Asymmetric Critical Level Theories∗

John Mori†

May 12, 2020

1 Introduction

Most normative theories agree that one ought to promote happiness and prevent suﬀering of conscious beings, all other things being equal. However, when actions bring conscious beings into existence or preclude them from coming into existence, intuitions concerning their moral status differ widely. Jan Narveson’s Procreation Asymmetry (henceforth, the “Asymmetry”) captures a set of intuitions about bringing lives into existence [Narveson, 1967], and critical level theories provide useful distinctions concerning various levels of wellbeing. I present several theories with features from both the Asymmetry and critical level theories.

The development of my theories is primarily motivated by the failures of simpler theories. That is, for each theory I examine, I propose a thought experiment for which the theory implies a potentially unintuitive conclusion, and I present another theory that avoids the same conclusion, while preserving intuitive conclusions from preceding theories. My paper proceeds as follows: ﬁrst

I state a few preliminary assumptions necessary for my analysis. I then introduce the Asymmetry and distinctions related to the Asymmetry, and I brieﬂy argue for supporting wide, soft theories.

I evaluate a speciﬁc Wide, Soft Theory proposed by [Thomas, 2019], and I demonstrate that the theory produces undesirable conclusions between options of the same population size. I present a

Critical Level Theory which makes distinctions to avoid the previous unintuitive conclusion but

∗Term paper for EP&E 471 Directed Reading and Research under the supervision of Shelly Kagan. †I am grateful to Shelly Kagan for the exacting but helpful conversations and comments, Stephen Darwall for ﬁrst suggesting that I read Derek Parﬁt’s Reasons and P ersons (I did not think twice about population ethics until years later), and friends and family who endlessly heard me talk about population ethics and encouraged me to pursue the topic more formally. Email: [email protected]

1 also implies an undesirable conclusion when population sizes vary across options. I then marry the distinctions made by the Asymmetry and by critical level theories, and present two theories — the Tolerant and Strict Asymmetric Critical Level theories, which resolve preceding challenges but also have caveats of their own. I extend these theories to allow them to make judgments on option sets of more than two options with the beatpath method. However, this method does not satisfy the Independence of Irrelevant Alternatives, which has undesirable implications. I suggest ways to resolve these diﬃculties and also future directions for reﬁning asymmetric theories.

1.1 Preliminaries

First, I brieﬂy state assumptions that are suﬃcient for discussing the cases and theories I present.

I assume welfarism — that wellbeing is all that matters morally. I am agnostic with regards to the correct theory of wellbeing, although theories may oﬀer diﬀerent intuitions in the thought experiments that I propose, in which case, I invite readers to revise aspects of the thought experiments to adequately capture wellbeing under various theories.

I do assume that a person’s wellbeing is cardinal — that there exists a function that maps the state of wellbeing onto numbers. I remain agnostic with regards to whether levels of wellbeing are continuous — that there exists infinite levels of wellbeing between any two levels — or discrete — that there exists only finitely many levels between any two levels. If wellbeing is discrete, then I only assume that the smallest unit of wellbeing is fine-grained enough such that the differences in wellbeing in the thought experiments I describe exist.

I also assume that the individual instances of wellbeing that a person experiences across the course of their life can be represented by a single value of cumulative wellbeing (think of the value of an integral of a function of wellbeing over time). Of course a life in which one experiences both great joy and great suffering is qualitatively different from a life in which one experiences constant boredom, but I assume that the value of pleasure and pain can be compared such that these two different lives can be appropriately represented by the same value of cumulative wellbeing. In cases that I present in which a person’s cumulative wellbeing is positive but low, the possibility for high variability in one’s experienced wellbeing may influence intuitions about the case, but the thought experiments merely require that a life at that constant positive low wellbeing is possible.

Henceforth, I refer to a person’s wellbeing as the single cumulative value of experienced wellbeing.

2 I assume that a life of great ﬂourishing and happiness has positive wellbeing and a life of enormous suﬀering has negative wellbeing. Let these lives be good lives and bad lives, respectively.

I assume there exists a function with arbitrary units of wellbeing that maps a level of very high wellbeing to 100 and a level of very low wellbeing (a life of great suffering) to −100. I do not assume that there exists either upper and lower bounds to wellbeing. My examples still hold even if there exists bounds to wellbeing, as long as the range of wellbeing contains the levels of wellbeing in my thought experiments. The existence of positive and negative wellbeing suggests that there exists a zero level of wellbeing, defined such that all else being equal, one ought not bring a bad life — a life of wellbeing strictly below zero — into existence, while the status of bringing neutral or good lives — lives with zero or positive wellbeing, respectively — into existence depends on the theory. Beliefs can differ regarding where the zero point exists on the wellbeing continuum — how a person’s life should go for it to be assigned zero wellbeing. The empirical characteristics of bad, neutral, and good lives will be better illustrated in thought experiments that I present later.

I assume that wellbeing is interpersonally comparable — that the wellbeing of one person can be compared to the wellbeing of another person in terms of how much each contributes to the total wellbeing of the population.

Theories in population ethics broadly fall into two categories: axiological, which concern the betterness/worseness of states of aﬀairs; and deontic, which concern what one ought to do, what is permissible, etc. Many authors have demonstrated impossibility theorems in population axiology

— that there does not exist a theory which satisﬁes a set of desirable axioms [Arrhenius, 2000]. The theories that I propose are deontic, although analogies can be made between the deontic judgments of “ought to choose x over y” to axiological relations of “x better than y”. I only consider cases that have neither risk nor uncertainty — one can extend my theories to options with uncertainty with Thomas’ supervenience principle [Thomas, 2019]. To clarify deontic judgments, an option is required if one ought to choose that option and is not required if it is not the case that one ought to choose that option. An option is impermissible if it is the case that one ought not to choose it and is permissible if it is not the case that one ought not to choose it.

The theories when they are ﬁrst presented all take pairs of options as inputs and make pairwise deontic judgments. I later discuss a method that takes these pairwise judgments to imply deontic conclusions in cases where there are more than two options.

3 2 The Asymmetry

The Asymmetry is the deontic1 view that one ought not to bring a bad life into existence, but it is not the case that one ought to bring a good life into existence, all other things equal. Consider my rendition of one of Parfit’s thought experiments that illustrates the Asymmetry [Parfit, 1984]: person A and their partner are considering whether to conceive a child who, if born, would be person B. Person A expects that from the joys and hardships of child rearing, their wellbeing would remain the same whether they decide to have the child or not. However, upon visiting their doctor, person A learns that they have a rare medical condition such that if they were to have a child, the child would live a short, bad life in constant suffering. Fortunately, the doctor can easily administer a treatment for the condition, such that if person A were to have a child after receiving the treatment, the child would live a good life. Person A could choose one of three options: not have the child (x), have the child without receiving the treatment (y), or have the child after receiving the treatment (z). For now, suppose that our theory of personal identity deems that the child is the same child whether they are born with or without the treatment — I will later discuss cases in which the identities of people differ across options.

We can represent this case in the following table:

Case 1: Basic Asymmetry

Option

x y z

A 50 50 50 Person B − −50 50

Each person’s wellbeing under each option is given in the table. The dash for B’s wellbeing under option x indicates that B does not come into existence under x. Let a necessary person be a person who exists in all options. On the other hand, let a contingent person be one who exists in at least one option and does not exist in at least one other option. In this case A is a necessary person and B is a contingent person. Recall that we ﬁrst consider the Asymmetry (and other theories) in pairwise options and extend the theories to larger option sets later.

1The Asymmetry can also be presented under axiologies [Frick, 2014]. I only examine the deontic version.

4 First, let’s consider the choice between not having a child and having the child without the treatment — options x and y. As it is wrong to cause someone to suffer, all else equal, it also seems wrong to bring someone into existence who would suffer. Thus, many have a strong intuition that it is impermissible have the child without the treatment. Second, let’s consider the choice between having the child with the treatment and doing so without — options y and z. Since the child comes into existence under both options, clearly one ought to prevent the child from suffering and give the child a good life by choosing option z.

The interesting pair is options x and z — not having a child and having a child who would live a good life. It certainly seems permissible to have the child — after all, the total wellbeing in the world would increase with the child’s existence. However, would it be obligatory to have the child, or generally to bring a good life into existence? By not having the child, no one is harmed — the child does not come into existence. But since the child would live a good life if born, not having the child would be an instance of failure to do good.

According to the Asymmetry, the negative wellbeing in the contingent bad life implies that one ought not bring the child into existence, whereas the positive wellbeing in the contingent good life does not imply that one ought to bring the child into existence — there exists an asymmetry between how one ought to treat negative and positive wellbeing of contingent people. Thus under the Asymmetry, choosing either option x or z is permissible. I will not discuss the normative foundations of the Asymmetry2, since for many, I included, the intuition alone makes the theory quite plausible.

The basic Asymmetry as presented is restricted to giving deontic judgments to options in which only one contingent person can come into existence and the wellbeing of necessary people remain constant. Of course, cases can vary greatly in the wellbeing of existing people and that of people who can come into existence, and in the literature, the Asymmetry has been extended to theories that deontic judgments in a large domain of cases. But these theories disagree on certain basic cases — most notably these theories diﬀer on the choice points between hard and soft theories, and between narrow and wide theories. 2For those interested, see [Frick, 2014]

5 2.1 Hard vs. Soft

Suppose that bringing an additional good life into existence does not leave necessary people equally well off, but rather comes at a cost to necessary people. Returning to the earlier thought experiment, person A decided to have the child after the treatment (person B), whom they are now taking care of. Person A is considering whether to have another child (person C). Since person A is cured from their condition, the second child would live a good life. However, taking care of two children will be a greater financial hardship for person A, and their first child would have a worse life if they were to have their second child than if they were not. Again, person A expects that their wellbeing will be constant across options.

Case 2: Costly Addition

Option

x y

A 50 50 Person B 50 40

C − 40

I design this thought experiment (and other experiments hereafter) such that the person who is making the decision — Person A — is not the person who is harmed — Person B — by bringing another person into existence. If person A is sacrificing their own wellbeing to bring another child into existence, this case may carry different moral intuitions. Perhaps there are purely welfarist normative considerations that come to play whether the person making the decision is better or worse off by the decision. For simplicity, I choose not to address these distinctions and assume that the wellbeing of the person making the decision is always unaffected by the decision.

In this case, having the child would harm person B. Also person C is not harmed by not bringing

C into existence, since C does not exist in option x. Let a theory be hard with respect to set A of people inferior to set B of people iﬀ (if and only if) it is impermissible to increase the total wellbeing of set A if doing so also decreases the total wellbeing of set B. Thus a theory that is hard with respect to contingent good lives inferior to necessary people and contingent people with bad lives

(henceforth, just “hard” unless speciﬁed otherwise) iﬀ it is impermissible to bring additional good lives into existence (person C) if it comes at any cost to the total wellbeing of necessary people

6 and contingent people with bad lives (persons A and B) (although this case does not contain any contingent people with bad lives, hard theories avoid bringing bad lives into existence). Thus, under a hard theory, person x should not have their second child.

However, we can put this intuition to the test by supposing extreme values in a thought experiment — minimal harms to necessary people and contingent people with bad lives, and potentially large amounts of wellbeing for contingent people. Suppose that scientists discover that a highly contagious virus has already infected the entire world population. Fortunately, the virus does not cause any pain, but tragically, it has sterilized everyone. Everyone except person A, that is, whose rare condition made them immune to the virus. Person A is taking care of their ﬁrst child (person

B) and is considering whether to have a second child, to whom person A would pass down their immunity and ability to reproduce (the sterility is of the sort such that two sterile people cannot reproduce, but person A and their partner — who is sterile — can still reproduce). Since everyone else is sterilized from the virus (assume that their wellbeing remains constant whether person A chooses to have the child or not), the human species would come to an end if person A chooses not to have the second child, and having the child would eventually repopulate the Earth with people with good lives (kindly ignore moral qualms about incest — the human race is at stake). Of course, since we believe that the universe will eventually come to an end, we expect the potential number of people who can come into existence to be finite, albeit potentially very large. Similarly to before, having a second child would take some resources from the first child, leading the first child to have a slightly worse life.

Case 3: Possible Extinction

Option

x y

Person A 50 50

Person B 50 49

Many contingent people - 50

Note that possible extinction is simply a case of costly addition with more contingent people with good lives at stake. Since B is harmed by A having the second child, a hard theory requires A to choose option x — not have the child and let the human species come to an end. Many would

7 ﬁnd this unintuitive 3Even though under the Asymmetry, it is not required to bring additional good lives into existence all else equal, perhaps the total wellbeing of good contingent lives can be great enough to overrule the impermissibility of harming necessary people (and bringing in contingent bad lives).

A theory is soft with respect to set A inferior to set B if it is permissible to increase the total wellbeing of set A if doing so also decreases the total wellbeing of set B. Thus, a theory that is soft with respect to contingent people with good lives inferior to necessary people and contingent people with bad lives — if it is permissible to bring additional good lives into existence at a cost to the average wellbeing of necessary people and contingent people with bad lives (thus, theories that deem bringing additional good lives into existence not only permissible, but also require doing so, are also soft theories). My theories fall into a particular subset of soft theories according to which bringing contingent people with good lives into existence is permissible if their total wellbeing oﬀsets the loss in wellbeing to necessary people and contingent people with bad lives, i.e. the magnitude of positive wellbeing of contingent people with good lives is equal to or greater than the magnitude of the loss in wellbeing of necessary people and the negative wellbeing of contingent people with bad lives (henceforth, a soft theory falls into this subset of general soft theories). I will formalize this notion of oﬀsetting later. Under a soft theory, since the wellbeing of contingent people with good lives is greater in magnitude to the loss of necessary people, choosing either option x or y is permissible in both the costly addition and possible extinction.

A hard theory is “hard” in the sense that it does not consider the wellbeing of contingent good lives at all if necessary people suﬀer or contingent bad lives are brought into existence. On the other hand, a soft theory is “soft” in the sense that it considers the wellbeing of contingent good lives in such a situation and deems it permissible to make that trade-oﬀ if that wellbeing is great enough.

3One may protest that the intuitive force for rejecting the hard claim that option x is required is not merely the product of the wellbeing of many contingent people at stake, but also the continued existence of humanity on the line. However, in a welfarist framework, the value of a potential future is merely the wellbeing of people who come into existence. And assuming that there is no time discount rate for wellbeing, the addition of the same many contingent people at the end of humanity or at another point in time count the same in welfarist theories. The thought experiment could also be revised so that the existence of humanity is not at stake, but person A has the option to bring many people into existence.

8 2.2 Narrow vs. Wide

Every action has the potential to bring diﬀerent contingent people into existence — a slight change of timing to the events in someone’s day can change the conception — and thus genetic makeup

— of a child, who then in their life make small changes in timing in other people’s lives. Thus, many actions can change the identity of many people in the future [Parﬁt, 1984]. While I remain agnostic with regards to theories of personal identity, I assume that actions can determine whether a particular person comes into existence or does not, and that it is possible for the same person to come into existence across options. When an action brings the same number of contingent people into existence but merely changes the identities of those people, perhaps these contingent people can be treated more like necessary people.

Consider my rendition of Parﬁt’s Non-Identity Problem [Parﬁt, 1984]: Person A and their partner already decided to have a child. However, due to fertility issues, Person A is looking for a sperm donation. After research and consultation, they select two donations to choose one from.

However, person A is notified that one of their donors was diagnosed with a rare condition, such that if person A were to choose that donation, their child would live a middling good life, but still a good life (option x). They are also told that if they were to pick that donation, there luckily exists an costless treatment to the sperm that would let the child live a happier life (option y) than that without a treatment. The other donor does not have this condition, so if person A chooses the other donation, the child would live a life equally happy (option z) as that with the first donation and treatment. Since the two donations contain different genetic material, the identity of the child born would be different in option z than in options x and y. Person A expects their own wellbeing to be constant across the options.

Case 4: Basic Non-Identity

Option

x y z

A 50 50 50

Person B 10 50 −

C − − 50

Again, we initially consider this case in pairwise comparisons, and consider the entire option set

9 later. If person A were to choose the first donation, between having the treatment and not (between option x and y), they ought to choose option y to improve the child’s wellbeing. Between options y and z, even though the child who is born in one option has a different identity than the child who is born in the other option, since the wellbeing levels of the two children are the same, choosing either y and z is permissible. The crux of the non-identity arises between options x and z — how should the wellbeing of different contingent people be treated compared to necessary people?

Parfit suggests a “No Difference View” ([Parfit, 1984] p. 367), in which options x and z should be compared the same way as options x and y — there should be no difference between comparisons of options in which the distribution of wellbeing is the same and only the identities of persons differ. Under the No Difference View, one ought to choose option z over option x. In this case, the wellbeing of different contingent people across options are simply compared as if they were necessary people in both options. This direct comparison is straightforward in cases in which the number of contingent people are the same in each option and if we consider the total wellbeing of necessary and contingent people to make our deontic judgment — simply consider the total wellbeing of all people in each option. When options have different numbers of contingent people (and hence different numbers of people in general), if we want to compare the wellbeing of contingent people in the same way as necessary people, we can consider more particular methods.

2.2.1 Saturating Counterpart Relations

Meacham suggests saturating counterpart relations as a method of comparing the wellbeing of contingent people across options [Meacham, 2012]. Since I do not endorse Meacham’s speciﬁc theory,

I define the method of comparison slightly differently:4 between two options that have different numbers of people, let a saturating counterpart relation (SCR) be a set of pairs of contingent people such that each contingent person in the option with fewer people is uniquely paired with one contingent person in the option with more people. If the two options contain the same number of people, every contingent person in one option is paired with a contingent person in the other option. For a particular SCR, let the set of paired contingent people be the non-excess people. If the options differ in number of people, the option with a larger population also contains excess people

4Speciﬁcally, the saturating counterpart relation that I deﬁne does not have the “minimization” condition, but keeps the other three conditions: one-to-one function, before-t match, and saturation.

10 — contingent people who are not paired. Also, if the options diﬀer in number of people and both contain contingent people, whether a contingent people in the option with the larger population is assigned as a non-excess or excess person depends on the speciﬁc SCR. Consider:

Diﬀerent SCRs

Option

x y

A 50 50

B 40 − Person C − 50

D 60 −

Person A is a necessary person, and there exists two possible SCRs. If persons B and C are paired, they are non-excess people and person D is an excess person. If persons C and D are paired, they are non-excess people and person B is an excess person. Distinguishing excess and non-excess people through a SCR is merely method to compare the wellbeing of contingent people — there does not exist metaphysical characteristics that imply a speciﬁc SCR for any case.

Theories that pair contingent people in a SCR can count the wellbeing of non-excess people and that of excess people differently. Since any specific SCR is arbitrary, any plausible theory that pairs contingent people in a SCR should not produce conflicting deontic conclusions — a theory satisfies the Weak Independence of Saturating Counterpart Relations iff for options x and y, it is not the case that the theory under one SCR requires option x, and the theory under another SCR requires y.

Finally, a theory is wide with respect to non-excess people with good lives if the wellbeing of non-excess people with good lives is treated the same way as necessary people and contingent people with bad lives. Thus between options x and z in basic non-identity, a wide theory would pair persons B and C together as if they were one necessary person and consider the diﬀerence in their wellbeing in the same way as necessary people persons Since person C has higher wellbeing than person B, person A ought to choose option z. A theory is narrow with this respect if the wellbeing of non-excess people with good lives is not treated the same way as necessary people and

11 contingent people with bad lives5 . In the basic non-identity, a narrow theory can follow (but not necessarily) this reasoning regarding options x and z: since persons B and C are diﬀerent contingent people with good lives and since there is no obligation to bring good lives into existence under the

Asymmetry, there is not more reason to choose one additional good life over the other, and so choosing either option is permissible. Generally, a theory can be wide or narrow with respect to diﬀerent subsets of non-excess people with good lives, an instance of which we will see later.

3 Theories

The theories that I present all take as arguments the total wellbeing of diﬀerent groups of people and from these values produce deontic conclusions. I borrow the choice function R(x/y) from

[Thomas, 2019]. In a pairwise comparison between x and y, when R(x/y) is positive, one ought to choose x and when R(x/y) is negative, one ought to choose y. When R(x/y) is 0, choosing either x or y is permissible.

Before considering several specific theories, let us formalize the notion of offsetting mentioned in the soft view. The offset by is a relation between two values: r1 is offset by r2. I refer to r1 as the robust value and r2 as the offsetting value. The theories I consider generally have choice functions of this form:

R(x/y) = r1 oﬀset by r2.

Here, r1 and r2 are the robust and offsetting value, respectively, for choosing option x over option y. The robust value for choosing option x over y is the difference between the robust value in option x and that in option y, and similarly, the offsetting value for choosing option x over y is

x y x y the difference between that in option x and that in option y: r1 = r1 − r1 and r2 = r2 − r2 . Each theory that I will discuss assigns the wellbeing of different types of people to constitute the robust value and offsetting value in that option. Formally, define the “offset by” relation as:6:

5My definition of narrow and wide roughly follows Thomas (2019), except I explicitly define the definitions to be exhaustive over theories that consider contingent people. Different notions of narrow and wide exist in the literature. 6I modify the definition of the offset by relation such that the conditions are mutually exclusive. The two definitions are functionally equivalent. Thomas defines the relation as:

r r , r > 0 or r , r < 0  1 1 2 1 2 r1 oﬀset by r2 := max{0, r1 + r2} r1 ≥ 0 ≥ r2  min{0, r1 + r2} r2 ≥ 0 ≥ r1

12   r1 r1, r2 ≥ 0 or r1, r2 ≤ 0   r1 oﬀset by r2 := max{0, r1 + r2} r1 > 0 > r2    min{0, r1 + r2} r2 > 0 > r1

To illustrate this relation, suppose our choice function has the form:

R(x/y) = r1 oﬀset by r2

x y Intuitively, if both the robust and offsetting value are better in option x than y (r1 > r1 and x y r2 > r2 , so r1, r2 > 0), then one ought to choose x (R(x/y) > 0). Similarly, if both are worse in x, one ought to choose y. If the robust value is equal in x and y (r1 = 0), then choosing either option is permissible. Henceforth, I refer to betterness and worseness with respect to that of option x being better or worse than that of option y. If the robust value is better (r1 > 0) and the offsetting value is worse (0 > r2), then one ought to choose x (R(x/y) > 0) if the betterness of the robust value is greater in magnitude than the worseness of the offsetting value (r1 + r2 > 0), or both options are permissible if the offsetting value is worse enough to offset the betterness of robust value (r1 + r2 ≤ 0). On the other hand, if the robust value is worse (0 > r1) and the offsetting value is better (r2 > 0), then one ought to choose y if the worseness of the robust value is greater in magnitude than the betterness of the offsetting value (r1 + r2 < 0), or both options are permissible if the offsetting value is better enough to offset the worseness of the robust value (r1 + r2 ≥ 0). Since I assume that individual wellbeing is cardinal and interpersonally comparable, we can consider the total wellbeing of a group of people as the sum of the each individual wellbeing of that group. Let Tnec(x) be the total wellbeing of necessary people in option x, and let T¬exc(x) and

Texc(x) be that of non-excess and excess people in option x, respectively.

Asymmetry Distinctions

13 Option

x y

Necessary Tnec(x) Tnec(y)

Non-excess T¬exc(x) − People Non-excess − T¬exc(y)

Excess Texc(x) −

When comparing options with populations of different sizes, let the option with a larger population be option x, such that only option x contains excess people. Each theory classifies the total wellbeing of different groups of people into the robust value and the offsetting value.

3.1 A Wide, Soft Theory

[Thomas, 2019]7proposes a theory in which the total wellbeing of excess peoples counts robust if negative, or can oﬀset diﬀerences in total wellbeing of necessary and non-excess people if positive.

Formally:

  +Texc(x) Texc(x) < 0 R(x/y) = (Tnec(x) − Tnec(y)) + (T¬exc(x) − T¬exc(y))  oﬀset by Texc(x) Texc(x) ≥ 0

Under this wide theory, both the wellbeing of necessary and that of non-excess people count robustly. The Wide, Soft Theory extends the intuition of the Asymmetry from the individual to the group of people — the wellbeing of excess people in option x contributes to the offsetting value if the total wellbeing of excess people is non-negative, but that wellbeing counts robustly if it is negative. Under this soft theory, if excess people include both good lives and bad lives, the wellbeing of the good lives can offset that of the bad lives and differences in total wellbeing for necessary and non-excess people. But if the negative wellbeing of excess bad lives is greater in magnitude than the positive wellbeing of excess good lives, then the total wellbeing of excess people (which would be negative) counts robustly. Note that the theory requires option z between

7[Thomas, 2019] describes four theories (“views” in his words) that correspond with diﬀerent choices in the hard/soft and narrow/wide distinctions. He considers a “narrow, soft view” the “most theoretically natural” of the four. For reasons discussed in the previous sections, I only consider and build upon his Wide, Soft Theory.

14 x and z in the non-identity case (as wide theories do) and deems both options permissible in the potential extinction case (as soft theories do).

Recall that theories that distinguish contingent people into non-excess people and excess people through saturating counterpart relations may imply diﬀerent conclusions given diﬀerent SCRs.

Thomas’s Wide, Soft Theory satisﬁes the Weak Independence of Saturating Counterpart Relations.

I describe the intuition behind the proof here and provide a formal proof in the Appendix.

If (A) the total wellbeing of all people in option x is higher than that in option y, then either

(i) the total wellbeing of necessary and non-excess people in option x is higher than that of option y and the choice function requires option x, or (ii) the total wellbeing of necessary and non-excess people in option x is lower than or equal to that of option y, but since (A), the total wellbeing of excess people oﬀsets this diﬀerence if any (but not outweigh) and either option is permissible.

If (B) the total wellbeing in x is lower than that in y, then either (i) the total wellbeing of necessary and non-excess people in option x is higher than that of option y but the total wellbeing of excess people is negative to a greater extent (for (B) to be true), and thus the choice function requires option y, or (ii) the total wellbeing of necessary and non-excess people in option x is lower than that of option y and excess people in x cannot successfully oﬀset this diﬀerence (for (B) to be true), and thus the choice function requires option y.

Since the Wide, Soft Theory satisfies the Weak Independence of Saturating Counterpart Re- lations, the theory cannot require different options under different SCRs, but the theory can still deem one option required under one SCR and the same option merely permissible under another

SCR. Thomas suggests to “consider all saturating counterpart relations that extend transworld identity, and proceed as if uniformly uncertain which one is correct” [Thomas, 2019]. Given this advice, does a certain probability of requiring one option also require the option ex ante or deem the option merely permissible? Cases can be constructed to support either view, and I hesitantly suggest that for an option that is required or merely permissible under diﬀerent SCRs, if the clear majority of SCRs requires that option, then the theory requires that option, and the option is merely permissible otherwise.

The Wide, Soft Theory captures the desired intuitions in the non-identity case and potential extinction case, but since the wellbeing of necessary and non-excess people count robustly, the

Wide, Soft Theory faces the same diﬃculties as total utilitarian theories when the sizes of the

15 populations are equal across options. Consider the following case: a social planner considers which of two societies they should bring about in the future. In one potential society a small subset of the population — say, a hundred people — live flourishing lives. Also a much larger population live lives with very low but still positive wellbeing — lives that are barely worth living. As Parfit describes, their lives may consist of listening to muzak and eating potatoes [Parfit, 1984]. On the other hand, if the planner brings about the other society, the same small subset of the population suffers immense torture. Also a much larger population has the same size as the large population in the first society, but every member has a distinct identity from the first society. Members of the larger population also live lives similar to that of the larger population in the first society, except they have a few more potatoes and selections of muzak than those in the first society. Option x brings about the first society and option y brings about the second.

Case 5: Same Number Very Repugnant Conclusion

Option

x y

100 100 −100

Number of People 20001 1 −

20001 − 2

If we take these particular values of total wellbeing as arguments in the choice function,

R(x/y) = (10000 − (−10000)) + (20001 − 40002) = −1, the Wide, Soft Theory requires option y, which brings about a society of enormous suﬀering and marginally better lives. Of course, in the thought experiment, the population of necessary people can be arbitrarily large, and there would always exist a yet larger population of contingent people to produce the same result. All the human

flourishing in option x and suffering in option y is outweighed by the meager increase in wellbeing for a larger population in y, members of which do not even exist in option x. I offer this case to suggest a plausible motivation to refine our theory. Some may not find the conclusion above unintuitive, or choose to accept the conclusion. But perhaps the theories presented below would produce more intuitive conclusions.

16 3.2 A Critical Level Theory

Traditionally, critical level theories posit that the different ranges of positive wellbeing — below and above a critical level — count differently in axiological theories. I introduce critical levels to my deontic theories to make these same distinctions of positive wellbeing in the robust and offsetting value of a theory. Before I present theories that include distinctions from both the Asymmetry and critical level theories, I will present the Critical Level Theory without Asymmetry distinctions, which some may find independently plausible.

Before we consider the Critical Level Theory, let us introduce a new notion of offsetting in which the offsetting value can outweigh the robust value when the robust value is 0. In the “offset by” relation previously defined, the offsetting value cannot outweigh the robust value even when the robust value is zero. For the Wide, Soft Theory, this implies that one is indifferent between bringing an excess good life into existence and not doing so all else equal. However, we will consider theories in which changes to the offsetting value should determine the deontic judgments when the robust values of the two options are equal. Let us consider a variant of the offsetting relation —

“strongly oﬀset by”8 as deﬁned:

 r r , r > 0 or r , r < 0  1 1 2 1 2    r2 r1 = 0 r1 strongly oﬀset by r2 :=  max{0, r1 + r2} r1 > 0 ≥ r2    min{0, r1 + r2} r2 ≥ 0 > r1

Again, to illustrate this new relation, suppose that:

R(x/y) = r1 strongly oﬀset by r2

The strongly offset by relation differs from the offset by relation only when the robust value is

0. In the oﬀset by relation, the choice function would take the robust value of 0 and choosing either

8I deem the new relation strong since the offsetting value determines the decision when the robust value of two options are equal. If R(x/y) = r1 strongly offset by r2 and r1 = 0, r2 > 0, then one ought to choose x, which implies that x is permissible, which is the conclusion if r1 is offset by r2, and the converse is not true. Note that this relationship does not hold for option y, which the strongly offset by relation deems impermissible, and which the offset by relation deems permissible.

17 option is permissible. The strongly offset by relation instead defers to the offsetting value when the robust value is 0. Thus, one ought to choose option x if the robust value is 0 and the offsetting value is positive, and y if the offsetting value is negative, and choosing either option is permissible if the offsetting value is 0. The utility of the strongly offset by relation becomes apparent as theories become more nuanced.

The critical level is a level of wellbeing that demarcates flourishing lives and those that are worth living but just so.. Let people with wellbeing equal to and above the critical level be flourishing people, those with wellbeing between the critical level and zero (inclusive) be humdrum people, and those with wellbeing below zero be wretched people. I present a theory in which the total wellbeing of humdrum people can strongly offset that of flourishing and wretched people.

R(x/y) = (Tflo(x) − Tflo(y)) + (Twre(x) − Twre(y)) strongly oﬀset by (Thum(x) − Thum(y))

This theory captures the intuition that people with ﬂourishing or wretched lives count robustly, while humdrum lives — lives of merely muzak and potatoes — count merely in the oﬀsetting value. Why might this be the case? In the same number very repugnant conclusion, the loss of

ﬂourishing and the addition of suﬀering is required by a meager increase in wellbeing to humdrum lives. Suppose the critical level is above 2 — a life of merely muzak and potatoes — and below

100, a life of great ﬂourishing. Taking the values of total wellbeing as arguments, R(x/y) =

(10000 − 0) + (0 − (−10000)) weakly offset by (20001 − 40002) = 0. Here the small increase to the wellbeing of humdrum lives can only offset, but not outweigh, the happiness and suffering of the

flourishing or wretched people. Thus choosing either option is permissible, and we avoid requiring option y. The strongly offset by relation (instead of the offset by relation) implies that one ought to increase and ought not decrease the wellbeing of humdrum people all else equal.

While the Critical Level Theory introduces useful distinctions, there is reason to reintroduce the Asymmetry distinctions. Consider the following case, where the critical level is 10 (recall that people at the critical level live ﬂourishing lives): the social planner considers which one of two societies they should bring about. Similarly to the same number very repugnant conclusion, a small subset of the population either live ﬂourishing or wretched lives. Unlike before, no one else

18 exists in the ﬁrst society, while a large population comes into existence in the second society with lives that are right at the critical level.

Case 6: Critical Level Very Repugnant Conclusion

Option

x y

100 100 −100 Number of People 2001 − 10

By the Critical Level Theory choice function, R(x/y) = (10000−20010)+(0−(−10000)) strongly oﬀset by (0−

0) = −10, and one ought to choose option y. Thus, the loss of great ﬂourishing and the addition of great suﬀering is required by bringing people who live lives at the critical level into existence.

Similarly to the previous case, one may not ﬁnd this conclusion unintuitive, or choose to accept this conclusion.

On the other hand, under the Wide, Soft Theory, the wellbeing of the excess people with good lives contributes to the oﬀsetting value, which oﬀsets but does not outweigh the robust value, and thus choosing either option is permissible. I now present theories that incorporate distinctions from both the Critical Level Theory and the wide soft theory to produce intuitive judgments in both previous cases.

3.3 The Tolerant Asymmetric Critical Level Theory

The Asymmetry distinguishes people as necessary, non-excess, and excess, and the critical level distinguishes people as flourishing, humdrum, and wretched. How should we sort the total wellbeing of each group of people between the robust value and offsetting value in a theory? Suppose that r1 is the robust value and r2 is the offsetting value in a choice function: R(x/y) = r1 offset by r2. Consider the following assignment:

Robust and Oﬀsetting Value of People by Asymmetry and Critical Level Distinctions

19 Asymmetry Distinctions

Necessary Non-excess Excess

Flourishing r1 r1 r2

Critical Level Distinctions Humdrum r2 r2 r2

Wretched r1 r1 r1

First, let’s consider the bottom row — the wellbeing of wretched people. Intuitively, the amount of suﬀering in each option matters robustly — the basic Asymmetry case deems that bringing a wretched child into existence is impermissible all else equal, and critical level theories count negative wellbeing robustly. Thus the wellbeing of wretched people should be counted robustly, regardless of the Asymmetry distinction. Second, let’s consider the humdrum people — the middle row.

Since humdrum people do not experience great flourishing, critical level theories assign humdrum wellbeing to the offsetting value. For now, let’s assume that the Asymmetry distinction does not change this assignment (a later theory — the Strict Asymmetric Critical Level Theory — questions the status of humdrum excess people as offsetters). Third, let’s considering the flourishing people.

Critical level theories deem that all ﬂourishing lives count robustly. The Asymmetry maintains that necessary people count robustly, and through a saturating counterpart relation, non-excess contingent people count robustly. However, under the Asymmetry, the wellbeing of excess people with good lives counts only in the oﬀsetting value. The categorization above suggests the tolerant asymmetric Critical Level Theory (TACL):

R(x/y) = [(Tflo∩(nec∪¬exc)(x)−Tflo∩(nec∪¬exc)(y))+(Twre∩(nec∪¬exc)(x)−Twre∩(nec∪¬exc)(y))+Twre∩exc(x)

strongly oﬀset by (Thum∩(nec∪¬exc)(x) − Thum∩(nec∪¬exc)(y))] oﬀset by Tflo∩exc(x) + Thum∩exc(x)

One can verify that this theory avoids the unintuitive conclusions in the earlier cases — under this theory choosing either option is permissible in the same number very repugnant conclusion and in the critical level very repugnant conclusion. This theory utilizes both the strongly oﬀset

20 by relation and the offset by relation. The wellbeing of necessary and non-excess humdrum lives constitute the strongly offsetting value, so that they can determine the deontic judgment when the robust value is 0, and the wellbeing of excess good lives constitute the offsetting value, so that it is never required to bring excess good lives into existence.

However, indiﬀerence between options itself can be unintuitive in certain cases. Consider the following case: the thought experiment is similar in content to previous cases.

Case 7: Very Repugnant Conclusion

Option

x y

100 100 −100 Number of People 20001 − 1

Here I deem TACL “tolerant” since the wellbeing of humdrum excess people can oﬀset that of every other group, such that in this case, choosing either option is permissible. When so much

flourishing is sacrificed and so much suffering is caused merely for the bringing into existence of people with lives barely worth living, perhaps the permissibility of option y seems unacceptable.

Some may have the intuition that a desirable theory should deem option y impermissible — that humdrum excess people should not be able to oﬀset the wellbeing diﬀerences of other people.

3.4 A Strict Asymmetric Critical Level Theory

The strict asymmetric Critical Level Theory (SACL) challenges the assignment of the wellbeing of humdrum excess people to the offset value. Instead, it deems that the wellbeing of humdrum excess people cannot offset the wellbeing in the original robust value, but it can offset that of humdrum necessary and non-excess lives, which is itself in the offsetting value. Formally:

R(x/y) = [(Tflo∩(nec∪¬exc)(x)−Tflo∩(nec∪¬exc)(y))+(Twre∩(nec∪¬exc)(x)−Twre∩(nec∪¬exc)(y))+Twre∩exc(x)

strongly offset by [(Thum∩(nec∪¬exc)(x)−Thum∩(nec∪¬exc)(y)) offset by Thum∩exc(x)]] offset by Tflo∩exc(x)

21 Like TACL, the wellbeing of humdrum necessary and non-excess people can determine the choice through the strongly offset by relation if the robust value is 0, and the wellbeing of flourishing excess people can offset the robust value. In the very repugnant conclusion, unlike TACL which allows choosing either option, SACL requires option x: R(x/y) = [(10000 − 0) + (0 − (−10000)) +

0 weakly offset by [(0 − 0) offset by 10000]] offset by 0 = 20000. Since the wellbeing of humdrum excess people can only offset the strongly offsetting value of other humdrum people and not offset the robust value, SACL is hard with respect to humdrum excess people inferior to people whose wellbeing is in the robust value. Due to the offset by relation between humdrum people (rather than the strong offset by relation), this holds even when the wellbeing of humdrum necessary and non-excess people are the same in both options (Thum∩(nec∪¬exc)(x)−Thum∩(nec∪¬exc)(y) = 0), which is the case in the very repugnant conclusion.

Note that while TACL is a soft theory with respect to excess people with good lives inferior to people whose wellbeing is in the robust value (in that the positive wellbeing of excess people can offset differences in the robust value), SACL is a soft theory with respect to flourishing excess people, but is a hard theory with respect to humdrum excess people — no amount of wellbeing of humdrum excess lives can offset differences in the robust value. Let us revisit the possible extinction, except in which the possible future only contain humdrum lives.

Case 8: Possible Extinction from Humdrum Existence

Option

x y

Person A 50 49

Many contingent people - 9

SACL require option x — that the wellbeing of many humdrum excess people cannot even offset a small loss to one necessary flourishing person. Again, intuitions may differ on this case.

4 Technical Diﬃculties

Thus far, we have considered theories that produce deontic judgments for pairwise comparisons of options. Now we consider how these theories produce judgments for option sets of more than

22 two options. But ﬁrst, whichever method we consider, it should resolve intransitivity of pairwise deontic judgments.

4.1 Intransitivity

A theory is transitive with respect to the ought relation iﬀ the fact that one ought to choose option y over x and ought to choose option z over y implies that one ought to choose option z over x.

Let’s consider a case that demonstrates that intransitivity of the Wide, Soft Theory, TACL, and

SACL. Again, assume that the critical level is 10.

Case 10: Intransitivity with Respect to the Ought Relation

Option

x y z

Person A 50 60 40

Person B − 20 50

TACL and SACL would give the same deontic judgments for this case since there are no humdrum excess people. Recall that both theories give deontic judgments in pairwise comparisons.

Between options x and y, one ought to make person A better oﬀ and choose option y over x.

Between options y and z, one ought to increase the total wellbeing of flourishing necessary people and choose option z over y. Between options x and z, although person A is worse off in option z, the wellbeing of person B is high enough to offset person A’s loss, and thus choosing either option is permissible. Thus TACL and SACL are intransitive with respect to the ought relation.

Although the Critical Level Theory is transitive with respect to the ought relation, it is intransitive with respect to the permissible relation. A theory is transitive with respect to the permissible relation iﬀ the fact that between option x and y both options are permissible and between y and z both options are permissible implies between option x and z both options are permissible.

Case 11: Intransitivity with Respect to the Permissibility Relation

Option

x y z

Person A 50 49 50

Person B 1 9 5

23 The Critical Level Theory deems that between options x and y both are permissible and between options y and z both are permissible. However, between options x and z, one ought to choose option z. Thus the Critical Level Theory is also intransitive. This same case demonstrates that TACL and SACL are intransitive with respect to the permissible relation (it can easily be shown that the

Wide, Soft Theory is intransitive with respect to the permissible relation by changing person A’s wellbeing in option y from 60 to 40 in Case 10).

There exists many methods from social choice theory to aggregate pairwise deontic judgments to judgments on the whole option set. I discuss one such plausible method that resolves intran- sitives. [Schulze, 2011] proposed the beatpath method to determine permissible options from a set of individual preferences — analogous to pairwise comparisons for our purposes. The method applied to the ought relation is as follows: option x is connected to option y if there exists a path

n−1 between them, i.e. there exists a sequence of pairs of options {(zi, zi+1)}i=1 where option x is z1 and option y is zn such that for each pair (zi, zi+1), one ought to choose zi+1 (the choice function

R(zi+1/zi) > 0). Let the strength p of the path be the minimum value of the choice function over n−1 pairs in that path: p := min{R(zi+1/zi)}i=1 . The strength of strongest path p(x, y) is the highest strength of the path(s) connecting option x to y. If x is not connected to y, then p(x, y) = 0.

Option y is beatpath better than option x iﬀ p(x, y) > p(y, x). Option x is beatpath permissible iﬀ there does not exist another option that is beatpath better than it, i.e. for any other option z, p(z, x) ≥ p(x, z).

The beatpath better relation is transitive, so for ﬁnite option sets, there always exist at least one option that is beatpath permissible (inﬁnite option sets may not have beatpath permissible options). If we equip TACL and SACL with the beatpath method in case 10 above, both option x and y are connected to option z (the path that connects option x to z contains option y), and the strengths of both paths are 10. Option z is not connected to option x or option y, and thus option z is beatpath better than options x and y. Thus option z is beatpath permissible, and since in this set it is the only beatpath permissible option, option z is required. In case 11, option x is connected to z, and no other pairs are connected. Thus, only option x is not beatpath permissible, and options y and z are both permissible.

24 4.2 Independence of Irrelevant Alternatives

The beatpath method has several weakness, one of which is especially undesirable in a deontic theory — the beatpath method does not satisfy Independence of Irrelevant Alternatives (IIA). A theory satisﬁes IIA with respect to permissibility iﬀ for any option x that is permissible but not required in option set O, then it is not the case that option x is required in option set O ∪ {z}, where option z is not in O. In other words, the addition of an option should not change the status of an option from merely permissible to required, unless the added option is ultimately required.

The addition of an irrelevant alternative can change the deontic judgment of a theory, sometimes with undesirable conclusions. Recall that we introduced distinctions made by the Asymmetry to the

Critical Level Theory to avoid requiring option y in the Critical Level Very Repugnant Conclusion

— that one ought to cause immense suffering to necessary flourishing lives in order to bring many barely flourishing lives into existence. Consider the same case with an additional option x+:

Case 12: Critical Level Very Repugnant Conclusion with Irrelevant Alternative

Option

x x+ y Person A 100 105 −100

99 People 100 100 −100

2001 People − 0 10

Note that Person A is just one person of the 100 people in the original case and that options x and y are the same as they are in the original. Recall that under both TACL and SACL, choosing either option is permissible. Between options x and x+, since person A — a flourishing necessary person — is better off in option x+ and since the 2001 excess people all live neutral lives in option x+,one ought to choose option x+. Between options x+ and y, since everyone is a necessary person and since the total wellbeing for flourishing and wretched people is higher in option y than in x+, one ought to choose option y. By the beatpath method, option y is the only beatpath permissible option, and thus one ought to choose option y — the very conclusion that TACL and SACL are meant to avoid.

Again, intuitions diﬀer . We can accept that these theories with the beatpath method don’t satisfy IIA and face the unintuitive conclusions when the option set includes an unfortunate ir-

25 relevant option. Even if the option set doesn’t contain the unfortunate irrelevant option, we may be bothered by the knowledge that such an option could exist. Perhaps instead we should accept the Critical Level Theory with the beatpath method, which does not fall prey to not satisfying

IIA with respect to permissibility, and accept the unintuitive conclusion in the Critical Level Very

Repugnant Conclusion.

Perhaps we can preserve the permissibility of option x under the beatpath method. Let the expanded beatwise method be the beatwise method with one additional rule. Let a option be beatwise admissible iﬀ in a pairwise comparison between that option and a beatwise permissible option, choosing either option is permissible. In an option set, let beatwise admissible options be permissible (in addition to the beatwise permissible options). Then, under TACL and SACL with the expanded beatwise method, both options x and y are permissible in the option set, and we achieve the desired deontic judgment.

Is it undesirable that a beatwise admissible option — an option that is beatwise worse than at least one other option — is still permissible? Only to the extent that the beatwise admissible option is permissible in the pairwise comparison with the beatwise permissible option. For instance, those who are satisﬁed with option y being permissible in the Critical Level Very Repugnant Conclusion

(under both TACL and SACL) and in the Very Repugnant Conclusion (under TACL) would also be satisﬁed with beatwise admissible options. The extended beatwise method merits further analysis in future works.

5 Conclusion

For each theory I discussed, I proposed a case in which the theory produces an potentially unintuitive conclusion, and intuitions may differ widely regarding the plausibility of each theory. Given these conflicting intuitions, and given the impossibility theorems in population axiology, Parfit’s theory X — a theory that satisfies all moral intuitions about population ethics — may very well be impossible. But by considering bizarre thought experiments and designing sometimes inelegant theories, we eliminate implausible theories and approximate the most plausible theory. In a world where the existence and wellbeing of factory farmed animals are at our whim, and where the risks of existential catastrophes increase with technological growth, progress in population ethics helps

26 us understand the value of — and our moral obligations to — beings who may or may not inhabit the longterm future.

6 Appendix

6.1 Proof of Weak Independence of Saturating Counterpart Relations for Wide, Soft Theory

Without loss of generality, let option x be the option with the larger population (if both options have the same number of people, then no contingent person is excess and every SCR would produce the same value in the choice function). I will prove by contradiction. Assume that there exists two saturating counterpart relations α and β such that SCR α requires option x and SCR β requires option y, Rα(x/y) > 0 and Rβ(x/y) < 0:

 +T α (x) T α (x) < 0 α α α α  exc exc 0 < (Tnec(x) − Tnec(y)) + (T¬exc(x) − T¬exc(y))  α α oﬀset by Texc(x) Texc(x) > 0

 +T β (x) T β (x) < 0 β β β β  exc exc 0 > (Tnec(x) − Tnec(y)) + (T¬exc(x) − T¬exc(y))  β β oﬀset by Texc(x) Texc(x) > 0

Since necessary people are the same across SCRs, their total wellbeing across SCRs is the same:

α β α β Tnec(x) = Tnec(x) and Tnec(y) = Tnec(y). Similarly, since the non-excess people in y are the same α β across SCRs, their total wellbeing across SCRs is the same: T¬exc(y) = T¬exc(y). SCRs α and β classify diﬀerent contingent people as excess and non-excess people. However, since contingent people remain contingent, the sum total wellbeing for contingent people in x is

α α β β the same across SCRs: T¬exc(x) + Texc(x) = T¬exc(x) + Texc(x). α α α α α β β β For simplicity, let T = (Tnec(x)−Tnec(y))+(T¬exc(x)−T¬exc(y)) and T = (Tnec(x)−Tnec(y))+ β β (T¬exc(x) − T¬exc(y)). Let’s consider the total wellbeing of excess people in option x under the two SCRs. There are four cases:

α β α β α α (i) Suppose Texc(x) < 0 and Texc(x) < 0. Then, since R (x/y) > 0 and R (x/y) < 0, T +Texc(x) >

27 β β T + Texc(x). This is a contradiction, since the inequality should be an equality. α β α (ii) Suppose Texc(x) > 0 and Texc(x) > 0. Then, since R (x/y) > 0 and the offsetting value, α α Texc(x) > 0, T > 0. α α α Since Texc(x) > 0, T + Texc(x) > 0. β β β β Since R (x/y) < 0 and Texc(x) > 0, from the offset relation we have min{0,T + Texc(x)} < 0, β β and thus T + Texc(x) < 0. α α β β Then, T + Texc(x) > T + Texc(x), and we arrive at the same contradiction as (i). α β (iii) Suppose Texc(x) < 0 and Texc(x) > 0. α α α α Since R (x/y) > 0 and Texc(x) < 0 , T + Texc(x) > 0. β β β β β β Since R (x/y) < 0 and Texc(x) > 0, min{0,T + Texc(x)} < 0, and thus T + Texc(x) < 0. α α β β Then, T + Texc(x) > T + Texc(x), and we arrive at a contradiction. α β α (iv) Suppose Texc(x) > 0 and Texc(x) < 0.Then, since R (x/y) > 0 and the offsetting value, α α Texc(x) > 0, T > 0. α α α Since Texc(x) > 0, T + Texc(x) > 0. β β β β Since R (x/y) < 0 and Texc(x) < 0, T + Texc(x) < 0 α α β β Then T + Texc(x) > T + Texc(x), and we arrive at a contradiction.

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