2. TWO PARITY VIEWS

The first parity view is best understood as a version of the ‘rough equality’ view and I shall refer to it henceforth as the rough equality view. It is motivated by examples. Consider three excellent meals: f is an excellent French meal; i is an excellent Italian meal; and f+ is an excellent French meal which is slightly better than f. Suppose that neither f nor i is better than the other. Are excellent French and Italian meals exactly as good? ‘Exactly as good as’ is a transitive relation so that for any items x, y and z in the set of items, if x is exactly as good as y and y is exactly as good as z then x is exactly as good as z. If excellent French meals and excellent Italian meals are exactly as good, then given that ‘exactly as good as’ is transitive, f is exactly as good as i and i is exactly as good as f+ imply that f and f+ are exactly as good. Yet f+ is better than f. So while an excellent French meal is not better than an excellent Italian meal, they are not exactly as good either.Footnote ¹ But one might hold that they are nonetheless ‘comparable’. James Griffin (Reference Griffin1986: 80–2) sometimes refers to a distinct relation which arguably holds in examples of this sort as ‘rough equality’ while he also allows for the possibility that there may also be vagueness rather than a distinct relation in examples of this sort (see Qizilbash Reference Qizilbash2000); Parfit (Reference Parfit1984: 451) uses the term ‘rough comparability’ in such cases; and Ruth Chang invokes the relation of ‘parity’. I do not exclude the possibility that the examples which are used to motivate a distinct relation of parity might also be explained in terms of vagueness (see Sugden Reference Sugden2009; Qizilbash Reference Qizilbash2012). Rather I only take it, for the purposes of this paper, that the relation of parity can be motivated by these examples. While the rough equality view is influenced by Griffin's and Parfit's works, it follows Chang in using ‘parity’ and treats ‘parity’ as a term of art. Nonetheless, because of the influence of Griffin's work it is a version of the ‘rough equality’ view, where ‘roughly equal’ is understood as a distinct relation. ‘Parity’ is simply used as a term of art to refer to rough equality where this is understood as a distinct relation (Qizilbash Reference Qizilbash2005: 415).

To define parity formally, on the rough equality view, we need further definitions. ‘At least as good as’ is a primitive relation. x and y are exactly as good if and only if x is at least as good as y and y is at least as good as x. x is better than y if and only if x is at least as good as y and it is not the case that y is at least as good as x. x and y are incommensurate if and only if it is false that x is at least as good as y or y is at least as good as x. The rough equality view allows for the possibility that while x and y are incommensurate one is nonetheless comparable with the other. ‘Comparable with’ is taken to be a primitive relation. Parity is then defined as follows: x and y are on a par if and only if they are incommensurate and comparable. On this view, when parity holds between items – such as excellent Italian and excellent French meals – while some slight change in value may not tilt the balance in favour of one of them, any significant increase in the value of one of the items will make it better, and any significant reduction in the value of one will make it worse, than the other. This feature of parity is the ‘mark of parity’ on the rough equality view.Footnote ² While this mark does not follow from the formal definition of parity, it is implied by a central component of the rough equality view: the assumption that while parity is a form of equality and a distinct relation, it is not precise equality. The mark of parity also distinguishes parity from ‘incomparability’. If there were ‘incomparability’, on this view, while one is not better than the other even some significant increase (or reduction) in the value of one option would not make it better (worse) than the other. On the rough equality view, ‘exactly as good as’ and ‘better than’ are transitive.Footnote ³ Transitivity of ‘better than’ ('exactly as good as’) means that for any states of the world x, y and z in the set of states of the world X, if x is better than (exactly as good as) y and y is better than (exactly as good as) z then x is better than (exactly as good as) z. Furthermore, it is assumed that ‘BE transitivity’ holds so that for any states of the world x, y and z in X, if x is better than y and y is exactly as good as z then x is better than z.

The second parity view is based on a ‘fitting-attitudes analysis of value’ and I refer to it henceforth as the ‘fitting attitudes view’ – or ‘FA view’ for short. On this view value relations are defined as follows: ‘[a]n object is better than an another iff one is required to prefer it’; ‘[t]wo items are equally good iff they ought to be equi-preferred, i.e. if one is required to be indifferent between them’; and ‘[t]wo items, x and y, are on a par iff it is (i) permissible to prefer x to y, and (ii) permissible to prefer y to x’ (Rabinowicz Reference Rabinowicz2009 a: 402). ‘Radical’ incomparability is defined as follows: ‘x and y are incomparable if and only if it is required not to prefer one to the other or to be indifferent’ (Rabinowicz Reference Rabinowicz2009b : 84). In early statements of this view, ‘preference’ refers to a disposition to choose (Rabinowicz Reference Rabinowicz2009b : 83). However, on a more recent version it is conceived of in terms of degrees of favouring, so that if one item is preferred to another the first is favoured to a higher degree (Rabinowicz Reference Rabinowicz2012: 152). The definition of incommensurateness in this parity view is then: x and y are incommensurate if and only if they are not equally good and neither is better than the other. This definition of ‘incommensurate’ coincides with the definition of ‘incommensurate’ on the rough equality view if ‘equally as good’ means ‘exactly as good’. The more formal development of the FA view uses an ‘intersection model’ which involves a class of all permissible preference orderings K. The formal definitions of the relevant value relations are then: x is better than y if and only if x is preferred to y in every ordering in K; x and y are equally good if and only if they are equi-preferred in every K-ordering; and x and y are on a par if and only if K contains an ordering in which x is preferred to y and an ordering in which y is preferred to x (Rabinowicz Reference Rabinowicz2009b : 86).

Before proceeding, it is worth noting a difference between the two parity views. Because the rough equality view invokes the mark of parity insignificant changes in the value of two states which are on a par do not necessarily make one better or worse than the other, while significant changes in value do. This is only plausible if options which are on a par are roughly equal in value. The rough equality view may also differ here from Chang's view as it is expressed in her discussion of the relationship between her views and Parfit's. In that discussion, Chang contrasts parity and imprecise equality (Chang Reference Chang2016: 198–205). She also writes that: ‘imprecise equality is not a form of equality, but rather appears to be a distinct relation’ (Chang Reference Chang2016: 201) whereas on the rough equality view parity is rough equality understood as a distinct relation. If parity is a distinct relation which contrasts with exact equality what is distinctive about it as a form of equality is that when items are on a par insignificant changes in the value of one item need not make it either better than or worse than the other as they would in cases of exact equality. By contrast, the FA view does not invoke the mark of parity, nor does it claim that parity is a form of equality.

I've stressed the mark of parity as a feature that distinguishes the two parity views. It is worth considering a variation of the rough equality view. On this variation of the rough equality view, value relations are formally defined just as they are on the rough equality view using ‘at least as good as’ and ‘comparable to’ as primitives. But on this variation one would weaken the mark of parity. On this weakening, when parity holds between items, while some slight change in value may not tilt the balance in favour of one of them, either some significant increase in the value of one of the items will make it better or some significant reduction in the value of one will make it worse, than the other. This variation of the rough equality view would, however, not be a ‘rough equality’ view, since even some significant change in the value of one item need not necessarily change the comparative value of items when they are on a par. It does not have the mark of parity. I term this variation of the rough equality view, the ‘adjusted first parity view’.Footnote ⁴ On this variation of the first parity view, the distinction between parity and incomparability can be made as follows: when two items are incomparable, whilst they are incommensurate, no change in the value of one item makes it better or worse than the other.

3. MERE ADDITION AND THE INTUITION OF NEUTRALITY

In cases of mere addition, extra people exist: (i) who have lives worth living; (ii) who affect no one else; and (iii) whose existence does not involve any social injustice. Discussions of the ‘mere addition paradox’ sometimes drop (iii) (see, for example, Temkin Reference Temkin1987) and I follow this precedent for simplicity. To illustrate the rough equality view, I begin with a variation of one of Parfit's examples. In presenting the example I use notation Broome has used in his Weighing Lives and elsewhere (e.g. Broome Reference Broome1999: 230). In this notation each possible state is represented by a vector. Each place in the vector stands for a person who lives in at least one of the states of affairs being compared. The corresponding place in each vector compared stands for the same person. In a state where she does not exist, her place contains an Ω. If she exists, her place contains a number which indicates her lifetime well-being. Well-being is assumed to be measurable on a cardinal scale which is comparable between people. Any positive level of well-being is taken to be good rather than bad. Consider the following example involving three states of the world:

EXAMPLE 1.

• a: (10, 10, Ω, Ω)

• a+: (10, 10, 1, 1)

• b: (8, 8, 8, 8).

a+ adds two extra people to the population in a without affecting existing people's well-being. The lives of the extra people are good and so are worth living. a+ is also not worse than a since it would be hard to claim that it would have been better had these extra people not lived. Now consider b – here everyone who exists in a+ has a well-being level of 8 and aggregate welfare is higher than in a+. Welfare is also more equitably distributed in b than it is in a+. So it can be argued that b is better than a+. However, Parfit (Reference Parfit1984: 388, 425–32) thinks that if we accept that b is better than a we might accept:

The Repugnant Conclusion: for any possible population of at least 10 billion people, all with a very high quality of life, there must be some much larger imaginable population whose existence, if other things are equal would be better, even though its members have lives that are barely worth living.

This is because by repeatedly applying this logic in comparing states while doubling the size of the population one might endorse the view that a very large population living lives barely worth living is better than a small population enjoying a high quality of life. Because Parfit thinks that moral theories should not lead us to accept the Repugnant Conclusion, he thinks that these should judge that b is not better than a. However, if b is better than a+ and if a+ is not worse than a means that a+ is at least as good as a, then there are two possibilities. Either b is better than a+ and a+ is better than a so that by transitivity of ‘better than’ b is better than a. Or b is better than a+ and a+ is exactly as good as a so that by BE transitivity b is better than a. Either way, it follows that b is indeed better than a. Yet to avoid the Repugnant Conclusion we have judged that b is not better than a. So we have a contradiction. This is one version of the mere addition paradox. One potential ‘way out’ of this paradox is to drop the assumption that ‘better than’ is transitive (see Temkin Reference Temkin1987) but, as we have seen, the rough equality view explicitly assumes that this relation is transitive. An alternative potential ‘way out’ is to suppose that while a and a+ are comparable, they are incommensurate. If so, on the rough equality view they are on a par. In Parfit's own terms they are ‘roughly comparable’.

John Broome's discussions of mere addition focus on what he calls the ‘intuition of neutrality’. This intuition can be traced to Jan Narveson's path-breaking contributions (Narveson Reference Narveson1967, Reference Narveson1973). Narveson (Reference Narveson1973: 63) thinks that utilitarians are ‘in favour of making people happy, but neutral about making happy people. Or rather, neutral as a public policy, regarding it as a matter for private decision’. But this intuition can be filled out in different ways. One way in which Narveson (Reference Narveson1967: 65) fills it out is this: ‘[i]f an action would have no effects whatever on the general happiness, then it would be morally indifferent: we could do it or not, just as we pleased.’ A key element of the intuition, on my reading, is that – at the level of public morality – we are indifferent between certain acts in the sense that either is morally permissible if they affect nobody's welfare. In the context of adding people to the world, Narveson (Reference Narveson1967: 65) adds that: ‘[h]aving children, in other words, is normally a matter of moral indifference’. He qualifies this claim by arguing that ‘it does follow from utilitarian principles that, if we can predict that a child would be miserable if born, then it is our duty not to have it’ (Narveson Reference Narveson1967: 69).

It is a version of this intuition which Broome has in mind much of the time. What Broome (Reference Broome2004: 143) means by ‘neutral’ is ‘ethically neutral’. And for Broome (Reference Broome2004: 142) a level of well-being is neutral if, as regards a life lived at that level of well-being, ‘it is neither better nor worse that this life is lived than that it is not lived’. To explain Broome's position, let us first suppose that there is just one neutral level of well-being: adding lives above this level makes the world better and adding lives below this level makes it worse. In this case, what Narveson calls ‘moral indifference’ would be construed in terms of exact equality of goodness and there is only one level of well-being such that we are indifferent about adding someone to the world at that level as regards public policy. Broome thinks that according to the neutrality intuition there is a ‘neutral range’ such that adding someone at any level in the range is ethically neutral. The existence of such a range is what earlier I termed the ‘core’ neutrality intuition. Above this range it is better to merely add people to the world; and below it, it is worse to do so.

In characterizing the neutral range, the rough equality view follows a hint in the work of Blackorby, Bossert and Donaldson. While they do not say much about this range (or ‘interval’), at one point Blackorby et al. (Reference Blackorby, Bossert and Donaldson1997: 218) suppose rather tentatively that lives at the top end of the neutral range are ‘short of flourishing’, while those at the bottom of the range are ‘barely satisfactory’.Footnote ⁵ The rough equality view maps this intuition as follows. Lives in the neutral range are merely satisfactory. A mere addition of people with merely satisfactory lives is on a par with the original situation. Lives lived at a significantly better (worse) level than merely ‘satisfactory’ are flourishing (miserable). Furthermore, the borderline between lives which are significantly better (worse) than merely satisfactory and those which are not is imprecise. So the borderlines of the neutral range are imprecise on the rough equality view. For expository purposes in further explaining the rough equality view I focus exclusively on cases where one person is merely added to the world. I write the level of well-being of the added person as μ. Consider some level of μ in the neutral range where lives are merely satisfactory. On this view, any significant increase (decrease) in μ makes the world with the added person significantly better (worse). So the mere addition of flourishing (miserable) lives makes the world better (worse). This view is clearly consistent with Narveson's view that adding people whose lives are miserable makes the world worse, though it is inconsistent with the view that adding people with happy lives is neutral, if those lives are flourishing.

The FA view of mere addition (Rabinowicz Reference Rabinowicz2009 a: 403) is also related to Blackorby, Bossert and Donaldson's view. Because choice of some level of well-being or subinterval in the neutral range is central to this parity view, the relevant levels of well-being in the range are neutral. I write the well-being of person i in state of the world x as W_i (x) and π for a neutral level of well-being. For all states x in X, neutral range utilitarianism ranks states of the world according to the following formula:

$$\begin{equation*} {\Sigma _{i\,{\rm exists}\,{\rm{in}}\,x}}[{W_i}(x)-\pi ] \end{equation*}$$

On the FA view a preference ordering P_π on the set of states of the world is induced by a neutral well-being level π if and only if for any state of the world x, the position of x in P_π is determined by this formula. On this view, a preference ordering P belongs to the class of permissible orderings K if and only if there is a subinterval I of the neutral range such that P is the intersection of all complete orderings P_π induced by different well-being levels within I (see Rabinowicz Reference Rabinowicz2009 a: 404).

It is easiest to explain the FA view by considering an example (Rabinowicz Reference Rabinowicz2009 a: 405). In this example there are three people – one of whom may or may not exist – and four states of the world:

EXAMPLE 2.

• e = (3,4,Ω)

• f = (3,4,1)

• g = (3,3,3)

• h = (3,3,Ω)

Suppose that 1 and 3 are both in the neutral range so that preference orderings induced by these levels of well-being belong to the class of permissible orderings K on neutral range utilitarianism. Now consider the formula for neutral range utilitarianism. If π>2, e is preferred to g and if π<2 g is preferred to e. Different choices of π induce different preference orderings. Given that 1 and 3 are both in the neutral range, it follows that it is permissible to prefer e to g and also permissible to have the opposite preference. This is an instance of incommensurateness on the FA view since neither e nor g is preferred in each K ordering, nor are they equally good in each K ordering. The FA view then answers the question: ‘[h]ow can we explain that mere additions result in incommensurateness?’ as follows: ‘[i]ncommensurateness is explained by the permissibility of different preference orderings (which correspond to different choices of subintervals within the neutral range)’ (Rabinowicz Reference Rabinowicz2009 a: 405). This is because e and g are on a par, since in one K ordering e is preferred to g while in another g is preferred to e. For this reason, neutral range utilitarianism is a version of the parity view of mere addition.

The difference between the two parity views which was earlier noted leads to differences in the way they characterize the intuition of neutrality. On the rough equality view the intuition that ‘neutrality is parity’ hinges on the fact that parity is a form of rough equality so that when the mere addition of extra people is on a par with the original situation it is a matter of moral indifference. On the FA view, by contrast, when states of the world are on a par, a preference for either is permissible so that there is indifference about the preference one has. Furthermore, because the rough equality view invokes the mark of parity, it characterizes the neutral range in a way which the FA view need not. These differences between the two parity views are illustrated in the following sections, through a discussion of John Broome's objections. However, because it does not invoke the mark of parity the FA view is not open to the first objection I discuss.

4. DOES THE ROUGH EQUALITY VIEW IMPLY A NARROW NEUTRAL RANGE?

Broome's interpretation of the neutrality intuition is that it is very often neutral to add someone to the world (Broome Reference Broome2004: 143). One of Broome's objections to the rough equality view is that ‘[m]ost people think the neutral range is wide’ (Broome Reference Broome2007 a: 153). On his view, the intuition of neutrality implies a neutral range which is ‘wide’ in the sense that it can contain two levels of well-being, with one being much (and thus significantly) better than the other (Broome Reference Broome2007 a: 154). Broome (Reference Broome2007 a: 154) claims that the rough equality view ‘does not meet the problem that concerns me, since it implies that the zone is insignificantly wide’.

The neutral range is certainly ‘narrow’ in a particular sense on the rough equality view. Consider the question of whether to add a person to the world at some level of well-being μ which is in the neutral range. Suppose we define a ‘narrow’ neutral range as follows: a neutral range is ‘narrow’ if and only if for all levels of well-being in the neutral range, any significant increase in μ makes the world with the added person better than the world without addition or ‘status quo’ and any significant decrease in μ makes the status quo better than the world with the added person. A neutral range is ‘wide’ if it is not narrow. The rough equality view assumes that any significant increase (decrease) in μ makes the state with the added person significantly better (worse) (Qizilbash Reference Qizilbash2005: 422–5; Reference Qizilbash2007 a: 136). If we accept this assumption, this view implies that if mere addition of lives in the neutral range is on a par with the status quo, the neutral range is narrow. To see this, suppose that the neutral range is a zone of parity and it is not narrow (i.e. wide). Then there is some μ in the neutral range such that some significant increase (or decrease) in the well-being level of the added person would not make the state with the added person better (worse) than the status quo. On the rough equality view, for all μ in the neutral range the status quo is on a par with the state with the added person. Furthermore, if the status quo is on a par with the state with the added person then by the mark of parity, while some slight improvement (worsening) in either state may not make it better (worse) than the other, any significant improvement (worsening) will. As we saw, the rough equality view assumes that any significant increase (decrease) in μ makes the world with the added person significantly better (worse). So if, for all levels of μ in the neutral range, the status quo is on a par with the world with the added person then any significant increase (or decrease) in the well-being of the added person would make the state with the added person significantly better (worse) and would make the state with the added person better (worse) than the status quo. Earlier we concluded from the neutral range not being narrow that there is some level of well-being in the neutral range and some significant increase (or decrease) in this level would not make the state with the added person better (or worse) than the status quo. This is a contradiction. So according to the rough equality view, the zone is narrow.

One key question which is relevant in interpreting the result that the neutral range is narrow is: what makes an increase in μ ‘significant’? One way of answering this question is to define a ‘significant’ increase (decrease) in μ as one which is sufficiently large that it makes the world in which the person is added significantly better (worse). If we adopt this definition then the assumption made in deriving the result that the range is narrow would hold by the definition of a ‘significant’ increase or decrease in μ. Yet, if the range is narrow in this sense, it is far from clear that it is ‘narrow’ in any ordinary sense. And it is something like the ordinary senses of a ‘narrow’ or ‘wide’ range of lives which is implicit when Broome appeals to the intuition of neutrality and the width of the neutral range implied by that intuition. For example, he writes that ‘[w]e think intuitively that adding a person to the world is very often neutral’ (Broome Reference Broome2004: 143) which suggests that the neutral range is ‘wide’. The rough equality view may be consistent with this intuition if the neutral range consists of lives which are merely satisfactory and there is, in some ordinary sense, a ‘wide’ range of lives which are merely satisfactory so that adding someone to the world is very often neutral.

If one defends the rough equality view along these lines, a reasonable question to ask is whether one can say more in characterizing a ‘merely satisfactory life’. One way of doing this is to return to Parfit's own text. In his discussion of mere addition, Parfit at one point discusses an (appeal to) the ‘valueless level’ of well-being. Parfit tells us that below this level if ‘lives are worth living, they have personal value – value for the people whose lives they are. But the fact that such lives are lived does not make the outcome better’ (Parfit Reference Parfit1984: 412). The quality of life of the people who are merely added must be below the valueless level, since if they were above this level then their addition would make the outcome better. Nonetheless, the lives of those who are merely added must be worth living. So one way to characterize ‘merely satisfactory lives’ is that while such lives are worth living, and have value for the people whose lives they are, they are below the valueless level. The vagueness at the boundaries of the zone of parity is vagueness – at the bottom end – about boundary cases of a life worth living – and at the top end – about the point at which addition of life makes the outcome better. When one considers the range of lives which fall between those which are definitely not worth living so that on this reading their addition would make the world worse and those which are definitely of such high quality that their addition makes the world better, this range may also be ‘wide’ in the ordinary sense that a wide variety of lives would fall in this range. Indeed, in an ordinary sense of ‘significantly’, the quality of life of someone at the bottom (top) of the range could be significantly worse (better) than the quality of life of someone at the top (bottom).

An alternative way of answering the question of what makes an increase in μ ‘significant’ might drop the assumption that any significant increase (decrease) in μ necessarily makes the world with the added person significantly better (worse). Dropping this assumption would allow the rough equality view to capture the intuition that the zone is ‘wide’ even in the sense used in deriving the result that the zone is narrow, which is the sense Broome is using in objecting to the rough equality view. It is thus worth asking: is this assumption plausible? To answer this question, we need to revisit what makes an increase or decrease in the well-being of the added person ‘significant’. On the rough equality view, adding a person who has a (merely) satisfactory life is on a par with the status quo, while adding a person with a flourishing (miserable) life is better (worse) than the status quo. Should we stand by this position? Some might want to resist the view that increasing (decreasing) the quality of the added life from merely satisfactory to a flourishing (miserable) life necessarily makes the world with the added person significantly better (worse). One might hold the intuition that the quality of life of the added person should be more than merely flourishing (miserable) – it may need to be ‘superlative’ (‘utterly miserable’) for the addition of one person at that level to make the world with the added person significantly better (worse). If so, the derivation of the result that the rough equality view implies a narrow neutral range no longer follows and this view allows for a wide neutral range. If we find this intuition persuasive, it would be unnecessarily restrictive to rule out the possibility that even if a flourishing life is significantly better than one which is merely satisfactory, the world with the added person whose life is flourishing is not better than the world without that person.

There are, nonetheless, reasons for hesitating before accepting a wide neutral range along these lines. If we allow for addition of people at a level which is miserable (if not utterly miserable) to be on a par with the initial situation, the rough equality view is not consistent with that part of Narveson's intuition which suggests that ‘if we can predict that a child would be miserable if born, then it is our duty not to have it’. Furthermore, a miserable life may not be worth living, so that addition of people at this level may not count as mere addition. These considerations restrict any ‘widening’ on the lower, but not at the upper, end of the neutral range. They suggest the following view: the quality of life of the added person should be more than merely flourishing for addition at that level to make the world with the added person better than the status quo, while addition of someone at a miserable level would make the world with the added person worse.Footnote ⁶ This view would, nonetheless, violate the assumption we made in deriving the result that the zone is narrow: it implies a wide zone.

Finally, as discussed earlier, the adjusted first parity view discussed in the previous section might drop the mark of parity, so that when two states are on a par some (rather than any) significant change in the value of one state makes one better or worse than the other. If that is so, then given the other assumptions made on the rough equality view, one can accept the view that adding a miserable life – which is significantly worse than a merely satisfactory life – makes the outcome worse, even if adding a flourishing life which is significantly better than a merely satisfactory life does not necessarily do so. The adjusted first parity view thus allows for the possibility that there is a wide neutral range and no upper bound to the neutral range.

5. IS NEUTRALITY AS PARITY A ‘FUDGE’?

Broome thinks that ‘neutrality is most naturally understood as equality’ and that resorting to incommensurateness ‘looks like a fudge unless we can offer some reason why the neutrality of existence really amounts to incommensurateness rather than equality’ (Broome Reference Broome2004: 168–9). This ad-hocness objection applies to parity views because parity is a form of incommensurateness. Given my interpretation of the intuition this objection only has force in relation to the rough equality view if one naturally interprets the notion of ‘moral indifference’ in terms of exact equality of goodness. As we have seen, on the rough equality view parity is a form of equality – but it is not precise equality (Qizilbash Reference Qizilbash2007 a: 142; Reference Qizilbash2007 b: 111). This is an unsurprising claim since one meaning of the word ‘parity’ is equality – indeed this is the first meaning of the word offered in the Oxford English Dictionary. On this parity view the obvious attitude to take to states which are on a par is a form of indifference. For this reason, Griffin (Reference Griffin1986: 97) observed that: ‘[w]here we have rough equality, we treat the items, when it comes to choice, simply as equals. We are indifferent between them, even though our indifference in this case has an uneasiness about it absent in cases of strict equality.’ So when it comes to choosing between having a child and not doing so, if the level of well-being of the child is within the neutral range, it is a matter of moral indifference whether or not to have the child.

In responding to the rough equality view, Broome also suggests that incommensurateness might be relevant in contexts where there is more than one value involved. But in cases of mere addition, he thinks that this is not so. He writes that his remark that ‘[n]eutrality is most naturally understood as equality of value’ (Broome Reference Broome2007 b: 121) was meant to apply only to a particular context. In cases where there are choices involving more than one dimension it is natural to think that the values may be ‘incommensurable’. But when only one value is at stake this is not a natural understanding: it is more natural, Broome suggests, to think that the options – say two glasses of lemonade which are valued because they realise one value (pleasure) – are equally good. And in adding people to the world there is only one value ‘if it is a value’ at stake: the ‘number of people’. So it is not clear why there is ‘incommensurability’ in this case. Broome (Reference Broome2007 b: 121) suggests that the rough equality view provides the beginnings of a response to this worry by claiming that comparisons involving states in which some people are alive and others in which they are not are ‘complex’ in a way that cases involving a fixed population are not.

How might the rough equality view respond? The first point to note is that in the variation of the mere addition paradox provided in Example 1 (and indeed even in Broome's variation on the paradox in Example 3 below) there are at least two values involved in comparisons between states: (aggregate) well-being and equality. Nonetheless, suppose that we concede that there is only one value involved. The relation of parity is motivated by examples in the rough equality view. In the example involving excellent French and Italian meals given earlier there need be only one value (such as pleasure) involved when comparing the goodness of meals. The same can be said about the example Parfit offers in motivating his ‘rough comparability’ view. He (Parfit Reference Parfit1984: 430–2) compares the relative value of two Poets and a Novelist. In a comparison between the value of a Poet and a Novelist ‘rough comparability’ may hold because the items being compared – e.g. the achievements of the relevant authors – are rather different in nature or kind. This point clearly applies even if there is only one value – achievement – involved in the comparison. Of course, one might object here that multiplicity is still relevant to the example that Parfit has in mind. It may be that the Novelist exemplifies different virtues or kinds of achievement to the Poet, and this multiplicity of virtues or types of achievement is what generates the roughness in comparisons (Griffin Reference Griffin1986: 80). But even if that is so, the underlying comparison can still be of just one value (such as achievement) even if one acknowledges different virtues or kinds of achievement which can be displayed in realizing that value. Here it is helpful to distinguish between ‘value-types’ and ‘value-tokens’. ‘Accomplishment’ is a value-type and distinct realizations of this value can count as ‘value-tokens’. The ‘complexity’ involved in the comparison may relate not to the number of values involved, but to something else such as the distinct virtues exemplified in realizing the value or the distinct kinds of value-token involved in a comparison. The ‘complexity’ invoked in the rough equality view is arguably of this sort. It might relate to the number of dimensions involved in a comparison – there is an added dimension in the case where some people exist in one state and not another which is not present in cases where the same people are alive in each – which does not necessarily relate to the number of values involved. I should add that the mere fact that comparisons between different states can sometimes be complex in this way does not necessarily imply that comparisons between states where different people exist need invariably involve roughness in comparisons, just as multidimensional choices need not necessarily involve parity or imprecision.Footnote ⁷ Nonetheless, if the rough equality view insists there is imprecision or parity in population comparisons only in cases of mere addition, it would certainly be ad hoc. If this is how parity views are construed then parity views of mere addition fall foul of the objection. But parity views need not insist that parity in population comparisons only arises in cases of mere addition.

Ruth Chang (Reference Chang2016) has advanced a different rationale for invoking parity in population ethics. Might her intervention help to address the ad-hocness objection? Chang's discussion is related to Parfit's recent development of his imprecise equality view (Parfit Reference Parfit2016: 120–6). The crucial point in Chang's discussion relates to the ‘slide’ to the Repugnant Conclusion. If we begin with a small population all living at the same high quality of life, and repeatedly increase the size of the population while decreasing the quality of life of those alive, given transitivity of ‘better than’, the Repugnant Conclusion only follows if, when we compare each pair of states of the world in the sequence, the world with the larger population is better than the world with the smaller population. Parfit claims that if there is a pair of worlds in the sequence which are, in his terms, imprecisely equally good, the Repugnant Conclusion does not follow. Chang distinguishes ‘imprecise equality’ and ‘parity’. But she makes a point similar to Parfit's by suggesting that in the sequence ‘since there is a range of items that are on a par with one another, we break the chain of betterness . . . and the slide to the Repugnant Conclusion is halted’ (Chang Reference Chang2016: 210). Chang's explanation for why there is parity in this case is that as we move along the sequence of states of the world ‘a qualitative difference in successive outcomes will begin to manifest such that a successor is no longer better than its predecessor but on a par with it’ (Chang Reference Chang2016: 211). This qualitative difference might arise because as we move down the sequence and people's lives are less good, certain values (or value-tokens) can no longer be realized. At some point in the sequence, Chang would suggest, a larger population at some level of well-being is not better than a smaller population at a higher level: it is on a par. This way of introducing parity in discussions of the Repugnant Conclusion does not invoke mere addition. So it does not help address the ad-hocness objection to parity views of mere addition.

How does the FA view of mere addition respond to the objection that ‘parity as neutrality’ may be a ‘fudge’? To explain how it does, recall that in Example 2, on this view, it is permissible to prefer one state to the other – e to g in the relevant example – and it is also permissible to have the opposite preference. On this view, the answer to the question: ‘[h]ow can we explain that mere additions result in incommensurateness?’ is that: ‘[i]ncommensurateness is explained by the permissibility of different preference orderings (which correspond to different choices of subintervals within the neutral range)’ (Rabinowicz Reference Rabinowicz2009 a: 405). This is because e and g are on a par, since in one K ordering e is preferred to g while in another g is preferred to e so that on the FA view ‘[t]here is no need to appeal to heterogeneous values . . . to arrive to bona fide cases of incommensurate alternatives’ (Rabinowicz Reference Rabinowicz2009 a: 405). This response only has force, of course, if the only form of incommensurateness involved in mere addition is parity as is the case on the FA view. Nonetheless, one might still ask why either preference is permissible in cases of mere addition. The answer to this question might invoke some sort of multi-dimensionality or complexity in cases of mere addition as it does in the case of the rough equality view.

Broome is unconvinced by the FA view's characterization of neutrality as parity. He writes that on this view:

To defend the FA view, one might plausibly suppose instead that public morality requires certain preferences or attitudes and that – in the case of mere addition – there is no such requirement, so that any preference is permitted. In this specific sense, whether or not to have a child is a matter of ‘moral indifference’ and, to this degree is ‘ethically neutral’, at the level of public morality in cases of parity. Or at least the FA view might interpret the neutrality intuition in this way. And if it does so, it characterizes the intuition in a way which is different to the rough equality view.

6. IS NEUTRALITY AS PARITY ‘IMPLAUSIBLY GREEDY’?

Another doubt about parity views relates to Broome's view that incommensurateness involves a sort of ‘greedy neutrality’. Because parity is a form of incommensurateness, this doubt is an objection to both parity views. Broome (Reference Broome2004: 156) illustrates this point with a version of the mere addition paradox which involves three states. Consider:

EXAMPLE 3.

• k = (4,4, . . . 4, 6, Ω)

• l = (4,4, . . . 4, 6, 1)

• m = (4,4, . . . 4, 4, 4).

Suppose that 4 and 1 are in the neutral range. Broome's version of the paradox runs as follows. First, we judge that m is better than l because m has greater aggregate well-being, and a more equal distribution, than l does. Because 1 and 4 are in the neutral range, k and l are incommensurate and it cannot be true that k is better than m. If it were, then, by transitivity of ‘better than’, k is better than l. However, we know that k and l are incommensurate. So we would be led to contradiction. But Broome insists that, if one accepts the neutrality intuition, k is better than m. He thinks this because in moving from k to m there are two changes. Firstly, one person has come into existence and the change is neutral because 4 is in the neutral range. Secondly, another person's well-being has fallen from 6 to 4. This is a bad thing. Broome thinks that the combined effect of a neutral change and a change for the worse implies that m is worse than k. If k and m are incommensurate, he thinks that neutrality is not what it should be because it ‘swallows up’ (i.e. in some way compensates for) the badness of reducing one person's well-being from 6 to 4. He thinks that this makes the neutrality involved in a neutral range implausibly ‘greedy’.

Broome's intuition that ‘neutrality is not implausibly greedy’ goes well beyond any version of the neutrality intuition that I have discussed so far.Footnote ⁸ Indeed, in the FA view, there is no attempt to defend the intuition that neutrality is not greedy and ‘adding people is (axiologically) neutral simply means that it on its own makes the world neither better nor worse’ (Rabinowicz Reference Rabinowicz2009 a: 399). So also, in the FA view, there is no attempt to defend a version of the neutrality intuition where: ‘by a neutral thing we mean one that makes no change to the value of the world’ (Rabinowicz Reference Rabinowicz2009 a: 398). On the FA view, even if the world is neither better nor worse with the mere addition of people it is not true that there is no change to the value of the world, since the world with the added people is not ‘equally as good’ as the status quo: the two worlds are incommensurate (Rabinowicz Reference Rabinowicz2009 a: 399) and on a par. Here a significant difference between the two parity views arises because on the FA view parity is not a form of equality. Nonetheless, even on the rough equality view, addition of people to the world in the neutral range does not leave the world with the added people exactly as good as before: the two states of the world are incommensurate.

While the rough equality view insists that this intuition about neutrality not being ‘greedy’ extends beyond cases of mere addition, it tentatively advanced a ‘hunch’ about one way in which it might be extended to cases where the addition of extra people can make existing people worse off. The hunch would extend the rough equality view beyond cases of mere addition so that if addition of a person (or people) is on a par with the status quo it cannot justify any significant sacrifice of other values (Qizilbash Reference Qizilbash2007 b: 114). The thought here is that even if neutrality is greedy on this way of extending it, it may not be ‘implausibly’ greedy. But Broome interprets some of his examples of ‘greedy neutrality’ in terms of global warming and in these examples he thinks that the addition of people does involve significant sacrifice of other values (Broome Reference Broome2007 a: 155). Clearly, if the parity view were extended beyond cases of mere addition in the way tentatively entertained in the rough equality view it would not treat addition as neutral in those cases. Nonetheless, it is not obvious that it makes sense to extend the rough equality view of mere addition beyond cases of mere addition.

Broome further develops this line of argument in his work on climate change. For two worlds A and B he defines the neutrality intuition as follows:

This intuition also goes beyond the core neutrality intuition which the parity view defends: it requires more than the existence of a neutral range.Footnote ⁹ Indeed, Broome treats this later formulation of the intuition as capturing an essential feature and part of the intuition. He thinks that refuting ‘part of it is enough to refute the intuition as a whole’ (Broome Reference Broome2004: 176). He goes on to present a counter-example which he thinks demonstrates that the intuition is false. It is not obvious that parity views of mere addition should contest Broome's counter-example because I doubt that the intuition at work here is the one that parity views seek to defend. Nor does either parity view treat this intuition as part of one larger intuition which extends beyond cases of mere addition. So even if Broome has convincingly demonstrated that (what he thinks is) a feature of the neutrality intuition should be rejected, his demonstration may not count against parity views of mere addition. For this demonstration to count against parity views, Broome would need to show that parity views of mere addition imply or defend the relevant ‘feature’ or ‘part’ of the intuition.