ONT Re: Differential Logic
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DLOG. Note D54
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Difference Map of Conjunction
| "It doesn't matter what one does", the Man Without Qualities said to himself,
| shrugging his shoulders. "In a tangle of forces like this it doesn't make a
| scrap of difference." He turned away like a man who has learned renunciation,
| almost indeed like a sick man who shrinks from any intensity of contact. And
| then, striding through his adjacent dressing-room, he passed a punching-ball
| that hung there; he gave it a blow far swifter and harder than is usual in
| moods of resignation or states of weakness.
|
| Robert Musil, 'The Man Without Qualities', [Mus, 8]
With the tacit extension map !e!J and the enlargement map EJ well in place,
the difference map DJ can be computed along the lines displayed in Table 41,
ending up, in this instance, with an expansion of DJ over the cells of [u, v].
Table 41. Computation of DJ (Method 1)
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| |
| DJ = EJ + !e!J |
| |
| = J<u + du, v + dv> + J<u, v> |
| |
| = (u, du)(v, dv) + u v |
| |
o-------------------------------------------------------------------------------o
| |
| DJ = 0 |
| |
| + u v (du) dv + u (v)(du) dv |
| |
| + u v du (dv) + (u) v du (dv) |
| |
| + u v du dv + (u)(v) du dv |
| |
o-------------------------------------------------------------------------------o
| |
| DJ = u v ((du)(dv)) + u (v)(du) dv + (u) v du (dv) + (u)(v) du dv |
| |
o-------------------------------------------------------------------------------o
Alternatively, the difference map DJ can be expanded over the cells of [du, dv]
to arrive at the formulation shown in Table 42. The same development would be
obtained from the previous Table by collecting terms in an alternate manner,
along the rows rather than the columns of the middle portion of the Table.
Table 42. Computation of DJ (Method 2)
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| |
| DJ = !e!J + EJ |
| |
| = J<u, v> + J<u + du, v + dv> |
| |
| = u v + (u, du)(v, dv) |
| |
| = 0 + u dv + v du + du dv |
| |
| = 0 + u (du) dv + v du (dv) + ((u, v)) du dv |
| |
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Even more simply, the same result is reached by matching up the propositional
coefficients of !e!J and EJ along the cells of [du, dv] and adding the pairs
under boolean sums (that is, "mod 2", where 1 + 1 = 0), as shown in Table 43.
Table 43. Computation of DJ (Method 3)
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| |
| DJ = !e!J + EJ |
| |
o-------------------------------------------------------------------------------o
| |
| !e!J = u v (du)(dv) + u v (du) dv + u v du (dv) + u v du dv |
| |
| EJ = u v (du)(dv) + u (v)(du) dv + (u) v du (dv) + (u)(v) du dv |
| |
o-------------------------------------------------------------------------------o
| |
| DJ = 0 . (du)(dv) + u . (du) dv + v . du (dv) + ((u, v)) du dv |
| |
o-------------------------------------------------------------------------------o
Jon Awbrey
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