A “population axiology” is a way of combining the welfare of many people into a single measure of the welfare of everyone. For instance, adding a numerical measure of everyone’s happiness together is one axiology; averaging their wellbeings is another; summing the monetary wealth of the 1,000,000 poorest people is a third.

Note that two things vary about these population axiologies: what kind of welfare is being measured (“happiness”, “wellbeing”, “monetary wealth”), and how these quantities are combined across the population (“total”, “average”, “sum for the 1,000,000 worst-off people”). Utilitarian ethical theories basically require a population axiology, though there are many non-utilitarian axiologies too.

Ahrrenius’ proof shows that any axiology which satisfies three basic sanity conditions¹ necessarily leads to at least one of three distressing conclusions. Each of these conclusions is deeply contrary to our ethical intuitions. Let us meet the three prongs of the dilemma:

Option one: The Repugnant Conclusion

For any population of very happy people, there exists a much larger population with lives barely worth living that is better than the group of very happy people (according to the population axiology).

Option two: The Sadistic Conclusion

Suppose we start with a population of very happy people. For any proposed addition of a sufficiently large number of almost-as-happy people, there is a small number of horribly tortured people that is (according the population axiology) a preferable addition.

Option three: The Very Anti-Egalitarian Conclusion

For any population of two or more people which has uniform happiness, there exists another population of the same size which has lower total and average happiness, and is less equal, but is better (according to the population axiology).

Utilitarianism needs to do a great deal of wriggling to escape these implications. For instance, some of the STM authors favour variants of utilitarianism that are limited to pre-defined populations: we can say what is best in a world containing Mary, Fred and Jane, but not whether the world is better if Abigail also lives in it. Such positions are coherent, but they are unable to address important ethical questions: Should we have children? Should we try to prevent overpopulation? When is it wrong to abort a fetus?

Topic for discussion: is Arrhenius’ impossibility theorem more serious for the project of using utilitarianism to answer questions of public policy than Arrow’s theorem is for the project of using voting systems to elect politicians?

[1] The basic sanity conditions are almost immune to reasonable disagreement. They are:

• The “dominance condition”: if population A is the same size as population B, and every person in A is happier than their equivalent in B, then A is better than B;
• The “addition principle”: if it is bad to add a group of people B to population A, where the people in B are worse off than those in A, then it is at least as bad to add a group of people C, where C is larger than B and those in C are worse off than those in B.
• The “minimal non-extreme priority principle”: there exists a number of people n such that adding a n extremely well-off/happy people and a single person of slightly negative welfare to an original population A is better than not adding anyone to A. It is notable that the Difference Principle advocated by John Rawls does not in general satisfy this principle and is therefore not subject to the Arrhenius theorem.

Do you favor or oppose the death penalty for persons convicted of murder?

Somewhat favor: 29%
Strongly favor: 36%

The second one found:

Some people say that the death penalty is unfair because most of the people who are executed are African-Americans. Do you favor or oppose the death penalty for persons convicted of murder?

Somewhat favor: 25%
Strongly favor: 52%

Total support jumped 12 points from 65% to 77%, while strong support jumped 16 points (36% to 52%).

• Read the paper

Additional results: Another condition replaced “mostly black” with “many innocents” with no significant effect. 50% of blacks favored the death penalty in the neutral phrasing, 38% in the “mostly black” condition, and 34% in the “many innocents” condition.

The largest branch of consequentialism is utilitarianism. Utilitarians hold that the “best” outcomes are those which are the best for people collectively: “the greatest good for the greatest number”, as Bentham put it.

Utilitarianism calls for two things: altruism and calculation. It tells us, “if you know that the benefit that you would get from this hundred dollar note is less than the benefit that your impoverished friend Susan would get from a second-hand bicycle, you should buy her the bicycle.” And it tells us, “if you know your $100 could save a life in Darfur, you should send it to an humanitarian organisation there instead”. In fact, if lives can be saved for such small amounts, maybe we should be sending more than $100.

A recent study by Deborah Small, George Lowenstein and Paul Slovic demonstrates that, although human beings are capable of altruism, our altruism is in some sense psychologically incompatible with the kind of rational calculation we’d need to perform to be good act-utilitarians.

The experiment by Small et al. shows clearly that human beings² donate significantly more money to help the victims of catastrophes when two conditions hold: (a) the victim is an identifiable individual, rather than an undetermined individual or a large group in need; and (b) the donor is reasoning emotionally.³

When the experimental subjects were told about the human tendency to donate to indentified individuals in need (rather than large groups in need), they stopped reasoning emotionally. That change halved donations to identified individuals, but did not affect the alread-low donations to groups!

When the authors “primed” some experimental subjects with emotion-based tasks (`how does the word “baby” make you feel?’), and others with mathematical tasks, they observed that the emotionally-primed subjects gave twice as much to identified individuals. Both groups gave similar, low amounts to groups in need.

There are some powerful logical arguments in favour of act-utilitarianism and similar ethical positions. But until we find a way to train, trick, or teach ourselves to live by them, these philosophies will remain incomplete.

• Read the article
• Read more about consequentialism and utilitarianism in the Stanford Encyclopedia of Philosophy

Thanks to Toby Ord for suggesting this paper.