Any sane moral theory will require agents, on occasion, to promote the good. If so, puzzles within value theory have a wide resonance: if some claim about value seems unintuitive, it will affect not only straightforward act-consequentialist moral theories, but any moral theory with even a minor interest in the good. The topic of this article is one such puzzle. Consider the following argument.

1. 1. A headache is bad.

2. 2. Bads can be aggregated across persons to form worse bads. (Aggregation)

3. 3. For every bad x, there is a bad of lesser weight y, enough of which will outweigh the disvalue of x. (Continuity)

4. 4. If A is better than B, and B is better than C, then A is better than C. (Transitivity)

Hence,

This, it seems to me, is an unintuitive and frankly embarrassing conclusion. But avoiding Lives for Headaches and conclusions like Lives for Headaches is remarkably tricky. Each premise of this argument is intuitively plausible, even the somewhat more abstract (2) and (3). This argument has led some to accept Lives for Headaches (or its analogues) despite its lack of intuitive support.Footnote ¹

The intuition against Lives for Headaches, however, is robust and should be respected.Footnote ² A solution to Lives for Headaches is an important project not just in value theory and consequentialist moral theory, but also in moral theory generally. To give just one example, Lives for Headaches is influential in Judith Jarvis Thomson's acceptance of the ‘High-Threshold Thesis’, that is, the thesis that some rights (including the right to life) are maximally stringent, such that no amount of good will morally outweigh them. This, indeed, seems to be an extreme principle; even if we could save the lives of (and, indeed, grant the best possible lives to) all present and future persons, this would not be enough to outweigh a single right to life, say. This position is extreme even within rights theory. However, according to Thomson, accepting the High-Threshold Thesis is required to avoid Lives for Headaches. Thomson writes: ‘Those who reject the High-Threshold Thesis must find some other account of that constraint, for surely it is on no view permissible to kill a person to save billions from a minor headache.’Footnote ³ It seems to me, and many others, that the High-Threshold Thesis is implausible. Thus a solution to Lives for Headaches is surely called for, even if one is not a consequentialist of any stripe.

The argument for Lives for Headaches, however, is valid. If it is to be defeated it must be shown unsound. But which premise should we deny? Each looks independently plausible. One might simply deny that a headache is bad for the person who has it. Of course, headaches can be bad instrumentally, in making our waking lives intolerable such that none of our desires are fulfilled, or none of our plans and projects completed, or none of our potential for human flourishing is achieved. Nevertheless, there is nothing intrinsically bad about a headache, or so one suggestion goes. However, it seems to me that such a view is hard to sustain. Imagine the following fantastical choice. You have the option of living one of two possible lives. The lives are identical, except that one has an extra headache. Though it would be a small blemish on a life that might otherwise be exceptional, if we have the choice of avoiding such a headache or not avoiding it, surely there is good reason to avoid it. And if so, it seems to me, headaches should be construed as intrinsically bad, though their badness is surely not overwhelming.

Second, one might deny that value of this sort can or should be aggregated interpersonally. Given that, of itself, a mild headache is a minor burden compared to a death, perhaps we should accept that a death is worse: summing the disvalue of headaches across persons is illegitimate. The denial of aggregation, however, comes at a cost. It seems as though aggregation is required to capture the clearly intuitive conclusion noted below:

Rejecting aggregation requires one to develop a strategy to respond to Two Rocks – that is, a strategy that would reject aggregationism and also accept the indubitable suggestion that fifty deaths are worse than one.Footnote ⁴ Aggregationism provides a natural answer: because bads aggregate across persons, the badness of one death is outweighed by the badness of fifty. In other words, fifty deaths, considered as a whole, are worse than one death. The denial of aggregation, at least at first glance, seems committed to the opposite conclusion. (And, I shall argue, without adopting something like the strategy employed in this paper, it is so committed.)

Another possibility, proposed by Stuart Rachels among others,Footnote ⁵ is to claim that the relation of ‘all-things-considered better-than’ is intransitive. The proposal runs like this. Continuity holds – for every two values, some finite amount of the lesser value outweighs the greater value. Nevertheless, this need not yield Lives for Headaches, because even though, for example, x beatings outweigh one death, and x ⁿ headaches outweigh one beating, that amount of headaches would not outweigh one death, because betterness is intransitive – though A is worse than B, and B is worse than C, A is (or can be) better than C. Let's call this the ‘nuclear option’.Footnote ⁶ Like John Broome, I consider the transitivity of comparatives (like ‘all-things-considered better-than’) to be, roughly, a ‘truth of logic’.Footnote ⁷ In our Quinean age, of course, any truth of logic can be revised, but it seems sensible to explore all other options first. My proposal is one such option that, I believe, allows us to avoid the nuclear option.

In this article, I argue that though headaches are bad, these bads can never outweigh the big bad of death. Trading in one life for any amount of headaches, no matter how large, never improves the value of a state of affairs. In my view, aggregation can be accepted and Lives for Headaches rejected by rejecting the continuity of values, i.e. premise 3. If my proposal succeeds, even the hardest-core consequentialist need not accept Lives for Headaches. My proposal here is not especially new. Nevertheless, it has been thought vulnerable to devastating objections. I will first motivate the rejection of continuity in the following section with an examination into human welfare. I will then show why my view is not subject to the standard objections.

1. STRONG SUPERIORITY AND WELL-BEING

Rather than intransitivity or the value-neutrality of headaches, I prefer the following suggestion. Both headaches and deaths are intrinsically bad. But death is worse. In fact, saving someone from death is lexically prior in value to the relief of headaches. In other words, though headaches are bad, no amount of headaches equal the badness of death. (Other things being equal, of course. If someone's life is not worth living, death might in fact be good for them. I am assuming here, however, that the deaths in question are the deaths of those whose lives are worth living.)

Call the superiority in value displayed by a human life ‘strong superiority’. There are many different versions of a strong superiority relation. The strongest version is the one I have implicitly assumed so far:

However, there is another form of strong superiority that some have found much more plausible than really strong superiority:

The intuition behind pretty strong superiority suggests that, though some tiny amount of A might not be strongly superior to B, there may be some threshold amount of A, such that the achievement of that threshold is superior to any amount of B. Though the value of one small Shakespeare passage might not be enough to outweigh the value of any amount of knock-knock jokes, perhaps there is some sufficient amount of Shakespeare that would.Footnote ⁸Lives for Headaches could be avoided on either version of strong superiority – for instance, one could claim that what matters in a life is some threshold of goodness, and once that threshold is met no amount of headaches could outweigh that threshold. It seems to me, however, that there is a plausible case to be made for really strong superiority, and this version is the one on which I will focus. For the sake of brevity, I will drop the specifying adjective.

A simplifying assumption before I begin. Some have argued that the good is distribution sensitive, i.e. that how value is distributed among persons makes a difference to the overall value of the resulting state of affairs.Footnote ⁹ If this distribution sensitivity is strong enough, it might provide a rejection of Lives for Headaches without defending lexical superiority between indices of value.Footnote ¹⁰ I am sympathetic to the distribution sensitivity of the good. However, I believe that no account of the good, whether distribution-sensitive or not, need succumb to Lives for Headaches. I will thus assume, for the remainder of the article, that the good is not distribution-sensitive; I will assume that the account of the value of states of affairs is something akin to the classical aggregative approach of utilitarianism.

With that in mind, the claim that continuity breaks down between lives and headaches is implausible on certain views about the nature of value. To take one example, the plausibility of such a relation depends in large part on whether one is a value pluralist rather than a value monist.Footnote ¹¹ In other words, it is less plausible to insist upon a lexical priority relation if there is only one index of value along which states of affairs are measured. If, for instance, a hedonist value monism is true it seems very difficult to take the claim that one death is worse than any finite number of headaches seriously. If pain and pleasure are measured by their intensity and duration, and death is the deprivation of a finite amount of pleasure (given that humans are mortal – everyone is going to die sooner or later), surely that finite amount of pleasure could be outweighed by some finite number of headaches relieved. Your two seconds of pain is worse than my one. If so, assuming the truth of aggregation, my death – which involves the deprivation of some finite number of seconds of pleasure – is better than the sum-total of many people's headaches, the number of seconds of pleasure deprivation of which is greater (depending on the number of headaches involved).

My claim here is not that strong superiority on a value monist view is incoherent or radically implausible. Rather, it is an uphill climb. If there really is one index of intrinsic value, it seems as though more of that index is better than less, whether that amount is spread out over many individuals or concentrated into one. Any claim of lexical dominance, then, must explain how the underlying theory of value might be compatible with such a strong claim of superiority. The following is a sketch of an account of well-being and its value, one that will be painted with broad brushstrokes. Nevertheless, it seems to me that such an account is plausible. This account has the added feature of avoiding the problematic implication that mild headaches can trade-off against human lives. If so, it seems to me, there should be at least some prima facie reason for taking it seriously.

On my view, human well-being is pluralist. The major element is made up of the global plans and projects that feature in an agent's conception of the good life. But an agent's conception of the good life may be incomplete, incoherent, or clouded by insufficient factual information. Thus a key to understanding the major aspect of well-being, on my view, is the concept of ‘genuine’ endorsement. An agent genuinely values or endorses x if she endorses that object, life, plan, project, etc., as good or valuable when she is of sound mind and fully aware of all relevant information, including all information about the consequences of adopting it. I leave the ‘sound mind’ modifier unanalyzed here. I mean this to exclude from the category of genuine endorsement forms of endorsement that are the result of mental pathology (such as Alzheimer's disease or various forms of sociopathy or psychopathy). Many kinds of actual endorsement might not count as genuine endorsement. Actual endorsement will approximate genuine endorsement to the extent that an agent possesses full awareness of the information and the consequences of adopting some plan, project, or living some life x or y. Genuine endorsement helps to fill in gaps in one's conception of the good, and to correct those places where our conceptions of the good are incoherent or are incompatible with factual data. Though this proposal is to some extent schematic, it seems to me sufficient for genuine endorsement that someone endorse the object in question given a state of ‘deliberative rationality’ as described by John Rawls.Footnote ¹² The statement of value that x is better than y, or z better than w, is genuine if and only if issued with full awareness of all relevant facts involved in w, x, y, or z. Thus actual endorsement is neither sufficient nor necessary for genuine endorsement. Someone's activities could fail to reflect their conception of the good when actually endorsed if that endorsement is based on false beliefs, or if they are in some other way not in the proper cognitive state. Someone could be living a genuinely valuable life without actually endorsing it if, for example, the process by which genuine endorsement occurs is never actually engaged.Footnote ¹³

Given this conception of genuine endorsement, then, human lives go better or worse depending on the fulfillment of global plans, projects and achievements that characterize the agent's conception of the good, or that would be genuinely endorsed.Footnote ¹⁴ Thus the global elements of one's life will include the major plans, projects and relationships that characterize an agent's whole life or significant segments of an agent's whole life, as opposed to those momentary desires or hedonic achievements that (I shall argue) improve one's well-being, but that do not shape an agent's life as a whole. For example, I might never actually achieve my goal of becoming a great jazz trombone player. But my life goes better for me when I become a philosopher or maintain a successful marriage, etc. These achievements reflect my conception of the good: they are features of a life I genuinely value living. Call the global features of one's life ‘deliberative projects’, in other words, those projects, plans, goals and achievements one genuinely values fulfilling.

These deliberative projects form the major part of human well-being. Certainly people value minutiae, like the avoidance of single headaches, for instance, but the avoidance of a mild headache will not count as a global achievement in this way. Deliberative projects form the global, long-term, features of a life as a whole. Individual headaches do not do this. Deliberative projects are different in kind from these momentary hedonic states or momentary, summative, desires.Footnote ¹⁵ That such projects are different in kind seems to me a plausible representation of common-sense thinking about human lives and their value. Being a philosopher is different in kind from relieving that pesky headache. I leave for another time the proper philosophic account of this distinction.Footnote ¹⁶

My proposal bears resemblance to the accounts of welfare given by John Rawls,Footnote ¹⁷ Joseph Raz,Footnote ¹⁸ T. M. ScanlonFootnote ¹⁹ and Simon Keller.Footnote ²⁰ But my account of human welfare is not value-monist. It accepts the perfectly plausible judgment that headaches are intrinsically worse. It accepts the reasonable view that those who do not maintain deliberative projects can fail to maintain them in better or worse ways. On my view, the paradigmatic example of the minor feature of well-being is that of hedonic goods. (There may be additional minor goods, but hedonic goods will certainly be the paradigmatic instance of such goods; I will focus on them here.) Thus, a headache is always worse for someone than the relief of a headache – but no amount of headaches should trade off against the achievement of deliberative projects. And it seems to me that, from an intrapersonal welfarist perspective, this is a straightforwardly plausible view. Headaches are certainly bad for those who have them. However, if I would value living a life in which I play the trombone, this should be worth whatever the cost in the momentary pain of mild headaches. If I genuinely endorse trombone playing – endorse it as valuable with full knowledge of the consequences of trombone playing – it seems quite unintuitive to believe that trading off trombone playing for relief of headaches is better for me, however many might be traded off. It would lead to a life I value living less. If so, it seems to me quite straightforward that deliberative projects are better for me than the avoidance of those headaches, no matter how many could be avoided.Footnote ²¹

One objection to this understanding of welfare might seek to exploit my reliance on the notion of genuine endorsement. After all, one way in which we might decide whether a project is worth pursuing is whether a byproduct or consequence of that project involves headaches. If so, it seems that headaches do become one of the central aspects of human well-being, and hence might in certain cases trade off against deliberative projects. Of course, headaches can influence the extent to which I value living some project or other – they are certainly an aspect of the consequences of that deliberative project that I have assumed full access to and awareness of. For instance, if I knew that becoming a trombone player would cause persistent headaches, this feature of the project might be relevant to my belief that that global feature of my life is valuable. But this does not show that headaches trade off against one's valued projects. All this shows is that sometimes pain – if included among the consequences of a particular project – can influence the extent to which I value living according to some project or other. Thus headaches can be instrumentally injurious to the achievement of deliberative projects in two ways. First, because pain sometimes makes us unable to function in certain ways, they may simply deprive us of our ability to live according to our deliberative projects. Second, because they are included as a consequence of global projects according to which we might live, they may cause some global project to fail to achieve the status of a deliberative project. But living according to deliberative projects is lexically prior to headaches, even if sometimes headaches can influence the extent to which one achieves deliberative projects (in either way). Headaches, on this view, might affect the extent to which we value living according to a given activity or project if those headaches are a consequence of that activity. Nevertheless, the intrinsic value of headaches does not trade off against the long-term, global projects according to which one values living. This seems to me a plausible view.

So plausible, in fact, that I might be accused of triviality. On my view, pain can count as a consequence of a deliberative project that could thereby affect the extent to which one values that project. If so, where is the bite in the suggestion that deliberative projects are lexically prior to headaches? The only case in which a choice scenario comes up between headaches and deliberative projects is a case in which I can trade off some project for the relief of mild headaches. But then the failure to obtain the relief of headaches counts as a consequence of adopting that project: it is included in the information upon which I can rely in making a judgment about the value of the project. If so, from an intrapersonal perspective, I can perfectly well trade off hedonic achievements for deliberative projects by simply considering the extent to which some deliberative project would end up costing me hedonic achievements. Thus, from my own perspective, headaches perfectly well trade off against deliberative projects by affecting the extent to which those projects are valued.

The first thing that I should note is that even if this account displayed this form of triviality, the thesis still stands. The intrinsic value of headaches is lexically prior to the achievement of deliberative projects one values. The only thing this objection notes is that sometimes headaches can be instrumentally important. Nevertheless, the deliberative projects, rather than the headaches, are intrinsically dominant in these comparisons. The lexical priority of the intrinsic value of deliberative projects to headaches still stands tall. However, I should also note that even though this particular thesis might look trivial in the intrapersonal case, it is certainly anything but when the story about welfare is translated into the interpersonal context. No matter how many headaches one might relieve (that are themselves headaches and are not instrumental to the achievement of valued projects), these headaches do not trade off against the achievement of deliberative projects. Assume that you could grant some deliberative project x to an agent at the cost of mild pain for others that is not instrumentally important in the deliberative projects of others. No matter how much pain of this sort you have, this trade-off is always justified. The achievement of the project is lexically prior to the value of headaches. Though, once again, this thesis might sound trivial in the intrapersonal case, it is anything but in the interpersonal case.

Of course, this yields the result that, in the interpersonal case, the relief of headaches will trade off against deliberative projects, assuming that the relief of these headaches is instrumentally important to the value of some deliberative project or other. But this is the right result. After all, it is certainly plausible to relieve headaches at some significant cost assuming that those headaches affect the extent to which agents live according to their ideal of the good life. But my view still avoids the problematic features of Lives for Headaches. Single mild headaches, aggregated across billions of people, are not enough to trade off against a human life. For any given person, a mild headache is not a global feature of any project that would affect the extent to which they live a life plan they value living. Literally, headaches might trade off against human lives, but only to the extent to which those headaches influence whether someone lives a life she values living more than alternatives. And certainly this is not an unintuitive result.

The thesis I am proposing is thus that the fulfillment of deliberative projects is strongly superior to hedonic values, though both are important for a complete account of human welfare. If I value trombone playing with full information of its consequences, no amount of headaches would make abandoning that project better for me than keeping it. This account of human well-being seems plausible on reflection. More needs to be said, including the full specification of what it means to be a global feature of one's life. In addition, this view cannot be construed as fully subjectivist: the lexical priority of deliberative projects requires some objective (i.e. agent-independent) backing – see note 21. I leave these topics for another time. Nevertheless, if the broad outline of this view is plausible, we have the beginnings of a response to the problem of headaches and human lives. Headaches, of themselves, are not worth the cost in human lives because in killing an innocent person, one is ridding him of deliberative projects that comprise a life he values living (assume for the purposes of the example) for a gain that is trumped by the value of the projects that make up such a life. If this view is correct, I have defeated Lives for Headaches by denying continuity. However, this gambit is thought subject to devastating objections. Responding to these objections will form the rest of this article.

2. BARE SKEPTICISM

Strong superiority has a chorus of critics. Also known as ‘trumping’, this relation between two goods has been cast aside almost out of hand by, for example, James Griffin,

John Broome,

and Richard Arneson, who has described one form of trumping as ‘worse than dubious’Footnote ²⁴ – the trouble springing from the relation's implication that the A good is worth any amount of B goods. As an aside, one wonders if the examples suggested by Griffin and Broome are the most plausible. Surely any remotely plausible view – even one that countenanced strong superiority – would reject the conclusion that a second of extra life would outweigh, say, Northern Lights viewing. Though these worries amount to little more than what I shall call ‘bare skepticism’, their point is well-taken: lexical priority relations are worthy of a healthy dose of skepticism, given the strength of the claim being made. Is it really plausible to believe that there is no finite number of headaches, no matter how large, that would outweigh the badness of a death? No number at all? Perhaps it is more plausible to believe that our values are not fixed so rigidly: that there is some point, perhaps beyond human comprehension, such that a finite number of headaches prevented might trade off against one death.

Bare skepticism, however, strikes me as implausible given the stakes involved. For those who share the intuition against Lives for Headaches, it is far more implausible to believe that values are continuous. Accepting continuity implies that there will be some finite number of mild headaches that outweigh a human life. Assume that there is some set of values {x, x ₁, . . . x _n}. Assume that there is no point at which values are lexically prior to others. All weightings are, at most, very heavy. Now if x is a human life, there will be some finite number of x ₁s that will trade off against an x. Similarly for x ₂. Assuming that we want to avoid the nuclear option, there will be some finite number of x _ns that trade off against an x. Thus Lives for Headaches. At some point, every continuous theory must say that that number of headaches trades off against a human life. We should reject continuity. Of course, it is more implausible to believe in lexical priorities if one is a value monist. But absent value monism, we have a choice about how to weigh various values. If one weighting leads to the view that headaches can be traded off against human lives this weighting is seriously problematic. Bare skepticism should not be applied simply to those who believe that values display lexical priorities. Any claim that implies headaches can be traded off against lives is worthy of a long, hard, incredulous stare of its own.

3. THE SEQUENCE

I have just introduced a sequence of values from lives to headaches to show that any continuous theory must claim that lives for headaches holds (somewhere). But one powerful argument against lexical dominance appeals to just this sort of sequence. Such an argument seeks to exploit two claims. First, that one can break down welfare benefits into a sequence from the dominated value to the strongly superior value. Second, at each point in the sequence, it is implausible to suggest that the successor is lexically dominant over the predecessor. If there is no such stopping point, one might think, it is implausible to say that one endpoint of the continuum lexically dominates another – if there is to be lexical dominance, there must be some point at which we say that the lexical division is here. But any such division might seem arbitrary or unmotivated. Alastair Norcross runs a version of this argument (where avoidance of death is supposed to be the strongly superior value, a headache the dominated value, and ‘Less’ the statement of this superiority relation):

The question is this. Is the sequence, as Norcross presents it, enough to establish that there is no stopping point such that a lexical priority could exist between adjacent values? The answer is no. Indeed, the mistake is easy to spot. Norcross considers only the intrinsic badness of these various states of bodily injury in and of themselves. He does not consider the instrumental effects of these various states on other indices of value. But doing this smuggles in a presumption that we should reject: value monism, as if the dimension of bodily injury (most plausibly, hedonism) is the only dimension along which these various states of bodily injury can or should be measured – as if there is only one underlying index of value that might operate in considering the badness of states of bodily injury. If so, it is difficult to claim that there is a reasonable stopping point on the sequence, a point at which one inserts a lexical priority. On the assumption of a single underlying index of value, Norcross' sequence works. But assume now (contrary to fact) that a broken ankle carries with it a certain degree of pain, while the next point in the sequence carries with it slightly more pain, but that a broken ankle and not the very next point in the sequence is compatible with the achievement of some other index of value. In other words, assume that at some point, these various states of bodily injury become instrumentally injurious to our deliberative projects. Though on a hedonist or other monist dimension, the two points look as though they could clearly be traded off against each other, when we import a further dimension of value to which these various hedonic achievements may be instrumental, this becomes far less plausible. Of course, we might attempt to construct a Norcross-like sequence while taking seriously the distinction between indices of value. But this is difficult. It is plausible to believe that any loss in deliberative projects will be lacking an obviously large aspect of the good life for an individual. If one wishes to construct a sequence running from projects to headaches, the sequence goes wrong in its very first step: trading deliberative projects for other goods is not allowed. Stopping the sequence at the loss of deliberative projects is neither arbitrary nor unmotivated.

Norcross might respond by insisting that, though in the intrapersonal case, it is plausible to always construe the loss of a deliberative project as a value loss, the interpersonal case is different. For instance, suppose I could give up some project I value living in order to bring millions (billions!) up to a level just below the achievement of some project – say, by granting extensive momentary hedonic goods to millions (billions!). Is this a value loss? Perhaps not: the proposal might be that doing so is better than the achievement of one project for a single individual. But my hunch is that it is not. Any trade-off intended to better a state of affairs involves costs for some and benefits to others. But if so, why not confine those costs to the class of goods that do not affect the long-term features of a life one values living (as stipulated in the example)? It seems precisely backwards to impose a cost on a single individual such that that person lives a less worthwhile life from their own perspective when we could have taken another route: we could have imposed only a comparatively minor cost (albeit on a huge number of people) in order to preserve what is truly important for the value of human life: the extent to which that life is valued by the agent who lives it. In doing so there is a net gain in terms of what really matters in human well-being. More people as opposed to fewer live lives that conform to their conceptions of the good.

4. A REDUCTIO

A further argument against strong superiority has been offered by Gustaf Arrhenius. This argument seeks to combine intuitive general principles with the transitivity of betterness to generate a reductio of lexical dominance. Arrhenius' suggestion, in paraphrase, is as follows. Assume that there is a continuum of marginal differences in welfare. Consider three intrapersonal alternatives, A, B and C, each with some level of three goods each, g1, g2, and g3. g1 is strongly superior to g3 and g2; g2 is weighed very heavily against, but is not strongly superior to, g3. The difference between g1 and g2 is marginal (more on this below). Now consider alternative A, which contains an amount of g1, but none of g2 or g3. B has the same level of g1 as A, but an additional amount of g3. C, however, has a very large amount of g2. If so, Arrhenius suggests, the suggestion that g1 is strongly superior to the others is seriously problematic.

Arrhenius' argument takes the form of a reductio. Given the value relations as stated, we have a contradiction, or so he thinks. Arrhenius believes that the proper way to avoid the contradiction is to deny the strong superiority relation between g1 and the others. But this argument fails. First, there is no contradiction. Because we have assumed that g1 is strongly superior to g2, it cannot be the case that B obtains g1 and C does not, and C is better than B. This simply follows from the strong superiority relation. Nevertheless, Arrhenius believes that he has strong intuitive reason for believing that C is better than B, hence the reductio. But there is more than one way to avoid this contradiction. Rather than denying that g1 is lexically superior, it is more plausible to deny that the value difference between g1 and g2 is marginal.

However, Arrhenius might claim, we have assumed that there is a continuum of marginal value differences. If so, at any point of lexical priority there must be a state that is marginally worse. But it is not enough simply to say that such a continuum exists. This claim must be argued for. And Arrhenius' argument for marginal differences in welfare tacitly makes the same problematic assumption that felled Norcross' sequence: value monism, or the assumption that in taking account of the full-range of human well-being, only a single index of value is relevant (at the very least, Arrhenius does not discuss the way in which value pluralism might cause problems for the thesis that the differences in welfare might be marginal):

This sequence is obviously possible. But what Arrhenius fails to consider is the extent to which the index of pin-pricks might affect another index, e.g. the extent to which one fulfills one's projects. Any old pin-prick is going to be worse by a marginal amount. But consider some deliberative project x. There may be some sequence of added pin-pricks such that one more pin-prick loses an agent x. But if that pin-prick comes at the cost of x, it is clear that the value loss of that pin-prick is huge: it costs the agent a project that they value living, one that forms a significant chunk of a valued life plan. Thus, if we are only considering pin-pricks of themselves, the value difference seems obviously marginal. But when we consider what those pin-pricks might mean on a value-pluralist account, we come to realize that the sequence of marginal differences Arrhenius tries to construct is implausible. The index of single pin-pricks manifestly does not form a continuum from the best possible life to the worst possible life. The lexical priority of g1 to the others is not defeated by reductio. And, indeed, Arrhenius seems to understand this point: ‘Consequently, a believer in Strong or Weak Superiority must deny one of the two compelling intuitions invoked in the reasoning above.’Footnote ²⁸ And, if I'm reading Arrhenius correctly, one of the two ‘compelling intuitions’ is the intuition that C is better than B – or, if you like, the intuition that the difference between g1 and g2 is or could be marginal. But this intuition should be rejected. If g1 is a project one values fulfilling, and g2 involves a lack of that project, the difference is large, indeed.

I might be accused of being unfair to Arrhenius. I have claimed that the loss of one's valued projects yields a non-marginal value difference. But this is not strictly speaking entailed by the idea of strong superiority: there may be a small value difference between a dominated value and the lexically dominant value. One way of construing strong superiority requires only that there is an asymptotic relation between the two values – the dominated value would simply display a diminishing marginal returns feature in relation to the asymptote. Thus, the more of the dominated value you have, the closer it gets to the value of the lexically dominant value, though never actually reaching the dominant value. Under this description of lexical priority marginal differences in value are allowed. But this would do no good for Arrhenius. On this interpretation, his argument fails just the same – the value curve is asymptotic! (If g1 is superior to but marginally better than g2, adding g2 will not outweigh g1.) But this needn't be the only way to specify strong superiority. One might, as I have, describe the situation differently. Given that the loss of one's valued projects is always a significant loss in value, one could (if one had to) suggest that there is a ‘gap’ in value between the asymptote and the value of the lexically dominant good. This would imply no marginal differences. Any such loss is non-marginally bad. This gap needn't be large; it would only have to be non-marginal which, surely, the loss of one's valued projects is. (Although it seems to me that the asymptotic proposal is implausible: it seems unlikely that elements of intrinsic value display a diminishing marginal returns feature in the way this proposal would require.) Alternatively, you could eschew an asymptotic understanding of Strong Superiority, and claim that there is always an infinite (or transfinite, or nonstandard ‘hyperreal’Footnote ²⁹) difference between the dominant and dominated good(s). I will not argue for one rather than the other, but whatever way you choose, it is possible to insist on a non-marginal difference between dominant and dominated goods. And isn't this what the friend of strong superiority was insisting on all along?

5. SHOULD WE ACCEPT LIVES FOR HEADACHES?

Perhaps there is reason for questioning the assumption that has so far been running through this article. Perhaps we should reconsider whether or not the supposedly embarrassing conclusion that some number of headaches could be enough to morally outweigh a human life is really all that embarrassing. Norcross suggests that perhaps we accept such trade-offs every day.

Norcross is soliciting an intuition that he believes many people accept, i.e. that it is not wrong to fail to impose a national speed limit of 50 mph, even though this would save many lives. In fact, he suggests, the reason for this is the overwhelming cost in convenience for many people. Not just convenience, but also avoidance of annoyance: driving slower on the freeway would certainly cause a great deal of annoyance for a great many people. Though this would certainly not affect the extent to which they genuinely endorse their lives (i.e. assume it would not – or would rarely – influence their deliberative projects), perhaps it is not morally required to have a lower speed limit. And if convenience for many at the moral cost of human lives is plausible, Norcross argues, there seems to be very little stopping the suggestion that headaches ought to trade off against human lives, in at least some cases.

The first thing to note is that Norcross' argument is not a claim about value theory, but rather a claim about the right: it would not be wrong to fail to impose a low speed limit. However, there could be many intuitions getting in our way here. It is certainly possible that our intuitions are not tracking claims about value, but rather are tracking claims about moral obligation that do not reflect an interest in promoting value. It could be the case that we believe that we should not lower speed limits on anti-paternalist grounds; that people have a right to assume greater risk on freeways should they choose to do so. Of course, this might morally mandate alternatives such as low-cost public transportation for those who must rely on freeway travel for daily essentials and for those who are transporting children and others who do not have the opportunity to assume such risk.Footnote ³¹ None of this seems to me beyond the moral pale. Indeed, it seems genuinely intuitive. However, leaving aside questions of the right, the value theory question is different. Regardless of whether we believe it would be morally required to lower the speed limit, do we think it would be better if the speed limit were lowered? Or, leaving aside the legal question, do we think it would be better if people traveled a little slower at a cost of some annoyance and inconvenience in order to save lives? Absolutely! It seems to me that this scenario is clearly better than sacrificing lives for the sake of convenience.

This intuition can more clearly be brought out by considering an alternative suggestion. Suppose that I could greatly increase the convenience and speed of freeway travel by murdering Jane. Would this be morally acceptable? The answer is surely not. Of course, in terms of the right we might explain the asymmetrical responses to these cases by appealing to the do/allow distinction or some other such distinction; intended/foreseen, say. But the theory of value has no such distinction at its disposal. Would it be better or worse for Jane to die to increase the convenience and speed of freeway travel? The answer, it seems quite clear, is worse. I see no reason to accept the evaluative – as opposed to the normative – proposal of Lives for Headaches.

Another argument against the view that headaches should never trade off against human lives concerns the trustworthiness of the evaluative intuition. There is always some skepticism about strong superiority when it comes to our intuitions about large numbers. In order for trumping to be adequately defended, we would have to believe that no amount of headaches could outweigh a valued life. But no amount? Can we actually have such intuitions? Are such intuitions trustworthy? John Broome suggests they are not:

Further, Broome writes with specific reference to lexical priorities:

Broome's point is quite reasonable. How can we have intuitions about billions and billions of headaches? Isn't there some point at which our understanding of such large numbers gives out, and our intuitions should simply be treated as subject to strong suspicion?

It seems to me that this argument is inconclusive. I doubt that the concern about large numbers cuts against only the lexical priority view. For instance, why accept the view that all values are fungible, especially if we're dealing with large numbers in both cases? If our intuitions about large numbers are simply thrown out, this should cut neither the no-lexical priorities way, nor the lexical priorities way. It is indeterminate, unless there is strong theoretic reason for going one way or the other. Of course, Broome believes he has some axiological reason for thinking that all values are fungible from a theoretic standpoint. In particular, he is confident that goods are continuous.Footnote ³⁴ But his only evidence for continuity is the rejection of lexical priorities on the basis of faulty intuitions. If Broome wants to embrace continuity, he must embrace intuitions about large numbers! And if so, this means letting in the original anti-Lives for Headaches intuition. Or, if Broome would prefer to reject the suspect intuitions, it seems to me this is grist for my mill, rather than his. My view, rather than his, has an independent theoretic motivation: it is the outcome of an independently plausible account of human well-being. If so, it seems to me, the concern about large numbers is far more likely to pull in my direction than in Broome's.

6. STRONGER VERSUS WEAKER DENIALS OF CONTINUITY

In this article, I have tried to articulate and motivate a strong view: that the loss of any deliberative project is always a value loss, no matter how many non-project goods one obtains, and no matter which projects those are (so long as they are genuinely endorsed). It seems to me that there is some intuitive support for such a view. Some, however, may disagree. Briefly, I should like to note here ways in which the view I propose could be weakened in a way compatible with the solution to Lives for Headaches. I'll mention only two here, although there are perhaps more.

One possible weakening notes the potential unintuitive consequences of the really strong superiority relation when one considers the relative place of beneficiaries in the overall distribution. For instance, it might be that we have a choice between headache relief for the very badly off and the achievement of a single deliberative project for a person who is already, even in terms of deliberative projects, doing quite well. The consequence of the view as stated is that the deliberative project, in this case, wins out – deliberative projects are lexically superior to the relief of headaches and other such goods. But strictly speaking this implication is not essential for a solution to Lives for Headaches. One might insist that not all deliberative projects are strongly superior. Perhaps the projects of the very well-off are not, or perhaps those projects that make only a minor difference in the extent to which a person lives a life she values living. Instead, the suggestion might be that before someone has reached some threshold level of deliberative projects, those projects are always better than the relief of headaches – however, when an individual has some sufficient level of well-being, her projects then become fungible against minor, momentary goods. This might be closer to common-sense intuition: we surely wouldn't want to trade the extent to which a given individual lives according to any valued project for any amount of hedonic goods. (Indeed, the threshold could be set at a single deliberative project; this would entail a denial of continuity, and would entail the defeat of Lives for Headaches.)

Though I disagree, the current view might also be thought unintuitive when it comes to the crazy projects of various eccentric individuals. Imagine, for instance, that I have a project that can be frustrated by a single headache (call this the ‘total headache avoidance’ project). Perhaps, it might be thought, the total headache avoidance project shouldn't trade off against other, more reasonable, projects that people might have: becoming a philosopher, trombone player, maintaining a successful marriage, etc. Thus the solution to Lives for Headaches is compatible with treating genuine endorsement only as a necessary, rather than as a necessary and sufficient, condition of establishing the value of global projects. Perhaps endorsement must also be accompanied by some sort of objective value, whether construed as objective value in the ‘objective list’ sense,Footnote ³⁵ or in a perfectionist sense.Footnote ³⁶ I prefer the more subjective construal of deliberative projects, but the solution to Lives for Headaches contained herein is compatible with these various alternatives.