On Dec 5, 5:06 am, fom <fomJ...@nyms.net> wrote: > On 12/2/2012 11:20 PM, William Elliot wrote: > > > On Fri, 30 Nov 2012, Zuhair wrote: > > <snip> > > >> 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. >
I'm really sorry that I didn't have the chance to look at all of your responses. I'd do once I have time. Anyhow for now, it is sufficient to note that my theory does prove Weak supplementation for collections of atoms that is if x is a proper part of y and y is a collection of atoms then there exist a part of y that do not overlap with x.
> 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. > > <snip>