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Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Part 6 (view source)
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, 20:00, 19 April 2013→6.42. Sign Processes : A Start: markup
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Treated in accord with these interpretations, the sign relations <math>L(\text{A})\!</math> and <math>L(\text{B})\!</math> 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.
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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.
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.