Changes

§ 21.   There is a general set of situations where the task arises to “build a bridge” between significantly different types of representation.  In these situations, the problem is to translate between the signs and expressions of two formal systems that have radically different levels of interpretation, and to do it in a way that makes appropriate connections between diverse descriptions of the same objects.  More to the point of the present project, formal systems for mediating inquiry, if they are intended to remain viable in both empirical and theoretical uses, need the capacity to negotiate between an ''extensional representation'' (ER) and an ''intensional representation'' (IR) of the same domain of objects.  It turns out that a cardinal or pivotal issue in this connection is how to convert between ERs and IRs of the same objective domain, working all the while within the practical constraints of a computational medium and preserving the equivalence of information.  To illustrate the kinds of technical issues that are involved in these considerations, I bring them to bear on the topic of representing sign relations and their dyadic projections in various forms.

§ 21.   There is a general set of situations where the task arises to “build a bridge” between significantly different types of representation.  In these situations, the problem is to translate between the signs and expressions of two formal systems that have radically different levels of interpretation, and to do it in a way that makes appropriate connections between diverse descriptions of the same objects.  More to the point of the present project, formal systems for mediating inquiry, if they are intended to remain viable in both empirical and theoretical uses, need the capacity to negotiate between an ''extensional representation'' (ER) and an ''intensional representation'' (IR) of the same domain of objects.  It turns out that a cardinal or pivotal issue in this connection is how to convert between ERs and IRs of the same objective domain, working all the while within the practical constraints of a computational medium and preserving the equivalence of information.  To illustrate the kinds of technical issues that are involved in these considerations, I bring them to bear on the topic of representing sign relations and their dyadic projections in various forms.

The next four sections (§§ 22–25) give examples of ERs and IRs, indicate the importance of forming a computational bridge between them, and discuss the conceptual and technical obstacles that will have to be faced in doing so.

The next four sections (§§ 22–25) give examples of ERs and IRs, indicate the importance of forming a computational bridge between them, and discuss the conceptual and technical obstacles that will have to be faced in doing so.

§ 22.   For ease of reference, this section collects previous materials that are relevant to discussing the ERs of the sign relations A and B, and explicitly details their dyadic projections.

&sect; 22. &nbsp; For ease of reference, this section collects previous materials that are relevant to discussing the ERs of the sign relations <math>L(A)\!</math> and <math>L(B),\!</math> and explicitly details their dyadic projections.

&sect; 23. &nbsp; This section discusses a number of general issues that are associated with the IRs of sign relations.  Because of the great degree of freedom there is in selecting among the potentially relevant properties of any real object, especially when the context of relevance to the selection is not known in advance, there are many different ways, perhaps an indefinite multitude of ways, to represent the sign relations A and B in terms of salient properties of their elementary constituents.  In this connection, the next two sections explore a representative sample of these possibilities, and illustrate several different styles of approach that can be used in their presentation.

&sect; 23. &nbsp; This section discusses a number of general issues that are associated with the IRs of sign relations.  Because of the great degree of freedom there is in selecting among the potentially relevant properties of any real object, especially when the context of relevance to the selection is not known in advance, there are many different ways, perhaps an indefinite multitude of ways, to represent the sign relations <math>L(A)\!</math> and <math>L(B)\!</math> in terms of salient properties of their elementary constituents.  In this connection, the next two sections explore a representative sample of these possibilities, and illustrate several different styles of approach that can be used in their presentation.

&sect; 24. &nbsp; A transitional case between ERs and IRs of sign relations is found in the concept of a "literal intensional representation" (LIR).

&sect; 24. &nbsp; A transitional case between ERs and IRs of sign relations is found in the concept of a ''literal intensional representation'' (LIR).

&sect; 25. &nbsp; A fully fledged IR is one that accomplishes some measure of analytic work, bringing to the point of salient notice a selected array of implicit and otherwise hidden features of its object.  This section presents a variety of these "analytic intensional representations" (AIRs) for the sign relations A and B.

&sect; 25. &nbsp; A fully fledged IR is one that accomplishes some measure of analytic work, bringing to the point of salient notice a selected array of implicit and otherwise hidden features of its object.  This section presents a variety of these ''analytic intensional representations'' (AIRs) for the sign relations <math>L(A)\!</math> and <math>L(B).\!</math>

Note for future reference.  The problem so naturally encountered here, due to the "embarassment of riches" that presents itself in choosing a suitable IR, and tracing its origin to the wealth of properties that any real object typically has, is a precursor to one of the deepest issues in the pragmatic theory of inquiry:  "the problem of abductive reasoning".  This topic will be discussed at several later stages of this investigation, where it typically involves the problem of choosing, among the manifold aspects of an objective phenomenon or a problematic objective, only the features that are:  (1) relevant to explaining a present fact, or (2) pertinent to achieving a current purpose.

'''Note for future reference.''' The problem so naturally encountered here, due to the embarrassment of riches that presents itself in choosing a suitable IR, and tracing its origin to the wealth of properties that any real object typically has, is a precursor to one of the deepest issues in the pragmatic theory of inquiry:  ''the problem of abductive reasoning''.  This topic will be discussed at several later stages of this investigation, where it typically involves the problem of choosing, among the manifold aspects of an objective phenomenon or a problematic objective, only the features that are:  (1) relevant to explaining a present fact, or (2) pertinent to achieving a current purpose.

&sect; 26. &nbsp; Differential Logic & Directed Graphs

&sect; 26. &nbsp; Differential Logic and Directed Graphs

&sect; 27. &nbsp; Differential Logic & Group Operations

&sect; 27. &nbsp; Differential Logic and Group Operations

&sect; 28. &nbsp; The Bridge : From Obstruction to Opportunity

&sect; 28. &nbsp; The Bridge : From Obstruction to Opportunity

&sect; 30. &nbsp; Connected, Integrated, Reflective Symbols

&sect; 30. &nbsp; Connected, Integrated, Reflective Symbols

The next seven sections (&sect;&sect;&nbsp;31&ndash;37) are designed to incrementally motivate the idea that a language as simple as propositional calculus, remarkably enough, can be used to articulate significant properties of n place relations.  The course of the discussion will proceed as follows:

The next seven sections (&sect;&sect;&nbsp;31&ndash;37) are designed to incrementally motivate the idea that a language as simple as propositional calculus, remarkably enough, can be used to articulate significant properties of n place relations.  The course of the discussion will proceed as follows: