apply to the elements of W. These properties and propositions are independently given entities. In other words, they are primitive elements in their own right, and cannot in general be defined in terms of points, but they exist in relation to these points, and their extensions can be represented as sets of points.
For a variety of foundational reasons that I do not fully understand, perhaps most of all because theoretically given structures have their real foundations outside the realm of theory, in empirically given structures, it is best to regard points, properties, and propositions as equally primitive elements, related to each other but not defined in terms of each other, analogous to the undefined elements of a geometry.
There is a foundational issue arising in this context that I do not pretend to fully understand and cannot attempt to finally dispatch. What I do understand I will try to express in terms of an aesthetic principle: On balance, it seems best to regard extensional elements and intensional features as independently given entities. This involves treating points and properties as fundamental realities in their own rights, placing them on an equal basis with each other, and seeking their relation to each other, but not trying to reduce one to the other.
The discussion is now specialized to consider the IR's of the sign relations A and B, their denotative projections as the digraphs Den (A) and Den (B), and their connotative projections as the digraphs Con (A) and Con (B). In doing this I take up two different strategies of representation:
1. The first strategy is called the "literal coding", because it sticks to obvious features of each syntactic element to contrive its code, or the "O(n) coding", because it uses a number on the order of n logical features to represent a domain of n elements.
2. The second strategy is called the "analytic coding", because it attends to the nuances of each sign's intepretation to fashion its code, or the "log(n) coding", because it uses roughly log2(n) binary features to represent a domain of n elements.
Fragments
In the formalized examples of IR's to be presented in this work, I will keep to the level of logical reasoning that is usually referred to as "propositional calculus" or "sentential logic" (SL).
The constrast between ER's and IR's is strongly correlated with another dimension of interest in the study of inquiry, namely, the tension between empirical and rational modes of inquiry.
This section begins the explicit discussion of ER's by taking a second look at the sign relations A and B. Since the form of these examples no longer presents any novelty, this second presentation of A and B provides a first opportunity to introduce some new material. In the process of reviewing this material, it is useful to anticipate a number of incidental issues that are on the point of becoming critical, and to begin introducing the generic types of technical devices that are needed to deal with them.
Therefore, the easiest way to begin an explicit treatment of ER's is by recollecting the Tables of the sign relations A and B and by finishing the corresponding Tables of their dyadic components. Since the form of the sign relations A and B no longer presents any novelty, I can use the second presentation of these examples as a first opportunity to examine a selection of their finer points, previously overlooked.
Starting from this standpoint, the easiest way to begin developing an explicit treatment of ER's is to gather the relevant materials in the forms already presented, to fill out their missing details and expand the abbreviated contents of these forms, and to review their full structures in a more formal light.
Because of the perfect parallelism that the literal coding contrives between individual signs and grammatical categories, this arrangement illustrates not so much a code transformation as a re interpretation of the original signs under different headings.
</pre>
===6.24. Literal Intensional Representations===
===6.24. Literal Intensional Representations===