Let us suppose that there is a proper part P of me such that every part of me beyond P could perish, while P would remain. Some think the soul is such a part. Others think the brain is. Yet others might think that the head, or the head plus soul, or my upper half are such. Suppose now that at t0, this part P is a proper part of me, but later, at t1, everything making me up outside of P perishes, and no new stuff accretes, so that P is my only part, at least not counting the subparts of P. Maybe I am a brain in a vat or a disembodied soul at t1.
What is my relationship to P at t1? It cannot be identity. For if I were identical to P at t1, then by transitivity, I would also be identical to P at t0, and thus at t0 I would be a proper part of myself, which is absurd. Yet at t1, there is a sense in which there is nothing to me but P.
It seems that if identity is not the relation, then the relation is constitution, or some other such relation that falls short of identity. Thus, I am constituted by P at t1. This is pretty standard. But it bothers me. Here's why. At t1, P is still a part of me--it didn't cease to be a part of me just because all the other parts of me have gone away (e.g., if I have a brain, then a brain is a part of me, even if nothing beyond it is). Is P a proper or improper part? If a proper part, then there ought to be other stuff beyond P making me up. But ex hypothesi, at t1, P is all that's left of me. So P is an improper part. But the only improper parts of something are the thing as a whole and, on some views, nothing (or an empty or trivial part). Plainly P is not nothing. So then P is the whole of me, which we've already seen isn't true. Either way, we have a problem.
To rephrase, suppose:
- Everything excepting some proper part of me could perish with nothing new accreting to me, and with that part not coming to be beyond me.
- If a part x of y survives and y survives, and x does not come to be beyond y, then x will still be a part of y.
- Identity is transitive.
- If x is a proper part of y, then there is stuff beyond x in y.
A slightly different argument is to note that at t0, both P and I are parts of me in the same sense--one a proper and the other an improper part. (One might question this: maybe proper and improper parts are "parts" by analogy or equivocation.) This shouldn't change at t1: both P and I should still be parts of me in the same sense. But it doesn't seem categorially right to suppose that x and y can both be parts of z in the same sense when x constitutes y (the atoms of my heart and my heart are parts of me in different senses).
I am inclined to say that these arguments push one to reject (1)--the idea that I have a proper part such that everything beyond that part could perish while I survived with that one part. However, I also think that if any object has proper parts, surely there will be some object with a proper part that is such that the object could survive being "reduced to" that part. Indeed, very plausibly, if any object has proper parts, likewise my mind is a proper part of me, and I could surivve being "reduced to" just the mind (regardless whether the mind is a brain or a soul).
And so we have another argument for the denial of the thesis that some objects have proper parts.