3.1 Six Desiderata

While I will keep the formalisms to a minimum in the main text, it is useful to set up the basic constituents of my formal framework at this point. Formally, I take a life to be a finite sequence of integers. The sequence index represents the time index, and the integer represents the level of well-being of the respective person-stage at this time. The well-being levels are each an element of a finite set of well-being levels $ \mathbb{W}$ :

$$ {\mathbb{W}}:\{\alpha, \alpha +1, \ldots, -2,-\mathrm{1,0},\mathrm{1,2},\ldots, \beta, \beta +1,\beta +2\}$$

where $ \alpha $ corresponds to an abominably low well-being level, 0 to a neutral level, and $ \beta $ and above to the best possible well-being levels. A prudential axiology of lives is a binary betterness relation over all possible lives. That is, we do not restrict the domain of lives, e.g. to equally long lives. The betterness relation is, as usual, built from the basic relation $ \succeq $ which is to be understood as at least as good as.Footnote ¹¹ The present section examines six desiderata that such an axiology of lives plausibly ought to meet.

Before moving on to these desiderata, let me address two objections that might be raised against the choice to formalize well-being levels with integers. First, why use the domain of integers, rather than, say the domain of real numbers? The simple answer to this question is that Arrhenius’ original theorem, which will be reproduced here, is often reconstructed using integers (cf. e.g. Thomas Reference Thomas2016). I keep in line with this tradition for formal convenience. However, insofar as the results depend only on the ordering between the elements of the domain, the same results apply to the domain of real numbers too.

Second, one may object that formalizing well-being levels with integers is implausible because it is controversial that there even is a sequence of adjacent well-being levels between which the difference is always marginal, as the formal structure of integers seems to suggest. This problem is well-known in population ethics and while it is a clear limitation of my framework, I cannot aim to solve it here. Instead, I will simply assume the following about levels of well-being: Individual units of well-being are meaningful, and we interpret adding an additional unit of well-being intuitively as simply ‘being overall just a little better off (with respect to well-being)’.Footnote ¹²

Let us now consider the first desideratum for an axiology of lives. In order for a binary betterness relation to give us a genuine axiology, the respective relation should ideally generate an ordering. This leads to the following desideratumFootnote ¹³ :

Transitivity: For any three lives $ x$ , $ y$ and $ z$ , if $ x\succeq y$ , and $ y\succeq z$ , then $ x\succeq z$ .

Transitivity is conceptually and practically desirable. Betterness itself is often understood to be transitive as a matter of conceptual necessity. Additionally, the threat of betterness-cycles leaves agents vulnerable to money-pumps and can lead to inconsistent guidance in choices between multiple lives. While there may be good reason to give up Transitivity, doing so generates substantive theoretical and practical costs.

The next two desiderata, Egalitarian Dominance and Dominance Addition, owe their names to analogous axioms in population ethicsFootnote ¹⁴ and can be grouped together as Dominance Principles.

Egalitarian Dominance: Take any two lives $ x$ and $ y$ with the following properties.

• • $ x$ and $ y$ are equally long.

• • Each life has a consistent level of well-being.

• • The well-being in $ x$ is consistently higher than the well-being in $ y$ .

If the above properties hold for $ x$ and $ y$ , then $ x$ is better than $ y$ .

More informally, Egalitarian Dominance holds that a life that is equally long but consistently and equally better than another ought to be judged as overall better. The matrix below illustrates this idea: $ A$ must be better than $ B$ , given that $ A$ has a consistently high well-being and $ B$ has a consistently low well-being. If a theory of prudence fails to satisfy even Egalitarian Dominance, it seems to me, then the concepts of ‘better’ or ‘worse’, for a temporally extended person lose their meaning altogether.

Egalitarian Dominance: $ A\succ B$

A similarly strong desideratum for a theory of prudence is the following.

Dominance Addition: Take any two lives $ x$ and $ y$ that have the following properties.

• • The same number of person-stages as exist in $ y$ also exist in $ x$ , where the latter have a higher level of well-being.

• • Additional person-stages with positive well-being exist in $ x$ .

If the above properties hold for $ x$ and $ y$ , then $ x$ is at least as good as $ y$ .

The intuitive idea of Dominance Addition is that adding positive well-being ought not to make the respective life worse. Consider the following example of this idea.

Dominance Addition: $ B\succeq A$

In the above case, Dominance Addition holds that $ B$ ought not to be strictly worse than $ A$ . That is, it cannot be the case that it is strictly the prudentially right choice to choose $ A$ . This, again, seems intuitively right. Increasing our counterfactual well-being and adding more positive well-being should at the very least be prudentially allowed.Footnote ¹⁵

The next two desiderata, General Non-Elitism and General Non-Extreme Priority, are likewise borrowed from population ethics. In the prudential context, they can be viewed as desiderata of non-myopia: They both hold that small well-being improvements for one person-stage in one life should not generally outweigh possibly large well-being improvements for possibly many other person-stages in another. Insofar as prudence is precisely concerned with balancing well-being over time, rather than deciding myopically at any particular time, they both seem to be very plausible constraints for prudence.

General Non-Elitism: Take any two lives $ x$ and $ y$ that have the following properties.

• • $ x$ and $ y$ are equally long.

• • There is some number (possibly 0) of person-stages in $ x$ and $ y$ with equal levels of well-being, i.e. they share a subpopulation of person-stages.

• • There is one person-stage in $ x$ whose level of well-being is exactly one unit higher than the well-being of one corresponding person-stage in $ y$ . Let $ a$ be the well-being level of the latter person-stage in $ y$ .

• • There is some number $ n$ of person-stages in $ y$ with well-being level $ a$ , and there is the same number $ n$ of person-stages in $ x$ with a well-being level that is lower than $ a-1$ (i.e. at least two units lower than $ a$ ).

If the above properties hold for $ x$ and $ y$ , then for any $ a$ there is some number $ n$ such that $ y$ is at least as good as $ x$ .

The intuition behind General Non-Elitism is the following: Just because there is a small comparative decrease in well-being at one time, this should not prudentially outweigh raising potentially very many well-being levels at other times to this same level. The following matrix illustrates an example of General Non-Elitism.

General Non-Elitism: $ B\succeq A$

The fact that at $ {t}_{2}$ , I am slightly better off if I choose $ A$ ought not to entirely outweigh the large well-being gains I receive from choosing $ B$ at $ {t}_{3}$ and $ {t}_{4}$ , where the well-being is raised to the same level I have at $ {t}_{2}$ in $ B$ .Footnote ¹⁶ As mentioned above, this seems to be a plausible constraint on prudence; arbitrarily holding the marginal gain of one specific person-stage as determining overall prudential goodness seems very counterintuitive.

General Non-Extreme Priority builds on a very similar intuition as General Non-Elitism. Its formal definition is as follows.

General Non-Extreme Priority: Take any two lives $ x$ and $ y$ with the following properties.

• • $ x$ and $ y$ are equally long.

• • There is some number (possibly 0) of person-stages in $ x$ and $ y$ with equal level of well-being, i.e. they share a subpopulation of person-stages.

• • There is one person-stage in $ x$ whose level of well-being is exactly one unit higher than the well-being of one corresponding person-stage in $ y$ .

• • There is some additional number $ n$ of person-stages in $ y$ whose level of well-being is at least as high as $ \beta $ (i.e. very high), while the same number $ n$ of additional person-stages in $ x$ have a well-being level barely above 0.

If the above properties hold for $ x$ and $ y$ , there is some number $ n$ such that $ y$ is at least as good as $ x$ .

The intuitive idea is as follows: Very small improvements at certain times cannot take categorical priority over raising the well-being of possibly many person-stages from very low positive levels to very high positive levels. The following matrix illustrates an example. For illustration purposes, we assume that $ \beta =1000$ .

General Non-Extreme Priority: $ B\succeq A$

Just because the person-stage at $ {t}_{3}$ in $ A$ is slightly better off than its counterpart in $ B$ , this should not take priority over the large well-being gains in $ B$ when comparing the person-stages at $ {t}_{4}$ and $ {t}_{5}$ .

Given the conceptual similarity to General Non-Elitism (GNE), similar reasons speak in favour of General Non-Extreme Priority (GNEP) as a condition for prudence. Arbitrarily holding the marginal gain of one specific person-stage as determining overall prudential goodness seems very counterintuitive.

Given their similarity, it is also worth briefly emphasizing the difference between GNE and GNEP. While both GNE and GNEP insist that a lone marginal edge should not swamp larger gains elsewhere, they are based on different patterns of compensation. Speaking informally, GNE says that if you “lose” one unit at a single stage, say dropping from 100 down to 99, you can make up for that loss by boosting a sufficient number of other stages up to that same level of 99 – each of those raises being at least two units – so that collectively those compensatory gains outweigh the lone drop. GNEP, by contrast, insists that if you only “lose” one unit at one stage, that tiny edge must not lexically override arbitrarily many very large jumps from very low positive levels to very high positive levels.

Lastly, a good theory of prudence should be able to avoid intrapersonal analogues of the Very Repugnant Conclusion.Footnote ¹⁷ The formal definition of this desideratum is as follows.

Non-Repugnance: Take any two lives $ x$ and $ y$ with the following properties.

• • $ x$ consists of some number of person-stages $ {n}_{x}$ with well-being levels of $ \beta $ (i.e. very high).

• • There is some number of person-stages $ {n}_{y1}$ in $ y$ with a well-being level that has a negative well-being level $ a$ .

• • There is an additional number $ {n}_{y2}$ of person-stages in $ y$ whose well-being level is barely above 0.Footnote ¹⁸

If the above properties hold for $ x$ and $ y$ , then it is not the case that for any $ b$ , $ {n}_{x}$ and $ {n}_{y1}$ , there is an $ {n}_{y2}$ such that $ y$ is better than $ x$ .

The intuition behind non-repugnance is the following: A long extended life that at times involves potentially significant suffering, while at best being barely worth living, should not generally be prudentially better than a shorter life with extremely high levels of well-being. Consider the following matrix as an example. Again, we assume for illustration purposes that $ \beta =1000$ .

Non-Repugnance: $ A\succeq B$

An objection to Non-Repugnance might be raised at this point: the interpersonal case may be sufficiently dissimilar that the repugnance does not carry over to the intrapersonal case. Indeed, one could argue that in the interpersonal case, no single being experiences the many instances of low well-being, while things differ in the intrapersonal case. In $ B$ , someone gets to live a very long life, albeit with a low level of well-being. I will return to the question of how repugnant the intrapersonal Very Repugnant Conclusion really is. For now it suffices to note that, prima facie, it is plausible that $ A$ should at least be weakly prudentially better. A theory of prudence that tells us that we must go through agonizing torture in order to secure a long life barely worth living, thereby foregoing a shorter but extremely happy life, seems to go wrong.

3.2 The Impossibility of Prudence

Having defined six desiderata for an adequate axiology of prudence, we must now ask the following question: Is there an axiology which meets them all? Unfortunately, the answer is no.

Theorem: There is no betterness relation which jointly satisfies Transitivity, Egalitarian Dominance, Dominance Addition, General Non-Elitism, General Non-Extreme Priority and Non-Repugnance.

The proof for the theorem is entirely analogous to the proof for Arrhenius’ fifth impossibility theorem (cf. e.g. Thomas Reference Thomas2016). In the Appendix, I show how Thomas’ particular reconstruction can be adapted to the context of prudence. Given this theorem, it becomes apparent that prudence does not only bear some superficial structural similarity to population ethics. It also inherits (at least some of) its substantive problems.Footnote ¹⁹ Insofar as an adequate axiology is (at least) characterized by the above desiderata, it is impossible to find one. We are thus tasked with finding escape routes out of the impossibility. This is what the rest of the paper is concerned with.

4.1 Giving up Egalitarian Dominance or Dominance Addition

We can reject giving up Egalitarian Dominance without much difficulty. Egalitarian Dominance seems almost definitional of prudence altogether: If one life outperforms the other in terms of well-being at every point in time, and there are no other relevant differences between them (e.g. they are exactly equally long), it must simply be prudent to choose it. If we cannot uphold this judgement, we seem to give up on the concept of prudence altogether.

Dominance Addition, on the other hand, is more controversial. In population ethics, Critical-Level views give up Dominance Addition (cf. Blackorby et al. Reference Blackorby, Bossert and Donaldson2002; Qizilbash Reference Qizilbash2007; Rabinowicz Reference Rabinowicz2009; Thornley Reference Thornley2022a). Applied to prudence, the structure of such views might be as follows: There is a certain level of positive well-being that a person-stage must exceed to contribute positively to the value of a life. That is, well-being levels between 0 and this critical level do not make the life better, but possibly even worse. More formally, this amounts to the following:

$ x$ is at least as good as $ y$ iff

$$ {\sum }_{i=1}^{n}\,({w}_{xi}-\gamma )\ge {\sum }_{i=1}^{m}\,({w}_{yi}-\gamma )$$

where $ {w}_{xi}$ refers to the person-stage at time $ i$ in $ x$ , $ {w}_{i}$ refers to the person-stage at time $ i$ in $ y$ , $ n$ refers to the number of person-stages in $ x$ , $ m$ refers to the number of person-stages in $ y$ , and $ \gamma $ is the relevant critical level.

In less formal terms, standard Critical-Level views would evaluate lives as follows: For each person-stage, subtract the critical level $ \gamma $ from its well-being, then sum over the resulting levels and compare the result. The higher total is the better life. Note that this violates Dominance Addition:

Dominance Addition: $ B\succeq A$

If the added person-stages have a positive well-being $ x$ but $ \gamma \gt x$ , then the addition of these person-stages will contribute negatively to the value of $ B$ . If enough such person-stages are added, $ A$ would come out to be better than $ B$ . This feature leads the standard version of the Critical-Level view into a very grave counterexample, namely what I call the Masochistic Conclusion (analogous to the Sadistic Conclusion, cf. Bossert (Reference Bossert, Arrhenius, Bykvist, Campbell and Finneron-Burns2022) for the inevitability of this conclusion). Say the critical level $ \gamma $ equals 20. Then the following is an instance of this conclusion:

Masochistic Conclusion: $ B\succ A$

With a critical level of 20, $ A$ ’s total amounts to −40 (given that (10–20)*4 = −40), while $ B$ ’s total amounts to −30 (given that −10–20 = −30). Hence, the standard Critical-Level view would imply that $ B$ is better than $ A$ , which seems on a par with giving up Egalitarian Dominance in terms of how counterintuitive it is. If we cannot say that $ A$ is better than $ B$ in the Masochistic Conclusion case, it seems we have basically given up on the conceptual core of the very meaning of prudence.

This problem is, of course, well-known in the population ethics literature. But it is worth noting that it may be even more troubling in the context of prudence. In population ethics, we may well argue that the lives which are positive but below the threshold may be good for the people who live them and yet not sufficiently good to make the population overall morally better. But in the prudential realm, one would have to say that the additional positive well-being is somehow not actually good for the person, or adds negatively to their life. That this leads to the judgement that sometimes a life with exclusively negative well-being could be better for a person than a life with exclusively positive well-being seems unacceptable in the context of prudence.

Critical-Level views have since been developed further. For instance, Critical-Range views introduce several critical levels. According to these views, a life is only better than another if it is better relative to all the critical levels – otherwise they are incommensurable (cf. Blackorby et al. Reference Blackorby, Bossert and Donaldson2002). I am not convinced that this proposal helps us: Even if our principle only implies that $ B$ is incommensurable (or even on a parFootnote ²⁰ ) with $ A$ in the Masochistic Conclusion case, this seems sufficient to reject the view in the prudential realm. It seems so overwhelmingly clear that a life which is consistently positive is prudentially better than a life which is consistently negative. In addition, though less decisive, such proposals also give up the axiom of completeness:

Completeness: For any two lives $ x$ and $ y$ , $ x\succeq y$ or $ y\succeq x$ (or both).

Giving up Completeness has powerful advocates (cf. Chang Reference Chang1997, Reference Chang2002), and is generally viewed to be easier to give up than for example Transitivity. Yet, just like intransitive preferences, incomplete preferences can be money-pumped too (cf. Gustafsson Reference Gustafsson2022) and powerful arguments from comparability have recently been mounted in favour of completeness (cf. Dorr et al. Reference Dorr, Nebel and Zuehl2021, Reference Dorr, Nebel and Zuehl2023). Given these problems, I will thus not consider giving up Dominance Addition in any more detail and turn to the other four escape routes.

4.2 Giving up Non-Repugnance

So-called Totalism gives up Non-Repugnance but meets the rest of the desiderata. It can be formally stated as follows:

$ x$ is at least as good as $ y$ iff

$$ {\sum }_{i=1}^{n}\,{w}_{xi}\ge {\sum }_{i=1}^{m}\,{w}_{yi}$$

where $ {w}_{xi}$ refers to the person-stage at time $ i$ in $ x$ , $ {w}_{y}$ refers to the person-stage at time $ i$ in $ y$ , $ n$ refers to the number of person-stages in $ x$ , and $ m$ refers to the number of person-stages in $ y$ .

It is easy to verify that this principle meets all the other desiderata. So what might speak in favour of giving up Non-Repugnance? Recall the objection we raised against Non-Repugnance. While the Very Repugnant Conclusion is generally rejected in population ethics, the intrapersonal case might be relevantly different. This is due to the following disanalogy between population ethics and prudence: Most views of persons and person-stages regard the two as relevantly different in terms of their separateness. That is, while separate person-stages are usually still understood to be part of the same subject, and often also the same moral patient, different people are generally regarded as distinct subjects and distinct moral patients. As noted in section 3, this may lead to important disanalogies. For instance, Rawls famously argued that the separateness of persons has significant moral import (Rawls Reference Rawls1971: 23 + 163), i.e. the fact that distinct people have fundamentally distinct subjective loci must be taken seriously by any adequate moral theory.

Indeed, Brink argues that it is precisely the fact that it is the very same person who can be compensated by later person-stages for the sacrifices of current person-stages which is the distinctive feature of prudence (Brink Reference Brink2003: 223).Footnote ²¹ In the interpersonal case, no single subject experiences the many instances of low well-being, while things are different in the intrapersonal case. Consider again our example above:

Non-Repugnance: $ A\succeq B$

In $B$ , someone gets to live a very long life, albeit with a low level of well-being. In population ethics, we find the Very Repugnant Conclusion very repugnant partially because no one actually benefits from the large population with low levels of well-being. But in the intrapersonal case, someone does benefit. Therefore, the intrapersonal version of the Very Repugnant Conclusion is not actually repugnant, or so the argument might go.

Above I already noted one possible counterargument. Even if we would want to prudentially allow for people to choose extreme suffering in order to secure the long extended life by judging $A$ and $B$ to be equally good, it seems odd to have it be a prudential requirement, i.e. insisting that $B$ is better. However, there is a further argument bolstering Non-Repugnance as a criterion.

The intuition leading us to hold that $B$ could be better than $A$ might at least partly be caused by the fact that a long life is itself valuable to us, such that we intuitively attribute a higher level of well-being to the individual person-stages because we imagine them being better off given the long life overall. To eliminate this confounder, consider the following interpretation of the above matrix: Each period of time lasts for one million years. In $A$ you exist for three million years at a very high level of well-being; your life is consistently filled with the best possible things. In $B$ , you exist for n million years, but the first three million are filled with agonizing torture and suffering, while the rest of the time you are at a level of well-being that is just barely worth living. Put this way, both lives end up being very long, such that our intuitions likely attribute much less value to the mere fact of temporal extension. I at least find my intuitions become clear at this point. It is extremely repugnant to judge $B$ to be better than $A$ .

This error theory is contentious of course. My proposed reinterpretation may not actually move our intuitions sufficiently. One may also reject fanciful cases with such long lives outright, and argue that our intuitions are likely to be led astray even more if the time horizons are longer than anything we have ever experienced. If either of these objections are sound, we may want to accept the intrapersonal Very Repugnant Conclusion after all, which allows us to escape the impossibility theorem by giving up on Non-Repugnance. The remaining desiderata could then be satisfied by Totalism. Taking this route would thus vindicate the traditional stance with a renewed sense of the associated theoretical costs and benefits. The costs have sometimes been understood to be acceptable even in the interpersonal domain, with some authors arguing that just one intuition should not drive all of population ethics (cf. Tannsjo Reference Tännsjö2002; Huemer Reference Huemer2008; Zuber et al. Reference Zuber2021).

4.3 Giving up Transitivity

Given Egalitarian Dominance and Dominance Addition, it is the iterative application of General Non-Extreme Priority and General Non-Elitism which leads via Transitivity straight to the Very Repugnant Conclusion. That is, General Non-Extreme Priority and General Non-Elitism enable us to construct a series of lives by consistently altering lives step by step, such that each individual alteration makes the lives better according to General Non-Extreme Priority or General Non-Elitism. But by the end of this procedure we are comparing two lives with big well-being differences, such that an overall transitive judgement between the first and the last life considered does no longer seem right. Together, these series lead us to the Very Repugnant Conclusion. At least one of the series constructed in this way may actually itself already be viewed as problematic. Consider the following construction of a series of lives, where $A$ is the first of a series of lives that can be constructed by an iterative application of GNE to eventually result in ${L_{nm}}$ :

Transitivity+General Non-Elitism: ${L_{nm}} \pm A$

By iteratively decreasing the well-being of the person-stages and making up for it with the well-being of a sufficient number of the remaining person-stages, a series of lives can be constructed such that by Transitivity, ${L_{nm}} \succeq A$ . That is, all the person-stages with a well-being level of 100 are first reduced to a well-being level of 99 by making up for it with a sufficient number of remaining person-stages whose well-being is increased from 1 to 99. Then, all of the resulting person-stages with a well-being-level of 99 are reduced to a well-being level of 98 by making up for it with a sufficient number of remaining person-stages whose well-being is increased from 1 to 98. This procedure can be repeated until it results in ${L_{nm}}$ . However, it does not seem intuitively plausible that ${L_{nm}}$ should be at least as good as $A$ , even if each judgement in each individual step is plausible. Such continuum cases are of course well known, and are often cited as examples of legitimate intransitive preferences (e.g. Temkin Reference Temkin1996). After all, denying Transitivity of the better-than relation enables us to block the continuum sketched above: While we can uphold each individual judgement generated by GNE, we can deny that therefore ${L_{nm}} \pm A$ . For instance, we may hold that $A \succ {L_{nm}}$ , or even earlier in the sequence we can hold that for some $x$ such that $nm \gt x \gt 2$ that $x \succ {L_{nm}}$ . However, this intransitivity is of course in need of explanation. When and why does this judgement flip?

The literature on spectrum cases and denying Transitivity as a way to solve them is vast and I can merely sketch how they apply to the case of prudence we are investigating. Temkin (Reference Temkin2012) lays out the most prominent compilation of arguments in favour of giving up the Transitivity of the better-than relation. He argues, among other things, that we should expect the better-than relation to be intransitive if the relata under consideration are sufficiently different such that different things matter to the comparison between the options.

Applying these insights to spectrum cases, Temkin notes that for comparisons between outcomes that are very close to each other on the spectrum and comparisons between outcomes that are far away from each other on the spectrum, different things might be relevant. After all, small incremental steps can lead an outcome to be so different from the initial starting point that new considerations might come into play (1996: 224ff). One branch of literature that develops a view along these lines is the literature on limited aggregation. The present section will sketch what I believe to be the most promising view of limited aggregation for the present context and apply it to prudence.

To do so, let us first dive into limited aggregation in the interpersonal context. Consider two different cases: In case 1, we can either save one person from death or some number $n$ people from paraplegia. In case 2, we can either save one person from death, or some number $m$ people from a mild headache. If, interpersonally, we adopt a totalist full aggregationist framework, there would be both some number $n$ and some number $m$ such that in each case, it is better to save the larger group of people (from paraplegia or headaches). However, proponents of limited aggregation wish to limit full aggregation: They argue that, while it is plausible that there is some number $n$ such that we should save the group of people from paraplegia in case 1, it is not plausible that there could be any number $m$ such that we should save the group of people from a mild headache in case 2. This intuitive verdict is usually justified using the concept of relevance. They argue that, while paraplegia is sufficiently similar in badness to death such that it is relevant to it, a mild headache is not. That is, if the harms are sufficiently far away from each other on a badness spectrum, relevance comes into play. This leads to overlapping relevance classes on the spectrum of harms, and one kind of harm can only aggregatively outweigh another if it is still within the reach of relevance. For instance, a sufficient number of claims by people threatened by paraplegia may be able to outweigh one claim by a person threatened by death, and a sufficient number of claims by people threatened by the loss of a hand may be able to outweigh one claim by a person threatened by paraplegia. Yet, no number of claims by people threatened by the loss of a hand could ever outweigh one claim by a person threatened by death.Footnote ²²

A first prominent attempt at developing such a theory came from Alex Voorhoeve (Reference Voorhoeve2014). Since then, various objections and further developments have led to more and more refined versions of limited aggregation (for a small selection cf. Privitera Reference Privitera2018; Horton Reference Horton2018; van Gils and Tomlin Reference van Gils, Tomlin, Sobel, Vallentyne and Wall2020; Rüger Reference Rüger2020; Hart Reference Hart2022). For the current purposes, I will apply what I believe to be the only version of limited aggregation which is both able to avoid obvious counterexamples as well as meet all of our desiderata except for Transitivity.Footnote ²³ This version was proposed by Rüger (Reference Rüger2020). Due to space constraints, I will directly apply his view to prudence, rather than sketch it first in the interpersonal domain.

Consider the choice between the following two lives:

Relevance

In the standard interpersonal rescue cases, we are comparing different claims to be rescued with one another. But what are we comparing when we are choosing between lives? Every person-stage has either positive well-being, negative well-being or neutral well-being. If the well-being is positive, then the fact that a life contains it constitutes a reason for choosing it, and we can posit that the strength of the reason is proportional to how positive it is. If the well-being is negative, then the fact that a life contains it constitutes a reason against choosing it, and the strength of the reason is proportional to how negative it is. We can thus translate claims to be rescued to reasons to choose a particular life. In the above case, we could thus note the following reasons:

Reasons to choose $A$ : 100, 1, 1, 1

Reasons to choose $B$ : 80, 10, 10, 10

There is one reason of strength 100 to choose life $A$ which derives from the positive well-being in life $A$ , and three reasons of strength 1 to choose life $A$ which derive from the negative well-being in $B$ which we would avoid by choosing $A$ . And there is one reason of strength 80 to choose life $B$ which derives from the positive well-being in life $B$ and three reasons of strength 10 to choose life $B$ which derive from negative well-being in life $A$ we would avoid by choosing $B$ . We can say that the reasons for $A$ are competing with the reasons for $B$ , but are not competing among one another. Furthermore, we can say that reasons can be relevant or irrelevant to each other, in the same way claims to be rescued can be relevant or irrelevant to each other. Let us postulate that the reason to choose a life based on a benefit/harm is relevant to another when it is within a reach of 20 well-being units. So for instance, a reason of strength 100 is relevant to reasons of strength of up to 120 (derived from either positive or negative well-being). A reason of strength of 90 is relevant to well-being levels up to 110 (derived from either positive or negative well-being). We can now formulate Rüger’s view (Reference Rüger2020: 461) applied to prudence:

$x$ is at least as good as $y$ iff

1. (a) There is no reason to choose $y$ to which no competing reason to choose $x$ is relevant and

2. (b) Totalism says that $x$ is at least as good as $y$ .

Put more simply, the view works as follows: If there is a reason to choose one life to which no competing reason is relevant, then this life is the better life. But if that is not the case, then we do the normal totalist aggregation and whichever life is better according to Totalism is the better life. For the above case, this view would yield the following: Since there is no reason to choose $A$ to which no other reason is relevant, nor is there a reason to choose $B$ to which no other reason is relevant, we aggregate in the standard totalist way which would yield that $B$ is better than $A$ (since 77 $ \gt $ 70). However, if we were to slightly alter the case, the judgement flips:

Relevance*

Now the reason to choose $A$ , which derives from the 100 units of well-being at ${t_1}$ , is so strong that no competing reason is relevant to it. Hence we now choose $A$ . In the interpersonal context, Rüger terms this view “Aggregation with Constraints”, and I will stick to that name in the prudential context.Footnote ²⁴ Aggregation with Constraints can meet all the desiderata except for Transitivity.Footnote ²⁵ It violates Transitivity due to its overlapping relevance classes. For instance, it generates intransitive judgements for the following three lives:

Intransitivity: $A \succ C \succ B \succ A$

Aggregation with Constraints yields the following cyclical judgements: $C \succ B \succ A \succ C$ . This is because while a level of 40 is relevant to a level of 60 and a level of 20 is relevant to a level of 40, a level of 20 is not relevant to a level of 60. This means that condition (a) takes hold in the comparison between $A$ and $C$ , but not in any other comparison. It is this feature which enables Aggregation with Constraints to flip the judgement on the spectrum initially considered at the start of this section.

Denying Transitivity is of course a substantive conceptual and decision-theoretic cost (cf. Nebel (Reference Nebel2018) for a strong argument against intransitivity in spectrum cases). Transitivity is often understood to be an integral property of any betterness relation as well as a plausible axiom in decision theory. Both of these are often justified by coherence considerations or pragmatic arguments, i.e. intransitive betterness judgements or choice behaviour is deemed incoherent or practically self-defeating in some sense. For instance, a prominent type of argument has been from money pumps, i.e. from the fact that intransitive preferences may lead the respective agent to lose an arbitrarily large amount of money. I cannot provide a comprehensive discussion of the plausibility of denying Transitivity, of course. It is worth noting, though, that there have been several recent advances in the exploration of value and decision theory without Transitivity (cf. e.g. McClennen Reference McClennen1990; Ahmed Reference Ahmed2017; Thoma Reference Thoma2020). Given this development, the escape route of denying Transitivity is at least worth taking seriously.

Apart from the bullet of denying Transitivity, Aggregation with Constraints also runs into another possibly troubling counterexample: In some cases, it deems a happier shorter life to be better than an arbitrarily long life that is less happy but still decently happy. Consider the following case:

Short vs Long: $A \succ B$

In this case, condition (a) of Aggregation with Constraints takes hold, since there is no reason to choose $B$ which is relevant to the one reason to choose $A$ , hence Aggregation with Constraints implies that $A$ is better than $B$ . Note also that we can make $n$ infinitely large and still preserve this judgement, which means that Aggregation with Constraints recommends a shorter life over an infinitely long and decently happy life, which seems rather counterintuitive. This is of course to be expected: The whole point of restricting aggregation is that there are some comparisons where no matter how much low well-being we add on one side, it cannot aggregate to outweigh the weighty reason the high well-being provides on the other side. Yet, it may still be particularly counterintuitive in the prudential realm.

We might defend Aggregation with Constraints against this objection by setting the parameters of the view such that Short vs Long cases are not too counterintuitive. For instance, if we set the relevance parameters such that the difference in well-being levels between the long and the short life is sufficiently large, then the shorter life has to be extraordinarily happy. And if the shorter life really is extraordinarily happy, the example may not be as troubling as it seems.

Additionally, recall the reinterpretation of the Repugnant Conclusion in section 4, where we eliminated the possible confounder of one life being much longer than another. I argued that the Repugnant Conclusion only seems less repugnant in the prudential context if we imagine one life to be very long and the other life quite short. This, I argued, is due to the fact that we might consider length of life itself to be valuable. And hence we misinterpret the cases intuitively if we do not adjust the time spans of the cases. Now, if we apply the same reasoning to the above case, it may no longer look so troubling. For instance, if we assume that each period of time lasts for one million years, choosing $A$ is less counterintuitive.

Furthermore, as we shall see in the following section, plausible ways of denying GNE instead run into very similar counterexamples. One might thus argue that Short vs Long at least does not favour giving up GNE over giving up Transitivity.

4.4 Giving up GNE

Giving up GNE to avoid the spectrum illustrated in the previous section amounts to the following rough picture: Somewhere on the spectrum of lives depicted above, GNE fails, such that a slight decrease in the well-being of one person-stage cannot be successfully outweighed by a respective increase of well-being for any number of person-stages. Call the level of well-being just above this switch $x$ and the level of well-being just below this switch $x - 1$ . The comparison between the following two lives would thus violate GNE (where $x - 1 \ge 3$ and ‘ $ \approx $ ’ means ‘is incommensurable with’):

Violation of GNE: $A \succ B$ or $A \approx B$

Depending on whether we hold that $A \succ B$ or $A \approx B$ , this strategy also requires us to give up Completeness. Either of these two judgements violates GNE and is sufficient to render the other desiderata consistent. Yet we must of course ask: Why is it that a marginal decrease in well-being for just one person-stage makes such a big difference in the evaluation of the lives?

The most promising approach to explaining this break, I believe, lies in postulating lexicalities in the nature of well-being itself. The idea that there are higher and lower types of well-being that are not only different in degree, but truly different in kind, goes at least as far back as to Mill (Reference Mill1861: ch. 2). A recent proposal by Nebel, intended as a view on population ethics, which he proposes in Totalism without Repugnance” (2022), develops this idea. In what follows, I will translate his promising view into the context of prudence.

Nebel postulates a weak lexical priority of high levels of well-being over low levels of well-being.Footnote ²⁶ He terms the high levels of well-being important well-being, and the low levels of well-being trivial well-being. In addition, he introduces certain thresholds limiting their mutual tradeoffs. Applying this theory to prudence yields the following proposal: There is a certain amount of important well-being ${\rm{\Delta }}$ , such that if the difference in important well-being between two lives exceeds ${\rm{\Delta }}$ , the life which contains more important well-being is always overall better than the other, no matter how much trivial well-being is present in either of the lives. This, of course, leaves the question of how to evaluate lives where the difference in important well-being does not exceed ${\rm{\Delta }}$ . Nebel proposes, first, that if one life has both more important and more trivial well-being than another, the former is better than the latter. Second, when important and trivial well-being do not favour the same lives, but the difference in important well-being does not exceed ${\rm{\Delta }}$ , Nebel compares the ratio of the differences in each category (where $\delta $ refers to the threshold of difference to be exceeded by the trivial well-being). If none of these conditions apply, the two lives are incommensurable. Overall, a prudential analogue of his view amounts to the following, where ${i_x}$ stands for the sum of important well-being in $x$ , and ${t_x}$ stands for the sum of trivial well-being in $x$ :

$x$ is at least as good as $y$ iff

1. 1. ${i_x} - {i_y} \gt {\rm{\Delta }}$ or

2. 2. ${i_x} \ge {i_y}$ and

   1. a. $\!\!\!{t_x} \ge {t_y}$ or

   2. b. $\!\!\displaystyle{{{i_x} - {i_y}} \over {{t_y} - {t_x}}} \gt {\Delta \over \delta }$

$x$ and $y$ are incommensurable iff none of the above conditions apply.

In population ethics, Nebel calls this the Lexical Threshold View (2022: 18).Footnote ²⁷ and I shall adopt this name for the analogous proposal in prudence. Now let us have a look at how this view violates GNE. Assume that any well-being above −50 and below +50 is considered trivial, while anything else is considered important well-being. Now consider the following comparison of lives:Footnote ²⁸

Violation of GNE: $A \succ B$ or $A \approx B$

$A$ contains 50 units of important well-being and $n - 1$ trivial well-being. $B$ contains 0 important well-being and $49n$ trivial well-being. We can thus note, first, that if ${\rm{\Delta }} \le 50$ , then $A \succ B$ , which would violate GNE. Second, the amount of important well-being favours $A$ while the amount of trivial well-being favours $B$ , such that condition 2.a. does not apply. Lastly, depending on how exactly we set $\delta $ , we again get $A \succ B$ , which would violate GNE, or none of the conditions apply, such that we get $A \approx B$ , which likewise violates GNE. Since, it is of course somewhat counterintuitive that $A$ would be better than $B$ , setting the parameters such that $A \approx B$ makes the violation less problematic.

Note that there is no analogous GNE violation in the negative domain, even if we postulate a lexical threshold in the negative domain as well, separating important from trivial suffering. This is because of an asymmetry in how single unit increases of well-being cross thresholds in the positive and negative domain. In the positive domain, an increase of one unit of well-being can only turn the positive well-being from trivial to important well-being. Thus, when we increase the other person-stages’ well-being for compensation in accordance with GNE (person-stages ${t_2} - {t_n}$ in the above case), we raise it from trivial levels (from 1 to 49). But in the negative domain, an increase of one unit of well-being can only turn the negative well-being from important to trivial suffering. This means that when we increase the other person-stages’ well-being for compensation in accordance with GNE, we increase it from lower, i.e. important levels of suffering. Consider the following negative analogous case:

Non-Violation of GNE: $B \succ A$

In the above case, $A$ has more important suffering than $B$ , and hence the Lexical Threshold View would recommend choosing $B$ , which is exactly what GNE demands. This asymmetry will become important in the subsequent section.

Overall, giving up GNE in the way Nebel proposes looks promising. It can meet all conditions except GNE. However, this proposal is faced with at least three problems: First, it denies Completeness. Nebel himself judges the cost of giving up Completeness to be quite high (cf. Dorr et al. Reference Dorr, Nebel and Zuehl2021, Reference Dorr, Nebel and Zuehl2023). Yet, as mentioned, completeness has generally been considered easier to deny than Transitivity.

Second, it denies the popular assumption that well-being does not exhibit such lexical breaks, which, at least prima facie, seems to be supported by common sense intuition. Lastly, as indicated earlier, it likewise implies that, sometimes, an arbitrarily (even infinitely long), decently happy life can never be better than a possibly much shorter very happy life. Let us illustrate this problem with an example. Assume that ${\rm{\Delta }}$ is the difference in important well-being which makes it such that the life which contains at least an amount of ${\rm{\Delta }}$ more important well-being than the other is always judged to be better. Further, assume that any well-being above −50 and below +50 is considered trivial, while anything below/above is considered important well-being. Now consider the following comparison of lives:

Short vs Long: $A \succ B$

In such a comparison, the Lexical Threshold View always judges $A$ to be better, regardless of how large $n$ is. Indeed, even if we set $n = \infty $ , the view still suggests that $A$ is better. The same responses given above to defend Aggregation with Constraints apply here too. First, just how counterintuitive these cases are depends on the exact parameters regarding ${\rm{\Delta }}$ and important and trivial well-being. That is, we might be able to spell out the parameters in a way that makes the example less counterintuitive. Second, reinterpreting the cases such that both lives are very long, thereby eliminating confounding intuitions regarding length of life, may likewise contribute to make Short vs Long cases more palatable.Footnote ²⁹

At this point, we have reached an interesting result, namely that giving up Transitivity and giving up GNE amount to very similar proposals when supplemented by plausible interpretations. Before comparing their relative advantages, it is worth exploring whether their shared problem may be overcome by giving up GNEP instead.

4.5 Giving up GNEP

If we do not want to accept Short vs Long as outlined in the previous two sections and instead hold on to the judgement that the long life should be better, there might be a way to respond to this challenge that will end up maintaining GNE but violating GNEP. Consider the following negative analogue of Short vs Long:

Short vs Long: $B \succ A$

If we assume that the lexicalities work the same way in the positive and negative domain, the Lexical Threshold View now suggests that the longer life $B$ is better. While this is of course still somewhat counterintuitive, insofar as the person is still suffering in $B$ all the way to ${t_n}$ , some may argue that it is still less counterintuitive than the positive version of Short vs Long. Crisp (Reference Crisp2022: 375) puts the intuition as follows: “Imagine you face a choice between a year of agony, and some number of years of pain from a hangnail. Here some will claim that any number of years of hangnail pain is less bad than the agony.” Additionally, if we hold that foregoing a long life is generally a bad thing, one may hold that choosing a very long life of trivial suffering over a short life of important suffering is less counterintuitive than choosing a short life of important positive well-being over a very long life of positive trivial well-being: At least the former does not involve foregoing an arbitrarily long life.

This view is further supported by both substantively descriptive as well as normative considerations. Klocksiem argues that there is a fundamental difference between what he calls mere discomforts and genuine suffering (Reference Klocksiem2016). That this kind of phenomenology could be real may be more intuitive in the negative domain than in the positive domain: the difference between mere discomforts, i.e. things we find negative but bearable, and genuine suffering, i.e. things we consider genuinely unbearable, has some intuitive appeal. In contrast, one may argue that there is no such intuitive difference between things we find mildly pleasurable and things we find very pleasurable. This sort of reasoning may be additionally supported by the fact that various moral theories treat minimizing very negative suffering as more important than maximizing very positive happiness (for reasons beyond the procreation asymmetry). This datum could be explained by a theory which posits a kind of asymmetry between negative and positive well-being itself. Some theories based on this proposal have recently been investigated more closely in the context of population ethics by Mogensen (Reference Mogensen2024) and Baker (Reference Baker2024). Mogensen and Baker end up rejecting such a view for reasons I will get back to. For now, it is worth sketching the version of such a view that I find most plausible, and which, to my knowledge, has not been proposed in the literature in this way.

Consider the following kind of view: The only point at which well-being exhibits a lexical threshold is between unbearable suffering and everything else. Apart from this threshold, well-being can be traded off against each other in the usual spirit of Totalism. Such a view may be even more plausible if it, like the Lexical Threshold View, only exhibits weak lexicality. That is, we may postulate that not just any amount of unbearable suffering is prioritized over any amount of other types of well-being, but that there is certain amount ${\rm{\Delta }}$ such that a difference in unbearable suffering which exceeds ${\rm{\Delta }}$ always makes the life with the higher amount of unbearable suffering worse. Indeed, we can adapt all of Nebel’s conditions to construct the Negative Lexicality View, where ${S_x}$ stands for the sum of unbearable suffering in $x$ and ${w_x}$ stands for the sum of well-being that is not unbearable suffering in $x$ :

$x$ is at least as good as $y$ iff

1. 1. ${S_x} - {S_y} \gt {\rm{\Delta }}$ or

2. 2. ${S_x} \ge {S_y}$ and

   1. a. $\!\!\!{w_x} \ge {w_y}$ or

   2. b. $\!\!\!\displaystyle{{{S_x} - {S_y}} \over {{w_y} - {w_x}}} \gt {\Delta \over \delta }$

$x$ and $y$ are incommensurable iff none of the above conditions apply.

Note that the Negative Lexicality View neither violates GNE nor Non-Repugnance, as defined above. It does not violate GNE since the GNE violations of the Lexical Threshold View only occur around the positive threshold and not the negative one, and the Negative Lexicality View removes the positive threshold. And it does not violate Non-Repugnance since we only took avoiding the Very Repugnant Conclusion as a desideratum, but not the Repugnant Conclusion. The lexicality in the negative domain of well-being avoids the Very Repugnant Conclusion, even if the absence of such a threshold in the positive domain does not avoid the ordinary Repugnant Conclusion. And, of course, the view has the additional advantage that it does not recommend a short life over an arbitrarily long life, as giving up Transitivity or GNE did.

However, as expected, the Negative Lexicality View violates GNEP. Consider the following case. Assume again that well-being of −50 or lower constitutes unbearable suffering, and assume again that $\beta = 1000$ .

Violating GNEP: $A \succ B$ or $A \approx B$

The sum of unbearable suffering in $A$ is 0, and the sum of other well-being in $A$ is $n - 50$ . The sum of unbearable suffering in $B$ is −50, and the sum of other well-being in $B$ is $1000n - 1000$ . First, it holds again that if ${\rm{\Delta }} \le 50$ , then $A \succ B$ , which violates GNEP. Second, the sum of unbearable suffering favours $A$ while the sum of other well-being favours $B$ , such that condition 2.a. does not apply. Lastly, depending on how exactly we set $\delta $ , we again get $A \succ B$ , which would violate GNEP, or none of the conditions apply, such that we get $A \approx B$ , which likewise violates GNEP. In contrast to the analogous case we examined when giving up GNE, it is even more implausible to set the parameters such that $A \succ B$ . After all, the difference in other well-being is vastly larger than the difference in unbearable suffering. This is fortunate since it is intuitively more plausible that $A \approx B$ than that $A \succ B$ . It is thus plausible that the Negative Lexicality View only ever exhibits incommensurability when violating GNEP. In particular, since GNEP only ever involves big well-being differences, while GNE can also involve small well-being differences (since for GNE the added well-being must just be higher by at least one unit), it is thus more plausible for the Negative Lexicality View to only ever violate GNEP based on incommensurability, than it is for the Lexical Threshold View to only ever violate GNE based on incommensurability. This may constitute an additional advantage of the Negative Lexicality View over the Lexical Threshold View. And insofar as avoiding the negative version of Short vs Long is indeed less important than avoiding the positive version, it looks like giving up GNEP constitutes a serious alternative escape route.

Before closing, let me address the objection raised against views like the Negative Threshold Lexicality View by Mogensen and Baker. As noted above, they consider similar views in the context of population ethics but ultimately end up rejecting them. For both of them, this is for the following reason (Baker Reference Baker2024: 14; Mogensen Reference Mogensen2024: 345f): Applied to population ethics, it looks as though a view exhibiting negative lexicality will almost certainly recommend the extinction of humanity. This is for the following reason: If humanity survives, it is very likely that the population of future minds that are morally relevant (i.e. humans, certain non-human animals and possibly digital minds) will be incredibly large (cf. Newberry Reference Newberry2021). Within this incredibly large number of people, it is very likely that a certain number of this vast population will experience the kind of suffering that is below the relevant threshold, such that it qualifies as unbearable suffering. Aggregated across people, the amount of unbearable suffering would likely be so large that a view with a negative lexical threshold will prioritize avoiding this suffering and recommend not to bring this population into existence in the first place, i.e. recommend human extinction. Mogensen and Baker take this to be too counterintuitive to accept (though e.g. Crisp (Reference Crisp2022) and Pettigrew (Reference Pettigrew2024) are less sure that this is a knockdown argument).

Whether we accept this conclusion as a reductio or not, it is important to see that the same argument cannot be applied as straightforwardly to prudence. This is for two reasons. First, Mogensen’s and Baker’s argument crucially relies on empirical premises. They do not claim that there is no possible future population for which it might be better if we opt for extinction instead. Rather, the argument says that given our current empirical reality, a view exhibiting negative lexicality would lead to recommending extinction, which is not intuitively plausible. These empirical premises are way less plausible for the intrapersonal case: While it is empirically incredibly likely that, notwithstanding premature extinction, the future of humanity will hold an enormous amount of unbearable suffering, it is way less likely for any given individual life that it will hold an enormous amount of unbearable suffering. For most actual lives, then, the Negative Threshold Lexicality View will not recommend suicide. But, second, even if for some possible or actual lives, the amount of unbearable suffering in the future does indeed seem to exceed the threshold such that suicide is recommended by views with negative lexicality, this seems to be a way less problematic bullet in the intrapersonal case than in the interpersonal case. Accepting that we ought to extinguish all sentient life seems much harder to do than accepting that for some particular lives, suicide would be better than to go through a future that involves a significant amount of horrible and unbearable suffering.Footnote ³⁰