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Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Part 6 (view source)
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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 ERs and IRs 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.
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 ERs and IRs 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 ERs and IRs 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 <math>X ~\operatorname{of}~ Y,\!</math> notated <math>X(Y)\!</math> or just <math>XY,\!</math> for any basic types <math>X\!</math> and <math>Y.\!</math> The constructed types of things involved in the ERs and IRs of sign relations are listed in Tables 47.2 and 47.3, respectively.
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Table 47.1 Basic Types for ERs & IRs of Sign Relations
Table 47.1 Basic Types for ERs & IRs of Sign Relations
Type Symbol
Type Symbol
Triple T
Triple T
Underlying Element U
Underlying Element U
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Table 47.2 Derived Types for ERs of Sign Relations
Table 47.2 Derived Types for ERs of Sign Relations
Type Symbol Construction
Type Symbol Construction
Relation R S(T(U))
Relation R S(T(U))
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Table 47.3 Derived Types for IRs of Sign Relations
Table 47.3 Derived Types for IRs of Sign Relations
Type Symbol Construction
Type Symbol Construction
Relation P(R) P(S(T(U)))
Relation P(R) P(S(T(U)))
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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.
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:
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. ERs of sign relations describe them as sets (Ss) of triples (Ts) of underlying elements (Us). 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 (SSs), signs for triples (STs), and signs for the underlying elements (SUs). 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.
1. ERs of sign relations describe them as sets (Ss) of triples (Ts) of underlying elements (Us). 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 (SSs), signs for triples (STs), and signs for the underlying elements (SUs). 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. IRs of sign relations describe them in terms of properties (Ps) that are taken as primitive entities in their own right. / refer to properties (Ps) of transactions (Ts) of underlying elements (Us).
2. [Variant] IRs of sign relations describe them in terms of properties (Ps) that are taken as primitive entities in their own right. / refer to properties (Ps) of transactions (Ts) of underlying elements (Us).
2. IRs of sign relations refer to properties of sets (PSs), properties of triples (PTs), and properties of underlying elements (PUs). 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 SPSs, SPTs, and SPUs.
2. [Variant] IRs of sign relations refer to properties of sets (PSs), properties of triples (PTs), and properties of underlying elements (PUs). 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 SPSs, SPTs, and SPUs.
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