Re: Information = Comprehension x Extension -- Discussion

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ICE.  Discussion Note 25

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There are two distinct types of lattice-like structure
that we may quite readily associate with a universe of
discourse like X% = [b, c, r].  Using the language that
we introduced last time, we may refer to the first type
as a "lattice of coordinates" (LOC) and the second type
as a "lattice of predicates" (LOP).  These two lattices
are very different from each other, in both quantity and
degree, that is, in the measure of their cardinalities or
dimensionalities and in their formal logical significance.
Our next order of business, then, is to distinguish these
two types of lattices by means of sufficiently clear marks
that we'll never be liable to get them mixed up in practice.

The first thing that distinguishes a LOC from a LOP is that
there are many LOC's but just one LOP for each non-trivial
universe of discourse X% = [x_1, ..., x_n], where n >= 1.

Figure 3 depicts one possible LOC for the universe X% = [b, c, r].
It is constructed by picking one cell, in this case, the one that
has the coordinates (b)(c)(r) of things ~Blue & ~Colored & ~Red,
and then ordering the remaining cells according to the minimal
path that it takes to get from the chosen root cell to the
cell in question.

o-----------------------------------------------------------o
|                                                           |
|                          b  c  r                          |
|                             o                             |
|                            /|\                            |
|                           / | \                           |
|                          /  |  \                          |
|                         /   |   \                         |
|                        /    |    \                        |
|                       /     |     \                       |
|                      /      |      \                      |
|            b  c (r) o    b (c) r    o (b) c  r            |
|                     |\     / \     /|                     |
|                     | \   /   \   / |                     |
|                     |  \ /     \ /  |                     |
|                     |   \       /   |                     |
|                     |  / \     / \  |                     |
|                     | /   \   /   \ |                     |
|                     |/     \ /     \|                     |
|            b (c)(r) o   (b) c (r)   o (b)(c) r            |
|                      \      |      /                      |
|                       \     |     /                       |
|                        \    |    /                        |
|                         \   |   /                         |
|                          \  |  /                          |
|                           \ | /                           |
|                            \|/                            |
|                             o                             |
|                         (b)(c)(r)                         |
|                                                           |
o-----------------------------------------------------------o
Figure 3.  Rank Three Lattice of Cell Coordinates

Jon Awbrey

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http://www.cs.bsu.edu/homepages/mighty/history.html
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