Say that we have two goods X and Y. We write if X >> Y if any (positive) amount of good X is better than any (positive) amount of good Y. For example, a speciesist might hold that humans >> chickens, meaning that increasing human welfare by any minuscule amount is more important than massive increases to chicken welfare.

If a set has no X and Y such that X >> Y we say that it is Archimedean. Otherwise, the set is Non-Archimedean.

Brian's latest post argues (roughly):

1. Archimedean ethics are preferred because they have fewer unintuitive consequences
2. Hedonistic utilitarianism seems more likely than Preference utilitarianism to be Archimedean
3. Therefore, we should be hedonistic utilitarians

I'm not as convinced about (1) as Brian is, but let's accept it and focus on (2). It doesn't seem to me more obvious that happiness is Archimedean than preferences.

For example, consider how bad it is to get poked vs. how bad it is to have your leg broken. Many people feel that it's better to get poked an unbounded number of times than to have your leg broken even once. So poking << leg being broken, which means that happiness is non-Archimedean.

There are a lot of other ways of testing your intuitions on this, for example with Temkin's "Spectrum Arguments" where you have something like:

1. 1 person being tortured for 10 years
2. 2 people being tortured for 9 years
3. 4 people being tortured for 8.1 years
4. ...

Most people think that (A) is better than (B) which in turn is better than (C) etc. But eventually you end up with some state (Z) where a huge number of people are being tortured for a microsecond, which people think is better than (A). One way of getting around this apparently cyclic preference is to claim that you never in fact reach (Z) - because pain is non-Archimedean, the years of torture are "infinitely" worse than seconds of torture, so a countable sequence can never reach (Z).

I actually think these arguments are wrong, that happiness is Archimedean, but they seem just as convincing to me as Brian's arguments that preferences are non-Archimedean. I will leave you with a quote from J. S. Mill describing his (non-Archimedean) "Hierarchy of Pleasures":

It is better to be a human being dissatisfied than a pig satisfied; better to be Socrates dissatisfied than a fool satisfied.

Most people think that (A) is better than (B) which in turn is better than (C) etc.

This reasoning is non-utilitarian, though, because it is not actually maximizing happiness (and minimizing suffering). Instead, it's just bad scope insensitivity.

Thanks for the post, Ben!

xodarap wrote:Brian's latest post argues (roughly)

Most of my essay is actually defending preference utilitarianism, but I did note this kind of idea in a throwaway comment.

xodarap wrote:Many people feel that it's better to get poked an unbounded number of times than to have your leg broken even once. So poking << leg being broken, which means that happiness is non-Archimedean.

To the contrary, I think people say they prefer to be poked unboundedly many times than break a leg, but if you asked them to assign actual suffering numbers to the two events, they wouldn't assign negative infinity to breaking a leg or zero to poking. (Ok, poking may be a bad example, because it's not clear that's negative at all and could even be positive. Pinprick might be better.)

xodarap wrote:Most people think that (A) is better than (B) which in turn is better than (C) etc. But eventually you end up with some state (Z) where a huge number of people are being tortured for a microsecond, which people think is better than (A).

My personal opinion is that it doesn't matter who's being tortured -- a microsecond of torture is equally bad regardless. What does matter is whether it's torture or a pinprick. I'm not sure pinpricks can outweigh torture, though I don't have a settled opinion, and probably enough of them could. Anyway, the good news is this doesn't matter, because for any numerical tradeoffs we might encounter in real life, there wouldn't be enough pinpricks to outweigh torture.