Re: Information = Comprehension x Extension -- Discussion

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ICE.  Discussion Note 30

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Each "lattice of coordinates" (LOC) for the universe W% = [c, r]
has four elements, which may be designated by the same collection
of cell coordinates:  "c r", "c (r)", "(c) r", "(c)(r)".  The only
difference between LOC's is in the arbitrary choice of one of these
four as the anchor point, base point, or bottom element.  Figure 7
shows the LOC for the universe [c, r] at the base point (c)(r).

o-----------------------------------------------------------o
|                                                           |
|                            c r                            |
|                                                           |
|                             o                             |
|                            / \                            |
|                           /   \                           |
|                          /     \                          |
|                         /       \                         |
|                        /         \                        |
|                       /           \                       |
|                      /             \                      |
|              r (c)  o               o  (r) c              |
|                      \             /                      |
|                       \           /                       |
|                        \         /                        |
|                         \       /                         |
|                          \     /                          |
|                           \   /                           |
|                            \ /                            |
|                             o                             |
|                                                           |
|                          (c) (r)                          |
|                                                           |
o-----------------------------------------------------------o
Figure 7.  Lattice of Coordinates for [c, r] at (c)(r)

But a LOC is a radically frame-dependent device and thus it
gives us no key to the commonwealth of invariant orderings
that we desire for our current realm of logical purposes.

So let's turn to another source of order.

The "lattice of predicates" (LOP) for the universe W% = [c, r]
has 2^4 = 16 elements, which puts it just beyond my ability to
draw in the current frame.  However, the quotient lattice that
we get by postulating the truth of the proposition (r (c)) has
just 16/2 = 8 elements, allowing me to draft it in full detail.

Figure 8 shows the lattice for the predicate set W^ = (W -> !B!)
modulo the equivalence relation that sets r (c) = 0 and adjusts
all of the other predicates accordingly.  In a logical frame of
mind one tends to think of quotient operations in terms of the
propositions that are equated to 1, or stipulated to be true,
so let us denominate this quotient lattice as [c, r]/(r(c)).
In the Figure, I label each point with the two propositions
in [c, r] that are equivalent mod (r(c)) in [c, r]/(r(c)).
By the way, () and (()) are alternate names for 0 and 1,
respectively, while (u, v) is the exclusive disjunction
of u and v, and ((u, v)) is the logical equivalence
of u and v.

o-----------------------------------------------------------o
|                                                           |
|                      (( ))  o  (r (c))                    |
|                            /|\                            |
|                           / | \                           |
|                          /  |  \                          |
|                         /   |   \                         |
|                        /    |    \                        |
|                       /     |     \                       |
|                      /      |      \                      |
|                     /       |       \                     |
|                    /        |        \                    |
|                   /         |         \                   |
|                  /          |          \                  |
|                 /           |           \                 |
|        (c r)   /        ((c) (r))        \    (c (r))     |
|          (r)  o             c             o  ((c, r))     |
|               |\           / \           /|               |
|               | \         /   \         / |               |
|               |  \       /     \       /  |               |
|               |   \     /       \     /   |               |
|               |    \   /         \   /    |               |
|               |     \ /           \ /     |               |
|               |      \             /      |               |
|               |     / \           / \     |               |
|               |    /   \         /   \    |               |
|               |   /     \       /     \   |               |
|               |  /       \     /       \  |               |
|               | /         \   /         \ |               |
|               |/           \ /           \|               |
|       (c, r)  o            (c)            o    r          |
|        c (r)   \         (c) (r)         /   c r          |
|                 \           |           /                 |
|                  \          |          /                  |
|                   \         |         /                   |
|                    \        |        /                    |
|                     \       |       /                     |
|                      \      |      /                      |
|                       \     |     /                       |
|                        \    |    /                        |
|                         \   |   /                         |
|                          \  |  /                          |
|                           \ | /                           |
|                            \|/                            |
|                      r (c)  o  ( )                        |
|                                                           |
o-----------------------------------------------------------o
Figure 8.  Quotient Lattice of Predicates for [c, r] mod (r (c))

It's a little bit late to be doing this sort of stuff,
so I'll see how many errors I can find in the morning.

NB AM.  I found six errors.  So glad I waited to send.

Jon Awbrey

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