Slightly more formally, the arguments look more or less like this:
(A1) The piece of clay existed on Monday
(A2) The statue did not exist on Monday
(A3) If the piece of clay and the statue are identical, then the piece of clay existed on Monday iff the statue existed on Monday
(AC) The piece of clay and the statue are not identical
(B1) The piece of clay would survive being squashed into a ball
(B2) The statue would not survive being squashed into a ball
(B3) If the piece of clay and the statue are identical, then the piece of clay would survive being squashed into a ball iff the statue would survive being squashed into a ball.
(BC) The piece of clay and the statue are not identical
To my mind, the most surprising feature of this puzzle is that it has mislead so many good philosophers into embracing the view that material constitution is a relation between distinct objects with all its implausible consequences despite the fact that a much more simple and plausible solution to the puzzle has been around for decades (as far as I can see the position I have in mind is the one developed and defended by Roderick Chisholm in the 1970s). So, I was wondering if readers could help me see what's wrong with the Chisholmian solution or explain why it is almost completely ignored in the literature (in fact I cannot even think of anyone truly engaging with it in the literature).
Let me start by putting aside my mereological nihilist sympathies (as I assume few would embrace mereological nihilism with its seemingly implausible consequences just for the sake of solving that puzzle) and assume that there are pieces of clay and statues. For the sake of simplicity, let me also assume that the one made on Tuesday is the only statue there is, was, and will ever be in the whole wide world. Given these assumptions, it seems that one could truly affirm that
(A1*) On Monday, there was a piece of clay (i.e. On Monday, there is an x such that x is a piece of clay),
(A2*) On Monday, there was no a statue (i.e. On Monday, there is no y such that y is a statue).
It is our inclination to accept something like (A2*), I suspect, that can be exploited to mislead us into assenting to (A2). However, accepting (A2*) does not amount to accepting anything like (A2), as one can easily concede that there was no statue on Monday and that there was one on Tuesday while denying that something new has come into existence between Monday and Tuesday (contrary to what (A2) surreptitiously suggests). One can do so simply by maintaining that, whereas our piece of clay (call it 'Clay') was not yet a statue on Monday, it became one on Tuesday, when the artist turned it into one. So, while there was no statue on Monday and there is one on Tuesday, the thing that became a statue on Tuesday (i.e. Clay) already existed on Monday, although on Monday it was not yet a statue, as it did not meet the conditions for satisfying 'x is a statue' (whatever these may be).
Consider now Argument B. Sure enough, if Clay were to be squashed into a ball, something would still be a piece of clay and nothing would be a statue. However, this does not imply that something would go out of existence in the process. It is simply that, under these counterfactual circumstances, Clay would no longer meet the conditions for satisfying 'x is a statue' (whatever these may be), while it would still meet the ones for satisfying 'x is a piece of clay'. So, one could truly affirm that:
(B1*) If Clay were to be squashed into a ball, there would still be something that is a piece of clay (i.e. there would be an x such that x is a piece of clay).
(B2*) If Clay were to be squashed into a ball, there would no longer be be something that is a statue (i.e. there would be no y such that y is a statue).
And that, I think, is all we really mean to assent to when we assent to (B1) and (B2).
Consider now a third variation on our puzzle.
(C1) The piece of clay would not survive the loss of any of its proper parts
(C2) The statue would survive the loss of some of its proper parts
(C3) If the piece of clay and the statue are identical, then the piece of clay would not survive the loss of any of its proper parts iff the statue would not survive the loss of any of its proper parts.
(CC) The piece of clay and the statue are not identical
Consider, for example, a piece of Clay that is neither too big nor too small--e.g. the piece that forms the nose of the statue (call it 'Nosy'). Here, the underlying intuition seems to be that, if Nosy came to be detached, the statue would remain the same statue as before (although deprived of its nose) but the piece of clay wouldn't be any longer the same. All this argument seems to show, however, is that the conditions for satisfying 'x is the same statue as y' are different from those for satisfying 'x is the same piece of clay as y'. Let's grant that, if Nosy were to be detached from Clay, Clay would cease to exist. In its place, we would have two smaller pieces of clay: Nosy and the rest of Clay (call it 'Clay Jr'). Each of them used to be a proper part of Clay and, so each of them, is partially identical with it (in the sense that part of each is identical with part of Clay) although not (wholly) identical with it. More importantly, one of them (i.e. Clay Jr) still meets the conditions for satisfying 'x is a statue'. So, after Clay ceases to exist, there still is a statue.
But what of the intuition that this statue is the same statue as the one that was there before? Since Clay and Clay Jr are not identical, how can the statue that Clay Jr is be the same statue as the one that Clay used to be? I think the answer should be that, despite the appearances, 'x is the same statue as y' (nor 'x is the same piece of clay as y' for that matter) expresses an identity relation. (Note that this position differs from the one (in)famously put forward by Peter Geach, as it maintains that Clay and Clay Jr are absolutely distinct, whether or note we take them to satisfy 'x is the same statue as y' or 'x is the same piece of clay as y'.) In other words, in order for Clay and Clay Jr