I was recently flattered to learn that John Broome had not only read a recent article of mine ("Headaches, Lives, and Value" in Utilitas from 2009, I think), but had also published a reply in Utilitas for the current issue. In the spirit of dialog, I thought I might try out a response to his critique here. Warning: this may be interesting only for those who have looked at my article or looked at Broome's critique. That category may well be empty.
In that article, I try to argue that we should accept lexical priority relations for two reasons, chief amongst them being the following argument:
1. A headache is bad
2. Bads can be aggregated across persons to form worse bads. (Aggregation)
3. For every bad x, there is a bad of lesser weight y, enough of which will outweigh the disvalue of x. (Continuity)
4. If A is better than B, and B is better than C, A is better than C. (Transitivity)
Hence, Lives for Headaches: There is some number of headaches such that the relief of those headaches is sufficient to outweigh the good life of an innocent person.
I claim this argument is valid, and that premises 1, 2, and 4, are solid. If, like me, you don't like the conclusion you should reject 3, and accept lexical priorities.
Broome objects, and claims that this argument is in fact invalid, and that this is recognizable by "anyone with a little knowledge of mathematical analysis." He shows this by considering the following principle of aggregation. Assuming that a single instance of death is badness 10, and a single headache is badness 1, we aggregate bads in the following way: the badness of n people's suffering bad of type t (Bnt) = the badness of 1 person's suffering bad of type t (B1t) multiplied by (2-1/n).
Put as Broome puts it: Bnt=B1t(2-1/n)
In other words, for each additional headache, the marginal badness of that headache is not as much as the previous headache (this is assuming that these headaches are equally bad for their possessors). (The badness of 1 headache is 1, of two headaches is 1.5, of three headaches is 1.66, of four headaches is 1.75, and on and on.) When adding these headaches up, we reach an asymptotic limit: the limit is always double the value of a single person's suffering of badness of type t. No amount of headaches will be more than badness 2, insufficient to outweigh the badness of a single death.
On this principle of aggregation, we can say that headaches are bad, we can say that more of them is worse than fewer, we can say that for every single instance of a death (B1death=10), enough of the next-worse bad (maimings, say: B1maiming=9, asymptote = 18) will outweigh a single instance of death, and that transitivity holds. So the argument as I present it is invalid.
This is a cool response, but I don't think it entirely works. One problem is really my fault. Premise 3 is ambiguous. Premise 3 was not intended to claim that for any bad type x, B1x could be outweighed by Bny, where B1x>B1y. Rather, that for any bad type x, Bnx can be outweighed by some instance of Bny, where B1x>B1y. If that's right, Broome's aggregative principle doesn't render my argument invalid. But my bad on that one.
Of course, I do claim that the argument is valid. But that's a bit cheeky. Even on my interpretation of (3), the argument is obviously valid only on reasonable background principles about aggregation. For instance, one could render my argument wholly invalid by accepting the following headache-aggregative-principle (HAP): two headaches are worse than one, but more than two headaches are never worse than two. If we accept HAP, we can accept (1)-(4), and reject the conclusion. So I'm assuming, in the background, that silly principles like HAP aren't in play.
I think Broome's principle is unreasonable (indeed, I note this of a similar principle in the article, p. 53). Imagine what we must say to accept it. First, we must say that every additional headache, say, in a collection of headaches is worth less than the headache that came before, even if they are, e.g., of equivalent pain, etc. But this is quite implausible. Why should there be not just a diminishing marginal utility of, say, resources or instrumental goods, but a diminishing marginal utility of utility?
Second, take the badness of death. Bndeath, for any value of n, must always be less than 20. But that means that the badness of a single death (10), is more than half as bad as the deaths of all humans past present and future. That sounds implausible to me.
Third, imagine two alternatives. First, you could eliminate the human species entirely, present and future. Second, you could kill one person, give one person a very bad chronic illness, give one person a broken leg, and one person a headache. How do we evaluate these alternatives on Broome's principle? Well, the badness of the first option is slightly less than 20, insofar as the aggregative limit of death is 20. But, if we assume, say that B1illness=7, and B1brokenleg=3, and B1headache=1, the collective badness of these four individual bads (one death+one chronic illness+one broken leg+one headache) is 21, worse than the total elimination of the human species. (Thanks to Ben Eggleston.)
So I think I'm on solid ground ignoring Broome's principle, just as I'm on solid ground ignoring HAP. On standard assumptions about aggregation, (1)-(4) imply Lives for Headaches. And that's all I really wanted to claim.
Does this make sense to y'all? Do let me know if your reading of Broome is different than mine, or if I've made any mistakes in the math.