Article 15409 of sci.logic:
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From: klein@math.wisc.edu (Moses Klein)
Newsgroups: sci.logic
Subject: Re: Democracy, a logical contradiction?
Date: 18 Oct 1995 21:47:29 GMT
Organization: University of Wisconsin, Madison
Lines: 34
Message-ID: <463slh$4bu@news.doit.wisc.edu>
References: <462rr8$54d@rc1.vub.ac.be>
NNTP-Posting-Host: conley.math.wisc.edu

In article <462rr8$54d@rc1.vub.ac.be>,
Antoon Pardon <apardon@rc1.vub.ac.be> wrote:
>I have heard that someone has started from five conditions
>which were generally accepted as being desirable/needed
>for a democarcy and has subsequently proven that those
>conditions result in a contradiction. Can someone here
>provide me with a pointer to such an argument or give an
>outline here?

Arrow's Theorem. A voting system is defined as a set of subsets (called 
winning coalitions) of a finite set of objects (called voters) such that:
1) if S is a winning coalition and S is a subset of T, then T is a 
winning coalition,
2) for any set S of voters, exactly one of S and the complement of S is a 
winning coalition,
3) no singleton is a winning coalition, and
4) if the set of voters preferring option A to option B is a winning 
coalition, and the set of voters preferring option B to option C is a 
winning coalition, then the set of voters preferring option A to option C 
is a winning coalition.

(This is essentially what's involved; offhand I don't recall precisely 
how he formulated it.) Astute and knowledgeable logicians will notice 
that this is essentially the definition of a nonprincipal ultrafilter on 
a finite set, and therefore cannot exist.

The idea is that the voting system should give a way of constructing a 
collective ranking of preferences as a function of the set of individual 
voters' preferences. Preferences should obviously be ranked in a linear 
order. The last postulate, which fails in ordinary voting systems, is 
intended to guarantee that the collective preferences are linearly ranked 
if every individual's preferences are.

Moses Klein (klein@math.wisc.edu)


