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.

Thank you for this helpful precis! But I don't quite follow your point about the supplementation principle. All that the principle (as you state it) seems to say is that for any proper part of a thing, there is another non-overlapping part of the same thing. So, if Descartes has a left leg, then he also has some other part, say his right hand, which does not overlap the left leg.

The principle doesn't seem to imply the stronger position that, if Descartes has a left leg as a proper part, then he also must have D-minus as a part. Indeed, it seems consistent with a position claiming that there is a mutually non-overlapping and jointly exhaustive division of Descartes' body into

nonarbitrary undetached organic parts.Posted by: Boram Lee | Thursday, September 09, 2010 at 09:00 PM

That the argument leads to such absurd conclusions ought to make us deeply suspicious. One line of attack might be to say that if we are to understand '=' as 'is identical to' then an expression either side of '=' must refer to an enduring object. If expressions like 'Descartes after t*' refer to anything at all then the obvious assignments are

D-minus before t* = D-minus after t* = D-minus

Descartes after t* = Descartes before t* = Descartes

This assignment makes 'D-minus after t* = Descartes after t*' false.

Another, perhaps equivalent, line might be to say that the verb 'is' inside '=' is ambiguously tensed in statements like 'D-minus before t* = D-minus after t*'. At just what time are we claiming this proposition to be true?, or indeed, is this statement grammatically well-formed?

Lastly, perhaps, these statements are universal claims about equality of properties. Thus, for example, 'D-minus before t* = D-minus after t*' is to be taken to mean 'the shape that D-minus had before t* equals the shape that D-minus had after t* and the stuff comprising D-minus before t* equals the stuff comprising D-minus after t* and...'. But under this interpretation 'Descartes after t* = Descartes before t*' turns out false because his shape and composition change across time t*.

William explains it rather better here, I think.

Posted by: David Brightly | Friday, September 10, 2010 at 03:40 AM

Boram,

That's an excellent and penetrating comment. The idea behind Supplementation is that you cannot have a whole with just one proper part. This seems intuitively obvious in the case of spatial wholes. There couldn't be a ball with a northern hemisphere but no southern hemisphere. The principle could be put as follows:

If x is a proper part of y, then there exists a z such that z is a proper part of y and z is disjoint from x.

But as you astutely point out, this seems to allow that z be a proper part of x that is not identical to y minus x. The principle seems to be satisfied if Descartes has a proper part such as his right hand which is disjoint from his left leg.

Whereas what Ineed for my criticism of PvI is that D-minus and L stand or fall together.

But wouldn't repeated applications of the principle give me the result I want? Suppose we have an object O which has exactly three proper parts a, b, c each of which is a simple. By Supplementation, we know that O cannot have just a as a proper part. It has to have either b or c as well. Suppose it is b. Then by Unrestricted Composition we get the sum (a + b). This sum is a proper part of O. But then by Supplementation again we know that O cannot just have (a + b) as its sole proper part. So it must have another. The only possibility is c. So by applying Supplementation twice we show that if O has one proper part a then it also has proper parts b, c. By Unrestricted Composition again we get (b + c). (b + c) is just O minus a.

In short, Supplementation tells that if an object is construed as a classical mereological sum, and it has a proper part z, then it has a proper part disjoint from z which has as parts the remaining proper parts of O.

Is that cogent or is that confused?

Posted by: Bill Vallicella | Friday, September 10, 2010 at 01:19 PM

I was about to jump in the 'bikes now and then' but David beat me.

Posted by: William | Friday, September 10, 2010 at 02:49 PM

Bill, you write:

"Then by Unrestricted Composition we get the sum (a + b). This sum is a proper part of O."

As far as I can see, the second quoted sentence does not follow from either Supplementation or Unrestricted Composition or both. All that Supplementation seems to say is that a composite object must have at least two non-overlapping proper parts (if it has

aas a proper part, then it must have eitherborcas a proper part). And Unrestricted Composition only says something about how parts compose into wholes, and nothing about how wholes decompose into parts (so it has nothing to say about whether the sum(a + b)is a part of O or not).Who knows, perhaps there is a more complicated way in which one can derive the second quoted sentence from Supplementation plus axioms of classical mereology. What I find valuable in your discussion is the insight that the denial of DAUP for some composite material object need not lead one to deny that there is a non-arbitrary way of decomposing that object into proper parts. I don't know how far that insight will carry us in solving the Problem of the Many and other related puzzles... there always seems to be some catch somewhere for any given approach (e.g., a proposed approach for a particular problem cannot address a generalized version of the same problem)... but it seems worth looking into.

Posted by: Boram Lee | Friday, September 10, 2010 at 03:23 PM

Correction. I wrote: "And Unrestricted Composition (UC) only says something about how parts compose into wholes, and nothing about how wholes decompose into parts." Actually I was wrong. If A and B compose into A+B by UC, then of course A and B are each proper parts of A+B. And again by UC, A+B and C compose into A+B+C, and by construction A+B and C are proper parts of A+B+C.

What I should deny, then, is that the mereological sum A+B+C could be identical to a living organism like Descartes or his body; equivalently, I should deny that living organisms decompose into arbitrary undetached parts (while accepting their decomposition into organic parts).

Posted by: Boram Lee | Friday, September 10, 2010 at 04:11 PM

Boram,

Peter Simons (Parts, p. 26) writes, "An individual which has a proper part needs other proper parts in addition to

supplementthis one to obtain the whole."This suggests to me that if L is a proper part of Descartes, then Suplementation is not satisfied if only D's right hand is admitted as a proper part.

So if L is proper part (in the sense of class. mereology) of D, then D minus is also a proper part of D. They stand and fall together, to repeat the pun.

You now seem to be saying that L is not an organic proper part of D. But a leg can be severed and reattached -- so wouldn't that make it an organic proper part?

Posted by: Bill Vallicella | Friday, September 10, 2010 at 05:21 PM

Bill,

Simons's remark seems to be an informal explanation of one of the intuitions for adopting Supplementation, but the statement of the principle itself (both here and in Varzi's SEP entry on "mereology" for instance) says nothing more than that a composite object must have at least two proper parts. Another intuition supporting Supplementation, perhaps, is that it helps to distinguish a proper part from an improper part. An improper part of an object is precisely one for which there is no non-overlapping part of the same object.

In any case, I am not suggesting that, besides D's left leg, only D's right hand is to be admitted as a proper part. I am assuming that, at one level of decomposition, the human body is such that it has one privileged non-arbitrary division into undetached body parts: into arms and legs, hands and feet, head, neck and torso, and so on. Take all these non-overlapping parts together, and you have the whole organism. So it recaptures Simons's supporting intuition, without admitting the existence of the L-complement as a material object in its own right, or in other words, without admitting the existence of a mereological sum of the enumerated body parts minus the left leg as a material object. When counting body parts, I aim to admit into my ontology only Nature's own carvings, and not any arbitrary ones whatever.

Posted by: Boram Lee | Friday, September 10, 2010 at 06:25 PM

Your argument clearly assumes the following:

(*) if A was identical with some B, then A is identical with the B.

Why should that be true? Surely it isn't. Descartes was identical with a soul combined with an 'able body'. Now Descartes is identical with a soul combined with a differently abled body. I don't see the problem.

Posted by: William | Saturday, September 11, 2010 at 01:43 AM