The problem of identity is a central metaphysical question, since on it depends our notions of objects, individuals and persistence. Without the ability to identify individuals, it is perhaps even the case that we could not individuate successfully, so it is obvious we need a successful criterion of identity. However, a large amount of effort has been devoted to deciding what the correct criterion of identity is, or even what can qualify as a valid criterion of identity. I think that this approach is flawed, in that it is setting out to arbitrarily decide what constitutes identity. My own view is that identity is not something fundamental to objects, but rather something that we as humans add to our conception of objects in order to be able to cognise them in meaningful ways. It is therefore possible that there is no completely successful criterion of identity, since it is not a logical necessity. However, should a criterion of identity be shown to fit all cases, I would be more than happy to accept it, provided that it also conforms to our understanding of an object, and does not seek to completely redefine it.
Starting from the belief that humans wish to be able to identify an object as the same individual over time, given certain circumstances, the task is to explain what it is about the object that allows us to say it is one and the same, although some of its attributes have changed. One problem that is going to occur is that we already believe we know which objects are persistent through time, and that in order to explain any system of identity, we are going to have to make recourse to objects that persist. Whilst I do not feel this actually does damage to the project, it is worth being aware of, and ensuring that no vicious circularity appears.
One possible way of dealing with identity through change is to build a permissible amount of change into the theory. One could allow that a small amount of change can occur, and yet the object is still itself, so long as no larger amount of change occurs. Taking the Ship of Theseus as the paradigm example, we could therefore allow a certain mass of the ship to be replaced, and be happy that the ship is still the same one. The question is, if more changes are carried out at a later date, does one include them in the same change for accounting purposes, or is it a new change. The twofold problem is that if one seeks to say that it is the same change, then only a tiny additional change can make the object no longer itself. However, if one seeks to make it a new change, i.e. reset the 'original' object, then one runs into a problem with the laws of identity. The first problem here is best expressed as follows: If one has a ship, and allows that five percent of its mass can be replaced and it remain one and the same ship, then suppose that when it is repaired, exactly five percent of the mass is replaced by new mass. So far, the ship is the same as the original. However, then suppose that it again is repaired, this time only changing a single nail. This would put the total change over five percent, and the ship is therefore a new ship, simply by changing a single nail. This seems ridiculous, given that we are attempting to incorporate some flexibility into identity. However, if one decides to take up the alternate proposition that change starts anew, then one runs into another identity crisis. The original ship, ship A, is repaired, replacing five percent of the mass, to form ship B. So far A and B are one and the same ship, and there is no problem. However, when B is repaired and the nail is replaced, forming ship C, a violation of the laws of identity occurs. For whilst A and B are identical, and B and C are identical, A and C cannot be, since more than five percent is different. Since we should be able to say that if A=B and B=C then A=B=C, a failure in this identity seems disastrous. So, if we seek to allow a limited amount of change, we seem to either get nowhere significant, since once the limit is reached, the same problem occurs, or we break what seems to be a fundamental law of identity. In addition to this, there is the question of how exactly one could determine what an acceptable level of change was. There seems to be no necessary level, and an arbitrary one seems ridiculous. I therefore find this 'quota of change' idea unworkable.
An alternative is to assert that the constitution of the object is irrelevant, that it has some form of essence that can persist through change. Thus, whatever is done to an object, it remains the same object, simply in a different form. Whilst this would perhaps solve some of the problems, it would also complicate an ontology, and leave many questions unanswered as to what form this 'essence' takes, how, if at all, it can be created and destroyed, and how we can come to have knowledge of this essence. To take the example of a bronze statue, it must once have been a lump of bronze, so at what point did it acquire the label of statue, or did it always have it, but the label remains dormant when not instantiated. A proposition such as this seems to me to unnecessarily complicate the issue, without really providing solutions.
There is perhaps a limited set of cases where the concept of an essence may in fact be applicable, that of inherent change, particularly in the case of organic life. There is a more intuitive understanding of what could persist through change in the case of an organism, or at least the higher organisms, in the form of the attribution of mental states. However, once more we encounter the difficulty of accessing that information, the old problem of how we can know another's mind. In short, we only believe there to be continuity of mental life in other beings, and such a presupposition seems unreasonable when one seeks the certainty of base concepts such as that of an object. However, this category, that of organisms, does seem to offer one paradigm example of identity through change, that of one's self. We consider ourselves to persist through change, and perhaps that could be analysed and applied in other circumstances. However, I will not be engaging in such a project here, as personal identity is a large enough topic on its own. It seems that there are too many difficulties inherent in an object possessing its own criterion of identity, and so perhaps such an approach must be abandoned. Perhaps there is no case for objects in themselves persisting through change.
An alternative to inherent identity is imposed identity. One could argue that we, as the beings which require the ability to identify and individuate, are in fact imposing non natural characteristics on the world in order to make sense of it. In that case, we need to supply some explanation of what we consider to be valid identity through change, as opposed to what is actually identity through change. For me, the prime candidate for this has to be Frege's theory of reference, or one of its variants. Under this system, each act of using a name would refer to one object, unless unsuccessful, and it is possible that at a later date the same name could apply to what we would intuitively call the same object, but is in fact a somewhat different one. In order to be able to accept a level of change in the reference, Frege has the mechanism of sense, which can be interpreted as our intention in referring. Thus whilst the original ship of Theseus and the somewhat modified one are not identical, and a single instance of naming cannot in fact apply to them both, two instances of naming, one intended to refer to each ship, can be allowed. It is true that this system would lead to some strange sentences being valid, such as "The ship of Theseus is not identical with the ship of Theseus", but one can quite easily show that this is simply a result of imprecise language usage. One is using the same name to refer to two non-identical objects, and therefore one is bound to generate some ambiguities or confusion. However, this position is not without its flaws. Not only are there problems with the theory in a wider range of places than apply here, but one can argue that it still begs the question somewhat. No matter that we can now talk of objects as though they persist through change, we wish to know if and how they can persist through change. It seems that while this position sounds good, it is in fact more of a psychological explanation to be added to a metaphysical theory.
Having examined what I consider to be the primary candidates for explaining how an object could persist through change, I find that I cannot accept any of them. For me, even though mereological essentialism has the drawback of removing some parts of an ontology that one would perhaps desire, and even being somewhat counterintuitive, I find it to be the most appealing position. Given that an object does not seem able to carry its identity with it, it is my opinion that any change does in fact entail the end of one object and the beginning of another. In fact, it renders the notion of an object logically superfluous, in that what is the point of complicating one's ontology with something that is little more than a truism. It seems to resemble the axiom 'whatever is, is', which whilst useful, hardly merits an addition to one's ontology. One possible argument against a lack of identity is that one can argue that there is necessarily one criterion, that every object is identical with itself. However, not only can one point out that objects are now superfluous, but the statement itself is nothing but a tautology in logical terms demonstrating nothing.
Of course, one must still seek to accommodate our common parlance of objects as not only existing but persisting, and I would seek to do this by deepening the divide between logic and natural language. Logic has to be careful what it says in order to maintain its privileged status, whereas natural languages simply need to be useful. Therefore, whilst the common notion of object as an entity that has spatio-temporal continuity and / or symmetry of properties is pragmatic, it cannot qualify as a logical precept, and this is why instances of identity paradoxes are numerous. Whilst it would be unreasonable to attempt to eliminate the concept of an object from all use, we must recognise that it can only be used as a 'best fit' tool, and will generate many paradoxes. We must be sure to maintain a weak sense of identity.