By Simon Knutsson
Written April 18, 2017; published November 17, 2017

This is the research proposal I included in my applications to PhD programs in philosophy. I got admitted to London School of Economics and Stockholm University.[1] As an applicant, it can be difficult to know how a research proposal is supposed to look. Universities’ application guidelines are often brief and unclear. It is also unfair that applicants with the right connections, for example, applicants who know students in top PhD programs, can get help with their proposals and thereby get an edge over those who lack such connections. Hopefully my proposal being public can help those who are planning to apply to PhD programs in philosophy and level the playing field.

Value superiority


Stockholm, SwedenAre some goods superior to others, such that any or some amount of a good g is better than any amount of another good g′ ? For example, Ross (1930, p. 150) finds it likely that virtue is better than any amount of pleasure. Similar ideas about bads have been discussed, for instance, that torture is worse than any amount of mild headaches. These ideas have been discussed under the labels “higher goods,” “higher values,” “value superiority,” “discontinuity,” “trumping” and “lexicality.” I will use the term ‘superiority’ for such relationships between goods or bads. Superiority has been endorsed by philosophers at least since Hutcheson in the 18th century,[2] perhaps by Aristotle,[3] and lately by, for example, Lemos (1993), Carlson (2000), Dorsey (2009), Rabinowicz  (Arrhenius and Rabinowicz 2015), Klocksiem (2016) and Parfit (2016).[4] Others have argued against it (e.g., Ryberg 2002; Arrhenius 2005; Norcross 2009). Value superiority is important for many areas of ethics, including medical ethics (Voorhoeve 2015), political priorities (Brülde 2010) and the impossibility theorems in population ethics (Carlson 2015).

Aim and research question

My main research question is, are there value superiorities? I plan to explore at least three aspects of this question: (1) the structure of value if there is superiority, (2) measurement of value in the case of superiority, and (3) sequence arguments and infinity. Tentatively, I expect to argue at least for the following: The structure of value is such that it can be indeterminate whether, for example, a good is superior to another, and this allows one to give convincing replies to challenges that have been made against superiority. Regarding measurement, I expect to argue that vectors can represent value in the case of superiority better than Carlson argues. Finally, a sequence argument against superiority by Arrhenius (and Rabinowicz) has a potential weakness: it is not clear that an analogous argument is successful against an adjusted notion of superiority (superiority*) formulated in terms of both finite amounts and an infinity of a good.

Project description

1.     The structure of superiority

Which is the most plausible account of the structure of value if there is superiority, and is such an account plausible? Attempts have been made to spell out the structure of value superiority; to explain why there would be superiority and how a theory of value would look like if there is superiority. Temkin and Rachels hold that superiority applies when there is a sufficiently large difference between the goods or bads in question; [5] Klocksiem (2016) defends “threshold lexicality,” according to which superiority between bads sets in at an absolute level of intensity between discomfort and pain; Dorsey (2009) explains superiority in terms of an account of well-being that involves global projects; Carlson (2000) and Rabinowicz (2003) refer to diminishing marginal value; Qizilbash (2005) points to the vagueness of categorical evaluations such as “serious” and “non-serious” illness; Parfit (2016) relies on indeterminacy and the relation “imprecisely equally good”; Lemos (1993) refers to organic unities and Carlson (2015) similarly points to a holistic non-Archimedean objective list theory of welfare.

My main hypothesis about the structure of value related to superiority is that value relations such as “better than” and “superior to” allow for indeterminacy in the sense that it can be indeterminate, for example, whether an amount of some value bearer is better or worse than an amount of another value bearer. Similarly, it can be indeterminate whether a good is superior to another. If this is correct, I hypothesize that it can be the basis for a convincing reply to the sequence argument against superiority that can be found in Arrhenius (2005) and Arrhenius and Rabinowicz (2015), as well as to the kind of spectra presented by Temin, Rachels and others, in which one is, for example, asked to contemplate gradually experiencing milder pain for longer durations.

There are some publications on ideas similar to superiority that also deal with indeterminacy or vagueness (Qizilbash 2005; Knapp 2007; Parfit 2016; Nebel forthcoming, sec. 4.2), but they essentially briefly propose indeterminacy or vagueness as a possible solution to challenges with superiority, without explaining in-depth how that would work and without assessing whether it is plausible when the implications of such a position are spelled out. In general, according to Dougherty (forthcoming, sec. 7), “the topic of ethical vagueness is a relatively neglected topic … on which there is still much work to be done.”

In one of my attached writings sample, which is work in progress, I have started investigating whether pointing to indeterminacy can amount to a convincing reply to sequence arguments. My tentative hypothesis is that rather than bringing in theories of vagueness, such as supervaluationism, epistemicism and many-valued logic, one should, more directly, give an account of evaluative indeterminacy. One plausible way to do that may be in terms of permissible preferences or appropriate attitudes (a “fitting attitudes”-analysis of value) as, for example, Rabinowicz (2008) does. One could introduce indeterminacy in such a model, as Rabinowicz (2009) mentions, by allowing for that it can be indeterminate whether it is permissible to have some preference, or that it is permissible to have indeterminate preferences (c.f., e.g., Aldred (2007) on vague preferences). If there is evaluative indeterminacy, one probably wants to avoid that it has sharp boundaries, similar to how philosophers working on vagueness often want there to be no sharp boundary between colors that are red and those that are borderline red. Here, what we could call higher-order evaluative indeterminacy seems key. For example, the second order of such indeterminacy could say about value bearers a and b that it is indeterminate whether it is indeterminate whether it is permissible to prefer a to b. I am not aware of any thorough philosophical work on such higher-order evaluative indeterminacy, although there is a literature on betterness and vagueness (e.g., Broome 1997; Carlson 2013; Qizilbash 2007, 2012, 2014), which sometimes briefly mentions higher-order vagueness. One should probably not give an account of higher-order evaluative indeterminacy directly in terms of, for example, a supervaluationist account of higher-order vagueness, but one could reason similarly to how philosophers have reasoned about the higher-order vagueness of vague terms such as ‘red.’ (See attached writing sample for more information.)

2.     Representational measurement of value

Which representations of value superiority are possible, and which operations, such as addition, do they allow? Some work on superiority deals with how to measure and compute value if there is superiority. This work can be seen as supporting superiority because one may hold that it is an advantage of a theory of value if it allows for a stronger degree of measurement, for example, on a ratio scale rather than on an ordinal scale. Brülde (2003, p. 24), for instance, lists measurability as an adequacy condition for a theory of well-being, and Klocksiem (2016, p. 1328) mentions the potential adequacy condition for a theory of value that value should be representable by single real numbers. Value can straightforwardly be represented by single reals if one assumes a form of superiority such as the one based on diminishing marginal value, but not if one assumes some other forms of superiority.

Attempts at solutions include Carlson (2007, 2010) and Feit (2001), who suggest representing value by lexical vectors or ordered pairs of real numbers,[6] and Klocksiem (2016, p. 1328), who says that his favored form of superiority can be represented by hyperreals. However, vector representation has been said to have problems and the suggestion to use hyperreals is not sufficiently developed.

Carlson (2001, sec. 3) argues against vector representation for more complex forms of superiority. One of his arguments is that vector representation does not work for weak superiority (i.e., the idea that a sufficient amount of a good g is better than any amount of another good g′), in part because if a conjunction of n (or more) g-objects is a superior good of a higher order than m g-objects, where m < n, then n g-objects would also, according to Carlson, be superior to a conjunction of conjunctions of m g-objects, even though that conjunction of conjunctions would have more than n g-objects. This would be absurd, the argument goes. I plan to argue that vector representation can overcome this argument, because such a conjunction of conjunctions (which is probably better understood as a concatenation of concatenations) would be a superior good of the same order as n g-objects. A main challenge, pointed out by Carlson in conversation, is how much of aggregation and additivity one can save if one assumes weak superiority and if one uses vectors to represent value.

Hyperreals have been used to represent non-Archimedean structures (e.g. Narens 1974), but that does not imply that they work well as a representation of value in the case of superiority, especially weak superiority, and philosophers have only pointed in passing to hyperreals as a possible solution (e.g. Klocksiem 2016, p. 1328). It remains to be explored to what extent superiority can be represented by hyperreals.

Finally, the measurement literature includes representation by other mathematical entities, such as intervals, sets and geometric objects, and I plan to investigate what the options are for representing value in the case of superiority by mathematical entities other than single reals, vectors or hyperreals, and whether such alternative representations are plausible.

3.     Superiority, sequence arguments and infinity

The best sequence argument against superiority can be found in Arrhenius (2005) and Arrhenius and Rabinowicz (2015). The argument is roughly that superiority between any two goods in a sequence of goods (a sequence g1, …, gn in which g1 is better than g2, which is better than g3, …, which is better than gn) implies that one good needs to be weakly superior to another adjacent good that is only marginally worse, which, according to the argument, is implausible. However, the argument has a potentially important weakness: it is not clear that an analogous conclusion can be established for a different notion of superiority, which we can call ‘superiority*,’ and which includes the idea of an infinity of a type of good. One could argue that superiority* is the more interesting notion. Roughly speaking, my point is that Arrhenius and Rabinowicz’s sequence argument is made in a framework of finite numbers, which allows the authors to derive potentially counterintuitive implications of superiority. But in the case of superiority, one may need to include infinities in the discussion. I plan to expand on and polish this idea.

The following is a more detailed description of my point. The sequence argument includes the step that if a good g1 is not weakly superior to another good gi-1 then there is some number k such that m g1-objects are not better than k gi-1-objects (Arrhenius 2005, p. 110; Arrhenius and Rabinowicz 2015, p. 242). This step can be taken because of their definition of weak superiority: an object g is weakly superior to an object g′ if and only if for some number m, m g-objects are better than any number of g′-objects (note my emphasis of ‘number’). We can modify the definition by, for example, adding infinity, and get weak superiority*: an object g is weakly superior* to an object g′ if and only if for some number m, m g-objects are better than any number of g′-objects and an infinity of g′-objects. Assuming infinity is not a number, the step above can no longer be taken: that a good g1 is not weakly superior* to another good gi-1 does not imply that there is some number k such that m g1-objects are not better than k gi-1-objects, because it could be that m g1-objects are not better than an infinity of gi-1-objects, while still being better than k gi-1-objects for any number k. My point does not avoid weak superiority between adjacent goods, but it avoids weak superiority* between adjacent goods, and it could be argued that weak superiority* is the more interesting relation.

An objection, mentioned by Erik Carlson in conversation, is that the sequence argument may still spell trouble for weak superiority* since one could argue that a “principle” such as weak superiority* should avoid implausible implications even if one considers only a subset of its scope (i.e., only finite amounts). This is an interesting objection that I would like to address in a part of my work on sequence arguments and infinities.

4.     Extra topic: the continuity axiom

A continuity axiom is commonly assumed in decision theory as a requirement of rationality. The following is a rough formulation of continuity: a strict preference relation ≻ is continuous if for all consequences a, b and c for which abc, there is a probability p, 0 < p < 1, such that b ∼ (a, p, c), where ∼ is indifference, and (a, p, c) is a lottery in which consequence a occurs with probability p and consequence c occurs with probability 1 – p. For example, if a is getting 1 SEK, b is getting 0 SEK and c is death, and assuming abc, there is a probability greater than zero that is sufficiently low so that rationality requires that an agent prefers a gamble between death and getting 1 SEK to certainly getting 0 SEK and survive.[7]

The question I am most interested in is whether the continuity axiom is a plausible requirement of rationality. There is a philosophical literature on the plausibility of the continuity axiom (e.g., Temkin 2001; Danielsson 2004; Arrhenius and Rabinowicz 2005; Jensen 2012), but, according to McCarthy (2016), more philosophical work on continuity is needed. For example, there is, as far as I know, nothing written about whether continuity is a plausible requirement of phenomena that are similar to rational preferences and relevant to axiology, such as appropriate pro-attitudes or appropriate emotive attitudes. Continuity is related to superiority, partly because if there is value superiority, it seems implausible that continuity would be a requirement of rationality or appropriate pro-attitudes. But one can reject continuity even if there is no value superiority. Investigating the plausibility of continuity does not obviously help answering my main research question (‘are there value superiorities?’), but it is related and similar arguments are used in both debates. I consider the continuity axiom to be a bonus topic that I would like to do research on if I have time.


Aldred, J. (2007). Intransitivity and Vague Preferences. Journal of Ethics, 11(4), 377–403.

Arrhenius, G. (2005). Superiority in Value. Philosophical Studies, 123(1–2), 97–114.

Arrhenius, G., & Rabinowicz, W. (2005). Value and Unacceptable Risk. Economics and Philosophy, 21(2), 177–197.

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. (1997). Is Incommensurability Vagueness? In R. Chang (Ed.), Incommensurability, Incomparability and Practical Reason. Harvard University Press.

Brülde, B. (2003). Teorier om livskvalitet. Lund: Studentlitteratur.

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. (2001). Organic Unities, Non-Trade-Off, and the Additivity of Intrinsic Value. Journal of Ethics, 5(4), 335–360.

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. (2013). Vagueness, Incomparability, and the Collapsing Principle. Ethical Theory and Moral Practice, 16(3), 449–463.

Carlson, E. (2015, January 30). On some impossibility theorems in population ethics. Draft.

Danielsson, S. (2004). Temkin, Archimedes and the transitivity of “Better.” Patterns of Value: Essays on Formal Axiology and Value Analysis, 2, 175–179.

Dorsey, D. (2009). Headaches, lives and value. Utilitas, 21(01), 36–58.

Dougherty, T. (forthcoming). Vagueness and indeterminacy in ethics. In T. McPherson & D. Plunkett (Eds.), The Routledge Handbook of Metaethics.

Feit, N. (2001). The Structure of Higher Goods. Southern Journal of Philosophy, 39(1), 47–57.

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.

Hutcheson, F. (1968). A system of moral philosophy (Vols. 1-1). New York: Augustus M Kelley.

Jensen, K. K. (2012). Unacceptable risks and the continuity axiom. Economics and Philosophy, 28(1), 31–42.

Klocksiem, J. (2016). How to Accept the Transitivity of Better Than. Philosophical Studies, 173(5), 1309–1334.

Knapp, C. (2007). Trading Quality for Quantity. Journal of Philosophical Research, 32(1), 211–233.

Lemos, N. M. (1993). Higher Goods and the Myth of Tithonus. Journal of Philosophy, 60(9), 482–496.

McCarthy, D. (2016). Probability in Ethics. In A. Hajek & C. Hitchcock (Eds.), The Oxford Handbook of Philosophy and Probability. Oxford University Press.

Narens, L. (1974). Measurement without Archimedean Axioms. Philosophy of Science, 41(4), 374–393.

Nebel, J. M. (forthcoming). The Good, the Bad, and the Transitivity of Better Than. Noûs.

Norcross, A. (2009). Two Dogmas of Deontology: Aggregation, Rights, and the Separateness of Persons. Social Philosophy and Policy, 26(1), 76–95.

Parfit, D. (2016). Can We Avoid the Repugnant Conclusion? Theoria, 82(2), 110–127.

Qizilbash, M. (2005). Transitivity and Vagueness. Economics and Philosophy, 21(1), 109–131.

Qizilbash, M. (2007). The Mere Addition Paradox, Parity and Vagueness. Philosophy and Phenomenological Research, 75(1), 129–151.

Qizilbash, M. (2012). Incommensurability or Vagueness? a Comment on Rabinowicz and Sugden. Proceedings of the Aristotelian Society, 112(3), 333–338.

Qizilbash, M. (2014). “Incommensurability” and Vagueness: Is the Vagueness View Defensible? Ethical Theory and Moral Practice, 17(1), 141–153. 

Rabinowicz, W. (2003). Ryberg’s Doubts About Higher and Lower Pleasures: Put to Rest? Ethical Theory and Moral Practice, 6(2), 231–237.

Rabinowicz, W. (2008). Value Relations. Theoria, 74(1), 18–49.

Rabinowicz, W. (2009). Incommensurability and Vagueness. Proceedings of the Aristotelian Society, Supplementary Volumes, 83, 71–94.

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. (2001). Worries about continuity, transitivity, expected utility theory, and practical reasoning. In D. Egonsson, J. Josefsson, B. Petersson, & T. Rønnow-Rasmusen (Eds.), Exploring Practical Philosophy (pp. 95–108). Aldershot: Ashgate Publishers.

Thrall, Robert 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. (2015). Why Sore Throats Don’t Aggregate Against a Life, but Arms Do. Journal of Medical Ethics, 41(6), 492–493.


[1] This is the proposal I sent to Stockholm University. The proposal I sent to London School of Economics was similar but had a few differences: it was 25% shorter, less developed, and the fourth sub-project was different—instead of an extra sub-project on the continuity axiom it was a regular (not “extra”) sub-project about the reliability of intuitions related to value superiority.

I applied to Stockholm, LSE, Cambridge, and Oslo. My research plans fit Stockholm and LSE better than Cambridge and Oslo. I would have applied to Uppsala, where my plans also fit very well, but Uppsala did not announce a PhD position at the time. The three universities that I targeted with my application were LSE, Stockholm, and Uppsala.

[2] Hutcheson (1968, pp. 117–118).

[3] According to Lemos (1993).

[4] For more references, see Arrhenius (2005, p. 97).

[5] According to Klocksiem (2016).

[6] Similar work has been done in economics, for example, on multidimensional utilities by Hausner (1954) and Thrall (1954).

[7] One can also formulate continuity in terms of the value of outcomes, as is done by Arrhenius and Rabinowicz (2005, p. 178).

PhD Philosophy Research Proposal (Admitted to LSE and Stockholm)