2. Framework, axioms and undesirable conditions

Blackorby et al. (Reference Blackorby, Bossert, Donaldson and Fleurbaey1998) make an important contribution by extending population ethics beyond anonymous frameworks. Their analysis shows that fundamental impossibility results can emerge even when individuals’ identities matter. Following the non-anonymous framework introduced by Blackorby et al. (Reference Blackorby, Bossert, Donaldson and Fleurbaey1998) and applied by Cato and Harada (Reference Cato and Harada2023), we consider a general setting with variable populations. Let $\mathbb{N}$ be the set of natural numbers and $\mathbb{R}$ the set of real numbers. Let $\cal{N}$ be the collection of non-empty and finite subsets of $\mathbb{N}$ . Each element N of $\cal{N}$ represents a population (a set of individuals). For each $i\in N$ , ${u}_{i}\in \mathbb{R}$ represents individual i’s utility level. For each $N\in \cal{N}$ , ${\mathbb{R}}^{N}$ denotes the subspace of ${\mathbb{R}}^{\mathbb{N}}$ for which the coordinate labels correspond to the elements in N. The collection of distributions is given as follows:

\begin{align}\mathrm{\Omega }={\bigcup }_{N\in \cal{N}}{\mathbb{R}}^{N}\mathrm{}.\end{align}

For each element of $\mathrm{\Omega }$ , there exists the corresponding set N of individuals and a profile of their utility levels, $({u}_{i}{)}_{i\in N}$ . For simplicity, $({u}_{i}{)}_{i\in N}$ is written as ${u}_{N}$ . A utility level of 0 corresponds to neutrality, which is equivalent to a life with no experience.

A population principle R is a reflexive, complete and transitive binary relation over $\mathrm{\Omega }$ . We note that ${u}_{M}R\mathrm{}{v}_{N}$ means ‘ ${u}_{M}$ is at least as good as ${v}_{N}$ ’. The asymmetric and symmetric parts of R are denoted by P and I, respectively.

Throughout this paper, we consider two axioms as fundamental properties of the population principle. The first axiom, known as Existence Independence, states that the ranking between two distributions, ${u}_{M}$ and ${v}_{N}$ , remains unchanged if a population L with distribution ${w}_{L}$ , which has no effect on either, is added to both. In other words, the ranking between $({u}_{M},{w}_{L})$ and $({v}_{N},{w}_{L})$ mirrors that between ${u}_{M}$ and ${v}_{N}$ . The converse also holds: if a subpopulation common to both ${u}_{M}$ and ${v}_{N}$ has no effect, then removing it leaves their relative ranking unchanged.

The following axiom is a version of the Archimedean property. Consider two distributions, ${u}_{N}$ and ${v}_{N}$ , with the same population N. It may be the case that ${v}_{N}$ is a distribution with extremely high utility levels, while ${u}_{N}$ is a distribution with extremely low utility levels, so that ${v}_{N}$ is ranked far above ${u}_{N}$ . Now suppose we add a large number of individuals to both populations: each added individual receives utility $\mu$ in one distribution and utility $\nu$ in the other, where $\mu \gt \nu$ . The axiom requires that the distribution $(\mu {1}_{M},{u}_{N})$ be ranked above $(\nu {1}_{M},{v}_{N})$ , provided that the number of individuals added is sufficiently large, regardless of how much better ${v}_{N}$ was than ${u}_{N}$ to begin with. Intuitively, this expresses a principle of aggregation: even a small advantage per person becomes decisive when it is shared by a sufficiently large number of individuals. For the statement of this axiom, we define

\begin{align}{\cal{N}}_{-N}=\{M \in \cal {N}\it|M\cap {N}=\mathrm{\varnothing }\}\mathrm{},\end{align}

being the collection of elements M of $\cal{N}$ such that each set M does not include any element of N.

This axiom is related to the Positive-Limit and Negative-Limit Properties in Blackorby et al. (Reference Blackorby, Bossert, Donaldson and Fleurbaey1998), and it strengthens the Limit Property axiom of Cato and Harada (Reference Cato and Harada2023). With Anonymity, Spears and Budolfson (Reference Spears and Budolfson2021), Bossert et al. (Reference Bossert, Cato and Kamaga2023) and Cato et al. (Reference Cato, Harada and Oshitani2024) offer impossibility results with versions of limit properties. A related Archimedean condition is also employed by Rusonik et al. (Reference Rusonik, Samuelson and Gilboa2026) in their position-based approach to population ethics.

We now define the Repugnant and Very Sadistic Conclusions. The Repugnant Conclusion states that even for a population with very high utility levels, there exists a much larger population with small but positive utility levels that is better (Parfit Reference Parfit1984). As alluded to above, this conclusion necessarily holds for total utilitarianism. The Repugnant Conclusion is widely considered something that any plausible population principle should avoid, and it has been extensively examined by many since its introduction by Parfit (Reference Parfit1984). It is formulated as follows:

The Repugnant Conclusion holds for the population principle R if, for all $\xi \in {\mathbb{R}}_{++}$ , all $N\in \cal{N}$ and all $\varepsilon \in (0,\xi )$ , there exists $M \supsetneq\, N$ such that $\varepsilon \mathbb{1}_{M}P\mathrm{}\xi \mathbb{1}_{N}$ .

The Very Sadistic Conclusion was introduced by Arrhenius (Reference Arrhenius2000 a, 73; forthcoming, Section 5.1)Footnote ¹ and states that, for any population consisting solely of negative utility levels, there exists a population consisting solely of positive utility levels that is nonetheless worse. As we will see below, there are several variants of the sadistic conclusion, but there is a special reason to avoid this ‘very’ version: even if there were an extremely large number of people living in hellish conditions, there would still exist an alternative consisting entirely of positive utility that is, implausibly, worse. It is formulated as follows:

The Very Sadistic Conclusion holds for the population principle R if, for all $N\in \cal{N}$ and all ${v}_{N}\in {\mathbb{R}}_{--}^{N}$ , there exists $M\in \cal{N}$ and ${u}_{M}\in {\mathbb{R}}_{++}^{M}$ such that ${v}_{N}P\mathrm{}{u}_{M}$ .

In our formulations above, these conclusions need not presume that population principles respect Anonymity. We say that a population principle raises a dilemma between repugnancy and sadism if it implies that either the Repugnant or the Very Sadistic Conclusion holds. Next, we introduce the standard Sadistic Conclusion, which was also proposed by Arrhenius (Reference Arrhenius2000 a,b, forthcoming). According to this conclusion, it can be the case that adding a population with negative utility levels is better than adding a population with positive utility levels. This does not imply that adding any hellish distribution is better than adding some population consisting solely of positive utility levels. Formally, the Sadistic Conclusion is defined as follows.

The Sadistic Conclusion holds for R if there exist M, N, $L\in \cal{N}$ , ${u}_{M}\in {\mathbb{R}}_{++}^{M}$ , ${v}_{N}\in {\mathbb{R}}_{--}^{N}$ and ${w}_{L}\in {\mathbb{R}}^{L}$ such that $L\cap M=\mathrm{\varnothing }$ , $L\cap N=\mathrm{\varnothing }$ and $({v}_{N},{w}_{L})\;\mathrm{}P\mathrm{}({u}_{M},{w}_{L})$ .

Another version, which can be called the Strong Sadistic Conclusion, states that there exists a population with positive utility levels and another with negative utility levels such that the former is worse than the latter. The contrast between the Strong Sadistic Conclusion and the Very Sadistic Conclusion is comparatively straightforward to understand. Under the Strong Sadistic Conclusion, the negative utility of the population that is better than one with positive utility may be extremely close to zero – that is, it may not be all that bad. It remains possible, for example, that a hellish distribution is worse than any population with solely positive utility levels. The Very Sadistic Conclusion is particularly extreme in that it rules out even this possibility.

The Strong Sadistic Conclusion holds for R if there exist M, $N\in \cal{N}$ , ${u}_{M}\in {\mathbb{R}}_{++}^{M}$ and ${v}_{N}\in {\mathbb{R}}_{--}^{N}$ such that ${v}_{N}P\mathrm{}{u}_{M}$ .

Note that the Very Sadistic Conclusion logically implies the Strong Sadistic Conclusion. On the other hand, in general, the Sadistic Conclusion is logically independent from the Very Sadistic Conclusion. However, as observed by Blackorby et al. (Reference Blackorby, Bossert, Donaldson, Tännsjö and Ryberg2004), the Very Sadistic Conclusion implies the Sadistic Conclusion if R satisfies the Strong Pareto principle, which is normally considered uncontroversial. This means that the Very Sadistic Conclusion is essentially more problematic than the Sadistic Conclusion. Notably, we do not impose the Strong Pareto principle. In our framework, which includes not only humans but also various things in the universe, the Strong Pareto principle may not be very plausible. This is because the Strong Pareto principle requires that any improvement of a single individual’s utility is good. We consider a non-anonymous setting, which may include ‘individuals’ that should be ignored in welfare evaluations. However, also in our setting, the Very Sadistic Conclusion is more problematic than the Sadistic Conclusion, as we impose Existence Independence, under which the Strong Sadistic Conclusion implies the Sadistic Conclusion. Thus, the Sadistic Conclusion is implied by the Very Sadistic Conclusion even without the Strong Pareto principle.

3. A utilitarian condition for the dilemma between repugnancy and sadism

We start by formally introducing CLU. A population principle R is critical-level generalized utilitarian (CLGU) if there exists a continuous and strictly increasing transformation function $g:\mathbb{R}\to \mathbb{R}$ such that $0\in g\left(\mathbb{R}\right)$ and, for all M, $N\in \cal{N}$ , all ${u}_{M}\in {\mathbb{R}}^{M}$ and all ${v}_{N}\in {\mathbb{R}}^{N}$ ,Footnote ²

\begin{align}{u}_{M}R{v}_{N}\,\,\ {\rm if\, and\, only\, if}\,\, \ {\sum }_{i\in M}g ({u}_{i})\ge {\sum }_{i\in N}g({v}_{i}).\end{align}

Note that $\alpha ={g}^{-1}\left(0\right)$ is the critical level. That is, adding a person with this level is always indifferent. Axiomatic characterizations of this class of principles have been established by a series of works of Blackorby and Donaldson (Reference Blackorby and Donaldson1984), Blackorby et al. (Reference Blackorby, Bossert and Donaldson1995) and Blackorby et al. (Reference Blackorby, Bossert and Donaldson2005). Existence Independence is a key axiom for their characterizations.

We now show that the following generalized utilitarian condition is sufficient for the dilemma.

It is easy to see that the class of critical-level utilitarian principles falls into a subclass of orderings that satisfy this Condition.

The following result establishes that the generalized utilitarian condition is sufficient for the dilemma.

Case 2: There exists $\mu \in {\mathbb{R}}_{++}$ such that ${\sum }_{i\in \mathbb{M}} \,{g}_{i}\left(\mu \right)\le 0$ . By the Condition, ${\sum }_{i\in \mathbb{M}} \,{g}_{i}\left(\varepsilon \right)=-\mathrm{\!\infty }$ for all $\varepsilon \in (0,\mu )$ . Fix $\varepsilon \in (0,\mu )$ and consider any ${v}_{N}\in {\mathbb{R}}_{--}^{N}$ . By letting $M\in \cal{N}$ be a sufficiently large subset of $\mathbb{M}$ , it follows that ${\sum }_{i\in M\cap \mathbb{M}} \,{g}_{i}\left(\varepsilon \right)\lt {\sum }_{i\in N\cap \mathbb{M}} \,{g}_{i}\left({v}_{i}\right)$ , implying that ${v}_{N}P\mathrm{}\varepsilon {1}_{M}$ . Hence, the Very Sadistic Conclusion holds for R.

The Proposition follows since Cases 1 and 2 are exhaustive.□

We then provide two examples of generalized utilitarian criteria that satisfy the Condition.

Example 1 (Some people do not matter). There exists an infinite subset $\mathbb{M}$ of $\mathbb{N}$ and an increasing and continuous function $g:\mathbb{R}\to \mathbb{R}$ such that, for all M, $N\in \cal{N}$ , all ${u}_{M}\in {\mathbb{R}}^{M}$ and all ${v}_{N}\in {\mathbb{R}}^{N}$ ,

\begin{align}{u}_{M}R\mathrm{}{v}_{N}\,\, \ \mathrm{}\mathrm{}\mathrm{}\mathrm{}{\rm{i}\mathrm{f}\,\mathrm{}\mathrm{a}\mathrm{n}\mathrm{d}\ \mathrm{}\mathrm{o}\mathrm{n}\mathrm{l}\mathrm{y}\ \mathrm{}\mathrm{i}\mathrm{f}}\ \,\, \mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}{\sum }_{i\in M\cap \mathbb{M}}g\left({u}_{i}\right)\ge {\sum }_{i\in N\cap \mathbb{M}}g\left({v}_{i}\right)\mathrm{}.\end{align}

If $\mathbb{M}=\mathbb{N}$ , then we obtain CLGU with critical level ${g}^{-1}\left(0\right)$ , presuming that $0\in g\left(\mathbb{R}\right)$ . If $\mathbb{M} \subsetneq \mathbb{N}$ , then people in $\mathbb{N}\backslash \mathbb{M}$ do not matter. In the latter case, the Strong Pareto principle is not satisfied.

In such a lexicographic population principle, people in ${\mathbb{M}}^{\mathcal{l}}$ matter infinitely more than people in ${\mathbb{M}}^{k}$ if $0\le {\cal{l}}\lt k\le m-1$ , while people in ${\mathbb{M}}^{m}$ do not matter at all. The Strong Pareto principle is satisfied if and only if ${\mathbb{M}}^{m}=\mathrm{\varnothing }$ .

It is straightforward to show that all population principles in the two classes considered in Examples 1 and 2 satisfy our two axioms, Existence Independence and the Strict Limit Property.

We note that a population principle that satisfies Existence Independence and Strict Limit Property does not necessarily satisfy the Condition.

\begin{align}{u}_{i}^{\mathrm{*}}={u}_{i}\, \,{\rm for\, all}\,\, i\in M\end{align}

and

\begin{align}{u}_{i}^{\mathrm{*}}=\alpha \, \,{\rm for\, all}\,\, i\in \{1,\ldots, r\}\backslash M\mathrm{}.\end{align}

Analogically, define ${v}_{N}^{\mathrm{*}}\in {\mathbb{R}}^{r}$ by letting

\begin{align}{v}_{i}^{\mathrm{*}}={v}_{i}\, \,{\rm for\, all}\,\, i\in N\end{align}

and

\begin{align}{v}_{i}^{\mathrm{*}}=\alpha \, \,{\rm for\, all}\,\, i\in \{1,\ldots, r\}\backslash N\mathrm{}.\end{align}

The set of individuals is $\{1,\ldots, r\}$ for ${u}_{M}^{\mathrm{*}}$ and ${v}_{N}^{\mathrm{*}}$ . Let ${R}_{T}^{r}$ be a lexicographic ordering that compares the r-dimensional distributions: for all $u\mathrm{\text{'}}$ , $v\mathrm{\text{'}}\in {\mathbb{R}}^{r}$ , $u\mathrm{\text{'}}{R}_{T}^{r}v\mathrm{\text{'}}$ if and only if

\begin{align}{u}{^\prime_{i}}={v{^\prime_{i}}}\, \,{\rm for\, all}\,\, i\in \{1,\ldots, r\}\mathrm{},\end{align}

or

\begin{align}\exists k\in \left\{1,\ldots, r\right\}\, \,{\rm {such}\, {that}}\,\, {u}{^\prime_{i}}={v}{^\prime_{i}}\, \,{\rm {for}\, {all}}\,\, i\in \left\{k+1,\ldots, r\right\}\,\, {\rm and}\,\, {u}{^\prime_{k}}\gt {v}{^\prime_{k}}.\end{align}

Define R by letting: for all ${u}_{M}\in {\mathbb{R}}^{M}$ and all ${v}_{N}\in {\mathbb{R}}^{N}$ ,

\begin{align}{u}_{M}\mathrm{}R\mathrm{}{v}_{N}\iff {u}_{M}^{\mathrm{*}}{R}_{T}^{r}\mathrm{}{v}_{N}^{\mathrm{*}}\mathrm{}.\end{align}

This R is a population principle satisfying Existence Independence and the Strict Limit Property without being in the class that satisfies the Condition.

4. General repugnancy–sadism dilemma

As shown by the Proposition, the class of generalized utilitarian orderings cannot avoid the repugnancy–sadism dilemma, even in a non-anonymous setting. The following result shows that any population principle satisfying the axioms of Existence Independence and the Strict Limit Property raises the dilemma between repugnancy and sadism. Hence, this dilemma holds for a larger class of population principles, including non-utilitarian principles, such as the one in Example 3.

Proof. Assume that R satisfies Existence Independence and the Strict Limit Property and, furthermore, avoids the Very Sadistic Conclusion. It is sufficient to show that the Repugnant Conclusion holds for R. Take any $\xi \gt 0$ and $N\in \cal{N}$ , and let $\varepsilon \in (0,\xi )$ . By the avoidance of the Very Sadistic Conclusion, there exist ${N}^{\mathrm{*}}\in \cal{N}$ and ${v}_{{N}^{\mathrm{*}}}\in {\mathbb{R}}_{--}^{{N}^{\mathrm{*}}}$ such that, for all $M\in \cal{N}$ and all ${u}_{M}\in {\mathbb{R}}_{++}^{M}$ ,

\begin{align}{u}_{M}R\mathrm{}\mathrm{}{v}_{{N}^{\mathrm{*}}}\mathrm{}.\end{align}

We distinguish two cases.

Case 1: $N\cap {N}^{\mathrm{*}}=\mathrm{\varnothing }$ . Let ${\varepsilon }^{-}\in (0,\varepsilon )$ . By the Strict Limit Property, there exists a non-empty set ${M}^{\mathrm{*}}\subseteq {\mathbb{N}}\backslash (N\cup {N}^{\mathrm{*}})$ such that, for all $M\subseteq {\mathbb{N}}\backslash (N\cup {N}^{\mathrm{*}})$ with $M\supseteq {M}^{\mathrm{*}}$ ,

\begin{align}(\varepsilon {\mathbb 1}_{M},{v}_{{N}^{\mathrm{*}}},\varepsilon {\mathbb 1}_{N})\,\mathrm{}P\mathrm{}({\varepsilon }^{-}{\mathbb 1}_{M},{\varepsilon }^{-}{\mathbb 1}_{{N}^{\mathrm{*}}},\xi {\mathbb 1}_{N})\mathrm{}.\end{align}

Note that $({\varepsilon }^{-}{\mathbb 1}_{M},{\varepsilon }^{-}{\mathbb 1}_{{N}^{\mathrm{*}}})\in {\mathrm{\Omega }}_{++}$ . Thus, since R avoids the Very Sadistic Conclusion,

\begin{align}({\varepsilon }^{-}{\mathbb 1}_{M},{\varepsilon }^{-}{\mathbb 1}_{{N}^{\mathrm{*}}})\,\mathrm{}R\,\mathrm{}\mathrm{}{v}_{{N}^{\mathrm{*}}}\mathrm{}.\end{align}

Existence Independence implies that

\begin{align}({\varepsilon }^{-}{\mathbb 1}_{M},{\varepsilon }^{-}{\mathbb 1}_{{N}^{\mathrm{*}}},\xi {\mathbb 1}_{N})\,\mathrm{}R\,\mathrm{}({v}_{{N}^{\mathrm{*}}},\xi {\mathbb 1}_{N})\mathrm{}.\end{align}

Hence, by transitivity we obtain

(1) \begin{align}(\varepsilon {\mathbb 1}_{M},{v}_{{N}^{\mathrm{*}}},\varepsilon {\mathbb 1}_{N})\mathrm{}\,P\,\mathrm{}({v}_{{N}^{\mathrm{*}}},\xi {\mathbb 1}_{N})\mathrm{}.\end{align}

This holds true for all $M\subseteq {\mathbb{N}}\backslash (N\cup {N}^{\mathrm{*}})$ with $M\supseteq {M}^{\mathrm{*}}$ .

Existence Independence implies

\begin{align}(\varepsilon {\mathbb 1}_{M},\varepsilon {\mathbb 1}_{N})\,\mathrm{}P\,\mathrm{}\left(\xi {\mathbb 1}_{N}\right)\mathrm{}.\end{align}

The Repugnant Conclusion holds.

Case 2: $N\cap {N}^{\mathrm{*}}\ne \mathrm{\varnothing }.$ Let $K=N\cap {N}^{\mathrm{*}}$ . Now, we take ${N}^{\mathrm{*}\mathrm{*}}\in \cal{N}$ such that ${N}^{\mathrm{*}}\cap {N}^{\mathrm{*}\mathrm{*}}=\mathrm{\varnothing }$ and $N\backslash K\subseteq {N}^{\mathrm{*}\mathrm{*}}$ . Let ${\varepsilon }^{\mathrm{*}}\in (0,\varepsilon )$ . By applying the argument in Case 1, (1) holds for ${\varepsilon }^{\mathrm{*}}$ and ${N}^{\mathrm{*}\mathrm{*}}$ . That is, there exists a non-empty set ${M}^{\mathrm{*}}\subseteq {\mathbb{N}}\backslash ({N}^{\mathrm{*}}\cup {N}^{\mathrm{*}\mathrm{*}})$ such that

\begin{align}({\varepsilon }^{\mathrm{*}}{\mathbb 1}_{M},{v}_{{N}^{\mathrm{*}}},{\varepsilon }^{\mathrm{*}}{\mathbb 1}_{{N}^{\mathrm{*}\mathrm{*}}})\mathrm{}\,P\,\mathrm{}({v}_{{N}^{\mathrm{*}}},\xi {\mathbb 1}_{{N}^{\mathrm{*}\mathrm{*}}})\end{align}

for all $M\subseteq {\mathbb{N}}\backslash ({N}^{\mathrm{*}\mathrm{*}}\cup {N}^{\mathrm{*}})$ with $M\supseteq {M}^{\mathrm{*}}$ . By applying Existence Independence, it follows that $({\varepsilon }^{\mathrm{*}}{\mathbb 1}_{M},{\varepsilon }^{\mathrm{*}}{\mathbb 1}_{{N}^{\mathrm{*}\mathrm{*}}})\mathrm{}P\mathrm{}\mathrm{}\xi {\mathbb 1}_{{N}^{\mathrm{*}\mathrm{*}}}$ and therefore, by another application of Existence Independence,

(2) \begin{align}({\varepsilon }^{\mathrm{*}}{\mathbb 1}_{M},\xi {\mathbb 1}_{{N}^{\mathrm{*}}},{\varepsilon }^{\mathrm{*}}{\mathbb 1}_{{N}^{\mathrm{*}\mathrm{*}}})\mathrm{}\,P\,\mathrm{}(\xi {\mathbb 1}_{{N}^{\mathrm{*}}},\xi {\mathbb 1}_{{N}^{\mathrm{*}\mathrm{*}}})\mathrm{}.\end{align}

This holds for all $M\subseteq {\mathbb{N}}\backslash ({N}^{\mathrm{*}\mathrm{*}}\cup {N}^{\mathrm{*}})$ with $M\supseteq {M}^{\mathrm{*}}$ .

Note that $N\subseteq {N}^{\mathrm{*}}\cup {N}^{\mathrm{*}\mathrm{*}}$ . Let $L=({N}^{\mathrm{*}}\cup {N}^{\mathrm{*}\mathrm{*}})\backslash N$ . Thus,

\begin{align}{N}^{\mathrm{*}}\cup {N}^{\mathrm{*}\mathrm{*}}=L\cup N\mathrm{}.\end{align}

By the Strict Limit Property, there exists a non-empty set ${M}^{\mathrm{*}\mathrm{*}}\subseteq {\mathbb{N}}\backslash ({N}^{\mathrm{*}}\cup {N}^{\mathrm{*}\mathrm{*}})$ such that, for all $M\subseteq {\mathbb{N}}\backslash ({N}^{\mathrm{*}}\cup {N}^{\mathrm{*}\mathrm{*}})$ with $M\supseteq {M}^{\mathrm{*}\mathrm{*}}$ ,

(3) \begin{align}(\varepsilon {\mathbb 1}_{M},\xi {\mathbb 1}_{L},\varepsilon {\mathbb 1}_{N})\mathrm{}\,P\,\mathrm{}({\varepsilon }^{\mathrm{*}}{\mathbb 1}_{M},\xi {\mathbb 1}_{{N}^{\mathrm{*}}},{\varepsilon }^{\mathrm{*}}{\mathbb 1}_{{N}^{\mathrm{*}\mathrm{*}}})\end{align}

because $\varepsilon \gt {\varepsilon }^{\mathrm{*}}$ . Note that (2) and (3) hold for all $M\subseteq {\mathbb{N}}\backslash ({N}^{\mathrm{*}}\cup {N}^{\mathrm{*}\mathrm{*}})$ with $M\supseteq {M}^{\mathrm{*}}\cup {M}^{\mathrm{*}\mathrm{*}}$ . Thus, by transitivity we obtain

\begin{align}(\varepsilon {\mathbb 1}_{M},\xi {\mathbb 1}_{L},\varepsilon {\mathbb 1}_{N})\mathrm{}\,P\,\mathrm{}(\xi {\mathbb 1}_{{N}^{\mathrm{*}}},\xi {\mathbb 1}_{{N}^{\mathrm{*}\mathrm{*}}})\mathrm{}.\end{align}

This implies

\begin{align}(\varepsilon {\mathbb 1}_{M},\xi {\mathbb 1}_{L},\varepsilon {\mathbb 1}_{N})\mathrm{}\,P\,\mathrm{}(\xi {\mathbb 1}_{L},\xi {\mathbb 1}_{N})\end{align}

since ${N}^{\mathrm{*}}\cup {N}^{\mathrm{*}\mathrm{*}}=L\cup N$ . Existence Independence implies that

\begin{align}(\varepsilon {\mathbb 1}_{M},\varepsilon {\mathbb 1}_{N})\mathrm{}\,P\,\mathrm{}\mathrm{}\xi {\mathbb 1}_{N}\mathrm{}.\end{align}

The Repugnant Conclusion holds.□

We note that this dilemma is not raised if either Existence Independence or the Strict Limit Property is dropped. The following does not raise the dilemma; it satisfies the Strict Limit Property and violates Existence Independence.

Example 4 (Variable-Value principle). There exists an infinite subset $\mathbb{M}$ of $\mathbb{N}$ and a non-decreasing and bounded function $f:\mathbb{N}\to \mathbb{R}$ such that, for all M, $N\in \cal{N}$ , all ${u}_{M}\in {\mathbb{R}}^{M}$ and all ${v}_{N}\in {\mathbb{R}}^{N}$ ,

\begin{align}{u_M}R{v_N}\,\,\,{\rm{if\,and\,only\,if}}\,\,\,{{f(|M \cap {\mathbb M}|)} \over {|M \cap {\mathbb M}|}} \cdot {\sum\nolimits _{i \in M \cap {\mathbb M}}}\,{u_i} \ge {{f(|N \cap {\mathbb M}|)} \over {|N \cap {\mathbb M}|}} \cdot {\sum\nolimits_{i \in N \cap {\mathbb M}}}{v_i}.\end{align}

If $\mathbb{M}=\mathbb{N}$ and f is constant, then we obtain the standard Average Utilitarian principle. If $\mathbb{M}=\mathbb{N}$ and f is increasing, then we obtain the standard Variable-Value principle, proposed by Hurka (Reference Hurka1983) and Ng (Reference Ng1986, Reference Ng1989).

The next example, which satisfies Existence Independence and violates the Strict Limit Property, does not raise the dilemma if the critical level $\alpha$ is positive.

\begin{align}{u}_{\left(M\right)}=({u}_{\left(1\right)},\ldots, {u}_{\left(m\right)})\in {\mathbb{R}}^{m}\end{align}

be a rank-ordered permutation of ${u}_{M}$ such that ${u}_{\left(1\right)}\le {u}_{\left(2\right)}\le \ldots \le {u}_{\left(m\right)}$ , with ties being broken arbitrarily. For all M, $M\mathcal{\text{'}}\in \cal{N}$ with $\left|M\right|=\left|M\mathrm{\text{'}}\right|=m$ , and all ${v}_{M\mathrm{\text{'}}}\in {\mathbb{R}}^{M\mathrm{\text{'}}}$ ,

\begin{align}{u}_{M}{R}_{L}^{m}\mathrm{}{v}_{M\mathrm{\text{'}}}\end{align}

if and only if

\begin{align}{u}_{\left(i\right)}={v}_{\left(i\right)}\, \,{\rm for\, all}\,\, i\in \{1,\ldots, m\}\mathrm{},\end{align}

or

\begin{align}(\exists k\in \left\{1,\ldots, m\right\}\, \,{\rm such\, that}\,\, {u}_{\left(i\right)}={v}_{\left(i\right)}\, \,{\rm for\, all}\,\, i\lt k\,\, {\rm and}\,\, \mathrm{}{u}_{\left(k\right)}\gt {v}_{\left(k\right)})\mathrm{}.\end{align}

Let $\alpha \in \mathbb{R}$ be the critical level. For all M, $N\in \cal{N}$ , all ${u}_{M}\in {\mathbb{R}}^{M}$ and all ${v}_{N}\in {\mathbb{R}}^{N}$ ,

\begin{align}{u}_{M}R\mathrm{}{v}_{N}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\iff \mathrm{}\mathrm{}\left({u}_{M},\alpha {\mathbb {1}}_{{M}^{\mathrm{\text{'}}}}\right){R}_{L}^{n}{v}_{N}\,\mathrm{}\mathrm{}{\rm{i}\mathrm{f}}\,\mathrm{}\left|N\right|=n\gt m=\left|M\right|\mathrm{},\end{align}

\begin{align}{u}_{M}R\mathrm{}{v}_{N}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\iff \mathrm{}\mathrm{}{u}_{M}{R}_{L}^{m}\left({v}_{N},\alpha {\mathbb {1}}_{{N}^{\mathrm{\text{'}}}}\right)\,\mathrm{}{\rm{i}\mathrm{f}}\,\mathrm{}\left|N\right|=n\lt m=\left|M\right|\mathrm{},\end{align}

\begin{align}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! {u}_{M}R\mathrm{}{v}_{N}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\iff \mathrm{}\mathrm{}{u}_{M}{R}_{L}^{m}\mathrm{}{v}_{N}\,\mathrm{}\mathrm{}{\rm{i}\mathrm{f}}\,\mathrm{}\left|N\right|=n=m=\left|M\right|\mathrm{},\end{align}

where $\left|M\mathrm{\text{'}}\right|=n-m$ , $M\mathrm{\text{'}}\cap M=\mathrm{\varnothing }$ , $\left|N\mathrm{\text{'}}\right|=m-n$ and $N\mathrm{\text{'}}\cap N=\mathrm{\varnothing }$ . This is an extension of the Leximin principle proposed by Blackorby et al. (Reference Blackorby, Bossert and Donaldson1996).

Finally, we consider Rank-Discounted Utilitarianism introduced by Asheim and Zuber (Reference Asheim and Zuber2014); see also Asheim and Zuber (Reference Asheim, Zuber, Arrhenius, Bykvist, Campbell and Finneron-Burns2022). This principle applies rank discounting to welfare evaluations, where an individual’s welfare contribution is weighted according to their relative position in the welfare ranking of the total population, with higher weights assigned to individuals with lower welfare. To be precise, R is rank-discounted generalized utilitarian if there exist $\beta \in \left(\mathrm{0,1}\right)$ and a continuous and strictly increasing transformation function $g:\mathbb{R}\to \mathbb{R}$ with $0\in g\left(\mathbb{R}\right)$ such that, for all M, $N\in \cal{N}$ , all ${u}_{M}\in {\mathbb{R}}^{M}$ and all ${v}_{N}\in {\mathbb{R}}^{N}$ ,

\begin{align}{u}_{M}R\mathrm{}{v}_{N}{\rm \,\,\, \,if\, and\, only\, if \,\,\,\, }{\sum }_{r=1}^{\left|M\right|}{\beta }^{r}g\left({u}_{\left(r\right)}\right)\ge {\sum }_{r=1}^{\left|N\right|}{\beta }^{r}g\left({v}_{\left(r\right)}\right).\end{align}

This population principle might be considered to be positioned between utilitarianism and lexicographic preferences. As shown by Asheim and Zuber (Reference Asheim and Zuber2014), Rank-Discounted Utilitarianism orderings avoid both the Repugnant Conclusion and Very Sadistic Conclusion, but at the cost of satisfying neither Existence Independence nor the Strict Limit Property.Footnote ³

5. Extensions of the dilemma

In this section, we present six observations that extend our main result. The first one follows from the proof of the Theorem, while the proofs of the five others are contained in the appendix. Figure 1 illustrates graphically our main result and the extensions provided in the present section.

Figure 1.

Graphical illustration of the main result and its extensions.

Figure 1 Long description

A diagram illustrating different principles in population ethics and their relationships. The diagram features multiple overlapping shapes, each representing a different principle. The central shape is labeled CLU, surrounded by other shapes labeled RDU, CLL, AU, and VarVal. Annotations indicate various properties and conditions, such as 'Existence Independence of the Worst-Off' and 'Strict Limit Property for the Worst-Off'. The diagram also highlights areas where only certain principles avoid specific dilemmas, marked by dashed lines and text annotations. The relationships between the principles are depicted through the overlapping areas and the directional flow indicated by arrows.

5.2. Relaxing transitivity and completeness

In the previous arguments, we assumed the population principle to be reflexive, complete and transitive. However, both completeness and transitivity have each been criticized. For example, Sen (Reference Sen2004) argues that it is not plausible to force all states of affairs to be comparable, and it is rather natural to assume that there are some states that cannot be ranked (‘assertive incompleteness’). A powerful challenge to the transitivity of indifference is illustrated by the classic coffee–sugar example (Raiffa Reference Raiffa1968).Footnote ⁴ Suppose that adding $0.1$ grams of sugar to a cup of coffee is too small a difference to affect taste and is therefore considered indifferent. Then, by transitivity, adding $k\cdot 0.1$ grams and adding $(k+1)\cdot 0.1$ grams must also be indifferent for all $k\in \mathbb{N}$ . This would imply, absurdly, that zero grams and 200 grams of sugar are indifferent. The core issue revealed by this example is the assumption that the symmetric part I of R is transitive.

Armstrong (Reference Armstrong1939) offers an alternative explanation: indifference occurs when the difference between utilities is below a certain threshold. Hence, if the difference in the value between A and B is $a\gt 0$ , and between B and C is $b\gt 0$ , where both a and b are below the threshold $\theta \gt 0$ that triggers preference, the difference between A and C would be $a+b$ , which could exceed $\theta$ and thus trigger a preference for A over C. This violates the transitivity of the symmetric part I of R.

It does not, however, violate the transitivity of the asymmetric part P of R. Hence, we can still impose quasi-transitivity of R in the sense that for all M, N, $K\in \cal{N}$ , all ${u}_{M}\in {\mathbb{R}}^{M}$ , all ${v}_{N}\in {\mathbb{R}}^{N}$ and all ${w}_{K}\in {\mathbb{R}}^{K}$ ,

\begin{align}{u}_{M}P\,\mathrm{}{v}_{N}\, \,{\rm and}\,\, {v}_{N}P\,\mathrm{}{w}_{K}\Rightarrow {u}_{M}P\,\mathrm{}{w}_{K}.\end{align}

It turns out that there is no dilemma between repugnancy and sadism if the transitivity of R is weakened to quasi-transitivity of R, as the following example shows.

Example 6 (Same-People Utilitarian Extension principle). For all M, $N\in \cal{N}$ , all ${u}_{M}\in {\mathbb{R}}^{M}$ and all ${v}_{N}\in {\mathbb{R}}^{N}$ ,

\begin{align}{u}_{M}R\mathrm{}{v}_{N}\iff \left[\left(M=N\,\,\, \, {\rm and}\,\,\, \, {\sum }_{i\in M}{u}_{i}\ge {\sum }_{i\in N}{v}_{i}\right)\,\,\mathrm{}{\rm or}\,\,\mathrm{}N\ne M\right].\end{align}

The relation R of Example 6 satisfies Existence Independence and the Strict Limit Property, while avoiding the dilemma between repugnancy and sadism. It satisfies quasi-transitivity but violates the following weakening of transitivity, which has been widely examined; see, for instance, Luce (Reference Luce1956): Semi-transitivity of R requires that, for all M, N, K, $L\in \cal{N}$ , all ${u}_{M}\in {\mathbb{R}}^{M}$ , all ${v}_{N}\in {\mathbb{R}}^{N}$ , all ${w}_{K}\in {\mathbb{R}}^{K}$ and all ${s}_{L}\in {\mathbb{R}}^{L}$ ,

\begin{align}{u}_{M}P\,\mathrm{}{v}_{N}\, \, {\rm{and}}\, {v}_{N}P\,\mathrm{}{w}_{K}\Rightarrow {s}_{L}P\,\mathrm{}{w}_{K}\, \, {\rm{or}}\, \, {u}_{M}P\,\mathrm{}{s}_{L}\mathrm{}.\end{align}

Any reflexive, complete and transitive binary relation R satisfies this property. On the other hand, any reflexive and semi-transitive relation R is quasi-transitive. Hence, in these senses, semi-transitivity is a weakening of transitivity but a strengthening of quasi-transitivity. Armstrong’s approach fails transitivity but is compatible with both semi-transitivity and quasi-transitivity.

We make the observation that the dilemma between repugnancy and sadism, while non-existent under quasi-transitivity, remains under semi-transitivity. For the purpose of this observation, say that a relation R is a semi-transitive population principle if R satisfies semi-transitivity and reflexivity. Note that the semi-transitive population principle R is not necessarily complete. Hence, both completeness and transitivity are relaxed to some degree.

Observation 2. If the semi-transitive population principle R satisfies Existence Independence and the Strict Limit Property, then R raises a dilemma between repugnancy and sadism.

This follows from Proposition A in the appendix.

5.4. Weaker Existence Independence axioms

As alluded to above, Existence Independence is violated by rank-discounted utilitarian population principles, which avoid the dilemma between repugnancy and sadism. However, Rank-Discounted Utilitarianism satisfies some types of independence, and thus, it is of interest to explore weaker versions of Existence Independence.

First, we examine a way of violating Existence Independence under Rank-Discounted Utilitarianism. Consider the following two populations:

\begin{align}{u}_{N}=\left(\mathrm{0,4}\right)\, \,{\rm and}\,\, {v}_{N}=\left(\mathrm{1,1}\right)\end{align}

where $\left|N\right|=2$ . Now, assume that $g\left(x\right)=x$ and $\beta =0.5$ . Note that the corresponding values under Rank-Discounted Utilitarianism are

\begin{align}{\sum }_{r=1}^{\left|N\right|}{\beta }^{r}{u}_{\left(r\right)}=4{\beta }^{2}=1\, \, \, \, \rm{and}\,\, \, \, {\sum }_{{\it r}=1}^{\left|{\it N}\right|}{\beta }^{\it r}{\it v}_{\left({\it r}\right)}=\beta (1+\beta )=0.75\mathrm{}.\end{align}

Hence, the former is better than the latter. Let us add a singleton, M, to both populations, where $N\cap M=\mathrm{\varnothing }$ and the population size of M is the same as N. Consider the following distributions:

\begin{align}{\hat{u}}_{N\cup M}=\left({u}_{N},1\cdot {\mathbb {1}}_{M}\right)\, \,{\rm and}\,\, {\hat{v}}_{N\cup M}=({v}_{N},1\cdot {\mathbb {1}}_{M})\mathrm{}.\end{align}

The total number of individuals is 3. Now, the corresponding values under Rank-Discounted Utilitarianism are

\begin{align}{\sum }_{r=1}^{\left|N\right|+m}{\beta }^{r}{\hat{u}}_{\left(r\right)}={\beta }^{2}+4{\beta }^{3}=0.75\,\, \, \, \rm{and} \,\,\,\, {\sum }_{{\it r}=1}^{\left|\it N\right|}{\beta }^{\it r}{\hat{\it v}}_{\left({\it r}\right)}=(\beta +{\beta }^{2}+{\beta }^{3})=0.875\mathrm{}.\end{align}

The latter is better than the former. Thus, Existence Independence fails. This occurs because the newly added individuals belong to the middle class, in contrast to those who already exist, especially for ${u}_{N}$ .Footnote ⁵

Therefore, it is of interest to consider restricted versions of Existence Independence that apply only to cases involving the addition or removal of individuals who would be the best-off or worst-off. The following two axioms are non-anonymous versions of axioms offered by Asheim and Zuber (Reference Asheim and Zuber2014).

Axiom (Existence Independence of the Best-Off*) For all M and $L\in \cal{N}$ such that $M\cap L=\varnothing$ , all ${u}_{M}$ , ${v}_{M}\in {\mathbb{R}}^{M}$ and all ${w}_{L}\in {\mathbb{R}}^{L}$ satisfying ${\rm min}\,{w}_{L}\ge {\rm max}\{{\rm max}\,{u}_{M},{\rm max}\,{v}_{M}\}$ , ${u}_{M}R{v}_{M}$ if and only if $({u}_{M},{w}_{L})R({v}_{M},{w}_{L})$ .

Axiom (Existence Independence of the Worst-Off) For all M, N, $L\in \cal{N}$ such that $(M\cup N)\cap L=\varnothing$ , all ${u}_{M}\in {\mathbb{R}}^{M}$ , all ${v}_{N}\in {\mathbb{R}}^{N}$ and all ${w}_{L}\in {\mathbb{R}}^{L}$ satisfying ${\rm max}\, {w}_{L}\le {\rm min}\{{\rm min}\, {u}_{M},{\rm min}\,{v}_{N}\}$ , ${u}_{M}R{v}_{N}$ if and only if $({u}_{M},{w}_{L})R({v}_{N},{w}_{L})$ .

It is easy to confirm that these axioms are satisfied by Rank-Discounted Utilitarianism. However, we have not been able to determine whether any population principle R satisfying Existence Independence of the Best-Off*, Existence Independence of the Worst-Off and the Strict Limit Property raises a dilemma between repugnancy and sadism. We cannot prove this, and we have no counterexample to offer. However, the following strengthening of Existence Independence of the Best-Off* is considered, where the population M in ${u}_{M}$ is allowed to differ from the population N in ${v}_{N}$ , leading to the following Existence Independence axiom.

Axiom (Existence Independence of the Best-Off) For all M, N, $L\in \cal{N}$ such that $(M\cup N)\cap L=\varnothing$ , all ${u}_{M}\in {\mathbb{R}}^{M}$ , all ${v}_{N}\in {\mathbb{R}}^{N}$ and all ${w}_{L}\in {\mathbb{R}}^{L}$ satisfying ${\rm min}\,{w}_{L}\ge {\rm max}\{{\rm max}\,{u}_{M},{\rm max}\,{v}_{N}\}$ , ${u}_{M}R{v}_{N}$ if and only if $({u}_{M},{w}_{L})R({v}_{N},{w}_{L})$ .

We can establish the following result, which demonstrates that Existence Independence of the Best-Off and Existence Independence of the Worst-Off, combined with the Strict Limit Property, are sufficient for raising the dilemma. Note that Existence Independence of the Best-Off and Existence Independence of the Worst-Off are logically implied by Existence Independence, which applies regardless of the relative positions of the individuals being added or removed.

Observation 5. If the population principle R satisfies Existence Independence of the Best-Off, Existence Independence of the Worst-Off and the Strict Limit Property, then R raises a dilemma between repugnancy and sadism.

This follows from Proposition A in the appendix. There is actually little need to modify the proof of the original theorem, since it only relies on Existence Independence in cases involving the best-off and worst-off individuals.

One might argue that this observation suggests that a violation of the Strict Limit Property is a more essential reason why the dilemma is not raised by Rank-Discounted Utilitarianism, since this class of population principles satisfies Existence Independence of the Worst-Off and Existence Independence of the Best-Off*. We warn against this conclusion as the weakening of Existence Independence of the Worst-Off to Existence Independence of the Worst-Off* is not innocent; note in particular that the class of Variable-Value principles considered in Example 4 satisfy Existence Independence of the Worst-Off* but not Existence Independence of the Worst-Off.

7. Concluding remarks

Our analysis demonstrates that the fundamental dilemma associated with the Repugnant Conclusion and the Very Sadistic Conclusion extends far beyond CLU to encompass a broad class of population principles. Any population principle that satisfies the fundamental axioms of Existence Independence and the Strict Limit Property inevitably faces this dilemma. This result holds without requiring Anonymity, which has significant implications for evaluating populations that differ in temporal, spatial or probabilistic dimensions, or when including non-human populations. The non-anonymous nature of our framework allows for greater flexibility in addressing ethical problems where the identity of individuals may matter, such as intergenerational justice or multi-species welfare considerations.

The unavoidable nature of this dilemma suggests that population ethics may require reconsidering some deeply held intuitions. While we have shown that abandoning either Existence Independence or the Strict Limit Property can help avoid the dilemma, as demonstrated by the Variable-Value principle and the Critical-Level Leximin principle, respectively, these alternatives come with their own limitations. Future research might develop hybrid approaches that develop new axioms that better capture our moral intuitions while avoiding the repugnancy–sadism dilemma. Otherwise, it strengthens the position that avoiding the Repugnant Conclusion should not be a prerequisite for any acceptable population principle (Zuber et al. Reference Zuber, Venkatesh, Tännsjö, Tarsney, Stefánsson, Steele, Spears, Sebo, Pivato, Ord, Ng, Masny, MacAskill, Lawson, Kuruc, Hutchinson, Gustafsson, Greaves, Forsberg, Fleurbaey, Coffey, Cato, Castro, Campbell, Budolfson, Broome, Berger, Beckstead and Asheim2021).