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Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Part 6 (view source)
Revision as of 17:45, 15 May 2013
, 17:45, 15 May 2013→6.43. Reflective Extensions: markup
It may be observed that <math>S\!</math> overlaps with <math>O\!</math>O in the set of first-order signs or second-order objects, <math>S^{(1)} = O^{(2)},\!</math> exemplifying the extent to which signs have become objects in the new sign relations.
It may be observed that <math>S\!</math> overlaps with <math>O\!</math>O in the set of first-order signs or second-order objects, <math>S^{(1)} = O^{(2)},\!</math> exemplifying the extent to which signs have become objects in the new sign relations.
To discuss how the denotative and connotative aspects of a sign related are affected by its reflective extension it is useful to introduce a few abbreviations. For each sign relation <math>L \in \{ L(\text{A}), L(\text{B}) \}\!</math> the following operations may be defined.
To discuss how the denotative and connotative aspects of a sign related are affected by its reflective extension it is useful to introduce a few abbreviations. For each sign relation <math>L\!</math> in <math>\{ L(\text{A}), L(\text{B}) \}\!</math> the following operations may be defined.
{| align="center" cellspacing="6" width="90%"
|
<math>\begin{array}{lllll}
\operatorname{Den}^1 (L)
& = &
(\operatorname{Ref}^1 (L))_{SO}
& = &
\operatorname{proj}_{OS} (\operatorname{Ref}^1 (L))
\\[4pt]
\operatorname{Con}^1 (L)
& = &
(\operatorname{Ref}^1 (L))_{SI}
& = &
\operatorname{proj}_{SI} (\operatorname{Ref}^1 (L))
\end{array}</math>
|}
<pre>
<pre>
The dyadic components of sign relations can be given graph theoretic representations, namely, as "digraphs" (directed graphs), that provide concise pictures of their structural and potential dynamic properties. By way of terminology, a directed edge <x, y> is called an "arc" from point x to point y, and a self loop <x, x> is called a "sling" at x.
The dyadic components of sign relations can be given graph theoretic representations, namely, as "digraphs" (directed graphs), that provide concise pictures of their structural and potential dynamic properties. By way of terminology, a directed edge <x, y> is called an "arc" from point x to point y, and a self loop <x, x> is called a "sling" at x.