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Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Part 6 (view source)
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, 03:48, 3 May 2012→6.10. Higher Order Sign Relations : Examples: markup
Tables 42 and 43 show one way that the sign relations <math>L(A)\!</math> and <math>L(B)\!</math> can be extended in a reflective sense through the use of quotational devices, yielding the ''first order reflective extensions'', <math>\operatorname{Ref}^1 L(A)\!</math> and <math>\operatorname{Ref}^1 L(B).\!</math> These extensions add one layer of HA signs and their objects to the sign relations <math>L(A)\!</math> and <math>L(B),\!</math> respectively. The new triples specify that, for each <math>{}^{\langle} x {}^{\rangle}\!</math> in the set <math>\{ {}^{\langle} \text{A} {}^{\rangle}, {}^{\langle} \text{B} {}^{\rangle}, {}^{\langle} \text{i} {}^{\rangle}, {}^{\langle} \text{u} {}^{\rangle} \},\!</math> the HA sign of the form <math>{}^{\langle\langle} x {}^{\rangle\rangle}\!</math> connotes itself while denoting <math>{}^{\langle} x {}^{\rangle}.\!</math>
Tables 42 and 43 show one way that the sign relations <math>L(A)\!</math> and <math>L(B)\!</math> can be extended in a reflective sense through the use of quotational devices, yielding the ''first order reflective extensions'', <math>\operatorname{Ref}^1 L(A)\!</math> and <math>\operatorname{Ref}^1 L(B).\!</math> These extensions add one layer of HA signs and their objects to the sign relations <math>L(A)\!</math> and <math>L(B),\!</math> respectively. The new triples specify that, for each <math>{}^{\langle} x {}^{\rangle}\!</math> in the set <math>\{ {}^{\langle} \text{A} {}^{\rangle}, {}^{\langle} \text{B} {}^{\rangle}, {}^{\langle} \text{i} {}^{\rangle}, {}^{\langle} \text{u} {}^{\rangle} \},\!</math> the HA sign of the form <math>{}^{\langle\langle} x {}^{\rangle\rangle}\!</math> connotes itself while denoting <math>{}^{\langle} x {}^{\rangle}.\!</math>
Notice that the semantic equivalences of nouns and pronouns referring to each interpreter do not extend to semantic equivalences of their higher order signs, exactly as demanded by the literal character of quotations. Also notice that the reflective extensions of the sign relations <math>L(A)\!</math> and <math>L(B)\!</math> coincide in their reflective parts, since exactly the same triples were added to each set.
<pre>
<pre>
Table 42. Higher Ascent Sign Relation Ref1(A)
Table 42. Higher Ascent Sign Relation Ref1(A)
Object Sign Interpretant
Object Sign Interpretant
<i> <<i>> <<i>>
<i> <<i>> <<i>>
<u> <<u>> <<u>>
<u> <<u>> <<u>>
</pre>
<pre>
Table 43. Higher Ascent Sign Relation Ref1(B)
Table 43. Higher Ascent Sign Relation Ref1(B)
Object Sign Interpretant
Object Sign Interpretant
<i> <<i>> <<i>>
<i> <<i>> <<i>>
<u> <<u>> <<u>>
<u> <<u>> <<u>>
</pre>
<pre>
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.
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.