2.1 The Claims Model

Let O = {x, y, …} be any set of outcomes. An axiology provides a moral ranking of O and an explanation as to why this is the ranking. I’ll abbreviate the moral ranking as ≽ ^M , and will assume that it is transitive and reflexive (but not necessarily complete).Footnote ¹⁴

When I speak of one outcome being ‘better’ than another or ‘equally good’ as another, this is always shorthand for ‘morally better’ or ‘morally equally good’: how the two outcomes are ranked by ≽ ^M . ‘x ≻ ^M y’ means that x is morally better than y, and ‘x ∼ ^M y’ that the two outcomes are equally morally good.

Let I be the set of individuals of moral concern (‘the population’) each of whom exists in at least one of the outcomes in O. Individual well-being is central to any NPA^C axiology. Proponents of such axiologies may well disagree about the nature of well-being – about whether it is reducible to mental states such as pains and pleasures, to preference-satisfaction, to objective goods, etc. – and this article will be agnostic on the issue. I do assume that the account of well-being incorporated in a given NPA^C axiology, whatever its content, makes comparisons of well-being levels and well-being differences. Well-being level and difference comparisons are formalized using the concept of a ‘history’. A ‘history’ is a bundle of properties. More specifically, the history of a given individual in a given outcome is the property bundle comprised of all the well-being-relevant properties that the individual has in that outcome. The symbol ‘h _i (x)’ denotes the history of individual i in outcome x.

A given outcome set O and set of individuals I yields a set of histories H. H includes all and only those histories that arise by pairing each outcome x in O with each of the individuals who exists in x.Footnote ¹⁵ Well-being level comparisons are expressed as a ranking of H (this ranking denoted as ≽ ^L ); well-being difference comparisons are expressed as a ranking of H×H, the set of pairs of histories (this ranking denoted as ≽ ^D ).

I assume that the rankings of H and H×H conform to a variety of structural axioms capturing truisms about well-being level and difference comparisons.Footnote ¹⁶ These axioms do not include a stipulation that well-being level and difference comparisons are complete or measurable.Footnote ¹⁷ None of the arguments presented in this article regarding the implications of an NPA^C axiology require an assumption of well-being completeness or measurability – the one exception being the argument in section 5 to the Repugnant Conclusion. The incomparability of some histories with respect to well-being precludes completeness and, hence, measurability too; even with completeness, the lexical priority of certain sources of well-being over others can preclude measurability. The analysis here (again, leaving aside the Repugnant-Conclusion portion of section 5) allows for both incomparability and lexicality with respect to well-being.

Even without completeness or measurability, it is meaningful to refer to a particular well-being level (level W*) or difference (difference ΔW*). A well-being level is an equivalence class with respect to the ranking of histories, and a well-being difference is an equivalence class with respect to the ranking of history pairs.Footnote ¹⁸ I will use symbols such as ‘W*’ or ‘ΔW*’ to denote levels and differences, respectively. ‘W ≻ ^L W*’ means that W is a higher well-being level than W*, and ‘W∼ ^L W*’ that the two well-being levels are the same. Similarly, ‘ΔW ≻ ^D ΔW*’ means that ΔW is a larger well-being difference than ΔW*, and ‘ΔW ∼ ^D ΔW*’ that the two well-being differences are the same.

For most of the article, in referring to a given well-being difference ΔW, I assume that ΔW is a positive well-being difference: a difference between some well-being level W ⁺⁺ and a lower level W ⁺ .Footnote ¹⁹ If ΔW is not positive, I will say so explicitly.

The well-being at issue here is lifetime well-being. One can imagine a narrowly person-affecting axiology grounded, instead, in time-slice well-being or dimensional well-being. However, such an approach is, I believe, significantly less plausible than making lifetime well-being the currency of moral assessment. By NPA^C axiology, I mean an axiology such that the moral ranking of outcomes is grounded in gains and losses to persons with respect to lifetime well-being. For the remainder of the article, ‘well-being’ and cognate terms always have the implicit qualifier ‘lifetime’.

As I’ve already stated, an NPA^C axiology is such that the moral comparison of any two outcomes depends upon the magnitude and weight of individuals’ well-being gains and losses between the two (with well-being gains counting towards moral betterness and well-being losses against). I will precisify this idea via a specific model of how individual well-being gains and losses give rise to the moral comparison: the claims-across-outcomes model. In the remainder of this section, I present the claims-across-outcomes model (for short, ‘claims’ model) for the fixed-population case. Section 5 generalizes the model to the variable-population case.

In the fixed-population case, the population I is a finite set of individuals each of whom exists in all of the outcomes in O. N is the number of such individuals. A claim-across-outcomes (‘claim’) is a relation between a given individual i and a given pair of outcomes x, y, with four possible valences: i has a claim to x over y, a claim to y over x, a null claim, or an incomparable claim. The valence of i’s claim is determined by their well-being: i has a claim to x over y iffFootnote ²⁰ they are better off in x than y; a claim to y over x iff they are better off in y than x; a null claim iff they are equally well off in the two outcomes; and an incomparable claim if their well-being levels in x and y are incomparable. Non-null claims also have a strength.

I use the shorthand ‘welfare-unaffected’ as follows: an individual is ‘welfare-unaffected’ as between two outcomes iff they are equally well off in the two, and is ‘welfare-affected’ if not welfare-unaffected.

The grounding of moral betterness in individual well-being gains and losses is expressed, in the claims framework, via the following rules:

• Supervenience. If the pattern of claims (in terms of valence and strength) between x and y is the same as between x* and y*, x is at least as good as y iff x* is at least as good as y*.

• Claims as Pro Tanto Moral Considerations. If there is a claim to x over y, then x is better than y unless there is a claim to y over x or an incomparable claim.Footnote ²¹

• Two Person Conflicts. If one person has a claim to x over y, and a second person has a claim to y over x, with everyone else welfare-unaffected, then: x is better than y if the first person’s claim is stronger; y is better than x if the second person’s claim is stronger; the two outcomes are equally good if the two claims are of equal strength; and the two outcomes are incomparable iff the two claims are of incomparable strength.

• Equal Balance. If two persons have incomparable claims between outcomes x and y, with everyone else welfare-unaffected, then: if the first person’s well-being level in x is the same as the second’s in y, and vice versa, the two outcomes are equally good.Footnote ²²

These rules use the strength of non-null claims to help determine the outcome ranking (claim strength is invoked by both Supervenience and Two Person Conflicts), but don’t themselves explain how the strength of a non-null claim is assigned. An account of the assignment of claim strength will be developed as the article proceeds.

Additional rules might be added to the claims framework, but no more are needed for my purposes. What is remarkable is that these rules alone, combined with a plausible view about the assignment of claim strength, suffice to show that an NPA^C axiology’s moral ranking satisfies both the Pareto and Pigou-Dalton axioms; is insensitive to desert considerations; and (with the rules generalized to the variable-population case) satisfies No Difference but yields the Repugnant Conclusion (with none of these implications, except the last, conditional on well-being completeness and measurability).Footnote ²³

NPA^C axiologies respect the separateness-of-persons in two senses.Footnote ²⁴ (1) The well-being gains and losses of each person are a distinct determinant of the moral ranking of outcomes. If April is one person and Barry a different person, then April’s well-being is one factor that determines how outcomes morally compare, while Barry’s well-being is a second, separate factor. In the claims model, this is captured by making each person a separate holder of claims. With N persons in the population, there are N claims between each pair of outcomes.

(2) How changes in well-being along multiple temporal or substantive dimensions figure into the moral ranking depends upon whether these are changes to the same person or to different persons. For example, imagine that x and y differ with respect to someone’s well-being both at time t and at time t*. Denote the first difference as ΔW and the second as ΔW*.Footnote ²⁵ Then how these well-being changes figure into the moral comparison of x and y depends on whether these are changes at different times to the same person, or rather changes to different persons. The same is true if there are multiple substantive dimensions of well-being (e.g. multiple well-being goods), and ΔW is a change to one dimension while ΔW* is a change to a second.

NPA^C axiologies respect the separateness of persons in the second sense because it is each person’s lifetime well-being that is registered as a distinct determinant of the moral ranking of outcomes. In the claims model, this is captured by making the valence and strength of each person’s claim depend upon their lifetime well-being. Assume that ΔW and ΔW* are both changes to the well-being of one person, April. Then the two changes are inputs into the x/y ranking via the very same channel, namely April’s claim. They trade off against each other as ingredients in April’s lifetime well-being, which in turn determines the valence and strength of that single claim. (For example, imagine that April is better off as a young adult in x than y, but worse off in middle age. Or, she is happier in x than y, but accomplishes less. Then whether April has a claim to x over y, to y over x, a null claim, or an incomparable claim depends on the net effect of these two temporal or substantive changes on April’s lifetime well-being.) By contrast, if ΔW is a change to the well-being of one person, April, and ΔW* to the well-being of a second, Barry, the two changes are inputs into the x/y ranking via two channels. The first change is reflected in April’s claim, the second in Barry’s. The changes trade off not as ingredients in the lifetime well-being of one person, but in a different way: at the level of claims. (For example, if April is happier in x than y, while Barry accomplishes less, with no other differences between the outcomes, then whether x is better than y depends on whether April’s claim to x in light of her greater happiness in x is stronger than Barry’s claim to y in light of his greater accomplishment there.)

2.2 Should NPA^C Axiology be Defined More Restrictively?

NPA^C axiology, as I have defined it, is such that the moral comparison of two outcomes depends upon the magnitude and weight of individuals’ well-being gains and losses between the two, with well-being gains counting towards moral betterness and well-being losses against.

A more restrictive definition would stipulate that the comparison depends only upon the magnitude of individuals’ well-being gains and losses. Losses and gains figure into the comparison only via the size of these losses and gains, with no upweighting or downweighting of these magnitudes by other considerations. As precisified by the claims framework, NPA^C axiology defined in this restrictive way would mean that the strength of an individual’s non-null claim between two outcomes depends upon their well-being difference between the two – and not, in addition, their well-being level or desert.

I see no reason to define NPA^C axiology so restrictively. Note, first, that the restrictive and less restrictive versions are both linked to the narrow in-a-respect and all-things-considered person-affecting principles. Both versions imply these principles if well-being is complete.Footnote ²⁶

Second, insisting on the restrictive definition of NPA^C axiology just begs the question whether a moral concern for equity (the Pigou-Dalton axiom) or desert can flow from a narrowly person-affecting account of how the outcome ranking should be justified. Rather than rule out such concerns from the get-go (as does the restrictive definition), we should adopt a wider definition of NPA^C axiology, which makes the moral role of equity and desert a matter for substantive argument. The role of well-being levels and desert as determinants of claim strength should be open for discussion, rather than definitionally precluded.

3.1 Axioms: Pareto, Pigou-Dalton, Anonymity

With a fixed population and undifferentiated desert, NPA^C axiology supports three distinct axioms regarding the moral ranking (or so I’ll argue here): Pareto (meaning the combination of Pareto Indifference and Strong Pareto); Anonymity; and, plausibly, Pigou-Dalton.

Of course, none of these axioms follows from the minimal formal supposition that ≽^M is transitive and reflexive. Further, the axioms are logically independent: none implies the other, and indeed no two imply the third. What NPA^C axiology does is to provide a common justificatory framework that argues for the first two and, plausibly, for the third as well.

I’ll show this using the claims model. Pareto: Pareto Indifference. Note that any outcome z is equally good as itself. Further, in the z/z comparison, all individuals have null claims. Consider now any two outcomes x, y, that meet the antecedent condition of Pareto Indifference: each person is equally well off in x as in y. Thus each person has a null claim between x and y. Note now that the pattern of claims between x and y is the same as between z and itself (all null in both cases). By Supervenience, because z is equally good as itself, it follows that x is equally good as y.

Pareto: Strong Pareto. If outcomes y and x meet the antecedent condition of Strong Pareto, every person in the population falls into one of two groups. One group (with at least one member) are individuals who are better off in y than x. A second group (perhaps empty) are individuals who are welfare-unaffected as between the two outcomes. Individuals in the first group have claims to y over x; individuals in the second group have null claims. By Claims as Pro Tanto Moral Considerations, y is better than x.

Anonymity. Let’s say that any two outcomes z and zz are a ‘two person well-being swap’ if there is some individual i who is at well-being level W in z and well-being level W* in zz; a second individual j who is at well-being level W* in z and W in zz; and everyone else is welfare-unaffected. Given undifferentiated desert, the claims model argues that z and zz are equally good.

Why? (1) If W ∼ ^L W*, then z and zz are equally good by Pareto Indifference.

(2) Assume instead that W* ≻ ^L W. If so, i has a claim to zz over z, and j has a claim to z over zz, with everyone else welfare-unaffected. The Two Person Conflicts rule comes into play. In general, when an individual is better off in a second outcome than a first, and therefore has a claim to the second outcome over the first, there are three factors that might determine the strength of their claim: their starting point well-being level, their well-being difference, and their desert. With undifferentiated desert, the last factor drops away. It follows that the two individuals, i and j, have equally strong claims. Each starts at the same well-being level (W) and moves up by the same amount (the well-being difference between W* and W). See Table 1.

Table 1. A Two-Person Well-Being Swap

Note: Outcomes z and zz are a two-person well-being swap. If W* ≻ ^L W, then Ike has a claim to zz over z, and Jake an equally strong claim to z over zz. Alternatively, if W ≻ ^L W*, then Jake has a claim to zz over z, and Ike an equally strong claim to z over zz.

Thus i has a claim to zz over z, while j has a conflicting, equally strong, claim to z over zz; and everyone else has null claims. By Two Person Conflicts, z is equally good as zz.Footnote ²⁸

If, alternatively, W ≻ ^L W*, it will be j with a claim to zz over z, and i a claim to z over zz; but the very same reasoning will show that the two claims are of equal strength, and thus z is equally good as zz.

(4) The final possibility is that W* and W are incomparable. Then, by the Equal Balance rule, x and y are equally good.

(5) Assume that y and x meet the antecedent condition for Anonymity, namely: the well-being levels in y are a permutation of those in x. Note that every permutation of an ordered list of N well-being levels (N finite) can be expressed as a finite sequence of two-person well-being swaps (Hall Reference Hall1959: 60). Thus, there exists a finite sequence of outcomes x ¹, …, x ^M , such that x ¹ and x are a two-person well-being swap; each outcome and the one that succeeds it in the sequence are a two-person well-being swap; and y and x ^M are a two-person well-being swap.Footnote ²⁹ Therefore x ¹ is equally good as x; x ² is equally good as x ¹; …; y is equally good as x ^M . By the transitivity of ≽ ^M , y is equally good as x.

Pigou-Dalton. If the antecedent conditions for the Pigou-Dalton axiom are met, individual i has a claim to x over y, while j has a claim to y over x, and everyone else has null claims. So the Two Person Conflicts rule applies. What are the well-being levels and differences at issue in the conflicting claims of i and j? Let W** and W′ denote the well-being levels of i and j, respectively, in x. Let W* denote the well-being level of i in y, and W′′ the well-being level of j in y. ΔW is the well-being difference between W** and W*, and also the well-being difference between W′′ and W′. One of the antecedent conditions for Pigou-Dalton is that i in y is better off than j in x; this means that W* ≻ ^L W′.

In short: i’s claim is a claim to move up from well-being level W* by difference ΔW, while j’s claim is a claim to move up from well-being level W′ by the same difference (ΔW), with W* ≻ ^L W′. See Table 2.

Table 2. The Pigou-Dalton Axiom

Note: Outcomes x and y meet the antecedent conditions for the Pigou-Dalton axiom. Thus W** ≻ ^L W′; W** ≻ ^L W*; W′′ ≻ ^L W′; the difference between W** and W* (ΔW) is equal to the difference between W′′ and W′; and W* ≻ ^L W′.

By Two Person Conflicts, y is better than x iff j’s claim is stronger than i’s. As above in the discussion of Anonymity, we observe: With undifferentiated desert, the strength of a claim is some function of the starting-point well-being level and the well-being difference.

In order to determine the comparative strength of the two claims in the Pigou-Dalton case, we don’t need a full account of how well-being levels and differences interact to determine claim strength. All we need is a ceteris paribus account, explaining how two claims compare in strength when two individuals start at different well-being levels, one level higher than the other, and the differences are the same. There are four possibilities, here: (a) the claims are equally strong; (b) the individual who starts at the higher well-being level has the stronger claim; (c) the individual who starts at the lower well-being level has the stronger claim; or (d) the claims are incomparable in strength.

Of these four possibilities, I find (c) to be the most plausible. It is supported by two intuitions – call these the ‘Weak-Priority-of-the-Worse-Off Intuition’ and the ‘Structural Intuition’. (I have these intuitions and the reader may as well.) The Weak-Priority-of-the-Worse-Off Intuition is this: if we have to choose which of two equally deserving individuals to receive a benefit, and the two individuals stand to benefit by the same amount in well-being terms, the worse-off individual should receive the benefit. The Structural Intuition is this. Comparisons of levels are at least as fundamental to the structure of well-being as comparisons of differences.Footnote ³⁰ It therefore seems arbitrary to make claim strength wholly a function of well-being differences, and to deny well-being levels any role – even the minimal, tie-breaker role of determining who has the stronger claim when differences are equal.

The proponent of account (a) might try to appeal to the following intuition (call it the Bigger-Benefit Intuition): if we have to choose which of two equally deserving individuals to receive a benefit, and the first stands to benefit by more in well-being terms than the second, the first individual should receive the benefit.Footnote ³¹ I don’t have the Bigger-Benefit Intuition. In any event, this intuition does not support account (a). What the Bigger-Benefit Intuition does support is a different proposition about claim strength, namely this: in a two-person conflict in which the well-being differences are unequal, the individual who stands to gain more has a stronger claim, regardless of the two individuals’ well-being levels.Footnote ³² Account (a) says that, even in the Pigou-Dalton case, in which the well-being differences are the same, the two individuals’ well-being levels are irrelevant to the strength of their claims.

Perhaps account (a) can be supported in a different way. I do not go so far as to assert that account (a) is implausible.Footnote ³³ What I do assert is that account (c) is also quite plausible, in light of the Weak-Priority-of-the-Worse-Off Intuition and the Structural Intuition.Footnote ³⁴

Account (b) is implausible. It is consistent with the Structural Intuition, but flies in the face of the Weak-Priority-of-the-Worse-Off Intuition and gains no support from the Bigger-Benefit Intuition. Account (b) yields a reverse Pigou-Dalton axiomFootnote ³⁵ that is nowhere defended in the contemporary philosophical literature. Finally, account (d) is also hard to justify: since the two individuals’ starting point well-being levels are comparable, and the differences are comparable (indeed, the same), how does incomparability in the strength of the two claims arise?

If account (c) is adopted, j’s claim to y over x (a claim to move up by difference ΔW from starting-point well-being level W′) is stronger than i’s claim to x over y (a claim to move up by difference ΔW from starting-point well-being level W*, with W* ≻ ^L W′). Hence, by Two Person Conflicts, y is better than x.

If account (a) is adopted rather than account (c) – that is, the proposition that well-being differences are the sole determinant of claim strength with undifferentiated desert is endorsed – the upshot is to reject the Pigou-Dalton axiom and instead endorse the ‘Difference Comparison’ axiom in the case of undifferentiated desert.

3.2 Eligible Rankings

The previous section argued that an NPA^C axiology endorses the Pareto and Anonymity axioms and plausibly the Pigou-Dalton axiom as well. Consider the universe of transitive, reflexive, moral rankings of a given outcome set O, with a fixed population I. Which satisfy this combination of axioms?

The class of Paretian, Anonymous, and Pigou-Dalton-respecting rankings is an inclusive class. It includes not only prioritarianism but also a number of other well-known rankings, listed immediately below. In describing these various rankings, I assume well-being completeness and measurability; this will make the statement of the ranking rule much easier.Footnote ³⁶

By contrast, the class of Paretian, Anonymous, and Pigou-Dalton-respecting rankings excludes those that violate Pareto (such as Strong Egalitarianism); those that violate Anonymity (such as Weighted Utilitarianism); and those that satisfy Pareto and Anonymity but violate Pigou-Dalton (such as Utilitarianism or Sufficientism).Footnote ³⁷

In short, endorsing the combination of Pareto, Anonymity and Pigou-Dalton has significant implications for the form of the moral ranking – by ruling out strong egalitarianism, weighted utilitarianism, utilitarianism, and sufficientism – but does not get us all the way to a specific ranking. Making further progress within the confines of NPA^C axiology as precisified by the claims-across-outcomes model will require additional stipulations regarding claim strength – a matter beyond the scope of the present article.

If the Difference Comparison axiom is adopted in lieu of Pigou-Dalton, we do end up with a specific ranking. It can be shown that Pareto, Anonymity, and Difference Comparison imply the utilitarian ranking.Footnote ³⁸

4.1 Axioms: Pareto, Desert-Modulated Anonymity, Desert-Modulated Pigou-Dalton, Priority for the More Deserving

With differentiated desert, the claims framework supports four distinct axioms regarding the moral ranking: Pareto, Desert-Modulated (DM) Anonymity, Desert-Modulated (DM) Pigou-Dalton, and Priority for the More Deserving. Pareto is exactly the same as above, and is restated here for convenience.

The argument for Pareto is exactly the same as above. The arguments for the remaining three axioms are as follows.

DM Anonymity. Let’s say that any two outcomes z and zz are a ‘two person well-being/desert swap’ if there is some person i whose well-being/desert pairs in z and zz, respectively, are (W, D) and (W*, D*); another person j whose well-being/desert pairs in z and zz respectively are (W*, D*) and (W, D); and every other person is welfare-unaffected and has the same desert levels in the two outcomes. If so, z and zz are equally good.

Why? (1) If W ∼ ^L W*, then by Pareto Indifference z and zz are equally good.

(2) Assume instead that W* ≻ ^L W. If so, i has a claim to zz over z, and j has a claim to z over zz, with everyone else having null claims. The rule for Two Person Conflicts comes into play. By symmetry, the two individuals have equally strong claims, and hence z is equally good as zz. Parallel reasoning shows that the two outcomes are equally good if W ≻ ^L W*.

(3) The other possibility is that W* and W are incomparable. Then, by a modified version of the Equal Balance rule, z and zz are equally good.Footnote ⁴⁰

(4) If y and x meet the antecedent conditions for DM Anonymity, y can be reached from x by a finite sequence of two-person well-being/desert swaps. By the transitivity of ≽ ^M , y is equally good as x.

DM Pigou-Dalton. If the antecedent conditions for the DM Pigou-Dalton axiom are met, individual i has a claim to x over y, while j has a claim to y over x, and everyone else has null claims. So the Two Person Conflicts rule applies. As with the discussion in section 3.1 of the straight Pigou-Dalton axiom, let W** and W′ denote the well-being levels of i and j, respectively, in x. Let W* denote the well-being level of i in y, and W′′ the well-being level of j in y. ΔW is the well-being difference between W** and W*, and also the well-being difference between W′′ and W′. Since i in y is better off than j in x, it follows that W* ≻ ^L W′.

In short: i’s claim is a claim to move up from well-being level W* by difference ΔW, while j’s claim is a claim to move up from well-being level W′ by the same difference (ΔW), with W* ≻ ^L W′.

Which individual has the stronger claim? In section 3.1, defending the straight Pigou-Dalton axiom, I argued that an individual at a lower starting-point well-being level plausibly has a stronger claim, ceteris paribus. If the reader accepts this argument, then they should conclude that individual j has the stronger claim in the case at hand. Why? Individual j starts at a lower level than individual i, and the well-being differences are equal. The antecedent conditions for DM Pigou-Dalton posit that j’s desert level in each of the two outcomes is at least as high as i´s desert level in each of the two.

Assume, first, that the four desert levels are equal. If so, the two claims do not differ with respect to either well-being differences or desert, and (if indeed the individual at a lower starting-point well-being level has a stronger claim, ceteris paribus) it follows immediately that j has the stronger claim. Assume, next, that j’s desert level in at least one of the outcomes is higher than i’s in at least one. Surely this strengthens, rather than weakens, j’s claim relative to the case in which the four desert levels are equal.

Priority for the More Deserving. If the antecedent conditions for Priority for the More Deserving hold true, individual i has a claim to x over y, while j has a claim to y over x, and everyone else has null claims. See Table 3. So the Two Person Conflicts rule applies. Note now that both i and j have a claim to move from W to W*. Thus, with respect to two potential determinants of claim strength – starting-point well-being level and well-being difference – the two individuals are identical. However, with respect to the third potential determinant, desert, the individuals are different. Individual j in each of the two outcomes is at a higher desert level than individual i in each of the two.

Table 3. Priority for the More Deserving

Note: Outcomes x and y meet the antecedent conditions for Priority for the More Deserving. Thus W* ≻ ^L W; D′ ≻ ^DE D ⁺ , D′ ≻ ^DE D ⁺⁺ ; and D ^′′ ≻ ^DE D ⁺ , D ^′′ ≻ ^DE D ⁺⁺ .

If an NPA^C axiology includes desert as a determinant of claim strength, then desert must at least have a ceteris paribus role. At the very least: if two claims do not differ with respect to either starting-point well-being level or well-being difference, the comparative strength of the claims is determined by the individuals’ comparative deserts. If so, j has the stronger claim.Footnote ⁴¹

4.2 Inconsistencies

By saying that desert is ‘intrapersonally fixed’, I mean this: each individual is at a constant desert level in all of the outcomes in O, the set of outcomes being ranked. Amy is at level D in all the outcomes; Bob is at level D* in all the outcomes (which may or may not be the same as D); and so forth for each of the individuals. Desert is ‘intrapersonally variable’ if it is not intrapersonally fixed.

In what follows, I assume (as throughout the article) that the moral ranking ≽ ^M is transitive and reflexive.

If desert is intrapersonally fixed, the four axioms – Pareto, DM Anonymity, DM Pigou-Dalton, and Priority for the More Deserving – are jointly consistent. It is possible for a transitive, reflexive ≽ ^M to satisfy all four. To illustrate, consider a desert-modulated version of prioritarianism. There is some function f(·), which assigns each individual a number in a given outcome as a function of their well-being and their desert. f(·) is strictly increasing and strictly concave in well-being and satisfies the ‘slope condition’ in desert. See Figure 1. Each outcome is assigned a score equaling the sum of individuals’ f(·) values, and the outcomes are ranked in the order of these scores. It is not difficult to see that desert-modulated prioritarianism will yield a reflexive, transitive, ranking of outcomes that satisfies all four of the above axioms, if desert is intrapersonally fixed.Footnote ⁴²

Figure 1. Desert-Modulated Prioritarianism.

Source: Adler (Reference Adler2018). Note: Well-being and desert are both measurable, by w and d numbers respectively. Desert-modulated prioritarianism assigns each outcome the score $$\sum\nolimits_{i = 1}^N {f({w_i},{d_i})} $$ and ranks outcomes according to these scores. The figure illustrates f(·) as a function of well-being for two different levels of desert, with d* > d. Note that f(·) is not merely strictly increasing and strictly concave in w for each given desert level, but satisfies the slope condition; at each level of w, f(w, d*) has a greater slope than f(w, d). The dashed lines illustrate that the $$\sum\nolimits_{i = 1}^N {f({w_i},{d_i})} $$ formula satisfies DM Pigou Dalton and Priority for the More Deserving with desert intrapersonally fixed.

However, with intrapersonally variable desert, problems emerge. First, Priority for the More Deserving is internally inconsistent. That is, Priority for the More Deserving on its own, plus the basic premise of a reflexive, transitive ≽ ^M , leads to a logical contradiction – even without positing additional axioms. See Table 4.

Table 4. Priority for the More Deserving is Internally Inconsistent

Note: The well-being levels are such that W** ≻ ^L W* ≻ ^L W. The desert levels are such that D ⁺⁺⁺ ≻ ^DE D ⁺⁺ ≻ ^DE D ⁺ ≻ ^DE D. Priority for the More Deserving yields an intransitivity: it requires that x ≻ ^M zz ≻ ^M z ≻ ^M y ≻ ^M x.

The internal inconsistency of Priority for the More Deserving might be remedied by shifting to a restricted version of Priority for the More Deserving, as follows:

Restricted Priority for the More Deserving differs from Priority for the More Deserving by adding the condition that each person’s desert level in x is the same as that person’s desert level in y.Footnote ⁴³

Restricted Priority for the More Deserving is internally consistent, even in the case of intrapersonally variable desert. However, with intrapersonally variable desert, Restricted Priority for the More Deserving is inconsistent with Pareto Indifference. See Table 5.

Table 5. Restricted Priority for the More Deserving is Inconsistent with Pareto Indifference

Note: The well-being levels are such that W* ≻ ^L W. The desert levels are such that D* ≻ ^DE D. Pareto Indifference requires that x ∼ ^M z and y ∼ ^M zz. Restricted Priority for the More Deserving requires that y ≻ ^M x and z ≻ ^M zz. Satisfying both requirements is inconsistent with the transitivity of ≽ ^M .

What about Strong Pareto? If we assume that well-being and desert are measurable and that the moral ranking satisfies DM Anonymity and a continuity axiom, Restricted Priority for the More Deserving is inconsistent with Strong Pareto. See Table 6.Footnote ⁴⁴

Table 6. Restricted Priority for the More Deserving and Strong Pareto

Note: w* and w are well-being numbers, and d* and d desert numbers, with w* > w and d* > d. With well-being and desert both measurable, each outcome becomes a vector of individual well-being and desert numbers. DM Continuity requires that if one such vector is ranked above or below a second, then this ranking holds true for a sufficiently small region around the first vector. By Restricted Priority for the More Deserving, y ≻ ^M x. By DM Continuity, z ≻ ^M x for ε > 0 sufficiently small. By DM Anonymity, zz ∼ ^M z. By the transitivity of ≽ ^M , zz ≻ ^M x, which contradicts Strong Pareto.

These results and related ones are summarized in Table 7.

Table 7. A Summary

Note: The results in Table 7 presuppose the formal properties of ≽ ^M , namely that it is transitive and reflexive. An axiom or axiom combination is “internally inconsistent” or “internally consistent” if, respectively, inconsistent or consistent with the formal properties of ≽ ^M .

4.3 An Objection

I have argued that NPA^C axiology with differentiated desert supports both the Pareto axiom and the axiom of Priority for the More Deserving (as well as DM Anonymity and, plausibly, DM Pigou-Dalton). Pareto and Priority for the More Deserving, in turn, lead to the inconsistencies summarized in Table 7, which arise when desert is intrapersonally variable.

It might be objected that NPA^C axiology with differentiated desert does not in fact support the Pareto axiom, but only a restricted Pareto axiom (Restricted Pareto Indifference and Restricted Strong Pareto). ‘Restricted’ signals that these axioms apply only if each person in the population has the same desert level in the two outcomes under comparison.

The objection runs as follows.Footnote ⁴⁵ The claims framework stipulates that the valence of an individual’s claim depends upon their well-being – an individual has a claim to x over y iff they are better off in x than y; a claim to y over x iff they are better off in y than x; a null claim iff they are equally well off in the two outcomes; and an incomparable claim iff their well-being levels in x and y are incomparable – and that non-null claims have a strength. With differentiated desert, claim valence should be understood a different way, allowing for the valence of someone’s claim to depend both on their well-being and their desert. (For example, if Jorge is equally well off in x and y, but at a higher desert level in y, we might wish to say that he has a claim to y over x – a claim to become more deserving – rather than a null claim.) With claim valence thus understood, the claims framework no longer implies Pareto but merely Restricted Pareto.

First, with claim valence allowed to depend upon both well-being and desert, Supervenience no longer implies Pareto Indifference, but merely Restricted Pareto Indifference. Why? Supervenience implies that, if the pattern of claims (in terms of valence and strength) between x and y is the same as that between any outcome z and itself, then x and y are equally good (since z is equally good as itself). Imagine that everyone is welfare-unaffected as between x and y but some individuals do not have the same desert level in y as in x. Then the pattern of claims, in term of valence and strength, is not necessarily the same between y and x as between z and itself. While everyone’s claim as between z and itself is null, that need not be the case as between y and x. If some individuals have non-null claims between y and x, because their desert levels vary even though they are welfare-unaffected, then Supervenience does not imply that x and y are equally good. It would imply that only if everyone were welfare-unaffected as between x and y and no one’s desert differed between the two – the antecedent condition for Restricted Pareto Indifference.

Second, with claim valence allowed to depend upon both well-being and desert, Claims as Pro Tanto Moral Considerations no longer implies Strong Pareto, but merely Restricted Strong Pareto. Why? Imagine that at least one person is better off in y than x, and everyone is at least as well off in y as in x. However, some individuals do not have the same desert level in y as in x. If so, it is possible that some of these individuals have claims to x over y or incomparable claims – in which case Claims as Pro Tanto Moral Considerations would not imply that y is better than x. Claims as Pro Tanto Moral Considerations only implies that y is better than x if at least one person is better off in y than x, and everyone is at least as well off in y as in x, and no one’s desert differs between the two outcomes – the antecedent condition for Restricted Strong Pareto.

An analogous objection can be made to Priority for the More Deserving. With claim valence allowed to depend upon both well-being and desert, the claims framework does not support that axiom – only Restricted Priority for the More Deserving.Footnote ⁴⁶

With Pareto and Priority for the More Deserving replaced by Restricted Pareto and Restricted Priority for the More Deserving, respectively, the inconsistencies summarized in Table 7 go away. Restricted Priority for the More Deserving is both internally consistent and consistent with Restricted Pareto, whether desert is intrapersonally fixed or intrapersonally variable.Footnote ⁴⁷

However, I believe that the objection fails. An NPA^C axiology, as I understand it, is an account of the moral ranking of outcomes such that the moral comparison of any two outcomes depends upon the magnitude and weight of individuals’ well-being gains and losses between the two, with well-being gains counting toward moral betterness and well-being losses against. This is precisified, via the claims framework, by making the valence of an individual’s claim depend upon their well-being – not their well-being and desert. If claim valence can depend upon both well-being and desert, so that Sue lacks a claim in favour of y over x even though better off in y (as supposed by the objection above to Strong Pareto), then Sue’s well-being gain no longer counts towards the moral betterness of y. If claim valence can depend upon both well-being and desert, so that Cheng has a non-null claim between x and y even though equally well off in the two outcomes (as supposed by the objection above to Pareto Indifference), then there is something other than well-being gains and losses counting, respectively, towards moral betterness and worseness: namely, the change in Cheng’s desert.

The response to the objection can also be framed in terms of the narrow all-things-considered and narrow in-a-respect person-affecting principles. NPA^C axiology, as I have stated it, is meant to articulate the thinking about the grounds of moral betterness underlying those principles. Leaving aside complications relating to well-being incompleteness, the claims framework as I have precisified it does imply those principles.Footnote ⁴⁸ The claims framework, understood instead to have claim valence depend upon both well-being and desert, would not do so. Imagine that everyone is welfare-unaffected as between x and y, but some have different desert levels in the two outcomes. Suppose that these individuals end up with claims in favour of y over x. Then, by Claims as Pro Tanto Moral Considerations, y is better than x even though not better for any person – contradicting the narrow all-things-considered person-affecting principle.

A claims-across-outcome framework with claim valence dependent upon both well-being and desert could well be a good precisification of some axiology. But it is not a good precisification of NPA^C axiology – or so I would argue. And thus the objection to my contention that NPA^C axiology supports the axioms of Pareto and Priority for the More Deserving does not succeed.

4.4 The Upshot: NPA^C Axiology is Best Understood to be Desert-Neutral

How might the proponent of NPA^C Axiology respond to the inconsistencies summarized in Table 7?

One possibility is to abandon the basic premise that ≽ ^M is transitive. Without transitivity, Priority for the More Deserving is consistent with the remaining formal properties of ≽ ^M and derivatively, ≻ ^M and ∼ ^M , and with the combination of Pareto Indifference, Strong Pareto, DM Pigou-Dalton and (a weakened version of) DM Anonymity.Footnote ⁴⁹ It is well beyond the scope of this article to engage the debate about the transitivity of moral betterness and equal goodness. Suffice it to say that I find the case for transitivity to be compelling.

A second possibility is to posit that desert is intrapersonally fixed. The various difficulties summarized in Table 7 arise only with intrapersonally variable desert. But positing intrapersonally fixed desert is, substantively, very implausible. Desert is a non-well-being property of an individual that helps to determine the strength of their claim. Stated without reference to the claims framework: desert for purposes of an NPA^C axiology is an individual property that helps to determine the moral weight of that individual’s well-being gain or loss between two outcomes, and that does so independently of helping to determine that individual’s well-being level (or anyone else’s well-being level) in those outcomes. Desert, thus defined, could be intrapersonally fixed. For example, a ‘caste’ conception of desert might suppose that there is a hierarchy of castes; each person is born into their caste and remains in it (whatever they might do) for their entire life; higher castes have higher levels of desert. But such a conception of desert (and any other that posits a fixed individual desert, which remains the same regardless of the individual’s choices and efforts) is very unattractive. It is implausible that the moral weight of a well-being gain or loss to some person depends on a non-well-being factor that is wholly beyond the person’s control. Individual prudence, moral deservingness, and other standard conceptions of desert in the philosophical literature certainly do allow that an individual, by their actions, can change how deserving they are.

A third is to restrict the Pareto axiom. I argued in the preceding section that doing so is inconsistent with NPA^C axiology. In any event, Priority for the More Deserving is inconsistent with the transitivity of ≽ ^M ; this difficulty is not resolved by shifting from the full Pareto axiom to Restricted Pareto.

A fourth possibility is to maintain the transitivity of ≽ ^M ; to allow for intrapersonally variable desert; to continue endorsing the full Pareto axiom; but to drop Priority for the More Deserving and replace it with Restricted Priority for the More Deserving. Restricted Priority for the More Deserving is ad hoc.Footnote ⁵⁰ In any event, it conflicts with Pareto Indifference.

A fifth and final possibility is to drop both Priority for the More Deserving and Restricted Priority for the More Deserving, while retaining desert as a relevant consideration in other axioms (in particular, DM Pigou-Dalton and DM Anonymity). Again, however, this seems quite ad hoc.

I conclude that we should drop desert. NPA^C axiology is best understood not to include a desert component: a property of an individual that helps to determine the moral weight of that individual’s well-being gain or loss between two outcomes, and that does so independently of helping to determine that individual’s well-being level (or anyone else’s well-being level) in those outcomes.

But the reader might object: Many philosophers have argued for the moral relevance of individual desert. And not only nonconsequentialists. A substantial and vibrant body of scholarship argues for the moral relevance of desert with respect to the moral ranking of outcomes (Feldman Reference Feldman1995; Carlson Reference Carlson1997; Arneson Reference Arneson, Holtug and Lippert-Rasmussen2007; Hurka Reference Hurka2001; Arrhenius Reference Arrhenius2007; Kagan Reference Kagan2012; Skow Reference Skow2012; Kershnar and Tooley Reference Kershnar and Tooley2022).

My conclusion does not undercut this literature. Rather, what I have argued is that desert is irrelevant to the outcome ranking if that ranking is transitive, desert is intrapersonally variable, and the ranking is grounded in NPA^C axiology. The theorist who wishes to retain desert as a factor helping to determine the outcome ranking – without denying transitivity or insisting (implausibly) that desert is intrapersonally fixed – can do so by rejecting NPA^C axiology.

5.1 NPA^C Axiology and the Claims Model in the Variable-Population Case

The same ‘history’ setup that was used to explain well-being level and difference comparisons in the fixed-population case also applies to the variable-population case.Footnote ⁵¹ A given history h is a bundle of well-being-relevant properties, with h _i (x) denoting the bundle including all of individual i’s well-being-relevant properties in outcome x. H is the set of histories. H includes all and only those histories that arise by pairing each outcome x in O with each of the individuals who exists in x. H×H is the set of history pairs. Well-being level comparisons are formalized as a ranking of H, and difference comparisons as a ranking of H×H.

In what follows, I use ‘well-being better’ as shorthand for ‘better with respect to well-being’ – and similarly for ‘well-being worse’, ‘well-being equally good’, and ‘well-being incomparable’.

An NPA^C axiology claims that well-being can be compared to nonexistence. What, more precisely, does this mean? It doesn’t mean this: for each history h _i (x) of individual i, h _i (x) is either well-being better than i’s nonexistence, well-being worse than i’s nonexistence, or well-being equally good as i’s nonexistence. If the well-being ranking of H is incomplete,Footnote ⁵² then there will be pairs of histories that are well-being incomparable. If the proponent of NPA^C axiology allows for this, they should surely allow that a history h _i (x) may be well-being incomparable with the nonexistence of individual i.

What distinguishes NPA^NC and NPA^C axiology is this: the former insists, while the latter denies, that every history is well-being incomparable with nonexistence. NPA^NC axiology insists: for every combination of a set of outcomes O and population I, resulting in a set of histories H, and for every history h _i (x) in H, h _i (x) is well-being incomparable with the nonexistence of individual i. NPA^C axiology denies that this is the case.

The assertion that a given history h _i (x) is well-being comparable with i’s nonexistence – that it is well-being better, well-being worse, or well-being equally good as i’s nonexistence – is, to be sure, controversial. Many population ethicists reject such an assertion as incoherent – as confused about the metaphysics of existence, the nature of well-being, or both.Footnote ⁵³ One important critique of well-being comparisons to nonexistence might be termed the ‘subject-relativity’ critique. Well-being is goodness for a subject. History h _i (x) is well-being better than history h _i (y) iff outcome x is better for individual i than outcome y. But consider now the proposition that history h _i (x) is well-being better than i’s nonexistence. If so, outcome x is better for individual i than any outcome y in which i doesn’t exist. That in turn implies that y is worse for individual i than outcome x. But this is incoherent. Nonexistent beings don’t have properties, and so it can’t be the case that individual i in outcome y (an outcome in which they don’t exist) has the relational property of being worse off than in x.

On the other hand, some population ethicists defend well-being comparisons to nonexistence. For example, Gustaf Arrhenius, Wlodek Rabinowicz, and Nils Holtug have proposed to answer the subject-relativity critique along the following lines (Holtug Reference Holtug2010: Ch. 5; Arrhenius and Rabinowicz Reference Arrhenius, Rabinowicz, Hirose and Olson2015). Consider any two outcomes, x and y, and individual i. If i exists in both outcomes, the proposition that ‘x is better for i than y’ should be understood as follows: (1) if x were to obtain, x would be better for i than y (and y would be worse for i than x); and (2) if y were to obtain, x would be better for i than y (and y would be worse for i than x). However, if i exists only in outcome x, not y, the proposition ‘x is better for i than y’ should be understood only to mean (1), not (1) and (2). To say that well-being involves goodness for a subject (subject-relativity) merely requires ascribing relational properties to individuals in outcomes where they exist, and not also in outcomes where they don’t exist (which would be incoherent).Footnote ⁵⁴

I am persuaded by the Arrhenius/Rabinowicz/Holtug analysis and, thus, find it plausible that well-being can be compared to nonexistence. But the metaphysical and axiological issues at stake in the debate between those who accept and those who reject well-being comparisons to nonexistence are very thorny. This article does not attempt to engage those issues in detail and to take a definitive position in that debate. Rather, the approach here is provisional. An NPA^C axiology allows for well-being comparisons to nonexistence. Although such comparisons may turn out to be incoherent (confused about the metaphysics of existence, the nature of well-being, or both), such incoherence is not sufficiently clear that we should reject NPA^C axiology from the get-go. It is epistemically reasonable to endorse an NPA^C axiology and thus reasonable to explore its implications.

What, then, are the implications of NPA^C axiology in the variable-population case? The claims model can be extended to this case, as follows. For a given individual i, let a ‘zero history’ of that individual be a history which is equally good for them as their nonexistence. A bit more formally: let Z _i be the subset of the outcome set O such that (1) individual i exists in all of these outcomes, and (2) each of the outcomes in Z _i is equally good for that individual as i’s nonexistence. Z _i might be empty. If Z _i is not empty, then every history h _i (z), with z an outcome in Z _i , is a zero history of individual i. By the transitivity of well-being, all of i’s zero histories are at the same well-being level.Footnote ⁵⁵

Assume now that the set of outcomes O is sufficiently ‘rich’ that, for each individual i in the population of individuals I, H includes at least one zero history of that individual. We now have an easy way to define the well-being level of i’s nonexistence. Let y be any outcome in which i does not exist. Then i’s well-being level in y – the well-being level of i’s nonexistence – is the well-being level of i’s zero histories.

Note that this is an indirect definition of the well-being level of nonexistence. For any history h _i (x), we can directly define the well-being level of that history, with reference to the well-being ranking of histories. A well-being level is an equivalence class of H with respect to ≽ ^L ; the well-being level of a particular history, h _i (x), is the equivalence class to which it belongs. The well-being level of a given individual i’s nonexistence cannot be defined in this manner – since nonexistence is not a history of i. However, i’s zero history is a history of theirs; all of i’s zero histories do belong to an equivalence class of H with respect to ≽ ^L . Thus, since i’s nonexistence is equally good for them as each of their zero histories, we can indirectly define the well-being level of i’s nonexistence as the well-being level of i’s zero histories.

A parallel strategy can be followed to arrive at an indirect definition of well-being differences involving nonexistence. Recall that a well-being difference is an equivalence class of H×H with respect to the ranking of history pairs, ≽ ^D . The well-being difference between i’s nonexistence and a history h _j (x) is just the well-being difference between any zero history of i’s and h _j (x). For example, consider a difference comparison involving i’s nonexistence and three histories: h _j (y), h _k (z), and h _l (zz). Then the well-being difference between i’s nonexistence and h _j (y) is greater than/less than/equal to/incomparable with the well-being difference between h _k (z) and h _l (zz) iff the well-being difference between a zero history of i’s and h _j (y) is greater than/less than/equal to/incomparable with the well-being difference between h _k (z) and h _l (zz).

We can also define the notions of being ‘better off’, ‘worse off’, ‘equally well off’ and ‘incomparably well-off’ for the variable-population case. In the fixed-population case, it is a truism that individual i is better off/worse off/equally well off/incomparably well off in outcome x as compared to y iff i’s well-being level in x is higher than/lower than/equal to/incomparable with i’s well-being level in y. With our definition of the well-being level of i’s nonexistence in hand, we can readily analyse better off/worse off/equally well off/incomparably well-off in the variable-population case. Individual i in outcome x is better off/worse off/equally well off/incomparably well off as nonexistence iff i’s well-being level in x is higher than/lower than/equal to/incomparable with the well-being level of i’s nonexistence (i.e. the well-being level of any zero history of i’s).

Having defined well-being levels and well-being differences involving nonexistence, and having leveraged this definition to explain what it means to characterize a person as better off/worse off/equally well off/incomparably well-off as nonexistence, it is straightforward to state the claims framework for the variable-population case. For any two outcomes x, y, let I _xy be the set of ‘x-or-y persons’: those individuals who exist in x, in y, or in both outcomes. Each individual i in I _xy has a claim between x and y, with four possible valences: i has a claim in favour of x over y, in favour of y over x, a null claim, or an incomparable claim. The valence of i’s claim is determined by their well-being: i has a claim in favour of x over y iff they are better off in x than y; a claim in favour of y over x iff they are better off in y than x; a null claim iff they are equally well off in the two outcomes; and an incomparable claim iff their well-being levels in x and y are incomparable. Non-null claims also have a strength.

In the fixed-population case, various rules were stated to capture how the moral ranking depends upon the pattern of claims, in terms of claim valence and strength. These rules were: Supervenience; Claims as Pro Tanto Moral Considerations; Two Person Conflicts; and Equal Balance. The very same rules apply to the variable-population case. These rules take as given the valence and strength of claims. In order to fix a claim’s valence for the variable-population case, we need to explicate the well-being level of nonexistence; in order to fix a claim’s strength for the variable-population case, we need to do so, and also to explicate well-being differences involving nonexistence. Having provided these explications, we are in a position to assign claim valence and strength – and with that assignment in hand, we can apply the very same rules as in the fixed-population case, namely Supervenience; Claims as Pro Tanto Moral Considerations; Two Person Conflicts; and Equal Balance. These rules themselves do not need to change.

5.2 Axioms

Section 4 analysed the serious difficulties that arise from using desert as a determinant of claim strength. The analysis was undertaken for the fixed-population case; I concluded that claim strength in that case should depend only on well-being levels and differences, and not also desert. The difficulties demonstrated in section 4 would clearly carry over to the variable-population case, and so its conclusion applies to that case too.

If indeed claim strength in the variable-population case depends only on well-being levels and differences, the claims model argues for generalized versions of the Pareto and Anonymity axioms and, plausibly, for a generalized version of the Pigou-Dalton axiom as well.

The arguments from the claims framework for Pareto, Anonymity, and Pigou-Dalton in the fixed-population case can be readily reconfigured to yield arguments for, respectively, Generalized Pareto, Generalized Anonymity and Generalized Pigou-Dalton in the variable-population case. The only difference is that the fixed-population arguments make reference to the claim valences and strengths of individuals in I (the individuals each of whom exists in all of the outcomes, including the outcomes x and y referenced by the axiom); while the variable-population arguments make reference to the claim valences and strengths of individuals in I _xy (the individuals who exist in one or both of the outcomes x and y referenced by the axiom).

Section 3.2 set forth an inclusive list of outcome-ranking rules that satisfy Pareto, Anonymity, and Pigou-Dalton in the fixed-population case: prioritarianism, leximin, prioritarianism with a lexical threshold, relative prioritarianism, and weak utilitarianism. Each such rule has multiple variable-population extensions. Some, but not all, of these extensions will satisfy Generalized Pareto, Generalized Anonymity and Generalized Pigou-Dalton. For example, prioritarianism in the fixed-population case ranks outcomes by summing well-being numbers plugged into a strictly increasing, strictly concave transformation function. Prioritarianism can be extended to the variable-population case via ‘total prioritarianism’, ‘critical-level prioritarianism’ or ‘average prioritarianism’, among other possibilities.Footnote ⁵⁶ ‘Total prioritarianism’ satisfies the combination of Generalized Pareto, Generalized Anonymity and Generalized Pigou-Dalton, while ‘critical-level prioritarianism’ and ‘average prioritarianism’ do not.

5.3 Implications with Respect to Standard Population-Ethics Axioms

In this section, I assume that everyone’s zero histories are at the same well-being level – a well-being level that I’ll denote W ^zero. This premise is required for some if not all of what follows,Footnote ⁵⁷ and in any event seems very plausible.

A moral ranking ≽ ^M that satisfies the Generalized Pareto, Generalized Anonymity and Generalized Pigou-Dalton axioms has various further properties with respect to axioms often discussed in the population-ethics literature. First, ≽ ^M can be shown to satisfy axioms that are widely seen as attractive (Blackorby et al. Reference Blackorby, Bossert and Donaldson2005; Greaves Reference Greaves2017; Arrhenius Reference ArrheniusForthcoming), all of which are implied by Generalized Pareto: Mere Addition, Negative Mere Addition, Avoidance of the Sadistic Conclusion and Priority for Lives Worth Living.Footnote ⁵⁸

Second, ≽ ^M satisfies No Difference: the ranking of two outcomes in which the very same number of individuals exist does not depend upon whether those who exist in the first outcome are identical to those who exist in the second. No Difference is implied by Generalized Anonymity. Parfit endorses No Difference (Parfit Reference Parfit1987: Ch. 16; Reference Parfit2011: 217–31). It is foundational for much formal work on population ethics.Footnote ⁵⁹

Third, however, if we assume that well-being is measurable, the Repugnant Conclusion (RC) holds true of ≽ ^M .

The RC is expressed in various different ways in the literature (see e.g. Parfit Reference Parfit1987: Ch. 17; Blackorby et al. Reference Blackorby, Bossert and Donaldson2005: 162; Parfit Reference Parfit2016; Greaves Reference Greaves2017; Arrhenius Forthcoming). With the concept of W ^zero in hand, it can be stated as follows.

Assume that well-being is measurable. If so, the combination of Generalized Pareto, Generalized Anonymity, and Generalized Pigou-Dalton implies the RC.

• Proof. Let w ^high, w ^low, and w ^zero be the well-being numbers corresponding to levels W ^high, W ^low, and W ^zero, respectively. Pick N large enough that Nw ^low > Mw ^high + (N − M)w ^zero. Let outcome x* be such that the same M individuals who exist in x also exist in x*, all with well-being w ^high; there are (N – M) additional individuals, all at w ^zero. By Generalized Pareto Indifference, x and x* are equally good.

• Let w ^lower be such that w ^low > w ^lower > w ^zero and Nw ^lower = Mw ^high + (N – M)w ^zero. Let outcome x** be such that there are N existing individuals, the same individuals who exist in x*, and all are at w ^lower. By Pigou-Dalton and the transitivity of the moral ranking, x** is better than x*. (Note that any mean-preserving equalization of well-being can be achieved by a sequence of Pigou-Dalton transfers.Footnote ⁶⁰ )

• Let x*** be such that there are N existing individuals, the same who exist in x* and x**, and all are at w ^low. By Generalized Anonymity, y and x*** are equally good. By Strong Pareto, x*** is better than x**. Thus we have that y is equally good as x***, x*** is better than x**, x** is better than x*, and x* is equally good as x – hence by transitivity y is better than x.Footnote ⁶¹

If Generalized Pigou-Dalton is dropped and replaced with a generalized version of the Difference Comparison axiom, the RC is still implied.Footnote ⁶²

To see why the RC may not follow from the combination of Generalized Pareto, Generalized Anonymity, and Generalized Pigou-Dalton if well-being is not measurable, consider the following toy example, adapted from work by Teruji Thomas (Reference Thomas2018).Footnote ⁶³ Assume that there are two dimensions to well-being: knowledge and pleasure. A given history is assigned a pair of numbers, (k, p), the first its knowledge number, the second its pleasure number. Well-being levels are tracked by these pairs, as follows – with knowledge taking lexical priority over pleasure.Footnote ⁶⁴ If one history has a higher knowledge number than a second, it is better; if the knowledge numbers are equal, then the histories are ranked according to their pleasure numbers.

Assume now that outcomes are ranked using a two-step total-prioritarian rule: x is better than y if the total prioritarian value of x, calculated using its knowledge numbers, is greater than the total prioritarian value of y, calculated using its knowledge numbers; if the outcomes are ranked equal at this step, then x and y are compared according to their total-prioritarian values calculated using pleasure numbers.Footnote ⁶⁵

It can be shown that this two-step total-prioritarian rule will satisfy Generalized Pareto, Generalized Anonymity, and a dimension-specific version of Generalized Pigou-Dalton.Footnote ⁶⁶ However, it does not imply the Repugnant Conclusion. For example, let (1, 1) be the knowledge/pleasure numbers of the zero histories. Let W ^high be histories with the numbers (10, 10), and W ^low be histories with the numbers (1, 2). Note that the two-step total-prioritarian rule will rank any outcome in which the M existing individuals are all at W ^high better than any outcome in which the N existing individuals are all at W ^low , regardless of the magnitudes of M and N. So the RC does not hold true.

Further research is needed to establish the conditions under which NPA^C axiology does imply the RC even though well-being is not measurable.

5.4 NPA^NC Axiology?

NPA^NC axiology rejects well-being comparisons to nonexistence. A substantial number of philosophers believe that it is metaphysically impossible for an outcome in which someone exists to be better for, worse for, or equally good for that person as their nonexistence. NPA^NC axiology concurs in this view and therefore posits that: for every combination of a set of outcomes O and population I, resulting in a set of histories H, and for every history h _i (x) in H, h _i (x) is well-being incomparable with the nonexistence of individual i.

In the Introduction, I mentioned two highly counterintuitive implications of the narrow all-things-considered person-affecting principle. NPA^NC axiology, as precisified through the claims model, seems to have just these implications. Consider, first, a fixed-number case such that the same number of individuals exist in x and y, but there are individuals who exist in x but not y (the ‘x individuals’); and individuals who exist in y but not x (the ‘y individuals’). If each individual who exists in both outcomes is equally well off in both, then NPA^NC axiology implies that the outcomes are neither better nor worse than each other, regardless of the well-being levels of the x individuals and the y individuals – even if, for example, all of the x individuals have wonderful lives and all of the y individuals have terrible lives.

Why this implication? Each of the x individuals has an incomparable claim between the two outcomes, as does each of the y individuals; those who exist in both outcomes have null claims. Could the claims of the x and y individuals somehow balance against each other such that x is, on balance, better or worse than y?

When well-being incomparability arises as the intersection of a set of admissible complete rankings,Footnote ⁶⁷ incomparable claims can plausibly balance out in the manner just described. To illustrate, imagine that there is a set W of pairs of admissible complete rankings of the sets H and H×H, respectively, with (≽ ^L* , ≽ ^D* ) one such pair. The rankings ≽ ^L and ≽ ^D arise as follows: history h is at least as good as history h* iff h is at least as good as h* according to every ≽ ^L* in W; and the difference between histories h and h* is at least as large as the difference between histories h ⁺ and h ⁺⁺ iff the first difference is at least as large as the second according to every ≽ ^D* in W.

Imagine now that two individuals, i and j, are each incomparably well off in x and y, with everyone else equally well off. It could be the case that, for every (≽ ^L* , ≽ ^D* ) pair, if one individual has a claim in favour of y, the other individual has a stronger claim in favour of x. That is, on each of the complete well-being rankings giving rise to ≽ ^L and ≽ ^D , the balance of claims is in favour of x. If so, x would be the better outcome even though i and j have incomparable claims and everyone else null claims.

But the incomparability between histories and nonexistence, according to NPA^NC axiology, does not arise as the intersection of a set of admissible complete rankings. If i exists in x but not y, then there is no metaphysically possible well-being ranking such that h _i (x), i’s history in x, is better than, worse than, or equally good as i’s nonexistence in y. It’s not that there are a multiplicity of eligible, conflicting, well-being comparisons between h _i (x) and i’s nonexistence. Rather, h _i (x) and i’s nonexistence simply are not the kinds of items that can be compared with respect to well-being. Thus, in the fixed-number case described four paragraphs above, it’s very hard to see how the incomparable claims of the x individuals and y individuals could somehow balance out so that x is, on balance, better or worse than y.

A similar analysis suggests that NPA^NC axiology has a second highly counterintuitive implication. Consider now a different-number case such that there are individuals who exist in x but not y; none exist in y but not x. If each individual who exists in both outcomes is equally well off in both, then NPA^NC axiology seems to imply that the outcomes are neither better nor worse than each other – even if the x individuals have wonderful lives or, instead, terrible ones. Note that the only non-null claims between x and y are the incomparable claims of the x individuals. Once more, it is hard to see how these could somehow balance out such that x is, on balance, better or worse than y.Footnote ⁶⁸

If NPA^NC axiology indeed has these implications, as seems plausible, then we should reject it. The implications, together, amount to the following: outcome x is better or worse than outcome y only if there is at least one individual who exists in both and is not equally well off in the two. This is too counterintuitive to be believed. In short, the project of grounding the moral ranking in well-being gains and losses to individuals must, it seems, be abandoned if coupled with the premise that well-being cannot be compared to nonexistence.