On 12/2/2012 11:20 PM, William Elliot wrote: > On Fri, 30 Nov 2012, Zuhair wrote: >
>> ll. Supplementation: x P y & ~ y P x -> Exist z. z P y & ~ x P z. > > x subset y, y not subset x -> some z subset y with x not subset z. > x proper subset y -> some z subset y with x not subset z > x proper subset y -> y\x subset y, x not subset y\x > > Oh my, no empty set.
You have made an incorrect step here.
In mereology there is no reason to take y\x as substantive.
Supplementation is supposed to enforce existence of a proper part of y in y\x.
In this case, z could be a proper part of x. Then zPy and -xPz is satisfied.
This is not a supplementation axiom in the classical sense.
As for no empty set, Zuhair may have seen this axiom in a formulation of mereology where the axiom was intended to preclude existence of a null part. This is a standard ontological position among those individuals who investigate and reflects a position once taken by Frege in criticizing the likes of Hausdorff and Cantor:
"... a forest without trees."
Moreover, Zuhair's construction is similar to Zermelo's 1908 paper on set theory. Heijenoort's translates Zermelo's "Teil" -- that is, subsets of nonvoid sets -- as "parts", and, the null set is introduced separately.
This is precisely what Zuhair has attempted to do.