Changes

===6.22. Extensional Representations of Sign Relations===

===6.22. Extensional Representations of Sign Relations===

Up to this point, the concept of a sign relation has been discussed largely in terms of ERs.  The sign relations <math>L(\text{A})\!</math> and <math>L(\text{B})\!</math> were initially described as collections of transactions among three participants and formalized as sets of triples of underlying elements.

Up to this point, the concept of a sign relation has been discussed largely in terms of ERs.  The sign relations A and B were initially described as collections of transactions among three participants and formalized as sets of triples of underlying elements.

Other examples of ERs are widely distributed throughout the foregoing discussion of A and B.  The extensional mode of description is prevalent, not only in the presentation of sign relations by means of relational data tables, but also in the presentation of dyadic projections by means of digraphs.  This manner of presentation follows the natural order of acquaintance with abstract relations, since the extensional mode of description is the category of representation that usually prevails whenever it is necessary to provide a detailed treatment of simple examples or an exhaustive account of individual instances.

Other examples of ERs are widely distributed throughout the foregoing discussion of <math>\text{A}\!</math> and <math>\text{B}.\!</math> The extensional mode of description is prevalent, not only in the presentation of sign relations by means of relational data tables, but also in the presentation of dyadic projections by means of digraphs.  This manner of presentation follows the natural order of acquaintance with abstract relations, since the extensional mode of description is the category of representation that usually prevails whenever it is necessary to provide a detailed treatment of simple examples or an exhaustive account of individual instances.

Starting from a standpoint in concrete constructions, the easiest way to begin developing an explicit treatment of ERs is to gather the relevant materials in the forms already presented, to fill out the missing details and expand the abbreviated contents of these forms, and to review their full structures in a more formal light.  Consequently, this section inaugurates the formal discussion of ERs by taking a second look at the interpreters A and B, recollecting the Tables of their sign relations and finishing up the Tables of their dyadic components.  Since the form of the sign relations A and B no longer presents any novelty, I can exploit their second presentation as a first opportunity to examine a selection of finer points, previously overlooked.  Also, in the process of reviewing this material it is useful to anticipate a number of incidental issues that are reaching the point of becoming critical within this discussion and to begin introducing the generic types of technical devices that are needed to deal with them.

Starting from a standpoint in concrete constructions, the easiest way to begin developing an explicit treatment of ERs is to gather the relevant materials in the forms already presented, to fill out the missing details and expand the abbreviated contents of these forms, and to review their full structures in a more formal light.  Consequently, this section inaugurates the formal discussion of ERs by taking a second look at the interpreters <math>\text{A}\!</math> and <math>\text{B},\!</math> recollecting the Tables of their sign relations and finishing up the Tables of their dyadic components.  Since the form of the sign relations <math>L(\text{A})\!</math> and <math>L(\text{B})\!</math> no longer presents any novelty, I can exploit their second presentation as a first opportunity to examine a selection of finer points, previously overlooked.  Also, in the process of reviewing this material it is useful to anticipate a number of incidental issues that are reaching the point of becoming critical within this discussion and to begin introducing the generic types of technical devices that are needed to deal with them.

The next set of Tables summarizes the ERs of A and B.  For ease of reference, Tables 48.1 and 49.1 repeat the contents of Tables 1 and 2, respectively, the only difference being that appearances of ordinary quotation marks ("...") are transcribed as invocations of the so called "arch operator" (<...>).  The reason for this slight change of notation will be explained shortly.  The denotative components Den A and Den B are shown in the first two columns of Tables 48.2 and 49.2, respectively, while the third column gives the transition from sign to object as an ordered pair <s, o>.  The connotative components Con A and Con B are shown in the first two columns of Tables 48.3 and 49.3, respectively, while the third column gives the transition from sign to interpretant as an ordered pair <s, i>.

The next set of Tables summarizes the ERs of A and B.  For ease of reference, Tables 48.1 and 49.1 repeat the contents of Tables 1 and 2, respectively, the only difference being that appearances of ordinary quotation marks ("...") are transcribed as invocations of the so called "arch operator" (<...>).  The reason for this slight change of notation will be explained shortly.  The denotative components Den A and Den B are shown in the first two columns of Tables 48.2 and 49.2, respectively, while the third column gives the transition from sign to object as an ordered pair <s, o>.  The connotative components Con A and Con B are shown in the first two columns of Tables 48.3 and 49.3, respectively, while the third column gives the transition from sign to interpretant as an ordered pair <s, i>.