[EM] Poor Richard's Education

[Joe Weinstein]

Recently (5/13/02) Don Davison wrote:

"For [Richard Moore's] information and education, the words `Condorcet 
candidate' or `Condorcet winner' are code words which mean or imply that 
Condorcet is the standard for all single-seat election methods including 
Condorcet itself.  Whenever anyone uses the code words `Condorcet Winner' 
they are attempting to establish Condorcet to this high position as the 
standard of all single-seat election methods including Condorcet itself.  I 
do not accept Condorcet as the standard, but of more importance I say it is 
dishonest to regard any method to be a standard of all methods including 
itself.  This dishonesty will create junk mathematics and I said as much in 
my reply to Alex's use of the code words."

Actually, from my perspectives -  on word usage, rhetoric, logic and 
mathematics, Don's argument here is dishonest and junk.

     The argument first tries to exploit and magnify a readily dissolvable 
possible confusion among at least three quite different sorts of things - a 
confusion possible only because they share the common label 'Condorcet':  
Condorcet Winner (a possible feature of an election), Condorcet Criterion (a 
possible feature of an election method), and Condorcet Methods (certain 
methods explicitly designed to have the feature).

     Don's main claim seems to be that there is a 'Condorcet' conspiracy to 
institute a Condorcet 'standard'.  Even if you were determined to be a bit 
confused, his claim still would not follow.

     Concerning ANY election, by ANY method in which pairwise preference may 
be expressed (or in a well-defined way inferred), we may ask - as a simple 
matter of interest and curiosity about facts - whether there is or is not a 
Condorcet Winner, and if yes then who.

     Our interest and curiosity need have NOTHING at all to do with whether 
we like or dislike or praise or blame the election method, or use it as a 
'standard', or have another 'standard', or desire some kind of magical 
'Condorcet' standard, or in fact have or want a 'standard' at all.

     So why should I or Richard or anyone assume (or need 'education') that 
there's a hidden 'code' or hidden agenda?  If you're ready to assume that, 
then you're ready, like the 1950's conspiracy theorists, to spend life 
worrying about the Communist (Condorcist?) potentially hidden under each and 
every mattress.

     Or, if you are a Communist yourself, you are ready to worry endlessly 
about the lurking capitalist agents who supposedly use the code word 
'dollar' to sneak mercenary and capitalist propaganda and standards into as 
many conversations as possible.

Don deplores taking  "any method to be a standard of all methods including 
itself"  Allegedly this procedure is a "dishonesty" which "will create junk 
mathematics".  Not at all.

Non-junk math often takes a given object as standard for many objects, 
necessarily including itself.  For instance, given a coordinatized plane, 
the origin serves as a standard of location for all points, including 
itself.

Don of course speaks here not of standard 'objects' but of  standard 
'methods'.  All right, let's give another example (much more long-winded), 
this time of a method which serves as a standard for many methods, including 
itself.  The example comes in fact from the mathematics developed to study 
logic - and build computers.

     An important kind of mathematical structure is called a Boolean algebra 
(in brief, BA).  The methods we consider are those which can be used to 
define and create BAs.

     [Extended digression for the technically minded but not technically 
schooled: a BA is defined by a set of elements, and by three operations 
(often signified by symbols v, & and -) which are defined on these elements. 
  v and & are two-argument operations (like ordinary addition), and - has 
one argument (like the - used to signify a negative number such as -17.)  
These respective operations must satisfy relationships found among usual set 
or logical operations known familiarly as 'or', 'and' and 'not'.  From the 
operations one may also derive a comparison relation, and in terms of this 
comparison there are two special elements: a least or 'zero' element and a 
greatest or 'universal' element.]

     A simplest and most evident general way to create a BA is by method 
FOS, i.e. as a so-called 'field of sets':  take a given set S of objects, 
take certain subsets (by design or at whim) of S, form all repeated unions 
and intersections and complements (in S) of these subsets to get yet more 
subsets, and then define the BA as follows:  elements = all defined subsets, 
v = operation of union of two subsets, & = operation of intersection of two 
subsets, - = operation of complement in (subtraction from) the universal 
element S.

     So FOS is an obviously workable method for creating many BAs.   But a 
truly ideal 'standard' method would enable you, using that method, to create 
an isomorphic copy of every possible BA.

     The greatest triumph of early modern study of BAs was Stone's 
representation theorem, which showed that FOS is indeed such a standard:  
EVERY BA is isomorphic to one which is created by FOS.  In fact, knowing the 
elements and the workings of the operations of a given BA, Stone's proof 
shows how to use FOS to create another BA isomorphic to it.  This is true in 
every case, including the case that the original BA itself was earlier 
created by FOS.

     So FOS is indeed a standard for all BA-creating methods, including 
itself.

Don goes on to argue:

"It is not proper to use any of the considered methods as a standard ... It 
would be as if in the American Beauty Contest Miss California (or Miss 
Condorcet) was regarded to be the standard by which all the contestants are 
to be judged including Miss California - and the winner is Miss California - 
of course, who else - more deceit of the Condorcet people."

     In other words, you should not be able to choose a method as standard 
and then enter it in a contest whose goal is to best fit the standard's own 
salient features.  This argument appeals to our desire for 'fairness' to the 
other contestants;  but the argument is itself illogical and unfair.  If 
your standard was chosen as the most perfect available fit to desired 
behavior or performance of some kind, then any fully open contest based just 
on such behavior or performance SHOULD logically and fairly choose your 
standard, not reject it.

     For instance, suppose that you are King Midas and that - thanks to its 
desirable (to you) material properties - you have chosen 24K gold as the 
'standard' substance.  Then you run a fully open contest among all 
materials, with the prize to the one which best fulfills these properties.  
The winner of your contest SHOULD be 24K gold.

     It's really quite OK to set up a standard and then use it to win 
contests.  What IS improper is different:  namely, to claim that your 
standard is universal, after nonetheless having derived the standard based 
on criteria which fail to be relevant for all uses.

     In particular, for election methods we lack sufficiently limiting 
criteria which are relevant for all purposes.  That's the real reason why no 
one method serves as the unqualified universal standard.

By the way, I myself am not too concerned about Condorcet Winners, nor 
focused on the Condorcet Criterion, nor have I much interest in comparisons 
and intracacies of various Condorcet Methods; and I certainly don't perceive 
any such method as somehow a 'standard'.

But I'm tired of being - along with innocent others - subjected to illogical 
and uninformed (when not also deliberately misleading) rants about alleged 
Condorcet codes and conspiracies; about alleged standards and 
pseudostandards; and about allegedly junk mathematics.

Joe Weinstein
Long Beach CA USA

_________________________________________________________________
Get your FREE download of MSN Explorer at http://explorer.msn.com/intl.asp.

----
For more information about this list (subscribe, unsubscribe, FAQ, etc), 
please see http://www.eskimo.com/~robla/em