ONT Re: Sign Relations
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Sign Relation SIG,
I continue with the discussion of sign relations in the medium
of concrete examples, as illustrated by the "Story of A and B".
This episode sketches a variety of graph-theoretical pictures
that can aid the imagination in thinking about sign relations.
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| Document History:
|
| Subject: Inquiry Driven Systems: An Inquiry Into Inquiry
| Contact: Jon Awbrey <jawbrey@oakland.edu>
| Version: Draft 8.61
| Created: 23-Jun-1996
| Revised: 04-Sep-2001
| Advisor: M.A. Zohdy
| Setting: Oakland University, Rochester, Michigan, USA
| Excerpt: Subsection 1.3.4.4 (Graphical Representations)
1.3.4.4 Graphical Representations
The dyadic components of sign relations can be given graph-theoretic
representations, as "digraphs" (or "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 denotative components Den(A) and Den(B) can be represented as digraphs on the
six points of their common world set W = O |_| S |_| I = {A, B, "A", "B", "i", "u"}.
The arcs are given as follows:
1. Den(A) has an arc
from each point of {"A", "i"} to A and
from each point of {"B", "u"} to B.
2. Den(B) has an arc
from each point of {"A", "u"} to A and
from each point of {"B", "i"} to B.
Den(A) and Den(B) can be interpreted as "transition digraphs" that chart the
succession of steps or the connection of states in a computational process.
If the graph is read this way, the denotational arcs summarize the "upshots"
of the computations that are involved when the interpreters A and B evaluate
the signs in S according to their own frames of reference.
The connotative components Con(A) and Con(B) can be represented as digraphs on
the four points of their common syntactic domain S = I = {"A", "B", "i", "u"}.
Since Con(A) and Con(B) are SER's, their digraphs conform to the pattern that
is manifested by all digraphs of equivalence relations. In general, a digraph
of an equivalence relation falls into connected components that correspond to
the parts of the associated partition, with a complete digraph on the points of
each part, and no other arcs. In the present case, the arcs are given as follows:
1. Con(A) has the structure of a SER on S,
with a sling at each of the points in S,
two-way arcs between the points of {"A", "i"}, and
two-way arcs between the points of {"B", "u"}.
2. Con(B) has the structure of a SER on S,
with a sling at each of the points in S,
two-way arcs between the points of {"A", "u"}, and
two-way arcs between the points of {"B", "i"}.
Taken as transition digraphs, Con(A) and Con(B) highlight the associations that are permitted
between equivalent signs, as this equivalence is judged by the interpreters A and B, respectively.
The theme running through the last three subsections, that associates different
interpreters and different aspects of interpretation with different sorts of
relational structures on the same set of points, heralds a topic that will
be developed extensively in the sequel.
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