The issue is, on the assumption that Tibbles had a tail yesterday, and that he lost it overnight, and now has no tail, what is it that became the cat without a tail? Was it
(a) The part of Tibbles yesterday which had no tail, i.e. Tibbles yesterday minus tail.
(b) The whole of Tibbles, i.e. Tibbles yesterday with tail.
Brandon seems to be saying (rather absurdly in my view) that it is the proper part of Tibbles, i.e. Tibbles-minus-tail that becomes Tibbles today. I say 'rather absurdly' because the proper part, whatever it is, is something that always lacks a tail. So how can it change into precisely what it already is. I maintain, by contrast, that the subject of change is Tibbles himself, i.e. a thing which yesterday had a tail, but became something without a tail.
It should be said that this is not what I intend to convey. In fact, my argument against maximality turns on exactly the point Ocham notes here: Tibbs does not change. On the maximality view, however, Tibbs was originally not a maximal part; therefore Tibbs was not a cat, and now Tibbs, without changing, has become a cat (because Tibbles lost its tail). So the maximality view commits us to saying that things that are not cats can become cats without any intrinsic change. This is due to the fact that maximality requires that all and only maximal parts of cats are cats. Maximality requires us to make the absurd claim that the proper part of Tibbles becomes the cat today.
But it does appear that I've misunderstood Ocham's position, since I took him to be holding the maximality view. The above summary, however, is not maximality at all, but something different. On a maximality view, we have a tension once we recognize that, despite their differences from spatial parts, there are temporal parts. Is the cat the maximal spatial part or the maximal spatiotemporal part? If we accept the former, we have thus committed ourselves to saying that there is a different cat on the mat from moment to moment (now Tibbles, now Tibbs, now something else). If we accept the latter, we have to hold that none of these is a cat at all -- the only cat is the whole of which all of these are merely proper parts. (In something roughly like the sense that the movie, Goonies, is not identifiable with any one scene but is the whole of which all the scenes of the movie are parts.) Either way we get into weird territory.
But Ocham's view is not actually either of these, because, in effect, he posits some subject (let's call it S) which was Tibbles (when it had a tail) and is now Tibbs (when it does not have a tail). So is this a different view from the ones I noted before, or does it fall into one of them? To clarify this, it might be helpful to look at another puzzle Ocham proposes in a later comment. Suppose we take a sentence like:
(A) S was Tibbles and is now Tibbs.
Of this sort of sentence Ocham asks:
In what way does this assert or imply any kind of 'transtemporal identity'? Clearly the pronoun 'he', if it is to make any sense, must refer back to the proper name Tibbles. So this implies an identity, between the object referred to by 'Tibbles', and what is referred to by 'he'. This further implies an identity, if the statement is true, between what satisfies the predicate '- is now lost his tail' and what satisfies '- had a tail'. But how is this identity 'trans temporal'? It's not at all. It's an identity between an object which exists now, an object which once had a tail, and an object which exists now, and has lost a tail.
But while it's true that the identity is between an object that exists now and an object that exists now, this isn't all that (A) implies. For (A) to be true, the object that exists now has to have been an object then (otherwise the object now couldn't be said to have lost a tail). Then it had different properties than it does now -- (A), in fact, gives us one such difference of properties. So for (A) to be true, or even to make any sense, S has to have endured through time, undergoing changes (it is different now than it was), but such that it is the same S. For (A) to be true, S has to have had two mutually exclusive properties. Of course, there is no contradiction here, because it has them at different times. Still there is no (classical) identity between S as it was and S as it now is, even though they are the same S. A relative identity theorist will say that S was identical to Tibbles at that time, and now S is identical to Tibbs; Tibbles and Tibbs are identical qua S, and thus there is a (relative) identity between S as it was and S as it now is. But since Ocham denies relative identity, we see clearly where his position fits: it's a sameness without identity view, at least for temporal parts. Tibbles and Tibbs are the same S, but treating them as the same S is not to treat them as identical (although S is identical to S, of course). And this, I suspect, is why I was completely puzzled by his position; I thought he was putting forward a maximality view, but he kept saying things that suggested sameness without identity.
If we're going to go this route for temporal parts, however, why wouldn't we do the same for spatial parts? Tibbles and Tibbs are the same S, even though what has a tail and what does not have a tail are not identical. There is no need to appeal to maximality and no need to say that only Tibbles is a cat or that Tibbs is not a cat -- in fact, Tibbles is just S considered as a totality of parts, and Tibbs is just S considered without its tail. In all these cases S is the cat, so Tibbles is the cat considered in all its parts, and Tibbs is the cat considered in all its parts except the tail. Tibbles and Tibbs are the same cat, even though they are not identical, just as the cat before its amputation is the same cat as the cat after its amputation, even though they are not identical.