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Re: Final clubs (was: Sets etc.)



doi djan, I think there is something wrong with this example:

>Membership in Club A does not preclude membership in any club.
>Membership in Club B precludes membership in every other club.
>Membership in Club C precludes membership in Club D.
>Membership in Club D precludes membership in Club A.
>Membership in Club E precludes membership in Clubs A and C.
>There are no other clubs.
>
>Then:  Club B is trivially final.

Ok.

>Club A cannot be final because it is compatible with all other clubs.

This is (1)False and (2)Not enough reason not to be final.

It is false because A is incompatible with all but C.
Also, being compatible with all other clubs would still
allow it to be final if all other clubs were not final.
So A is not final only if C is final, which we
still don't know.

>Club C is
>incompatible with every other club but A, so it is final.

But what if A were final? A is final if C is not, C is final
if A is not. How do we decide which is it?

>Club D is incompatible with A, B, and C, but not E, so it is
>final if E is not final.  The same applies to Club E,
>mutatis mutandis.  Therefore neither D nor E is final.

Again, how do you decide that neither of them is final,
instead of choosing one as final? Just to be fair?

So your example is just a more complicated version of my
example, a situation where it is undecidable which clubs
are final and so the "definition" does not define anything.

I think the following definition works for situations
where final clubs are well defined, but it obviously
does not work in most situations, because in most situations
final clubs are not well defined:

    A club is a final club if membership in the club
    precludes membership in some other club.

Now I suspect that is not what Quine had in mind. Perhaps
if membership in A precludes membership in B, then we
should not assume that membership in B precludes membership
in A (i.e. the bylaws of club B don't preclude its members
from joining club A, even though in practice they won't
be able to due to the bylaws of club A). If we take this
view of preclusion, then I think the definition should be:

     A club X is a final club if membership in club X
     precludes membership in any club Y such that
     membership in Y precludes membership in X.

In other words, any club that reciprocates preclusions
is a final club.

Again the definition gives wrong answers in situations
where finality of clubs is undefined.

Well?

co'o mi'e xorxes

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