(p.558) Appendix G Book Summary
(p.558) Appendix G Book Summary
G.1 Chapter 1
In chapter 1, I presented an overview of the book and a guide to the material. I discussed the role that intuitions play in my book, and noted that I will be presenting a number of impossibility arguments throughout the book, some of which span several pages, and some of which span many chapters. I observed that I find myself in the position of someone who is juggling a number of very important or valuable items, and who realizes that he cannot hang on to them all, but is loath to let any of them drop. I suggested that others may also find themselves in my position, in reading this book.
I introduced some important terminology, including the distinction between a relation's being transitive, intransitive, and nontransitive, where saying that a relation R is nontransitive is neutral between the case where R is intransitive, and so we might say that transitivity fails for R, and the case where instead we would merely want to say that transitivity fails to apply across different sets of alternatives to which R applies, and to which we might think transitivity should apply.
I pointed out that many of this book's arguments are relevant both to judgments about the goodness of outcomes and to judgments about the goodness of individual lives. Thus, since virtually everyone accepts the intelligibility of the latter sort of judgments, I claimed that even those who are suspicious about the intelligibility of the former sort of judgments—a view that I barely understand and have little sympathy with—need to take account of this book's arguments.
I expressed my conviction that in the normative domain, as elsewhere, one should be careful not to apply tools that are useful and powerful for certain questions and domains to other questions and domains for which they are ill suited. (p.559) Likewise, I warned against being content with simple, clean, clear answers in contexts where complex, messy, and murky answers are, unfortunately, required.
I acknowledged that I am not, myself, up to solving the various problems that this book raises. But I expressed the hope that perhaps others will find my problems interesting and important, and have better success than I in addressing them. Regardless, I suggested that coming to terms with this book's arguments may require us to substantially rethink and revise our understanding of the good, moral ideals, and the nature of practical reasoning.
G.2 Chapter 2
In chapter 2, I introduced and discussed two Standard Views about aggregation that focus on how we should make trade-offs involving benefits or burdens between quality and number in assessing outcomes involving different individuals. These views offer answers to such questions as whether it would improve or worsen an outcome more if a large number of people were benefited or harmed a little, or if a small number of people were benefited or harmed a lot.
The First Standard View—Trade-offs between Quality and Number Are Sometimes Desirable involves an additive-aggregationist approach to assessing outcomes that allows trade-offs between quality and number, while the Second Standard View—Trade-offs between Quality and Number Are Sometimes Undesirable Even When Vast Numbers Are at Stake involves an anti-additive-aggregationist approach to assessing outcomes that prohibits trade-offs between quality and number. I claimed that each of the Standard Views is extremely plausible in different contexts, and that many people are firmly committed to them for making certain comparisons.
I noted that most people firmly believe that “all-things-considered better than” is a transitive relation, which means that for any three alternatives A, B, and C, if all things considered A is better than B, and all things considered B is better than C, then all things considered A is better than C. I then presented several Spectrum Arguments, each of which involves a spectrum of outcomes, such that the First Standard View applies for comparing certain, nearby outcomes along the spectrum, while the Second Standard View applies for comparing certain other, far apart outcomes along the spectrum. I then showed that, together, the rankings of outcomes generated by the Standard Views are incompatible with the view that “all-things-considered better than” (in my wide reason-implying sense) is a transitive relation. It follows that unless one can show that there couldn’t be a spectrum of cases of the sort I presented—a claim that will be very hard to defend—one will have to give up at least one of the Standard Views about aggregation or the view that “all-things-considered better than” (in my wide reason-implying sense) is a transitive relation. I noted that, for many, it will be extremely difficult to give up any of the views in question, and that giving up any of them would have serious practical and theoretical implications.
(p.560) In addressing these issues, I discussed a number of anti-additive-aggregationist examples, including, among others, Derek Parfit's Repugnant Conclusion, Thomas Scanlon's case of Jones receiving electrical shocks from a television transmitter, and my own Lollipops for Life case. I also introduced J. Ross's Principle, which I returned to throughout the book. According to J. Ross's Principle, there are cases where it would be most rational to act as if the Second Standard View was true, even if we have most reason to believe that the First Standard View is true, and vice versa. Roughly, this would be the case if there was little or no difference between two alternatives according to one of the Standard Views, but there was a big difference between two alternatives according to the other Standard View. I also noted that we should distinguish between two notions that are often conflated, the notions of incommensurability and incomparability. Finally, I discussed the view that there can be discontinuities in value, suggesting that the view is both deeply plausible and coherent, but that in some cases for which we might invoke it, it is inconsistent with other deeply held views.
G.3 Chapter 3
In chapter 3, I presented and assessed a “new” principle of aggregation, an anti-additive-aggregationist view which I called the Disperse Additional Burdens View. I noted that, like many principles, the Disperse Additional Burdens View is incomplete, applying to some alternatives but not others. I also noted that the view is a natural extension of chapter 2's Second Standard View, and that, like the Second Standard View, it has great plausibility.
Unfortunately, the Disperse Additional Burdens View faces a serious problem of iteration. Specifically, I argued that, in principle, if one repeatedly follows the Disperse Additional Burdens View in those cases where it seems both relevant and plausible, one might end up making a series of choices each of which, individually, produces the best possible available outcome, but which together produce an outcome that is clearly inferior to the outcome that one would have produced had one consistently refused to allow oneself to be guided by the Disperse Additional Burdens View in the individual cases to which it applies. I suggested that the problem of iteration facing the Disperse Additional Burdens View resists a stable solution and challenges the transitivity of the “all-things-considered better than” relation (in my wide reason-implying sense).
In discussing the Disperse Additional Burdens View, I noted that the View may help account for some of the intuitive attractiveness of the antiegalitarian's Levelling Down Objection, as well as certain prevalent attitudes toward charitable giving. I also claimed that the View may give rise to practical problems for national or international organizations in such areas as health care and famine relief. In addition, I suggested that my analysis of the Disperse Additional Burdens View and its implications challenges Derek Parfit's claim that agent-neutral theories can’t face moral analogues of Prisoner's Dilemmas, and requires us to reevaluate Parfit's claims about, and our understanding of, the nature and scope of Each-We Dilemmas.
(p.561) G.4 Chapter 4
In chapter 4, I argued that anti-additive-aggregationist reasoning applies within lives as well as between lives, and that, accordingly, most people accept a view analogous to chapter 2's Second Standard View and chapter 3's Disperse Additional Burdens View for assessing individual lives. Specifically, I argued that most people accept the Fourth Standard View—Even within Lives, Trade-offs between Quality and Duration Are Sometimes Undesirable Even When Vast Differences in Duration Are at Stake, a view that prohibits trade-offs, in certain contexts, between benefits and duration within a life.
I suggested that the source and scope of anti-additive-aggregationist views may have been obscured by two important factors: one, compensation's being possible within lives but not between lives; and two, the substantial influence on contemporary thought of John Rawls's and Robert Nozick's claims about the separateness of individuals. Although the facts in question provide some important reasons to treat certain cases involving multiple lives differently than analogous cases involving a single life, I argued that they do not ultimately support the common assumption that trade-offs are always permissible within lives, even though they are not always permissible between lives.
In addition to presenting a number of examples involving a single life which were analogues of examples from chapters 2 and 3 involving multiple lives, I also presented a number of examples showing that, in assessing lives, many people care about such factors as the shape, pattern, and direction of lives, as well as the sum total of individual units of good possessed at each moment of the lives.
I discussed an analogue of Derek Parfit's Repugnant Conclusion, which I called the Single Life Repugnant Conclusion. I noted that a variation of my Single Life case was first discussed by J. M. E. McTaggart, and considered and rejected his various arguments for the view that we should judge an extraordinarily flourishing human life lasting a million years as worse than a barely contented oyster-like life if the latter life persists long enough.
Together, chapter 4's considerations suggested that we should reject Henry Sidgwick's conception of individual self-interest, which measures the goodness of an individual life as a simple additive function of how much utility it has at each moment, treating each moment equally. Arguably, versions of Sidgwick's conception have dominated Western thought from Plato through the present.
G.5 Chapter 5
In chapter 5, I claimed that, in certain contexts, most people accept an additive-aggregationist approach for assessing trade-offs within lives. Specifically, I claimed that for certain kinds of comparisons, most people accept the Third Standard View—Trade-offs between Quality and Duration Are Sometimes Desirable, a view that allows trade-offs within lives between benefits and duration. I then showed (p.562) that analogous to the inconsistency highlighted in chapter 2, there is an inconsistency between the Third Standard View's additive-aggregationist approach within lives, the Fourth Standard View's anti-additive-aggregationist approach within lives, and the transitivity of the “all-things-considered better than” relation (in my wide reason-implying sense). At least, there is an inconsistency between those views and the plausible assumption that there could be a spectrum of cases such that the Third Standard View was plausible and appropriate for comparing nearby members of the spectrum, but the Fourth Standard View was plausible for comparing other, far apart members of the spectrum. Thus, I showed that problems of consistency arise with respect to trade-offs between quality and duration within a life, for the same reasons that problems of consistency arise with respect to trade-offs between quality and number across lives, namely, that for some such comparisons additive-aggregationist reasoning seems plausible, but for others anti-additive-aggregationist reasoning seems plausible.
I then considered a particularly forceful example, owing to Stuart Rachels,1 illuminating the power, appeal, and inconsistency of the views that most people have regarding trade-offs between quality and duration within a life. Following that, I discussed numerous objections that might be raised to my example. This discussion canvassed a host of issues, including whether there might be sharp boundaries between lives worth living and lives not worth living; whether there might be unpleasant experiences so slight that it wouldn’t matter at all how long they lasted; whether some of our firm intuitions about when trade-offs are permissible have to be limited in scope; whether we must distrust our intuitions about cases involving countless years of life; whether there may be interaction effects between certain unpleasant experiences within a life that are relevant to our judgments; whether principles of decomposition, additive aggregation, and recombination apply to the normative realm; and whether the badness of an unpleasant experience of any given duration is proportional to the length of the life in which that experience obtains. Ultimately, I argued that none of the objections discussed is telling against the worries raised by my example or the general problem of inconsistency presented in chapter 5.
I concluded that as with the case of trade-offs between lives, the issue of trade-offs within lives is deeply problematic. I noted that, almost certainly, most people will have to give up a view that they find deeply plausible to avoid inconsistency in their thinking. I also noted that doing this would not be easy, and that it is far from clear what view should be given up.
(p.563) G.6 Chapter 6
In chapter 6, I began a careful exploration of the notion of transitivity. I looked at a number of different notions that are, or are not, transitive, in order to illuminate the nature and foundation of transitive relations. This clarified the conditions that would have to obtain for the “all-things-considered better than” relation to be either transitive or nontransitive, where the notion “nontransitive” is a technical one, defined in section 1.5.
My discussion canvassed a number of topics, including whether all “…er than” relations are transitive; the connection between a property's being gradable in the linguists’ sense, the possibility of ranking objects on a single linear scale in terms of the degree to which they possess a gradable property, and transitivity; the fact that incommensurable or incomparable objects cannot be placed on a common scale; and that transitivity fails to apply in cases where different alternatives are compared in terms of different relations, and hence by appeal to different scales.
In discussing these issues, I argued that a relation, R, would be intransitive for the very same reason that transitivity fails to apply across different relations. I first noted that if aR 1 b, and bR 2 c, nothing follows about how a and c compare with respect to R 1, R 2, or in any other respect. This is because transitivity simply fails to apply to comparisons involving different relations. I then pointed out that if there were a relation, R, such that the comparison of certain alternatives with respect to R depended on how those alternatives ranked on one scale, while the comparison of other alternatives with respect to R depended on how those alternatives ranked on a different, independent scale, then R might be intransitive, and would certainly be nontransitive, for the very same reason that transitivity fails to apply to comparisons involving different relations. That is, I claimed that R could be nontransitive whenever the factors that were relevant and significant for comparing a given outcome with respect to R varied depending on the alternatives with which that outcome was compared. So, for example, if the relevance and significance of the factors for determining whether a stands in relation R to b differed from the relevance and significance of the factors for determining whether a stands in relation R to c, then R might be a nontransitive relation.
Chapter 6 included an extended discussion of the notion of “not worse than” understood as the category of “rough comparability,” “imprecise equality,” or “being on a par with.” I noted that many people accept the category of rough comparability, and that those who do, accept that “not worse than” is an intransitive relation. I then argued that once one has fully understood the conditions that would account for the possibility that some alternatives stand in the “not worse than” relation, and for the subsequent possibility that the “not worse than” relation is intransitive, one would see that those same conditions would open up the possibility that “all-things-considered better than” (in my wide reason-implying sense) might be an intransitive relation. Thus, I suggested that if one is willing to accept that there is a “not worse than” relation which is intransitive, as (p.564) many are, one should also be willing to accept that the “all-things-considered better than” relation (in my wide reason-implying sense) is intransitive.
I also pointed out that one reason that many people have resisted the notion that “all-things-considered better than” could be intransitive is because they have recognized and worried about the possibility that if that were the case, then perfectly rational people could, at least theoretically, be money pumped. That is, if “all-things-considered better than” were an intransitive relation, it could be rational, at least in theory, to pay to go around in a circle—constantly paying to move from one alternative to another available better alternative, only to end up where one started. However, I suggested that some who reject the view that “all-things-considered better than” could be intransitive for this reason nevertheless accept that “not worse than” is an intransitive relation, and I argued that the theoretical possibility of being money pumped arises in the latter situation as well as the former. So, if the theoretical possibility of being money pumped is not enough to rule out the possibility that “not worse than” is an intransitive relation, it can’t be enough to rule out the possibility that “all-things-considered better than” is an intransitive relation.
I concluded chapter 6 by granting that global and strategic reasoning may offer a practical solution to the possibility of being money pumped in either case, but I suggested that one should not conflate a practical solution with a theoretical one, in situations where a theoretical problem is posed and for which a theoretical solution is, ideally, desired.
G.7 Chapter 7
In chapter 7, I continued my exploration of transitivity. I began by considering an argument of Frances Kamm's purporting to show that the “permissible to do rather than” relation is intransitive.2 I noted that Kamm's argument may not be as worrisome as some of my other purported counterexamples to transitivity, and actually shows that the “permissible to do rather than” relation is nontransitive rather than intransitive, but that Kamm's analysis of why the “permissible to do rather than” relation seems not to be transitive is important and instructive. Kamm rightly saw that different factors are relevant to the permissibility of an action depending on what alternatives are available to that action, and that it is this feature that ultimately accounts for the fact that it can be permissible to do A rather than B, and permissible to do B rather than C, and yet it might not be permissible to do A rather than C.
I next discussed the “moral obligatoriness” relation, “ought to be done rather than,” suggesting that it, too, may be nontransitive for the same reason (p.565) that the “not worse than” and “permissible to do rather than” relations are nontransitive. Specifically, I contended that it could be the case that A ought to be done rather than B, and B ought to be done rather than C, and yet it might not be the case that A ought to be done rather than C, and that this is because the relevance and significance of the factors for determining whether I ought to do an action can vary depending on the alternatives with which it is compared.
This naturally raised the following thought. If, indeed, as I argued and many accept, there are already three significant normative relations that are nontransitive—namely, the “not worse than,” the “permissible to do rather than,” and the “ought to be done rather than” relations—on reflection, is it not plausible to believe that the fourth normative relation, “all-things-considered better than” (in my wide reason-implying sense), might also be nontransitive, despite many people's initial conviction that this couldn’t be so?
I then argued that if, as many believe, the right is at least relevant to the good—in the sense that acting rightly is itself a good-making feature of outcomes—there is good reason to suspect that the nontransitivity of the “ought to be done rather than relation” will be inherited by the “all-things-considered better than” relation. Indeed, I observed that this would be a particular instance of a significant general truth: that if an important aspect of a practical notion is nontransitive, the notion itself is likely to be nontransitive.
Chapter 7 is where I first introduced the terminology of the Internal Aspects View and the Essentially Comparative View, corresponding to two different views we might have about how to assess and compare different outcomes. Ultimately, I claimed that whether the “all-things-considered better than relation” is transitive or not is correlated with which, if either, of these two views is correct. On the Internal Aspects View, every outcome can be assessed by reference to a common scale reflecting how desirable that outcome is, and the value of an outcome is unchanging, and entirely dependent on its internal features. On this view, “all-things-considered better than” will, indeed, be a transitive relation. On the Essentially Comparative View, there is no single fact of the matter as to how valuable an outcome is; rather, there are a plurality of such facts, as different factors can be relevant and significant for assessing an outcome's value depending on the alternatives with which it is compared. On this view, “all-things-considered better than” may not be a transitive relation.
Together, the considerations of chapters 6 and 7 illuminated the nature of the conflicts presented in chapters 2 through 5, showing, among other things, that the Essentially Comparative View is implicit in our maintaining both of the differing views about additive-aggregation discussed in chapters 2 through 5. Thus, chapters 6 and 7 clarified why, ultimately, we must either reject the view that in assessing the value of a given outcome an additive-aggregationist approach can be appropriate for some comparisons, but an anti-additive-aggregationist approach can be appropriate for others, or abandon the view that the “all-things-considered better than” relation is transitive.
(p.566) G.8 Chapter 8
In chapter 8, I offered a brief characterization of Expected Utility Theory, and an analogous theory which I called Expected Value Theory. I then noted that many people reject the Completeness assumption which underlies these theories, accepting the category of rough comparability, or “not worse than,” discussed in chapter 6. I suggested that most people probably assume that it should be easy enough to modify Expected Utility Theory and Expected Value Theory to accommodate Incompleteness, but I argued that this assumption is dubious. In particular, I argued that it would be difficult or impossible to reconcile Incompleteness with all of a number of other very important assumptions of Expected Utility Theory and practical reasoning, including various Principles of Equivalence, a State-by-State Comparison Principle, a Reflection Principle, a Pareto Principle, and the Sure-Thing Principle.
I also raised questions about another assumption underlying Expected Utility Theory and Expected Value Theory, the Principle of Continuity. I noted that many people find the Principle of Continuity extremely plausible for a wide range of cases, including a set of cases that I called “easy” cases. But I also noted that many find the Principle of Continuity extremely implausible for certain other cases, that I called “extreme” cases. I suggested that rather than accept a simple “pure” view, which would insist that continuity be either accepted for all cases or rejected for all cases, many would believe that the Principle of Continuity is plausible for many, but not all, cases. But I pointed out that this seemingly reasonable “compromise” position is incompatible with the Principle of Substitution of Equivalence and the Axioms of Transitivity.
Together, my arguments in this chapter suggested that one will not be able to easily modify Expected Utility Theory and Expected Value Theory in order to capture the widely held views that the Completeness Axiom is implausible and that the Principle of Continuity is implausible for “extreme” cases. Once one makes such concessions, other important positions will also have to be given up that are central to our understanding of the theories in question and practical reasoning. Given this, I suggested, that one should be dubious of any claims to the effect that we can be confident that “all-things-considered better than” (in my wide reason-implying sense) must be a transitive relation, because it is one of various Axioms of Transitivity that underlie Expected Utility Theory, and because of the role and success that Expected Utility Theory plays in other important areas, such as game theory, decision theory, and large parts of economics.
G.9 Chapter 9
Some of the most natural and important worries that people have about my Spectrum Arguments are anticipated, and responded to, when I first develop them. However, in chapter 9, I dealt with the most serious remaining objections.
(p.567) The first objection responds to my arguments by appealing to the significance of there being different kinds of alternatives along my spectrums. According to this objection, cases at one end of one of my spectrums are different in kind from cases at the other end, so there must be some point along the spectrum where there is a break, or discontinuity, from one kind to another. It is claimed that this fact undermines one of the key premises underlying my Spectrum Arguments.
The second objection claims that my arguments are versions of the Standard Sorites Paradox, familiar examples of which purport to “prove” that hairiness is the same as baldness and that a heap of sand is the same as a grain of sand. Although there is much dispute about exactly where Sorites Paradoxes go wrong, there is no dispute that they do go wrong, and must be rejected. Hence, it is claimed that my Spectrum Arguments should also be rejected, along with the other obviously fallacious Standard Sorites Paradoxes.
The third objection suggests that my arguments rest on certain heuristics and similarity-based reasoning schemes well known to produce intransitive judgments. But psychologists have amply demonstrated that the heuristics and reasoning schemes in question can lead our intuitions astray, and, in particular, that they do so precisely in those cases where they generate intransitive judgments. Hence, it is concluded, there is good reason to reject my arguments.
Convinced that there must be something wrong with my Spectrum Arguments, many people are attracted to some version or other of these different objections. I argued that none of them succeeds. If we ultimately decide to reject my arguments, we will have to do so on grounds other than those considered in chapter 9.
G.10 Chapter 10
In chapter 10, I presented a Standard Model for Utility, which expresses one natural way of understanding and valuing utility. On this model, utility is noninstrumentally and intrinsically valuable; neutral with respect to sentient beings, places, and times; and impartial. In addition, how good an outcome is regarding utility is a function of how much total utility the outcome has. I also presented a Standard Model for Combining Ideals, which holds that there is no limit on how good an outcome can be regarding various ideals, like utility, and that how good an outcome is all things considered is a simple additive function of how good it is with respect to each ideal relevant to assessing outcomes. I suggested that the two Standard Models are intuitively plausible and implicitly accepted by many. I also noted that, together, the Standard Models capture and express our additive-aggregationist views.
I next showed that, together, the Standard Models entail Parfit's Repugnant Conclusion and, more generally, that we need another model for capturing our anti-additive-aggregationist views. I presented and explored one schema for doing this, which I called the Capped Model for Ideals. The Capped Model can share the element of the Standard Model for Combining Ideals which holds that (p.568) how good an outcome is all things considered is a simple additive function of how good it is with respect to each ideal relevant to assessing outcomes. However, on the Capped Model, there will be some cases where in comparing one outcome with another we must impose an upper limit on how good an outcome can be regarding any particular ideal, and so an upper limit on how good the alternatives can be all things considered. Thus, for example, I noted that on the Capped Model, after a point, mere increases in the amount of utility will not substantially increase the outcome's value, even regarding utility. Accordingly, such increases, however great, will not be sufficient to outweigh significant losses in other morally important respects. I showed how this kind of model would enable us to capture our intuitions about the Repugnant Conclusion, as well as our anti-additive-aggregationist views more generally.
I explored the points of agreement and disagreement between the two Standard Models and the Capped Model. I also noted a number of ways in which one might try to flesh out the details of a Capped Model. I acknowledged that the Capped Model faced numerous serious problems, and that ultimately we may need to pursue another model entirely, including one more sensitive to holistic considerations, to adequately capture our anti-additive-aggregationist views.
Among other things, my discussion raised important questions about the sense in which we should be neutral with respect to sentient beings, places, and times, casting doubt on the plausibility of standard assumptions about such issues. In addition, I argued that moral ideals cannot be fully and adequately characterized in isolation from each other, that, for certain comparisons, at least, some ideals must share certain formal or structural features. In particular, in addition to those considerations already presented—which showed that, for certain comparisons, at least, just as there may be an upper limit on how good an outcome can be regarding equality, there may also have to be an upper limit on how good an outcome can be regarding other ideals, like utility—I presented further considerations which suggested that if numbers count for utility, as most believe, they may also have to count for equality, contrary to what many have assumed.
G.11 Chapter 11
In chapters 11 and 12, I explored the key question raised by my Spectrum Arguments: whether different factors may be relevant and significant for assessing and comparing outcomes, depending on the alternatives with which those outcomes are compared. According to the Essentially Comparative View, the answer to that question is “yes”; according to the Internal Aspects View the answer to that question is “no,” as the assessment of an outcome will depend solely on the internal features of that outcome, and hence be independent of any other alternatives to which that outcome is compared.
(p.569) Chapter 11 began with an extended discussion of Derek Parfit's Mere Addition Paradox. I showed that, in presenting his paradox, Parfit implicitly relied on an Essentially Comparative View. I noted that if one accepts an Essentially Comparative View the three judgments constituting Parfit's Mere Addition Paradox will no longer be incompatible, since, on such a view, “all-things-considered better than” (in my wide reason-implying sense) will not be a transitive relation.
I then explored the Internal Aspects View, which Parfit has since come to accept. I claimed that the Internal Aspects View has great intuitive appeal and showed that it has many advantages. It entails the view that “all-things-considered better than” is a transitive relation; it licenses the Principle of Like Comparability for Equivalents and it captures a particularly plausible version of an Independence of Irrelevant Alternatives Principle, according to which for any two alternatives A and B, neither adding to, nor subtracting from, a set of alternatives whose members include A and B should affect how A and B compare with each other. I also showed how the Internal Aspects View would enable us to plausibly respond to the Mere Addition Paradox, by rejecting one or more of the three judgments constituting the paradox.
I claimed that the real paradox illuminated by the Mere Addition Paradox lies not in the inconsistency between the three particular judgments that technically produce the paradox, but in the inconsistency between the two conflicting ways of understanding ideals. The role that “Mere Addition” plays in our judgment about how A+ compares to A in Parfit's Paradox speaks to the tremendous power and appeal of the Essentially Comparative View. But the firm conviction, that many hold, that “all-things-considered better than” must be a transitive relation, speaks to the tremendous power and appeal of the Internal Aspects View. It is, I claimed, really these two conflicting views that underlie the Mere Addition Paradox, and that ultimately make it so significant and deeply problematic.
G.12 Chapter 12
In chapter 12, I explored the Essentially Comparative View. I argued that the Essentially Comparative View has great plausibility, and that, for many, it would be extremely difficult to abandon. I noted that, for many, the most plausible versions of maximin and utility are essentially comparative. I also argued that if one wants to give independent weight to the Pareto Principle, as many do, one has to accept an Essentially Comparative View. In addition, I presented an extended discussion of a Narrow Person-Affecting View, showing that such a principle is also essentially comparative. I considered numerous objections and worries that could be raised to the Narrow Person-Affecting View, but showed that, ultimately, it is extremely plausible to think that the Narrow Person-Affecting View is both relevant and significant for making certain comparisons, even if it is not the only ideal that is relevant and significant for making such comparisons.
(p.570) I noted that many people have thought that certain principles are restricted in scope, such that they are relevant and significant for comparing certain outcomes but not others. I then showed that such a view typically involves a commitment to the Essentially Comparative View. I further showed that one needed to appeal to an Essentially Comparative View to capture our judgments about certain kinds of cases, including one which I called How More Than France Exists. I also noted that appealing to an Essentially Comparative View could help us make sense out of the many instances of intransitive judgments that are made on a regular basis by people who are seemingly clear-thinking, well-informed, and perfectly rational.
Together, the considerations presented in chapters 11 and 12 support this book's main lesson. Many of us have beliefs that are deeply plausible but incompatible with each other. Moreover, the beliefs are such that giving up any of them would have grave implications for practical reasoning as we currently understand and engage in it.
G.13 Chapter 13
In chapter 13, I considered four different approaches to trying to preserve the transitivity of the “all-things-considered better than” relation in the face of this book's arguments. The first approach adopts a fine-grained individuation of the different alternatives in my examples. This approach enables one to claim that transitivity doesn’t fail in my examples, but merely fails to apply. I argued that such a move might enable one to defend transitivity as a technical feature of certain outcomes, but would leave one open to being rationally money pumped. More important, I argued that such a move would basically undermine the fundamental role that transitivity plays in practical reasoning.
The second approach adopts something akin to the “time trade-off method” often employed in the health literature in order to evaluate the different outcomes presented in my Spectrum Arguments. I readily granted that this approach would result in a transitive ranking of alternatives, but argued that this approach simply begs the question against the many arguments and considerations offered in support of the Essentially Comparative View. More important, I pointed out that the costs of such a move would, inevitably, be great, as they would involve all of the costs involved in giving up the Essentially Comparative View.
The third approach is to adopt a Sports Analogy, whereby one would accept an Essentially Comparative View for assessing outcomes, but contend that in order to rank any two outcomes one would need to compare them not only with each other, but with all of the other outcomes. I observed that this view committed one to rejecting one of the most plausible versions of the Independence of Irrelevant Alternatives Principle. More important, I argued that this approach raised a host of difficulties, especially in the moral domain, some of which were theoretical and some of which were practical. I suggested that the practical difficulties might be (p.571) insuperable, but that even if they were not, such an approach would be similar to the fine-grained approach, in that it would effectively eviscerate the role and importance of transitivity for the purposes of practical reasoning.
The fourth approach is to argue that one can preserve the transitivity of the “better than” relation if one engages in a back-and-forth process of reevaluating and revising one's views, as necessary, so as to arrive at a state of reflective equilibrium with respect to one's overall set of beliefs. I argued that such a process will only guarantee the result in question if, from the outset, one builds into the process a “consistency” requirement that the Axiom of Transitivity be met. I argued that building such a requirement into the process from the outset would beg the question against any arguments challenging the Axiom of Transitivity, and would be contrary to the spirit with which one should seek to arrive at a state of reflective equilibrium with respect to one's beliefs. I also noted that no process can avoid the costs that such a result would ensure, which is partly what this book is intended to highlight.
G.14 Chapter 14
In my conclusion, chapter 14, I began by taking stock of the main topics presented. I then listed some of the key lessons learned and in doing this indicated a number of the different combinations of views that stand or fall together. I then noted some of the work that remains to be done if one hopes to arrive at a full understanding of how to assess outcomes, suggesting, among other things, that more attention would need to be paid to the issues of holism and infinity.
I returned to a topic first broached in chapter 1 and argued that we were now in a better position to understand why, even if the “all-things-considered better than” relation is transitive, it is not so merely in virtue of the meanings of the words or the logic of “goodness,” at least, not given my wide reason-implying sense of “all-things-considered better than.”
I considered a number of possible responses to this book, including whether we should abandon considerations of goodness entirely in the practical domain, or abandon the view that one can intelligibly compare outcomes in terms of goodness, or instead opt for a monistic approach to assessing goodness like that offered by total utilitarianism. I expressed my antipathy to each of these approaches, but acknowledged that they may seem more attractive in light of my results than they did previously.
I broached the question of whether it might be appropriate to embrace incredible or even inconsistent views. Considering a host of examples drawn from science, math, and philosophy, I suggested that it might well be more rational to embrace a set of inconsistent views, at least temporarily, than to forsake any of the views merely on the basis of the inconsistency that obtains between them. My remarks implied that at this stage of our understanding, it might be more rational (p.572) to accept both the Internal Aspects View and the Essentially Comparative View in thinking about the normative realm, than to abandon either; and that similarly, for now, at least, it might be more rational to continue to appeal to additive-aggregationist reasoning, anti-additive-aggregationist reasoning, and various Axioms of Transitivity, in our practical deliberations, rather than to choose between them simply because we recognize that they are inconsistent.
I discussed the topic of moral and practical dilemmas. I argued that contrary to what has long been assumed, if this book's arguments are correct, then even consequentialist theories might face moral and practical dilemmas. Specifically, I noted that any consequentialist might face a series of finite alternatives, such that no matter what alternative he chose he would have acted wrongly according to consequentialism, as there would have been another available alternative that he might have chosen instead which would have been better than the alternative he chose. I also suggested that the kinds of moral and practical dilemmas which might arise for consequentialist theories would also arise for any moral theory which granted that in some contexts, at least, one would be acting wrongly if one chose a worse outcome over another available outcome that was better than it. I further suggested that, at least prior to this book's results, we would have thought that all plausible moral theories would have made such a claim, and hence would face moral and practical dilemmas.
I next raised the question of whether this book supports a form of skepticism. I acknowledged that it might, insofar as it seems to call into question the possibility of plausibly ranking outcomes in terms of goodness. But I noted that this form of skepticism is very different from other familiar forms of skepticism. My view doesn’t challenge the reality of moral facts; it questions whether the moral facts that exist permit a very important kind of judgment that we previously thought we could make. This raises a serious crisis regarding how we ought to make choices in the normative domain, but it isn’t a version of nihilism.
I ended my conclusion with some final remarks, regarding the position I find myself in with respect to this book's claims. I noted that I remain very perplexed, recognizing that many of the views I care most deeply about are inconsistent, but not seeing how to give any of them up. I also noted that, on my view of the normative realm, it isn’t up to me to just choose which views to keep and which to give up. I suggested that pessimists may claim that this book forces us to seriously rethink and revise our understanding of the good, moral ideals, and the nature of practical reasoning. And that optimists, like me, may agree.
In appendix A, I showed that the same sort of considerations that apply to trade-offs between quality and number in cases involving different people also apply to trade-offs between duration and number in cases involving different people.
(p.573) In appendix B, I explored the relations between quantity, quality, duration, and number. I noted that there is a sense of quantity, which I employ, according to which the quantity of pleasure or pain in an outcome includes considerations of the quality, duration, and number of pains in that outcome. So, at least as I understand the notion, quantity is not in fact distinct from quality, as is often supposed. I also showed that the lines between the notions of quality, duration, and number are blurred, though one can’t simply dispense with one or two of the notions in favor of the other two or one.
In appendix C, I noted how, given certain assumptions, one could revise the standard Sorites Paradox of the Heap along the same lines that I revised the standard Hairiness/Baldness Sorites Paradox in chapter 9, so as to convert it into the sort of Spectrum Argument that I presented in chapters 1 through 5.
In appendix D, I considered and rejected three further objections to my Spectrum Arguments. The first objection appeals to claims about vagueness and indeterminacy, but I show that it amounts to a version of chapter 9's “different kinds” objection, and should be rejected for the same reason already given against that objection in chapter 9. The second objection claims that my Spectrum Arguments are a version of Zeno's Paradox, and fail for the same reason that Zeno's Paradox fails. I argued that this is not the case, and is based on a misinterpretation of my argument. I pointed out that this objection, which was originally lodged against an early formulation of my argument, doesn’t apply to this book's arguments, and that if I had spelled out my original premises more carefully, as I now do, this misinterpretation would not have arisen. The third objection claims that we are, in fact, able to find a uniquely best alternative along my spectrums. I pointed out that this solution dubiously favors one particular ordering of the spectrum over alternative orderings that might be equally legitimate. I further argued that the position in question faces many of the same problems facing the Sports Analogy, discussed in chapter 13, and that as such it faces a wide array of serious difficulties that have significant implications for practical reasoning. Finally, I pointed out that the proposed solution to my Spectrum Arguments has no traction against my other worries, raised in chapters 11 and 12, regarding the transitivity of the “all-things-considered better than” relation (in my wide reason-implying sense).
In appendix E, I considered an argument for the claim that the Narrow Person-Affecting View should be restricted in scope so as to apply only to alternatives involving the very same people. I pointed out that this argument is based on a misleading example that cannot support the conclusion in question. It is true, I admitted, that the Narrow Person-Affecting View can render deeply implausible judgments in alternatives involving large populations where there is only a single person in common in the two populations. But this hardly shows, nor, I argued, is it plausible to believe, that the Narrow Person-Affecting View should be abandoned entirely in comparing any two populations where some people are the same and some people are not.
(p.574) In appendix F, I considered whether one might appeal to lexical priority to preserve the Axiom of Transitivity in the face of this book's arguments. I showed that the initial reason for thinking that this might be possible is mistaken and unhelpful. I further expressed my doubts that appealing to lexical priority would, in fact, be of any relevance to the arguments I have presented, or would help to avoid the worries I have raised.
Appendix G summarizes the book, but of course, if you are reading this line you already know that! Finally, I am pleased to add that I have nothing more to add.
(1) Rachels's original example appeared in his unpublished Philosophy, Politics, and Economics thesis, “A Theory of Beneficence” (Oxford University, 1993). Rachels later modified his example, as well as offered other arguments for intransitivity in “Counterexamples to the Transitivity of Better Than,” Australian Journal of Philosophy 76 (1998): 71–83.
(2) Kamm's example appeared in “Supererogation and Obligation,” Journal of Philosophy 82 (1985): 118–38.