Value Lexicality

First written: 22 Nov. 2016; Last update: 22 Nov. 2016


Many philosophers have endorsed the idea that some goods are lexically better than others, in the sense that some goods are better than any amount of some other goods. For example, it has been claimed that virtue is better than any amount of pleasure. Several philosophers have defended analogous lexicality between bads, which I find plausible. For instance, that torture is worse than any amount of mild discomfort. Some philosophers have advocated lexicality of bads over goods, which is the idea that some bads, such as some forms of torture, cannot be counterbalanced by goods. This is also plausible in my view, and it is important partly because it is a way to end up with a suffering-focused ethic.

These kinds of value lexicality are often described simplistically, and with a lack of appreciation for the nuance and depth of the work in this field. This essay is an introduction to value lexicality and the related philosophical literature. Key topics that this essay explains include strong versus weak lexicality, value aggregation, views on large numbers, sequence arguments and different accounts of the structure of lexicality.

Lexicality between goods, between bads or of bads over goods

I use the phrase “value lexicality” broadly, and synonymously with “value superiority.”1

Lexicality between goods

Lexicality between goods is the idea that some goods are superior to others, such that any or some amount of a good g is better than any amount of another good g′.2 For example, Ross (1930, p. 150) held that virtue is better than any amount of pleasure. Lexicality between goods has been endorsed by philosophers at least since Hutcheson in the 18th century,3 perhaps even by Aristotle,4 and more lately by Lemos (1993), Carlson (2015), Rabinowicz (Arrhenius & Rabinowicz, 2015) and Parfit (2016). These later endorsements tend to not refer to goods like virtue, but rather, for example, to that even if pleasure is good, some number of rich lives with pleasure, purpose, knowledge, freedom and so on is better than any number of lives with only, say, minor simple pleasures.

Lexicality between bads

Lexicality between bads is that any or some amount of a bad b is worse than any amount of another bad b′. Examples that have been defended are that torture, premature death or pain is lexically worse than (i.e. worse than any amount of) mild headaches, annoyance or discomfort. Supporters of such lexicality include Temkin (1996), Rachels (1998), Carlson (2000), Dorsey (2009) and Klocksiem (2016). I find this kind of lexicality plausible, especially the idea that some of the worst forms of torture are worse than any amount of mild discomfort.

Lexicality of bads over goods

Some philosophers have defended lexicality of bads over goods, which I also find plausible. The idea is that some bads, such as torture, cannot be counterbalanced by any amount of the goods in question. We can distinguish between a restricted and an unrestricted version of this idea. The restricted version is that some bads are lexically worse than some, but not all, goods. For example, even if some suffering cannot be counterbalanced by happiness, perhaps it can be counterbalanced by some other good. The unrestricted version, which is still plausible in my view, is that some bads are lexically worse than all kinds of goods; in other words, that the bads cannot be counterbalanced by any goods.

Mayerfeld (1999, p. 178) endorses at least the restricted version. He holds that the lifelong torture of one person cannot be counterbalanced by the lifelong bliss of others, regardless of how many would experience the bliss. In that passage, he only considers one kind of bad, suffering, and one kind of good, bliss, and I don’t know whether he would be open to that some goods other than bliss could counterbalance the suffering. Similarly, Brülde (2010, p. 577) says that there may well be sufferings that cannot be counterbalanced by happiness. Schopenhauer seems to have held the unrestricted version.5 More recently, the unrestricted version has been endorsed by, among others, Pearce (1995, sec. 2.7 Why Be Negative?) and Hedenius (1955, pp. 100–102).6

Strong and weak lexicality

The distinction between strong and weak lexicality is important. I will state the distinction in terms of bads, but one can state analogous formulations for lexicality between goods, or between bads and goods.

Strong lexicality (between bads) is roughly that any amount of a bad b, however small, is worse than any amount of another bad b′, however large.

Weak lexicality (between bads) is roughly that there is an amount of a bad b that is worse than any amount of another bad b′, however large.7

The amount of b in weak lexicality could be required to be very big, such as all the horrors occurring in all war zones and torture chambers in the world. The following is an illustration of how a form of lexicality between bads and goods could be formulated:

Weak lexicality (of bads over goods) is roughly that there is an amount of a bad b that is worse than any amount of a good g is good, so that a whole consisting of the amount of b and the amount of g is bad.8

There are additional weaker forms of lexicality.9

Value aggregation

Some think that value can be aggregated into a sum (I’m doubtful), in which case the question can arise how value could be aggregated if there is lexicality. A standard idea of aggregation without lexicality would be that values can be represented by real numbers, such as 1 and −0.5, and then simply added together. Lexicality need not hinder such aggregation. For example, a form of lexicality, suggested by Carlson (2000) and Rabinowicz (2003), is that some goods or bads have diminishing marginal value, such that as there is more of them, their value approaches some limit, say 100. This form of lexicality is compatible with simply representing values with real numbers than can be added together.

lexical-vectorFor some other kinds of lexicality, one can aggregate value using vector addition and subtraction. The following is a simplified example:10 Assume that some bad b is strongly lexically worse than another bad b′. If we have one scenario with 2 instances of b and 4 instances of b′, we could represent that as (−2, −4). Similarly, if we, in another scenario, have 1 instance of b and 5 of b′, that would be (−1, −5). By comparing the first elements in the two vectors, we can conclude that the first scenario (−2, −4) is worse than the second (−1, −5), because −2 is lower than −1. If we would have had the same number of b, say 1, in the two scenarios, we could write that as (−1, −5) and (−1, −4), we would then need to compare the second elements to find out which scenario is worse. We would conclude that (−1, −5) is worse than (−1, −4), since −5 is lower than −4. Assuming value is additive, we could also add such vectors together, and conclude that the aggregate value of, for example, (−2, −4) and (−1, −5) is (−3, −9).

I doubt that values can be added and subtracted into sums, at least not across individuals, regardless of whether there is lexicality, but my point is that if one believes in value aggregation by addition and subtraction, lexicality need not be a problem for it.

Views on large numbers

Value lexicality ties into questions about very large amounts, durations or numbers of affected individuals. For example, if there is lexicality such that torture cannot be counterbalanced by any amount of happiness, that holds even if we are talking about trillions of happy individuals existing during millions of years. One can question whether our views or intuitions about such large numbers should be relied upon. There has been a debate about this for at least the last couple of decades. Temkin (1996) and Rachels (1998), who favor lexicality, bring up and try to rebut that objection. On the other hand, Norcross (1997, p. 147) argues that our views are unreliable, as do Broome (2004, pp. 56–58) and Voorhoeve (2013). But Parfit (2016, p. 111), who defends lexicality between goods, says that “even if we cannot adequately imagine very large numbers, we understand the belief that there is no number of people whose existence would be better” than another population.

Sequence arguments

A type of argument against lexicality is a sequence argument, also known as a spectrum argument or continuum argument. Arrhenius (2005) is a nice publication presenting such an argument against lexicality. His argument is directed against both strong and weak lexicality among goods. To repeat, strong lexicality is roughly that any amount of a good g is better than any amount of another good g′, and weak lexicality is roughly that some amount of g is better than any amount of g′. Since the sequence argument applies, with the appropriate changes, also to bads, I will rephrase it in terms of bads.

Simplified, the argument goes as follows: Assume a finite sequence of bads b1, …, bn in which b1 is worse than b2, which is worse than b3, …, which is worse than bn. One can picture that b1 is something really bad, such as torture or getting one’s life ruined, and that bn is some minor bad, such as mild discomfort. Any such sequence in which b1 is strongly or weakly lexically worse than bn must contain an element bi-1 that is weakly lexically worse than the element bi that immediately follows it. The reason is that there must exist some last element bi-1 that b1 is not weakly lexically worse than, while b1 is weakly lexically worse than the following element bi. So there is some amount of b1 that is worse than any amount of bi, but since b1 is not weakly lexically worse than bi-1, there is some amount of bi-1 that is at least as bad as the amount of b1. Since the amount of b1 is worse than any amount of bi, and since the amount of bi-1 is at least as bad as the amount of b1, the amount of bi-1 is worse than any amount of bi. In other words, bi-1 is weakly lexically worse than bi. If the sequence is chosen such that each element (b1, b2, b3 and so on) is only marginally better than the preceding element, it is implausible, the argument goes, that bi-1 would be weakly lexically worse than the only marginally better bi, and hence, in such cases, strong and weak lexicality is implausible.

There are several replies to this argument. First, whether weak lexicality between adjacent elements is plausible is open to debate. Arrhenius finds it implausible, while Rabinowicz is not worried about it.11 Second, one could plausibly argue that the sequence argument needs to be reformulated so that it only implies ‘minimal’ lexicality between adjacent elements, which is weaker and easier to accept than weak lexicality.12 Third, Carlson (2015) argues that the sequence argument’s assumption that the value difference between adjacent elements is marginal begs the question against lexicality. I would reply, on behalf of Arrhenius, that one should then see if the sequence argument can be reformulated in terms of marginal factual steps so that it becomes non-question-begging, but I don’t think that it would save his sequence argument. Fourth, one could argue, as Qizilbash (2005) seems to do, that lexicality need not set in between adjacent elements because of the vagueness of categorical evaluations such as “serious” and “non-serious” illness. He seems to say that although serious illness is plausibly weakly lexically worse than non-serious illness, since “serious” and “non-serious” are vague, no illness need to be lexically worse than a slightly milder illness.

The structure of lexicality

Attempts have been made to spell out the structure of lexicality to explain why there would be lexicality and how a theory of value would look if there is lexicality. I have mentioned some of these already, but here I mention more and elaborate. Temkin and Rachels hold that lexicality sets in when there is a sufficiently large difference between the goods or bads in question.13 Klocksiem (2016) defends a threshold lexicality, according to which lexicality between bads sets in at an absolute level of intensity between discomfort and pain. He says that

the phenomenology of pleasure and pain is consistent with threshold lexicality and helps to explain how it occurs. We regard certain negative experiences, such as the negative experiences involved with hangnails or mosquito bites, as mere “discomfort”; whereas we describe others, such as the negative experience involved with an improperly yanked-out hangnail, or a skinned knee, or a broken bone, as genuine “pain.” Discomforts are negative experiences, and are intrinsically bad, but they are bad in a trivial and insignificant way. It would be better in itself to endure an arbitrarily long episode of discomfort than to undergo a sufficiently long-lasting episode of genuine pain.

Dorsey (2009) explains lexicality in terms of an account of well-being that involves the importance of life projects. As mentioned, Rabinowicz (2003) and Carlson (2000) refer to diminishing marginal value of different goods or bads, and Qizilbash (2005) points to the vagueness of categorical evaluations such as “serious” and “non-serious” illness. Lemos (1993) relies on organic unities—that is, that the value of a whole need not be the same as the value of its parts. Carlson (2015) points to a holistic non-Archimedean objective list theory of welfare when defending lexicality among goods. According to his proposed theory

pleasure, knowledge, friendship, freedom, appreciation of beauty, development of one’s talents, purposeful activity, and so on, are positive welfare components. It seems plausible to claim that for a life to have very high welfare, it must contain all or most of these things, to some degree. A life containing just one or two types of welfare components will likely be of an impoverished kind, having at best moderately high welfare. Nevertheless, it appears that an impoverished life can always be slightly improved, by adding more welfare components of the type or types already included in the life. However, such additions can never result in a life with very high welfare. Any finite series of improvements that takes us from an impoverished life to a life with very high welfare must include at least one step at which a welfare component of a new type is added. Such a step, arguably, means an improvement that cannot be arbitrarily slight (p. 20).

Further reading

The best overview of value lexicality is Arrhenius and Rabinowicz (2015), but it is behind a paywall. Instead, I recommend Arrhenius (2005) and Arrhenius and Rabinowicz (2003), which are both ungated, and similar to the gated text. For objections to lexicality, see, among others, Norcross (1997, 1998, 2009), Ryberg (2002), Arrhenius (2005), Voorhoeve (2008) and Klocksiem (2016, sec. 6). A defense of lexicality not already mentioned is Ridge (1998), which replies to an example against lexicality by Norcross (1997).


  • Arrhenius, G. (2005). Superiority in Value. Philosophical Studies, 123(1–2), 97–114. Ungated.
  • Arrhenius, G., & Rabinowicz, W. (2003). On Millian Discontinuities. Patterns of Value - Essays on Formal Axiology and Value Analysis, 1, 1–8. Ungated.
  • Arrhenius, G., & Rabinowicz, W. (2005). Millian Superiorities. Utilitas, 17(2), 127–146.
  • Arrhenius, G., & Rabinowicz, W. (2015). Value Superiority. In I. Hirose & J. Olson (Eds.), The Oxford Handbook of Value Theory (pp. 225–248). Oxford: Oxford University Press.
  • Broome, J. (2004). Weighing lives. Oxford: Oxford University Press.
  • Brülde, B. (2010). Happiness, morality, and politics. Journal of Happiness Studies, 11(5), 567–583.
  • Carlson, E. (2000). Aggregating harms - Should we kill to avoid headaches? Theoria, 66(3), 246–255.
  • Carlson, E. (2007). Higher Values and Non-Archimedean Additivity. Theoria, 73(1), 3–27.
  • Carlson, E. (2010). Generalized extensive measurement for lexicographic orders. Journal of Mathematical Psychology, 54(4), 345–351.
  • Carlson, E. (2015, January 30). On some impossibility theorems in population ethics. Draft. Ungated.
  • Dorsey, D. (2009). Headaches, lives and value. Utilitas, 21(1), 36–58.
  • Hausner, M. (1954). Multidimensional utilities. In R. M. Thrall, C. H. Coombs, & R. L. Davis (Eds.), Decision processes (pp. 167–180). New York: John Wiley.
  • Hedenius, I. (1955). Fyra dygder. Stockholm: Albert Bonniers Förlag.
  • Hick, J. (1973). Evil and the infinite future good. In P. Edwards & P. Arthur (Eds.), A modern introduction to philosophy. New York: Free Press.
  • Hutcheson, F. (1968). A system of moral philosophy (Vols. 1–1). New York: Augustus M Kelley.
  • Klocksiem, J. (2016). How to Accept the Transitivity of Better Than. Philosophical Studies, 173(5), 1309–1334.
  • Lemos, N. M. (1993). Higher Goods and the Myth of Tithonus. Journal of Philosophy, 60(9), 482–496.
  • Mayerfeld, J. (1999). Suffering and moral responsibility. New York: Oxford University Press.
  • Norcross, A. (1997). Comparing Harms: Headaches and Human Lives. Philosophy and Public Affairs, 26(2), 135–167.
  • Norcross, A. (1998). Great Harms From Small Benefits Grow: How Death Can Be Outweighed by Headaches. Analysis, 58(2), 152–158.
  • Norcross, A. (2009). Two Dogmas of Deontology: Aggregation, Rights, and the Separateness of Persons. Social Philosophy and Policy, 26(1), 76–95. Ungated.
  • Parfit, D. (2016). Can We Avoid the Repugnant Conclusion? Theoria, 82(2), 110–127.
  • Pearce, D. (1995). The hedonistic imperative.
  • Qizilbash, M. (2005). Transitivity and Vagueness. Economics and Philosophy, 21(1), 109–131.
  • Rabinowicz, W. (2003). Ryberg’s Doubts About Higher and Lower Pleasures: Put to Rest? Ethical Theory and Moral Practice, 6(2), 231–237.
  • Rachels, S. (1998). Counterexamples to the Transitivity of Better Than. Australasian Journal of Philosophy, 76(1), 71–83.
  • Ridge, M. (1998). How to Avoid Being Driven to Consequentialism: A Comment on Norcross. Philosophy & Public Affairs, 27(1), 50–58.
  • Ross, W. D. (1930). The right and the good. Oxford: Clarendon Press.
  • Ryberg, J. (2002). Higher and Lower Pleasures – Doubts on Justification. Ethical Theory and Moral Practice, 5(4), 415–429.
  • Temkin, L. S. (1996). A Continuum Argument for Intransitivity. Philosophy and Public Affairs, 25(3), 175–210.
  • Thrall, R. M. (1954). Applications of multidimensional utility theory. In R. M. Thrall, C. H. Coombs, & R. L. Davis (Eds.), Decision processes (pp. 181–186). New York: John Wiley.
  • Voorhoeve, A. (2008). Heuristics and Biases in a Purported Counter-Example to the Acyclicity of “Better Than.” Politics, Philosophy and Economics, 7(3), 285–299.
  • Voorhoeve, A. (2013). Vaulting Intuition: Temkin’s Critique of Transitivity. Economics and Philosophy, 29(3), 409–425.


  1. Some, such as Wlodek Rabinowicz, would likely object to this usage of mine and say that value lexicality is a more narrow concept than value superiority: value lexicality is a kind of value superiority, but not vice versa. For example, consider a good g and another good g′ that both have diminishing marginal value so that as the amount of either good becomes larger the value of the amount approaches a limit. If the value limit of g is higher than that of g′, then g is superior to g′, because there is some amount of g that is better than any amount, however large, of g′. But, they may say, g is still not lexically better than g′. Nevertheless, I will say about cases like this that g is lexically better than g′, because “lexicality” is a more widely known term than “value superiority.”  (back)
  2. Lexicality between goods has also been discussed under the labels “higher goods,” “higher values,” “discontinuity” and “trumping.”  (back)
  3. Hutcheson (1968, pp. 117–118).  (back)
  4. According to Lemos (1993). See Arrhenius (2005, p. 97) for more historical supporters of lexicality between goods.  (back)
  5. See Lemos (1993, p. 485).  (back)
  6. Another possibility is lexicality of goods over bads; that is, that some goods are so good that an outcome is good regardless of the amount of some bads it contains. I have not seen that position being endorsed in the philosophical literature. The closest I am aware of is Hick’s (1973, p. 471) remark, quoted in Lemos (1993, p. 486), that Christian theodicy claims that the Kingdom of God is so good that it outweighs all finite evils.  (back)
  7. These formulations of strong and weak lexicality are essentially taken from Arrhenius and Rabinowicz (2015), but with some minor changes.  (back)
  8. This formulation draws on Carlson (2007, p. 4), which formulates lexicality of bads over goods in terms of the value of wholes.  (back)
  9. For example, Arrhenius and Rabinowicz (2015, appendix 12.6) talks of ‘minimal superiority’ among goods, which, in terms of bads, would be defined as follows: A bad b is minimally lexically worse than another bad b′ if and only if for some number m, there is no number k such that k b′-objects are worse than m b-objects. Roughly speaking, if b is weakly lexically worse than  b′, there is some amount of b that is worse than any amount of b′. If b is minimally lexically worse than b′, there is some amount of b such that no amount of b′ is worse than it. Minimal lexicality is interesting, for example, if the ordering is not complete (see note 9 below).  (back)
  10. See Carlson (2010) for a sophisticated treatment of measurement for lexicographic orders, and the literature that has followed Hausner (1954) and Thrall (1954) on multidimensional utilities and multidimensional utility theory.  (back)
  11. Arrhenius and Rabinowicz (2005, p. 138, 2015, p. 238).  (back)
  12. See note 6 above for the definition of minimal lexicality. As Arrhenius (2005, p. 112) says, his sequence argument relies on a completeness assumption. Completeness means roughly that for any two amounts of bads b and b′ under consideration, either the first is at least as bad as the second, or the second is at least as bad as the first. Dropping the completeness assumption results in a weaker form of the sequence argument, which says that a sequence in which b1 is strongly, weakly or minimally lexically worse than bn, must contain an element bi-1 that is minimally lexically worse than the element bi that immediately follows it (Arrhenius & Rabinowicz, 2015, appendix 12.6).  (back)
  13. According to Klocksiem (2016).  (back)

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