Philip Kitcher

Columbia University

1. The Basic Plot

In the brilliant final section of Reasons and Persons, Derek Parfit presents a puzzle about how the goodness of states of affairs relates to the quality of the lives led by people in those states. Stripped to barest essentials, the puzzle runs as follows: if the value of a state is obtained simply by aggregating the quantity of whatever makes life worth living, then a world in which a significant number of people (say ten billion) enjoy lives of very high quality would be inferior to a world in which a vastly greater number of people have lives that are barely worth living (the Repugnant Conclusion); if the value of a state is measured by the average quality of the lives led in that state, then a world in which a few (even one or two) lead lives of exceptional quality would be superior to a world in which billions lead lives whose quality is just slightly lower; neither of these consequences is acceptable, and hence neither aggregation nor averaging captures the relation of the quality of a world to the quality of its constituent lives; so what does? Simplifying the puzzle as I have done ignores the richness of Parfit’s discussion, and, in particular, the ways in which he devises scenarios that block obvious and not-so-obvious attempts at solution.

Questions of this general type ¾ How do we understand collective utility in terms of the utility of individuals? ¾ have, of course, been systematically studied in a number of contexts, and there are well-known results. Nonetheless, as philosophical economists (and economically literate philosophers) are well aware, ‘utility’ is a slippery term, and it is important to think carefully about what it means in the particular context of investigation, so that the problem to be addressed is set up with the right conditions. Parfit’s puzzle is inspired by concerns about overpopulation (381), and thus falls within "population axiology" (rather than, say, the theory of voting procedures). The pertinent notion of utility can’t therefore be construed in terms of the preferences of existing people but is, rather, grounded in the quality of the lives led by generations yet unborn.

Parfit declares that he is in search of "Theory X", which we can think of as a function that takes representations of the quality of individual lives into some measure of the overall goodness of the world which contains exactly those lives. In subsequent discussions, various authors have proposed candidates, offered counterexamples to the suggestions of others, and, more recently, started to argue that the problem is insoluble. Some of the literature proceeds relatively informally; other contributions examine principles that are at some remove from Parfit’s actual arguments. What I hope to show is that a natural extension of Parfit’s reasoning, when formalized, establishes an interesting impossibility result. As already noted, this is not Parfit’s own assessment of his discussion (see, for example, 443), but it is, I think, the view to which he is committed.

Before undertaking this project, it’s worth underscoring the importance of Parfit’s puzzle. Foes of consequentialism might conclude that the difficulties only arise because of the hopelessness of maintaining that the good is prior to the right. They would be quite wrong to do so. Most views of moral deliberation assume that deliberators are able to assess the value of the states that might be produced by their actions. If they suggest that value is affected by the moral quality of the decisions and actions that produce those states, that is itself a view about the form of the function that assigns value: the function can’t be sensitive only to the numbers of individuals who lead lives of particular qualities, but must take into account how the activities that produce and develop those lives accord with prior moral principles. Opponents of consequentialism will thus agree from the start that the two simplest functions (aggregation and averaging) will not do, but they’ll still face the challenge of saying what the correct function is. One of the great merits of Parfit’s own elaboration of his puzzle is that he demonstrates how a number of prominent non-consequentialist ideas don’t help us solve it (see, for example, 422-25).

2. A Simple Formalism

I’ll start by supposing that each person’s life can be associated with a real number that measures how well it goes. The measure applies to the whole life, so that it’s a timeless truth about the individual p that p’s life has quality Q(p). The better p’s life, the higher the value of Q(p). 0 marks the boundary at which lives become worth living: with respect to people for whom Q(p) < 0, we can truly say of them that it would have been better if they had not existed. I’ll set a scale for the quality of lives by supposing that the best lives we know (from history, contemporary acquaintance, or even from fiction) have quality 100.

There are some serious assumptions here, and it’s worth noting them explicitly. First is the Single Number Assumption: Even if the quality of a life is determined by many different kinds of factors, it’s possible to weigh these against one another and identify a single numerical measure. Parfit’s diagrammatic representations of parts of his argument effectively embody this assumption, for he displays the relative qualities of lives by drawing lines that extend more or less far up a vertical axis (see 385, 388, 419 for examples). I’ll return to the Single Number Assumption at the very end of this essay.

Second is the Ratio-Scale Assumption: If Q is an adequate measure of the quality of lives and if Q* is also an adequate measure of the quality of lives, then there’s a strictly positive number k such that, for all p, Q*(p) = kQ(p). It is not plausible to assume that there are "natural units" of the quality of lives, so that we could talk of a unique number that "really" measures the quality of a life. The Ratio-Scale Assumption assures us that, nonetheless, we can meaningfully speak of the ratio of the qualities of lives (as Parfit does: see 385, 389 for examples), independently of the measurement scale we choose. Now in the context of many aggregation problems about collective "utility", this would be a very strong supposition, but I believe that it is defensible in the present case. Parfit’s puzzle starts from recognizing that we can judge that some lives go extremely well, while others are barely worth living. Given that there’s an invariant zero, we know that, if Q and Q* are both adequate representations, then Q*(p) = g(Q(p)).Q(p), where g is some strictly positive function. Making the relatively uncontroversial assumptions that transformations that take adequate representations to adequate representations can be iterated and/or inverted, consider the ratio g(Q(p⁺))/g(Q(p^-), where p⁺ is a person whose life goes exceptionally well and p^- is a person whose life is barely worth living. If this ratio is less than 1, then iterating sufficiently often will lead us from an initial representation of the relative qualities of the two lives which is very large to a supposedly adequate representation that makes that ratio as small as you please. If the ratio is greater than 1, then we can consider the inverse transformation and iterate that to produce just the same effect. Either route to the same end spells doom for the supposition that we can talk of one kind of life going vastly better than another. Thus the ratio ought to be 1, so that g(Q(p)) reduces to a positive constant.

My third assumption is the Full Comparability Assumption: For any adequate representation of the quality of one person’s life, there’s a unique adequate representation of the quality of any other person’s life. Again, this is far stronger than suppositions made in the context of other utility aggregation problems, where interpersonal comparisons are often treated with suspicion. In the case at hand, however, many of the normal sources of concern are absent ¾ for the task is not to assess how much better state X would be for person p (an actual person with actual preferences) than state Y would be for person q (another actual person with actual preferences) ¾ but to consider the lives that would be lived by possible future people and to evaluate them according to the presence or absence of features we regard as crucially important to our own well-being. We do make such judgments (and it’s arguable that responsible people ought to make such judgments), so that some interpersonal comparability is required. Parfit himself hedges a bit, by talking of "rough or partial comparability", an idea that we could express by supposing that fixing the quality of one person’s life selected a restricted set of assignments of quality of life to others, but this would endanger the comparisons of lives that are well worth living with those that are barely worth living (so that a situation that counted as Repugnant under one representation might become acceptable under a different redescription). I suspect that there are ways to relax the assumption of Full Comparability that will generate Parfit’s puzzle, but I won’t explore them here. Not only is the treatment I’ll offer simpler, but recent discussions of interpersonal comparisons of utility provide at least the beginnings of accounts about how such comparisons might be justified.

With these assumptions explicitly noted, we can proceed by taking there to be a single real-numbered scale on which the quality of different people’s lives can be measured, and we can forget, for the rest of this essay, about the infinite number of rival scales that would multiply our "units" by a positive real number. Any world can now be represented as an ordered pair of vectors n, Q such that n_i (the i-th component of n) is exactly the number of people in that world whose lives have quality Q_i (the i-th component of Q). Call such representations "thin descriptions". A fundamental thesis of consequentialism suggests that only numbers and the measure of the good count in ascribing value. Our problem, then, is to try to find a function that maps thin descriptions of worlds to measures of the value of the worlds so represented.

In fact, this is not quite right, because we can represent any world in infinitely many ways, and so worth must be assigned by any of a family of functions. Thin descriptions have a number of components that depends on the degree of stratification of the world, that is, on the number of distinct levels of quality that the lives in it achieve. Some worlds are homogeneous: they contain a number of people, all of whose lives have exactly the same measure of whatever makes lives worth living. Parfit mainly considers homogeneous worlds and the simplest heterogeneous worlds, those with just two classes of lives. But there’s no limit to the number of distinct classes there might be. Consequently, we can’t operate with a single function whose arguments are 2k-vectors (for some appropriate number k) ¾ V = V(n₁,.......,n_k, Q₁,.......,Q_k) ¾ since we could always envisage worlds for which there were more divisions among the qualities of lives.

Instead, we have to work with an infinite set of value functions, V_k , each defined on 2k-tuples. When we are comparing a pair of worlds we have to employ the function that accommodates the more heterogeneous, and this means that we must use an inflated thin description of the less heterogeneous world. Suppose, for example, that we want to compare w = < 10, 100 > with w* = < <9, 1>, <200, 1> >. We proceed by adopting an inflated representation of w as < <10, 0>, <100, 1> > and comparing the values of V₂(10,0,100,1) and V₂(9,1,200,1). This trick will work perfectly generally so long as we require that inflated measures reduce to uninflated measures.

More precisely, let’s introduce some obvious principles that our value functions ought to satisfy:

(1) tells us that we can always reshuffle the order of the arguments as long as we maintain the correspondence between numbers and lives measured by a particular quality. (In other words, the order in which we list the quality strata doesn’t matter.) (2) tells us that we can throw into our representation zero lives of any quality whatsoever without changing the value assigned to the world.

I’m going to exploit the reshuffling principle, (1), to propose a canonical thin representation for a world: we’ll always order the numbers and quality-measures by proceeding from highest to lowest quality; when we compare two worlds with different stratifications, we’ll insert zeros for the more homogeneous world and associate these with the corresponding quality measures in the comparison world. Hence, instead of < <5,1>, <100, 200> >, we’ll prefer < <1,5>, <200, 100> >, and in comparing this world with < <1,2,1,6>, <300,200,100,20> >, we’ll adopt the representation < <0,1,5,0>, <300,200,100,20> >.

A more substantive principle than either of those so far introduced officially commits us to the same number quality principle:

It also seems plausible to advance a disaggregated version of the principle: if a group that’s homogeneous in one world corresponds to two groups in another world, one of which has exactly the same measure of quality and the other of which has a higher measure, then the second world has greater value than the first. In fact, this principle follows from (3) provided that we allow the possibility that Q_i = Q_j when i ¹ j, and also suppose that

(4) V_2k(n₁,...,n_k, Q₁,...,Q_k) = V_2(k+1)(n₁,...,n_i-r,...,n_k,r Q₁,...,Q_k,Q_i).

We can now see very clearly how Parfit’s arguments against Simple Aggregation and against Averaging work. The Simple Aggregation Principle is:

(5) V_2k(n₁,...,n_k, Q₁,...,Q_k) = S n_iQ_i.

(5) accords with (1)-(4), and thus meets the constraints on value functions so far introduced. As Parfit sees, however, (5) has an implausible consequence. Consider two homogeneous worlds, w₁ = <10¹⁰,100> and w₂ = <10²⁰,10^-5>. In the second world, there are an enormous number of people whose lives are barely worth living; in the first world, there are many people who lead lives of very high quality. We suppose that w₁ is better than w₂. Hence, V₂(10¹⁰,100) should be greater than V₂(10²⁰,10^-5). But (5) takes the former value to be 10¹² and the latter to be 10¹⁵. So (5) yields the wrong result.

The values chosen represent a very large class of cases. The fundamental point is that we have a new constraint.

(6) embodies Parfit’s sense of repugnance at his conclusion, and it is incompatible with (5). (Plainly, (6) has analogues for other functions in our family which take a larger number of arguments.)

The Averaging Principle is easy to formulate.

As with Simple Aggregation, this will accord with (1)-(4). To see why (7) is problematic, we need only compare w₃ = <1,100> with w₄ = <1,10¹⁰,100,99.99999>. Intuitively, w₄ in which a large number of people live extraordinarily well is preferable to w₃ in which only one person lives at the very highest level attained in w₄. Thus we should have V₂(1,100) = V₄(1,0,100,99.99999) < V₄(1,10¹⁰,100,99.99999). (7) reverses this inequality, and thus should be rejected. In fact, we can extend the point, recognizing that it’s counterintuitive to rank <1, 100> ahead of <10¹⁰, 99.99999>. But, according to (7), we ought to accept that.

Once again the numbers are representative of a class of values, and the important point is to acknowledge a constraint.

A solution to Parfit’s Puzzle must satisfy (1)-(4), (6) and (8). Plainly, there may also be other constraints, as yet unrecognized. Parfit believes that there are. But we can actually satisfy the ones he offers.

3. Paradise Regained ¾ The Compromise View

Simple Aggregation goes astray because, as the numbers increase, aggregate quality matters less than average quality. Averaging is wrong because it gives no importance to numbers at all. One natural response is to compromise, giving weight to aggregation when the numbers are small and increasing the weight of averaging as the population becomes larger. The general form of the function we seek would then be

Here, N can be identified with aggregate numbers (N = S n_i) or with aggregate quality (N = S n_iQ_i), and L is a constant dependent on the level at which increasing numbers cease to count. The function f should be 1 (or close to 1) when N is small, and should go to 0 as N becomes large.

There are, of course, infinitely many functions of this general form. To understand the successes of the approach (as well as to identify important constraints on any adequate function) it will help to consider a very simple one

In other words, below a certain population size, N, V is just simple aggregation; at or above that level, it’s N times the average quality of life; thus, above N, V behaves like an averaging function (more precisely it responds to the average quality and multiplies it by a parameter so as to be continuous at the threshold population size).

It’s elementary to verify that (10) satisfies our earlier adequacy conditions, (1)-(4), (6), and (8). The significant point is to see how (10) evades the Repugnant Conclusion. Let’s focus on the version of the puzzle that Parfit presents in elaborating his "Mere Addition Paradox". Here he describes a sequence of worlds that we can represent as follows:

Recall that (10) favors simple aggregation when the population size is beneath a value N. If 2.10²⁰ < N, then the value assigned to worlds increases as we go down the alphabet (this just reflects the fact that while we are stuck with simple aggregation we are forced to the Repugnant Conclusion). If, however, N < 2.10²⁰, the Repugnant Conclusion will not necessarily obtain. Just where the alphabetically increasing value of worlds ends depends on the value of N. If N < 4.10⁷/3, then A is preferable to A⁺, and the conclusion is blocked at the start; if 4.10⁷/3 < N < 2.10⁷, then A⁺ is preferable to A, and also (markedly!) to New A. As N approaches 10¹⁰ from below, New A becomes preferable to A⁺, and, while N remains below 10¹⁰, New A is preferable to New B. Plainly, however, if N is above 2.10¹⁰, New B will be assigned higher value than New A (since simple aggregation will favor New B). There are values of N that halt the march to the Repugnant Conclusion between New B and New C, others that do so between New C and New D, and so on.

This raises an obvious question: which is the right value of N, and which point in the sequence does it mark out as the right point to halt the assignment of increasing value? Consulting intuitions on this point is of no use. That’s because there’s no definite point at which aggregation gives way to averaging, any more than there’s a definite number of grains of sand at which we achieve a heap.

My formalisation motivates an obvious strategy for coping with this aspect of the puzzle, akin to the method of supervaluations. Let’s suppose that we can exhaustively divide values of N into those that are definitely admissible and those that are definitely inadmissible (allowing for the possibility that admissibility itself is vague would complicate the treatment but wouldn’t change its essential features). Inadmissible values of N consist of those that are too small and those that are too large. Now we can define an ordering of worlds in terms of their values as follows:

(11) allows for the possibility that some worlds aren’t ranked as more valuable than others (because they are ordered differently by different admissible functions); (12) allows us to treat this range of incomparable pairs of worlds in terms of a degreed notion of relative value (provided that the pertinent values of N form a connected subinterval of the interval of admissible values).

Functions of form (10) are special cases of the more general form (9), and it seems highly likely that the admissible functions for ranking worlds will make up a much vaster set. For (10) proposes that aggregation stops and averaging begins all at once, whereas there are infinitely ways of producing a gradual shifting of the balance between these considerations (Parfit recognizes this; see 403-4). Hence we ought to generalize to refer to all admissible functions in (11), and to compute degrees in terms of measures of sets of functions in (12).

Do (11) and (12) (or their more general versions) subvert any of our earlier conclusions? No. (1) - (4), (6) and (8) are still satisfied. We avoid the Repugnant Conclusion (in the particular case of the puzzle we’ve been considering) if all admissible values of N are less than 10²⁰, which is plainly plausible. And, as we intended, we block the march from A to New Z without committing ourselves to the view that there is some definite point at which worlds start getting worse.

So have we solved Parfit’s puzzle? In the form in which he presents it, we have. Building on (9) and (10), and deploying (11) and (12), we can avoid the Repugnant Conclusion, without generating counterintuitive consequences that doom simple averaging ¾ Parfit’s scenarios for undermining the appeal to averaging trade on the idea that the number of lives in the superior world is small (i.e. below the threshold value), and, of course, our version of the Compromise View, to wit (10), uses aggregation for small worlds. Parfit himself thinks that the Compromise View encounters troubles of particular kinds, but, when his scenarios are formalized, it’s not hard to see that (10) gives the right results.

Yet one might worry that our exhibition of a function that satisfies Parfit’s own constraints is an artefact of particular arbitrary choices we have made about population sizes and the values of the lives that people lead. In setting up the sequence of worlds that is supposed to drive us to the Repugnant Conclusion I followed Parfit’s own account of the numbers. The range of values of N that generate admissible functions of form (10) was a direct response to those particular numbers. One might suppose, however, that the same puzzle could arise with different levels of quality (on the same fixed scale) and/or with different population sizes, provided that the same structure was preserved. Imagine that, instead of the sequence given above we were offered:

I’ll assume that this new sequence also gives rise to a Repugnant Conclusion, so that we ought to be able to show that Diminished New Z is inferior to Diminished A. If all admissible functions of form (10) take N to be less than 10¹⁸, this result will follow. Nonetheless, as we move from one version of the problem to another, the region of the sequence in which the Repugnant Conclusion is blocked may vary. For smaller population sizes, we may have to go further down the alphabet to discover the places at which worlds no longer increase in value.

Some people may be prepared to accept this feature of the solution ¾ or even the further idea that there are similar sequences for which the Conclusion is no longer Repugnant (consider the case in which the analogue of A is <1,0, 100,1> and the analogue of Z is <10⁴,10¹⁰,100,1>). Others may hold that the region of the sequence in which the Conclusion is blocked should remain the same. They endorse a requirement of population-size invariance:

It’s easy to see that (13) is incompatible with any approach based on (10). For suppose that admissible values of N lie between 10¹⁰ and 10¹². Let w₁ = <10⁴, 100>, w₂ = <10¹²,0.1>. Then, for all admissible values of N, V₂(w₂) > 10⁹, while V₂(w₁) = 10⁶. When we consider 10¹⁰w₁ and 10¹⁰w₂, however, we get V₂(10¹⁰w₁) > 10¹² > V₂(10¹⁰w₂), reversing the inequality between V₂(w₁) and V₂(w₂).

I’m not sure if one ought to accept the invariance requirement (13). Those whose intuitions accord with (13) can’t develop the Compromise View along the lines of (10). Instead, they need a different way of identifying the point at which aggregation gives way to averaging (or, more generally, the way in which averaging comes to have greater weight in the assignment of value). It can’t be a matter of sheer numbers in the populations, but the distribution of numbers across quality levels has to count.

This idea is hard to formalize in general, but special cases are straightforward. When we consider worlds with just two kinds of people, the average quality of life can be used as an index of the extent to which value accrues by aggregation. Thus

It’s not hard to check that (14) will satisfy the basic demands (1)-(4), (6) and (8). We evade the Repugnant Conclusion, since, at some stage in the alphabetical sequence, the average quality of life falls below R and the aggregate quality is then multiplied by the factor (1-R+Q^#), assuming that Q^# remains positive. More concretely, suppose that all admissible values of R are greater than or equal to 2; since Q^# approaches 1 as we go down the alphabet, the multiplier is either negative or approaches zero.

This is only a sketch of a rather different way of developing the Compromise View. As I’ve already noted, I think that the invariance requirement is controversial. It does, however, represent a general type of consideration that Parfit discusses, namely the idea that some worlds are less valuable because the distribution of quality within them is unequal. Egalitarian concerns can be introduced into an account of the value of worlds either by focusing on acts which cause inequalities (which would involve descriptions that are not acceptable to consequentialists) or by considering features of the distribution, no matter how it is caused. The general challenge for egalitarian consequentialists is to measure the degree of inequality in worlds with arbitrarily many levels of stratification and to show how the degree of inequality affects the values of these worlds. An approach along the lines of (14) is a very crude way of meeting this challenge in a very simple case. Since I believe that the degree of inequality of a distribution does affect the value of a state, I think that the general problem would have to be solved if we were to find the correct family of value-assignment functions. But our present task is to see if there are functions meeting Parfit’s constraints. For the moment, we seem to have done precisely that: the Compromise View appears to provide a defensible reconstruction of our intuitions.

4. Paradise Relost: Troubles for the Compromise View

But only for the moment. I’ll start with an example that is easily fixed, but that points the way to less tractable scenarios.

The Two Large Hells. Let w₃ = <N, 0, -100, -90>, where N is above the threshold for (10); let w₄ = <N, M, -100, -90>, where M > 0. Intuitively w₄ is worse than w₃ ¾ or, at best, no better than w₃ ¾ since it contains extra people who live appalling lives. According to (10), however, w₄ is better than w₃, since the size of both worlds is beyond the point at which we start to measure by average quality, and the average quality in w₄ is higher (V₄(N,0, -100,-90) = -100N < N{(-100N -90M)/(N+M)} = V₄(N,M, -100,-90)).

To fix this, we need only recognize the asymmetry between the sum of positive contributions to the quality of life and the sum of suffering, (albeit in a way quite different from that considered by Parfit, 407 ff.). (10) should give way to:

(10*) If S n_i < N or if S n_iQ_i < 0, V_2k(n₁,...,n_k, Q₁,...,Q_k) = S n_iQ_i;

It’s not hard to verify that (10*) preserves all the advantages of (10) for the cases considered in the last section, and that it gives the appropriate ranking to w₃ and w₄.

The Really Absurd Conclusion. Unfortunately, (10*) doesn’t guard against a different scenario in which a small amount of intense suffering figures in an otherwise wonderful world. As before, let N be above the threshold for (10) (and (10*)), and consider the worlds <N,m,100,-100> and <N,M,100,90>. Assuming that m is small in comparison with N, the aggregate quality in the former world will be positive and (10) and (10*) assign the same value. Thus we have: V₄(N,m,100,-100) = N{100(N-m)/(N+m)}, V₄(N,M,100,90) = N{(100N+90M)/(N+M)}. Now surely a world containing N people living best-quality lives and m people living lives of intense misery is worse than a world in which the fortunate N are joined by some large number M people whose lives, while not quite of best quality are, nonetheless, extremely good. Hence we want V₄(N,m,100,-100) < V₄(N,M,100,90). But, given the values computed from (10)/(10*), this inequality can easily be reversed. Assuming, as we have done throughout, that N is at least 10¹⁰, (10) and (10*) will lead us to prefer the world <N,10,100,-100> to the world <N,1000,100,90>. This is a really absurd conclusion.

What has gone wrong? Parfit argued, cogently, that averaging fails at small numbers. But averaging breaks down at large numbers too, and it does so dramatically when we compare different ways of reducing the average quality of life ¾ by adding a few examples of intense misery and by adding a larger number of high quality lives. Because the latter additions can reduce the average more substantially than the former, averaging will lead us to prefer worlds containing pockets of extreme suffering. Once this point has been appreciated, we should reject efforts to justify anything like the Compromise View.

The Numbers Always Count. Consider two worlds containing enough people to lie above the threshold for (10)/(10*). The first world, w₅ = <N,0,100,100-e > where e << 1; the second, w₆ = <1000N,1,100,100-e >. According to (10)/(10*)

V₄(1000N,1,100,100-e ) = N{(10⁵N +100-e )/(1000N+1)} < 100N = V₄(N,0,100,100-e )

so that w₅ >_V w₆. This means that a world in which a large number of people live best lives is superior to one in which a thousand times as many people lead best lives and one person lives a life of extraordinarily high quality, just slightly below the best. In my judgment, this conclusion is counterintuitive, and can only be sustained if one thinks that, for sufficiently large worlds, averaging (multiplied by a threshold value) is the appropriate measure. Once we have seen the Real Absurdity of this idea, we can endorse the intuitive verdict. Averaging works neither for worlds of abysmally low quality (recall the Two Large Hells) nor for large worlds of very high quality. For all its promise, the Compromise View is unsustainable.

This moral echoes themes in Parfit. Parfit is fond of imagining worlds in which the population of a world divides into two disconnected groups (perhaps on different galaxies), so that there can be no question of an attempt to remedy any differences in the quality of the lives they lead. We can extend the comparison of w₅ and w₆ to worlds in which the quality classes are spatially separated in some fashion that constitutes an insuperable barrier, and invite comparison of w₇ = <N,0,100,99> with w₈ = <N,M,100,99>. I claim that, no matter how large (or small) M may be, w₈ is preferable to w₇, on the grounds that the extra lives in w₈ are of such high quality that a world in which they were the only lives would be extremely valuable. Surely one cannot maintain that a world containing two populations of lives of extraordinary quality is inferior to one that simply contains one such population, unless one is in the grip of a commitment to averaging, which, we’ve already seen, can’t be defended.

If this conclusion seems dubious, imagine replacements for w₈ that increase the quality of the slightly suboptimal lives. There’s a range of values between 99 and 100, and there will be some x strictly between 99 and 100 such that <N,M,100,x> is superior to w₇ for all values of M > 0. If we consider ourselves as world-changers within w₇, and ask whether it would be worth creating M new people who would live lives just like those of the N people already living except for one slight flaw ¾ a mild attack of dyspepsia perhaps or an unfortunate sequence of coughs at a symphony concert, then the answer surely ought to be affirmative.

This point is very much in the spirit of some of Parfit’s arguments against averaging in the small. It will be important later when I offer my attempt at an Impossibility Theorem.

Since averaging in the large turns out to have counterintuitive consequences, an obvious way to attack Parfit’s puzzle is to try to refine the ordinary notion of aggregation. Perhaps what goes wrong is that, although lives of positive quality are worth living, there’s a critical level and it’s only when lives reach this level that a world containing them has a positive value. The most straightforward version of this approach, developed by the economists Charles Blackorby, Walter Bossert and David Donaldson, proposes the function

where a is the level (> 0) at which life quality makes a positive impact on the value of a world. To fix ideas, let’s suppose for the moment that a = 20.

A Non-Repugnant Conclusion. Simple aggregation fell victim to the Repugnant Conclusion. How does Critical-Level Utilitarianism handle the alphabetical sequence of worlds? Plainly, there are stages of the sequence at which the value of the lives drop below the threshold ¾ worlds <0,M,100,Q> where Q < 20. For such worlds, the value assigned by (15) will be negative (M(Q-20), in fact). So the original Repugnant Conclusion is blocked.

Nevertheless, there’ll be a last world in the sequence for which the quality of lives is above the threshold ¾ a world w₉ = <0,M,100,20+s> where s is small. Suppose that the first world in the sequence is w₁₀ = <n,0,100,20+s>. If M is sufficiently large ¾ M > 80n/s ¾ there’ll be an analogue of the Repugnant Conclusion, in that a world of many people (w₁₀ with n large) who lead best lives will rank as inferior to a world in which an enormous number of people have lives just above the critical level (w₉). Critical-Level Utilitarians hold that this conclusion isn’t Repugnant. After all, they suggest, the lives in w₉ are well worth living, and there are an enormous number of them.

The Ludicrous Conclusion. But we should now consider two further worlds, w₁₁ = <n,M,100,20-s> and w₁₂ = <0,1,100,20+s>, where n, M, s are as in w₉ and w₁₀. According to (15), V₄(n,M,100,20-s) = 80n - Ms < 0, and V₄(0,1,100,20+s) = s > 0, so that w₁₁ is ranked as inferior to w₁₂. This means that a world in which a large number of people (n) lead best lives and in which an enormous number of people (M) live lives just below the critical threshold is inferior to a world in which just one person lives a life just above the critical threshold. But the difference between 20+s and 20-s could be truly minute ¾ a matter of the fact that a few great musical works are missing or traffic congestion is slightly more severe. Nevertheless, Critical-Level Utilitarians are committed to believing that although enough 20+s lives could outweigh a large number of best lives, 20-s lives are so bad that adding the same enormous number of them to a world full of best lives would make that world worse than one containing one life just above the critical threshold (and also worse than the null world in which nobody exists).

To see how bizarre this is, consider two further worlds, w₁₃ =

<2n/3,0,-100,20-s> and w₁₄ = <0,M,100,20-s>. (15) assigns values of -80n and -Ms respectively. Since Ms > 80n, it follows that w₁₄ is worse than w₁₃. This means that the lives just below the critical threshold are so bad that an enormous number of them would generate a world worse than one in which a large number of people lead lives of intense suffering. This is to be consistent with the claim that a slight increase in the quality of those lives ¾ relief from minor dyspepsia, the rescue of some important musical works, a decrease in traffic congestion ¾ would transform them so that the same enormous number of transformed lives would be superior to a large number of best lives. I contend that this is a Ludicrous Conclusion.

It may be useful to consider the nature of the ordering of the worlds we’ve been examining. The following relations hold:

The ordering asks us to think that an enormous number of 20+s lives is better than a large number of best lives, that both of these worlds are vastly preferable to a world containing a large number of best lives and the enormous number of 20-s lives, and vastly preferable to worlds vastly preferable to worlds containing just the enormous number of 20-s lives. Just focusing on w₉ and w₁₄, we see that a tiny change ¾ the loss of a few great works of art, say ¾ can change the world from one that is better than a situation in which a large number of people live best lives to one that is worse than a situation in which that large number of people suffer intensely. No adequate account of the value of worlds can have this as a consequence.

Fuzziness to the Rescue. There’s an obvious reply to my indictment of Critical-Level Utilitarianism. We generate silly comparisons by supposing that there’s a single critical level. But Critical-Level Utilitarians should take their cue from Parfit, who is quite clear that talk of levels ought to recognize a band across which "the personal value of a life increasingly diverges from the value that this life contributes to the outcome" (412). In section 3, I offered (11) and (12) as a way of defending the Compromise View against charge of arbitrary discontinuities. Something similar can be introduced here to ward off the objections so far presented.

Instead of a single critical level, we now operate with a range of levels. One world is ranked higher than another just in case every choice of a level from the range would yield a substitution instance of (15) in which the former world had higher value than the latter. Thus:

For concreteness, let [a ,b ] = [20,25].

The analogue of the Repugnant Conclusion now consists in comparing w₁₅ =

<n,0,100,25+s> with w₁₆ = <0,M,100,25+s>, where M > 80n/s. For all x in [20,25], the value ascribed to w₁₆ exceeds that of w₁₅ ¾ for the former is M(25+s-x) > Ms, and the latter is n(100-x), which is at most 80n < Ms. If we now compare w₁₁ and w₁₂, we find that they are roughly equivalent (w₁₁ » _V w₁₂), so that we avoid the previously troublesome conclusion that ranked w₁₁ behind w₁₂. The best we can manage by way of a surrogate is to compare w₁₁ with w₁₇ = <0,1,100,25+s>. Plainly, for every x in [20,25], we have

Thus w₁₇ clearly counts as better than w₁₁. But it’s no longer so clear that this gives rises to counterintuitive consequences, since the difference between the lives in the worlds is no longer 2s but is much greater (5+2s), and so cannot be glossed in terms of the relatively trivial differences we cited earlier.

Why Fuzziness Fails. Our revised version of Critical-Level Utilitarianism evades the original form of the Ludicrous Conclusion. Yet consider two further worlds, w₁₈ = <0,M,100,20+s> and w₁₉ = <0,M,100,25-s>. w₁₈ no longer counts as superior to w₁₀, the world with n best lives, but it’s viewed as roughly equivalent. If the new approach is to succeed, we must agree that this conclusion isn’t Repugnant. But now

Hence w₁₈ is vastly preferable to w₁₁, despite the fact that the former contains an enormous number of lives whose counterparts in the latter are only slightly worse (being shadowed in any of the trivial ways canvassed earlier), and the latter also contains a large number of best lives. This is the same counterintuitive result we had earlier.

Furthermore, it’s not hard to show that w₁₉ is roughly comparable to w₁₀, the world with just n best lives and that w₁₉ is vastly preferable to w₁₈. In fact, the difference between the values assigned to w₁₉ and w₁₈ is 5M - 2s, which is truly enormous in comparison with 80n, the maximum value assigned to w₁₀. We thus have the anomaly of two worlds, each roughly equivalent to a world containing n best lives, but which differ in value by an amount orders of magnitude greater than the value assigned to the world with n best lives. However flexible our ideas about rough equivalence among worlds, this seems too great a variation between "equivalent" worlds.

As in the earlier argument, it’s not difficult to derive counterintuitive comparisons between worlds which Critical-Level Utilitarianism treats as having negative value ¾ consider, for example, <0,M,100,20-s> and <0,16n/25,100,-100>; the value of the latter is greater than -80n, while the value of the former is at best -Ms; thus a world with an enormous number of 20-s lives is counted as worse than a world with a large number of lives of intense suffering. This result is not only counterintuitive on its own, but also fares poorly when we recognize that the same enormous number of 20+s lives count as roughly equivalent to the same large number of best lives.

I conclude that the inadequacies of Critical-Level Utilitarianism can’t be remedied by considering a range of thresholds. I take this to show that the approach is beyond repair. This conclusion will be reinforced in the next section.

The approaches already explored and the scenarios on which they founder suggest some conditions that a value-ascribing function ought to satisfy. In this section, I’ll try to assemble the principles. Later, I’ll argue that they are jointly unsatisfiable. Although I’ll try to motivate the constraints as I specify them, the attempted proof may provide reasons for questioning one or more of them; hence I’ll probe them further later (section 8).

I begin by supposing that our conditions (1)-(4) are uncontroversial. It will simplify the subsequent discussion if we concentrate (as indeed we have done for most of this essay) on worlds that have at most two quality classes. Hence I’ll formulate our constraints as conditions on the function V₄. Further, we’ll be concerned mainly with the behavior of this function for worlds with a very large population, N; in this I follow Parfit, who, of course, formulates the Repugnant Conclusion for a world containing a large number of best lives. We’ll also suppose that s is a small number, s << 1.

Start with the moral of the Repugnant Conclusion. This tells us that if s ³ x > 0, then, for any number of people M, no matter how large, a world with M people leading lives of value x is worse than a world of N people leading best lives. In other words:

Next we have the moral of the anti-averaging arguments, not only those offered by Parfit but also the more general considerations urged in critique of the Compromise View. These tell us that, however many people the world contains leading extremely high quality lives, adding more people whose lives are of slightly lower quality doesn’t make the world worse. In particular, if we compare a world with N best lives to one that adds a number of lives of quality just s below best (100-s lives), a world in which the two groups of lives never interact, then the second world is at least as good as the first.

Parfit’s own discussion also motivates a Replacement Principle. Suppose that <N,M,100,100-x> is at least as good as <N,0,100,100-x>. Then there are k, y such that 0 < y < x, k > 0, and a world with kN people leading lives of quality 100-y would be better than a world of N people leading best lives. In other words

More general versions of [NAV] and [REP], in which the specific reference to best lives gives way to lives of "sufficiently high quality Q", seem plausible; but we shan’t need stronger principles and so I won’t try to defend them here.

Together [NAV] and [REP] yield a condition that will serve as the basis step in an inductive argument.

In my judgment, [BAS] is easily motivated directly by the considerations against averaging, but, in case these don’t seem sufficiently powerful, I’ve decomposed its rationale into the simple entailment by [NAV] and [REP].

We now need to draw on the discussion of Critical-Level Utilitarianism to capture the essential point of the scenarios that attacked the idea of a critical level. I imagined instances in which a world of N best lives was equivalent or inferior to a world of M people leading lives of lesser quality Q, and then considered worlds containing only lives of quality Q-s. The intuitive judgment was that Q lives must be so good, if they can outweigh N best lives (provided that there are enough of them), that Q-s lives can’t be so awful that no world containing just Q-s lives has positive value. After all, the Q-s lives could be very similar to the Q lives, differing only in some rather trivial discomfiture. How can it be that a world containing the mildly discomfited lives have at most zero value, while a world containing Q lives should be superior to a world with a large number of best lives? So I advance a principle to the effect that there are no critical levels.

The next principle I’ll articulate takes up a point that emerged in considering the Compromise View. Suppose that Q-lives have so low a quality that no amount of them can ever outweigh a world of a large number of best lives. Then, adding such lives to a world of high quality would inevitably diminish the quality of the world. This is intended to capture the attraction of the Compromise View, by taking account of the idea that lives that are too low in quality don’t add to the value of an already high quality world. Notice that this is much weaker than the claim urged by the Compromise View that, when the population is big enough, lives that are below the average quality lessen the quality of the world. It only claims that something like this holds when the extra lives aren’t just below the average but of such poor quality that they could never add up to outweigh a large number of best lives. We can think of this as a principle of the diminution of value through addition.

The last constraint embodies the idea that the context-dependence of value is limited. Again, this is inspired by the discussion of the Compromise View, where we found it hard to see how adding a galaxy of lives that, if it were to exist independently would constitute a valuable world, could lower the value of a world, even a world of high value. I don’t suppose that a life always make the same contribution to the value of a world, whatever the distribution of the qualities of other lives in the world ¾ that would be to formulate some variant on the simple aggregation view ¾ but I do think that there’s a limit to the variant impact that a life can have. So I propose

We’ll return to the motivation for [CIV] again in section 8, since it seems the most controversial of the constraints I’ve articulated.

For the moment, however, it suffices to note that [DVA] and [CIV] directly entail a principle about the negative value of lives that I’ll use in the proof:

We’re now ready for the impossibility result.

There’s no way of resolving Parfit’s Puzzle in its full form. For the full form of the puzzle demands the specification of a function jointly satisfying [NRC], [BAS], [NCL] and [NEG]. But this contradicts the

Outline of the Proof Strategy. The proof is by induction. We divide up the interval [0,100] into subintervals of length s. We then show that [BAS], [NCL] and [NEG] yield the Repugnant Conclusion, and, specifically, contradict [NRC]. [BAS] tells us that there’s a world of 100-s lives that is preferable to the original world with its N best lives. Now assume that there’s a world of 100-is lives (i a positive integer) that is preferable to the original world. We can now use [NCL] and [NEG] to show that there’s a world of 100-(i+1)s lives that’s preferable to the original world. Proceeding by induction, we show that there’s a world of lives of quality s that is preferable to the original world. This, of course, directly contradicts [NRC].

Proof. We assume that, for some integer K, 100 = Ks. We know from [BAS] that there’s a k > 0 such that V₄(N,0,100,100-s) < V₄(0,kN,100,100-s). Suppose now that there’s a k > 0 such that V₄(N,0,100,100-is) < V₄(0,kN,100,100-is) but that there’s no k > 0 such that V₄(N,0,100,100-(i+1)s) < V₄(0,kN,100,100-(i+1)s). If i+1 = K, then our supposition entails that V₄(N,0,100,s) < V₄(0,kN,100,s) which contradicts [NRC]. So it must be that i+1 < K, so that 100-(i+1)s > 0.

It follows from [NCL] that there’s some n for which V₄(0,n,100,100-(i+1)s) > 0. Hence, contraposing [NEG], there’s an M > 0 for which V₄(N,0,100,100-(i+1)s) < V₄(0,M,100, 100-(i+1)s). So it follows that there’s a k > 0 such that

V₄(0,kN,100,100-(i+1)s) > V₄(N,0,100,100-(i+1)s)

contradicting our supposition. Therefore we can conclude that if there’s a k > 0 such that V₄(N,0,100,100-is) < V₄(0,kN,100,100-is) then there’s a k > 0 such that

V₄(N,0,100,100-(i+1)s) < V₄(0,kN,100,100-(i+1)s).

Using this result and our earlier consequence of [BAS], we know, by induction, that there’s a k > 0 such that

V₄(N,0,100,100-(K-1)s) < V₄(0,kN,100,100-(K-1)s).

(Notice that we can’t apply the argument based on [NEG] any further, since (K-1)s + s = Ks = 100, so that 100-(K-1)s-s = 0, and the condition on Q in [NEG] is no longer satisfied.) But 100-(K-1)s = Ks-(K-1)s = s. So we have shown that

For some k > 0, V₄(N,0,100,s) < V₄(0,kN,100,s).

This contradicts [NRC].

Hence, there’s no function V₄, jointly satisfying [NEG], [BAS], [NCL] and [NRC].

I envisage two responses to the argument of section 7. One is that Parfit’s Puzzle is shown to be another instance of the familiar sorites problem, and that what goes wrong is that [NCL] isn’t fully true. Now if [NCL] asserted directly something analogous to the crucial premises in the paradoxes of the bald man and the heap ¾ "If a man with n hairs isn’t bald, then a man with n- 1 hairs isn’t bald" ¾ there might be a basis for saying that [NCL] isn’t fully true. Notice, however, that [NCL] doesn’t assert that if enough Q lives can outweigh N best lives then sufficiently many Q-s lives can outweigh N best lives (provided, of course, that Q > s); that, I agree, could be attacked for being less than fully true if it were offered without support from premises that are fully true. Instead, [NCL] says that if life at level Q is so good that sufficiently many lives at that level could outweigh N best lives, then life at level Q-s can’t be so bad that no combination of lives at that level has positive value. In the objections to Critical-Level Utilitarianism, I’ve tried to show why this counts as fully true: imagine a world of lives at some level below 100, with sufficiently many to outweigh N best lives; now imagine those lives slightly diminished in one of the trivial ways canvassed; the result can’t be a world that is no better than the empty world (i.e. no better than a world of zero value).

Thus efforts to reveal Parfit’s Puzzle as another sorites problem are not going to be successful. The alternative diagnostic strategy is to identify one of the constraints that ought to be rejected. I’ve already argued that [NCL] should be retained. We’ll now consider the other conditions in turn.

[NRC] is extremely well-motivated by Parfit’s own discussions. I believe that he is completely correct in maintaining that no number of lives barely above subsistence level could compensate for a large number N of best lives. So I’ll focus on the conditions enunciated in the generalization of the Puzzle.

Parfit argues against averaging by considering a scenario ¾ "Only France Survives" ¾ in which he envisages the world population being reduced to the nation with the highest quality of life. I claim that [NAV] can be defended by considering an expanded version of the same scenario. Suppose that the population is truly enormous, with a large number of people living best lives and the rest (a vastly greater number) leading lives that are best lives except for some trivial discomfiture. Let the people who live best lives inhabit SuperFrance. Just as the actual world is superior to one in which only France survives, so too the original vast-population world is preferable to one in which only SuperFrance survives. Hence it isn’t plausible to give up [NAV].

Nor can we abandon [REP]. If there are lives sufficiently close to best that they would increase the value of a world containing a large number of best lives, then enough of the mildly suboptimal lives could surely compensate for the best lives. This is to reiterate the moral of the argument in section 4 that the numbers always count. We can use the considerations urged there, either to motivate [BAS] directly or to defend [REP]. So it isn’t possible to evade the argument by giving up [BAS].

Let’s turn now to [DVA]. Here the guiding intuition is that if Q is so low that no number of Q lives could outweigh a large number of best lives, then adding Q lives to a world with a large number of best lives would inevitably diminish the quality of that world. One of the points that motivates averaging is the thought that bringing new lives of very low quality into a world in which a large number of people were leading high quality lives would make the world worse. How low is "very low"? [DVA] proposes, plausibly I think, that any lives that are so mean that enough of them couldn’t outweigh a large number of best lives are low enough to fall below the "very low" threshold. Thus [DVA] represents a kind of compromise between aggregation and averaging. Some lives, it suggests, are sufficiently good that an enormous number of them could outweigh a large number of best lives ¾ this is a nod towards aggregation. Other lives do not have this property because their quality is too poor, and adding lives like this to a world full of best lives would lower the quality of that world ¾ this is a gesture in the direction of averaging. In my judgment, [DVA] captures what’s plausible about attempts (like the Compromise View) to combine aggregation and averaging.

I’ve saved for last [CIV] because I think it’s the most likely to be questioned. As already noted in section 6, [CIV] embodies a limited context-independence of value. Lives that diminish the value of worlds full of best lives must be sufficiently mean that no amount of them could have positive quality.

[CIV] can be defended by the kinds of scenarios urged against averaging at high numbers at the end of section 4. Suppose that some number of Q lives gives a world positive value. Now take a world with a large number of best lives. Suppose we add ¾ in a newly formed galaxy ¾ the appropriate number of Q lives. The resulting world can be no worse. For it is, effectively, two subworlds, the old high-quality world plus a world that would, by itself, have positive value. Hence, any Q such that there’s some combination of Q lives with positive value is such that adding that number of Q lives to a large number of best lives doesn’t diminish the value of the world. Contraposing, we get [CIV].

This intuitive claim can be buttressed with a more elaborate argument. The general thought that the impact of lives on worlds depends on the other lives in the world ¾ the context-dependence of value ¾ supposes that there’s some measure of the distribution of the qualities of the other lives and the impact of the new lives depends on their quality and on this measure. More formally, this type of context-dependence assumes that

The simplest distributional measure is the mean, and the most obvious proposal would take j to subtract the value of the first argument from Q_k, so that the impact of the lives would be negative if the value of Q_k fell below the mean. Not only does this violate [NAV] but it’s also at odds with our anti-averaging arguments.

Let’s concentrate, as we have done throughout, on the function V₄. We want to know if it’s possible simultaneously to satisfy the conditions:

If we can, then, of course, we can evade [CIV]. We know, of course, that if Q = 100-s, (a) and (b) can’t hold together. But I see no basis for an inductive argument to the effect that (a) and (b) are unsatisfiable for all values of Q > 0. In this instance, there may be critical levels, between which (a) and (b) are jointly satisfiable. Intuitively, as we diminish Q we continue to satisfy (a) even though the lives fall below the level at which they continue to augment a world originally full of best lives.

The analysis so far suggests that a function V₄ that avoids [CIV] should take the following form. We want

(18) V₄(0,n, 100,Q) = nQ

expressing the idea that the top quality lives in a world are given full value, while those of lesser quality have their impact adjusted by a function of their distance from the top lives in the world. Intuitively, to circumvent [CIV] and satisfy both (a) and (b), you have to think that a homogeneous world of Q lives works by something like aggregation but that, in a heterogeneous world, where Q lives belong to the poorer class, the value of the Q lives is discounted with a larger discount the further they are below the higher value.

Notice, however, that the combination of (17) and (18) leads to immediate peculiarities. For consider w₂₀ = <N,M,100,Q*-s > and w₂₁ = <0,M,100,Q*-s>. (17) assigns w₂₀ a value less than 100N; (18) assigns w₂₁ the value M(Q*-s). If M is sufficiently large w₂₁ will be superior to w₂₀, which is absurd, since the latter world contains an extra N best lives.

Perhaps it’s a mistake to believe that (17) and (18) are the right ways to articulate the idea that the top quality lives in a world serve as the standard against which other lives in that world are measured and by which their impact is assessed. I conclude the discussion by claiming the following: [CIV] can be supported by a powerful intuitive argument (one drawn from the anti-averaging considerations) and an analysis of ways of circumventing [CIV] shows that the obvious strategies engender absurd consequences. [CIV] is hard to reject.

So where did we go wrong? Maybe at the very beginning, when we started to talk about the quality of lives. There I made the Single Number Assumption, adopting a very simple view of the structure of that kind of good we conceive of as "quality of life". Once that assumption has been made, the Archimedean property of the reals takes over: for any real number however small, r, and for any real number however big, R, there’s an integer N such that Nr > R. With numbers you can get from something tiny to something vast in tiny steps, provided that you take enough of them. Can you do the same with the quality of lives?

Think about the way in which the impossibility proof works. We start with lives of high quality, and then imagine small changes ¾ a cough at a symphony here, mild indigestion there ¾ arguing that such changes conform to particular algebraic relations. It’s supposed that enough small changes could lead us from best lives to lives that are barely above the point at which lives are worth living. Parfit’s own elaboration of the Repugnant Conclusion, especially in the sequence of worlds we considered in section 3, embodies just that supposition (which is one reason why I take the proof of section 7 to distill what’s crucial to his puzzle). But is this right?

We could think of things differently. Lives have many dimensions, and one can’t represent the quality of a life by a single real number. Instead, we should think of the quality of a life as a vector. That doesn’t doom attempts to compare the qualities of lives. But maybe it will enable us to tame Parfit’s puzzle.

Suppose then that Q(p), the quality of p’s life, is an ordered pair <I, T>, where I reflects the important things (p’s relationships with others, achievements, appreciation of beauty and so forth) and T measures the trivia (the absence of coughs at symphony concerts, free-flowing traffic, unprotesting digestion etc.). T is a real number; I’ll leave open for the moment what I might be.

Now one might propose that the impossibility theorem really shows that you can’t satisfy my proposed principles along the T dimension. But that’s no problem, because when you recognize that the result is restricted to this dimension, you see immediately what to do: abandon [NRC]. For it’s not Repugnant to conclude that a large enough number of lives whose quality is <I*, 0> could outweigh a world full of lives whose quality is <I*, 100> ¾ provided that I* reflects flourishing with respect to the important things. Suppose that two people live lives that are full of rewarding relationships, genuine achievements, appreciation of natural and artistic beauties, that one of these is encumbered by minor problems and that the other is free. The encumbered life is worse than the unencumbered life. Yet the important things go right in both, and this allows us to say that a world with a sufficient number of encumbered lives would be better than a world with a smaller number of unencumbered lives.

But this is only half a solution. If I is a real number then we’ve made no genuine gain, for the argument of section 7 can now be run along the first dimension. So to escape Parfit’s puzzle, one must propose that measurement of the quality of human lives, along the dimension (or dimensions) that really counts (count) doesn’t conform to the structure of the real numbers. Specifically, you can’t descend by small steps from flourishing lives to lives that are barely worth living.

So how does one exhibit the structure of the crucial dimensions? Perhaps discretely. In the simplest case we’d take all the aspects of our lives that really matter to us and suppose that each corresponds to a dimension on which one of two values ¾ 1 or -1, say ¾ is assigned. Lives that score 1 on enough of these dimensions go well, even if they receive low (even zero) values on the continuous dimensions that reflect the trivia.

Here’s a formal solution to Parfit’s Puzzle. The quality of a life is a pair <I, T> where I Î {-1, 1} and T is a real number between 0 and 100. Let w be a world containing N lives, for n of which the I-value is 1. Then the value of w is given by

The value-ordering of worlds is generated by comparing the first components; only when these are the same does the second come into play. Thus if V(w) = <m, R> and V(w^*) = <m^*, R^*>, then

The Repugnant Conclusion is now blocked because no number of lives that succeed in trivial ways can compensate for lives that are successful in the important respects.

Yet this may just be a formal solution, one that violates well-motivated views about the goodness of states. Can it survive detailed confrontation with scenarios as imaginative as those that Parfit offers us? Could it be developed to allow a larger number of different dimensions? Would it require consequentialists radically to reconceive their ideas about the good (or, more specifically, about the quality of lives)? I don’t know the answers to these questions. I hope, however, to have shown that Parfit’s Puzzle is deep and important, that Parfit’s arguments are best seen as steps towards an impossibility proof, and that that proof may force us to abandon some attractive assumptions about the structure of the good.