ONT Re: Differential Logic
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DLOG. Note D71
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| The past and present wilt . . . . I have filled them and
| emptied them,
| And proceed to fill my next fold of the future.
|
| Walt Whitman, 'Leaves of Grass', [Whi, 87]
Taking Aim at Higher Dimensional Targets
In the next Subdivision I consider a logical transformation F that has the
concrete type F : [u, v] -> [x, y] and the abstract type F : [B^2] -> [B^2].
From the standpoint of propositional calculus, the task of understanding such
a transformation is naturally approached by parsing it into component maps with
1-dimensional ranges, as follows:
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| |
| F = <f, g> = <F_1, F_2> : [u, v] -> [x, y] |
| |
| where f = F_1 : [u, v] -> [x] |
| |
| and g = F_2 : [u, v] -> [y] |
| |
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Then one tackles the separate components, now viewed as propositions
F_i : U -> B, one at a time. At the completion of this analytic phase,
one returns to the task of synthesizing all of these partial and transient
impressions into an agile form of integrity, a solidly coordinated and deeply
integrated comprehension of the ongoing transformation. (Very often, of course,
in tangling with refractory cases, one never gets as far as the beginning again.)
Jon Awbrey
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