I read the paper and I believe the same objections applies: the reasoning only works if you assume "pain" to be a constant number subject to the additive property.

If we have to use math, I'd say: the headaches are temporal - the effect of all the good you've done today is effectively gone tomorrow one way or another. But killing a person means, to quote "Unforgiven", that "you take away everything he's got and everything he's ever gonna have". So the calculation needs at least a temporal discount factor.

I also believe that the examples are too contrived to be actually useful. Comparing a room with one person to another with five million is like comparing the fine for a person traveling at twice the speed limit with that of someone traveling at 10% the speed of light - the results of such an analysis are entertaining to think about, but not actually useful.

No, that isn't true. We can consider some metric like being at some temperature for an hour. Start with some truly torturous pain like being at 500 degrees for an hour (you'd die quickly, ofc). One person being at 500 degrees is less bad than 10 at 499 degrees which is less bad than 100 at 498 degrees...which is less bad than some number at 85 degrees (not torture, just a bit unpleasant).

I think OP's objection is that - even granting that a "badness value" can be assigned to headaches and that 3 people with headaches is worse than 2 - there's no clear reason to suppose that 3 is exactly half again as bad as 2. It may be that the function mapping headaches to badness is logarithmic, or even that it asymptotes towards some limit. In mathematical terms it can be both monotonic and bounded.

Thus, when comparing headaches to a man being tortured, there's no clear reason to suppose that there is a number of headaches that is worse than the torture.

That's reversed. The number of people can be mapped linearly, but not the intensity of the pain.

(Intuitively, it's hard to say saving 100 people is 100x as good as saving 1, because we can't have 100 best friends, but it doesn't affect the math at all)

Right but Norcross gives an argument against that. You either have to think that slightly reducing pain and making it 1000 times as prevalent doesn't always make a worse state of affairs or deny that worse than is transitive.

What are the arguments for "worse than" being transitive? It's not immediately clear to me that it ought to be.

No; thanks for bringing it to my attention. The first page is intriguing... I'll see if I can locate a free copy somewhere.

Here's a copy I found: https://philosophysmith.com/wp-content/uploads/2018/07/alist...

It's pretty short, I liked it. Was surprised to find myself agreeing with it at the end of my first read.

Thanks. It's very clever how he uses probabilistic variants to move between scenarios. I've read to the end and whilst I'm not convinced, it's definitely given me food for thought. I'm stuck on two bits so far:

* He slides between talking about personal decisions vs decisions about someone else. The argument for Headache is couched in terms of whether an average person would drive to the chemist. Whilst the argument for shifting from Headache to Many Headaches is couched in terms of decisions made by an external party. This feels problematic to me. There may be some workaround.

* He describes rejecting transitivity as being overwhelmingly implausible. Is that obvious? Ethical considerations ultimately boil down to subjective evaluations, and there seems no obvious reason why those evaluations would be transitive.