[Inquiry] Re: Differential Logic -- Series B

[Jon Awbrey]

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DLOG.  Note B19

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Let's collect the various ways of representing the structure
of a universe of discourse that is described by the following
cactus expressions, verbalized as "just 1 of x, y, z is true".

o-------------------------------------------------o
|                                                 |
|                     x  y  z                     |
|                     o  o  o                     |
|                     |  |  |                     |
|                     o--o--o                     |
|                      \   /                      |
|                       \ /                       |
|                        @                        |
|                                                 |
o-------------------------------------------------o
|                  ((x),(y),(z))                  |
o-------------------------------------------------o

Table 12 shows the truth table for the existential
interpretation of the cactus formula ((x),(y),(z)).

Table 12.  Existential Interpretation of ((x),(y),(z))
o-----------o-----------o-----------o-------------o
|     x     |     y     |     z     |  (x, y, z)  |
o-----------o-----------o-----------o-------------o
|                                   |             |
|     0           0           0     |      0      |
|                                   |             |
|     0           0           1     |      1      |
|                                   |             |
|     0           1           0     |      1      |
|                                   |             |
|     0           1           1     |      0      |
|                                   |             |
|     1           0           0     |      1      |
|                                   |             |
|     1           0           1     |      0      |
|                                   |             |
|     1           1           0     |      0      |
|                                   |             |
|     1           1           1     |      0      |
|                                   |             |
o-----------------------------------o-------------o

Figure 13 shows the same data as a 2-colored 3-cube,
coloring a node with a hollow dot (o) for "false"
or a star (*) for "true".

o-------------------------------------------------o
|                                                 |
|                     x  y  z                     |
|                        o                        |
|                       /|\                       |
|                      / | \                      |
|                     /  |  \                     |
|                    /   |   \                    |
|                   /    |    \                   |
|                  /     |     \                  |
|                 /   x (y) z   \                 |
|       x  y (z) o       o       o (x) y  z       |
|                |\     / \     /|                |
|                | \   /   \   / |                |
|                |  \ /     \ /  |                |
|                |   \       /   |                |
|                |  / \     / \  |                |
|                | /   \   /   \ |                |
|                |/     \ /     \|                |
|       x (y)(z) *       *       * (x)(y) z       |
|                 \  (x) y (z)  /                 |
|                  \     |     /                  |
|                   \    |    /                   |
|                    \   |   /                    |
|                     \  |  /                     |
|                      \ | /                      |
|                       \|/                       |
|                        o                        |
|                    (x)(y)(z)                    |
|                                                 |
o-------------------------------------------------o

Figure 14 repeats the venn diagram that we've already seen.

o-----------------------------------------------------------o
| U                                                         |
|                                                           |
|                      o-------------o                      |
|                     /```````````````\                     |
|                    /`````````````````\                    |
|                   /```````````````````\                   |
|                  /`````````````````````\                  |
|                 /```````````````````````\                 |
|                o`````````````````````````o                |
|                |``````````` X ```````````|                |
|                |`````````````````````````|                |
|                |`````````````````````````|                |
|                |`````````````````````````|                |
|                |`````````````````````````|                |
|             o--o----------o```o----------o--o             |
|            /````\          \`/          /````\            |
|           /``````\          o          /``````\           |
|          /````````\        / \        /````````\          |
|         /``````````\      /   \      /``````````\         |
|        /````````````\    /     \    /````````````\        |
|       o``````````````o--o-------o--o``````````````o       |
|       |`````````````````|       |`````````````````|       |
|       |`````````````````|       |`````````````````|       |
|       |`````````````````|       |`````````````````|       |
|       |``````` Y ```````|       |`````` Z ````````|       |
|       |`````````````````|       |`````````````````|       |
|       o`````````````````o       o`````````````````o       |
|        \`````````````````\     /`````````````````/        |
|         \`````````````````\   /`````````````````/         |
|          \`````````````````\ /`````````````````/          |
|           \`````````````````o`````````````````/           |
|            \```````````````/ \```````````````/            |
|             o-------------o   o-------------o             |
|                                                           |
|                                                           |
o-----------------------------------------------------------o
Figure 14.  Venn Diagram for ((x),(y),(z))

Figure 15 shows an alternate form of venn diagram for the same
proposition, where we collapse to a nullity all of the regions
on which the proposition in question evaluates to false.  This
leaves a structure that partitions the universe into precisely
three parts.  In mathematics, operations that identify diverse
elements are called "quotient operations".  In this case, many
regions of the universe are being identified with the null set,
leaving only this 3-fold partition as the "quotient structure".

o-----------------------------------------------------------o
| \                                                       / |
|   \                                                   /   |
|     \                                               /     |
|       \                                           /       |
|         \                                       /         |
|           \                 X                 /           |
|             \                               /             |
|               \                           /               |
|                 \                       /                 |
|                   \                   /                   |
|                     \               /                     |
|                       \           /                       |
|                         \       /                         |
|                           \   /                           |
|                             o                             |
|                             |                             |
|                             |                             |
|                             |                             |
|                             |                             |
|              Y              |              Z              |
|                             |                             |
|                             |                             |
|                             |                             |
|                             |                             |
|                             |                             |
|                             |                             |
|                             |                             |
|                             |                             |
|                             |                             |
o-----------------------------o-----------------------------o
Figure 15.  Quotient Structure Venn Diagram for ((x),(y),(z))

Jon Awbrey

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inquiry e-lab: http://stderr.org/pipermail/inquiry/
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