At least since Derek Parfit presented his “mere addition paradox”,1 the difficulties of formulating a viable population axiology have been widely recognized. Building on Parfit’s work, a number of philosophers have proved impossibility theorems, showing that certain plausible adequacy conditions are mutually inconsistent.2 To the best of my knowledge, the most impressive and important such results are due to Gustaf Arrhenius.3 His theorems involve more compelling adequacy conditions and weaker assumptions of measurement than earlier work in this area. On the basis of these theorems, Arrhenius is inclined to deny the existence of a satisfactory population axiology.4
The aim of this paper is to show that Arrhenius’s impossibility results are not inescapable. I shall mainly focus on his “sixth” theorem, which he considers to be his strongest result. Arrhenius’s proof of this theorem requires a certain assumption, as regards the order of welfare levels, which is more contentious than he recognizes. This assumption rules out “non-Archimedean” theories of welfare. If such theories are not excluded, there are, as I shall show, population axiologies that satisfy all the adequacy conditions of Arrhenius’s sixth theorem. In the penultimate section of the paper I shall argue, moreover, that my objection pertains to all of Arrhenius’s axiological impossibility theorems.5 Since non-Archimedean theories of welfare are far from obviously false, Arrhenius’s results fail to show that there is no acceptable population axiology.
1. Arrhenius’s Sixth Impossibility Theorem and the Crucial Assumption
Arrhenius’s sixth theorem states that no population axiology satisfies five adequacy conditions, which are informally rendered as follows:
Egalitarian Dominance: If population A is a perfectly equal population of the same size as population B, and every person in A has higher welfare than every person in B, then A is better than B, other things being equal.
General Non-Extreme Priority: For any welfare level A and any population X, there is a number n of lives such that a population consisting of the X-lives, n lives with very high welfare, and one life with welfare A, is at least as good as a population consisting of the X-lives, n lives with very low positive welfare, and one life with welfare slightly above A, other things being equal.
Non-Elitism: For any triplet of welfare levels A, B, and C, A slightly higher than B, and B higher than C, and for any one-life population A with welfare A, there is a population C with welfare C, and a population B of the same size as A∪C and with welfare B, such that for any population X consisting of lives with welfare ranging from C to A, B∪X is at least as good as A∪C∪X, other things being equal.
Weak Non-Sadism: There is a negative welfare level and a number of lives at this level such that [for any population X] an addition of any number of people with positive welfare [to X] is at least as good as an addition of the lives with negative welfare [to X], other things being equal.
Weak Quality Addition: For any population X, there is a perfectly equal population with very high positive welfare, a very negative welfare level, and a number n of lives at this level, such that the addition of the high welfare population to X is at least as good as the addition of any population consisting of the n lives with negative welfare and any number of lives with very low positive welfare to X, other things being equal.6
Although some of these conditions may not be entirely perspicuous, I think they are, on reflection, very plausible. In any case, they will not be questioned here.
Arrhenius’s proof proceeds by formulating more exact (and more technical) versions of these conditions, and showing that these exact conditions are mutually inconsistent. He thus arrives at
The Sixth Impossibility Theorem: There is no population axiology which satisfies Egalitarian Dominance, General Non-Extreme Priority, Non-Elitism, Weak Non-Sadism, and Weak Quality Addition.7
The main background assumptions of this theorem are as follows. A life is individuated by the person whose life it is and the kind of life it is. A population is a finite set of lives in a possible world. A population axiology is a quasi-order of all possible populations, with respect to their value. (A quasi-order is a reflexive and transitive, but not necessarily complete relation.) The relation “has at least as high welfare as”, denoted ≿, quasi-orders the set L of possible lives. The relations “has higher welfare than”, denoted ≻, and “has equally high welfare as”, denoted , are the asymmetric and the symmetric parts, respectively, of (L, ≿). Some possible lives have positive welfare, some have negative welfare, and some have neutral welfare. Which of these categories a life belongs to is independent of scales of measurement. We assume that L includes a wide range of lives, from ones with very high positive welfare to ones with very low negative welfare. Further, a welfare level is an equivalence class of L under . Let L be the set of welfare levels, let (L, ≿) be the -reduction of (L, ≿), and let ≻ denote the asymmetric part of ≿.8
By assuming only that (L, ≿) is a quasi-order, Arrhenius leaves open the possibility that some lives, and the corresponding welfare levels, are incomparable. Thus, there may be welfare levels X and Y, such that neither X ≿ Y nor Y≿X. In order to simplify the discussion, I shall mostly ignore this possibility, and assume that (L, ≿) is a complete order. If the sixth impossibility theorem can be shown to be escapable, assuming that (L, ≿) is complete, it is a fortiori escapable if (L, ≿) is allowed to be incomplete.
A final assumption that Arrhenius makes is crucial for the purposes of this paper. This assumption is somewhat complex, and not very precisely stated by Arrhenius. First, he presumes that (L, ≿) is either “discrete” or “dense”, according to the following definitions:
Discreteness: For any pair of welfare levels X and Y, there are only finitely many welfare levels between X and Y.9
Denseness: There is a welfare level between any pair of distinct welfare levels.10
Let us refer to this notion of discreteness as Arrhenius discreteness, or A-discreteness, for short. Second, Arrhenius claims that (L, ≿) is “fine-grained”. An A-discrete order of welfare levels is taken to be fine-grained iff the difference between any two adjacent levels is “merely slight”, in an undefined, intuitive sense.11 (How fine-grainedness should be understood when it comes to dense orders will be discussed in the next section.) Third, he claims that if (L, ≿) is fine-grained and dense, it has a fine-grained A-discrete suborder which is comprehensive, in the sense that for any X in (L, ≿), there is a Y in the suborder, such that the difference between X and Y is merely slight.12 In what follows, “suborder” refers to such comprehensive suborders.
These three claims together imply
The Crucial Assumption: (L, ≿) has a fine-grained A-discrete suborder.13
This assumption is necessary for Arrhenius’s proof of the sixth theorem, since the proof requires that welfare levels can be represented by integers. There must be an integer-valued function f, such that, for any welfare levels X and Y, f(X) ≥ f(Y) iff X ≿Y.14 Integer representability presupposes A-discreteness, since there are only finitely many integers between any given pair of integers. Arrhenius’s proof shows that no population axiology satisfies the exact versions of his five adequacy conditions, given that the order of welfare levels assumed in the proof is fine-grained and A-discrete. The Crucial Assumption implies that, even if (L, ≿) is not itself A-discrete, the sixth theorem can be proved relative to an A-discrete suborder of (L, ≿). This suffices to establish the theorem, since for a population axiology to quasi-order the set of finite subsets of L, it must quasi-order every subset of this set.
Suppose, to the contrary, that the Crucial Assumption does not hold. That is, no A-discrete suborder of (L, ≿) is fine-grained. This implies that every A-discrete suborder (L*, ≿*) contains at least one pair of adjacent welfare levels X and Y, such that X ≻* Y, and X is not merely slightly higher than Y. In Arrhenius’s exact version of General Non-Extreme Priority, the locution “one life with welfare slightly above A” is substituted by the assumption that the one life is at the higher welfare level adjacent to A. Similarly, in the exact version of Non-Elitism, the assumption that A is “slightly higher” than B is replaced by the assumption that A is higher than and adjacent to B. Hence, these exact conditions, on which Arrhenius’s proof is based, do not reflect the content of the informal conditions, and fail to be intuitively compelling, unless (L*, ≿*) is fine-grained.
The Crucial Assumption is thus essential for Arrhenius’s purposes. I shall argue that this assumption is insufficiently supported by Arrhenius, and that there are positive reasons for doubting it.
2. Denseness, Fine-grainedness, and Two Notions of Discreteness
The above criterion for fine-grainedness, in terms of adjacent welfare levels, is not applicable to dense orders, since such orders do not contain any pairs of adjacent elements. Arrhenius does not say what fine-grainedness amounts to, as regards dense orders. He argues, however, that all dense orders have a fine-grained A-discrete suborder:
If Denseness is true of [(L, ≿)], then we can form a [suborder (L*, ≿*) of (L, ≿)] such that [A-discreteness] is true of [(L*, ≿*)], and such that all the conditions that are intuitively plausible in regard to populations which are subsets of L are also intuitively plausible in regard to populations which are subsets of L*. Given that Denseness is true of [(L, ≿)], one cannot plausibly deny that there is such a [suborder (L*, ≿*)] since the order of the welfare levels in [(L*, ≿*)] could be arbitrarily fine-grained even though [A-discreteness] is true of [(L*, ≿*)].15
This argument is faulty. Suppose, for example, that (L, ≿) is isomorphic to the natural order of the real numbers strictly smaller than 0, followed by the real numbers strictly greater than 10. Suppose also that differences between numbers mirror differences between corresponding welfare levels, and that a difference represented by a numerical difference of 10 or more is not merely slight. Under these assumptions, (L, ≿) is dense, but lacks a fine-grained A-discrete suborder. Any A-discrete suborder will include a “gap”, in the form of a welfare level represented by a number smaller than 0, immediately followed by a level represented by a number greater than 10.
Intuitively, the dense order just discussed is not fine-grained. This is because there is a “hole” between welfare levels represented by numbers smaller than 0, and levels represented by numbers greater than 10.16 Hence, it might still be true that every fine-grained dense order has a fine-grained A-discrete suborder. An obvious suggestion is that a dense order is fine-grained if it lacks holes. Figuratively speaking, a dense order without holes is like an unbroken line, with each element of the order corresponding to a point on the line. This seems to constitute a high degree of fine-grainedness. Absence of holes, in a dense order, is equivalent to “Dedekind completeness”.17 Applied to (L, ≿), this amounts to the following. An upper bound of a suborder (L*, ≿*) of (L, ≿) is a welfare level in (L, ≿) which is at least as high as every level in (L*, ≿*). A least upper bound of (L*, ≿*) is an upper bound of (L*, ≿*) that is at least as low as every upper bound of (L*, ≿*). (L, ≿) is Dedekind complete iff every suborder of (L, ≿) with an upper bound has a least upper bound.
Let us, then, make the plausible assumption that all Dedekind complete dense orders should be classified as fine-grained.18 This puts us in a position to determine whether the Crucial Assumption follows from Arrhenius’s supposition that (L, ≿) is either fine-grained and A-discrete, or fine-grained and dense. The answer is negative. To see this, suppose that (L, ≿) is isomorphic to the Cartesian product of the real numbers, Re × Re, ordered lexicographically. That is, an ordered pair (x, y) of realsis at least as great as an ordered pair (z, w) iff x > z, or x = z and y ≥ w. This means that (L, ≿) is dense and Dedekind complete.19 Suppose, further, that the difference between two welfare levels, X and Y, is merely slight, in the relevant intuitive sense, only if the first number in the ordered pair representing X is the same as the first number in the pair representing Y. This implies that the Crucial Assumption does not hold. That is, (L, ≿) has no fine-grained A-discrete suborder. For example, any A-discrete suborder will contain a highest level among those represented by an ordered pair (1, x), for some x, immediately followed by a level represented by an ordered pair (y, z), y > 1. The suborder thus has a gap at this point, in the sense of two adjacent levels with a not merely slight difference.20
Arrhenius might respond that denseness is an unlikely property of (L, ≿), anyway. In fact, he regards A-discreteness as more probable than denseness:
My own inclination is that [A-discreteness] rather than Denseness is true. If the latter is true, then for any two lives p1 and p2, p1 with higher welfare than p2, there is a life p3with welfare in between p1 and p2, and a life p4 with welfare in between p3 and p2, and so on ad infinitum. It is improbable, I think, that there [is] such fine discrimination between the welfare of lives, even in principle. Rather, what we will find at the end of such a sequence of lives is a pair of lives in between which we cannot find any life or only lives with roughly the same welfare as both of them.21
This argument disregards the possibility that (L, ≿) is neither A-discrete nor dense. In particular, Arrhenius does not distinguish between A-discreteness and a logically weaker property that we may call standard discreteness, or S-discreteness, for short:
S-discreteness: For any non-maximal (non-minimal) welfare level X, there is a higher (lower) welfare level Y, such that there is no welfare level between X and Y.22
Arrhenius’s argument from limited discrimination does not support A-discreteness in particular. At most, it supports S-discreteness.23
To see that S-discreteness does not imply A-discreteness, consider the Cartesian product of the integers, I × I, ordered lexicographically. This order obviously satisfies the analogue to S-discreteness. For any ordered pair P, there is a greater ordered pair P*, such that there is no ordered pair between P and P*. For example, (1, 2) is greater than (1, 1), and there is no ordered pair between these two. The analogue to A-discreteness, on the other hand, is not satisfied. For instance, there is an infinite number of ordered pairs between (1, 1) and (2, 1), viz., the pairs (1, 2), (1, 3), (1, 4) …., followed by the pairs … (2, -2), (2, -1), (2, 0).
What, then, if we suppose (L, ≿) to be fine-grained and S-discrete, but not necessarily A-discrete? Presumably, the same criterion for fine-grainedness applies, as in the case of A-discreteness. That is, (L, ≿) is fine-grained iff the difference between any two adjacent levels is merely slight. If so, fine-grainedness and S-discreteness of (L, ≿) do not imply the Crucial Assumption. Suppose that (L, ≿) is isomorphic to the lexicographic order of I × I, and, again, that the difference between X and Y is merely slight only if the first number in the ordered pair representing X is the same as the first number in the pair representing Y. This implies that any A-discrete suborder has a gap.
To sum up, Arrhenius fails to support the assumption that (L, ≿) is either A-discrete or dense, since he does not consider other possibilities. Further, his argument against denseness does not support A-discreteness, but only S-discreteness. Fine-grainedness and S-discreteness do not imply the Crucial Assumption. Moreover, he is wrong in claiming that the Crucial Assumption follows from denseness and fine-grainedness. Hence, the Crucial Assumption is poorly defended.24
3. Archimedeanness and the Crucial Assumption
The Crucial Assumption may of course be true, even though Arrhenius’s arguments in its favour are weak. The question, then, is whether there are any positive reasons to doubt its truth. I think that there are, in fact, such reasons. The Crucial Assumption immediately implies the following principle:
Proto-Archimedeanness: For any welfare levels X and Y in (L, ≿), such that X≻Y, there is a finite sequence of welfare levels Vn≻ ... ≻V1 in (L, ≿), such that Vn ≿X, Y≿ V1, and, for each pair Vi, Vi-1, 1 < i ≤ n, the difference between Vi and Vi-1 is merely slight.
That is, the difference between any two welfare levels is bridged in a finite number of small steps.
Moreover, a stronger condition follows if a difference is taken to be merely slight only if it is not significantly greater than some other difference between welfare levels in (L, ≿). It appears that Arrhenius’s criterion for fine-grainedness, as regards A-discrete orders, presupposes such an interpretation of “merely slight”.25 An A-discrete order is not sufficiently fine-grained to be used in Arrhenius’s proof, if some differences between adjacent levels are much greater than other such differences. Given that all merely slight differences are roughly equal in size,26 the Crucial Assumption implies
Archimedeanness: For any welfare levels X, Y, Z, and U in (L, ≿), such that X≻Y and Z≻U, there is a finite sequence of welfare levels Vn≻ ... ≻V1 in (L, ≿), such that Vn ≿X, Y≿ V1, and, for each pair Vi, Vi-1, 1 < i ≤ n, the difference between Vi and Vi-1 is roughly equal to the difference between Z and U.
That is, the difference between any two welfare levels is bridged in a finite number of arbitrarily small steps.27
Many philosophers have expressed views which are incompatible with Archimedeanness, and arguably also with proto-Archimedeanness. Parfit claims that a “century of ecstasy”, 100 years of extremely high quality of life, is better than a “drab eternity”, an infinitely long life containing nothing bad, and nothing good except muzak and potatoes. “Though each day of the Drab Eternity would have some [constant] value for me, no amount of this value could be as good for me as the Century of Ecstasy.”28 This view contradicts Archimedeanness, since it implies that the welfare difference between the century of ecstasy and, say, a life consisting of one day of muzak and potatoes cannot be bridged by adding a finite number of days of muzak and potatoes to the latter life, although each such addition brings a constant increase in welfare level.29
In a similar vein, James Griffin suggests that there are examples of
a positive value that, no matter how often a certain amount is added to itself, cannot become greater than another positive value, and cannot, not because with piling up we get diminishing value [...] , but because they are the sort of value that, even remaining constant, cannot add up to some other value.30
Thus, Griffin finds it
plausible that, say, fifty years at a very high level of well-being—say, the level which makes possible satisfying personal relations, some understanding of what makes life worth while, appreciation of great beauty, the chance to accomplish something with one’s life—outranks any number of years at the level just barely worth living—say, the level at which none of the former values are possible and one is left with just enough surplus of simple pleasureover pain to go on with it.31
Arrhenius is, of course, well aware that many philosophers hold views like those just cited.32 Nevertheless, he does not seem to recognize the inconsistency between such views and the Crucial Assumption. I suspect that this is partly due to the fact that he does not distinguish between A-discreteness and S-discreteness.
4. A Non-Archimedean Toy Theory of Welfare
In order to construct a simple theory of welfare that satisfies fine-grainedness and S-discreteness, but violates the Crucial Assumption, we may suppose that (L, ≿) is isomorphic to the lexicographic order of I × I, and that all differences between adjacent levels are merely slight. There are, let us imagine, two kinds of welfare components. One of these kinds is “superior” to the other, in the sense that any amount of the superior positive (negative) components contributes more to the total positive (negative) welfare in a life than any amount of the “inferior” positive (negative) components. Assume, further, that superior and inferior welfare is measured on separate additive ratio scales. Any welfare level can then be represented by an ordered pair of integers, (h, l), where h (for “higher”) represents a net amount of superior welfare, and l (for “lower”) represents a net amount of inferior welfare. We make the following assumptions:
(1) A welfare level X, represented by (hX, lX), is at least as high as a welfare level Y, represented by (hY, lY), iff hX > hY, or hX = hY and lX ≥ lY.
(2) A welfare level X, represented by (hX, lX), is
positive iff hX > 0, or hX = 0 and lX > 0,
negative iff hX < 0, or hX = 0 and lX < 0,
neutral iff hX = 0 and lX = 0,
very high iff hX ≥ m, for a particular positive integer m,
very low but positive only if hX = 0 and lX > 0,
slightly negative only if hX = 0 and lX < 0, and
very negative iff hX ≤ n, for a particular negative integer n.
(3) A welfare level X, represented by (hX, lX), is merely slightly higher than a welfare level Y, represented by (hY, lY), only if hX = hY and lX = lY + r, r > 0.
Under these assumptions, the Crucial Assumption is violated. Any A-discrete suborder of (L, ≿) contains some pair of adjacent welfare levels X and Y, such that X is very high, whereas Y is not. By (3), therefore, the difference between X and Y is not merely slight. Similarly, any A-discrete suborder of (L, ≿) contains a pair of adjacent levels Z and U, such that U is very negative, while Z is not. Again, (3) implies that the difference between Z and U is not merely slight.
Moreover, it is easy to see that S-discreteness of (L, ≿) is not an essential feature of this example. We may instead suppose that (L, ≿) is isomorphic to Re × Re, and hence dense and fine-grained, but retain the other assumptions of the theory. The Crucial Assumption is violated in this case, as well.