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Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Part 6 (view source)
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, 18:34, 14 April 2013→6.43. Reflective Extensions: segment
This section takes up the topic of reflective extensions in a more systematic fashion, starting from the sign relations A and B once again and keeping its focus within their vicinity, but exploring the space of nearby extensions in greater detail.
This section takes up the topic of reflective extensions in a more systematic fashion, starting from the sign relations A and B once again and keeping its focus within their vicinity, but exploring the space of nearby extensions in greater detail.
Tables 77 and 78 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).
Tables 80 and 81 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).
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Table 77. Reflective Extension Ref1(A)
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Table 80. Reflective Extension Ref1(A)
Object Sign Interpretant
Object Sign Interpretant
A <A> <A>
A <A> <A>
<i> <<i>> <<i>>
<i> <<i>> <<i>>
<u> <<u>> <<u>>
<u> <<u>> <<u>>
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Table 78. Reflective Extension Ref1(B)
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Table 81. Reflective Extension Ref1(B)
Object Sign Interpretant
Object Sign Interpretant
A <A> <A>
A <A> <A>
<i> <<i>> <<i>>
<i> <<i>> <<i>>
<u> <<u>> <<u>>
<u> <<u>> <<u>>
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The common "world" = {objects} U {signs} of the reflective extensions Ref1 (A) and Ref1 (B) is the set of 10 elements:
The common "world" = {objects} U {signs} of the reflective extensions Ref1 (A) and Ref1 (B) is the set of 10 elements:
Against this background, other varieties of reflective extension can be specified by means of semantic equations (SEQs) that are considered to be imposed on the elements of F. Taking the reflective extensions Ref1 (A) and Ref1 (B) as the first orders of a "free" project toward reflective closure, variant extensions can be described by relating their entries with those of comparable members in the standard sequences Refn (A) and Refn (B).
Against this background, other varieties of reflective extension can be specified by means of semantic equations (SEQs) that are considered to be imposed on the elements of F. Taking the reflective extensions Ref1 (A) and Ref1 (B) as the first orders of a "free" project toward reflective closure, variant extensions can be described by relating their entries with those of comparable members in the standard sequences Refn (A) and Refn (B).
A variant pair of reflective extensions, Ref1(A|E1) and Ref1(B|E1), are presented in Tables 79 and 80, respectively. These are identical to the corresponding "free" variants, Ref1(A) and Ref1(B), with the exception of those entries that are constrained by the system of semantic equations:
A variant pair of reflective extensions, Ref1(A|E1) and Ref1(B|E1), are presented in Tables 82 and 83, respectively. These are identical to the corresponding "free" variants, Ref1(A) and Ref1(B), with the exception of those entries that are constrained by the system of semantic equations:
E1: <<A>> = <A>, <<B>> = <B>, <<i>> = <i>, <<u>> = <u>.
E1: <<A>> = <A>, <<B>> = <B>, <<i>> = <i>, <<u>> = <u>.
??? Redo F(S) over W ??? Use WF = O U F ???
??? Redo F(S) over W ??? Use WF = O U F ???
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Table 79. Reflective Extension Ref1(A|E1)
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Table 82. Reflective Extension Ref1(A|E1)
Object Sign Interpretant
Object Sign Interpretant
A <A> <A>
A <A> <A>
<i> <i> <i>
<i> <i> <i>
<u> <u> <u>
<u> <u> <u>
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Table 80. Reflective Extension Ref1(B|E1)
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Table 83. Reflective Extension Ref1(B|E1)
Object Sign Interpretant
Object Sign Interpretant
A <A> <A>
A <A> <A>
<i> <i> <i>
<i> <i> <i>
<u> <u> <u>
<u> <u> <u>
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Another pair of reflective extensions, Ref1(A|E2) and Ref1(B|E2), are presented in Tables 81 and 82, respectively. These are identical to the corresponding "free" variants, Ref1(A) and Ref1(B), except for the entries constrained by the following semantic equations:
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Another pair of reflective extensions, Ref1(A|E2) and Ref1(B|E2), are presented in Tables 84 and 85, respectively. These are identical to the corresponding "free" variants, Ref1(A) and Ref1(B), except for the entries constrained by the following semantic equations:
E2: <<A>> = A, <<B>> = B, <<i>> = i, <<u>> = u.
E2: <<A>> = A, <<B>> = B, <<i>> = i, <<u>> = u.
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Table 81. Reflective Extension Ref1(A|E2)
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Table 84. Reflective Extension Ref1(A|E2)
Object Sign Interpretant
Object Sign Interpretant
A <A> <A>
A <A> <A>
<i> A A
<i> A A
<u> B B
<u> B B
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Table 82. Reflective Extension Ref1(B|E2)
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Table 85. Reflective Extension Ref1(B|E2)
Object Sign Interpretant
Object Sign Interpretant
A <A> <A>
A <A> <A>