[Inquiry] Re: Differential Logic -- Series B

[Jon Awbrey]

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DLOG.  Note B20

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Let's now look at the last type of clause that we find in my
transcription of the Jets and Sharks data base, for instance,
as exemplified by the following couple of lobal expressions:

   ( jets ,
    ( art    ),( al   ),( sam  ),( clyde ),( mike  ),
    ( jim    ),( greg ),( john ),( doug  ),( lance ),
    ( george ),( pete ),( fred ),( gene  ),( ralph ))

   ( sharks ,
    ( phil ),( ike  ),( nick ),( don ),( ned  ),( karl ),
    ( ken  ),( earl ),( rick ),( ol  ),( neal ),( dave ))

Each of these clauses exhibits a generic pattern whose logical properties
may be studied well enough in the form of the following schematic example.

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

The proposition (u, v, w) evaluates to true
if and only if just one of u, v, w is false.
In the same way, the proposition (x,(y),(z))
evaluates to true if and only if exactly one
of x, (y), (z) is false.  Taking it by cases,
let us first suppose that x is true.  Then it
has to be that just one of (y) or (z) is false,
which is tantamount to the proposition ((y),(z)),
which is equivalent to the proposition ( y , z ).
On the other hand, let us suppose that x is the
false one.  Then both (y) and (z) must be true,
which is to say that y is false and z is false.

What we have just said here is that the region
where x is true is partitioned into the regions
where y and z are true, respectively, while the
region where x is false has both y and z false.
In other words, we have a "pie-chart" structure,
where the genus X is divided into the disjoint
and X-haustive couple of species Y and Z.

The same analysis applies to the generic form
(x, (x_1), ..., (x_k)), specifying a pie-chart
with a genus X and the k species X_1, ..., X_k.

Jon Awbrey

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