Changes

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 &ldquo;external world&rdquo; to signs in the &ldquo;internal world&rdquo;.  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.

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 &ldquo;external world&rdquo; to signs in the &ldquo;internal world&rdquo;.  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.

The effect of reflection on the original sign relation <math>L \subseteq O \times S \times I\!</math> can be analyzed as follows.  Suppose that a step of reflection creates higher order signs for a subset of <math>S.\!</math> Then this step involves the construction of a newly extended sign relation:

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:

R' c O'xS'xI', where O' = O U O1  and S' = S U S1.

| <math>L' \subseteq O' \times S' \times I', ~\text{where}~ O' = O \cup O_1 ~\text{and}~ S' = S \cup S_1.\!</math>

In this construction O1 c S is that portion of the original signs S for which HO signs are created in the initial step of reflection, thereby being converted into O1 c O'.  The sign domain S is extended to a new sign domain S' by the addition of these HO signs, namely, the set S1.  Using the arch quotes (<...>), the mapping from O1 to S1 can be defined as follows:

In this construction O1 c S is that portion of the original signs S for which HO signs are created in the initial step of reflection, thereby being converted into O1 c O'.  The sign domain S is extended to a new sign domain S' by the addition of these HO signs, namely, the set S1.  Using the arch quotes (<...>), the mapping from O1 to S1 can be defined as follows: