Thanks, yeah I may have gotten slightly confused when writing.
1) VNM
Wikipedia screenshot:
Let P be the thing I said in the post:
If A ≻ B ≻ C, there’s some probability p ∈ (0, 1) where a guaranteed state of the world B is ex ante morally equivalent to “lottery p·A + (1-p)·C”
or, symbolically
P≡(A≻B≻C⇒∃p∈(0,1)[B∼pA+(1−p)C])
and let
Q≡(L≺M≺N⇒∃ε∈(0,1)[(1−ε)L+εN≺M≺εL+(1−ε)N])
I think (P and Independence)⟹Q but notP⟹Q in general.
So my writing was sloppy. Super good catch (not caught by any of the various LLMs iirc!)
But for the purposes of the argument everything holds together because you need independence axiom for VNM to hold. But still, sloppy.
2) “grows without bound” bit
Me: “This arbitrariness diminishes somewhat (though, again, not entirely) when viewed through the asymptotic structure. Once we accept that compensation requirements grow without bound as suffering intensifies, some threshold becomes inevitable. The asymptote must diverge somewhere; debates about exactly where are secondary to recognizing the underlying pattern.”
You:
“Grow without bound” just means that for any M, we have f(X) > M for sufficiently large X. This is different from there being a vertical asymptote so a threshold is not inevitable. For instance one could have f(X) = X or f(X) = X^2.
Straightforward error by me, I will change the wording. Not sure how that happened
3) “continuity”
It would be confusing to call this behavior continuous, because (a) the VNM axiom you reject is called continuity and (b) we are not using any other properties of the extended reals, but we are using real-valued probabilities and x values.
Yeah, idk, English and math only provide so many words. I could have spent more words more driving home and clarifying this point or inventing and defining additional terms. My intuition is that it’s clear enough as is (evidently we disagree about this) but if a couple other people say “yeah this is misleading and confusing” then I’ll concede that I made a bad choice about clarity vs brevity as a writing decision.
4) “write down”, “compute”, and “physically instantiate” till end
Ngl I am pretty confused about everything starting here. I think I’m just reading you wrong somehow. Like the difference in those magnitudes is huge, point taken, but I don’t see why that matters for my argument.
Moving from 10^10^10 to infinity, we would then believe that suffering has a threshold t where t + epsilon intensity suffering cannot be offset by removing t—epsilon intensity suffering
Confused here because yeah clearly adding t+epsilon and removing t-epsilon gives you a net change below zero. But I sense you might be getting at the (very substantive and important) cluster of critiques I respond to in this comment (?)
also need to propose some other mechanism like lexicographic order for how to deal with suffering above the infinite badness threshold.
Yeah I’m ~totally agnostic about this in the post. There are many substantively different possibilities about what the moral world might be like when dealing above that threshold, I agree! Could be distinct levels of lexicality, perhaps some literal integer like 13 levels or perhaps arbitrarily many. Probably other solutions/models as well
Maybe I should just remove/modify the “write down”, “compute”, and “physically instantiate” bit of rhetorical flourish because it might be doing more harm than good.
(Note that it may take me some time to update the post to reflect sections 1 and 2 in this comment)
Thanks, yeah I may have gotten slightly confused when writing.
1) VNM
Wikipedia screenshot:
Let P be the thing I said in the post:
or, symbolically
P≡(A≻B≻C⇒∃p∈(0,1)[B∼pA+(1−p)C])
and let
Q≡(L≺M≺N⇒∃ε∈(0,1)[(1−ε)L+εN≺M≺εL+(1−ε)N])
I think (P and Independence)⟹Q but not P⟹Q in general.
So my writing was sloppy. Super good catch (not caught by any of the various LLMs iirc!)
But for the purposes of the argument everything holds together because you need independence axiom for VNM to hold. But still, sloppy.
2) “grows without bound” bit
Me: “This arbitrariness diminishes somewhat (though, again, not entirely) when viewed through the asymptotic structure. Once we accept that compensation requirements grow without bound as suffering intensifies, some threshold becomes inevitable. The asymptote must diverge somewhere; debates about exactly where are secondary to recognizing the underlying pattern.”
You:
Straightforward error by me, I will change the wording. Not sure how that happened
3) “continuity”
Yeah, idk, English and math only provide so many words. I could have spent more words more driving home and clarifying this point or inventing and defining additional terms. My intuition is that it’s clear enough as is (evidently we disagree about this) but if a couple other people say “yeah this is misleading and confusing” then I’ll concede that I made a bad choice about clarity vs brevity as a writing decision.
4) “write down”, “compute”, and “physically instantiate” till end
Ngl I am pretty confused about everything starting here. I think I’m just reading you wrong somehow. Like the difference in those magnitudes is huge, point taken, but I don’t see why that matters for my argument.
Confused here because yeah clearly adding t+epsilon and removing t-epsilon gives you a net change below zero. But I sense you might be getting at the (very substantive and important) cluster of critiques I respond to in this comment (?)
Yeah I’m ~totally agnostic about this in the post. There are many substantively different possibilities about what the moral world might be like when dealing above that threshold, I agree! Could be distinct levels of lexicality, perhaps some literal integer like 13 levels or perhaps arbitrarily many. Probably other solutions/models as well
Maybe I should just remove/modify the “write down”, “compute”, and “physically instantiate” bit of rhetorical flourish because it might be doing more harm than good.
(Note that it may take me some time to update the post to reflect sections 1 and 2 in this comment)
Again, sharp eye, thanks for the comment!