ONT Re: Differential Logic -- Series B

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DLOG.  Note B9

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Another way of looking at this situation is by letting the (first order)
differential features du, dv, dw be viewed as the features of another
universe of discourse, called the "tangent universe to X with respect
to the interpretation c" and represented as dX.c.  In this setting,
Dq.c, the "difference proposition of q at the interpretation c",
where c = uvw, is marked by the shaded region in Figure 4.

o-----------------------------------------------------------o
| dX.c                                                      |
|                                                           |
|                      o-------------o                      |
|                     /               \                     |
|                    /                 \                    |
|                   /                   \                   |
|                  /                     \                  |
|                 /                       \                 |
|                o                         o                |
|                |           dU            |                |
|                |                         |                |
|                |                         |                |
|                |                         |                |
|                |                         |                |
|             o--o----------o   o----------o--o             |
|            /    \``````````\ /``````````/    \            |
|           /      \````2`````o`````3````/      \           |
|          /        \````````/`\````````/        \          |
|         /          \``````/`1`\``````/          \         |
|        /            \````/`````\````/            \        |
|       o              o--o-------o--o              o       |
|       |                 |```````|                 |       |
|       |                 |```````|                 |       |
|       |                 |```````|                 |       |
|       |       dV        |```4```|       dW        |       |
|       |                 |```````|                 |       |
|       o                 o```````o                 o       |
|        \                 \`````/                 /        |
|         \                 \```/                 /         |
|          \                 \`/                 /          |
|           \                 o                 /           |
|            \               / \               /            |
|             o-------------o   o-------------o             |
|                                                           |
|                                                           |
o-----------------------------------------------------------o
Figure 4.  Tangent Venn Diagram for Dq.c

Taken in the context of the tangent universe to X at c = uvw,
written dX.c or dX.uvw, the shaded area of Figure 4 indicates
the models of the difference proposition Dq.uvw, specifically:

   1.  u v w  du  dv  dw

   2.  u v w  du  dv (dw)

   3.  u v w  du (dv) dw

   4.  u v w (du) dv  dw

Jon Awbrey

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http://www.cs.bsu.edu/homepages/mighty/history.html
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