On Feb 4, 2010, at 12:07 PM, Matthew West wrote: Dear Chris, Well I agree that anything defined by axioms is affected when those axioms change. However, at least some primitives just are – membership would be one of them. Are you therefore arguing that in using membership in the axioms that define other things, that that affects the meaning of membership itself (as an example)?
I would say this, for one :-) Seems to me that this position is forced upon one, in fact. If you want to maintain the opposite view, you have to say that some formal symbols or names have a meaning which is fixed regardless of the axioms that they are used in. But how is this meaning that transcends any axiomatic theory to be expressed?
I can see that there are clearly subsets of membership that are say set membership and type membership, but I’m not sure that that changes the meaning of membership.
In cases like this the meanings may be very close, but that is because the axiomatic theories are also very close. But indeed, yes, if types and sets are different, then we have to at least contemplate the possibility that membership in a set, and membership in a type, are different. And indeed, I think they are, and that this has to do with the very difference, between type and mere set, that your axiomatic distinction is trying to capture. Membership in a set is a casual matter, requiring no special affinity between the thing and the set; whereas membership in a type is in virtue of having some property or aspect which is characteristic of that type.
Pat H.
Regards Matthew West Information Junction Tel: +44 560 302 3685 Mobile: +44 750 3385279 This email originates from Information Junction Ltd. Registered in England and Wales No. 6632177. Registered office: 2 Brookside, Meadow Way, Letchworth Garden City, Hertfordshire, SG6 3JE. I copy in John’s comment ( he uses ‘elementOf’ where you use ‘membership’) : The inconsistencies lie in the choice of axioms. All versions of set theory are based on two dyadic relations: subsetOf and elementOf. The differences lie in the axioms that are asserted in each theory. You could call subsetOf and elementOf primitives, but they don't behave the way that you have been claiming for the kinds of primitives you want. In particular, their "meaning" is determined by the axioms and each version of set theory has a different set of axioms. That is one of the main reasons why I keep saying that this search for primitives is misguided. It's totally irrelevant what set of words (or predicates or relations or types or concepts or whatever) you start with -- because all the serious work is done by the axioms. As soon as you add more axioms to a theory, the "meaning" of the so-called "primitives" changes. Dear Chris, Could you elaborate please. Though, as I think John Sowa pointed out in general (apologies if it was someone else), the ‘root primitive’ membership has different senses / meanings in the two cases – so it is not exactly the same. Regards Matthew West Information Junction Tel: +44 560 302 3685 Mobile: +44 750 3385279 This email originates from Information Junction Ltd. Registered in England and Wales No. 6632177. Registered office: 2 Brookside, Meadow Way, Letchworth Garden City, Hertfordshire, SG6 3JE. _________________________________________________________________ Message Archives: http://ontolog.cim3.net/forum/ontolog-forum/ Config Subscr: http://ontolog.cim3.net/mailman/listinfo/ontolog-forum/ Unsubscribe: mailto:ontolog-forum-leave@xxxxxxxxxxxxxxxxShared Files: http://ontolog.cim3.net/file/Community Wiki: http://ontolog.cim3.net/wiki/ To join: http://ontolog.cim3.net/cgi-bin/wiki.pl?WikiHomePage#nid1JTo Post: mailto:ontolog-forum@xxxxxxxxxxxxxxxx
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