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Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Part 6 (view source)
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The next two pairs of reflective extensions demonstrate that there are ways of achieving reflective closure that do not generate infinite sign relations.
The next two pairs of reflective extensions demonstrate that there are ways of achieving reflective closure that do not generate infinite sign relations.
As a flexible and fairly general strategy for describing reflective extensions, it is convenient to take the following tack. Given a syntactic domain <math>S,\!</math> there is an independent formal language <math>F = F(S) = S \langle {}^{\langle\rangle} \rangle,\!</math> called the ''free quotational extension of <math>S,\!</math>'' that can be generated from <math>S\!</math> by embedding each of its signs to any depth of quotation marks. Within <math>F,\!</math> the quoting operation can be regarded as a syntactic generator that is inherently free of constraining relations. In other words, for every <math>s \in S,\!</math> the sequence <math>s, {}^{\langle} s {}^{\rangle}, {}^{\langle\langle} s {}^{\rangle\rangle}, \ldots\!</math> contains nothing but pairwise distinct elements in <math>F\!</math> no matter how far it is produced. The set <math>F(s) = s \langle {}^{\langle\rangle} \rangle \subseteq F\!</math> that collects the elements of this sequence is called the ''subset of <math>F\!</math> generated from <math>s\!</math> by quotation''.
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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).