[Inquiry] Re: Differential Logic

[Jon Awbrey]

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

DLOG.  Note D77

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

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
o------------o---------------------------------o------------o
|     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    |
|            |          |           |          |            |
o------------o----------o-----------o----------o------------o

Jon Awbrey

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o