This puzzle, similar to Peter Geach's Tibbles the Cat in content, is unlike it in vintage. Its origin is attributed by Philo of Alexandria (30 B.C. – 45 A. D.) to Chrysippus the Stoic (c. 280 B.C. – c. 206 B. C.) What follows is my take on the puzzle. I draw heavily upon Michael B. Burke, "Dion and Theon: An Essentialist Solution to an Ancient Puzzle," The Journal of Philosophy, 1994, pp. 129-139.
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:
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.
Ad (A). 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.
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.
Ad (B). This option is so counterintuitive that no one has maintained it.
Ad (C). It seem obvious that Dion survives the amputation. But what about Theon? Can we say that Theon ceases to exist after the amputation? Michael Burke (p. 134 ff.) thinks we can. He gives something like the following argument:
1. The concept of a person is maximal: the proper parts of persons are not themselves persons.
Therefore
2. Theon before the amputation was not a person.
3. Persons are essentially persons, and hence nonpersons are essentially nonpersons.
Therefore
4. Theon before the amputation was essentially a nonperson.
Therefore
5. Theon could not have survived a change that would have made it, if it survived, a person.
6. The amputation of Dion's left foot is a change that would have made Theon, if it survived, a person.
Therefore
7. Theon did not survive the amputation of Dion's left foot, and so does not exist after the amputation.
Ad (D). If one adopts mereological essentialism, then one can say that Theon survives but Dion does not. Mereological essentialism is the doctrine that wholes have their parts essentially. Accordingly, for any whole W and part x of W, if x is a part of W, then necessarily x is a part of W. This implies that no whole can survive the gain or loss of a part. If so, then Dion ceases to exist when he loses his left foot. But Theon continues to exist. Unfortunately, mereological essentialism applied to ordinary continuants such as cats and cabbages and cars is highly counterintuitive. My car does not cease to exist when the antenna is snapped off.
None of the solutions is satisfying as far as I can see. One might conclude that reason in us is weak and dialectical in Kant's sense.