[Inquiry] Re: Differential Logic B -- Discussion

[Jon Awbrey]

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DLOG B.  Discussion Note 4

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HT = Hugh Trenchard

Re: http://stderr.org/pipermail/inquiry/2004-February/001223.html

HT: Ah -- looks like my initial interpretation was almost
    precisely backwards!  Things are definitely clearer now.
    So the origin, or 000 is, so to speak, the remainder of
    the universe of discourse.  (For example, in expropriation
    law, an area of law I was working in for a while -- I am
    employed as a paralegal -- one speaks of a portion of land
    expropriated from an owner as the land "taken" and the area
    still owned by the land owner as the "remainder").

"Adverse possession" !!!  Slowly I turned ...

Yes, all of those "cells", the undivided regions, are really equal citizens
in the universe of discourse, and it's only the particular form of planar
projection that gives them such different shapes in the venn diagram.
Another popular way to represent three logical dimensions would be
in the form of a 3-cube, like so:

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

In this variety of picture, the cells are the nodes of the cube
and the propositions are all the different ways of coloring the
nodes of the cube in just 2 colors, corresponding to "indicated"
and "undicated", or true and false, under the given proposition.
In general, for an n-cube, there are 2^n nodes for the singular
elements that the logician calls "interpretations", and 2^(2^n)
possible 2-colorings that represent the set of propositions.

HT: And the common elements of the three overlapping circles
    is the point where all three are true, and the "petals"
    not overlapping with anything are the points where one
    of three is true.  Have I got it right now?

Yes, that's the ticket.

HT: That still leaves my other question though -- if there is
    a common region where all three are true, then aren't there
    also three regions where two cells overlap -- which I guess
    now means "both are true" (rather than one of two is true,
    which was my previous interpretation)?

Here we come to the question that is sometimes described
as the distinction between contemplation and conviction,
or the difference between considering a proposition and
asserting it.  This is also bound up with the difference
between mention and use.  Historically speaking, a whole
lot more noise than signal has been emitted on this score.

Consider what we are doing when we draw a venn diagram like this:

o-----------------------------------------------------------o
| U                                                         |
|                                                           |
|                      o-------------o                      |
|                     /```````````````\                     |
|                    /`````````````````\                    |
|                   /```````````````````\                   |
|                  /`````````````````````\                  |
|                 /```````````````````````\                 |
|                o````````````X````````````o                |
|                |`````````````````````````|                |
|                |`````````````````````````|                |
|                |`````````x (y)(z)````````|                |
|                |`````````````````````````|                |
|                |`````````````````````````|                |
|             o--o----------o```o----------o--o             |
|            /````\          \`/          /````\            |
|           /``````\ x  y (z) o  x (y) z /``````\           |
|          /````````\        / \        /````````\          |
|         /``````````\      /   \      /``````````\         |
|        /````````````\    /x y z\    /````````````\        |
|       o``````````````o--o-------o--o``````````````o       |
|       |`````````````````|       |`````````````````|       |
|       |`````````````````|       |`````````````````|       |
|       |````````Y````````|       |```````Z`````````|       |
|       |`````````````````|(x)y z |`````````````````|       |
|       |`````````````````|       |`````````````````|       |
|       o````(x) y (z)````o       o```(x)(y) z``````o       |
|        \`````````````````\     /`````````````````/        |
|         \`````````````````\   /`````````````````/         |
|          \`````````````````\ /`````````````````/          |
|           \`````````````````o`````````````````/           |
|            \```````````````/ \```````````````/            |
|             o-------------o   o-------------o             |
|                                                           |
|                         (x)(y)(z)                         |
|                                                           |
o-----------------------------------------------------------o
Figure 11.  Venn Diagram for ((x),(y),(z))

(As always, these are best viewed in equal-width character formats.)

One thing that we are doing is according additional attention to
the gravely accented areas that we have shaded in like so [```].
That is done in order to represent one particular proposition
out of the 2^(2^3) = 2^8 = 256 possible propositions that are
available to us in this 3-dimensional universe of discourse.
Our practical reasons for representing such a proposition
may be various.  We may just be thinking about it, we may
believe that it's true, we may wish that it were true,
we may wish to convince somebody else that it's true
whether we believe it ourselves or not, and so on.
I'll bet you can think of a slew of other cases.

The mathematical way of handling the issue is in terms
of concepts that are known as "fibers" and "quotients".

But I'll need more coffee before I can even contemplate that ...

Jon Awbrey

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