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Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Part 6 (view source)
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, 03:36, 3 May 2012→6.10. Higher Order Sign Relations : Examples: markup
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.
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.
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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>
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Notice that the semantic equivalences of nouns and pronouns referring to each interpreter do not extend to semantic equivalences of their HO signs, exactly as demanded by the literal character of quotations. Also notice that the reflective extensions of the sign relations A and B coincide in their reflective parts, since exactly the same triples were added to each set.
Notice that the semantic equivalences of nouns and pronouns referring to each interpreter do not extend to semantic equivalences of their HO signs, exactly as demanded by the literal character of quotations. Also notice that the reflective extensions of the sign relations A and B coincide in their reflective parts, since exactly the same triples were added to each set.