Dion and Theon

Bill has a post out on the puzzle of Dion and Theon.  Here is his introduction:

Yesterday, Dion was a whole man, but today he had his left foot successfully amputated. Yesterday, 'Theon' was introduced as a name for that proper part of Dion that consisted of the whole of Dion except his left foot. (To keep the formulation of the puzzle simple, let us assume that dualism is false and that Dion is just a living human organism.) It is clear that yesterday Dion and Theon were numerically distinct individuals, the reason being that yesterday Theon was a proper part of Dion.  (By definition of 'proper part,' if x is a proper part of y, then x is not identical to y.  And if x and y are not identical, then x and y are distinct.  Two items can be distinct without being wholly distinct.)  Now the question is which of the following is true today, after the amputation:

A. Both Dion and Theon exist.
B. Neither Dion nor Theon exist.
C. Dion exists but Theon does not.
D. Theon exists but Dion does not.

The problem is to justify one of these answers. If none of the answers can be rationally justified, then we have a tetralemma which might be taken to suggest that there is something deeply problematic about our ordinary talk and thought about material particulars and their persistence. Given my conception of philosophy as at once both aporetic and revisionist, this would be a welcome result if I could support it.

Bill has arguments against all four limbs.  Against (A) he says,

Because Dion and Theon both existed yesterday, you might think they both exist today. There is, however, a reason to think that it cannot be true that both Dion and Theon exist today after the amputation. The reason is that it is impossible both that (i) Dion and Theon be numerically distinct and that (ii) Dion and Theon occupy exactly the same place and be composed of exactly the same matter arranged in exactly the same way, as is the case today after the amputation.

If Dion and Theon were numerically distinct objects then I'd have to agree with Bill that it's difficult to see how one could so change as to occupy exactly the same space, etc, etc.  However, we can resolve the puzzle by taking Theon to be not an object but to be a parcel of matter.  The two concepts are related but in a sense orthogonal.  We can say that at some moment a certain parcel of matter constitutes an object.  But as time passes the object, especially if it's a living thing, will lose some matter and gain other matter.  Thus at different times it may be constituted by different parcels.  This is to see an object as a process through which matter flows.  But a parcel, by definition,  always contains the same matter.  And while at any moment the matter of an object occupies a connected region of space, a parcel of matter can find itself distributed over multiple disconnected regions of space.  So we can say that yesterday two disjoint parcels of matter jointly constituted Dion:  Theon, as defined above, and the matter of Dion's left foot, Pedro, say.  Today, Pedro has become detached and Theon alone constitutes Dion.  There is no problem with Dion and Theon occupying the same space.  That is what it means for a parcel of matter instantaneously to constitute an object.  Bill continues,

Could we say that Dion and Theon both survived the operation but are now one and the same? This is impossible given the Indiscernibility of Identicals. For today, after the operation, something is true of Dion which is not true of Theon, namely, that he, Dion, once had two feet. So Dion and Theon cannot be or have become identical.

We do not have to say that today Dion and Theon are the same, only that Theon alone now constitutes Dion.  There is no problem with an object and the parcel of matter that currently constitutes it having distinct pasts and distinct futures.