From jjllambias@hotmail.com Fri Mar 3 14:36:34 2000
X-Digest-Num: 383
Message-ID: <44114.383.2185.959273826@eGroups.com>
Date: Fri, 03 Mar 2000 14:36:34 PST
From: "Jorge Llambias"
Subject: Re: final clubs
> > >Suppose we have the following situation:
> > >Club A imposes no conditions on its members
> > >Club B requires members to swear a loyalty oath,
> > > and to swear no other oaths
> > >Club C requires members to swear a loyalty oath,
> > > and to swear no other oaths
> > >
> > How can you tell in that example that it is B and C that
> > are final, and not that A is the only final club?
>
>B and C are final because they fit the recursive definition
>of 'final club': membership in one member of the set precludes
>membership in any other member of that set. The set of
>final clubs is a mutually exclusive set.
Only if you start by assuming that B and C are final!
I agree you get a consistent answer.
But start by assuming that A is the only final one.
Then neither B nor C are final (they don't fit the
definition) while A does.
The two answers are possible.
>A is not final because A does not preclude membership in
>any other club, and therefore does not meet the definition.
The definition calls for precluding membership in other
*final* clubs. There are no other final clubs, so A trivially
meets the definition.
>(Also, the definition of 'final club' implies that there must
>be more than one final club so that there is something to
>preclude!)
Well, we can have the debate on whether universals
have existential import all over again! :)
If the definition requires at least two final clubs,
then I agree that not all clubs need be final.
Then I must modify my definition. The situation
must be such that all clubs that preclude membership
must preclude membership in every club that precludes
membership.
A final club is then defined as any club with some
preclusion.
The proof is similar to the case without existential import.
Obviously clubs with no preclusions can only be final if
they are the only final club.
A counterexample in this case would require a situation
in which there is a non-final club with some preclusion rule.
co'o mi'e xorxes
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