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Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Part 6 (view source)

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→‎6.10. Higher Order Sign Relations : Examples: markup

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| <math>\operatorname{Nom} \subseteq \operatorname{Den}_L \subseteq O \times S ~\text{such that}~ \operatorname{Nom} : O \to S.\!</math>

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Part of the task of making a sign relation more reflective is to extend it in ways that turn more of its signs into objects. This is the reason for creating higher order signs, which are just signs for making objects out of signs. One effect of progressive reflection is to extend the initial naming function <math>\operatorname{Nom}\!</math> through a succession of new naming functions <math>\operatorname{Nom}',\!</math> <math>\operatorname{Nom}'',\!</math> and so on, assigning unique names to larger allotments of the original and subsequent signs. With respect to the difficulties of construction, the ''hard core'' or ''adamant part'' of creating extended naming functions resides in the initial portion <math>\operatorname{Nom}\!</math> that maps objects of the “external world” to signs in the “internal world”. The subsequent task of assigning conventional names to signs is supposed to be comparatively natural and ''easy'', perhaps on account of the ''nominal'' nature of signs themselves.

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Part of the task of making a sign relation more reflective is to extend it in ways that turn more of its signs into objects. This is the reason for creating HO signs, which are just signs for making objects out of signs. One effect of progressive reflection is to extend the initial Nom through a succession of new naming functions Nom', Nom'', and so on, assigning unique names to larger allotments of the original and subsequent signs. With respect to the difficulties of construction, the "hard" core or the "adamant" part of creating an extended naming function is in the initial portion Nom that maps objects of the "external world" into signs in the "internal world". The subsequent task of assigning conventional names to signs is supposed to be comparatively natural and "easy", perhaps on account of the "nominal" nature of signs themselves.

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The effect of reflection on the original sign relation R c OxSxI can be analyzed as follows. Suppose that a step of reflection creates HO signs for a subset of S. Then this step involves the construction of a newly extended sign relation:

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