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Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Part 6 (view source)
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===6.42. Sign Processes : A Start===
===6.42. Sign Processes : A Start===
<pre>
To articulate the dynamic aspects of a sign relation, one can interpret it as determining a discrete or finite state transition system. In the usual ways of doing this, the states of the system are given by the elements of the syntactic domain, while the elements of the object domain correspond to input data or control parameters that affect transitions from signs to interpretant signs in the syntactic state space.
Working from these principles alone, there are numerous ways that a plausible dynamics can be invented for a given sign relation. I will concentrate on two principal forms of dynamic realization, or two ways of interpreting and augmenting sign relations as sign processes.
One form of realization lets each element of the object domain O correspond to the observed presence of an object in the environment of the systematic agent. In this interpretation, the object X acts as an input datum that causes the system Y to shift from whatever sign state it happens to occupy at a given moment to a random sign state in [X]Y. Expressed in a cognitive vein, "Y notes X".
Another form of realization lets each element of the object domain O correspond to the autonomous intention of the systematic agent to denote an object, achieve an objective, or broadly speaking to accomplish any other purpose with respect to an object in its domain. In this interpretation, the object X is a control parameter that brings the system Y into line with realizing a target set [X]Y.
Tables 75 and 76 show how the sign relations for A and B can be filled out as finite state processes in conformity with the interpretive principles just described. Rather than letting the actions go undefined for some combinations of inputs C O and states C S, transitions have been added that take the interpreters from whatever else they might have been thinking about to the SEC's of their objects. In either modality of realization, cognitive or control oriented, the abstract structure of the resulting sign process is exactly the same.
Table 75. Sign Process of Interpreter A
Object Sign Interpretant
A "A" "A"
A "A" "i"
A "i" "A"
A "i" "i"
A "B" "A"
A "B" "i"
A "u" "A"
A "u" "i"
B "A" "B"
B "A" "u"
B "i" "B"
B "i" "u"
B "B" "B"
B "B" "u"
B "u" "B"
B "u" "u"
Table 76. Sign Process of Interpreter B
Object Sign Interpretant
A "A" "A"
A "A" "u"
A "u" "A"
A "u" "u"
A "B" "A"
A "B" "u"
A "i" "A"
A "i" "u"
B "A" "B"
B "A" "i"
B "u" "B"
B "u" "i"
B "B" "B"
B "B" "i"
B "i" "B"
B "i" "i"
Treated in accord with these interpretations, the sign relations A and B constitute partially degenerate cases of dynamic processes, in which the transitions are totally non deterministic up to semantic equivalence classes but still manage to preserve those classes. Whether construed as present observation or projective speculation, the most significant feature to note about a sign process is how the contemplation of an object or objective leads the system from a less determined to a more determined condition.
On reflection, one observes that these processes are not completely trivial since they preserve the structure of their semantic partitions. In fact, each sign process preserves the entire topology ("family of sets closed under finite intersections and arbitrary unions") generated by its semantic equivalence classes. These topologies, Top(A) and Top(B), can be viewed as partially ordered sets, Pos(A) and Pos(B), by taking the inclusion ordering (c) as (<). For each of the interpreters A and B, as things stand in their respective orderings Pos(A) and Pos(B), the semantic equivalence classes of "A" and "B" are situated as intermediate elements that are incomparable to each other.
Top(A) = Pos(A) = { {}, {"A", "i"}, {"B", "u"}, S }.
Top(B) = Pos(B) = { {}, {"A", "u"}, {"B", "i"}, S }.
In anticipation of things to come, these orderings are germinal versions of the kinds of semantic hierarchies that will be used in this project to define the "ontologies", "world views", or "perspectives" corresponding to individual interpreters.
When it comes to discussing the stability properties of dynamic systems, the sets that remain invariant under iterated applications of a process are called its "attractors" or "basins of attraction".
More care needed here. Strongly and weakly connected components of a digraph?
The dynamic realizations of the sign relations A and B augment their semantic equivalence relations in an "attractive" way. To describe this additional structure, I introduce a set of graph theoretical concepts and notations.
The "attractor" of X in Y.
Y@X = "Y at X" = @[X]Y = [X]Y U { Arcs into [X]Y }.
In effect, this discussion of dynamic realizations of sign relations has advanced from considering SEP's as partitioning the set of points in S to considering attractors as partitioning the set of arcs in SxI = SxS.
</pre>
===6.43. Reflective Extensions===
===6.43. Reflective Extensions===