Thomas Kwa🔹 comments on Utilitarians Should Accept that Some Suffering Cannot be “Offset”

[Thomas Kwa🔹]

There are a few mistakes/​gaps in the quantitative claims:

Continuity: 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” (i.e., p chance of state of the world A, and the rest of the probability mass of C)

This is not quite the same as either property 3 or property 3′ in the Wikipedia article, and it’s plausible but unclear to me that you can prove 3′ from it. Property 3 uses “p ∈ [0, 1]” and 3′ has an inequality; it seems like the argument still goes through with 3′ so I’d switch to that, but then you should also say why 3 is unintuitive to you because VNM only requires 3 OR 3′.

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.

“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.

To be clear, whether we call this behavior ‘continuous’ depends on mathematical context and convention. In standard calculus, a function that approaches infinity exhibits an infinite discontinuity. [...]

[1] In the extended reals with appropriate topology, such a function can be rigorously called left-continuous.

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.

Once you’ve accepted that some suffering might require a number of flourishing lives that you could not write down, compute, or physically instantiate to morally justify, at least in principle, the additional step to “infinite” is smaller in some important conceptual sense than it might seem prima facie.

This may seem like a nitpick, but “write down”, “compute”, and “physically instantiate” are wildly different ranges of numbers. The largest number one could “physically instantiate” is something like 10^50 minds, the most one could “write down” the digits of is something like 10^10^10.

Not all large numbers are the same here, because if one thinks the offset ratio for a cluster headache is in the 10^50 range, there are only 50 ‘levels’ of suffering each of which is 10x worse than the last. If it’s over 10^10^10, there are over 10 billion such ‘levels’, it would be impossible to rate cluster headaches on a logarithmic pain scale, and we would happily give everyone on Earth (say) a level 10,000,000,000 cluster headache to prevent one person from having a (slightly worse than average) level 10,000,000,010 cluster headache. 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, and also need to propose some other mechanism like lexicographic order for how to deal with suffering above the infinite badness threshold.

So it’s already a huge step to reject numbers we can “physically instantiate” to ones we can barely “write down”, and another step from there to infinity; at both steps your treatment of comparisons between different suffering intensities changes significantly, even in thought experiments without an unphysically large number of beings.

[Aaron Bergman]

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

$P\equiv (A\succ B\succ C\Rightarrow \exists p\in (0,1)[B\sim pA+(1-p\left)C\right])$

and let

$Q\equiv (L\prec M\prec N\Rightarrow \exists \epsilon \in (0,1)\left[\right(1-\epsilon )L+\epsilon N\prec M\prec \epsilon L+(1-\epsilon \left)N\right])$

I think $\left(P\text{and Independence}\right)⟹Q$ 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:

“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)

Again, sharp eye, thanks for the comment!