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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 <math>\underline{S}\!</math> be the type of signs, <math>S\!</math> the type of sets, <math>T\!</math> the type of triples, and <math>U\!</math> 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.

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.

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).

[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. [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 (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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