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ONT Re: Differential Logic


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DLOG.  Note D49

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Tacit Extension of Conjunction (cont.)

Figures 37-a through 37-d present several pictures of the proposition J and
its tacit extension !e!J.  Notice in these Figures how !e!J in EU% visibly
extends J in U%, by annexing to the indicated cells of J all of the arcs
that exit from or flow out of them.  In effect, this extension attaches
to these cells all of the dispositions that spring from them, in other
words, it attributes to these cells all of the conceivable changes
that are their issue.

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 37-a.  Tacit Extension 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 37-b.  Tacit Extension of J (Bundle)

o---------------------------------------------------------------------o
|                                                                     |
|                                                                     |
|            o-------------------o   o-------------------o            |
|           /                     \ /                     \           |
|          /                       o                       \          |
|         /                       / \                       \         |
|        /                       /   \                       \        |
|       /                       /     \                       \       |
|      /                       /       \                       \      |
|     /                       /         \                       \     |
|    o                       o (du).(dv) o                       o    |
|    |                       |   -->--   |                       |    |
|    |                       |   \   /   |                       |    |
|    |              dv .(du) |    \ /    | du .(dv)              |    |
|    |      u      <---------------@--------------->      v      |    |
|    |                       |     |     |                       |    |
|    |                       |     |     |                       |    |
|    |                       |     |     |                       |    |
|    o                       o     |     o                       o    |
|     \                       \    |    /                       /     |
|      \                       \   |   /                       /      |
|       \                       \  |  /                       /       |
|        \                       \ | /                       /        |
|         \                       \|/                       /         |
|          \                       |                       /          |
|           \                     /|\                     /           |
|            o-------------------o | o-------------------o            |
|                                  |                                  |
|                               du . dv                               |
|                                  |                                  |
|                                  V                                  |
|                                                                     |
o---------------------------------------------------------------------o
Figure 37-c.  Tacit Extension of J (Compact)

o-----------------------------------------------------------o
|                                                           |
|                         (du).(dv)                         |
|                          --->---                          |
|                          \     /                          |
|                           \   /                           |
|                            \ /                            |
|                           u @ v                           |
|                            /|\                            |
|                           / | \                           |
|                          /  |  \                          |
|                         /   |   \                         |
|                        /    |    \                        |
|               (du) dv /     |     \ du (dv)               |
|                      /      |      \                      |
|                     /       |       \                     |
|                    /        |        \                    |
|                   /         |         \                   |
|                  v          |          v                  |
|                 @           |           @                 |
|               u (v)         |         (u) v               |
|                             |                             |
|                             |                             |
|                             |                             |
|                             |                             |
|                          du . dv                          |
|                             |                             |
|                             |                             |
|                             |                             |
|                             |                             |
|                             v                             |
|                             @                             |
|                                                           |
|                          (u).(v)                          |
|                                                           |
o-----------------------------------------------------------o
Figure 37-d.  Tacit Extension of J (Digraph)

Jon Awbrey

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