ONT Re: Differential Logic
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DLOG. Note D51
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Enlargement Map of Conjunction
| No one could have established the existence of any details
| that might not just as well have existed in earlier times
| too; but all the relations between things had shifted
| slightly. Ideas that had once been of lean account
| grew fat.
|
| Robert Musil, 'The Man Without Qualities', [Mus, 62]
The enlargement map EJ is computed from the proposition J by making
a particular class of formal substitutions for its variables, in this
case "u + du" for "u" and "v + dv" for "v", and subsequently expanding
the result in whatever way happens to be convenient for the end in view.
Table 38 shows a typical scheme of computation, following
a systematic method of exploiting boolean expansions over
selected variables, and ultimately developing EJ over the
cells of [u, v]. The critical step of this procedure uses
the facts that (0, x) = 0 + x = x and (1, x) = 1 + x = (x)
for any boolean variable x.
Table 38. Computation of EJ (Method 1)
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| |
| EJ = J<u + du, v + dv> |
| |
| = (u, du)(v, dv) |
| |
| = u v J<1 + du, 1 + dv> + |
| |
| u (v) J<1 + du, 0 + dv> + |
| |
| (u) v J<0 + du, 1 + dv> + |
| |
| (u)(v) J<0 + du, 0 + dv> |
| |
| = u v J<(du), (dv)> + |
| |
| u (v) J<(du), dv > + |
| |
| (u) v J< du , (dv)> + |
| |
| (u)(v) J< du , dv > |
| |
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| |
| EJ = u v (du)(dv) |
| + u (v)(du) dv |
| + (u) v du (dv) |
| + (u)(v) du dv |
| |
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Table 39 exhibits another method that happens to work quickly in
this particular case, using distributive laws to multiply things
out in an algebraic manner, arranging the notations of feature
and fluxion according to a scale of simple character and degree.
Proceeding this way leads through an intermediate step which,
in chiming the changes of ordinary calculus, should take on
a familiar ring. Consequential properties of exclusive
disjunction then carry us on to the concluding line.
Table 39. Computation of EJ (Method 2)
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| |
| EJ = <u + du> <v + dv> |
| |
| = u v + u dv + v du + du dv |
| |
| EJ = u v (du)(dv) + u (v)(du) dv + (u) v du (dv) + (u)(v) du dv |
| |
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Jon Awbrey
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