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Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Part 6 (view source)
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The rest of this section considers the development of sign relations that have moderate capacities to reference their own signs as objects. In each case, these extensions are assumed to begin with sign relations like <math>L(A)\!</math> and <math>L(B)\!</math> that have disjoint sets of objects and signs and thus have no reflective capacity at the outset. The status of <math>L(A)\!</math> and <math>L(B)\!</math> as the reflective origins of the associated reflective developments is recalled by saying that <math>L(A)\!</math> and <math>L(B)\!</math> themselves are the ''zeroth order reflective extensions'' of <math>L(A)\!</math> and <math>L(B),\!</math> in symbols, <math>L(A) = \operatorname{Ref}^0 L(A)\!</math> and <math>L(B) = \operatorname{Ref}^0 L(B).\!</math>
The rest of this section considers the development of sign relations that have moderate capacities to reference their own signs as objects. In each case, these extensions are assumed to begin with sign relations like <math>L(A)\!</math> and <math>L(B)\!</math> that have disjoint sets of objects and signs and thus have no reflective capacity at the outset. The status of <math>L(A)\!</math> and <math>L(B)\!</math> as the reflective origins of the associated reflective developments is recalled by saying that <math>L(A)\!</math> and <math>L(B)\!</math> themselves are the ''zeroth order reflective extensions'' of <math>L(A)\!</math> and <math>L(B),\!</math> in symbols, <math>L(A) = \operatorname{Ref}^0 L(A)\!</math> and <math>L(B) = \operatorname{Ref}^0 L(B).\!</math>
The next set of Tables illustrates a few the most common ways that sign relations can begin to develop reflective extensions. For ease of reference, Tables&40 and 41 repeat the contents of Tables 1 and 2, respectively, merely replacing ordinary quotes with arch quotes.
<pre>
<pre>
Table 40. Reflective Origin Ref0(A)
Table 40. Reflective Origin Ref0(A)
Object Sign Interpretant
Object Sign Interpretant
B <u> <B>
B <u> <B>
B <u> <u>
B <u> <u>
</pre>
<pre>
Table 41. Reflective Origin Ref0(B)
Table 41. Reflective Origin Ref0(B)
Object Sign Interpretant
Object Sign Interpretant
B <i> <B>
B <i> <B>
B <i> <i>
B <i> <i>
</pre>
<pre>
Tables 42 and 43 show one way that the sign relations A and B can be extended in a reflective sense through the use of quotational devices, yielding the "first order reflective extensions", Ref1(A) and Ref1(B). These extensions add one layer of HA signs and their objects to the sign relations for A and B, respectively. The new triples specify that, for each <x> in the set {<A>, <B>, <i>, <u>}, the HA sign of the form <<x>> connotes itself while denoting <x>.
Tables 42 and 43 show one way that the sign relations A and B can be extended in a reflective sense through the use of quotational devices, yielding the "first order reflective extensions", Ref1(A) and Ref1(B). These extensions add one layer of HA signs and their objects to the sign relations for A and B, respectively. The new triples specify that, for each <x> in the set {<A>, <B>, <i>, <u>}, the HA sign of the form <<x>> connotes itself while denoting <x>.