Non-Archimedean population axiologies

Abstract

Non-Archimedean population axiologies – also known as lexical views – claim (i) that a sufficient number of lives at a very high positive welfare level would be better than any number of lives at a very low positive welfare level and/or (ii) that a sufficient number of lives at a very low negative welfare level would be worse than any number of lives at a very high negative welfare level. Such axiologies are popular because they can avoid the (Negative) Repugnant Conclusion and satisfy the adequacy conditions given in the central impossibility result in population axiology due to Gustaf Arrhenius. I provide a novel argument against them which appeals to the way that good and bad lives can intuitively outweigh one another.

1. Introduction

By a ‘heavenly life’ I mean a life at a very high positive welfare level. By a ‘barely good life’ I mean a life at a very low positive welfare level. This allows us to state the

Repugnant Conclusion.

For any number of heavenly lives, there is a number of barely good lives that would be better. ¹

Similarly, by a ‘hellish life’ I mean a life at a very low negative welfare level. And by a ‘barely bad life’ I mean a life at a very high negative welfare level. This allows us to state the

Negative Repugnant Conclusion.

For any number of hellish lives, there is a number of barely bad lives that would be worse.

Pace Zuber et al. (2021), many continue to believe that we must avoid the (Negative) Repugnant Conclusion. Non-Archimedean population axiologies – also called lexical views – can do so. For they accept at least one of the following two claims:

The Weak Superiority of Heavenly Lives.

A sufficient number of heavenly lives would be better than any number of barely good lives. ²

The Weak Inferiority of Hellish Lives.

A sufficient number of hellish lives would be worse than any number of barely bad lives. ³

The Weak Superiority of Heavenly Lives is inconsistent with the Repugnant Conclusion; and the Weak Inferiority of Hellish Lives is inconsistent with the Negative Repugnant Conclusion. The former captures the intuition that sufficiently large losses in quality (welfare) cannot be compensated for by gains in quantity (number), no matter how large; and the same goes for the latter, mutatis mutandis. Thus, non-Archimedean axiologies are prima facie compelling.

Nevertheless, the purpose of this paper is to offer a novel argument against non-Archimedean axiologies. Here’s the plan. Section 2 introduces non-Archimedean axiologies in greater detail, showing why they have great initial appeal. Section 3 offers a novel argument against them. To preview, the argument will be that the non-Archimedean solution to the Negative Repugnant Conclusion is untenable, and that without it, the remaining non-Archimedean view entails a different unacceptable claim, which I call the Repugnant Elitist Conclusion. Sections 4 and 5 explore the possibilities that are left open if one insists on retaining the non-Archimedean solution to the Negative Repugnant Conclusion. One option, which I call the Somber View, turns out to be deeply pessimistic, in that it pushes us towards favouring extinction. The remaining option, which I call Heavy Tails, entails a weaker version of the Repugnant Elitist Conclusion. Section 6 considers a final escape route from the argument in section 3. Section 7 concludes.

2. The Simple Additive Picture

This section introduces non-Archimedean population axiologies in greater detail. A population is a set of lives, each of which has a lifetime welfare level. A population axiology is an ‘is all-things-considered better than’ relation over the set of possible populations. As stated in the Introduction, non-Archimedean axiologies accept at least one of the Weak Superiority of Heavenly Lives and the Weak Inferiority of Hellish Lives. Moreover, as I’ll understand them, non-Archimedean axiologies endorse an attractive understanding of the structure of value that I call the Simple Additive Picture. The Simple Additive Picture is the conjunction of three principles – Addition, Separability and Transitivity – which I introduce presently.

I begin by building to the principle of Addition. It’s plausible that adding a good life – a life at a positive welfare level – to a population makes the population better. Why? Huemer (2008: 923) provides a simple rationale: ‘Worthwhile lives are good. More of a good thing is better. Therefore, increasing the number of worthwhile lives makes the world better.’ ⁴ Similarly, it’s plausible that adding a bad life – a life at a negative welfare level – to a population makes the population worse. We can capture these thoughts precisely by introducing some notation. Following Nebel (2023), for all populations A and B, let ‘A ∪ B’ refer to the union of disjoint populations A* and B*, where A* is identical to A in size and welfare distribution, and likewise for B* and B. We can now state the principle of

Addition.

For any population X and good (bad) life l, X ∪ l is better (worse) than X.

Here’s another plausible claim: when we’re evaluating a change to a population, people who are entirely unaffected by that change should not factor into our evaluation. Imagine, for example, that we’re thinking about adding a new person to Earth’s present population. Intuitively, how good or bad it would be to add this person doesn’t depend on how many people lived in the Achaemenid Empire (ca. 550–330 BCE), or on how well-off these ancient Persians were. This thought can be captured more precisely in the principle of

Separability.

For all populations X, Y, and Z, X is better than Y iff X ∪ Z is better than Y ∪ Z. ⁵

Finally, it’s plausible that the ‘is all-things-considered better than’ relation is transitive. Call this claim Transitivity.

Addition, Separability and Transitivity comprise the Simple Additive Picture. Now, the most well-known population axiology that endorses the Simple Additive Picture is Standard Totalism. According to Standard Totalism, (i) welfare is a scalar quantity (which means that every welfare level can be represented by a single real number) and (ii) for every population A and B, A is better than B just in case total welfare in A is greater than total welfare in B. Standard Totalism infamously entails both the Repugnant Conclusion and the Negative Repugnant Conclusion. Non-Archimedean views consequently enjoy great initial appeal, for, like Standard Totalism, they endorse the Simple Additive Picture, but, unlike Standard Totalism, they are able to avoid the (Negative) Repugnant Conclusion. ⁶ Moreover, non-Archimedean views can avoid Gustaf Arrhenius’s (2011) influential impossibility result in population axiology – at least in cases of choice under certainty. ⁷ For these reasons, I used to believe that non-Archimedeanism was the best game in town. I have recently come to believe that this was a mistake. The next several sections explain why.

3. The Repugnant Elitist Conclusion

This section lays out my main argument against non-Archimedean axiologies. The argument proceeds in three stages. Firstly, I argue that hellish lives and barely bad lives are exchangeable. Loosely, this means that we can trade these types of lives off against each other. (A precise definition is given below.) I argue for this exchangeability claim primarily by arguing against the Weak Inferiority of Hellish Lives. Secondly, I introduce a relation that I call outweighing and suggest that non-Archimedeans should accept that barely good lives and barely bad lives can outweigh each other. Finally, given the claims reached thus far, I derive the Repugnant Elitist Conclusion, which I regard as a reductio.

3.1. Preliminaries

Two preliminaries before we begin. Firstly, I assume that a population is good (bad) iff it is better (worse) than the empty population, which contains zero lives. Secondly, it will be useful to prove up front that the Simple Additive Picture entails a claim that I call Scaling. To state Scaling, we’ll need to introduce some further notation. For any population X, let ‘n(X)’ refer to a population composed of the union of X and (n-1) copies of X, where a copy of a population is a disjoint population identical to the original in size and welfare distribution. Thus, for example, ‘2(X)’ refers to X ∪ X*, where X* is identical to X in size and welfare distribution. We can now state

Scaling.

For all populations A and B, A is better than B iff for every natural number q, q(A) is better than q(B). ⁸

We’ll start by showing that for all populations A and B, if A is better than B, then for every natural number q, q(A) is better than q(B). ⁹ Take an arbitrary pair of populations A and B and assume that A is better than B. We want to show that it follows that for every natural number q, q(A) is better than q(B). We’ll do so by induction. We already have the base case, where q = 1: we’ve assumed that A is better than B. Here’s the inductive step: for every natural number q, if q(A) is better than q(B), then (q + 1)A is better than (q + 1)B. To establish the inductive step, assume that q(A) is better than q(B). It follows from Separability that q(A) ∪ B is better than (q + 1)B. And, given that A is better than B, it also follows from Separability that (q + 1)A is better than q(A) ∪ B. Since (q + 1)A is better than q(A) ∪ B and q(A) ∪ B is better than (q + 1)B, by Transitivity, (q + 1)A is better than (q + 1)B. This establishes the inductive step, thereby completing the left-to-right half of the proof. Going from right to left is straightforward. Assume that for every natural number q, q(A) is better than q(B). Then in particular A is better than B (this is the special case where q = 1).

3.2. Hellish lives and barely bad lives are exchangeable

We now turn to stage one of the argument. The argument will be that hellish lives and barely bad lives are exchangeable.

Exchangeability.

One bad (good) is exchangeable with another bad (good) iff for any quantity of the former, there is a quantity of the latter that would be worse (better), and vice versa.

So, if hellish lives and barely bad lives are exchangeable, then for any number of hellish lives, there is a number of barely bad lives that would be worse, and vice versa.

3.2.1. Against the Weak Inferiority of Hellish Lives

To argue that hellish lives and barely bad lives are exchangeable, I first argue against the Weak Inferiority of Hellish Lives. Here’s the argument in a nutshell: in conjunction with the Simple Additive Picture, the Weak Inferiority of Hellish Lives entails

Strong Nonsuperiority Across Adjacent Levels.

For any finite, ascending sequence of negative welfare levels beginning with a level that corresponds to a hellish life and ending with a level that corresponds to a barely bad life, there are two welfare levels that are (i) adjacent to one another in the sequence and (ii) such that no number of lives at the higher level would be worse than any number of lives at the lower level.

Strong Nonsuperiority Across Adjacent Levels is false, so, since we’re holding the Simple Additive Picture fixed, we must reject the Weak Inferiority of Hellish Lives.

I’ll begin by deriving Strong Nonsuperiority Across Adjacent Levels; then I’ll argue against it. ¹⁰ It will be useful to have the following two definitions at hand for easy reference:

Weak nonsuperiority.

x is weakly nonsuperior to y iff some quantity of x is not better than any quantity of y.

Strong nonsuperiority.

x is strongly nonsuperior to y iff any quantity of x is not better than any quantity of y.

To save space, I’ll refer to a finite, ascending sequence of negative welfare levels beginning with a level that corresponds to a hellish life and ending with a level that corresponds to a barely bad life as a ‘negative sequence’. Consider an arbitrary negative sequence S. Call the first level in S ‘l ₁’ and the final level ‘l _n ’.

We begin by showing that there are two welfare levels in S that are (i) adjacent to one another in S and (ii) such that weak nonsuperiority holds across lives at these levels. ¹¹ Assume for contradiction that there are no such welfare levels. Then for every pair of welfare levels that are adjacent in S, each number of lives at the lower (i.e. worse) welfare level is better than some number of lives at the higher (i.e. better) welfare level. It follows by Transitivity that each number of lives at l ₁ is better than some number of lives at l _n . But this contradicts the Weak Inferiority of Hellish Lives, which says that some number of hellish lives (i.e. lives at l ₁) is worse than any number of barely bad lives (i.e. lives at l _n ). So there are (at least) two welfare levels that are (i) adjacent to one another in S and (ii) such that weak nonsuperiority holds across lives at these levels. Call these welfare levels ‘l _j ’ and ‘l _k ’.

We’ll now show that lives at l _j are strongly nonsuperior to lives at l _k . ¹² Assume for contradiction that lives at l _j are not strongly nonsuperior to lives at l _k . This means that for some natural numbers m and n, n lives at l _k would be worse than m lives at l _j . Now take an arbitrary natural number x and consider x lives at l _j . There must be a multiple of m – say, qm – such that qm is greater than x. By Addition, qm lives at l _j would be worse than x lives at l _j . However, since n lives at l _k would be worse than m lives at l _j , by Scaling, qn lives at l _k would be worse than qm lives at l _j . Since qn lives at l _k would be worse than qm lives at l _j and qm lives at l _j would be worse than x lives at l _j , by Transitivity, qn lives at l _k would be worse than x lives at l _j . Since x was arbitrary, we have that for any natural number of lives at l _j , there is a natural number of lives at l _k that would be worse. But that contradicts the fact that lives at l _j are weakly nonsuperior to lives at l _k . So lives at l _j are strongly nonsuperior to lives at l _k . And since S was an arbitrary negative sequence, we have Strong Nonsuperiority Across Adjacent Levels. ¹³

I’ll now give a counterexample to Strong Nonsuperiority Across Adjacent Levels. Here’s the setup. We construct a finite sequence of lives that contain nothing but pain. The pain in each life is constant and qualitatively uniform. The lives differ with respect to two variables: duration and intensity of pain. The first life is relatively long and the pain in it is agonizing. It’s a hellish life. The final life is one second long and the pain in it is barely noticeable. It’s a barely bad life. We can get from the duration of the first life to the duration of the final life via a finite number of small decreases in length, e.g. in increments of 0.5 seconds. Plausibly, we can also get from the intensity of pain in the first life to that in the final life via a finite number of small decreases in pain intensity. We can therefore construct a finite sequence of lives beginning with the first life and ending with the final life such that the sole descriptive difference between lives that are adjacent to one another in the sequence is either a small decrease in duration or a small decrease in pain intensity (but not both). Thus, as I imagine the sequence, the second life contains pain of equal intensity to that in the first life, but is 0.5 seconds shorter; the third life is equal to the second life in duration, but contains pain that is slightly less intense; and so on.

Now consider the sequence of welfare levels that map one-to-one onto the lives in this sequence. This sequence of welfare levels – S’ – is a negative sequence. So, according to Strong Nonsuperiority Across Adjacent Levels, there are two welfare levels adjacent to one another in S’ such that strong nonsuperiority holds across the lives at these levels. I submit that this is false. For (1) if strong nonsuperiority holds across lives at two different negative welfare levels, then the welfare difference between the lives must be very large, in a straightforward intuitive sense. But (2) for every pair of lives that are adjacent in our sequence, it’s not the case that the welfare difference between them is very large. So strong nonsuperiority does not hold across any pair of lives that are adjacent in our sequence. Therefore, Strong Nonsuperiority Across Adjacent Levels is false.

I’ll now defend (1) and (2), in order. Consider an arbitrary pair of negative welfare levels, which we’ll call ‘level 1’ and ‘level 2’. If lives at level 1 are strongly nonsuperior to lives at level 2, then no number of lives at level 2 would be worse than any number of lives at level 1. In particular, 10 billion lives at level 2 would not be worse than one life at level 1. For it to be plausible that 10 billion lives at one negative welfare level would not be worse than one life at another negative welfare level, the difference between the welfare levels must be very large, in a straightforward intuitive sense. To illustrate, it is plausible that 10 billion barely bad lives would not be worse than one hellish life. In contrast, it is not plausible that 10 billion terrible-but-not-quite-hellish lives would not be worse than one hellish life. They would be worse.

Turning now to (2): by hypothesis, the descriptive difference between any two lives that are adjacent in our sequence is small. It therefore seems that the welfare difference between any two lives that are adjacent in our sequence is small. However, one might object that small descriptive differences in pain intensity can make a difference to the type of pain at issue, which itself is welfare-relevant. ¹⁴ For instance, one might hold that although the descriptive difference between agony and pain that is terrible-but-not-quite-agony is small, the welfare difference between agony and pain that is terrible-but-not-quite-agony is not small. There would then be two lives that are adjacent in our sequence – one containing agony and the other containing pain that is terrible-but-not-quite-agony – such that the welfare difference between them is not small. Is it then plausible to claim that strong nonsuperiority holds across these lives? No. For even if this gambit gives us a rationale for resisting the inference from small descriptive difference to small welfare difference, it does not support the further claim that the welfare difference is so large as to ground the holding of strong nonsuperiority across the lives in question. To continue with our example for concreteness: even granting arguendo that the welfare difference between the life containing agony and the adjacent life containing pain that is terrible-but-not-quite-agony is not small, it does not follow that the welfare difference is very large in the sense intended in (1).

Finally, it’s worth considering the following partners-in-guilt defence of Strong Nonsuperiority Across Adjacent Levels. ¹⁵ Consider a finite, ascending sequence of welfare levels that solely contains some negative welfare levels and a neutral level (=_df a welfare level such that adding a life at this level does not make the population better or worse). Suppose that the final two levels in the sequence are a barely negative level and the lone neutral level in the sequence. Many will agree that strong nonsuperiority holds across lives at these two levels, for intuitively, no number of bad lives would be better than any number of neutral lives. What’s more, many will retain this judgement even if it is stipulated that the descriptive difference between life at the barely negative level and life at the neutral level is small. By way of illustration, return to the sequence of pain-filled lives that we recently constructed. The final life in this sequence is one second long and contains pain that is barely noticeable. Append to this sequence a life that is one second long and hedonically neutral. Plausibly, this is a neutral life; and the descriptive difference between it and its predecessor is small. Thus, the holding of strong nonsuperiority across lives that are descriptively similar is not a unique commitment of non-Archimedeans who endorse the Weak Inferiority of Hellish Lives.

I am not convinced by this partners-in-guilt defence for two reasons. First, the non-Archimedean must claim that strong nonsuperiority holds across different bad lives – e.g. across hellish lives and lives that are terrible-but-not-quite-hellish. In contrast, in claiming that bad lives are strongly nonsuperior to neutral lives (as we just did), our view is that strong nonsuperiority holds across lives that are bad and lives that aren’t. This is a particular instance of the more general – and plausible – principle that bad things are strongly nonsuperior to not bad things (i.e. that no amount of a bad thing would be better than any amount of a not bad thing). Second, to avoid the Negative Repugnant Conclusion, non-Archimedeans must claim that there are at least two instances of strong nonsuperiority holding across bad lives. ¹⁶ As Jensen (2020: 308) observes, this has the unpalatable consequence of creating a ‘zone’ of incommensurability within the welfare levels, which appears to ‘dissolve the difference in value between the members of the zone’. ¹⁷ It therefore seems to me false that non-Archimedeans are partners in guilt with those of us who simply wish to claim that bad things are strongly nonsuperior to not bad things.

This concludes my case against Strong Nonsuperiority Across Adjacent Levels. ¹⁸ Since the Weak Inferiority of Hellish Lives, in conjunction with the Simple Additive Picture, entails Strong Nonsuperiority Across Adjacent Levels, and we are keeping the Simple Additive Picture fixed, we must reject the Weak Inferiority of Hellish Lives. ¹⁹

3.2.2. In favour of exchangeability

If the Weak Inferiority of Hellish Lives is false, does it follow that hellish lives are exchangeable with barely bad lives? No. Handfield and Rabinowicz (2018) and Rabinowicz (2022) prove (assuming Transitivity) that Trilemma holds for all goods (bads) that are quantifiable and such that more is better (worse). ²⁰

Trilemma.

For all such goods (bads) x and y such that x is better (worse) than y, exactly one of the following is true: x is weakly superior (inferior) to y, x and y are exchangeable, or x is radically incommensurable with y.

Radical incommensurability.

x is radically incommensurable with y iff there is a quantity k of x such that for every quantity k⁺ of x at least as great as k, there is some quantity k’ of y such that k⁺ x is incommensurable with every quantity of y at least as great as k’.

Could hellish lives be radically incommensurable with barely bad lives? If they were, then there would be some natural number k such that for every natural number k ⁺ at least as great as k, there is a natural number k’ such that k ⁺ hellish lives would be incommensurable with every natural number of barely bad lives at least as great as k’. For concreteness, suppose that k = 100 and consider 101 hellish lives. It would follow that there is a natural number – say, 10,000 – such that 101 hellish lives would be incommensurable with every natural number of barely bad lives at least as great as 10,000. Could this be true? I believe the answer is No.

Suppose that hellish lives are radically incommensurable with barely bad lives. Now consider an arbitrary negative sequence and suppose for contradiction that lives at every pair of levels that are adjacent in the sequence are exchangeable. It follows by Transitivity that for any number of lives at the first level (which corresponds to a hellish life), there is a number of lives at the final level (which corresponds to a barely bad life) that would be worse. But that contradicts the supposition that hellish lives are radically incommensurable with barely bad lives. So, it’s not the case that lives at every pair of levels that are adjacent in the sequence are exchangeable. It follows by Trilemma that either weak inferiority or radical incommensurability holds across lives at at least one pair of adjacent levels. I assume the former option is a non-starter, so, since the sequence was arbitrary, we have

Radical Incommensurability Across Adjacent Levels.

In any negative sequence, radical incommensurability holds across lives at at least one pair of welfare levels that are adjacent in the sequence.

This is implausible for essentially the same reasons that Strong Nonsuperiority Across Adjacent Levels is implausible. Moreover, the claim that hellish lives are radically incommensurable with barely bad lives does not even enjoy the intuitive appeal of the Weak Inferiority of Hellish Lives. I conclude against the radical incommensurability of hellish and barely bad lives. By Trilemma, we are left with exchangeability.

3.3. Outweighing

We now proceed to stage two of the argument. Here, I introduce the outweighing relation and suggest that non-Archimedeans should accept that barely good lives and barely bad lives can outweigh each other.

Outweighing.

x can outweigh y iff for any quantity k of y that is good (bad), there is a quantity k’ of x such that ky together with k’x is bad (good).

Non-Archimedeans should accept that barely bad lives can outweigh barely good lives, because denying this claim is intuitively intolerable. Here’s why. Suppose that barely bad lives could not outweigh barely good lives. Then there would be some natural number n such that n barely good lives along with any number of barely bad lives would not be a bad population. That is unacceptable. n barely good lives along with 10 ⁿ barely bad lives would be a bad population. So barely bad lives can outweigh barely good lives.

Can barely good lives outweigh barely bad lives? If they could not, then there would be some natural number m such that m barely bad lives along with any number of barely good lives would not be a good population. To me, this does not register as obviously false in the way that the corresponding claim about good lives does. However, we can show a stronger result: if m barely bad lives along with any number of barely good lives would not be a good population, then a single barely bad life along with any number of barely good lives would not be a good population. To see how, assume for contradiction that there is some natural number x such that x barely good lives ∪ one barely bad life would be a good population, i.e. would be better than the empty population. It follows from Scaling that for every natural number q, qx barely good lives ∪ q barely bad lives would be better than the empty population. But that is inconsistent with the claim that barely good lives can’t outweigh barely bad lives – i.e. that there is a natural number m such that m barely bad lives ∪ any number of barely good lives would not be good, i.e. would not be better than the empty population. So, if barely good lives can’t outweigh barely bad lives, then a single barely bad life cannot be taken together with any number of barely good lives to form a good population.

However, it seems internally ill-motivated for non-Archimedeans to deny that there is some number of barely good lives that can be taken together with a single barely bad life to form a good population. Despite being barely so, barely good lives are good. They are positively valuable for the persons leading them. By Addition, the addition of each barely good life makes the world better. In virtue of what, then, would the goodness of enough such lives – we can make the number as large as we like – fail to overcome the badness of a single barely bad life? For now, I’ll assume that non-Archimedeans lack a compelling answer to this question, but we’ll revisit this assumption in section 6. ²¹

3.4. The Repugnant Elitist Conclusion

So far, I’ve argued that hellish lives and barely bad lives are exchangeable (stage 1) and that non-Archimedeans should accept that barely good lives and barely bad lives can outweigh one another (stage 2). We can now derive the

Repugnant Elitist Conclusion.

Some number of heavenly lives along with any number of hellish lives would be a good population.

We begin by showing that for all good populations X and Y and every bad population Z, if X is better than Y and Y ∪ Z is good, then X ∪ Z is good too. To do so, assume that we have two arbitrary good populations X and Y such that X is better than Y and an arbitrary bad population Z such that Y ∪ Z is good, i.e. better than the empty population. Since X is better than Y, by Separability, X ∪ Z is better than Y ∪ Z. Since X ∪ Z is better than Y ∪ Z and Y ∪ Z is better than the empty population, by Transitivity, X ∪ Z is better than the empty population – i.e. X ∪ Z is good. Using the same strategy, we can also show that for all bad populations X and Y and every good population Z, if X is worse than Y and X ∪ Z is good, then Y ∪ Z is good too. The proof is omitted for brevity.

Now for the Repugnant Elitist Conclusion:

1. 1. There is a natural number t such that t heavenly lives would be better than any natural number of barely good lives. (Weak Superiority of Heavenly Lives)

2. 2. For every natural number of barely bad lives, there is a natural number of barely good lives such that the barely bad lives ∪ the barely good lives would be a good population. (Barely good lives can outweigh barely bad lives)

3. 3. For all good populations X and Y and every bad population Z, if X is better than Y and Y ∪ Z is good, then X ∪ Z is good.

4. 4. t heavenly lives ∪ any number of barely bad lives would be a good population. (1–3)

5. 5. For any natural number of hellish lives, there is a natural number of barely bad lives that would be worse. (Exchangeability)

6. 6. For all bad populations X and Y and every good population Z, if X is worse than Y and X ∪ Z would be good, then Y ∪ Z would be good.

7. 7. t heavenly lives ∪ any number of hellish lives would be a good population. (4–6)

(7) is the Repugnant Elitist Conclusion, which I regard as a reductio. (Intuitively, for any natural number of heavenly lives, there is some natural number of hellish lives such that the heavenly and hellish lives together would not be a good population.) To drive a final nail into the coffin, we can also show that one heavenly life ∪ any natural number of hellish lives would not be a bad population. To see how, assume for contradiction that there is a natural number n such that one heavenly life ∪ n hellish lives would be bad, i.e. worse than the empty population. Then, by Scaling, for every natural number q, q heavenly lives ∪ qn hellish lives would be worse than the empty population. But that contradicts the Repugnant Elitist Conclusion.

3.5. Argument recap

Non-Archimedean population axiologies endorse the Simple Additive Picture and the Weak Superiority of Heavenly Lives and/or the Weak Inferiority of Hellish Lives. We can name and taxonomize the resulting positions as follows in Table 1.

Table 1. Non-Archimedean population axiologies

Here’s the main argument against non-Archimedeanism in a nutshell: in conjunction with the Simple Additive Picture, the Weak Inferiority of Hellish Lives entails Strong Nonsuperiority Across Adjacent Levels. That’s false, so we should reject the Weak Inferiority of Hellish Lives. That rules out the Somber View and Heavy Tails but leaves open the Rosy View. However, the Rosy View entails the Repugnant Elitist Conclusion, which is false. So all non-Archimedean positions fail.

The next few sections proceed as follows: in due recognition of the fact that not everyone will find Strong Nonsuperiority Across Adjacent Levels as damning as I do, I consider the Somber View in the next section and Heavy Tails in the one after. Then, in section 6, I consider the option of denying that barely good lives can outweigh barely bad lives.

4. The Somber View

The Somber View is the mirror image of the Rosy View. It accepts the Weak Inferiority of Hellish Lives and the claim that barely good lives and barely bad lives can outweigh one another. And, just as the Rosy View holds that hellish lives and barely bad lives are exchangeable, the Somber View holds that heavenly lives and barely good lives are exchangeable. Thus, just as the Rosy View bites the bullet on the Negative Repugnant Conclusion, the Somber View bites the bullet on the Repugnant Conclusion.

We can run an argument parallel to the one given for the Repugnant Elitist Conclusion to show that the Somber View implies the following: any population with a sufficient number of hellish lives is bad, no matter the number of heavenly lives it also contains (see Appendix 2 for proof). ²² I find this implication prima facie plausible, unlike the Repugnant Elitist Conclusion. For it is plausible that the extremes of suffering are so horrific that an outcome is irredeemably marred if it contains enough of them. ²³ That is the chief reason why the Somber View is worthy of note and discussion. However, in light of the grim inductive evidence from the historical record and the many different ways in which the future could play out, there is, disturbingly, a large expected number of hellish lives in the future. ²⁴ Now, strictly speaking, the Somber View does not say anything about the future, for, as stated, it does not say anything about the evaluation of risky prospects. But a natural synthesis of the Somber View with expected value reasoning will yield the result that the future is bad in expectation. ²⁵ So, on the Somber View, it would be expectedly better if we succumbed to an immediate, painless extinction. That is a difficult pill to swallow. ²⁶ Moreover, the Somber View implies that no number of heavenly lives along with a single hellish life would be a good population (see Appendix 2 for proof). The Somber View therefore has two options: either a single hellish life along with any number of heavenly lives would constitute a bad population, or it would constitute a population that is neither good, nor bad, nor neutral – i.e. a population that is incommensurable with the empty population. Neither option seems right. For a population containing one hellish life and 10 trillion heavenly lives seems good overall. ²⁷

5. Heavy Tails

The preceding observations suggest to me that the Somber View is false. It remains for us to examine Heavy Tails. Heavy Tails affirms the Weak Superiority of Heavenly Lives; the Weak Inferiority of Hellish Lives; that barely good lives and barely bad lives can outweigh each other; and that heavenly and hellish lives can outweigh each other. ²⁸ Heavy Tails thus avoids both the Repugnant Elitist Conclusion and the Somber View’s proclivities towards extinction – in addition to both the Repugnant Conclusion and the Negative Repugnant Conclusion. On this basis, Heavy Tails seems to me the relatively most plausible non-Archimedean view, despite the fact that it bites the bullet on Strong Nonsuperiority Across Adjacent Levels. However, Heavy Tails implies that one heavenly life along with any number of barely bad lives would not be a bad population (see Appendix 2 for proof). This implication – call it the Weak Repugnant Elitist Conclusion – is rather counterintuitive. For it seems that a population containing one heavenly life and 10 trillion barely bad lives would be bad (for instance). Heavy Tails must deny this. To be thorough, though, it is also worth noting that Heavy Tails has the parallel implication that one hellish life along with any number of barely good lives would not be a good population. I lack a strong intuition about this implication, though I appreciate that some will find it welcome.

6. Outweighing Redux

Might the preceding arguments be taken as a reductio of the claim that barely good lives and barely bad lives can outweigh each other (§3.3)? I’ll take the claim that barely bad lives can outweigh barely good lives as unassailable, since it’s extremely plausible that one barely good life can be taken together with some number of barely bad lives to form a bad population. In contrast, the claim that one barely bad life can be taken together with some number of barely good lives to form a good population enjoys less intuitive support. So, the place to push is against the claim that barely good lives can outweigh barely bad lives.

The non-Archimedean could alternatively claim that one barely bad life along with any number of barely good lives would be neither good, nor bad, nor neutral (i.e. that it would be neither better than, nor worse than, nor equally as good as the empty population, but rather incommensurable with the empty population). This move continues to strike me as ad hoc, given that, by the non-Archimedean’s own lights, the world just gets better and better as we add (barely) good lives. ²⁹ Still, it does have the welcome effect of blocking the arguments for the Repugnant Elitist Conclusion and the Weak Repugnant Elitist Conclusion; and for this reason, I expect that many non-Archimedeans will go in for it. What, then, is the best view for the non-Archimedean who takes this route?

Denying that barely good lives can outweigh barely bad lives does not help the Somber View, so we’ll examine revisions of Heavy Tails and the Rosy View. Revised Heavy Tails is the same as Heavy Tails, except that it denies that barely good lives can outweigh barely bad lives – thereby avoiding the Weak Repugnant Elitist Conclusion. Unfortunately, Revised Heavy Tails – alongside Heavy Tails – implies that there is a natural number n such that n heavenly lives ∪ any natural number of barely bad lives would be a good population (see Appendix 2 for proof). ³⁰ This claim – call it Heavenly Dominance – is counterintuitive. For instance, it seems that for every natural number q, q heavenly lives ∪ 10^100q barely bad lives would not be a good population. As Mogensen (Forthcoming) remarks, ‘it may well strike us most repugnant of all to assert that there are some lives so good that, for their sake, we should be willing to accept that arbitrarily many individuals may have to have lives bad enough that, for their sake, we should wish that they had never been born’. It’s also worth noting, though, that (Revised) Heavy Tails has the parallel implication that there is a natural number m such that m hellish lives ∪ any natural number of barely good lives would be a bad population. I lack a clear intuition about this implication, but I expect that many will find it welcome.

A final option to consider is the Revised Rosy View. The Revised Rosy View is the same as the Rosy View, except that it denies that barely good lives can outweigh barely bad lives – thereby avoiding the Repugnant Elitist Conclusion. Moreover, unlike Revised Heavy Tails, the Revised Rosy View does not imply Strong Nonsuperiority Across Adjacent Levels or Heavenly Dominance. It is the ability of the Revised Rosy View to avoid these two unattractive implications of Revised Heavy Tails that makes it a live option. (Otherwise, one might wonder about the motivation for a non-Archimedean view that avoids the Repugnant Conclusion but fails to avoid the Negative Repugnant Conclusion.)

7. Conclusion

The main goal of this paper has been to offer a novel argument against non-Archimedean population axiologies, which I did in section 3. A secondary goal has been to assess the relative plausibility of competing non-Archimedean axiologies, which I attempted to do by identifying their most counterintuitive implications. Results are summarized in Tables 2 and 3.

Table 2. Non-Archimedean population axiologies (expanded)

Table 3. Itemized billing for non-Archimedean axiologies

To close, I’d like to make three conjectures that I hope will be productively provocative. Firstly, if there is such a thing as the correct abductive methodology for ethical theorizing, and if this methodology gives significant weight to simplicity – as abduction seems to in scientific theorizing – then we ought to prefer Standard Totalism to non-Archimedeanism. Standard Totalism is much simpler than non-Archimedeanism. Still, it has seemed to many that non-Archimedeanism is more extensionally adequate than Standard Totalism, ³¹ with the chief worry levelled against the former being its implications for choice under risk. ³² I take the results shown in this paper to narrow the gap in extensional adequacy between non-Archimedeanism and Standard Totalism – even in cases of choice under certainty. With a sufficiently narrow gap in extensional adequacy, considerations of simplicity can then carry the all-things-considered judgement of which view to prefer.

Secondly, if Standard Totalism is true, then familiar forms of maximizing welfarist consequentialism are false. According to Standard Totalism, all good lives are exchangeable, all bad lives are exchangeable, and good and bad lives can all outweigh each other. Standard Totalism implies that for any number of hellish lives, there is a number of barely good lives such that it would be better to add the barely good lives to the world than to alleviate the suffering contained in the hellish lives (Temkin 2012: 413). Although I am prepared to entertain that this is true as a matter of impersonal axiology – i.e. as a matter of what would be better ‘from the point of view of the universe’ – I am unprepared to accept that we morally ought to bring into existence some large number of people with barely good lives, rather than alleviating the extreme suffering of people who already exist.

Disturbingly, however, we can pose the paradoxical arguments in population axiology that lead us to Standard Totalist repugnance in deontic terms (Arrhenius 2004). These deontic arguments constitute a major challenge to nonconsequentialist moral theories. As I see things, then, one of the most important open questions in ethical theory is whether nonconsequentialists can meet this challenge. If they could not, then I would be inclined towards the view that the impossibility of population ethics supports either moral scepticism or metaethical error theory. ³³ The error theory would, of course, undermine this essay, for it would imply that population ethics has no subject matter. Nevertheless, reasoning through the paradoxes of population ethics would still have been worthwhile, insofar as it was instrumental in allowing us to grasp the metaethical truth or to better understand our own (mind-dependent) values.