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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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There are many ways to extend sign relations in an effort to develop their reflective capacities. The implicit goal of a reflective project is to reach a condition of ''reflective closure'', a configuration satisfying the inclusion <math>S \subseteq O,\!</math> where every sign is an object. It is important to note that not every process of reflective extension can achieve a reflective closure in a finite sign relation. This can only happen if there are additional equivalence relations that keep the effective orders of signs within finite bounds. As long as there are higher order signs that remain distinct from all lower order signs, the sign relation driven by a reflective process is forced to keep expanding. In particular, the process that is ''freely'' suggested by the formation of <math>\operatorname{Ref}^1 L(A)\!</math> and <math>\operatorname{Ref}^1 L(B)\!</math> cannot reach closure if it continues as indicated, without further constraints.

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Tables 44 and 45 present ''higher import extensions'' of <math>L(A)\!</math> and <math>L(B),\!</math> respectively. These are just higher order sign relations that add selections of higher import signs and their objects to the underlying set of triples in <math>L(A)\!</math> and <math>L(B).\!</math> One way to understand these extensions is as follows. The interpreters <math>A\!</math> and <math>B\!</math> each use nouns and pronouns just as before, except that the nouns are given additional denotations that refer to the interpretive conduct of the interpreter named. In this form of development, using a noun as a canonical form that refers indifferently to all the <math>(o, s, i)\!</math> triples of a sign relation is a pragmatic way that a sign relation can refer to itself and to other sign relations.

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There are many ways to extend sign relations in an effort to develop their reflective capacities. The implicit goal of a reflective project is to reach a condition of "reflective closure", a configuration satisfying the inclusion S c O, where every sign is an object. It is important to note that not every process of reflective extension can achieve a reflective closure in a finite sign relation. This can only happen if there are additional equivalence relations that keep the effective orders of signs within finite bounds. As long as there are HO signs that remain distinct from all LO signs, the sign relation driven by a reflective process is forced to keep expanding. In particular, the process that is "freely" suggested by the formation of Ref1(A) and Ref1(B) cannot reach closure if it continues as indicated, without further constraints.

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Table 44. Higher Import Sign Relation HI1(A)

Object Sign Interpretant

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<B, , >

<B, , >

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Tables 44 and 45 present "HI extensions" of A and B, respectively. These are just HO sign relations that add selections of HI signs and their objects to the underlying set of triples in A and B. One way to understand these extensions is as follows. The interpreters A and B each use nouns and pronouns just as before, except that the nouns are given additional denotations that refer to the interpretive conduct of the interpreter named. In this form of development, using a noun as a canonical form that refers indifferently to all the <~~o, s, i~~> ~~triples of a sign relation is a pragmatic way that a sign relation can refer to itself and to other sign relations.~~

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Table 45. Higher Import Sign Relation HI1(B)

Object Sign Interpretant

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<B, , >

<B, , >

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Several important facts about the class of HO sign relations in general are illustrated by these examples. First, the notations appearing in the object columns of HI1(A) and HI1(B) are not the terms that these newly extended interpreters are depicted as using to describe their objects, but the kinds of language that you and I, or other external observers, would typically make available to distinguish them. The agents A and B, as extended by the transactions of HI1(A) and HI1(B), respectively, are still restricted to their original syntactic domain {"A", "B", "i", "u"}. This means that there need be nothing especially articulate about a HI sign relation just because it qualifies as HO. Indeed, the sign relations HI1(A) and HI1(B) are not very discriminating in their descriptions of the sign relations A and B, referring to many different things under the very same signs that you and I and others would explicitly distinguish, especially in marking the distinction between an interpretive agent and any one of its individual transactions.

Jon Awbrey

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