Reply to Broome

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.

79 Replies to “Reply to Broome

  1. I found this very interesting, but had a bit of a problem following it.
    I don’t have access to the article, but I’m pretty sure that you can’t have written down Broome’s formula correctly. You say Bnt=B1t(2-1/B1t). But surely the formula for Bnt must have an n in there somewhere. This is also true of your verbal explanation of Broome’s account.
    If Broome takes this approach to aggregating the same kind of bad over different people, would he aggregate different kinds of bads in the way you suggest in objection three? I suspect that there may be a Broome style way of aggregating different kinds of bads that does not mean that a death, a chronic illness, a broken leg and a headache are worse than the destruction of the human race.
    Of course, this comment does not touch your other two objections.

  2. “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?”
    But Broome isn’t claiming diminishing marginal (dis)utility of (dis)utility. Rather, he’s claiming diminishing marginal disutility of headaches. Granted, each additional headache must be of the same precise type as the others. But to say that two headaches are of the same type doesn’t commit one to the claim that they are of the same disutility. They might instead be of the same type when, for example, they involve the same number of firing c fibers.
    It’s not implausible to maintain that headaches of this latter type have diminishing marginal utility when the headaches are each experienced by the same person. This person might, after all, become more accustomed to, and less bothered by, such headaches.
    When, however, the additional headaches of this type are experienced by different people, I’m with you, Dale, in seeing no grounds for claiming that their marginal utility diminishes. Here’s why (and here I quote from fn. 31 of an article I wrote called ‘Saving Lives, Moral Theory, and the Claims of Individuals’):
    “Unlike an intrapersonal case, where there may be a rationale for discounting the value of additional benefits of a given specific type (e.g., a given pleasurable sensation) when the same person receives this benefit a repeated number of times, there is no rationale for discounting the value of additional benefits of a given specific type to additional people. It is not as if a given benefit such as a pleasurable sensation is of less value to a person on account of the high number of others who have received this benefit. Hence we cannot claim that the same type of small benefit to infinitely many people may sum to a finite number because of the diminishing marginal utility of benefits to additional people. Rather, these benefits to different people are all of equal utility, and even very small benefits of the same positive value to an infinite number of people will sum to infinity.”

  3. To show that Broome’s idea lacks significance in this context, you need not show that there is no respect in which there is diminishing marginal utility of headaches in interpersonal cases. There may be some respect in which there is. You only need to show that there is a quality that headaches have that is not subject to diminishing marginal utility. For example, headaches may be painful and frightening. Plausibly, the more people that suffer headaches the less frightening they become. In other words, there may be an interpersonal equivalent of Mike’s idea that a person can be come moral accustomed to headaches. But badness of the painfulness of headaches is not plausibly subject to interpersonal diminishing marginal utility and that is enough to show that Broome’s argument lacks application in this context.

  4. Larry Temkin and Binmore and Voorehoeve have a very similar exchange about the intrapersonal case. Larry has premises like yours, but involving a single person with a given intensity of pain lasting a given amount of time, and claims that those premises are inconsistent. Then Binmore and Voorehoeve make essentially the point that Broome does, showing that the principles are not inconsistent. (The papers Binmore and Voorhoeve “Defending Transitivity against Zeno’s Paradox”, Philosophy and Public Affairs, Vol. 31, No. 3 (Summer, 2003), pp. 272-279; and Temkin,”A Continuum Argument for Intransitivity,” Philosophy & Public Affairs 25 (1996): 175-210). I take it that the main issues are the same in these two cases.
    Like you, I find their counterexample to express a very implausible view. It would be nice to show that all counterexamples to the inconsistency of these principles express an implausible view. You can do this by minimally strengthening #3 (you can e-mail me if you’re curious about the details).
    For what it’s worth, I find it more odd to reject 3 than to accept that enough headaches can be worse than death. To reject 3 involves claiming the following sort of thing:
    (Implausible Claim): There is some headaches such that if some number (say 1 million) suffered from that headache, it would be better for any number of people (including 10 trillion, or 10^80) to suffer from a headache that was only barely noticeably less bad.
    While I can see a rationale for the counter-intuitive claim that enough headaches is worse than death, and why one might be forced to believe it, I find the above violation inexplicable and bizarre. It might not strike the man on the street with as much surprise, but I think that’s just because the claim is a little more complicated. (Note that the Implausible Claim is the key to identifying what many regard as the most implausible consequences of maximin.)

  5. Hi all. Thanks for the comments, which I take to be generally encouraging. Just a few brief comments.
    @Fiona – O my gosh! Yes, you’re right. I screwed that one up. I’ve now changed the post to reflect the right answer. But I’m not sure how Broome would be aggregating different bads in a way that could avoid the conclusion here; I suppose I don’t want to say it’s impossible, but in the article he seems to suggest that there is a diminishing marginal disutility of single instances of a single bad, and hence single bads aggregate simply by addition, which would leave open that problem. But I’d be happy to consider alternatives.
    @Mike – I myself find the intrapersonal case a little hard to swallow. If it’s the same amount of pain, I don’t see that the nth headache is any less bad than the first. But this is, I think, a tangential issue insofar as this argument is explicitly pitched at the interpersonal case, and hence Broome’s account must be offering an interpersonal aggregative principle rather than an intrapersonal aggregative principle. Which, as you rightly say (by the way, thanks for that citation, I’ll be sure to note it!), is wild.
    @Victor – We have to assume for this argument that a single headache for one person generates the same disutility as a single headache for another person. I treat headaches as simply shorthand for “a minor amount of disutility n”, or something like that. And you’re right that in that case, Broome’s principle seems odd.
    @Nick – Broome accuses me in the article of not heeding the lessons of Binmore and Voohoeve, but I think the issues are totally distinct, and the distinction is between the intrapersonal case and the interpersonal case. I think it’s implausible in both cases, but even if we accept it in the intrapersonal case (as B&V discuss), we should reject it in the interpersonal case for the reasons noted here (and by Mike, for instance). So the relatedness of these issues is superficial only. BTW, I agree with you that Implausible Claim is, will, implausible, but I don’t think I have to accept it to deny 3. All that’s required is that there is some Bnx that cannot be outweighed by any Bny. Broome’s own principle denies this, and I explore a different way to do it in the article.

  6. I guess I find it plausible that if there were many more Van Gogh paintings of irises that eventually the value of the next one would be less than the value of the first. But I don’t find it very plausible that the value of different people’s headaches works like that. Part of what is good about paintings is there being a new kind of way of seeing things or having our attention drawn to a new corner of the world. What is bad about headaches is the way it feels and the badness of that does not seem to diminish as more people get them. While it does not sound odd at all to say that as lots of other van Gogh paintings are destroyed that the value (not monetary worth) of this one increases, it sounds incredibly odd to me to say that as Headache Nation is wiped out by a asteroid that the disvalue of my headache increases. Admittedly there is no argument there. I am just wondering if people really have a different intuition in such cases.

  7. @Dale – I don’t think Mike’s point gets the heart of the issue. Sure, you could get used to the pain. But you could tweak the example in various ways that get around his complaint. Examples:
    1. Suppose the person’s memory of the last hour is wiped every hour, and they are subjected to the same pain each hour.
    2. Imagine the person just doesn’t get used to the pain.
    3. Imagine that the pain is increased at a rate that exactly offsets the extent to which they get used to the pain.
    The basic issue here is about aggregation of badness at time for a person and duration into total badness for a person. This issue is formally very similar to aggregation of badness for a person and the number of people affected into total badness. Mike’s point doesn’t show otherwise.
    The reason I said you were committed to the Implausible Claim was this. Let’s think of headache intensity as the result of aggregating pain of the headache at a time with the duration of the headache. The following propositions are inconsistent:
    1*. A sufficiently intense headache (such as one causing great pain for your whole life) would be worse than death.
    2*. For any number of people and any headache intensity, a much larger number of people suffering a slightly less bad headache would be worse. (Denial of Implausible Claim)
    3*. No number of headaches is worse than death.
    4*. Better than is transitive.
    5*. Any headache, no matter how intense, can be transformed into a very mild headache by slightly decreasing its intensity a finite number of times.
    You seem to accept 3* and 4*. I thought 1* and 5* were obvious. So I assumed you were going to reject 2*, and thereby accept the Implausible Claim. But I should admit that there are other ways out here.

  8. Non-linear principles of aggregation raise an interesting problem of retributive justice. Suppose my inane comments cause headaches throughout the PEASoup community, where the badness of one headache is justly recompensed by $1. Now, if I have to pay you all back individually, I am potentially out a lot of money. On the other hand, if I can contrive to pay back “the community” in aggregate, I can just pay $2 and get change.

  9. Hi Nick –
    At heart, I agree with you about the intrapersonal case. But I just wanted to note that even if we accepted that equivalent pains are of non-equivalent intrapersonal disutility, that doesn’t say anything about how we aggregate pains of equivalent disutility interpersonally. I was, and still am, assuming that the headaches in (1) and the conclusion are of equivalent intrapersonal disutility.
    I see! Yes, I do accept Implausible Claim! I was thinking you had a different principle in mind (i.e., a minor headache versus a slightly more minor headache). But in the article (esp. against a challenge from Norcross) I argue that we should accept this sort of a claim in some cases. So you got me.

  10. @David: I think we agree, here.
    @Heath: That’s an interesting case! I think this is additional evidence that Broome’s purported counterexample is a non-starter.

  11. Nick —
    I’m puzzled by your remarks on my remarks. What ‘complaint’ of mine have you got in mind that is got around by introducing assumptions that render the marginal disutility of headaches nondiminishing in an intrapersonal context? And what is ‘the issue’ whose ‘heart’ I have failed to arrive at?
    I took my complaint to be that Dale was being uncharitable to Broome in maintaining that he was assuming that (dis)utility has diminishing marginal (dis)utility. I said that Broome was instead assuming that headaches have diminishing marginal disutility. I then went on to note that, while headaches suffered by a single person might have diminishing marginal disutility because this person might become accustomed to them, I couldn’t see grounds for affirming the diminishing marginal disutility of headaches suffered by different people. In other words, I was saying that Broome wasn’t incoherently assuming that disutility has diminishing marginal disutility. Rather, he was making an assumption that was plausible in certain intrapersonal contexts but implausible in the interpersonal context of Dale’s argument.

  12. Dale: yes, I should have said that I took myself to be in agreement with you and wondering if others actually disagreed with the thought as I described it.
    I take Mike to be right that Broome is not best thought of as arguing for the diminishing marginal utility of utility but rather, perhaps inplausibly, arguing for the diminishing marginal utility of headaches across persons.

  13. David and Mike –
    You’re right that that claim was a little strong. But the way I’d put his claim is that disutility (i.e., a pain of a certain intensity and duration) has diminishing marginal interpersonal badness. Which I take, given your suggestions, to be very implausible for the reasons offered (though, perhaps, not literally incoherent).

  14. Dale, I think I don’t understand premise 3 now. You tried to clarify it, but it’s written in the funny notation with some things that look like variables embedded in character strings, and those are hard to understand. (Like, what is the ‘n’ in your revised version, where you’ve italicized it? A variable? Is it a bound variable?)
    Let me say more generally why I’m a little puzzled by the original argument. (I haven’t read your paper.) Why isn’t this a counterexample to the argument?

    J1. Any death can be outweighed by a huge number of very slightly less bad deaths.
    J2. But no death can ever be outweighed by huge numbers of headaches.

    Because of the first, premise 3 is true. But because of the second, the conclusion is false.
    Are the other premises supposed to rule this out, or what?

  15. I don’t think Broome intends to actually advance the claim that headaches have diminishing marginal interpersonal badness. From what I know of his work**, he would deny this. The point of Broome’s paper, on one reading, is merely to show that 1-4 and the denial of “Conclusion” in Dale’s paper are not strictly inconsistent. This is due to — as Broome calls it in the context of Temkin’s version of the argument — a “technical hitch” (*Weighing Lives*, p. 55). The claim that some counterexample provides *a* way to avoid inconsistency in the face of 1-4 and not-Conclusion is distinct from the claim that it provides a *plausible* way to do so. I’d be inclined to read Broome, in his response to Dale, as only registering the former point. Once this technical difficulty is fixed (presumably by suitably revising 3 — cf. Erik Carlson’s response to Binmore and Voorehoeve on behalf of Temkin in his “Intransitivity without Zeno’s Paradox”) so that 1-4 and not-Conclusion are strictly inconsistent, Broome’s move would be to deny not-Conclusion, or to accept the Conclusion. Dale’s move, I take it, would be to deny the revised 3.
    ** For example, in a paper called “All Goods are Relevant” Broome writes,
    “I believe that a lot of small benefits can add up to be as important as one large benefit. When a patient in a United Kingdom hospital gets a headache, he or she is given an analgesic. Over a few years, the UK Health Service gives out a few million analgesics to cure headaches. The cost of all these pills adds up, and eventually it will amount to more than enough to save someone’s life. Evidently, the Health Service thinks that curing all those headaches is as valuable as saving a life. I agree.”
    He makes similar remarks elsewhere.

  16. @Jamie – Sorry about the obscure nomenclature. This is what I have in mind. Broome’s interpretation of Premise 3 holds that *one* instance of any bad x can be outweighed by enough of a lesser bad y. My version of premise three claims that *any amount* of any bad x can be outweighed by enough of a lesser bad y. So, on Broome’s view, it’s true that 1 death can be outweighed by enough of a lesser harm. But it’s also true that there is some number of deaths that couldn’t be outweighed by a lesser harm, in denial of P3.
    WRT your argument, the key is the “for every bad x” clause in P3. Assuming that there are only three bads, deaths1, deaths2, and headaches, Premise 3 implies that any amount of deaths1 could be outweighed by some amount of deaths2. But it also implies that any amount of deaths2 can be outweighed by some amount of headaches. By transitivity, this entails that some amount of headaches outweighs a death1.
    @Theron: I think my Premise 3 as I wanted to understand it is sufficient to rule out Broome’s claim. But also, there’s a funny thing here. Broome claims that “the primary function of continuity principles is to rule out lexical orderings” (I’m paraphrasing from p. 28 of Weighing Lives). But if he wishes to accept a genuine continuity principle, he can’t claim that no amount of headaches will trade-off against a death, insofar as this just is a lexical ordering.
    And you’re right that he doesn’t advance this in his own voice. But he also claims that “no argument for discontinuity has emerged”. But this is very uncharitable, as I’ve shown here.

  17. @Mike – I didn’t mean to be combative by using the word “complaint”. Dale was using a claim of yours, I assume the claim that having a headache might get less and less bad for you over time, to attempt to draw a disanalogy between aggregating harms to one person over time and aggregating harms to different people. The ground for thinking that the headaches might get less and less bad was that you would (could?) get used to them. As I understand Dale’s use of your point, this was supposed to illustrate a fundamental difference between the two kinds of aggregation. I identified possible cases where you don’t get used to the headache, so it remains equally bad for you over time. (By this, I mean that it illustrated some difference that would not allow us to carry over the lessons from the Temkin/Binmore and Voorhoeve exchange.) Since this is possible, the point about headaches getting less bad for you over time does not illustrate a fundamental difference between the two kinds of aggregation. (We can, after all, consider the alternate cases.) When I said that your point about headaches didn’t get to the heart of the matter, this is what I had in mind. It would have been better to say that Dale’s use of your remarks didn’t get to the heart of the matter. Hope that clears things up.

  18. The above comment got mixed up. It should read:
    @Mike – I didn’t mean to be combative by using the word “complaint”. Dale was using a claim of yours, I assume the claim that having a headache might get less and less bad for you over time, to attempt to draw a disanalogy between aggregating harms to one person over time and aggregating harms to different people. The ground for thinking that the headaches might get less and less bad was that you would (could?) get used to them. As I understand Dale’s use of your point, this was supposed to illustrate a fundamental difference between the two kinds of aggregation. I identified possible cases where you don’t get used to the headache, so it remains equally bad for you over time. Since this is possible, the point about headaches getting less bad for you over time does not illustrate a fundamental difference between the two kinds of aggregation. (By this, I mean that it did not illustrate some difference that would not allow us to carry over the lessons from the Temkin/Binmore and Voorhoeve exchange. We can, after all, consider the alternate cases.) When I said that your point about headaches didn’t get to the heart of the matter, this is what I had in mind. It would have been better to say that Dale’s use of your remarks didn’t get to the heart of the matter. Hope that clears things up.

  19. Dale: I agree that 3, on the revised reasonable interpretation of it (which you likely had in mind all along), rules out the claim that 1-4 and not-Conclusion are consistent. But 3, as stated in the paper, leaves room for other interpretations. You could read Broome as simply saying “please make 3 more precise, as one interpretation of it, as it is here formulated, would render 1-4 and not-Conclusion consistent.”
    Interesting point about Broome’s use of “continuity”. Against you, he claims that continuity is strictly consistent with lexical orderings (e.g., not-Conclusion). But you point out that in WL he claims that continuity rules out lexical orderings. How to resolve this putative tension?
    Perhaps one possibility is that there are different senses of “continuity” and “lexical priority/dominance” that could be in play here.
    Here’s how Broome defines lexical dominance: “One value dominates another lexically if *any* increase in the quantity of value, *however little*, is better than some particular increase in the quantity of another value” (p. 24, emphasis added). This is distinct from (more extreme than) the claim that there is some finite amount of B that is better than any amount of A. The latter is inconsistent with what Broome identifies as Griffin’s definition of continuity, “enough A outranks any amount of B.” Broome then proceeds to indicate that discontinuity in Griffin’s sense does not imply what he calls “mathematical discontinuity” (roughly, “an ordering is [mathematically] continuous if and only if, whenever some distribution A is better than another B, then any distribution that is very similar to A is also better than B, and any distribution that is very similar to B is also worse than A” pp. 27-8).
    Suppose one says that X amount of B is better than any amount of A. Discontinuity in Griffin’s sense (there is no amount of A that could outrank some amount of B) is consistent with mathematical continuity insofar as an amount of B “very similar” to X cannot be outranked by any amount of A.
    In other words, the claim you point to on p.28 is that mathematical continuity rules out lexical dominance (in the extreme sense defined on p. 24), but it does not rule out weaker forms of lexical dominance, e.g., that some finite (perhaps large) amount of B is better than any amount of A. He might then interpret your not-Conclusion (i.e., “Lives for Headaches”) as a weak lexical dominance view (like Griffin’s), rather than a strong/strict one, as def. on p. 24. (For instance, you might claim that a life of quality Q is better than preventing any number of headaches. But a “very similar” life — of only slightly different quality — is also better than preventing any number of headaches, insofar as mathematical continuity holds).
    So, while your argument may put pressure on a stronger sense of continuity (given by the reasonable interpretation of your premise 3), it does not challenge the more uncontroversial notion of mathematical continuity (given by another interpretation of your premise 3).
    Does this help clear up what Broome might have in mind? Hopefully I’m not getting you and/or Broome wrong.

  20. Nick — Thanks. That clears things up.
    Theron @ 2:05 pm — Your interpretation of Broome strikes me as right. He’s not endorsing the claim that the disutility of headaches is diminishing but simply assuming this for the sake of showing that Dale’s argument (on Broome’s understanding of its premises) isn’t valid. (I was careless in my first post in saying that Broome claims that the marginal disutility of headaches diminishes and was more careful in my second post to label this an assumption.)

  21. Dale,
    Okay, I understand premise 3 now.
    I don’t think your conclusion does follow from the premises. In your response to me, you said, “Assuming that there are only three bads…”. With that assumption, I agree that your conclusion follows. But the conclusion doesn’t follow without the assumption, and, obviously, the assumption is false.
    So, I’m thinking you must be appealing to a tacit premise.

  22. Dale and Jamie,
    I don’t know how helpful this will be, but here is how I’d imagine Erik Carlson would deal with Broome’s objection, if he were in Dale’s shoes. (Based on the paper of his I cited previously).
    Dale’s initial formulation of P3 reads, “For every bad X, there is a bad of lesser weight Y, enough of which will outweigh the disvalue of X.”
    We could interpret this as either P3* or P3**:
    P3*: “For every bad X, there is a bad Y less weighty than X by a (small) degree E, enough of which will outweigh the disvalue of X.”
    P3**: “There is (small) degree E such that for every bad X there is another bad Y only E less weighty than it, and enough of Y will outweigh the disvalue of X.” (Technical point: we might qualify this to exclude the “least weighty bad” — a bad so mild that something E less weighty than it is not actually a bad).
    On P3*, Dale’s argument is invalid, for the reason Broome (following Binmore and Voorehoeve) provided in his paper. As Broome puts it, “Starting from the death of an innocent person, Premise 3 tells us there is an infinite sequence of bads, each less bad than the one before, such that enough of each will outweigh the previous bad in the sequence. But the premises do not entail that this sequence will get as far as the slight bad of a headache.” This is because P3* is consistent with E getting smaller and smaller. Similarly, if I stand 10 feet from a wall, take a step that moves me one foot closer to the wall, another step toward the wall half as long at the previous step toward the wall, another one half that, etc., an infinite number of such steps won’t get me to the wall.
    On P3**, Dale’s argument is valid. P3** guarantees that there the relevant (long) sequence of bads that will eventually get down to that of a headache. If the degree E is the same each time, we can eventually move from a great bad to a slight bad (just as if each time I step closer to the wall, I move only 1 millimeter closer, I will nonetheless eventually reach it).
    Dale: I assume you had something like P3** in mind all along. Am I right? Jamie: does the version of the argument with P3** strike you as valid?
    If it is valid, then in order to avoid inconsistency, we are forced to abandon 1, 2, 3**, 4, or not-Conclusion. There is no reason offered to abandon 3*. And though giving up 3** is one way to avoid inconsistency, my own view is that this is not the best way to do so. But I guess that’s another topic.

  23. @Theron – I agree that there’s a distinction to be made between these types of superiority (I call these “Really Strong” and “Pretty Strong” in the article). And you’re right that Broome’s version of continuity rules out only Really Strong. But this doesn’t help. On his suggested aggregative principle, any life saved (i.e., any increase in quantity of lives saved) is better than any amount of headaches relieved (i.e., any increase in headaches relieved). That’s a Really Strong relation, which is rule out on Broome’s account of continuity. Gustaf Arrhenius makes the point I’m making: a sequence of goods that bear the Pretty Strong relation to each other, after a sufficient number, the first and last good will display the Really Strong relation. So Broome’s principle is pretty clearly incompatible with his own account of continuity, let alone mine.
    Incidentally, mine rules out both forms of superiority, insofar as any amount of any particular good must be outweighed by some amount of a lesser good. To claim that enough of A is better than any amount of B, it is not the case that for any amount of A (nA) some amount of B (mB) is better than nA. This holds across the spectrum, too. You might claim that enough of A (nA) is always better than any amount of C. But then (by Premise 3) you must claim that there is some middle-good (B) such that enough of B (mB) is better than nA. But if mB is better than nA, and no amount of C is better than nA, then it must be that no amount of C is better than mB. Again, denying Premise 3.
    @Jamie – I’m a bit confused. I hope that’s not true!! Anyway, here’s why I think it isn’t true. Premise three says that for any amount of a bad A (nA), there is some amount of a lesser bad B (nB) such that nB is worse than nA. So let’s say there are a lot of bads. A is death, Z is a headache. For every nA, there must be some lesser good B such that some amount (mB) is worse than nA. So now there’s got to be some C such that n^xC is worse than mB, and so on. Eventually you’re either going to find that some n^z amount of headaches are going to be, by transitivity, better than nA, or you’re going to find some bad less significant than a headache, some amount of which is worse than a death. But if that’s right, then surely some amount of a worse bad will also be worse than a death. Have I screwed up somewhere? Not sure where, if so. But at this point that’s just a hope!

  24. @Theron. I’m a little loopy, so forgive me for any rambling. For starters, I agree that we should accept your technical point (I hope this is mostly obvious).
    Yeah, if that’s Broome’s interpretation of P3*, it works. But I’m not sure why I have to posit any sort of continuum or any value of E. Premise 3 only says that, however many goods there are, and however more or less significant the are compared to one another, for every amount of a weightier one, there is some amount of a less weighty one that is better than the weightier one. It doesn’t say anything, as far as I can tell, about the comparative significance of these bads. Premise 3 rejects both Really and Pretty Strong superiority, and hence is incompatible with Broome’s view.
    Have I missed something?

  25. Dale: good points on how to classify your take of Lives For Headaches in terms of Really/Pretty Strong Superiority, and Arrhenius on these types of Superiority. What you say here leaves me a bit perplexed (as you perhaps are!) about Broome’s usage(s) of “continuity” in WL and in his response to you. I’ll let you know if somehow I become un-perplexed. And sorry if I made you more perplexed.
    Here’s a different interpretive story regarding Broome and “continuity”: he took *your* label for P3 (continuity), interpreted P3 as P3*, showed P3* to be consistent with 1, 2, 4, and not-Conclusion, and then concluded that you’ve given no reason to deny P3* (aka “continuity”). It is true that rejecting P3* expresses “a sort of discontinuity in value” and that “no argument for [this sort of discontinuity] has emerged.” Moreover, Broome does not make any reference, in his response to you, to any of his own technical definitions of continuity (e.g., those in WL).
    Your latest formulation of P3 goes, “however many goods there are, and however more or less significant they are compared to one another, for every amount of a weightier one, there is some amount of a less weighty one that is better than the weightier one.” But this is still technically consistent with P3*. You want to phrase P3 so that it is clearly not consistent with P3*. You do not have to set a specific level of E to do this; the point is that if (not only if) E is a constant, you’ll have a valid argument. (An aside: presumably P3** is more irresistible, the smaller E is and the larger the number of smaller bads in each step). In contrast, if the degree to which to the difference in badness of adjacent bads is allowed to vary (as it is in P3*), then this is consistent with this degree getting smaller and smaller, falling prey to the Broome/Binmore/Voorehoeve/Zeno worry.

  26. Here’s one further point regarding continuity. Even if the view underlying Broome’s counterexample to your argument (assuming the P3* reading of P3) is not in accord his views about continuity or aggregation, I thought we agreed that he’s not actually espousing these counterexamples as plausible ones (see my first post and Mike’s last post). Remember, Broome himself accepts the Conclusion (and all of your premises). He’s just offering a technical loophole that renders P1, P2, P3*, P4, and not-Conclusion consistent, without presupposing any particular view (including his own) in doing so.

  27. Also, I misspoke. I said, “You want to phrase P3 so that it is clearly not consistent with P3*.” I meant, “You want to phrase P3 so that it is clearly distinct from P3*.”
    Sorry about all the posts.

  28. Dale,

    Premise three says that for any amount of a bad A (nA), there is some amount of a lesser bad B (nB) such that nB is worse than nA. So let’s say there are a lot of bads. A is death, Z is a headache. For every nA, there must be some lesser good B such that some amount (mB) is worse than nA. So now there’s got to be some C such that n^xC is worse than mB, and so on. Eventually you’re either going to find that some n^z amount of headaches are going to be, by transitivity, better than nA, or you’re going to find some bad less significant than a headache, some amount of which is worse than a death.

    I think everything is true up to the last sentence. (I’m not 100% sure about this, because you don’t say what ‘x’ is, and don’t see what the exponentiation is doing in the expressions.) The (now bolded) sentence starting with “Eventually” does not follow from what came before it.
    I think maybe Theron has now explained this well enough. But Theron, I would prefer not to formulate the principle in terms of degrees of good and bad. For one thing, there might be lots of different ways of imposing a measure on good, and it could turn out that on some of them P3** is true but not on others.

  29. Dale, you say:

    I myself find the intrapersonal case a little hard to swallow. If it’s the same amount of pain, I don’t see that the nth headache is any less bad than the first.

    But the principle suggested by Broome, as I understand it, doesn’t entail that the nth headache is less bad than the first. It does entail that n headaches are less than n times as bad as 1 headache (where n > 1). But that seems a different proposition.

  30. Broome on lexical orderings
    Dale writes: “On [Broome’s] suggested aggregative principle, any life saved (i.e., any increase in quantity of lives saved) is better than any amount of headaches relieved (i.e., any increase in headaches relieved).” Dale maintains that this, however, is a lexical ordering, which Broome claims to be ruled out by a continuity principle.
    But Broome would not, I think, describe this as a lexical ordering. Here’s why. Broome writes: “As I use the term, one value dominates another lexically if any increase in the quantity of the one value, however little, is better than some particular increase in the quantity of the other value.” (Weighing Lives, pp. 23-24) Broome immediately goes on to say the following:
    “In this case, the lexical view [under discussion] is that extending a person’s life by any length of time, however short, is always better than improving a person’s life by some particular amount, without lengthening it. … A discrete-time model cannot accommodate this view, because it cannot recognize any extension to life that is shorter than its quantum of time.”
    Broome would, I think, maintain, more generally, that any value that is quantifiable only in discrete units cannot lexically dominate any other value.
    Since, moreover, the value of lives saved is quantifiable only in discrete units (nothing smaller than the quantum of 1 life saved), this value cannot lexically dominate any other value.

  31. Campbell – If the nth headache in a temporal sequence is never less bad than the first, then the badness of a given headache one suffers in a temporal sequence will always vary depending on the number of other headaches of the exact same type one will go on to suffer. That strikes me odder than what struck Dale as odd. “How bad was your headache today?” “I’ll have to wait and see. If I don’t go on to suffer the same type of headache in the future, then it was pretty bad. But if it recurs day after day, then it wasn’t so bad!”

  32. I think Theron has very helpfully shown that even if we interpret premise 3 of Dale’s argument in the manner that Dale had in mind (i.e., “*any* amount of any bad x can be outweighed by enough of a lesser bad y”) rather than in the way that Broome interpreted it (“*one* instance of any bad x can be outweighed by enough of a lesser bad y”), Dale’s argument can be shown to be invalid by offering a different token of the very same Zeno-esque type of counterexample that Broome constructed.
    Based on his original post, it appears that Dale’s response to this difficulty would be to claim that any such counterexample will have to assume an unreasonable principle of aggregation. This claim, however, stands in need of argumentation, which hasn’t yet been provided, as far as I can tell.
    Another response, much more effective, in my mind, is to follow Erik Carlson’s lead by replacing Dale’s Premise 3 with Theron’s Premise 3**.

  33. Mike, I believe the ‘bad’ in “How bad was your headache?” is not the same bad as the one that Broome, Campbell, etc., are talking about. To see this, notice that it would be equally inappropriate to answer, “I’m not sure, because it’s possible that Buddhism is correct and pain is not bad at all.”

  34. Mike,

    If the nth headache in a temporal sequence is never less bad than the first, then the badness of a given headache one suffers in a temporal sequence will always vary depending on the number of other headaches of the exact same type one will go on to suffer.

    Can you explain why you think this is true? Consider two temporal sequences of headaches: (a1, a2) and (b1, b2, b3). Suppose that a1 and a2 are equal in badness; and that b1, b2, and b3 are equal in badness. Are you claiming that, say, a2 and b2 must therefore differ in badness? I don’t see how that follows.

  35. @Theron – As long as somewhere along the line, there’s some next-worse bad, i.e., so long as the continuum is broken at least somewhere, Broome’s aggregative suggestion is incompatible with P3. If we allow that, say, deaths are badness 10, and the next worse bad is 9, then there is some amount of deaths that cannot be outweighed by the next-worse bad, denying P3, even if there is a continuum of bads below 9. Now, such a continuum might also yield that the argument doesn’t follow from my P3. (See comment to Jamie.) But my only point was that any break in the chain yields that Broome’s suggestion must reject P3.
    Re: continuity, see my comment to Mike, below.
    @Jamie – I went with exponentials just because I wanted to note that these were *really big numbers*. Sorry about the confusion. But I should note that the truth or falsity of P3 is neither here nor there. It’s whether P3 is acceptable on Broome’s own proposal; that’s the only claim I wanted to make. Regarding the validity of the argument, I now see the problem, I think: I need to rule out the total-continuum-of-bads hypothesis as well as weird aggregative principles. Incidentally, I argue against this hypothesis in the paper, but my level of success is up for grabs, I guess. But you’re right that this should be noted in the arg itself for its validity. (BTW, Broome says that P3 implies the total continuum of bads hypothesis, but I don’t see that. My P3 seems totally neutral as it stands WRT the continuum hypo. But you rightly note that its neutrality is a problem.)
    @Campbell – could I say a “less significant marginal contribution to the aggregate”? That’s all I really meant (but thanks for noting the distinction). Ultimately I think this is about which principle we use to aggregate headaches and treat headaches as single. Looking back on a life with one headache per day, we could either say that a person had a single headache every day, which would entail a significant amount of badness. Or we could say that the person had an aggregate x number of headaches, in which case it wasn’t so bad: less than 2. Which do we use? Is there a principled reason to go one way rather than the other?
    @Mike – Right on both counts! Re: lexical. Don’t know how I missed that. I read “any increase in the quantity of one value however small” to mean “the smallest possible increase” rather than “increase of a magnitude of the smallest possible positive real number”. For what it’s worth, I take my account to be basically standard in these discussions. But I guess that’s neither here nor there. You’ve convinced me that Broome and I have a different understanding of continuity; on my account his principle doesn’t work, but on his it does. Thanks for that.
    Re: Zeno. See above comment to Jamie.

  36. Campbell,
    I was assuming, as Broome was, that the badness of headaches is 2 – 1/n.
    If, for whatever number of headaches one ends up experiencing, the nth in that sequence is never less bad than the first, and therefore the headaches one experiences in this sequence are equally bad, then:
    If one ends up suffering a sequence of two equally bad headaches, the badness of each will be:
    (2 – .5)/2 = .75
    If one ends up suffering a sequence of three equally bad headaches, the badness of each will be:
    (2 – .3333333)/3 = .55
    And therefore the second in each of these sequences will differ in value.
    Does what I said now seem right to you? Or have I screwed up somewhere?

  37. could I say a “less significant marginal contribution to the aggregate”?

    Suppose H is a (finite) set of headaches of which h is a member. Then h’s marginal contribution to the aggregate value of H, I assume, may then be defined as the value of H minus the value of H\{h}. Broome’s principle, I think, implies that every headache in H makes an equal marginal contribution, so defined.

  38. Oh, well, I only meant that I don’t think the word ‘bad’ means the same thing in each. And then I gave an example that’s supposed to show that it couldn’t mean the same.
    I think in the question (“How bad was it?”) one might ask about someone’s headache, ‘bad’ just means ‘painful’. But there is no question of whether six headaches had by six people are six times as painful as one headache. So that’s not what ‘bad’ means in the question about aggregation. So, when you pointed out that the answer to the question you imagined was a very odd answer, that was not relevant to what Campbell was suggesting. He was suggesting something about a different sense of ‘bad’. (I would say it’s the badness of a state of affairs, and what is contributed to it by various constituents, like headaches.)

  39. Mike, it seems that you’re assuming a sort of additivity: the aggregate badness of some headaches must equal the sum of the individual badness of these headaches. I would reject that assumption.

  40. @Campbell – Aha!! Yes, I see now. To make my claim one would have to assume that, for any given headache, the marginal contribution to the aggregate is constant no matter the number of headaches. Broome’s view is still possible on this assumption, but the marginal contribution of additional headaches goes down. But if you reject that, the “unfairness”, I suppose, disappears. That still leaves the other problems, tho.
    Thanks for all of your terrific comments. You’re really helping me refine my reply to Broome.

  41. Campbell, you’re right that I was assuming additivity. I think this assumption makes sense in the case of the aggregate badness of headaches, since I don’t see reason to be holistic rather than more atomistic about the badness of sets of headaches. In response to Jamie, I also think the badness of sets of headaches is derivable from the painfulness of individual headaches in a manner that is more straightforward than is captured by a badness function that would justify an ‘I’ll have to wait and see’ answer to the ‘How bad was your headache question?’. (Here I set aside bad future causal effects of a headache that might contribute to the badness of that headache.) I grant that these are just vague assertions of how things strike me.

  42. @ Mike (5:33am) – Thanks for the helpful clarification about Broome on lexical orderings. For what it’s worth, I agree with your “vague assertions” too (9:10am).
    @ Dale (7:09am) – I don’t see how Broome’s counterexample is inconsistent with P3. P3 states, “For every bad x, there is a bad of lesser weight y, enough of which will outweigh the disvalue of x.”
    You write, “As long as somewhere along the line, there’s some next-worse bad, i.e., so long as the continuum is broken at least somewhere, Broome’s aggregative suggestion is incompatible with P3. If we allow that, say, deaths are badness 10, and the next worse bad is 9, then there is some amount of deaths that cannot be outweighed by the next-worse bad, denying P3, even if there is a continuum of bads below 9.”
    But the fact that there is no number of bad-9s that can outweigh some number of bad-10s is perfectly consistent with P3. There may be some bad of lesser weight (bad-9) than another bad (bad-10) enough of which will outweigh the greater bad. But P3 only says there’s *a* bad of lesser weight enough of which will outweigh a greater bad; it does not say that enough of *any* bad of any lesser weight can outweigh it.
    Maybe you do have a way of showing Broome’s counterexample to be ruled out by your P3, but I don’t think I’ve seen it (or at least, appreciated it) yet. Sorry if I missed it.

  43. @Theron – I’m confused by your penultimate para. You say seem to agree that “there is no number of bad-9s that can outweigh some number of bad-10s”. But in your very next sentence you say “There may be some bad of lesser weight (bad-9) than another bad (bad-10) enough of which will outweigh the greater bad.” (Can I assume you’re missing a “not” between “may” and “be”?)
    Try this on for size. You’re right that P3 says that there’s *a* bad of lesser weight, not *any* bad of lesser weight. But if we assume that there is no bad between 10 and 9, i.e., the continuum is broken there (as I was assuming), there will be no amount of bad-9s that could outweigh some number of bad-10s (indeed, the number is 5; 5 deaths = badness 18, and the aggregative asymptote of bad-9s is 18). Furthermore, because any lesser bad than bad-9 will have an asymptote below 18, no amount of any lesser good, either, could outweigh that amount of bad-10s.
    I feel like we’re talking past each other. Remember that I’m assuming that the next worse-bad is 9. But this problem holds so long as the next-worse bad is not simply the next on a continuum.

  44. Dale: first, yes, sorry about the typo.
    Second, I did indeed fail to appreciate what was meant by “there’s some next-worse bad, i.e., so long as the continuum is broken at least somewhere.” Sorry. If there is a break in the continuum such that, e.g., “there is no bad between 10 and 9” and a lexical priority relationship holds between bad-10 and bad-9, then bad-10 will be an example of a bad such that there is *no* bad of lesser weight than it, enough of which would outweigh it. Hence inconsistent with P3.
    But I do not see why Broome must be committed to jumps in the continuum of that sort. Compare this with Binmore and Voorehoeve’s original point about Zeno’s paradox. They do not need to assume that there are gaps in the intensity of pains in order to provide a counterexample to Temkin. (Take a look at their graph on p. 277 of their article “Defending Transitivity against Zeno’s Paradox”). Analogously, I don’t see why Broome should have to be committed to gaps in the “intensity/weightiness” of bads in order to provide a counterexample to you (again, assuming a P3*-ish reading of your P3).

  45. You’re right that he doesn’t have to be committed to jumps in the continuum. In fact, he has to be committed to no such jumps. I was assuming (which has now been brought out as an eminently challengeable assumption) that there were gaps in the continuum, especially given my argument of the paper. (The response I sent you discusses this in fn10.) But we’ve now got an email conversation going, so I’ll say more there.

  46. Dale,
    thanks for the neat exchange with Broome. Just wanted to make a quick point about the implausibility of Broome’s aggregative principle. Now, it might have been that he just used that principle to show that you need further premises in the argument. As someone pointed out, this might make it slightly unfair to object to his principle on the grounds that it is not a plausible principle more generally.
    The interesting question is, however, could there be a principle or a set of principles that gave the intuitive results in all the aggregation cases that didn’t rely on lexical priorities? Here the project is slightly different. We begin from our intuitive judgments, and we then try build the aggregation principles to fit them in order to consequentialize our common sense morality. So far, I’m not convinced that this could not be done. You are right that Broome’s principle won’t probably work for our death intuitions. But, the consequence of this is just that deaths do not aggregate in the same ways as headaches. Mixed cases like your last one will of course pose even further problems. However, all of this just seems to be a reason to find more sophisticated aggregative principles rather than to think that no principles are possible at all. So, I guess it might turn out to be quite a bit more difficult to show that we should adopt lexical priorities.

  47. Hi Jussi –
    Sorry for the wildly delayed response. I think plausibility is of the essence. First, it does seem to me that, charitably speaking anyway, I should be granted some license to assume away implausible principles (like HAP). If someone thinks this principle is more plausible than I do, I’d certainly be happy to be convinced otherwise. But more importantly, Broome claims that “no argument for discontinuity has emerged”. But if the only alternative to LFH is some form of asymptotic view (whether horizontal (Broome) or vertical (Zeno)), then it seems to me the argument remains, because these are so implausible even if he’s got me nailed that the argument is strictly speaking invalid without a further premise: 4.5. Asymptotic views fail, or something of that sort.
    I don’t know what to say about the second suggestion. I look forward to seeing it played out in more detail. I worry about the suggestion that deaths don’t aggregate like headaches, but I’d be happy to take a wait-and-see approach.

  48. Hi Dale,
    thanks for the reply. I think I’m with Broome here – sorry about this. Recall that it’s you who are giving an argument for discontinuity. For that argument to be successful, you’ll need to show that all potential sets of asymptotic aggregation principles would have some counter-intuitive consequences. Unfortunately, I don’t quite see that anything like has been established. So, I don’t yet see an argument that would remain.
    Here’s a thought for the death aggregation – each death is just as bad. That’s all we need to deal with your two cases. So the combination of Broome’s dummy principle for headaches and standard summing for deaths already gives the right results in all your cases. Now, you might find some counter-examples to this combination and as a result I’d need to make them more sophisticated. But, nothing seems to show that in principle there wouldn’t be a set of principles without lexical priorities to give the intuitive results. So, I think that the ball is pretty much in your court at the moment.

  49. Let me suggest an argument for the view that badness of headaches is non-linear, i.e. for rejecting this view:
    (LH) For any positive numbers n and m, the aggregate badness of n headaches is n/m times as great as the aggregate badness of m headaches.
    LH implies, e.g., that two headaches are twice as bad as 1, and two thirds as bad as three, and so on.
    What does it mean to say that X is n/m times as bad as Y? Here’s one thing it might mean. Suppose that Y is worse (more bad) than X, and X is worse than N, where N is some neutral event (neither bad nor good). Then X is n/m times as bad as Y iff a lottery which has a n/m chance of Y and a 1-(n/m) chance of N is equal in badness with X. Assume this is correct.
    Now, consider two options:
    (A) Seven people suffer headaches.
    (B) I toss a fair coin three times. If it lands heads every time, then no one suffers a headache. Otherwise, eight people suffer headaches.
    Assuming that the event of zero people suffering headaches is neutral, LH implies that A and B are equally bad. But I’m tempted to say that A is worse. If so, this suggests that headaches have diminishing marginal badness.

  50. Here’s another argument against linear badness of headaches. You might find this one more compelling.
    Consider two options:
    (C) I will toss a fair coin repeatedly until it comes up heads. If I toss the coin 100 times (i.e. 100 tails in a row), then one person dies. Otherwise, some neutral event happens.
    (D) One person suffers a headache.
    I say that C is better than D. This means that one headache is more than 1/(2^100) times as bad as one death (assuming the definition of ratios of badness I gave above). If badness of headaches is linear, then this implies that 2^100 headaches are collectively worse than one death.
    So I’m inclined to think that you can’t have it both ways. Either you reject linear badness of headaches (as in Broome’s suggestion), or you accept what you call ‘Lives for Headaches’.

  51. Campbell (9:17am):
    I do not share your temptation to say that A is worse than B. I think presenting the probability of no one suffering headaches (versus 8 suffering headaches) alongside B may help reduce that temptation, though maybe not.
    But I could imagine revised cases (say, where B* involves a very low probability of many headaches being realized) where my intuitions might say that A is worse.
    Still, I don’t think that LH has to be committed to the claim “X is n/m times as bad as Y iff a lottery which has a n/m chance of Y and a 1-(n/m) chance of N is equal in badness with X.” One could accept LH as a claim about badness per se, but reject standard expected utility theory about expected good/bad. So even if you were right that A is worse than B, this would not suggest that headaches have diminishing marginal badness, though it might suggest some claim about diminishing marginal expected badness. Of course, we’d have to look at more cases before concluding that.
    Campbell (9:42am):
    You write, “I say that C is better than D. This means that one headache is more than 1/(2^100) times as bad as one death (assuming the definition of ratios of badness I gave above). If badness of headaches is linear, then this implies that 2^100 headaches are collectively worse than one death.”
    But we can say that C is better than D because we think that lives get lexical priority over headache prevention. Hence, claiming this does not mean that “one headache is more than 1/(2^100) times as bad as one death” and we can avoid the implication that “2^100 headaches are collectively worse than one death” even if we accept that the badness of headaches is linear.
    In other words, LH plus lexical priorities about lives/headaches would say: the badness of headaches increases linearly (and so approaches infinity as the number of headaches indefinitely increases), but no amount of *this kind of badness* could be worse than the loss of a good life.
    You’re of course right that accepting Lives for Headaches is also an option. And I’ve not here claimed that it’s a worse option than accepting lexical priorities. But I think retaining LH and combining it with lexical priorities is a better way of avoiding Lives for Headaches than your suggestion that we reject LH.

  52. I’m not sure I understand all this now. Perhaps we need to know more about which qualities of bads tend to result in diminishing marginal utilities and which do not.
    The problem of aggregation will arise if there is some bad, B, such that
    1) B is trivially bad for the person who suffers it.
    2) Badness overall increases at the same rate as the number of people who suffer B.
    For in that case if n is very large badness overall is very large, and then it is difficult to see why the badness of very many people suffering B cannot be greater than the badness of a single death.
    Now we need to know whether there are any examples of B. I am not clear about how we should establish that there are. Headaches seem a good example, but could you say any more about why? Perhaps it’s because the badness of an headache is in the experience of it. My headache is experienced only by me, and yours only by you. There may be no way for the experiences to affect each other in the interpersonal case (say if we are unaware of each other’s existence). This is less common in the intrapersonal case (where the experience of the second headache is affected by the experience of the first).
    But there are some cases where we do not aggregate value where there is no interaction. Compare
    World 1: there are 10 people with identical lives on this world who never meet each other.
    World 2: there are 20 people with identical lives on this world who never meet each other.
    It may be that World 2 is < twice as good as World 1. The value of each life increases the closer to extinction we become. Is there anything similar to say in the case of headaches? I doubt it. We might think that the more people that have them, the more they become part of the human condition, diminishing their badness. But that sounds fishy.

  53. Campbell,
    I’m sorry. I misread your (9:42am) post. Disregard my comment, “But we can say that C is better than D because we think that lives get lexical priority over headache prevention.”
    In response to your (9:42am) post, I should instead repeat the point about badness per se versus expected badness (as I said in reply to your 9:17am post). This blocks your inference to the claim that “one headache is more than 1/(2^100) times as bad as one death.”
    Another option, in addition to rejecting LH (as you do) or accepting LH and lives for headaches (as Broome does) is to deny that C is better than D. This seems to me quite controversial.

  54. Theron,
    I don’t understand this thing you said:

    But we can say that C is better than D because we think that lives get lexical priority over headache prevention.

    C is better with respect to headache prevention (it prevents one headache), and D is better with respect to saving lives (it saves someone a very small chance of dying). So how does the view that ‘lives get lexical priority over headache prevention’ support the conclusion that C is better than D?

  55. Theron,
    Sorry, I missed your last reply before posting mine.
    Can you tell me what you mean by ‘badness per se’? The question of linearity depends on ratios of badness. Personally, I’m sceptical about taking such ratios as primitive, undefined. One way to define them is in terms of expected badness of lotteries, as I did above. If you reject this definition, then what would you replace it with? If you can’t define ratios, then I’m not sure I understand what you mean when you say badness is linear.

  56. @Jussi – I take my argument to have generalized to all asymptotic principles, unless you’re willing to defend the claim that one headache is more than half as bad as the aggregate of all headaches ever experienced, and that four headaches are better than a collection of bads q, r, s, t, such that each is slightly less significant than a headache, and q>r>s>t, (where “>” means “slightly more significant than”) which is strange, but also seems to reject sensible Pareto principles.
    @Victor – I’m a bit confused by your question. I’m sure it’s me not you.
    @Theron and Campbell – What Theron said. As I argue at length in the paper, value is pluralist. There are “project-like” goods and “satisfaction-like” goods. The aggregation of each is linear. But one then defines a betterness ordering between them which holds that no improvement of the latter could outweigh an improvement of the former. (Incidentally, I’ve altered my view on this in the subsequent years. I prefer a slightly more complex betterness ordering, but that’s neither here nor there for this discussion.) That requires a reinterpretation of expected utility theory, but we should welcome this, insofar as any theory of rational choice is neither plausible nor implausible without a background axiology in place. I’m not sure I see the last question tho. Are you asking for a semantic analysis?

  57. Campbell,
    I was using ‘badness per se’ to refer to the badness of some outcome, as opposed to the (expected) badness of a prospect. But I am not sure I can tell you what it *means* for an outcome to be bad per se. (Though of course we can offer debatable first-order substantive accounts of badness).
    LH says, “For any positive numbers n and m, the aggregate badness of n headaches is n/m times as great as the aggregate badness of m headaches.”
    This implies that more (equally intense) headaches is always worse than fewer. But LH does more than *order* headaches in this way. It says things like two headaches is *twice* as bad as one headache. This, you claim, should not be taken as primitive, but defined somehow (I am inclined to agree). You offer one definition in terms of expected badness of lotteries. (I take it that this is a fairly common definition, one that many economists, including Broome throughout much of his career, accept).
    Another way would be to introduce bads (headaches) of varying degrees of weightiness or intensity. Then we could say things like: one headache of intensity 10 is just as bad as two headaches of intensity 5. (Note that I’m using ‘intensity’ to refer to *how bad* a headache is, which may or may not coincide with the *descriptive* or psychological intensity of a headache. Perhaps ‘severity’ is more appropriate than ‘intensity’ to describe the badness of a headache.)
    Introducing this other (clearly relevant) dimension of the badness of headaches gives meaning to claims like LH without appealing to lotteries. There are presumably other dimensions of value (besides severity and number of headaches, and besides lotteries) that yield quantitative scales of goodness (as opposed to merely ordinal ones). I don’t see why we must be wedded to lotteries to do this. All we need to give meaning to quantities of goodness are multiple dimensions of goodness, which can be weighed against each other (cf., Broome, Weighing Lives, p. 90).
    In sum: I don’t deny that lotteries can provide a quantitative sense of goodness. But there are other ways to provide a quantitative sense of goodness. So we can make sense of quantitative claims about goodness without reference to lotteries. So we can interpret LH as a claim about outcomes and not prospects or lotteries. So your claim about A and B (mentioned at 9:17am), even if true, as a claim about prospects, does not show LH (as a claim about outcomes) to be implausible.

  58. Theron, do you think that you’ve given us an alternative to the lotteries method of imposing a measure? I don’t see this. You’ve just given a name, “intensities”, to the mysterious thing. We want to be told something about the mysterious thing other than its name.
    Measuring by lotteries is the famous way. There are other ways — looking at aggregation across time is one. But I don’t see that you’ve picked one out. Can you say more?

  59. Jamie,
    Maybe I’m confused about what’s needed for a quantitative scale of goodness, but I thought it was just (at least) two dimensions of goodness (x and y) that can be weighed against each other in a quantitative way. Campbell had already been using one dimension (number of headaches, which I take to be analogous to the temporal dimension), so I simply suggested another clearly relevant dimension (the severity of the headaches).
    If we had started with severity of headaches, I would have suggested that we also look at number (or duration) of headaches as another relevant dimension of goodness/badness (and I am unclear on how this would substantively differ from the “aggregation across time” proposal).
    If my response to you reveals that I am confused (a distinct possibility), would you kindly explain to me what is needed to sensibly formulate quantitative claims like LH without appealing to lotteries? (And/or how the “aggregation across time” proposal might work). Thanks.

  60. Also, I did not aim to offer a measure or quantitative definition of goodness that compares to expected utility theory in its generality. I might not have such a general definition or analysis. I just wanted to be able to “make sense” of my interpretation of LH in terms of outcomes rather than prospects (so I offered the particular example, “one 10-headache is as bad as two 5-headaches). Do I need a general definition or a to do this, or would an example of the quantitative notion of goodness I have in mind suffice?

  61. Oh, I see, sort of.
    But look, suppose Campbell had said, “Let’s compare the number-of-headaches badness. But I don’t mean the actual number of headaches, as in twice as many headaches. I mean, the badness of headache quantities.” This would not be helpful.
    Let me try to explain how I understand the question. We start with some idea of which things (headache collections, deaths) are better and worse than which. We have an ordering. What we want to know is how to get something cardinal. We would like to know how to impose a measure on headaches, collections of headaches, deaths, and so on, so that we can give some meaning to the question, what is the ratio of the badness of this one to the badness of that one? [Note that the ratio is independent of choice of unit, and also that it’s a bit safer to do what Campbell has been doing, which is to inquire about ratios of intervals rather than ratios of badness.]
    The reason I tend to agree with Campbell that this measure shouldn’t be taken as primitive is that I suspect our ordinary notion of badness is rather vague and could be regimented in several different ways. (Loudness of sounds was once this way, until acoustic science regimented it.) A very well exercised method is the lotteries method of decision theory. As I see it, the idea that a 100% chance of a headache is twice as bad as a 50% chance of a headache is not a hypothesis about the nature of badness, but a suggestion for how to cardinalize. It’s a good, useful way. But there are other ways.
    Sorry, that was long-winded.
    Parts of Weighing Goods and Weighing Lives are about this topic, with a slight change of view from the first to the second.

  62. Jamie,
    Okay, I think I follow. Thanks for that.
    But I take it that you agree there are other ways to cardinalize goodness besides appealing to lotteries. And so my response to Campbell still seems available.

  63. Dale,
    thanks. I’m slightly sceptical whether those really are consequences of all asymptotic principle, but here we would need to get to the finer details of mathematics.

  64. Theron,
    Yes, I agree that there are other methods of cardinalization. And I don’t want to claim that one method is uniquely correct. But I do think that when one says things like ‘the badness of headaches is linear’, one needs to be careful about what sort of cardinalization one is assuming, because this may make a difference to what one has said.
    Suppose we cardinalize using lotteries. Then I’d say the view that headache-badness is linear is about as plausible as Lives for Headaches. I gave an argument for this above.
    Now, you’re perfectly free to use some other method of cardinalization, but then unless you tell me what it is, I’ll find it hard to assess the view that ‘badness of headaches is linear’, because I won’t be sure what it means.
    I hope that clarifies my position. (By the way, on reflection, I’m inclined to accept Lives for Headaches, because this can be supported by the sort of small-increment arguments that some think disprove transitivity.)

  65. Campbell,
    Thanks. Three things:
    i. I might send you and/or Jamie an email, and ask for help with literature about different methods of cardinalization. You guys seems to know what you’re talking about here, whereas I feel like I’ve got a bit of catching up to do.
    ii. Let me try to put my concern about your proposed argument against LH another (fairly rough) way. When I affirm LH, it’s because I think that the badness of headaches add up, and that it would be some violation of impartiality to count the nth headache as anyhow less significant than the 1st headache. I don’t have probabilities in mind. But I am told that, for (complicated) theoretical reasons, LH commits me to some claim about the expected badness of headaches that I *may* not want to accept. To the extent that I do not want to accept it, I wonder if and why my espousal of LH truly must commit me to that implication. But you’re right, I have not offered you a rival method of cardinalization.
    iii. I’m inclined to agree with what you say in parentheses. But I think Temkin (in his forthcoming book *Rethinking the Good*) does an impressive job defending the rejection of transitivity as a defensible alternative. I’d really like to see more compelling arguments against his view offered!

  66. Hi all –
    Sorry for being absent. And, holy crap, nearing 70 comments! Woohoo!
    @Jussi – I think you’re right, we’d have to get down to the nitty-gritty, but I have a pretty good hunch that when we do, these problems will generalize.
    Re: cardinalization. Thanks for the helpful clarification of the problem. You’re right that I don’t have anything helpful to offer, and so the following points are more or less moot until I do. But here I go anyway. Either it’s the case that headaches aggregate to a limit or it isn’t. The former suggestion seems to have all the problems I pointed out, and so seems to me a non-starter. But if it isn’t, one might choose from a variety of different non-linear formulae. Perhaps headaches aggregate harmonically. If that’s right, then we don’t have to say that headaches aggregate to a limit, but we also don’t have to say that headaches are linear, and so I don’t see that I’m forced to choose between LH and LFH. (Maybe this could support the lotteries method? Not sure.)
    Also, and this is really cheap of me, but it seems to me that any method of cardinalization had better be compatible with our best axiological judgments, judgments about welfare, etc., etc., and not the other way around. To be committed to an implausible aggregative theory, or perhaps the impossibility of lexical superiority, on grounds that the best theory of cardinalization can’t support them is to get the cart before the horse. This is cheap of me because we need some way to cardinalize, which I haven’t provided. But if the choice is between treating these things as primitive, and accepting implausible axiological theories, the first, it seems to me, is the lesser of two evils.

  67. Dale,
    Consider the following:
    (1) A sufficiently small chance of one death is better than one headache.
    (2) Badness of headaches is not bounded above, i.e. for any number x > 1, there is a number n such that n headaches are collectively at least x times as bad as one headache.
    (3) One death is worse than any number of headaches.
    Let me stipulate that I’m assuming cardinalization by lotteries, as discussed above. So by (2), I mean this:
    (2′) For any number x > 1, there is a number n such that a lottery that gives a 1/x chance of n headaches and a 1-(1/x) chance of a neutral event is at least as bad as one headache.
    I’m fairly confident that, given one or two other innocent assumptions, (1), (2), and (3) are jointly inconsistent. (I basically sketched the argument for this in one of my earlier comments.)
    Broome’s suggested principle violates (2), because, as you point out, it implies that, e.g., no number of headaches is at least twice as bad as one headache. But I think you’ve convinced me that this is implausible. So I’m inclined to accept (2).
    Intuitively, I find (1) very plausible, and I think there’s independent reason for rejecting (3) (i.e. transitivity arguments). So I’m inclined to say that (3) is the one to go; we should accept Lives for Headaches, in other words.
    Do you want to reject (1) instead?

  68. Not to complicate matters unnecessarily, I hope, but I think there are two separate issues that might be confused here regarding the badness of headaches.
    First, we might wonder whether additional headaches are ‘equally bad’ for the person who has them. Cardinalizing by lotteries, that is to say that the prospect of getting two (indistinguishable) headaches on toss of Heads is exactly as bad as the prospect of certainly getting one headache (indistinguishable from the two above).
    Second, we might wonder about whether badness for a person contributes linearly to badness. We can, I think, generally judge which of two states of affairs is better, where this is not simply the judgment of whether it is better for this person, better for that person. Just off hand, a fairly plausible view about this question seems to me to be that there is increasing marginal overall badness of personal badness. (This corresponds to decreasing marginal goodness of personal goodness.) What this entails will depend on some more theoretic choices. But suppose we’ve already decided that additional headaches are equally bad for a person. Then my unreflective judgment that marginal badness of personal badness increases would amount to this: better for two people two suffer one headache each than for one person to suffer two headaches.
    Sometimes we forget that there are these two independent questions and we can then lose track of what our test cases are testing. (And in closing, let me say that I basically agree with Dale that a method of cardinalizing should be compatible with our considered judgments, at least if these are purely comparative judgments, about which outcomes are better and worse. I guess I suspect that these considered judgments radically underconstrain the theory; lots of methods will be consistent with them.)

  69. Hi Campbell –
    As soon as I posted the above comment, I realized that I goofed: you’re right to say that the unboundedness issue is distinct from the LH v LFH issue. So strike the last sentence of that para.
    But you make an interesting challenge. I think anybody who likes lexical superiority between two goods has a very serious problem w/this sort of case. Let me elaborate on where I see the problem arising. I don’t like to say things like a “chance” of one thing is “better” than another thing. It seems to me that chances have no intrinsic value at all, and that if “better” means “intrinsically better” you’ve got a false claim in (1). (Really, headaches aren’t “intrinsically bad” strictly speaking, but I’ve been assuming throughout we’re talking about the intrinsically bad experience connected with it.) However, the challenge comes up again in a very similar way like this. Let’s say you have to make a choice about whether to get in your car to drive to Walgreen’s for some headache medicine. You want the medicine. You don’t want to get killed in a car accident. The chance of getting killed in a car accident is non-zero on the way to Walgreen’s. We might think it is still rational for you to get into the car and drive to Walgreen’s. But how is this compatible with the claim that lives trump things like headaches? Hence it would seem I must deny
    1′: It is rational to accept a very small chance of death in exchange for a very high chance of avoidance of a headache.
    1′ sounds plausible, but cannot be accepted, or so it would seem, on a view that accepts (2) and (3), like I do. (I take this to be very similar to your worry, but correct me if I’m wrong.)
    As I noted before, I think the problem here lies in the “innocent assumptions”, like the principles of choice that generate 1′ from 2 and 3. But don’t ask me how to reorganize those principles. Frankly, I haven’t the foggiest, and so I’m inclined to treat this as a very difficult problem that fans of my sort of axiology need to deal with. For now, that’s the best I can do, which I recognize is unsatisfactory.

  70. Like Dale, I am flattered when I find that someone has read my work. If I had known that my rather technical note was going to have so many thoughtful readers, I might have adopted a less compressed style. I apologize to Dale if I seemed uncharitable. I hope I have encouraged him to make explicit those implicit premises of his argument that I would have ascribed to him if I had been more charitable.
    Perhaps a few clarifications might help to remedy the defects of the compressed style. Several of them have already been made by other contributors.
    1. I aimed only to refute the claim that Premises 1, 2, 3 and 4 entail Conclusion. My method was to give a counterexample that satisfies the Premises but not Conclusion. Please do not think I endorse the theory of value embodied in the counterexample. As Theron mentioned, I don’t.
    2. I tried to interpret Dale’s premises in the way that best suits his purpose. In particular, I interpreted Premise 3 in the way he now tells us he meant it. As he explained on 5 November, 7.19 pm, he meant: ‘For any amount of a bad A (nA), there is some amount of a lesser bad B (mB) such that mB is worse than nA’. (He typed ‘n’ for ‘m’, but the rest of the paragraph shows he meant ‘m’.) My example satisfies this condition, as I said in the original note. I don’t think my charity lapsed there.
    3. Moral philosophy has perhaps not yet established an agreed meaning for ‘continuity’. But I think we should not call a betterness ordering discontinuous if it can be represented by a mathematically continuous function.
    4. I said that ‘anyone with a slight knowledge of mathematical analysis’ would see the invalidity of the argument. That was rude; I’m sorry. What I meant was this. Starting from some positive number greater than, say, 1, there are sequences of numbers, each less than the one before, such that every number in the sequence is bigger than 1. For instance, there is the sequence 3, 2.5, 2.25, 2.125 . . . Similarly, starting from a bad thing that is worse than a headache, there is a sequence of bad things, each less bad than the one before, such that every bad thing in the sequence is worse than a headache. Dale’s argument seemed not to recognize this point.
    5. If Premise 3 is replaced with a stronger premise, Conclusion can be derived. This one will do the trick:
    Finite 3: There is a finite sequence of bads B1, B2, B3, . . .Bk, where B1 is death and Bk is a headache, such that, for any member of the sequence Br and any number n, there is a number m such that m people having Br+1 is worse than n people having Br.
    I think Finite 3 is about the weakest premise that will do the trick. Since Dale accepts the other premises and rejects Conclusion, he will reject Finite 3, and this rejection is about the strongest conclusion he can draw from his argument. He will have to decide whether it is enough to get him where he aimed to go in his paper. It will not get him discontinuity of value. Finite 3 is false in my example (since Conclusion is false there). But my example has a continuous value function.
    I believe Theron’s premise P3** entails Finite 3, except possibly under some extreme representation of the worseness ordering. However, its formulation depends on a particular representation of this ordering, and like Jamie I’m not happy with that.

  71. Hello John (if I may) –
    Thanks for your comment! As you note, I was thrilled to see that you had read the work, and have now taken time to comment on the comment. (I myself am astounded by the number of comments here.) Just a few points.
    Re: 2. You’re right. In my original comment, I was insufficiently attending to the point you make in 4, and insufficiently attending to what I earlier called the “total continuum of bads hypothesis”. You state this in the note like this: “for any real number a between 0 and 10 inclusive, there is a type of bad t such that bt [i.e., a single instance of t] is equal to a”. Assuming that hypothesis, my version of 3 does not yield a valid argument, and that was my error.
    Re: 4. But I’m inclined, I suppose, to deny that “for any real number a between 0 and 10 inclusive, there is a type of bad t such that bt [i.e., a single instance of t] is equal to a”. And if one does that, my version of P3 is more-or-less equivalent to Finite 3, which you’re right that I wanted to reject. I take also the argument later in the paper about welfare to be really doing the heavy lifting, but I suppose that’s neither here nor there.

  72. Dale, the ‘innocent assumptions’ I alluded to before are these two:
    (a) Betterness is transitive.
    (b) If pX + (1-p)Y is better than pZ + (1-p)Y then X is better than Z (where ‘pX + (1-p)Y’ denotes a lottery that returns prize X with probability p, and prize Y otherwise).

  73. Hi Campbell –
    I suppose (b) is the culprit (I would replace the first instance of “better than” with “more rational to select than”, or something of that nature; I’m really uncomfortable describing chances as having that sort of value; point holds in any event, as I show above). But I don’t have any sort of replacement, which is surely required. I can only offer a philosophical IOU.

  74. –Dale (6:01am) and Jamie (6:52am):
    According to both of you, the method of cardinalization we choose should be compatible with our best considered judgments about betterness. I think that this is perhaps an underlying view I had in mind, but did a poor job expressing, in my replies to Campbell.
    –Jamie (Nov. 6, 3:37am) and John:
    I now see the problem with my P3**. Thanks for the correction.
    –To Campbell (and all others who accept transitivity):
    What is, in your view, the most convincing argument in favor of transitivity? And/or, why shouldn’t we regard the rejection of transitivity a (worth considering) solution to the puzzles discussed above? I’ve read many papers that accept transitivity, or claim that it’s obvious, but only a few that explicitly argue for it.
    (Note: you don’t have to post the argument here, just the reference would do). Thanks.

  75. theron,
    how about. . .
    x is better than y, if and only if, x is good to degree n, y is good to degree m, and n is greater than m.
    to fill out the argument a bit more.
    1. ‘is better than’ is analytically basic or not.
    2. it is not basic.
    3. so it can be analyzed.
    4. the above analysis is correct.
    5. if it is correct, then ‘is better than’ is transitive.
    the idea is that ‘is greater than’ is clearly transitive, and so ‘is better than’ is transitive because it can be analyzed in terms of ‘is greater than’.
    now, 2 is open to doubt. but suppose you think that we shouldn’t simply take as primitive anything we want. suppose you think that we need a good reason to take something as primitive. perhaps we should take as primitive only concepts that correspond to properties or relations with which we can be directly acquainted.
    in any case, the analysis above avoids taking ‘is better than’ as primitive and gives it a plausible analysis that entails that ‘is better than’ is transitive.

  76. Sorry if my last post sort of changed the topic away from Dale’s original post. If people would like to send me their favorite arguments for transitivity, I’d very much love to discuss them over email. My email is tpummer AT ucsd DOT edu
    But really quickly, Christian: people who deny transitivity — like Larry Temkin — will deny that ‘is better than’ is the comparative of the predicate ‘is good’ or ‘is good to degree n’. They don’t deny that, on that analysis of ‘is better than’, it would be transitive (since it would be), but they either reject that analysis and offer another, or they take ‘is better than’ as a primitive. Temkin might well contend that his alleged counterexamples to the transitivity of ‘is better than’ show that the above analysis is flawed. (Defenders of intransitivity might also claim that it’s not obvious that ‘is good to degree n’ is more justifiably taken as primitive than ‘is better than’).

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