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Date: Wed, 1 Mar 2000 11:43:25 -0800 (PST)
From: "James F. Carter" <jimc@MATH.UCLA.EDU>
Subject: Re: [lojban] Use and abuse of sets
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From: "James F. Carter" <jimc@MATH.UCLA.EDU>

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On Wed, 1 Mar 2000, Jorge Llambias wrote an extensive reply to a post
by lojbab...

Since we're stirring the pot about mathematical expressions (MEX), I
thought I'd toss in some comments about -gua!spi.  I also was not entirely
happy with the specialized grammar for MEX.  A few small changes from
Loglan/Lojban practice allowed an elegant solution.

First, following a common mathematical practice, numbers are defined as
equivalence classes of equal-count sets, e.g. "the number 2" means "the
class of all pairs".  The obvious extension is assumed to be feasible for
extension rings and fields (e.g. fractions and decimal numbers).  

Next, mathematical functions are named by predicate words /brivla/ of the
usual kind, defined to take such objects as arguments.  First place
occupants have the property of being in the result set; thus /lo'i/
(veridical set) should be the article used when an argument /sumti/ is made
from the expression.  "The sum of 2 and 3" is interpreted to mean "the
sum of (the class of all pairs) and (the class of all triplets)", which
comes out to be "the class of all pentads".  Getting this definition to
work right takes a little thought in the area of eliminating duplicate
members while also not mixing triplets and quads with the pentads, but
mathematicians have worked this out.

The language's usual syntax for embedded arguments is used when a complex
sub-expression occurs as an argument of the containing expression.  
- -gua!spi has a very simple syntax for embedded arguments.  The syntax of
Lojban is perfectly useable in this application.

Units of measure are defined to multiply the second place argument (a
number or other expression) by the unit.  They would normally be used in
restrictive subordinate clauses.  "I weigh 70 kilograms" in Lojban would
come out /mi tilju poi kilgrake lo'i mei ze no/.  Assuming I haven't
botched something in rusty Lojban, the unit of measure is actually
semantically valid with current (?) place structures.

I'm not presently able to say if this vision of MEX is presently 100% valid
usage in Lojban, or if the only thing you need to use it is changes to the
(non-baselined) place structures of the math functions.  But I offer it as
a simple and effective MEX grammar that's coherent with the core grammar of
Lojban.

For more on -gua!spi see http://www.math.ucla.edu/~jimc/guaspi/guaspi.txt
(it's really in TeX, sorry for not having html for you to read).  The
discussion of MEX is about 80% through.  The sample story that follows (550
words), intended to show that -gua!spi can express material from daily
life, actually turned out to include a MEX naturally but essentially.

James F. Carter        Voice 310 825 2897	FAX 310 206 6673
UCLA-Mathnet;  6115 MSA; 405 Hilgard Ave.; Los Angeles, CA, USA  90095-1555
Internet: jimc@math.ucla.edu (finger for PGP key)
UUCP:...!{ucsd,ames,ncar,gatech,purdue,rutgers,decvax,uunet}!math.ucla.edu!jimc

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