An argument put forth by Derek Parfit against future generations having rights, and therefore against our having to observe those rights towards them.
It goes something like this: if we don't do certain things (recycle, consume less, ZPG, etc.), people 100 years in the future will be worse off. But if we change our actions to help these future people,we will affect the makeup of the future populations. So we just did something to help some people (those existing in the future), but in doing so we have eliminated those people, and replaced them with others (a different future population). Any time you try to do something for these people, they disappear! How can people have rights if they cannot receive the benefits of those rights? (A right is something held by a specific entity.)
It may be that the entity that holds the right is 'the future of humanity', but that seems awfully vague -- particularly since we can't point to any specific entity within this group. And we still face the same dilemma; when we do something to benefit future of humanity x, we immediately destroy it, and create the future of humanity y. For this to work, we have to be working not for a future we expect, but a future we hope for -- a future that doesn't 'exist', even in the vague sense that any future can be said to exist.
It may be that the rights theory of ethics isn't what we should be focusing on in this case. But that just opens more problems; if you are a utilitarian that puts off personal utility to benefit unspecified future generations, wont future generations do exactly the same thing? The future you are trying to benefit, one where people enjoy the fruits of your restraint, will never exist; by endorsing this sort of utilitarianism you guarantee that no one who is moral (in your view) will ever benefit from the moral system. If you follow a duty-based system you might argue that you have a generalized duty to improve the future, with no corresponding right-holders, but without specified right-holders you fall into a new problem. What do you do to make the future 'better' without referring to the people living in it? Don't you have to refer to the people in the future to know what makes a good future? And if you, for example, reduce population growth, aren't you harming all those future people who were never born? We're right back to the beginning.
The term 're-population paradox' is also used to refer to this idea, and for good reason. Most of us understand instantly that this is not true, and that the rights (or something close akin to rights) of future generations are important. Finding a strict description of why they are important, however, is trickier than it looks.
The label 'non-identity problem' comes from Derek Parfit’s book Reasons and Persons (Oxford: Clarendon Press, 1987). The term 're-population paradox' came later, in Thomas Schwartz's paper On the Rights of Future Generations published in Upstream/Downstream: Issues in Environmental Ethics, ed. D. Scherer (Philadelphia: Temple University Press, 1990). While both terms are used, 'non-identity problem' remains the more popular, if non-descriptive, of the two.