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PrometheanFaun comments on A Pure Math Argument for Total Utilitarianism - Less Wrong Discussion
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This is a good point, what I was trying to say is slightly different. Basically, we know that (A < B) ==> (f(A) < f(B)), where f is our order embedding. So it is indeed true that f(A) > f(B) ==> ¬(A < B), by modus tollens.
Yeah, that's a pretty clever way to get around the constraint. I think my claim "If the inequitable society has greater total utility, it must be at least as good as the equitable one" would still hold though, no?
Well... .... yeah, technically. But for example in the model ( worlds={A, B}, f(W)=sum(log(felicity(e)) for e in population(W)) ), such that world A=(2,2,2,2), and world B=(1,1,1,9). f(A) ≥ f(B), IE ¬(f(A) < f(B)), so ¬(A < B), IE, the equitable society is also at least as good as the inequitable, higher sum utility one. So if you want to support all embeddings via summation of an increasing function of the units' QoL.. I'd be surprised if those embeddings had anything in common aside from what the premises required. I suspect anything that agreed with all of them would require all worlds the original premises don't relate to be equal, IE, ¬(A<B) ∧ ¬(B<A).
... looking back, I'm opposed to your implicit definition of a " "baseline" ", the original population partial ordering premises are the baseline, here, not total utilitarianism.