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Is there some kind and amount of badness such that an outcome that contains it is overall bad, regardless of the amount of good in the outcome?

Priority: 6/10


Lexicality among goods can be phrased as follows:

Some or any amount of good A is better than any amount of good B.1

For example, W. D. Ross stated that “no amount of pleasure is equal to any amount of virtue.”2 That is, roughly, that it is better for someone to have virtue than any amount of pleasure in her life. Similar views have been proposed by other philosophers at least since the 18th century.3

Similarly, lexicality among bads have also been discussed, which can be phrased as follows:

Some or any amount of bad A is worse than any amount of bad B.

Several philosophers have defended such kinds of lexicality.4 For example, Stuart Rachels has said that 1 year of excruciating agony is worse than 1050 years of mild pain.5

The most interesting kind of lexicality appears to be not lexicality among goods or among bads, but rather, claims of the kind that there is a

lexicality between good and bad: an outcome with some or any amount of bad A is overall bad even if the outcome has any amount of good.

Some theories of value imply this kind of lexicality in the sense that any amount of bad makes the world worse than (or not as good as) an empty world. For example, the view that only the satisfaction or frustration of preferences have value or disvalue, combined with antifrustrationism.6 Christoph Fehige, who proposed antifrustrationism, writes that one of his proposed views entails that “nothing can be better than an empty world (a world without preferences, that is).”7 One way to tackle whether there is a lexicality between good and bad is to focus on whether value theories such as Fehige’s are correct. But here we will not take that route, we will instead assume for the purpose of this research topic that there may be states of the world in which the world is better than an empty world.

For this research topic, we focus on the question: If the world can be better that an empty world because it has some good in it, is there some amount of badness such that a world with such badness is worse than an empty world, regardless of the amount of good in the world?8

This question has seemingly been discussed only little in the philosophical literature, most discussion seems to focus on lexicality with respect to good A vs. good B or bad A vs. bad B, but not bad A vs. good B. There are authors who have made claims resembling lexicality between good and bad, but most have not discussed it extensively. For example, in the context of a happiness index, Bengt Brülde says that

there may well be sufferings that are so intense that no trade-offs are possible, neither in the intrapersonal nor the interpersonal case.

The following statement was made by Swedish philosopher Ingemar Hedenius in 1964:

The worst in life, the fate of the completely unhappy, the uninterrupted, infernalistic suffering, the hopeless humiliation, a child who is slowly tormented to death—I cannot see that all beauty in the world or even the most exceptional thoughts can “counterbalance” such, and neither that other humans’ happiness or culture can. (Our translation)9

Jamie Mayerfeld considers two outcomes: (n) in which one person has a lifetime of torment, and many others have a lifetime of extreme bliss. And (1) in which all have a lifetime close to the hedonistic zero. He says about the outcomes that

Like William James, I find the conclusion that (n) is better than 1 unacceptable. The lifelong bliss of many people, no matter how many, cannot justify our allowing the lifelong torture of one.10

A critical discussion of lexicality between good and bad can be found in Toby Ord’s online essay “Why I'm Not a Negative Utilitarian” in the section on Lexical Threshold Negative Utilitarianism. On the other hand, Clark Wolf speaks favorably of lexicality and defends what he calls ‘negative critical level utilitarianism’ (NCLU), according to which “population choices should be guided by an aim to minimize suffering and deprivation.”11

Research topics

How is lexicality between good and bad different from lexicality among goods and among bads?

Which of the most important aspects of lexicality among goods and among bads translate simply to lexicality between good and bad? What is unique about lexicality between good and bad compared to the other two? What about lexicality between good and bad makes it easier or more difficult to defend compared to the other two?

Output: Online essay or an article in a philosophy journal.

Specifying concrete sequences

This is a research topic that we suggest if it is relevant to lexicality between good and bad, which remains to be investigated (see the previous research topic). A common type of argument that is used in several of the debates about lexicality and whether value relations such as ‘all things considered better than’ are transitive has been called sequence or spectrum arguments. One version in terms of ‘worse than’ goes as follows. Assume that a state A, for example some amount of intense suffering, is claimed to be worse than any amount (or an arbitrarily large finite amount) of minor pains, a state we can call Z. The argument says that there is a state B with slightly less severe harms but with a larger amount of them, for example with more individuals experiencing them or experiencing them for a longer period of time, such that B is worse than A. Similarly, there is a state C with slightly less severe harms than in B but a larger amount of them such that C is worse than B. And so on until we reach a state Z with very large amounts of minor harms that is worse than Y. By transitivity of ‘all things considered worse than,’ Z is worse than A. But this result that Z is worse than A contradicts the starting position which was that A is worse than Z. Such a sequence can, for example, be taken to support that the badness in A is not lexically worse than the badness in Z, or that ‘all things considered worse than’ is not transitive.

Alastair Norcross uses a sequence argument to argue that “there is some finite number of headaches, such that it is permissible to kill an innocent person to avoid them.“12 Erik Carlson objects that Norcross does not actually specify a sequence. Carlson says,

To convince the skeptic, the proponent of the Sequence Argument has to do better than merely pointing to the prima facie plausibility of there being a sequence of the kind his argument relies on. He must, it seems, actually specify such a sequence…. Until [such a sequence] is actually produced... [we] need not be much worried by the Sequence Argument.13

He highlights that, upon closer inspection, we may be unable to specify a sequence, or there may be specified sequences that are intuitive that run in both directions.14

Although Carlson’s argument is a reply to Norcross in particular, it seems to apply to other sequence arguments, including those presented by Larry Temkin and Stuart Rachels.15 That is, a weakness in these sequence arguments may be that they, to varying degrees, only point to the plausibility that there exists a sequence of the right kind, but they do not flesh out the sequences. The research topic would be to build on Carlson’s ideas and specify such sequences, and to figure out what can be concluded from such an exercise.

Output: Philosophy undergraduate or master’s thesis, or an article in a philosophy journal.

Reply to Toby Ord’s objection to Lexical Threshold Negative Utilitarianism

Toby Ord says that Lexical Threshold Negative Utilitarianism involves a “very strange discontinuity.”

If you believe in Lexical Threshold NU, i.e. that there are amounts of suffering that cannot be outweighed by any amount of happiness, then you have to believe in a very strange discontinuity in suffering or happiness. You have to believe either that there are two very similar levels of intensity of suffering such that the slightly more intense suffering is infinitely worse, or that there is a number of pinpricks (greater than or equal to one) such that adding another pinprick makes things infinitely worse, or that there is a tiny amount of value such that there is no amount of happiness which could improve the world by that level of value.

The argument goes like this. We can imagine a long sequence of levels of intensity of suffering from extreme agony down to a pinprick, each of which differs from the one before it by a barely detectable amount. If we want to avoid the discontinuity in the badness of the intensity of suffering, then the suffering of a million people at the extreme agony intensity level must be only finitely many times as bad as the suffering of a million people at the next intensity level down, which is only finitely many times as bad as the suffering at the next level, and so on all the way down to the level of the pinprick. This implies that suffering at the very high intensity level is only finitely times worse than the suffering at the pinprick level. [Figure.]

Moreover unless, there is to be a case where n+1 people getting a pinprick is infinitely worse than n people getting a pinprick (for n greater than or equal to 1), we can run a similar argument moving from a million people receiving a pinprick down to a single person and show that the former must only be finitely many times worse than the latter. We thus get the conclusion that while a million people in agony is terrible, it is still only finitely times as bad as a single person receiving a pinprick.16

This kind of spectrum, sequence, or step-by-step argument that Ord uses has been discussed to a fair extent and there are several possible replies to it, as well as counterarguments to the replies.17 A reply to Ord's version could describe the possible replies and the challenges with the replies, and it would likely conclude that lexicality in this case is not clearly “very strange,” at least not in a way that is especially problematic.

Output: Online essay.

More research topics regarding lexicality

Feel free to contact us for more research ideas regarding lexicality.

Related topics

Related to lexicality are topics such as vagueness, completeness, discontinuity, extensive structures (in measurement), Archimedeanness and non-Archimedean measurement, infinities, and transitivity of value relations such as ‘all things considered better than.’

Related works

  • Arrhenius, Gustaf. “Superiority in Value.” Philosophical Studies 123 (2005): 97–114.
  • Arrhenius, Gustaf, and Wlodek Rabinowicz. “Millian Superiorities.” Utilitas 17 (2005): 127–46.
  • ———. “Value Superiority.” In The Oxford Handbook of Value Theory, edited by Iwao Hirose and Jonas Olson, 225–48. Oxford: Oxford University Press, 2015.
  • Carlson, Erik. “Aggregating Harms – Should We Kill to Avoid Headaches?” Theoria 66 (2000): 246–55.
  • ———. “Generalized Extensive Measurement for Lexicographic Orders.”
    Journal of Mathematical Psychology 54 2010: 345–51.
  • ———. “Non-Archimedean Extensive Measurement with Incomparability.” Mathematical Social Sciences 62 (2011): 71–76.
  • ———.  “Organic Unities, Non-Trade-Off, and the Additivity of Intrinsic Value.” Journal of Ethics 5 (2001): 335–60.
  • ———. “Value Theory (Axiology).” In Handbook of Formal Philosophy, edited by Sven Ove Hansson and Vincent Hendricks. Springer, forthcoming.
  • Dorsey, Dale. “Headaches, Lives and Value.” Utilitas 21 (2009): 36-58.
  • Feit, Neil. “The Structure of Higher Goods.” The Southern Journal of Philosophy 39 (2001): 47–57.
  • Klocksiem, Justin. “How to Accept the Transitivity of Better Than.” Philosophical Studies 173 (2016): 1309–1334.
  • Norcross, Alastair. “Comparing Harms: Headaches and Human Lives.” Philosophy and Public Affairs 26 (1997): 135–67.
  • ———. “Great Harms from Small Benefits Grow: How Death Can Be Outweighed by Headaches.” Analysis 58 (1998): 152–58.
  • ———. “Two Dogmas of Deontology: Aggregation, Rights, and the Separateness of Persons.” Social Philosophy and Policy 26 (2009): 76–95.
  • Ord, Toby. “Why I'm Not a Negative Utilitarian.” (accessed July 16, 2015).
  • Quinn, Warren E. “The Puzzle of the Self-Torturer.” Philosophical Studies 59 (1990): 79–90.
  • Rabinowicz, Wlodek. “Ryberg's Doubts about Higher and Lower Pleasures – Put to Rest?” Ethical Theory and Moral Practice 6 (2003): 231–35.
  • Ryberg, Jesper. “Higher and Lower Pleasures–Doubts on Justification.” Ethical Theory and Moral Practice 5 (2002): 415–29.
  • Wolf, Clark. “Social Choice and Normative Population Theory: A Person Affecting Solution to Parfit’s Mere Addition Paradox.” Philosophical Studies 81 (1996):  263–82.


  1. This formulation draws on Arrhenius, “Superiority in Value,” 97. The claim can be stated either as ‘some amount of good A,’ which corresponds to what Arrhenius calls ‘Weak Superiority,’ or it can be stated as ‘any amount of good A,’ which corresponds to Arrhenius’s ‘Strong Superiority.’  (back)
  2. Ross, The Right and the Good, 150. Quoted in Arrhenius, “Superiority in Value,” 97.  (back)
  3. According to Arrhenius, “Superiority in Value,” “similar views have been proposed by, among others, Roger Crisp, Jonathan Glover, James Griffin, Rem Edwards, Noah Lemos, Derek Parfit, and John Skorupski” (97). See also Feit, “The Structure of Higher Goods.”  (back)
  4. For example, Rachels, “Counterexamples to the Transitivity of Better Than;” Carlson, “Aggregating Harms – Should We Kill to Avoid Headaches?” and Temkin, “A Continuum Argument for Intransitivity.”  (back)
  5. Rachels, “Counterexamples to the Transitivity of Better Than,” 73.  (back)
  6. Antifrustrationism is the view that a frustrated preference is bad, but that the existence of a satisfied preference is not better than if the preference didn’t exist in the first place.  (back)
  7. Fehige, Christoph. “A Pareto Principle for Possible People.” In Preferences, edited by Christoph Fehige and Ulla Wessels, 508–43. Berlin: Walter de Gruyter, 1998. Page 521.  (back)
  8. That is, lexicality in terms of that there is ‘some’ amount of badness that makes the outcome overall bad, rather than that ‘any’ amount of the badness makes the outcome overall bad. This formulation in terms of ‘some’ is related to what Arrhenius calls Weak Superiority and what Carlson calls Weak Lexicality in Arrhenius “Superiority in Value” and Carlson “Aggregating Harms.”  (back)
  9. Hedenius, Ingemar. Livets Mening. Stockholm: Aldus, 1964. Page 24.  (back)
  10. Mayerfeld, Jamie. Suffering and Moral Responsibility, New York: Oxford University Press, 1999. Page 178.  (back)
  11. Wolf, “Social Choice and Normative Population Theory,” 273.  (back)
  12. Norcross, “Comparing Harms,” 159.  (back)
  13. Carlson, “Aggregating Harms,” 251–52.  (back)
  14. Ibid. Note that one would treat the sequence arguments differently depending what one takes them to show (non-transitivity or the lack of lexicality), and they would be open to different counterarguments. For example, that there would be sequences in both directions do not affect an argument for non-transitivity because such an argument needs only to show that there is one sequence that implies non-transitivity. Many thanks to Jacob Nebel for making this point in conversation.  (back)
  15. See for example, Rachels, “Counterexamples to the Transitivity of Better Than” and Temkin, “A Continuum Argument for Intransitivity” and Rethinking the Good, section 5.3.  (back)
  16. Ord, “Why I'm Not a Negative Utilitarian.”  (back)
  17. For example, see Ryberg, “Higher and Lower Pleasures”; Rabinowicz, “Ryberg's Doubts about Higher and Lower Pleasures”; Arrhenius and Rabinowicz, “Millian Superiorities” and “Value Superiority”; Arrhenius, “Superiority in Value”; Carlson, “Aggregating Harms” and “Organic Unities”; Feit, “The Structure of Higher Goods”; Norcross, “Comparing Harms” and “Two Dogmas of Deontology”; and Temkin, Rethinking the Good.  (back)

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