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Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Part 6 (view source)
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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.
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.
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 — the family of sets closed under finite intersections and arbitrary unions — that is generated by its semantic equivalence classes. These topologies, <math>\operatorname{Top}(\text{A})\!</math> and <math>\operatorname{Top}(\text{B}),\!</math> can be viewed as partially ordered sets, <math>\operatorname{Poset}(\text{A})\!</math> and <math>\operatorname{Poset}(\text{B}),\!</math> by taking the inclusion ordering <math>(\subseteq)\!</math> as <math>(\le).\!</math> For each of the interpreters <math>\text{A}\!</math> and <math>\text{B},\!</math> as things stand in their respective orderings <math>\operatorname{Poset}(\text{A})\!</math> and <math>\operatorname{Poset}(\text{B}),\!</math> the semantic equivalence classes of <math>{}^{\backprime\backprime} \text{A} {}^{\prime\prime}\!</math> and <math>{}^{\backprime\backprime} \text{B} {}^{\prime\prime}\!</math> are situated as intermediate elements that are incomparable to each other.
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Top(A) = Pos(A) = { {}, {"A", "i"}, {"B", "u"}, S }.
Top(A) = Pos(A) = { {}, {"A", "i"}, {"B", "u"}, S }.
Top(B) = Pos(B) = { {}, {"A", "u"}, {"B", "i"}, S }.
Top(B) = Pos(B) = { {}, {"A", "u"}, {"B", "i"}, S }.