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ONT Re: Differential Logic


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DLOG.  Note D58

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Differential of Conjunction (concl.)

Figures 46-a through 46-d illustrate the proposition dJ, rounded out
in our usual array of prospects.  This proposition of EU% is what we
refer to as the (first order) differential of J, and normally regard
as 'the' differential proposition corresponding to J.

o---------------------------------------o
|                                       |
|                   o                   |
|                  / \                  |
|                 /   \                 |
|                /     \                |
|               o       o               |
|              /%\     /%\              |
|             /%%%\   /%%%\             |
|            /%%%%%\ /%%%%%\            |
|           o%%%%%%%o%%%%%%%o           |
|          /%\%%%%%/ \%%%%%/%\          |
|         /%%%\%%%/   \%%%/%%%\         |
|        /%%%%%\%/     \%/%%%%%\        |
|       o%%%%%%%o       o%%%%%%%o       |
|      / \%%%%%/%\     /%\%%%%%/ \      |
|     /   \%%%/%%%\   /%%%\%%%/   \     |
|    /     \%/%%%%%\ /%%%%%\%/     \    |
|   o       o%%%%%%%o%%%%%%%o       o   |
|   |\     / \%%%%%/ \%%%%%/ \     /|   |
|   | \   /   \%%%/   \%%%/   \   / |   |
|   |  \ /     \%/     \%/     \ /  |   |
|   |   o       o       o       o   |   |
|   |   |\     / \     / \     /|   |   |
|   |   | \   /   \   /   \   / |   |   |
|   | u |  \ /     \ /     \ /  | v |   |
|   o---+---o       o       o---+---o   |
|       |    \     / \     /    |       |
|       |     \   /   \   /     |       |
|       | du   \ /     \ /   dv |       |
|       o-------o       o-------o       |
|                \     /                |
|                 \   /                 |
|                  \ /                  |
|                   o                   |
|                                       |
o---------------------------------------o
Figure 46-a.  Differential of J (Areal)

                                                  o-----------------------------o
                                                  |                             |
                                                  |      o-----o   o-----o      |
                                                  |     /       \ /       \     |
                                                  |    /         o         \    |
                                                  |   /         / \         \   |
                                                  |  o         o   o         o  |
                                                  @  |   du    |   |    dv   |  |
                                                 /|  o         o   o         o  |
                                                / |   \         \ /         /   |
                                               /  |    \         o         /    |
                                              /   |     \       / \       /     |
                                             /    |      o-----o   o-----o      |
                                            /     |                             |
                                           /      o-----------------------------o
                                          /
o----------------------------------------/----o   o-----------------------------o
|                                       /     |   |                             |
|                                      @      |   |      o-----o   o-----o      |
|                                             |   |     /%%%%%%%\ /       \     |
|          o---------o   o---------o          |   |    /%%%%%%%%%o         \    |
|         /           \ /           \         |   |   /%%%%%%%%%/%\         \   |
|        /             o             \        |   |  o%%%%%%%%%o%%%o         o  |
|       /             /`\      @------\-----------@  |%% du %%%|%%%|    dv   |  |
|      /             /```\             \      |   |  o%%%%%%%%%o%%%o         o  |
|     /             /`````\             \     |   |   \%%%%%%%%%\%/         /   |
|    /             /```````\             \    |   |    \%%%%%%%%%o         /    |
|   o             o`````````o             o   |   |     \%%%%%%%/ \       /     |
|   |             |````@````|             |   |   |      o-----o   o-----o      |
|   |             |`````\```|             |   |   |                             |
|   |             |``````\``|             |   |   o-----------------------------o
|   |      u      |```````\`|      v      |   |
|   |             |````````\|             |   |   o-----------------------------o
|   |             |`````````|             |   |   |                             |
|   |             |`````````|\            |   |   |      o-----o   o-----o      |
|   o             o`````````o \           o   |   |     /       \ /%%%%%%%\     |
|    \             \```````/   \         /    |   |    /         o%%%%%%%%%\    |
|     \             \`````/     \       /     |   |   /         /%\%%%%%%%%%\   |
|      \             \```/       \     /      |   |  o         o%%%o%%%%%%%%%o  |
|       \      @------\-/---------\---------------@  |   du    |%%%|%%% dv %%|  |
|        \             o           \ /        |   |  o         o%%%o%%%%%%%%%o  |
|         \           / \           /         |   |   \         \%/%%%%%%%%%/   |
|          o---------o   o---------o \        |   |    \         o%%%%%%%%%/    |
|                                     \       |   |     \       / \%%%%%%%/     |
|                                      \      |   |      o-----o   o-----o      |
|                                       \     |   |                             |
o----------------------------------------\----o   o-----------------------------o
                                          \
                                           \      o-----------------------------o
                                            \     |                             |
                                             \    |      o-----o   o-----o      |
                                              \   |     /%%%%%%%\ /%%%%%%%\     |
                                               \  |    /%%%%%%%%%o%%%%%%%%%\    |
                                                \ |   /%%%%%%%%%/ \%%%%%%%%%\   |
                                                 \|  o%%%%%%%%%o   o%%%%%%%%%o  |
                                                  @  |%% du %%%|   |%%% dv %%|  |
                                                  |  o%%%%%%%%%o   o%%%%%%%%%o  |
                                                  |   \%%%%%%%%%\ /%%%%%%%%%/   |
                                                  |    \%%%%%%%%%o%%%%%%%%%/    |
                                                  |     \%%%%%%%/ \%%%%%%%/     |
                                                  |      o-----o   o-----o      |
                                                  |                             |
                                                  o-----------------------------o
Figure 46-b.  Differential of J (Bundle)

o---------------------------------------------------------------------o
|                                                                     |
|                                                                     |
|            o-------------------o   o-------------------o            |
|           /                     \ /                     \           |
|          /                       o                       \          |
|         /                       / \                       \         |
|        /                       /   \                       \        |
|       /                       /     \                       \       |
|      /                       /   @   \                       \      |
|     /                       /   ^ ^   \                       \     |
|    o                       o   /   \   o                       o    |
|    |                       |  /     \  |                       |    |
|    |                       | /       \ |                       |    |
|    |                       |/         \|                       |    |
|    |         u         (du)/ dv     du \(dv)         v         |    |
|    |                      /|           |\                      |    |
|    |                     / |           | \                     |    |
|    |                    /  |           |  \                    |    |
|    o                   /   o           o   \                   o    |
|     \                 /     \         /     \                 /     |
|      \               v       \ du dv /       v               /      |
|       \             @<----------------------->@             /       |
|        \                       \   /                       /        |
|         \                       \ /                       /         |
|          \                       o                       /          |
|           \                     / \                     /           |
|            o-------------------o   o-------------------o            |
|                                                                     |
|                                                                     |
o---------------------------------------------------------------------o
Figure 46-c.  Differential of J (Compact)

o-----------------------------------------------------------o
|                                                           |
|                            u v                            |
|                             @                             |
|                            ^ ^                            |
|                           /   \                           |
|                          /     \                          |
|                         /       \                         |
|                        /         \                        |
|               (du) dv /           \ du (dv)               |
|                      /             \                      |
|                     /               \                     |
|                    /                 \                    |
|                   /                   \                   |
|                  v                     v                  |
|           u (v) @<--------------------->@ (u) v           |
|                           du dv                           |
|                                                           |
|                                                           |
|                                                           |
|                                                           |
|                                                           |
|                                                           |
|                                                           |
|                                                           |
|                                                           |
|                                                           |
|                             @                             |
|                          (u) (v)                          |
|                                                           |
o-----------------------------------------------------------o
Figure 46-d.  Differential of J (Digraph)

Jon Awbrey

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