Changes

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&nbsp;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&nbsp;47.2 and 47.3, respectively.

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&nbsp;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&nbsp;47.2 and 47.3, respectively.

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  Underlying Element U

  Underlying Element U

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Relation R S(T(U))

Relation R S(T(U))

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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.

ERs of sign relations describe them as sets (<math>S\!</math>'s) of triples (<math>T\!</math>'s) of underlying elements (<math>U\!</math>'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 (<math>\underline{S}S\!</math>'s), signs for triples (<math>\underline{S}T\!</math>'s), and signs for the underlying elements (<math>\underline{S}U\!</math>'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.

ERs of sign relations describe them as sets (<math>S\!</math>'s) of triples (<math>T\!</math>'s) of underlying elements (<math>U\!</math>'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 (<math>\underline{S}S\!</math>'s), signs for triples (<math>\underline{S}T\!</math>'s), and signs for the underlying elements (<math>\underline{S}U\!</math>'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.

[Variant] IRs of sign relations describe them in terms of properties (<math>P\!</math>'s) that are taken as primitive entities in their own right.  / refer to properties (<math>P\!</math>'s) of transactions (<math>T\!</math>'s) of underlying elements (<math>U\!</math>'s).

[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).

[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.

[Variant] IRs of sign relations refer to properties of sets (<math>PS\!</math>'s), properties of triples (<math>PT\!</math>'s), and properties of underlying elements (<math>PU\!</math>'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 <math>\underline{S}PS\!</math>'s, <math>\underline{S}PT\!</math>'s, and <math>\underline{S}PU\!</math>'s.

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

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