II. Mere addition of better-off people

Let us first consider a different-number case from Arrhenius (Reference Arrhenius, Frithjof Norheim, Eyal, Hurst and Wikler2013):

In figure 1, the height of the vertical lines and of the rectangle represent people's welfare according to some suitable measure or proxy. Each vertical line represents one individual whereas the rectangle represents a group, the size of which is indicated by the width of the rectangle. All the lives in figure 1 have positive welfare, or, as we could also put it, have lives worth living, which is represented by positive numbers.Footnote ¹⁴ Population A consists of two persons, one with half of the welfare of the other, 10 and 20 units respectively. In Z, we have added another 998 best-off people with 20 units of welfare.Footnote ¹⁵

Figure 1. Mere additions of better-off people.

Intuitively, many would judge Z to be better than A with respect to egalitarian concerns since Z is very close to perfect equality whereas A, on the other hand, is a straightforwardly unequal population. This is also what several standard measures of inequality imply, as we shall see below.

One way of rationalising this intuitive judgment in a more precise manner is this: in A, there are no relations of equality but one relation of inequality. In Z, on the other hand, there are a staggering 498,501 relations of equality, which in comparison outweigh the puny 999 relations of inequality.Footnote ¹⁶ Hence, one might reasonably judge Z as better than A in regard to egalitarian concerns, and a natural way to account for this intuition is to claim that from an egalitarian perspective, we should not only care about relations of inequality but also about relations of equality, as suggested by positive egalitarianism.Footnote ¹⁷

Notice that it doesn't follow from positive egalitarianism that Z is better than A in respect of egalitarian concerns since it is compatible with any ranking of these populations. The ranking depends on what weight we give to the great increase in equal relations as compared to the relatively small increase in unequal relations. Interestingly, that Z is better than A in regard to egalitarian concern is implied by the Gini coefficient (and some other standard measures, for example, relative mean deviation).Footnote ¹⁸ The Gini coefficient is approximately 0.17 for A and only 0.0005 for Z.Footnote ¹⁹

Negative egalitarianism, on the other hand, implies that Z is worse than A with regard to inequality since in the change from A to Z, the number of unequal relations increases from 1 to 999, and the size of the gaps are the same.Footnote ²⁰ This shows, interestingly, that the Gini coefficient is ruled out by negative egalitarianism. The reason is that Gini not only takes into account the number of unequal pairs, but is also sensitive to the number of equal pairs in a population since they figure in the denominator. Hence, Gini isn't, as one might believe, a measure that only gives weight to unequal relations, but it rather belongs to the family of positive egalitarian theories that also give weight to equal relations.Footnote ²¹

Segall (Reference Segall2016) has considered this case and disagrees with the judgement explained above of positive egalitarians and others who rank Z as better than A in regard to egalitarian concerns:

However, positive egalitarianism doesn't claim that the ‘999 better-off individuals living in equality amongst themselves . . . detract from . . . [the] badness’ of the inequality between the person with welfare 10 and the people with welfare 20. On the contrary, positive egalitarianism implies that Z is worse in one respect that is relevant for egalitarian concerns, namely in respect to the number of unequal relations to which the worst-off person is subject. It is just that positive egalitarians also value equal relations, and since there are so many of them in Z, this may outweigh the badness of the unequal relations when it comes to egalitarian concerns. Again, in A, there are no relations of equality and one relation of inequality, whereas in Z there is a staggering 498,501 relations of equality, which in comparison outweigh the few 999 relations of inequality.

Secondly, let us again stress that it doesn't follow from positive egalitarianism that Z is better than A in respect to egalitarian concerns, but it is compatible with that judgement. It depends on what weight we give to the increase in unequal relations as compared to the increase in equal relations. Hence, Positive egalitarians could accommodate Segall's intuition by changing the relative weight of the value of equal and unequal relations. To make Segall's claim into an effective argument against positive egalitarianism, one would have to show that there are no cases of the type depicted in figure 1, that is, cases in which the positive value of an increase in equal relations outweighs the negative value of a much smaller increase in unequal relations, possibly involving very small inequalities (e.g. consider a version of the case in figure 1 where A = <19, 20> and Z = <19,999 × 20>). This is a rather tall order, we surmise, since it is tantamount to showing that irrespective of how small the inequalities are and how many more equal relations there are, the positive value of the equal relations cannot outweigh the negative value of the unequal relations.

A further issue is the ‘isolation’ of the worst-off individual that Segall appeals to in the previous quote. This point can be understood in different ways. It could be taken as a psychological claim regarding the welfare of the worst-off individual: that she suffers from being the only worst-off person. Taken this way, the objection misses its target since any effects on the welfare of the person are already taken into account in figure 1 and the worst-off has the same welfare in both outcomes.

More interestingly, Segall hints at the putative badness of the isolation of a single worse off individual in a society, in the quote above. This can be a further aspect to take into account when we rank A and Z with respect to egalitarian concerns. Mosquera (Reference Mosquera2017a) has a more developed proposal that also takes account of this intuition. Her idea is that the way in which relations of inequality are distributed should also matter when accounting for the egalitarian value of a population. It is for example pro tanto worse if the inequalities are concentrated on a single worse off individual rather than on many worse off individuals. Mosquera refers to this as ‘Distribution-sensitive pairwise egalitarianism’. Going back to the case with A and Z, there are more relations of inequality concentrated on the worse off individual in Z, as compared to A. In this respect, Z is worse than A.

However, the question is what weight should be assigned to those considerations as compared to other egalitarian considerations. Mosquera doesn't commit to any particular weighing and would not claim that her account implies that Z is worse than A, taking into account all the relevant egalitarian concerns. It is indeed possible for an egalitarian to assign such an extremely high weight to the putative badness of the ‘isolation’ of a single worse-off individual in a society, such that the ranking of A and Z in respect to egalitarian concerns would be reversed as compared to the ranking provided by positive egalitarianism and Gini. But it is hard to find a convincing argument for such a radical view, we surmise.

To sum up, we think that Segall's ranking of A and Z is partly based on an equivocation of inequality and egalitarian concerns. Furthermore, it overlooks the great number of equal relations in Z as compared to the small number of unequal relations. It is this difference that makes it intuitively attractive to rank Z as better than A in respect of egalitarian concerns and along the lines of reasonable versions of positive egalitarianism and other theories such as the Gini coefficient. The appeal to ‘isolation’ or ‘concentration’ seems insufficient to reverse the ranking in this case.

III. The progression case

As we showed above, negative and positive egalitarianism come apart in different-number cases. But as we also mentioned in the introduction, negative and positive egalitarianism can also diverge in same-number cases. In fact, Segall suggests another criticism against positive egalitarianism that is based on a same-number case where he claims that positive egalitarianism yields the wrong result. It might be instructive to look at it now. Consider the following two same-sized populations of 1,000 people each:

In figure 2, the numbers above the height of the vertical lines and the rectangle represent people's welfare. The number of people at each welfare level is indicated by the numbers below the neutral welfare line.

Figure 2. The progression case.

Segall claims that Z′ is better than Z with regard to egalitarian concerns:

Positive egalitarianism of course yields the opposite ranking. One might get the impression from Segall's discussion that what makes Z better than Z′ according to positive egalitarianism is the fact that Z contains a lot of relations of equality, while Z′ has no relations of equality. As we shall show below, this is wrong, since negative egalitarianism also ranks Z as better than Z′ in respect of egalitarian considerations. Positive egalitarianism gives extra reasons for this ranking because of the positive value of the relations of equality in Z, but it is not the main reason for why Z is better than Z′.

It might not be evident that both negative and positive egalitarianism imply that Z′ is worse than Z with respect to equality so let us informally demonstrate it. We can do a one-to-one mapping of all the 999 unequal pairs in Z to a corresponding unequal pair in Z′ with a greater gap. According to negative egalitarianism, this subset of unequal pairs from Z′ will be worse with respect to equality than the set of unequal pairs from Z. Since all the pairs in Z′ that remain unmapped after this are unequal, this will only further decrease the egalitarian value of Z′. The same demonstration holds for positive egalitarianism since the equal relations in Z will increase the egalitarian value of Z further, as compared to the egalitarian value of Z′.

Here is a figure illustrating the same point:

In figure 3, the two ellipses denoted pr(Z) and pr(Z’) represent the set of all pairwise relations of equality and inequality in populations Z and Z′ respectively. Relations of equality are represented by ‘=’ whereas relations of inequality are represented by ‘≠’. The horizontal arrows from pr(Z) to pr(Z′) represent the mapping of all the relations of inequality from pr(Z) to relations of inequality of greater size in pr(Z′). Hence, the circle in pr(Z) represents the set of all relations of inequality in pr(Z) whereas the circle in pr(Z′) represents some subset containing relations of inequality in pr(Z′) of size greater than 49. The latter subset will be worse with respect to equality than the former according to negative egalitarianism. As figure 3 shows, all the pairs in pr(Z′) that remain unmapped are unequal, and thus will further decrease the egalitarian value of Z′. Moreover, all the unmapped pairs in pr(Z) are equal and thus will increase the egalitarian value of Z further as compared to the egalitarian value of Z′.

Figure 3. A one-to-one mapping of the unequal relations in Z to unequal relations of greater size in Z′

Does Segall's case show that we have to reject both positive and negative egalitarianism? We don't think so. Segall's ranking of Z and Z′ overlooks the great magnitude of the inequalities in Z′. Firstly, as Segall concedes, population Z′ contains a greater number of relations of inequality than Z.Footnote ²⁴ However, it is important to notice how much greater the number of such relations is in Z′ as compared to Z. As Mosquera has pointed out, the latter population contains only 999 relations of inequality of size 49 whereas Z′ contains as many as 499,500 relations of inequality of sizes ranging from 1 to 999.Footnote ²⁵

Secondly, the vast majority of the relations of inequality in Z′ involve a bigger gap than the inequalities in Z. In Z′, 450,775 relations out of a total of 499,500 relations of inequality are of size greater than 50. In Z, on the other hand, the relatively puny 999 relations of inequality are all of size smaller than 50.Footnote ²⁶

As an illustration only of the magnitude of the inequality contained in Z′, consider individual 1 in Z′ who is subject to 949 relations of inequality greater than 49. As pointed out by Mosquera, these 949 relations of inequality range from size 50 to size 999, whereas, again, the relations of inequality in Z are of size 49.Footnote ²⁷

Hence, there are many more and greater inequalities in Z′ as compared to Z. Thus, pace Segall, it makes eminent sense for an egalitarian to rank Z′ as worse than Z in respect of egalitarian concerns.

Finally, let us here take the opportunity to point out a simple misunderstanding of positive egalitarianism on Segall's part. He writes:

Positive egalitarianism, however, does take into account the size of the inequalities in the distribution since it is an extension of negative egalitarianism. The set of all pairs of individual welfare levels in a population contains information about the size of the gaps and the number of unequal pairs. Negative and positive egalitarianism take into account both of these aspects so that the greater the gap, and the more of such unequal relations, the worse in regard to egalitarian concerns. This is rather clear from Arrhenius’ original paper in which there is extensive discussion on how to take into account the size of inequalities, but also on how to weigh them against the positive value of equal relations.Footnote ²⁹ For example, he writes that ‘it seems appropriate that both the number and the size of inequalities matter’.Footnote ³⁰

More importantly, negative and positive egalitarianism, as defined above, rule out egalitarian views according to which the egalitarian value of a population sometimes isn't affected by an increase in the gaps or by an increase in the number of unequal pairs. A simple example is the Range measure where the egalitarian value only depends on the gap between the best and worst-off in the population. On this view, the egalitarian value doesn't change with an increase in the number of pairwise inequalities as long as the distance between the best and worst-off is preserved (by, for example, adding more worse off people). Likewise, the view that Segall ascribes to Arrhenius above, which ignores the size of the inequalities and only counts the number of unequal pairs,Footnote ³¹ is actually incompatible with negative and positive egalitarianism.Footnote ³²

IV. Class division

Are there any other reasons to rank Z as worse that Z′ with respect to egalitarian concerns? Segall (Reference Segall2016) again hints at the putative badness of the isolation of a single worse-off individual in a society (see quote in the preceding section). But analogously to our reasoning above regarding the case in figure 1, this putative badness can hardly outweigh the greater number and the greater size of the inequalities in Z′ as compared to Z.Footnote ³³

More interestingly, one might argue that Z is worse than Z′ regarding another egalitarian consideration: class division. In Z, there is a clear class division with one group of better-off people and another group with (only) one worst-off person. In Z′, one might argue, there is no clear class division since there is only a small difference in wellbeing between each adjacent individual (see figure 2). Hence, there seem to be no distinct upper or lower classes in Z′, or so the argument goes.Footnote ³⁴

Several things can be said in response to this argument. As with the claim regarding the badness of isolation, this could be taken as an empirical claim regarding the negative welfare effects of class division. It has been shown that there are positive correlations between different positions in a social hierarchy and the degree of mental and physical health, the degree of happiness, the rate of deaths, etc., of that position in the hierarchy.Footnote ³⁵ As true as that is, this version of the objection misses its target in the same way as the ‘empirical’ version of the isolation argument above, since any such effects on people's welfare are already taken into account in the figure.

More interestingly, whether Z and Z′ can be interpreted as containing a class division depends on the conception of welfare involved in the case or whether an index or proxy for welfare is used. For some conceptions or proxies of welfare, welfare differences might very well track class differences, understood in terms of social or economic status, the most common elements used to define class.Footnote ³⁶ For others, there is no clear connection. For example, conceptions or proxies of welfare such as income and wealth are likely to track class differences.

On the other hand, with conceptions of welfare such as experientialist, desire satisfaction, and objective lists theories the connection between welfare and class is weaker. For example, given these conceptions, the reasons why the welfare of individual 1 in Z is significantly lower than the others’ may be due to a painful medical condition that others in that population don't suffer, but she might otherwise enjoy high social and economic status. She might, for example, be a member of a wealthy aristocratic family despite suffering from such a painful condition. Such interpretations are not ruled out by the very abstract description of the case that we have provided here. In conclusion, these conceptions of welfare differences don't always track class differences.

The notion of welfare used in this article is a broad one and is compatible with experientialist, desire satisfaction, and objective list theories.Footnote ³⁷ Given that we use a broad notion of welfare, one cannot infer from the description of Z and Z′ that those populations contain any particular social stratification, or that the individuals involved in those populations belong to any particular class.Footnote ³⁸

However, let's for the sake of the argument accept that Z and Z′ can be compared in terms of class division. Even so, it would not be true that Z is worse than Z′ from the point of view of class division. This becomes clearer with the following representation of the case:

Although the difference in welfare between each pair of adjacent people in Z′ is small, by looking at figure 4 we can clearly see that there are groups whose individuals are much better off than the individuals in other groups, such as the people in the 1st and 2nd classes as compared to the people in the 3rd class. Indeed, we can imagine that these groups represent the lower class, middle class, and the upper class in that population. And the fact that there are people in between these three classes who cannot be clearly classified as belonging to any of these classes doesn't refute the existence of this class division in Z′.

Figure 4. Possible class divisions in Z and Z′.

Perhaps one might object here that it is not clear how the grouping into classes would be made in a society that looked like Z′, or that there is no salient way of doing so, while such a grouping is rather natural in Z. As true as this might be, it is of no normative significance. That there are many ways of grouping people doesn't show that there wouldn't be some ways (or at least one) that are more relevant from the perspective of egalitarian concerns. And our point is that there is at least one such grouping that shows that Z′ is also a class society. Hence, it is not the case that Z is worse than Z′ regarding this egalitarian consideration.

V. Perfect and rough equality

At this point, one might worry that the question of whether equality can have positive value is only interesting if people can in fact have exactly the same level of the currency that ought to be distributed equally, in a wide range of cases. For example, in realistic cases, perfect equality of welfare will almost never hold, or so one might argue. So, if welfare is the correct egalitarian currency, and if it is so rare that two or more individuals will be perfectly equal in respect to welfare, what is then the point of determining the positive value of equality? Doesn't this make the focus on equality, albeit theoretically interesting, practically irrelevant? Larry Temkin has recently put forward this argument.Footnote ³⁹

There are at least two answers to Temkin's worry. Firstly, whether people can achieve perfect equality depends on how the egalitarian currency is specified. In realistic cases, perfect (formal) legal equality (e.g. the same laws for different ethnic groups, or gender-neutral laws) is arguably much easier to obtain than perfect equality of welfare and this equality has already been achieved among large groups of people in many countries, or so one might argue. Hence, if one cares about legal equality, then the positive value of equality is practically relevant. Moreover, one could care about inequality and equality for several currencies at the same time, for example both legal equality and equality of welfare. In that case, even if perfect equality of welfare is rare, the positive value of equality is still practically relevant since perfect legal equality isn't rare.

Secondly, Arrhenius (Reference Arrhenius, Frithjof Norheim, Eyal, Hurst and Wikler2013) has suggested a version of positive egalitarianism in terms of what he calls ‘rough equality’. Small differences in welfare, that is, small inequalities, can also be described as ‘rough equalities’. One might then ask whether it is only perfectly equal relations that have positive value or whether this is also true of roughly equal relations. A promising idea is that inequality has negative value when the inequality is sufficiently big (‘strict inequality’ as we can call it), but when it gets smaller we reach the border for rough equality, which is of neutral value, and when the welfare difference gets even smaller, rough equality has positive value, and this value increases the closer we get to perfect equality. So, for example, <5, 10> might have negative value from an egalitarian perspective; <8, 10> neutral value, since we have reached the border for rough equality; <9, 10> positive value since we are getting close to perfect equality; and lastly <10, 10> has maximal positive value regarding egalitarian concerns.

We can formulate rough positive egalitarianism in the following way:

This view, we suggest, can accommodate the worry raised by Temkin. Rough positive egalitarianism captures the idea that individuals don't need to be at the exact same level of welfare in order for there to be some positive value of equality in the relation that holds between them. There is a point at which individuals’ welfare is close enough to each other's to instantiate this value: when they are sufficiently equally well off so that they can, so to speak, look each other in the eyes as equals. This view, we believe, fits the way in which people think and talk about equality. When lay people and policy makers express pro-attitudes and make claims regarding equality, they rarely have in mind perfect equality but rather rough equality. Real policies with an egalitarian agenda seldom aim at the achievement of perfect equality of the welfare of the citizens in a given country (although they might do so when it comes to other currencies, like the achievement of perfect legal equality).

Again, this conception of equality is analogous to how the value of desert has been conceptualised. If someone deserves a certain amount of some good, it is not only good that the person receives exactly what she deserves, it is also good, although less so, that she receives close to what she deserves. A mismatch between desert and receipt is only bad when the difference between what is deserved and what is being received is sufficiently big.Footnote ⁴¹

Segall argues that there is a tension between positive egalitarianism and rough positive egalitarianism. It involves population Z and Z′ from the progression case discussed in the previous sections. According to Segall, rough positive egalitarianism judges Z as worse than Z′. And this, Segall argues, contradicts the ranking of positive egalitarianism for these populations:

And he continues, referring to the supposed implications of rough positive egalitarianism:

It is important to note, though, that it is not the case that ‘in Z′ we have almost half a million relations of small inequalities and hence neutral value’. Certainly, there are small inequalities between adjacent or close pairs in the progression and they might be of neutral value or even of positive value according to rough positive egalitarianism. The latter would count in favour of Z′ being better than Z from an egalitarian perspective. But, as we pointed out above, the vast majority of the relations of inequality in Z′ involve big and greater gaps than the inequalities in Z. Hence, we can conclude that also for those who adhere to rough positive egalitarianism, there are very strong reasons to hold that Z′ is worse than Z with respect to egalitarian considerations. The same reasons that support the ranking provided by negative and positive egalitarians also apply for rough positive egalitarianism.Footnote ⁴³

VI. The repugnant equality objection

Finally, one may think that upholding to a view like positive egalitarianism, which attributes positive value to relations of equality, might lead to a version of Derek Parfit's famous repugnant conclusion which, following Mosquera (Reference Mosquera2017b), we will call repugnant equality:

According to Segall, positive egalitarianism ‘probably’ ranks Z as better than A from the point of view of equality even when the lives in Z are barely worth living since Z ‘contains many more relations of equality’ as compared to A.Footnote ⁴⁵ Hence, positive egalitarianism implies that Z is better than A in one respect, namely in respect of egalitarian concerns, that is, repugnant equality. He finds this implication counterintuitive.Footnote ⁴⁶

Figure 5. Repugnant equality.

Repugnant equality is a variation of Derek Parfit's well-known repugnant conclusion.Footnote ⁴⁷ It is also structurally analogous to the levelling down objection in the following sense. According to the levelling down objection, one cannot make a situation better in any respect by only decreasing the welfare of the better-off (to the same level as the worse-off). Many conceptions of the value of equality, like those discussed here, have this implication, but, along with many egalitarians, we don't find the objection persuasive. It would be a different matter if a theory implied that it was all things considered better to level down, which of course would be highly counterintuitive.Footnote ⁴⁸ Likewise, it doesn't follow from repugnant equality that one has to accept that Z is better than A all things considered, but only that Z is better in one respect. We reject the former value judgement but, pace Segall, don't find the latter one counterintuitive.

Actually, the very idea that it can be a decisive argument against a theory that it implies that an outcome is better or worse in one respect is often question-begging. The reason is that this kind of objection can be launched against any theory according to which the value can vary in at least two dimensions. This is, however, an unsurprising feature of such theories. Take hedonistic total utilitarianism, according to which the value of an outcome is determined by the total sum of pleasure over pain, as an example. Consider the value of adding a person with negative lifetime total sum of pleasure and pain. Of course, this would be all things considered bad according to hedonistic total utilitarianism. However, assume that this life involves one minute of pleasure in addition to all the pain and suffering. Then it is true that in one respect, from the perspective of hedonistic total utilitarianism, it is good to add this life because one minute of pleasure is added to the world. Clearly, this isn't much of an objection to this version of utilitarianism. Likewise for the levelling down objection against egalitarian theories in general and the ‘in-one-respect-better objections’ raised above by Segall against positive egalitarianism. There is nothing counterintuitive about this feature of multidimensional axiologies for which outcomes can be both good and bad in different respects.

Moreover, positive egalitarians are not even committed to ranking Z as better than A in respect of egalitarian concerns, that is, they are not committed to repugnant equality. As Arrhenius has pointed out, ‘how valuable equal relations are might arguably depend on the level of welfare involved’.Footnote ⁴⁹ This is analogous to the commonly held belief among egalitarians that the negative value of unequal relations partly depends on the welfare levels of the involved individuals such that inequalities at low levels are worse than inequalities at high levels. Hence, it is open to positive egalitarians to give different values to equal relationships at different welfare levels such that, for example, no number of equal relationships at a very low positive welfare level is better with regard to egalitarian concerns than a certain number of equal relationships at a very high positive welfare level. This could, to take a simple example, be achieved by aggregating the value of equal relationships at high welfare levels linearly while the value of equal relationships at low positive welfare levels is aggregated by a strictly concave function with an upper limit.Footnote ⁵⁰ Of course, there are more elaborate ways of developing this idea, for example, by having concave functions with different limits for each level of welfare. Hence, pace Segall, positive egalitarianism doesn't imply repugnant equality.Footnote ⁵¹

Similar considerations hold for the following putative objection to positive egalitarianism raised first by Arrhenius:

As with repugnant equality and the levelling down objection, there are two possible answers here. One can bite the – in our view just imaginary – bullet and accept a defeasible egalitarian reason to add equal people with negative welfare. Or, in extending positive egalitarianism to negative welfare levels, one can assign neutral value to equal relationships of negative welfare. In such cases, it wouldn't follow that adding people with negative welfare to a population would make such a population in one respect better. We prefer the first option since we are not persuaded by the levelling down objection and since we find it hard to motivate the second option.Footnote ⁵³

A related and interesting objection to positive egalitarianism (which also could be directed at negative egalitarianism, mutatis mutandis) has to do with its ‘impersonal nature’. The idea is that it is too concerned with making equal people instead of making people equal, to paraphrase Jan Narveson's famous slogan (in analogy to the criticism of total utilitarianism implying that we should make happy people instead of making people happy).Footnote ⁵⁴ As we have stated of positive egalitarianism, it is true that it isn't a so-called person-affecting theory: it disregards modal and temporal factors in its evaluation of outcomes.Footnote ⁵⁵ However, one could formulate person-affecting versions of positive egalitarianism, along the lines of how people have tried to formulate person-affecting versions of total utilitarianism. This latter project has, however, proved to be unsuccessful. The person-affecting intuition is very attractive but all efforts to incorporate it have yielded theories with strongly counterintuitive consequences.Footnote ⁵⁶ It is likely that the project of formulating a person-affecting version of positive or negative egalitarianism would meet the same fate, we surmise, so we shall not pursue it further here.Footnote ⁵⁷