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===6.23. Intensional Representations of Sign Relations===
===6.23. Intensional Representations of Sign Relations===
The next three sections consider how the ERs of <math>L(\text{A})\!</math> and <math>L(\text{B})\!</math> can be translated into a variety of different IRs. For the purposes of this introduction, only “faithful” translations between the different categories of representation are contemplated. This means that the conversion from ER to IR is intended to convey what is essentially the same information about <math>L(\text{A})\!</math> and <math>L(\text{B}),\!</math> to preserve all of the relevant structural details that are implied by their various modes of description, but to do it in a way that brings selected aspects of their objective forms to light. General considerations surrounding the task of translation are taken up in this section, while the next two sections lay out different ways of carrying it through.
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The larger purpose of this discussion is to serve as an introduction, not just to the special topic of devising IRs for sign relations, but to the general issue of producing, using, and comprehending IRs for any kind of relation or any domain of formal objects. It is hoped that a careful study of these simple IRs can inaugurate a degree of insight into the broader arenas of formalism of which they occupy an initial niche and into the wider landscapes of discourse of which they inhabit a natural corner, in time progressing up to the axiomatic presentation of formal theories about combinatorial domains and other mathematical objects.
The larger purpose of this discussion is to serve as an introduction, not just to the special topic of devising IRs for sign relations, but to the general issue of producing, using, and comprehending IRs for any kind of relation or any domain of formal objects. It is hoped that a careful study of these simple IRs can inaugurate a degree of insight into the broader arenas of formalism of which they occupy an initial niche and into the wider landscapes of discourse of which they inhabit a natural corner, in time progressing up to the axiomatic presentation of formal theories about combinatorial domains and other mathematical objects.
