2. THE PARITY VIEW

To define parity I need some notation. I use ‘R’ to mean ‘better than or precisely as good as’. It is treated as a primitive relation. I also use: ‘∧’ for conjunction; ‘∼’ for negation; ‘∨’ for the inclusive ‘or’; ‘∈’, for ‘is a member of’; ‘⇒’ for the conditional ‘if . . . then’; ‘∀’ for the universal quantifier (‘for all’); ‘⇔’ for ‘if and only if’; and ‘≠’ for ‘is not identical to’. Various properties can be defined on a set of conceivable alternatives (or states of affairs), X. Any binary relation, O, may have the following properties: reflexivity: ∀x ∈X, xOx; transitivity: ∀x, y, z ∈X, (xOy∧yOz) ⇒ xOz; and completeness: ∀x, y ∈X, x≠y, xOy∨yOx. A relation which is reflexive and transitive but not necessarily complete is a quasi-ordering. I define ‘better than’ – which is written ‘B’ – and ‘exactly as good as’, which is written ‘E’ as follows: xBy ⇔[xRy∧∼(yRx)]; and xEy ⇔ (xRy∧yRx). I assume that R is a quasi-ordering, which implies transitivity of B inter alia. I also follow previous work (Qizilbash Reference Qizilbash2002, Reference Qizilbash2005) in which I defined Broome's notion of ‘incommensurateness’ as follows: x is B-incommensurate with y ⇔ ∼(xRy∨yRx).Footnote ¹ To define parity, I introduce another primitive relation: ‘comparable to’, which is written ‘C’. If this relation does not hold between alternatives, they are ‘incomparable’. I use P for ‘parity’ and define it as follows: xPy ⇔ xCy ∧ ∼(xRy∨yRx).Footnote ² Parity is a distinct relation: when it holds between options it cannot be the case that one is better than the other, or that they are exactly as good. There is also no vagueness about which relation holds when options are on a par. How do exact equality and parity differ? When two options are exactly as good, a slight improvement (worsening) in one option makes it better (worse) than the other. This is not the case when the options are on a par. When they are on a par, only a significant improvement (worsening) in one option will make it better (worse) than the other.

To relate parity to population problems I use diagrammatic repre-sentations. For simplicity, I focus exclusively on the case where one person is added to the world leaving everyone else unaffected. Suppose that we compare the well-being vector of one population, which I write, w^e, with an alternative situation where one person is added to this population leaving the levels of well-being of existing people unchanged. I write the vector for this latter situation as (w^e, μ), where μ is the well-being level of the extra person. Suppose that all possible levels of μ lie in increasing order on a line. As we move along the line to the right (left), the well-being of the extra person gets better (worse). Broome (Reference Broome and Rabinowicz:2001) has also used this representational device. I call it a ‘well-being configuration’. It is shown in Figure 1.

Figure 1.

A well-being configuration.

Suppose that there is a zone of levels of μ such that (w^e, μ) and w^e are B-incommensurate. The zone has a lower bound of π₁ and an upper bound of π₂. It is shown in Figure 2. If μ < π₁, μ is in the ‘worse zone’ and w^eB(w^e, μ) while if μ > π₂, (w^e, μ)Bw^e and μ is in the ‘better zone’. If π₁ ≤ μ ≤ π₂, furthermore, it is false that w^eE (w^e, μ). If it were true that w^eE (w^e, μ) and μ were slightly increased (decreased) it would be in the better (worse) zone. This is clearly not the case if there is a zone between the better and worse zones. So the zone between the better and worse zones is a zone of B-incommensurateness.

Figure 2.

B-Incommensurateness with exact borderlines.

One might relate parity to population issues by supposing that for all levels of μ in the B-incommensurate zone w^eP(w^e, μ) so that it is a zone of parity. I have argued elsewhere that this position cannot be characterised in terms of Figure 2. Consider a level of μ in the B-incommensurate zone bordering the better zone. Here a slight improvement in μ can mean that (w^e, μ)Bw^e. This cannot be the case if w^eP(w^e, μ) since if this were the case only a significant improvement in μ would imply that (w^e, μ)Bw^e. If one is to characterise the B-incommensurate zone as a zone of parity it must have inexact borderlines. The parity view can be characterised by an amended version of Figure 2 which allows the borderlines of the B-incommensurate zone to be inexact. It characterises the rough borderlines of this zone in terms of an account of vagueness.Footnote ³ The parity view clearly excludes recourse to incomparability.

3. INTUITIONS OF NEUTRALITY

In Weighing Lives, Broome's discussion of these issues focuses on what he calls the ‘intuition of neutrality’, which he finds attractive but eventually rejects. There are actually numerous intuitions in Broome's discussion of neutrality. Broome defines a level of well-being as ‘neutral’ if, at this level, ‘it is nether better nor worse that this life is lived than that it is not lived’ (Broome Reference Broome2004: 142). Here ‘neutral’ means ‘ethically neutral’. Broome's terminology and relation system are distinct from mine. He uses ‘equally as good as’ to refer to the relation I have called ‘exactly as good as’; and my use of ‘B-incommensurate with’ (‘B-incommensurateness’) is equivalent to his ‘incommensurate with’ (‘incommensurateness’). In my terms, Broome begins by assuming completeness of R. This means that each possible distribution of well-being is either better than, worse than or exactly as good as any other conceivable distribution. In the well-being configuration, this means there is only one neutral level of well-being, η. This situation can be presented in Figure 3. At the neutral level (w^e, μ)Ew^e. Completeness thus implies that ‘[n]eutrality marks a sharp boundary between lives that are better lived than not and lives that are better not lived than lived.’ (Broome Reference Broome2004: 142).

Figure 3.

One neutral level.

Broome suggests that this conclusion conflicts with intuition because ‘[w]e think intuitively that adding a person to the world is very often ethically neutral. We do not think that just a single level of wellbeing is neutral, and that a person's living at any other level is either better or worse than her nonexistence’ (Broome Reference Broome2004: 143). This is what he terms ‘the intuition of neutrality’. He considers various versions of it. One runs: ‘if a person is added to the population of the world, her addition has no positive or negative value in itself’ (Broome Reference Broome2004: 145–46). Broome rejects this version because if a couple had the choice of having a child whose life would be ‘short and full of suffering’ they should definitely not have the child: it would make matters worse. He also considers the possibility that the range of lives which is neutral – or ‘neutral range’ - has a lower, but no upper, boundary.Footnote ⁴ If we reject this possibility, we are left with what Broome (Reference Broome2004: 144) thinks is a ‘more moderate view’: the range of neutral levels has both an upper and lower boundary. This view is equivalent to that presented in Figure 2 above, where the neutral range was characterised as a B-incommensurate zone. In Broome's terms the parity view is a version of this ‘more moderate view’ in which the upper and lower boundaries are inexact.

Broome raises a series of doubts about characterizing neutrality in terms of B-incommensurateness. First, he thinks that ‘it is arbitrary to call on the idea of incommensurateness’ in this context. He suggests that ‘neutrality is most naturally understood as equality of value’ so that resorting to B-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). Here it is clear that Broome actually sees (exact) equality as the most intuitive way of understanding neutrality: this is, what I term, his ‘alternative intuition’.

If neutrality is characterised in terms of parity Broome's first doubt may be less worrying. One might argue that while parity is not exact equality, it is a form of equality nonetheless. Broome's doubt would persist since ‘exactly as good as’ is equivalent to his relation ‘equally as good as’: precision is part of his concept of equality. So parity is not equality, as Broome understands it. It might be argued, nonetheless, that parity is the only form of equality on offer when comparisons are, for some reason, complex. In the context of population problems, such complexity arises in comparisons between certain distributions because a person (people) exists (exist) in some distribution(s) and not in others.Footnote ⁵

Since Broome's first doubt may persist, I add one further comment. One intuition about neutrality is this: adding a person (people) to the world at a specific level of well-being is ‘ethically neutral’ in the sense that it is a matter of (moral) ‘indifference’ whether or not the person (people) is (are) added.Footnote ⁶ For this reason, it might be argued that when adding people is neutral it is exactly as good as not doing so. That may be the basis of Broome's ‘alternative intuition’ which sees neutrality most naturally in terms of (exact) equality. My reading of this ‘alternative intuition’ involves assuming that when one is indifferent between options they are exactly as good. How do things stand when the options are on a par, rather than exactly as good? The options are comparable, but one is not better than the other. It is plausible that given a choice between these options, it does not matter which option is chosen at any particular point in time (Qizilbash, Reference Qizilbash2002: 148). For this reason, Griffin (Reference Griffin1986: 97; 2000: 287) has suggested that, as regards choice, parity (‘rough equality’ in his terms) can be treated as exact equality.Footnote ⁷ My further suggestion is that, if I have correctly understood Broome's rationale for his ‘alternative intuition’, neutrality might also be characterised as parity since one can quite plausibly be indifferent between options which are on a par. The parity view is thus consistent with one interpretation of the ‘alternative intuition’. Since this view also meets Broome's ‘intuition of neutrality’ – because it implies a neutral range – it meets two of his intuitions.

Broome's second doubt relates to the incompatibility of B-incom-mensurateness and vagueness. I have discussed this elsewhere.Footnote ⁸ His third doubt is that B-incommensurateness does not ‘capture adequately the intuitive idea of neutrality’ (Broome Reference Broome2004: 169) because it makes neutrality implausibly ‘greedy’. Broome illustrates this point with a version of the mere addition paradox. To explain it, I introduce Broome's notation. In this notation (e.g., Broome Reference Broome1999: 230) 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. Broome takes well-being to be measurable on a cardinal scale that is interpersonally comparable. His version of the paradox involves four alternatives (Broome Reference Broome2004: 148):

a = (4,4, . . . 4, 6, Ω);

b = (4,4, . . . 4, 6, 1);

c = (4,4, . . . 4, 4, 4);

d = (4,4, . . . 4, 4, Ω).

Suppose that 4 and 1 are in the B-incommensurate zone (or neutral range). Broome's discussion of this example goes as follows. First, we judge that cBb because c has a greater aggregate of well-being, and a more equal distribution of well-being, than b does. Next, since 1 and 4 are in the B-incommensurate zone, a and b are B-incommensurate. Finally, it cannot be true that aBc. If it were, then, by transitivity of B, aBb, but we know that a and b are B-incommensurate. So we would be led to contradiction. But Broome insists that aBc. He argues as follows. In moving from a to c there are two changes. First, one person has come into existence and the change is neutral because 4 is in the neutral range. Second, 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 is that c is worse than a. If a and c are B-incommensurate, neutrality is not what it should be because it ‘swallows up’ the badness of reducing one person's well-being from 6 to 4. This makes the neutrality involved in a B-incommensurate zone implausibly ‘greedy’. Broome's intuition is that neutrality is not implausibly ‘greedy’ and he abandons the project of characterising it in terms of B-incommensurateness. He later suggests that this sort of ‘greedy’ neutrality has even more implausible implications in the context of examples which he interprets in terms of global warming. Broome's argument applies to the parity view because parity is a form of B-incommensurateness.

Does this third doubt pose a serious challenge to the parity view? Central to the argument is the claim that c is worse than a. The parity view cannot, on its own, endorse this claim because in the move from a to c one person is added while another's well-being falls. Broome's intuitions about neutrality extend beyond cases where addition leaves the well-being of existing people unaffected. It is not clear how one might extend the parity view to cases where addition affects existing people. Broome's own view is that while there is a unique neutral level, there is vagueness about it. This vagueness leads to a vague zone between the better and worse zones in the well-being configuration (see Qizilbash Reference Qizilbash2005 and forthcoming). As Broome (2004: 182, 204–5) notes, the unattractive implications of the ‘greediness’ problem – including the implications relating to global warming – arise for this view as well. Broome concludes his discussion of the subject by suggesting that while ‘[w]e have to give up the neutrality intuition . . . . we do not have to give up the view that the neutral level is vague’ (Broome 2004: 206). Since on his account the vague zone is not strictly a neutral range (Broome 2004: 206), the intuition that neutrality is not implausibly greedy does not apply to it. Nonetheless, Broome thinks that the problem raised by his version of the mere addition paradox and related implications in the context of global warming suggest that ‘vagueness must not be very extensive’ (Broome 2004: 264). That is, the vague zone must not be too wide.Footnote ⁹

Broome's intuition that neutrality should not be greedy is based on the view that if adding people is neutral, then such addition cannot justify any sacrifice of the well-being of existing people. As he puts it ‘neutrality cannot count against other values’ (Broome 2004: 203). One might extend the parity view by suggesting that if the addition of a person (or people) is on a par with the status quo it cannot justify any significant sacrifice of other values. While this way of extending the parity view clearly makes neutrality ‘greedy’ in Broome's terms, the resulting greediness may not be particularly ‘implausible’. To this degree, an extended parity view might capture the intuition that neutrality ought not to be implausibly greedy. Given that the parity view captures his two other intuitions (as I understand them), this extended version of the view seems to fare quite well in fitting Broome's intuitions. Indeed, since Broome's own vagueness view does not meet his own intuitions (see Broome 2004: 182), it is not obvious that his view satisfies those intuitions any better than the extended parity view does.