SUO: World Carvings & Ostenciled Glyphs -- *Date 06 Feb 2002
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CM = Chris Menzel
CM: No, the point, as Randall emphasized, is that,
when it comes to a choice of languages for the
SUO, there are NOT different frames of reference.
There's just the language of first-order (or perhaps,
as conditions warrant, a somewhat stronger) logic in
one more-or-less equivalent guise or another -- KIF,
CGs, take your pick. These do not "carve up the world"
in significantly different ways. The choice between them
is entirely pragmatic. To suggest, as you do, that something
conceptually profound hangs upon it is the sort of red herring
that ends up wasting everyone's time.
<post-peripathetic preamble deleted>
Okay, I begin to see that we are talking about at minimum
two different worlds here, and though I have no desire to
carve up any one of them, I do frequently contemplate the
templates that we project on them, highlighting some bits
and obscuring others.
I need this quasi-commutative diagram again:
| p
| X o------>o %B%
| \ ^
| r = <x_1, ..., x_k> \ / q
| v /
| o
| %B%^k
X is whatever it is that's depicted by the area
interior to the big rectangle of a venn diagram.
I use the letter "X" to remind me of how little
else I really know about it. The proposition p
is a function p : X -> %B%. The coordinate map
or coordinate representation r = <x_1, ..., x_k>
is a function r : X -> %B%^k whose k components
are the functions x_j : X -> %B% for j = 1 to k.
The derived proposition q is a boolean function
q : %B%^k -> %B% that is uniquely determined by
the previous data of the diagram, and it is the
sort of mathematical object for whom the proper
pons asinorum is a gate of the electrical stile.
I think that you think I'm thinking about the world of X
when I say the things I'm saying about diverse frames of
reference, and it's true I do refer to that diversity on
many occasions, but there is, aside from that, a diverse
pluralism of ways in which the objects q : %B%^k -> %B%
can be described. This is the realm of variation that
I am talking about when I criticize the "pentaglyphic"
syntax that is based on the set of pseudo-primitives
{~, &, v, =>, <=>} and all of the variants thereof.
There are, as a matter of adamant mathematical fact,
vast differences in the representational felicities
of various notational systems for boolean functions,
and the variations of their expressive facility are
profound in both conceptual and computational terms.
Now, maybe when you say things like "The choice between them
is entirely pragmatic" you mean to hint that you do not care
all that much about this range of differences, but these are
the brands of pragmatic choices that make all the difference
in the world when it comes to efficient computation and thus
to actually succeeding at all in a computational environment.
I think that I have achieved a practical truism here.
I personally associate it with questions about the "presentation"
of mathematical structures in terms of "generators and relations",
but I don't know if that is a familiar enough resource hereabouts.
If I could put it in those terms, I would say that
the pentalphabet {~, &, v, =>, <=>} does not make
a good set of "generators" because there are too
many "relations" to be found among its elements.
But I know that you know what it means to present an axiom system
in the form of independent axioms. Even though it can be hard to
find such a presentation and tough to prove it when you've got it,
most people sense that it's more than just elegance, but a matter
of cognitive economy or conceptual efficiency to go for that form.
And that style of elect élan is just what the pentaglyphic
pseudo-basis {~, &, v, =>, <=>} fails to achieve for logic,
because the independence of these operators is compromised
by the excessive complicity of their insider entanglements.
At the outset this factor affects only that part of logic that
has to do with indicating objects of the type q : %B%^k -> %B%.
But whatever affects the complexion of that face of indication
also redounds on the complex of descriptions like p : X -> %B%.
<gratuitous parting shot expleted>
If you really want to waste a huge lot of time,
just try not taking these factors into account.
Or, you could fritter away your days trying to
wrap conformance clauses around rank vaporware.
Jon Awbrey
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