Suppose that the world contains an infinite row of people, whom we can (if we don't mind doing such a thing at least in a thought experiment) number in order ...,-4,-3,-2,-1,0,1,2,3,4,.... All of these people are the same in all morally relevant, with one exception. The folks with negative numbers are all very miserable, with an equal amount of misery, and the folks with non-negative numbers are all blissfully happy, with an equal amount of happiness. A reliable genie offers you a choice: If you raise your left hand, person with number -1 will be made blissfully happy, like the people with numbers 0,1,2,3,4,...; if you don't raise your right hand, person number 0 will be made as miserable as the people with negative numbers.
What should you do? It's clear: lift your left hand. You clearly have decisive reason to do this. But notice that total utility need not be changed by your action (assume for simplicity your own and the genie's utility is not changed). In fact, the situation where persons numbered ...,-4,-3,-2 are miserable and those numbered -1,0,1,2,3,4,... are blissfully happy is isomorphic to the situation where those numbered ...,-4,-3,-2,-1,0 are miserable and those numbered 1,2,3,4,... are blissfully happy. So on utilitarian grounds, there is nothing to choose from between these two options.
Someone whose ethics is not centered on the maximization of utility will notice that even though the total utility in both cases is the same (whatever it is: it seems to be infinity minus infinity!), there is a difference for two specific people, namely those numbered -1 and 0. This is yet another way in which personal identity matters. Unless persons have an identity over time or between worlds (or both), we have a hard time making sense of the difference the two cases. Utilitarianism does not particularly care about the identities of persons, and that's why it has trouble with this case.
Utilitarianism can perhaps be fixed to account for this. One might supplement it with the idea that when comparing utilities between possible outcomes, we only compute differences in utility. When choosing between options A and B, we let u(x,F) be the utility that possible person x has if F is chosen, and then sum up u(x,A)-u(x,B) over the union of the possible persons in the relevant A-world and the relevant B-world. Notice, though, that looking at it this way emphasize the importance of personal identity between worlds—it matters which goods and bads befall whom. Once we agree that it matters which goods and bads befall whom, utilitarianism should seem significantly less plausible. And we may still be able to manufacture counterexamples. Suppose the genie adds that however you choose, an infinite number of equally blissful genies causally isolated from everybody else, will pop into existence, but these genies will be numerically different in the scenario where you lift your left hand from the ones who pop into existence in the scenario where you don't lift your left hand. Then, the above utility difference method will generate infinite minus infinity as the difference between the two scenarios, which doesn't allow for a decision.
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