Why Classical Utilitarianism is the only (Archimedean) Ethic

Probably the most famous graph in ethics is this one of Parfit's:

He's constructing a series of worlds where each one has more people, but those people have a lower level of welfare. The question is whether the worlds are equivalent, i.e. whether it's equivalent to have a world with a huge number of barely happy people or a world with a small number of ecstatic individuals.

Classical utilitarianism answers "Yes", but some recent attempts to avoid unpleasant results (such as the "repugnant conclusion") have argued "No". For example, Parfit says:
Suppose that I can choose between two futures. I could live for another 100 years, all of an extremely high quality. Call this the Century of Ecstasy. I could instead live for ever, with a life that would always be barely worth living. Though there would be nothing bad in this life, the only good things would be muzak and potatoes. Call this the Drab Eternity. I believe that, of these two, the Century of Ecstasy would give me a better future.

The belief that the "Century of Ecstasy" is superior to the "Drab Eternity", no matter how long that eternity lasts, has been called "Non-Archimedean" by Arrhenius, in reference to the Archimedean Property of numbers, which says roughly that there are no "infinitely large" numbers.1 Specifically, a group is Archimedean if for any $x$ and $y$ there is some $n$ such that $$\underbrace{x+x+\dots+x}_{\text{n times}}>y$$
The following remarkable fact is true:
Classical Utilitarianism is the only Archimedean ethic.
This means that if we don't accept that the briefest instant of a "higher" pleasure is better than the longest eternity of a "lower" pleasure, then we must be classical utilitarians.


First, define the terms. As always, we assume that there is some set $X$ which contains various welfare levels. There is an operation $\oplus$ which combines welfare levels; the statement $x\oplus y=z$ can be read as "A life with welfare $x$ and then welfare $y$ is equivalent to having a life with just welfare $z$."2 It is assumed that this constitutes a group, i.e. the operation is associative and inverses and an identity exist.

In order to make decisions, we need some ranking; the statement $x>y$ means "The welfare level $x$ is morally preferable to $y$." We require $>$ to agree with our operation, i.e. if $x>y$ then $x\oplus c > y\oplus c$ for all $c$.

With the stipulation that our group is Archimedean, this reduces to a theorem of Hölder's, which states that all Archimedean linearly ordered groups are isomorphic to a subgroup of the reals under addition, i.e. classical utilitarianism. The proof is rather involved, but a fairly readable version can be found here.∎


In order to be useful, non-Archimedean theories can't just say that there is some theoretical amount of welfare which is lexically superior - this level of welfare must exist in our day-to-day lives. Personally, when comparing a brief second of happiness on my happiest day to years of moderate happiness, I would choose the years. This leaves me with no choice but to accept classical utilitarianism.

  1. Ethics with this property have also been called "discontinuous" or having a "lexical" priority.
  2. Unlike in past blogs where I used $\oplus$ to be a population ethic, here I define it in terms of intra-personal welfare to fit more in line with Parfit's quote.


  1. It looks to me as if there are some holes here.

    1. You've tacitly assumed that if A and B are just as good as one another in isolation then A+C and B+C are likewise just as good as one another. This seems like an odd assumption to make, unless there's some guarantee that A/B and C are completely isolated from one another; it precludes valuing diversity or fairness or consistency, for instance.

    2. You've assumed that > is a total order, i.e. that you can never have two states of affairs without a definite answer to the question "which, if either, is better?". I don't know about you, but I find that my intuitive moral evaluations often fail to deliver such a definite answer; it seems a shame to forbid this by fiat. (Such a situation doesn't make it impossible to make decisions; it just makes it impossible to make *perfectly satisfying* decisions.)

    3. Your proof delivers a conclusion about how *welfare levels* combine. It doesn't say anything about the relationship between actual states of affairs and welfare levels. Classical utilitarianism involves both of those things; it says not only that one should add utilities, but that utility is (more or less nuancedly) about excess of pleasure over pain, and that everyone's utility should be counted equally. (It's notoriously difficult to say what that last bit actually means, but it seems to me that it has to involve the relationship between utility and the state of the world.)

    4. By couching everything in terms of levels of welfare and preferences between them, you've tacitly assumed consequentialism.

    If I may put it a bit too provocatively, what you've shown is that *if you assume almost all the things generally thought of as distinguishing classical utilitarianism from other ethical systems*, then the only ethic with an archimedean ordered group structure is classical utilitarianism.

    1. Thanks for your comment Gareth. I think you make some good points, and thinking about this has clarified my thoughts as well.

      1. The usual definition of LOGs is if $x_1\leq y_1$ and $x_2\leq y_2$ then $x_1\oplus x_2 \leq y_1 \oplus y_2$ - in plain English "two bad things are worse than two good things." This doesn't seem unreasonable to me; it's basically preventing a form of Arrhenius' "Sadistic Conclusion."

      2. It sounds like you agree $\leq$ is at least a partial order - if we consider an equivalence relation where incomparable states are equivalent, the resulting order over the equivalence classes is total. So I don't see this as a problem.

      3. That is a fair criticism - I call the set "utility" for lack of a better word but it doesn't necessarily have anything to do with happiness. Do you know of a term which means "isomorphic to (R,+)?"

      4. I use terms like "welfare" and "utility" because it's too long-winded to say "however you want to describe a state of being." The remarkable fact is that even if you have a very complicated way of describing a state of being (e.g. a long vector of goods), that way is equivalent to using just a single real number (assuming it's an Archimedean group).

      Obviously the only (epistemically closed) people who will accept the assumptions are also those who accept the conclusion (assuming I didn't make a mistake!) so any proof is somewhat begging the question. But I find the result surprising, and given Holder's theorem's importance in algebra, I think others find it surprising as well.

    2. 1. I like to eat chocolate cake. Other things being equal, at any time t I would prefer "eats chocolate cake at t" over "doesn't eat chocolate cake at t". However, I do not prefer "eats chocolate cake nonstop all day" over "eats no chocolate cake for a day". In other words, there's a bunch of x's (no cake at time t) and a bunch of y's (cake at time t) such that individually we have xi < yi, but sum(xi) > sum(yi).

      I think medicine is very important. It's a very good thing that there are doctors. On the other hand, we probably have too many lawyers. So for each person capable of being a doctor or lawyer (I think the required mental characteristics are fairly similar), all else being equal I would prefer that they be a doctor. But I wouldn't want a world in which there are no lawyers at all. Again: xi < yi but sum(xi) > sum(yi).

      You can get around this by saying that your combination and comparison operations apply only to welfare levels and not to events or states of the world, but then it's totally unclear what the combination operation is actually supposed to mean. To get any conclusion at all like "classical utilitarianism is correct" you need x*y to mean something like "x for one person, y for another" or "x first, then y", but this is exactly the sort of thing for which examples like the ones above show that the combination and comparison operations aren't compatible.

      2. In general you can't usefully turn a partial order into a total order by collapsing incomparable elements. For instance, take X = (real numbers) u {*} where * is incomparable to everything other than itself. Then the only equivalence relation on X that has the incomparability relation as a subset is the one that makes everything equivalent to everything else. Now, for sure the induced ordering on the quotient is total -- but only because it's trivial!

      3. Generally, if I want to say "isomorphic to (R,+)" I just say it :-).

      4. I do agree that Hölder's theorem is striking. The only things I'm not convinced of are (a) that it really tells us anything non-question-begging about ethics and (b) that what it tells us, if it does, can accurately be described as "that classical utilitarianism is correct".

      (My own ethical system is pretty close to classical utilitarianism, by the way; I'm not saying all this because I dislike your conclusion. I just don't think it follows.)