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ROL. Note 3 (02)
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JA = Jon Awbrey
JS = John Sowa (04)
Cf: http://ontolog.cim3.net/forum/ontolog-forum/2007-08/msg00190.html
Cf: http://ontolog.cim3.net/forum/ontolog-forum/2007-08/msg00194.html
CC: Arisbe List, Inquiry List, Ontolog Forum, SemWeb List (05)
John, (06)
Continuing from where I left off,
with current comments unindented. (07)
JA: Let's look again at the concept of "inter-operability"
that you outlined last time. I'm a little hesitant
about calling it that just yet, and would prefer
to call it "inter-translatability" until I know
more about it. (08)
JS: Consider the following three notations: (09)
JS: 1. The first-order subset of Peirce's Algebra of Logic of 1885. (010)
JS: 2. The first-order subset of Frege's Begriffsschrift of 1879. (011)
JS: 3. Any of the three concrete notations in Annex A, B, or C of
the Final Draft International Standard of Common Logic of 2007. (012)
JA: I am told by people who apparently understand these things that
having not just 2 but 3 distinct languages on the Rosetta Stone
was crucial to finding the key, but let me first consider a far
simpler example of the ilk that I know from practical endeavors. (013)
JA: Something that I spent a goodly portion of the (19)80's doing,
and in such primitive computing circumstances that I had to write
all of the necessary utilities myself, was to translate an articula
x_1 of one language, medium, or type L_1 (written x_1 : L_1) into
an articula x_2 of another language, medium, or type L_2 (written
x_2 : L_2), perform a computation on x_2 : L_2 that would yield
an articula y_2 : L_2, then translate y_2 : L_2 back into the
corresponding y_1 : L_1. (014)
JA: Here is a diagram of the process: (015)
x_1 : L_1 ----------> x_2 : L_2
| |
|
| |
|
| |
V V
y_1 : L_1 <---------- y_2 : L_2 (016)
JA: The more solid arrows indicate the actual computations.
The more dashing arrow, the road not taken, as it were,
suggests the virtual computation, in effect exchanging
x_1 : L_1 for y_1 : L_1 or transforming x_1 : L_1 into
y_1 : L_1. (017)
Why do we do this? Why such a roundabout calculation?
Well, it's important to note that the reason for this
detour is not just some equivalence between languages
but based on the existence of complex factors, namely,
that L_1 and L_2 are analogous in an abstract logical
or mathematical sense while departing from each other
in a pertinent class of concrete pragmatic properties. (018)
The computational archetype of this particular gambit
is probably the trick known as "logarithms", where we
convert what was once considered a "hard" computation,
namely, multiplication, into a relatively "easy" task,
namely, addition. The trick works because there is a
homomorphism log : (X,*) -> (Y,+) on suitably bounded
subsets X and Y of the real numbers R that enables us
to start with a problem presented in the form a*b and
to re-present it in the form log(a) + log(b), and all
the computations involved in this long way round used
to be in former times appreciably easier to carry out
than the corresponding multiplication task. (019)
As a general observation, then, the reason that we keep
a diversity of languages around is not because they are
indifferent in all of their characters but because they
provide us with different advantages at different times. (020)
Breaking here ... (021)
Jon Awbrey (022)
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