What goes wrong if we try to use lexical totalism again to avoid your new theorem? You can capture lexicality with a function taking values only in the real numbers, no vectors or anything.
Basically, you just need the maximum difference in the slighter values (which you use l to denote) to never exceed a finite sure difference in the higher value (which you use h to denote). But you can squash the whole real line into a finite interval with a function like arctan. Consider capturing lexical totalism with the function f:Z×R→R defined by
f(h,l)=h+1πarctan(l),
where you sum h=∑ihi, hi∈Z and l=∑ili,li∈R across individuals/instances before applying f, and then take the expected value of f for ordering prospects.
By using the integers for the h values, they’re spaced out enough that any sure difference in them will always dominate any difference in l, since the range of 1πarctan has length 1. If I understood correctly, this function should also satisfy your risky versions of general non-extreme priority and non-elitism, since for a fixed difference in h, letting the probability of that difference go to 0 makes the difference to the expected value of f go to 0, and so can be outweighed by a finite difference in l. f should also satisfy all of the other exact conditions, since it’s the same as lexical totalism in the exact cases.
Thanks! This is a really cool idea and I’ll have to think more about it. What I’ll say now is that I think your version of lexical totalism violates RGNEP and RNE. That’s because of the order in which I have the quantifiers. I say, ‘there exists p such that for any k...’. I think your lexical totalism only satisfies weaker versions of RGNEP and RNE with the quantifiers the other way around: ‘for any k, there exists p...’.
Hmm, and the population X also comes after, rather than having the m,p,k possibly depend on X. It does look like your conditions are more “uniform” than my proposal might satisfy, i.e. you get existential quantifiers before universal quantifiers, rather than existential quantifiers all last (compare continuity vs uniform continuity, and convergence of a sequence of functions vs uniform convergence). The original GNEP and NE axioms have some uniformity, too.
I think informal explanations of the axioms often don’t get this uniformity across, but that suggests to me that the uniformity itself is not so intuitive and compelling at all in the first place, and it’s doing a lot of the work in these theorems. Especially when the conditions are uniform in the unaffected background population X, i.e. you require the existence of an object that works for all X, that seems to strongly favour separability/additivity/the independence of unconcerned agents, which of course favours totalism.
Uniformity also came up here, with respect to Minimal Tradeoffs.
Yes, that all sounds right to me. Thanks for the tip about uniformity and fanaticism! Uniformity also comes up here, in the distinction between the Quantity Condition and the Trade-Off Condition.
What goes wrong if we try to use lexical totalism again to avoid your new theorem? You can capture lexicality with a function taking values only in the real numbers, no vectors or anything.
Basically, you just need the maximum difference in the slighter values (which you use l to denote) to never exceed a finite sure difference in the higher value (which you use h to denote). But you can squash the whole real line into a finite interval with a function like arctan. Consider capturing lexical totalism with the function f:Z×R→R defined by
f(h,l)=h+1πarctan(l),where you sum h=∑ihi, hi∈Z and l=∑ili,li∈R across individuals/instances before applying f, and then take the expected value of f for ordering prospects.
By using the integers for the h values, they’re spaced out enough that any sure difference in them will always dominate any difference in l, since the range of 1πarctan has length 1. If I understood correctly, this function should also satisfy your risky versions of general non-extreme priority and non-elitism, since for a fixed difference in h, letting the probability of that difference go to 0 makes the difference to the expected value of f go to 0, and so can be outweighed by a finite difference in l. f should also satisfy all of the other exact conditions, since it’s the same as lexical totalism in the exact cases.
I discuss this kind of thing more here.
Thanks! This is a really cool idea and I’ll have to think more about it. What I’ll say now is that I think your version of lexical totalism violates RGNEP and RNE. That’s because of the order in which I have the quantifiers. I say, ‘there exists p such that for any k...’. I think your lexical totalism only satisfies weaker versions of RGNEP and RNE with the quantifiers the other way around: ‘for any k, there exists p...’.
Hmm, and the population X also comes after, rather than having the m,p,k possibly depend on X. It does look like your conditions are more “uniform” than my proposal might satisfy, i.e. you get existential quantifiers before universal quantifiers, rather than existential quantifiers all last (compare continuity vs uniform continuity, and convergence of a sequence of functions vs uniform convergence). The original GNEP and NE axioms have some uniformity, too.
I think informal explanations of the axioms often don’t get this uniformity across, but that suggests to me that the uniformity itself is not so intuitive and compelling at all in the first place, and it’s doing a lot of the work in these theorems. Especially when the conditions are uniform in the unaffected background population X, i.e. you require the existence of an object that works for all X, that seems to strongly favour separability/additivity/the independence of unconcerned agents, which of course favours totalism.
Uniformity also came up here, with respect to Minimal Tradeoffs.
Yes, that all sounds right to me. Thanks for the tip about uniformity and fanaticism! Uniformity also comes up here, in the distinction between the Quantity Condition and the Trade-Off Condition.