Re: Information = Comprehension x Extension -- Discussion

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ICE.  Discussion Note 26

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Re: ICE Discussion 25.  http://suo.ieee.org/ontology/msg05310.html
In: ICE Discussion.     http://suo.ieee.org/ontology/thrd1.html#05274

The fact that we get a different "lattice of coordinates" (LOC)
for each base point that we choose, together with the fact that
this choice is arbitrary and itself ultimately dependent on the
particular basis that we have chosen, or been given, as a frame
of reference for depicting the universe buried by the discourse,
makes us regard a LOC as a highly perspectival concept, that is
to say, a non-invariant construct.  On the other hand, the fact
that every LOC has the same structure when regarded at the next
level of abstraction, that is, all LOC's are created isomorphic --
this is a measure of invariant structure that adds to its worth.

For the 3-dimensional universe X% = [x_1, x_2, x_3] = [b, c, r]
of Peirce's example, each LOC has 2^3 = 8 elements, being equal
to the number of cells in the coordinate space X = <|b, c, r|>.

Considering a "predicate" to be a boolean-valued function on X,
that is, having the type p : X -> !B!, there must be 2^8 = 256
predicates in our present 3-dimensional universe X% = [b, c, r].

Consequently, the "lattice of predicates" (LOP) for the present example
has 256 elements, exceeding the bounds of my ASCII artistry to draw for
you here in this present setting.  So let me just indicate the shape of
the lattice by marking the top, the bottom, and a few points in between.

o-----------------------------------------------------------o
|                                                           |
|                          b  c  r                          |
|                          b  c (r)                         |
|                          b (c) r                          |
|                          b (c)(r)                         |
|                         (b) c  r                          |
|                         (b) c (r)                         |
|                         (b)(c) r                          |
|                         (b)(c)(r)                         |
|                                                           |
|                             o                             |
|                            / \                            |
|                           /   \                           |
|                          /     \                          |
|                         /       \                         |
|                        /         \                        |
|          b  c (r)     /           \      b  c  r          |
|          b (c)(r)    /             \     b (c) r          |
|                     o      ...      o                     |
|         (b) c  r     \             /    (b) c (r)         |
|         (b)(c) r      \           /     (b)(c)(r)         |
|                        \         /                        |
|                         \       /                         |
|                          \     /                          |
|                           \   /                           |
|                            \ /                            |
|                             o                             |
|                                                           |
|                           empty                           |
|                            set                            |
|                                                           |
o-----------------------------------------------------------o
Figure 4.  Lattice of Predicates for the Universe [b, c, r]

Figure 4 depicts the LOP for the 3-dimensional universe X% = [b, c, r]
by way of a sample of four of its elements.  The top of the lattice is
the predicate typically notated as "1", denoting the constant function
1 : X -> !B!, and this predicate is taken to "indicate" all 8 elements
of the coordinate space X, whose coordinate conjuncts are listed above
the top node.  The bottom of the lattice is the predicate notated "0",
denoting the constant function 0 : X -> !B! that is taken to indicate
none of the elements of the coordinate space X, and so its content
is succinctly indicated as an empty set of coordinate conjuncts.
Toward the middle of the lattice I have shown the subsets of
coordinates that correspond to a couple of other predicates.
On the left there are listed the four coordinate conjuncts
that correspond to the predicate that is true when b and r
are different in their logical values and false otherwise.
On the right there are given the four coordinate conjuncts
that correspond to the predicate that is true when b and r
are equivalent in their logical values and false otherwise.
These two predicates are complements of each other in the
lattice ordering, making their GLB the bottom element and
their LUB the top element, as evidenced by the fact that
the corresponding sets of coordinates are complementary.
An ellipsis will have to do for the other 252 elements.

Jon Awbrey

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