Several philosophers have proposed “spectrum arguments”, intended to show that the relation “better than” is not generally transitive.Footnote 1 These arguments proceed by describing a sequence of states of affairs, involving two factors that affect a person’s level of well-being. These factors intuitively combine so as to make the first state in the sequence better than the second, the second better than the third, and so on. Nevertheless, the last state in the sequence is, intuitively, better than the first.Footnote 2

A representative and much discussed spectrum argument is Larry Temkin’s “Hangnails” argument, in which the two factors are intensity and duration of pain. Temkin considers a sequence of states A, B, …, Z. State A is two years of excruciating torture, while B is four years of slightly less intense pain. C, in turn, is eight years of pain slightly less intense than that in B. And so on down the alphabet. State Z is millions of years of very mild pain, such as a hangnail. According to Temkin, most would judge B worse than A, C worse than B, and so on down to Z. Transitivity of betterness would then imply that Z is worse than A. Temkin finds it clear, however, that A is worse than Z. Hence, he concludes, transitivity does not hold.Footnote 3

I shall suggest a simple and general objection to spectrum arguments. Suppose that there are n available states of the kind deployed in such arguments. A state could, for example, be to feel pain of a certain intensity for a certain duration. Suppose also that there are n persons who are alike in all well-being relevant respects (maybe they are perfect replicas), and that we must see to it that each person is in a unique state. That is, we must bring about a one-to-one correspondence between persons and states. Let us call such a one-to-one correspondence an “arrangement”.

The following principle seems difficult to deny:

Permutation. All arrangements of the n persons and states are equally good, as concerns general well-being.

Since the n persons are exactly alike, as regards capacity for feeling pain, and so on, it cannot matter, with respect to general well-being, which individual is in which state. A second extremely plausible principle is as follows:

Dominance. If all the n persons are better off in arrangement X than in arrangement Y, then X is better than Y, as concerns general well-being.

No matter what one’s views are about aggregation of well-being, one should accept that making everybody better off is an improvement, with respect to general well-being.

Now consider these two arrangements:

Arrangement 1. Person i is in state i, for all i, 1 ≤ i ≤ n.

Arrangement 2. Person j is in state j – 1, for all j, 2 ≤ j ≤ n. Person 1 is in state n.

Assume, for reductio, that transitivity of betterness is violated with regard to person P and the sequence of states, 1, 2, …, n. Thus, P would be better off in state k than in state k + 1, for every k < n, but better off in state n than in state 1. Given that all the n persons are relevantly like P, it follows that they would all be better off in Arrangement 2 than in Arrangement 1. By Dominance, therefore, Arrangement 2 is better, as concerns general well-being, than Arrangement 1. This contradicts Permutation.

The upshot is that defenders of spectrum arguments against transitivity must reject either Permutation or Dominance. Neither option looks appealing.Footnote 4

To forestall potential objections to Permutation and Dominance, it should be stressed that these principles only concern the comparative goodness of arrangements as concerns general well-being. They do not say anything about the all-things-considered goodness of arrangements. Thus, Permutation does not preclude that a certain arrangement may be all-things-considered better than another, due to factors such as, for example, justice, desert or rights. Similarly, Dominance does not rule out that such factors could make a dominated arrangement all-things-considered better than a dominating one.

In fact, Temkin provides a putative counterexample to a permutation principle about all-things-considered betterness.Footnote 5 The structure of his example is as follows. There are 100 people suffering from a disease. We have a choice between two outcomes, which will to different degrees alleviate each person’s suffering. In Outcome 1, person i will experience i units of suffering, for all i, 1 ≤ i ≤ 100. In Outcome 2, person 1 will experience 100 units of suffering, and person j will experience j – 1 units of suffering, for all j, 2 ≤  j  ≤ 100. Temkin finds “some reason to think” that Outcome 1 is all-things-considered better than Outcome 2.Footnote 6 Person 1 would suffer much more in Outcome 2 than in Outcome 1, whereas the other 99 people would only suffer very slightly more in Outcome 1 than in Outcome 2. Hence, Temkin argues, person 1 may have a claim on us to choose Outcome 1, which is stronger than the combined claims the other 99 people have on us to choose Outcome 2. Outcome 1 can therefore plausibly be regarded as all-things-considered better than Outcome 2.

Even if this conclusion is indeed plausible, Temkin’s example is not a counterexample to Permutation. Claims on agents, of the kind Temkin presumes, are clearly not factors which affect how the outcomes compare in terms of general well-being. In Temkin’s terminology, such claims are “narrow person-affecting” considerations, whereas general well-being is an “impersonal” or a “wide person-affecting” consideration.Footnote 7 And Temkin acknowledges that impersonal and wide person-affecting considerations support the conclusion that Outcomes 1 and 2 are equally good.Footnote 8