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DSherron comments on A Pure Math Argument for Total Utilitarianism - Less Wrong Discussion
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I think this is a great question, as people who accept the premises of this article are likely to accept some sort of utilitarianism, so a major result is that average utilitarianism doesn't work.
If we are average utilitarians, then we believe that (2) ~~ (1,2,3). But this must mean that (2,6) ~~ (1,2,3,6) to be order preserving, which is not true. (The former's average utility is 4, the latter's 3.)
...Or you could notice that requiring that order be preserved when you add another member is outright assuming that you care about the total and not about the average. You assume the conclusion as one of your premises, making the argument trivial.