ONT Re: Extension x Comprehension = Information
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Before I go on with Peirce's story of information,
I want to stop for a while, at least long enough
to redraw a favorite old picture of mine, that
illustrates what all of this has to do with
artificial and natural kinds, as they have
been classically and humorously typified
by the example that is commonly known
as the case of the "plucked chicken".
The following Figure is largely self-explanatory.
| Creature
| o
| / \
| / \
| / \
| / \
| / \
| / \
| Apterous o o Biped
| |\ /|
| | \ / |
| | \ / |
| | \ / |
| | \ / |
| | \ / |
| | o G |
| | / \ |
| | / \ |
| | / \ |
| | / \ |
| | / \ |
| |/ \|
| Human o o Plucked Chicken
|
| A = Apterous = featherless animal
| B = Bipedal = two-legged being
| C = Critter = creature, creation
| G = GLB(A, B) = A |^| B
| H = Human Being
| P = Plucked Chicken
|
| Figure 1. On Being Human
The way the joke goes, the straight man "defines" a human being H
as an "apterous biped" A·B, a two-legged critter without feathers,
and then the wiseguy hits him over the head with a plucked chicken,
and by dint of this koan, he achieves enlightenment about the marks
that distinguish kindness of the artless kind from the crasser kinds
of artificial kindness. Leastwise, anyways -- so I've heard it told.
Our focus at present is on the extra measure of constraint,
in other words, the information, that comes between Set(X),
the full lattice of all possible subsets of our universe X,
and Nat(X), the more constrained, determined, or informed
lattice of "natural kinds" that we commonly acknowledge in
our more practical outlooks on this universe of discourse.
The next two Figures present different ways of viewing the situation.
Think of the initial set-up as being cast in a lattice of arbitrary sets.
Within that setting, the "greatest lower bound" (glb) of the extensions
of A and B is their set-theoretic intersection, G = glb(A, B) = A |^| B.
This G covers the desired class H but also admits the risible category P.
Suppose that we are clued into the fact that not all sets in Set(X)
are admissible, allowable, material, natural, pertinent, or relevant
to the aims of the discussion in view, and that only some mysterious
'je ne sais quoi' subset of "natural kinds", Nat(X) c Set(X), is at
stake, a limitation that, whatever else it does, excludes the set P
and all of that ilk from beneath glb(A, B). Though it is difficult
to say exactly how we are supposed to apply this global information,
we know it in the sense of being able to detect its local effects,
for instance, giving us the more "natural" lattice structures that
are shown on the right sides of Figures 2 and 3. Relative to these
"natural orders", we can observe that H = glb(A, B), more precisely,
the result of the lattice operation associated with the conjunction,
glb, or intersection of A and B gives us just the lattice element H.
Thus in this more natural setting the proposed definition works okay.
| Set >>>--->>> Nat
|
| C C
| o o
| / \ / \
| / \ / \
| / \ / \
| / \ / \
| / \ / \
| / \ / \
| A o o B A o o B
| |\ /| | /
| | \ / | | /
| | \ / | | /
| | \ / | | /
| | \ / | | /
| | \ / | | /
| | o G | | /
| | / \ | | /
| | / \ | | /
| | / \ | | /
| | / \ | | /
| | / \ | | /
| |/ \| |/
| H o o P H o
|
| Figure 2. Arbitrary Kinds Versus Natural Kinds
An alternative way to look at the transformation in our views as we
pass from the arbitrary lattice Set(X) to the natural lattice Nat(X)
is presented in Figure 3, where the ditto marks (") suggest that the
nodes for G and H are logically identified with each other. In this
picture, the measure of the interval that previously existed between
G and H, now shrunk to nil, affords a rough indication of the local
quantity of information that went into forming the natural result.
| Set >>>--->>> Nat
|
| C C
| o o
| / \ / \
| / \ / \
| / \ / \
| / \ / \
| / \ / \
| / \ / \
| A o o B A o o B
| |\ /| \ /
| | \ / | \ /
| | \ / | \ /
| | \ / | \ /
| | \ / | \ /
| | \ / | \ /
| | o G | G o
| | / \ | "
| | / \ | "
| | / \ | "
| | / \ | "
| | / \ | "
| |/ \| "
| H o o P H o
|
| Figure 3. Arbitrary Kinds Versus Natural Kinds
Jon Awbrey
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