Repugnant conclusions

Abstract

The population ethics literature has long focused on attempts to avoid the repugnant conclusion. We show that a large set of social orderings that are conventionally understood to escape the repugnant conclusion do not in fact avoid it in all instances. As we demonstrate, prior results depend on formal definitions of the repugnant conclusion that exclude some repugnant cases, for reasons inessential to any “repugnance” (or other meaningful normative properties) of the repugnant conclusion. In particular, the literature traditionally formalizes the repugnant conclusion to exclude cases that include an unaffected sub-population. We relax this normatively irrelevant exclusion, and others. Using several more inclusive formalizations of the repugnant conclusion, we then prove that any plausible social ordering implies some instance of the repugnant conclusion. This understanding—that it is impossible to avoid all instances of the repugnant conclusion—is broader than the traditional understanding in the literature that the repugnant conclusion can only be escaped at unappealing theoretical costs. Therefore, the repugnant conclusion provides no methodological guidance for theory or policy-making, because it does not discriminate among candidate social orderings. So escaping the repugnant conclusion should not be a core goal of the population ethics literature.

Appendices

Appendix

Proofs

2.1 Proof of Theorem 1

The proof uses Aggregation twice. If $\succsim$ satisfies Axiom 5, then choose $\nu \in (0,\varepsilon )$; this will be used to construct a $\mathbf {v}$ without negative lives. If $\succsim$ satisfies Axiom 6 but not Axiom 5, choose $\nu < 0$, which implies $\nu < \varepsilon$. Choose $\delta \in (0, \min (\frac{\varepsilon - \nu }{3}, \frac{\varepsilon }{3}))$ and, if $\nu <0$, $\delta < -\nu$. For any m, construct $\left( u^h\mathbf {1}_{n^h} , \nu \mathbf {1}_m \right)$; if m is large enough then $(\nu + \delta ) \mathbf {1}_{n^h+m} \succ \left( u^h\mathbf {1}_{n^h} , \nu \mathbf {1}_m \right)$, by Aggregation. For any $n^\varepsilon$, construct $\left( \varepsilon \mathbf {1}_{n^\varepsilon }, u^\ell \mathbf {1}_{n^\ell } , \nu \mathbf {1}_m\right)$; if $n^\varepsilon$ is large enough (once m is fixed) then $\left( \varepsilon \mathbf {1}_{n^\varepsilon }, u^\ell \mathbf {1}_{n^\ell } , \nu \mathbf {1}_m\right) \succ (\varepsilon - \delta ) \mathbf {1}_{n^\varepsilon + n^\ell +m}$. Choose m large enough to satisfy Aggregation; then choose $n^\varepsilon$ large enough to satisfy Aggregation and such that $n^\varepsilon + n^\ell +m > n^h+m$. By one of the sign axioms and by the construction of $\delta$, $(\varepsilon - \delta ) \mathbf {1}_{n^\varepsilon + n^\ell +m} \succ (\nu + \delta ) \mathbf {1}_{n^h+m}$. Then by applying transitivity twice, $\left( \varepsilon \mathbf {1}_{n^\varepsilon }, u^\ell \mathbf {1}_{n^\ell } , \nu \mathbf {1}_m\right) \succ \left( u^h\mathbf {1}_{n^h} , \nu \mathbf {1}_m \right)$.

2.2 Proof of Proposition 1

Because the extended very repugnant conclusion implies the very repugnant conclusion, the proof of Theorem 1 applies for orderings that satisfy Aggregation and a sign axiom.

For social orderings that satisfy Aggregation but not a zero axiom, or for social orderings that merely satisfy Minimal Equality Preference, the proof by construction uses the $u_j = v_j + \varepsilon$ horn of the definition of $\varepsilon$-change. For Aggregation-satisfying orderings, simply include a very large base population; improve many lives by $\varepsilon$.

To begin a construction for Non-Aggregation social orderings, choose any $\varepsilon$, $u^\ell$, $u^h$, $n^\ell$, and $n^h$ according to the Extended VRC. Next, set $m^h$ and $m^\ell$ in the EVRC to both be the maximum of $n^h +1$ and $n^\ell + 1$. Then, let $\xi$ in the Axiom be $u^\ell - \varepsilon$ from the EVRC. Let $\delta$ in the axiom be $\varepsilon$ from the EVRC. Notice that $\xi + \delta$ in the Axiom now equals $u^\ell$ from the EVRC. Now, let $\mathbf {u}$ from the Axiom be $u^h \mathbf {1}_{m^h}$. Notice that $n(\mathbf {u})$ from the Axiom is now fixed at $n(\mathbf {u})=m^h=m^\ell$ from the EVRC.

The construction next uses the Minimal Equality Preference axiom. We have now specified a $\mathbf {u}$, $\xi$, and $\delta$. So there exists an $n^*$ such that if $n>n^*$ then $(\xi + \delta )\mathbf {1}_{n + n(\mathbf {u})} \succ \left( \xi \mathbf {1}_{n} , \mathbf { u} \right)$. Choose such an n and call it $\tilde{n}$. This construction fulfills the conditions of the Extended VRC. Note that we may choose any $\mathbf {v} \in \Omega$. Let $\mathbf {v} = \xi \mathbf {1}_{\tilde{n}}$. Now notice that $\left( \xi \mathbf {1}_{\tilde{n}} , \mathbf { u} \right)$ from the Axiom is $\left( \xi \mathbf {1}_{\tilde{n}} , u^h \mathbf {1}_{m^h}\right)$, which is $\left( \mathbf {v} , u^h \mathbf {1}_{m^h}\right) = \mathbf {v}^h$ from the EVRC. Let $\mathbf {v}^\ell = (\xi + \delta )\mathbf {1}_{\tilde{n} + n(\mathbf {u})}=(\xi + \delta )\mathbf {1}_{\tilde{n} + m^\ell }=\left( u^\ell \mathbf {1}_{\tilde{n}} , u^\ell \mathbf {1}_{m^\ell }\right) =\left( (\xi + \varepsilon )\mathbf {1}_{\tilde{n}} , u^\ell \mathbf {1}_{m^\ell }\right)$. Set $n^\varepsilon$ equal to $\tilde{n}$. Finally, we can see that $\left( (\xi + \varepsilon )\mathbf {1}_{\tilde{n}} , u^\ell \mathbf {1}_{m^\ell }\right)$ is separated by $n^\varepsilon$ $\varepsilon$-changes from $\left( \mathbf {v} , u^\ell \mathbf {1}_{m^\ell }\right)$.

Maximin and maximax both can be shown to imply the extended very repugnant conclusion by having $\mathbf {v}$ contain the least or greatest (respectively) utility level, and then increasing this with one $\varepsilon$-change.