ONT Re: Differential Logic
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DLOG. Note D77
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Transformations of Type B^2 -> B^2 (cont.)
Table 64 shows how the action of the transformation F
on cells or points is computed in terms of coordinates.
Table 64. Transformation of Positions
o-----o----------o----------o-------o-------o--------o--------o-------------o
| u v | x | y | x y | x(y) | (x)y | (x)(y) | X% = [x, y] |
o-----o----------o----------o-------o-------o--------o--------o-------------o
| | | | | | | | ^ |
| 0 0 | 0 | 1 | 0 | 0 | 1 | 0 | | |
| | | | | | | | |
| 0 1 | 1 | 0 | 0 | 1 | 0 | 0 | F |
| | | | | | | | = |
| 1 0 | 1 | 0 | 0 | 1 | 0 | 0 | <f , g> |
| | | | | | | | |
| 1 1 | 1 | 1 | 1 | 0 | 0 | 0 | ^ |
| | | | | | | | | |
o-----o----------o----------o-------o-------o--------o--------o-------------o
| | ((u)(v)) | ((u, v)) | u v | (u,v) | (u)(v) | 0 | U% = [u, v] |
o-----o----------o----------o-------o-------o--------o--------o-------------o
Table 65 extends this scheme from single cells to arbitrary regions of
the source and target universes, and illustrates a form of computation
that can be used to determine how a logical transformation acts on all
of the propositions in a universe of discourse, what is usually called
the "induced action" of the transformation from universe to universe.
Table 65. Induced Transformation of Propositions
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| X% | <--- F = <f , g> <--- | U% |
o------------o----------o-----------o----------o------------o
| | u = | 1 1 0 0 | = u | |
| | v = | 1 0 1 0 | = v | |
| f_i <x, y> o----------o-----------o----------o f_j <u, v> |
| | x = | 1 1 1 0 | = f<u,v> | |
| | y = | 1 0 0 1 | = g<u,v> | |
o------------o----------o-----------o----------o------------o
| | | | | |
| f_0 | () | 0 0 0 0 | () | f_0 |
| | | | | |
| f_1 | (x)(y) | 0 0 0 1 | () | f_0 |
| | | | | |
| f_2 | (x) y | 0 0 1 0 | (u)(v) | f_1 |
| | | | | |
| f_3 | (x) | 0 0 1 1 | (u)(v) | f_1 |
| | | | | |
| f_4 | x (y) | 0 1 0 0 | (u, v) | f_6 |
| | | | | |
| f_5 | (y) | 0 1 0 1 | (u, v) | f_6 |
| | | | | |
| f_6 | (x, y) | 0 1 1 0 | (u v) | f_7 |
| | | | | |
| f_7 | (x y) | 0 1 1 1 | (u v) | f_7 |
| | | | | |
o------------o----------o-----------o----------o------------o
| | | | | |
| f_8 | x y | 1 0 0 0 | u v | f_8 |
| | | | | |
| f_9 | ((x, y)) | 1 0 0 1 | u v | f_8 |
| | | | | |
| f_10 | y | 1 0 1 0 | ((u, v)) | f_9 |
| | | | | |
| f_11 | (x (y)) | 1 0 1 1 | ((u, v)) | f_9 |
| | | | | |
| f_12 | x | 1 1 0 0 | ((u)(v)) | f_14 |
| | | | | |
| f_13 | ((x) y) | 1 1 0 1 | ((u)(v)) | f_14 |
| | | | | |
| f_14 | ((x)(y)) | 1 1 1 0 | (()) | f_15 |
| | | | | |
| f_15 | (()) | 1 1 1 1 | (()) | f_15 |
| | | | | |
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Jon Awbrey
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