In order to get clear about Dion-Theon and related identity puzzles we need to get clear about the Doctrine of Arbitrary Undetached Parts (DAUP) and see what bearing it has on the puzzles. Peter van Inwagen provides the following statement of DAUP:
For every material object M, if R is the region of space occupied by M at time t, and if sub-R is any occupiable sub-region of R whatever, there exists a material object that occupies the region sub-R at t. ("The Doctrine of Arbitrary Undetached Parts" in Ontology, Identity, and Modality, CUP, 2001, 75.)
Suppose I am smoking a cigar. DAUP implies that the middle two-thirds of the cigar is just as much a concrete material object as the whole cigar. This middle two-thirds is an undetached part of the cigar, but also an arbitrary undetached part since I could have arbitrarily selected uncountably many other lengths such as the middle three-fourths. Applied to Tibbles the cat, DAUP implies that Tibbles-minus-one-hair is just as full-fledged a material object as Tibbles. Van Inwagen maintains that DAUP is false.
I will reconstruct van Inwagen's argument for the falsity of DAUP as clearly as I can. Consider Descartes and his left leg L. To keep it simple, we make the unCartesian assumption that Descartes is just a live body. DAUP implies that L is a material object as much as Descartes himself. DAUP also implies that there is a material object we can call D-minus. This is Descartes-minus-L. It is obvious that Descartes and D-minus are not the same. (For one thing, they are differently shaped. For another, they are 'differently abled' in PC jargon.) At time t, D-minus and L are undetached nonoverlapping proper parts of Descartes, and both are just as much full-fledged material objects as Descartes himself is.
Now suppose a little later, at t*, L becomes detached from D-minus. In plain English, Descartes at t* loses his leg. (To avoid certain complications, we also assume that the leg is not only removed but also annihilated.) Does D-minus still exist after t*? Van Inwagen thinks it is obvious that D-minus does exist after the operation at t*. DAUP implies that the undetached parts of material objects are themselves material objects. So D-minus prior to t* is a material object. Its becoming detached from L does not affect D-minus or its parts, and if the separation of L from D-minus were to cause D-minus to cease to exist, then, van Inwagen claims, D-minus could not properly be called a material object. Descartes himself also exists after the operation at t*. Surely one can survive the loss of a leg. So after t* both D-minus and Descartes exist. But if they both exist, then they are identical. For otherwise there would be two material objects having exactly the same size, shape, position, mass, velocity, etc., and that is impossible.
In sum, at time t, D-minus and Descartes are not identical, while at the later time t* they are identical. The result is the following inconsistent tetrad:
D-minus before t* = D-minus after t*
D-minus after t* = Descartes after t*
Descartes after t* = Descartes before t*
It is not the case that D-minus before t* = Descartes before t*
The first three propositions entail the negation of the fourth. From this contradiction van Inwagen infers that there never was any such thing as D-minus. If so, then DAUP is false. But as van Inwagen realizes, his refutation of DAUP has a counterintuitive consequence, namely, that L does not exist either: there never was any such thing as Descartes' left leg. For it seems obvious that D-minus and L stand or fall together, to repeat van Inwagen's pun.
That is, D-minus exists if and only if L exists, and D-minus does not exist if and only if L does not exist. D-minus is an arbitrary undetached proper part of Descartes if and only if L is an arbitrary undetached proper part of Descartes. At this point, I think it becomes clear that van Inwagen's solution to the Dion/Theon or Descartes/D-minus puzzle is not compelling. He solves the puzzle by denying that there was ever any such material object as D-minus. But if there was no D-minus, then there was never any such material object as Descartes' left leg. It is obvious, however, that there was such a material object as Descartes' left leg L. So how could it be maintained that there was no such object as Descartes-minus? Van Inwagen makes it clear (p. 82, n. 12) that he does not deny that there are undetached parts. What I take him to be denying is that, for any P and O, where P is an undetached part of material object O, there is a complementary proper part of O, O-minus-P. So perhaps van Inwagen can say that L is a non-arbitrary undetached part of Descartes and that this is consistent with there being no D-minus. If so, he would have to reject the following supplementation principle of mereology which seems intuitively sound:
For any x, y, z, if x is a proper part of y, then there exists a z such that z is a part of y and z does not overlap x , where x overlaps y =df there exists a z such that z is a part of x and z is a part of y.
What the above supplementation principle says is that you cannot have a whole with only one proper part. Every whole having a proper part has a second proper part that supplements or complements the first so as to constitute a whole. Now Descartes' leg is a proper part of Descartes. So the existence of D-minus falls out of the supplementation principle.
It seems, then, that van Inwagen's rejection of DAUP issues in a dilemma. If there is no such object as Descartes minus his left leg, then there is no such object as Descartes' left leg, which is highly counterintuitive, to put it mildly. But if van Inwagen holds onto the left leg, then it seems his must reject the seemingly obvious supplementation principle lately mentioned.
My interim conclusion is that van Inwagen's solution to the Descartes/D-minus puzzle by rejection of DAUP is not compelling.