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Thursday, September 09, 2010


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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.

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.


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?

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

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 a as a proper part, then it must have either b or c as 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.

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).


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


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.

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.

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